A Dynamic Analysis of Randomly Oriented Functionally Graded Carbon Nanotubes/Fiber-Reinforced Composite Laminated Shells with Different Geometries

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} div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1732884643"> <meta name="title" content="A Dynamic Analysis of Randomly Oriented Functionally Graded Carbon Nanotubes/Fiber-Reinforced Composite Laminated Shells with Different Geometries"> <meta name="description" content="The present study demonstrates the free vibration behavior of composite laminated shells reinforced by both randomly oriented single-walled carbon nanotubes (SWCNTs) and functionally graded fibers. The shell structures with different principal radii of curvature are considered, such as cylindrical, spherical, elliptical–paraboloid shell, hyperbolic–paraboloid shell, and plate. The volume fraction of the fibers has a linear variation along the shell thickness from layer to layer, while the volume fraction of CNTs is constant in all shell layers and uniformly distributed. The fiber-reinforced elements are distributed with three functions which are V-distribution, O-distribution, and X-distribution in addition to the uniform distribution. A numerical analysis was carried out systematically to validate the proposed solution. A new analytical solution is presented based on the Galerkin approach for shells and is exploited to illustrate the influence of some factors on the free vibration behavior of CNTs/fibe-reinforced composite (CNTs/F-RC) laminated shells, including the distributions and volume fractions, various boundary conditions, and geometrical properties of the reinforcement materials. The proposed solution is shown to be an effective theoretical tool to analyze the free vibration response of shells." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/mathematics-logo.png?8600e93ff98dbf14" > <meta name="dc.title" content="A Dynamic Analysis of Randomly Oriented Functionally Graded Carbon Nanotubes/Fiber-Reinforced Composite Laminated Shells with Different Geometries"> <meta name="dc.creator" content="Ammar Melaibari"> <meta name="dc.creator" content="Ahmed Amine Daikh"> <meta name="dc.creator" content="Muhammad Basha"> <meta name="dc.creator" content="Ahmed Wagih"> <meta name="dc.creator" content="Ramzi Othman"> <meta name="dc.creator" content="Khalid H. Almitani"> <meta name="dc.creator" content="Mostafa A. Hamed"> <meta name="dc.creator" content="Alaa Abdelrahman"> <meta name="dc.creator" content="Mohamed A. Eltaher"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Mathematics 2022, Vol. 10, Page 408"> <meta name="dc.date" content="2022-01-27"> <meta name ="dc.identifier" content="10.3390/math10030408"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="The present study demonstrates the free vibration behavior of composite laminated shells reinforced by both randomly oriented single-walled carbon nanotubes (SWCNTs) and functionally graded fibers. The shell structures with different principal radii of curvature are considered, such as cylindrical, spherical, elliptical–paraboloid shell, hyperbolic–paraboloid shell, and plate. The volume fraction of the fibers has a linear variation along the shell thickness from layer to layer, while the volume fraction of CNTs is constant in all shell layers and uniformly distributed. The fiber-reinforced elements are distributed with three functions which are V-distribution, O-distribution, and X-distribution in addition to the uniform distribution. A numerical analysis was carried out systematically to validate the proposed solution. A new analytical solution is presented based on the Galerkin approach for shells and is exploited to illustrate the influence of some factors on the free vibration behavior of CNTs/fibe-reinforced composite (CNTs/F-RC) laminated shells, including the distributions and volume fractions, various boundary conditions, and geometrical properties of the reinforcement materials. The proposed solution is shown to be an effective theoretical tool to analyze the free vibration response of shells." > <meta name="dc.subject" content="analytical solution" > <meta name="dc.subject" content="shell structures" > <meta name="dc.subject" content="different geometries" > <meta name="dc.subject" content="free vibration" > <meta name="dc.subject" content="CNTs/fiber-reinforced composite" > <meta name="dc.subject" content="higher-order shear deformation theory" > <meta name="dc.subject" content="Galerkin method" > <meta name ="prism.issn" content="2227-7390"> <meta name ="prism.publicationName" content="Mathematics"> <meta name ="prism.publicationDate" content="2022-01-27"> <meta name ="prism.volume" content="10"> <meta name ="prism.number" content="3"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="408" > <meta name="citation_issn" content="2227-7390"> <meta name="citation_journal_title" content="Mathematics"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="A Dynamic Analysis of Randomly Oriented Functionally Graded Carbon Nanotubes/Fiber-Reinforced Composite Laminated Shells with Different Geometries"> <meta name="citation_publication_date" content="2022/1"> <meta name="citation_online_date" content="2022/01/27"> <meta name="citation_volume" content="10"> <meta name="citation_issue" content="3"> <meta name="citation_firstpage" content="408"> <meta name="citation_author" content="Melaibari, Ammar"> <meta name="citation_author" content="Daikh, Ahmed Amine"> <meta name="citation_author" content="Basha, Muhammad"> <meta name="citation_author" content="Wagih, Ahmed"> <meta name="citation_author" content="Othman, Ramzi"> <meta name="citation_author" content="Almitani, Khalid H."> <meta name="citation_author" content="Hamed, Mostafa A."> <meta name="citation_author" content="Abdelrahman, Alaa"> <meta name="citation_author" content="Eltaher, Mohamed A."> <meta name="citation_doi" content="10.3390/math10030408"> <meta name="citation_id" content="mdpi-math10030408"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2227-7390/10/3/408"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2227-7390/10/3/408/pdf?version=1643526652"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2227-7390/10/3/408/pdf?version=1643526652"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2227-7390/10/3/408/pdf?version=1643526652"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2227-7390/10/3/408/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2227-7390/10/3/408/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2227-7390/10/3/408/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2227-7390/10/3/408/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2227-7390/10/3/408/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2227-7390/10/3/408/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/mathematics-logo-social.png?8600e93ff98dbf14" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2227-7390/10/3/408" /> <meta property="og:title" content="A Dynamic Analysis of Randomly Oriented Functionally Graded Carbon Nanotubes/Fiber-Reinforced Composite Laminated Shells with Different Geometries" /> <meta property="og:description" content="The present study demonstrates the free vibration behavior of composite laminated shells reinforced by both randomly oriented single-walled carbon nanotubes (SWCNTs) and functionally graded fibers. The shell structures with different principal radii of curvature are considered, such as cylindrical, spherical, elliptical–paraboloid shell, hyperbolic–paraboloid shell, and plate. The volume fraction of the fibers has a linear variation along the shell thickness from layer to layer, while the volume fraction of CNTs is constant in all shell layers and uniformly distributed. The fiber-reinforced elements are distributed with three functions which are V-distribution, O-distribution, and X-distribution in addition to the uniform distribution. A numerical analysis was carried out systematically to validate the proposed solution. A new analytical solution is presented based on the Galerkin approach for shells and is exploited to illustrate the influence of some factors on the free vibration behavior of CNTs/fibe-reinforced composite (CNTs/F-RC) laminated shells, including the distributions and volume fractions, various boundary conditions, and geometrical properties of the reinforcement materials. The proposed solution is shown to be an effective theoretical tool to analyze the free vibration response of shells." /> <meta property="og:image" content="https://pub.mdpi-res.com/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g001-550.jpg?1643526744" /> <link rel="alternate" type="application/rss+xml" title="MDPI Publishing - Latest articles" href="https://www.mdpi.com/rss"> <meta name="google-site-verification" content="PxTlsg7z2S00aHroktQd57fxygEjMiNHydKn3txhvwY"> <meta name="facebook-domain-verification" content="mcoq8dtq6sb2hf7z29j8w515jjoof7" /> <script id="Cookiebot" data-cfasync="false" src="https://consent.cookiebot.com/uc.js" data-cbid="51491ddd-fe7a-4425-ab39-69c78c55829f" type="text/javascript" async></script>  <script type="text/plain" data-cookieconsent="statistics"> (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); 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</div> </div> </div> </div> <article ><div class='html-article-content'> <span itemprop="publisher" content="Multidisciplinary Digital Publishing Institute"></span><span itemprop="url" content="https://www.mdpi.com/2227-7390/10/3/408"></span> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <h1 class="title hypothesis_container" itemprop="name"> A Dynamic Analysis of Randomly Oriented Functionally Graded Carbon Nanotubes/Fiber-Reinforced Composite Laminated Shells with Different Geometries </h1> <div class="art-authors hypothesis_container"> by <span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop6892822' data-options='is_hover:true, hover_timeout:5000'> Ammar Melaibari</div><div id="profile-card-drop6892822" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Ammar Melaibari</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/1859201?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a href="https://scilit.net/scholars?q=Ammar%20Melaibari" class="button button--color-inversed" target="_blank"> Scilit </a><a href="https://www.preprints.org/search?search1=Ammar%20Melaibari&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a><a href="https://scholar.google.com/scholar?q=Ammar%20Melaibari" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a></div></div><sup> 1</sup><span style="display: inline; margin-left: 5px;"></span><a class="toEncode emailCaptcha visibility-hidden" data-author-id="6892822" href="/cdn-cgi/l/email-protection#4d622e2329602e2a2462216228202c2421603d3f2239282e392422236e7d7d7d7d7b7c7d2e7d797d297d7d7d757d7e7d7d7c7e7d757f7c7d2c7d7d7c79792b7d797d787c79792b7c7f7d7d"><sup><i class="fa fa-envelope-o"></i></sup></a><a href="https://orcid.org/0000-0002-0967-6774" target="_blank" rel="noopener noreferrer"><img src="https://pub.mdpi-res.com/img/design/orcid.png?0465bc3812adeb52?1732884643" title="ORCID" style="position: relative; width: 13px; margin-left: 3px; max-width: 13px !important; height: auto; top: -5px;"></a>, </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop6892823' data-options='is_hover:true, hover_timeout:5000'> Ahmed Amine Daikh</div><div id="profile-card-drop6892823" data-dropdown-content 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href="/cdn-cgi/l/email-protection#4f602c212b622c28266023602a222e2623623f3d203b2a2c3b2620216c7f7f7f77792b7f7e7e767f2c7f7a7f777e297d2b7f797f2c7e777b7c7f777f767e777b7c7e2a7f2c"><sup><i class="fa fa-envelope-o"></i></sup></a><a href="https://orcid.org/0000-0003-3116-2101" target="_blank" rel="noopener noreferrer"><img src="https://pub.mdpi-res.com/img/design/orcid.png?0465bc3812adeb52?1732884643" title="ORCID" style="position: relative; width: 13px; margin-left: 3px; max-width: 13px !important; height: auto; top: -5px;"></a></span> </div> <div class="nrm"></div> <span style="display:block; height:6px;"></span> <div></div> <div style="margin: 5px 0 15px 0;" class="hypothesis_container"> <div class="art-affiliations"> <div class="affiliation "> <div class="affiliation-item"><sup>1</sup></div> <div class="affiliation-name ">Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, Jeddah 80204, Saudi Arabia</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>2</sup></div> <div class="affiliation-name ">Department of Technology, University Centre of Naama, Naama 45000, Algeria</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>3</sup></div> <div class="affiliation-name ">Laboratoire d’Etude des Structures et de Mécanique des Matériaux, Département de Génie Civil, Faculté des Sciences et de la Technologie, Université Mustapha Stambouli, B.P. 305, R.P., Mascara 29000, Algeria</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>4</sup></div> <div class="affiliation-name ">Mechanical Design and Production Department, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt</div> </div> <div class="affiliation"> <div class="affiliation-item"><sup>*</sup></div> <div class="affiliation-name ">Author to whom correspondence should be addressed. </div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Mathematics</em> <b>2022</b>, <em>10</em>(3), 408; <a href="https://doi.org/10.3390/math10030408">https://doi.org/10.3390/math10030408</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 10 January 2022</span> / <span style="display: inline-block">Revised: 25 January 2022</span> / <span style="display: inline-block">Accepted: 25 January 2022</span> / <span style="display: inline-block">Published: 27 January 2022</span> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-736181" aria-controls="drop-supplementary-736181" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-736181" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content aria-hidden="true" tabindex="-1"> <a class="UD_ArticlePDF" href="/2227-7390/10/3/408/pdf?version=1643526652" data-name="A Dynamic Analysis of Randomly Oriented Functionally Graded Carbon Nanotubes/Fiber-Reinforced Composite Laminated Shells with Different Geometries" data-journal="mathematics">Download PDF</a> <br/> <a id="js-pdf-with-cover-access-captcha" href="#" data-target="/2227-7390/10/3/408/pdf-with-cover" class="accessCaptcha">Download PDF with Cover</a> <br/> <a id="js-xml-access-captcha" href="#" data-target="/2227-7390/10/3/408/xml" class="accessCaptcha">Download XML</a> <br/> <a href="/2227-7390/10/3/408/epub" id="epub_link">Download Epub</a> <br/> </div> <div class="js-browse-figures" style="display: inline-block;"> <a href="#" class="button button--color-inversed margin-bottom-10 openpopupgallery UI_BrowseArticleFigures" data-target='article-popup' data-counterslink = "https://www.mdpi.com/2227-7390/10/3/408/browse" >Browse Figures</a> </div> <div id="article-popup" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g001.png?1643526744" title=" <strong>Figure 1</strong><br/> <p>The material properties, geometry, and coordinate system of the shell, (<b>a</b>) Geometry and coordinates, (<b>b</b>) Gradation types of material.</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g002.png?1643526744" title=" <strong>Figure 2</strong><br/> <p>Forms of various plate/shells.</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g003.png?1643526744" title=" <strong>Figure 3</strong><br/> <p>The volume-fractions of the fibers along the thickness of the shell (<math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g004.png?1643526744" title=" <strong>Figure 4</strong><br/> <p>The effect of the number of layers “<span class="html-italic">N</span>”: (<b>a</b>) Cross-ply fibers (<b>b</b>) Unidirectional fibers (Plate, SSSS, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g005.png?1643526744" title=" <strong>Figure 5</strong><br/> <p>The effect of the radius of curvature <span class="html-italic">R</span>/<span class="html-italic">a</span> on the dimensionless frequency of various shell types (Cross-ply, SSSS, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g006.png?1643526744" title=" <strong>Figure 6</strong><br/> <p>The effect of the volume fraction of fibers and CNTs (Plate, Cross-ply, SSSS, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>).</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g007.png?1643526744" title=" <strong>Figure 7</strong><br/> <p>The effect of the geometry parameters <span class="html-italic">b</span>/<span class="html-italic">a</span> and <span class="html-italic">a</span>/<span class="html-italic">h</span> on the dimensionless frequency of a spherical shell for various boundary conditions (<math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>). (<b>a</b>) inplane ratio <span class="html-italic">b</span>/<span class="html-italic">a</span>, (<b>b</b>) slenderness ratio <span class="html-italic">a</span>/<span class="html-italic">h</span>.</p> "> </a> <a href="https://pub.mdpi-res.com/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g008.png?1643526744" title=" <strong>Figure 8</strong><br/> <p>The effect of the modes of vibration “<span class="html-italic">m</span>” and “<span class="html-italic">n</span>” on the dimensionless frequency of a spherical shell (<math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>5</mn> <mo> </mo> </mrow> </semantics></math>, SSSS, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</p> "> </a> </div> <a class="button button--color-inversed" href="/2227-7390/10/3/408/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <div class="html-p">The present study demonstrates the free vibration behavior of composite laminated shells reinforced by both randomly oriented single-walled carbon nanotubes (SWCNTs) and functionally graded fibers. The shell structures with different principal radii of curvature are considered, such as cylindrical, spherical, elliptical–paraboloid shell, hyperbolic–paraboloid shell, and plate. The volume fraction of the fibers has a linear variation along the shell thickness from layer to layer, while the volume fraction of CNTs is constant in all shell layers and uniformly distributed. The fiber-reinforced elements are distributed with three functions which are V-distribution, O-distribution, and X-distribution in addition to the uniform distribution. A numerical analysis was carried out systematically to validate the proposed solution. A new analytical solution is presented based on the Galerkin approach for shells and is exploited to illustrate the influence of some factors on the free vibration behavior of CNTs/fibe-reinforced composite (CNTs/F-RC) laminated shells, including the distributions and volume fractions, various boundary conditions, and geometrical properties of the reinforcement materials. The proposed solution is shown to be an effective theoretical tool to analyze the free vibration response of shells.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=analytical+solution">analytical solution</a>; <a href="/search?q=shell+structures">shell structures</a>; <a href="/search?q=different+geometries">different geometries</a>; <a href="/search?q=free+vibration">free vibration</a>; <a href="/search?q=CNTs%2Ffiber-reinforced+composite">CNTs/fiber-reinforced composite</a>; <a href="/search?q=higher-order+shear+deformation+theory">higher-order shear deformation theory</a>; <a href="/search?q=Galerkin+method">Galerkin method</a></div> <div> </div> </div> </div> </p> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-mathematics-10-00408' type='intro'><h2 data-nested='1'> 1. Introduction</h2><div class='html-p'>A functionally graded material (FGM), which is an advanced composite material with a continuous gradation of materials through spatial directions, is broadly employed in various applications, ranging from macroscale (i.e., spacecraft, naval, nuclear structures, etc.) to micro/nano-scale electro-mechanical systems (MEMS/NEMS). Recently, the reinforcement of FGM by carbon nanotubes (CNTs) has been used in order to improve the mechanical, electrical, and thermal properties of composite structures due to the excellent properties of CNTs [<a href="#B1-mathematics-10-00408" class="html-bibr">1</a>,<a href="#B2-mathematics-10-00408" class="html-bibr">2</a>]. The characteristics of the CNTs/fiber-reinforced composite structure strongly depend on many factors such as the fractal contents of CNTs/fibers [<a href="#B3-mathematics-10-00408" class="html-bibr">3</a>,<a href="#B4-mathematics-10-00408" class="html-bibr">4</a>] and Fiber’s geometry [<a href="#B5-mathematics-10-00408" class="html-bibr">5</a>].</div><div class='html-p'>FG beams, plates, and shells have received substantial attention, and an extensive spectrum of beam and plate theories has been introduced, Boutahar et al. [<a href="#B6-mathematics-10-00408" class="html-bibr">6</a>]. Shen and Zhang [<a href="#B7-mathematics-10-00408" class="html-bibr">7</a>] investigated the thermal buckling/postbuckling behavior for supported functionally graded carbon nanotubes-reinforced composite (FG-CNTRC) plates subjected to in-plane temperature variation. Zhu et al. [<a href="#B8-mathematics-10-00408" class="html-bibr">8</a>] examined the static and vibration behaviors of FG-CNTRC plates using the finite element method with the first order shear deformation plate theory. Alibeigloo [<a href="#B9-mathematics-10-00408" class="html-bibr">9</a>] developed the 3D thermoelasticity solution for an FG-CNTRC plate embedded in a piezoelectric sensor and actuator layers. Liew et al. [<a href="#B10-mathematics-10-00408" class="html-bibr">10</a>] presented a review to identify and highlight topics relevant to FG-CNTRC and recent research works. Zhang et al. [<a href="#B11-mathematics-10-00408" class="html-bibr">11</a>] developed a computational solution for the vibration response of FG-CNTRC thick plates resting on elastic foundations. Phung-Van et al. [<a href="#B12-mathematics-10-00408" class="html-bibr">12</a>] presented the size-dependent impact on the nonlinear transient dynamic response of FG-CNTRC nonlocal nanoplates under a transverse uniform load in thermal environments. Daikh et al. [<a href="#B13-mathematics-10-00408" class="html-bibr">13</a>] investigated the static response of simply supported FG-CNTRC nonlocal strain gradient nanobeams under various loading profiles using a hyperbolic higher shear deformation beam theory. He et al. [<a href="#B14-mathematics-10-00408" class="html-bibr">14</a>] implemented the multi-parameter perturbation method to predict the static and stress distribution of FG thin circular piezoelectric plates. The mechanical response of piezoelectric FG plates via a simple first-order shear deformation theory was studied using the isogeometric analysis method [<a href="#B15-mathematics-10-00408" class="html-bibr">15</a>] and the generalized finite difference method [<a href="#B16-mathematics-10-00408" class="html-bibr">16</a>].</div><div class='html-p'>Daikh et al. [<a href="#B17-mathematics-10-00408" class="html-bibr">17</a>] presented the influence of thickness stretching on the mechanical responses of FG-CNTRC nanoplates based on the nonlocal strain gradient theory. Esen et al. [<a href="#B18-mathematics-10-00408" class="html-bibr">18</a>] studied analytically the vibration time response of an FG-CNTRC nanobeam under moving loads using the Navier Procedure. Employing the finite element method, Karamanli and Vo [<a href="#B19-mathematics-10-00408" class="html-bibr">19</a>] analyzed the mechanical behavior of carbon nanotube-reinforced composites and graphene nanoplatelet-reinforced composite beams incorporating both normal and shear effects. Karamanli and Aydogdu [<a href="#B20-mathematics-10-00408" class="html-bibr">20</a>] investigated the dynamic behavior of two directional functionally graded carbon nanotube-reinforced composite plates considering various boundary conditions. Boutahar et al. [<a href="#B6-mathematics-10-00408" class="html-bibr">6</a>] illustrated the impact of thickness stretching on the bending vibratory behavior of thick FG beams using the refined hyperbolic function shear theory. Daikh et al. [<a href="#B21-mathematics-10-00408" class="html-bibr">21</a>] examined the static response of sandwich FG nonlocal strain gradient nanoplates rested on variable Winkler elastic foundation based on a new quasi 3D hyperbolic shear theory. Daikh et al. [<a href="#B22-mathematics-10-00408" class="html-bibr">22</a>] inspected the buckling stability and static response of axially FG-CNTRC plates with temperature-dependent material properties. Rostami and Mohammadimehr [<a href="#B23-mathematics-10-00408" class="html-bibr">23</a>] determined the dynamic stability and bifurcation of an FG-CNTRC plate under lateral stochastic loads via the classical plate theory. Khadir et al. [<a href="#B24-mathematics-10-00408" class="html-bibr">24</a>] exploited the four-unknowns quasi-3D theory to analyze the mechanical responses of FG-CNTRC nonlocal strain gradient nanoplates. Babaei et al. [<a href="#B25-mathematics-10-00408" class="html-bibr">25</a>] studied the vibrational response of thermally pre-/post-buckled FG-CNTRC beams on a nonlinear elastic foundation. Duc and Minh [<a href="#B26-mathematics-10-00408" class="html-bibr">26</a>] predicted the free vibration behavior of cracked FG-CNTRC plates using the phase field theory and the finite element method. Adhikari and Singh [<a href="#B27-mathematics-10-00408" class="html-bibr">27</a>] illustrated the geometrical nonlinear dynamic response of the FG-CNTRC plate based on a novel shear strain function using the isogeometric finite element procedure.</div><div class='html-p'>For shell structures, Shen [<a href="#B28-mathematics-10-00408" class="html-bibr">28</a>,<a href="#B29-mathematics-10-00408" class="html-bibr">29</a>] studied the postbuckling of FG-CNTRC cylindrical shells in thermal environments under axial loads and pressure loads based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity by using the singular perturbation technique. Aragh et al. [<a href="#B30-mathematics-10-00408" class="html-bibr">30</a>] exploited the 3D elasticity theory to evaluate the vibrations of FG-CNTRC cylindrical panels with two opposite edges simply supported via a semi-analytical solution procedure. Shen [<a href="#B31-mathematics-10-00408" class="html-bibr">31</a>] studied the torsional postbuckling response of FG-CNTRC cylindrical shells in thermal environments. Mirzaei and Kiani [<a href="#B32-mathematics-10-00408" class="html-bibr">32</a>] investigated the thermal buckling of temperature-dependent FG-CNTRC conical shells under the assumption of the first order shear deformation shell theory, Donnell kinematic assumptions, and the von Kármán type of geometrical nonlinearity. Thomas and Roy [<a href="#B33-mathematics-10-00408" class="html-bibr">33</a>] examined the vibration analysis of FG-CNTRC Mindlin shell structures by using finite element modelling. The FG material properties were graded smoothly through the thickness. Pouresmaeeli and Fazelzadeh [<a href="#B34-mathematics-10-00408" class="html-bibr">34</a>] studied the frequency response of doubly curved FG-CNTRC panels via the first-order shear deformation theory and Galerkin’s method. Shojaee et al. [<a href="#B35-mathematics-10-00408" class="html-bibr">35</a>] studied the vibration of FG-CNTRC skewed cylindrical panels using a transformed differential quadrature method. Tohidi et al. [<a href="#B36-mathematics-10-00408" class="html-bibr">36</a>] presented the nonlinear dynamic buckling of an FG-CNTRC cylindrical shell via the modified strain gradient theory (the SGT and the Mori–Tanaka approach). Avramov et al. [<a href="#B37-mathematics-10-00408" class="html-bibr">37</a>] illustrated analytically the self-sustained vibrations of FG-CNTRC shells under a supersonic flow using the linear piston theory. Aminipour et al. [<a href="#B38-mathematics-10-00408" class="html-bibr">38</a>] presented the size-dependent wave propagation of FG doubly curved nonlocal nanoshells based on the higher order shear deformation theory.</div><div class='html-p'>Ahmadi et al. [<a href="#B39-mathematics-10-00408" class="html-bibr">39</a>] studied the nonlinear vibration response of stiffened FG double-curved shallow shells exposed to thermal and nonlinear elastic environments under hamonic excitation using the perturbation methodology. Babaei et al. [<a href="#B40-mathematics-10-00408" class="html-bibr">40</a>] developed a theoretical investigation for the frequency response of thermally pre/post buckled CNTRC pipes using a two-step perturbation method. Chakraborty and Dey [<a href="#B41-mathematics-10-00408" class="html-bibr">41</a>] and Chakraborty et al. [<a href="#B42-mathematics-10-00408" class="html-bibr">42</a>] developed a semi-analytical approach to explore the nonlinear stability characteristics of an FG-CNTRC cylindrical shell subjected to combined axial compressive loading and localized heating. Dai et al. [<a href="#B43-mathematics-10-00408" class="html-bibr">43</a>] studied the mechanical behaviors of a 3D poroelasticity FG-GPLRC open shell resting on a non-polynomial viscoelastic substrate involving friction force and residual stresses. Fares et al. [<a href="#B44-mathematics-10-00408" class="html-bibr">44</a>] developed a consistent layerwise/zigzag model to analyze the free vibrations of multilayered of FG-CNTRC conical shells using the Galerkin method. Fu et al. [<a href="#B45-mathematics-10-00408" class="html-bibr">45</a>] investigated the vibration instability of FG-CNTRC laminated conical shells surrounded by elastic foundations using first order shear deformation. Yadav et al. [<a href="#B46-mathematics-10-00408" class="html-bibr">46</a>] developed a semi-analytical solution for the nonlinear vibrations of FG-CNTRC circular cylindrical shells by a radial harmonic force and viscous structural damping. Mohammadi et al. [<a href="#B47-mathematics-10-00408" class="html-bibr">47</a>] exploited an isogeometric Kirchhoff–Love shell in free and forced vibration analyses of sinusoidally corrugated FG-CNTRC panels. Cong et al. [<a href="#B48-mathematics-10-00408" class="html-bibr">48</a>] studied the vibration and nonlinear dynamic responses of a temperature-dependent FG-CNTRC laminated double curved shallow shell with a positive and negative Poisson’s ratio. Babaei [<a href="#B49-mathematics-10-00408" class="html-bibr">49</a>] investigated analytically the nonlinear vibration and snap-buckling behaviors of FG-CNTRC arches using the perturbation method.</div><div class='html-p'>Based on the existing literature, the study of the free vibration behavior of composite laminated shells reinforced by randomly oriented SWCNTs remains unexplored, despite the interest for potential applications with different geometrical shapes such as cylindrical, spherical, elliptical–paraboloid, and hyperbolic–paraboloid shells. This motivated the present study. The volume fraction of fibers has a linear variation along the shell thickness from layer to layer, while the volume fraction of CNTs is constant in all shell layers and uniformly distributed. Three gradation functions, including V-distribution, O-distribution, and X-distribution, in addition to the uniform distribution, are considered. Analytical solutions with the Navier procedure are addressed in detail. The rest of the article will focus on the material and geometrical modelling, the governing equations of motion, and analytical solutions, validation, and parametric studies.</div></section><section id='sec2-mathematics-10-00408' type=''><h2 data-nested='1'> 2. Material and Geometrical Modeling</h2><section id='sec2dot1-mathematics-10-00408' type=''><h4 class='html-italic' data-nested='2'> 2.1. Mechanical Properties and Geometrics</h4><div class='html-p'>A rectangular multilayer shell in the Cartesian (<span class='html-italic'>x</span>, <span class='html-italic'>y</span>, <span class='html-italic'>z</span>) coordinate system is shown in <a href="#mathematics-10-00408-f001" class="html-fig">Figure 1</a>. The proposed structure has a curved length and width <span class='html-italic'>a</span> × <span class='html-italic'>b</span>, and thickness <span class='html-italic'>h</span>. The principal radii of curvature of the middle plane in the <span class='html-italic'>x</span> and <span class='html-italic'>y</span> directions are <span class='html-italic'>R<sub>x</sub></span> and <span class='html-italic'>R<sub>y</sub></span>, respectively. Various forms are analyzed by varying the principal radius of curvature <span class='html-italic'>R<sub>x</sub></span> and <span class='html-italic'>R<sub>y</sub></span> (See <a href="#mathematics-10-00408-f002" class="html-fig">Figure 2</a>). The shell is reinforced by randomly oriented carbon nanotubes (CNTs) and long fibers. All shell layers have the same thickness. The volume fraction of the fibers has a linear variation along the shell thickness from layer to layer, while the volume fraction of CNTs is constant in all shell layers and uniformly distributed. Four different patterns of fiber distribution are created in this study, a uniform distribution UD and three functionally graded distributions FG-X, FG-V, and FG-O.</div><div class='html-p'>The effective material properties of the CNTs/fiber-reinforced composite shell were obtained based on a micromechanical model as follows, [<a href="#B50-mathematics-10-00408" class="html-bibr">50</a>,<a href="#B51-mathematics-10-00408" class="html-bibr">51</a>]: <div class='html-disp-formula-info' id='FD1-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>E</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mi>f</mi> </msub> <msubsup> <mi>E</mi> <mrow> <mn>11</mn> </mrow> <mi>f</mi> </msubsup> <mo>+</mo> <msubsup> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> <msubsup> <mi>E</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div><div class='html-disp-formula-info' id='FD2-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mfrac> <mn>1</mn> <mrow> <msub> <mi>E</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mi>f</mi> </msub> </mrow> <mrow> <msubsup> <mi>E</mi> <mrow> <mn>22</mn> </mrow> <mi>f</mi> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> <mrow> <msubsup> <mi>E</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> </mfrac> <mo>−</mo> <msub> <mi>V</mi> <mi>f</mi> </msub> <msubsup> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> <mfrac> <mrow> <mfrac> <mrow> <msubsup> <mi>v</mi> <mi>f</mi> <mn>2</mn> </msubsup> <msubsup> <mi>E</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> <mrow> <msubsup> <mi>E</mi> <mrow> <mn>22</mn> </mrow> <mi>f</mi> </msubsup> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>ν</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> <msup> <mrow/> <mn>2</mn> </msup> <msubsup> <mi>E</mi> <mrow> <mn>22</mn> </mrow> <mi>f</mi> </msubsup> <mo>/</mo> <msubsup> <mi>E</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> <mo>−</mo> <mn>2</mn> <msub> <mi>v</mi> <mi>f</mi> </msub> <msubsup> <mi>ν</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> <mrow> <msub> <mi>V</mi> <mi>f</mi> </msub> <msubsup> <mi>E</mi> <mrow> <mn>22</mn> </mrow> <mi>f</mi> </msubsup> <mo>+</mo> <msubsup> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> <msubsup> <mi>E</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> </mfrac> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div><div class='html-disp-formula-info' id='FD3-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mfrac> <mn>1</mn> <mrow> <msub> <mi>G</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mi>f</mi> </msub> </mrow> <mrow> <msubsup> <mi>G</mi> <mrow> <mn>12</mn> </mrow> <mi>f</mi> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> <mrow> <msubsup> <mi>G</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> </mfrac> </mrow> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div><div class='html-disp-formula-info' id='FD4-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>υ</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mi>f</mi> </msub> <msub> <mi>υ</mi> <mi>f</mi> </msub> <mo>+</mo> <msubsup> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> <msubsup> <mi>υ</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div><div class='html-disp-formula-info' id='FD5-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>υ</mi> <mrow> <mn>21</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>E</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>E</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> </mfrac> <msub> <mi>υ</mi> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div><div class='html-disp-formula-info' id='FD6-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mi>f</mi> </msub> <msub> <mi>ρ</mi> <mi>f</mi> </msub> <mo>+</mo> <msubsup> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> <msubsup> <mi>ρ</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div></div><div class='html-p'>Here <math display='inline'><semantics> <mi>E</mi> </semantics></math>, <math display='inline'><semantics> <mi>G</mi> </semantics></math>, <math display='inline'><semantics> <mi>ρ</mi> </semantics></math>, and <math display='inline'><semantics> <mi>υ</mi> </semantics></math> are the Young’s modulus, the shear modulus, the mass density, and Poisson’s ratio, respectively. The subscripts/superscript <span class='html-italic'>f</span> and <span class='html-italic'>eff</span> and <span class='html-italic'>m</span> denote the fibers, the effective material properties, and the matrix, respectively. <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mi>f</mi> </msub> </mrow> </semantics></math> is the volume fraction of fibers and <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> </semantics></math> is the effective volume fraction of the matrix (Polymer/CNTs), where:<div class='html-disp-formula-info' id='FD7-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>V</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msubsup> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> </mrow> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div></div><div class='html-p'>In the case of the complete random orientation of CNTs throughout the polymer constituent, the composite is considered to be isotropic; therefore, the effective material properties of the mixture of CNTs and polymer can be expressed as:<div class='html-disp-formula-info' id='FD8-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi>E</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> <mo>=</mo> <mfrac> <mrow> <mn>9</mn> <mi>K</mi> <mi>G</mi> </mrow> <mrow> <mn>3</mn> <mi>K</mi> <mo>+</mo> <mi>G</mi> </mrow> </mfrac> </mrow> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div><div class='html-disp-formula-info' id='FD9-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi>ν</mi> <mrow> <mi>e</mi> <mi>f</mi> </mrow> <mi>m</mi> </msubsup> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo>−</mo> <mn>2</mn> <mi>G</mi> </mrow> <mrow> <mn>6</mn> <mi>K</mi> <mo>+</mo> <mn>2</mn> <mi>G</mi> </mrow> </mfrac> </mrow> </semantics></math> </div> <div class='l'> <label >(9)</label> </div> </div></div><div class='html-p'><math display='inline'><semantics> <mi>K</mi> </semantics></math> and <math display='inline'><semantics> <mi>G</mi> </semantics></math> are the effective bulk and shear moduli of the CNT/polymer composite and can be derived by the relations:<div class='html-disp-formula-info' id='FD10-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mi>K</mi> <mo>=</mo> <msub> <mi>K</mi> <mi>p</mi> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <mn>3</mn> <msub> <mi>K</mi> <mi>p</mi> </msub> <msub> <mi>α</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <msub> <mi>α</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </semantics></math> </div> <div class='l'> <label >(10)</label> </div> </div><div class='html-disp-formula-info' id='FD11-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mi>G</mi> <mo>=</mo> <msub> <mi>G</mi> <mi>p</mi> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>η</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mi>G</mi> <mi>p</mi> </msub> <msub> <mi>β</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <msub> <mi>β</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </semantics></math> </div> <div class='l'> <label >(11)</label> </div> </div></div><div class='html-p'><math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mi>p</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> </semantics></math> are the volume fractions of the polymer and CNTs and are related by:<div class='html-disp-formula-info' id='FD12-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>V</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> </semantics></math> </div> <div class='l'> <label >(12)</label> </div> </div></div><div class='html-p'>The other parameters appear in the above relations and have the following formulations:<div class='html-disp-formula-info' id='FD13-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>K</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>G</mi> <mi>p</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>k</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>l</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </semantics></math> </div> <div class='l'> <label >(13)</label> </div> </div><div class='html-disp-formula-info' id='FD14-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>β</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mi>p</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>k</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>l</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mi>p</mi> </msub> </mrow> <mrow> <msub> <mi>G</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <msub> <mi>K</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>G</mi> <mi>p</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>G</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <msub> <mi>K</mi> <mi>p</mi> </msub> <mo>+</mo> <mn>7</mn> <msub> <mi>G</mi> <mi>p</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>G</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <msub> <mi>K</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>G</mi> <mi>p</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>m</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <msub> <mi>K</mi> <mi>p</mi> </msub> <mo>+</mo> <mn>7</mn> <msub> <mi>G</mi> <mi>p</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(14)</label> </div> </div><div class='html-disp-formula-info' id='FD15-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>δ</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>l</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <msub> <mi>K</mi> <mi>p</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>G</mi> <mi>p</mi> </msub> <mo>−</mo> <msub> <mi>l</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>G</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(15)</label> </div> </div><div class='html-disp-formula-info' id='FD16-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>η</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>l</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mn>8</mn> <msub> <mi>G</mi> <mi>p</mi> </msub> <msub> <mi>p</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>G</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>8</mn> <msub> <mi>m</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <msub> <mi>G</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <msub> <mi>K</mi> <mi>p</mi> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>G</mi> <mi>p</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>3</mn> <msub> <mi>K</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>G</mi> <mi>p</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>G</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mrow> <mn>7</mn> <msub> <mi>m</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>G</mi> <mi>p</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>l</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msub> <mi>G</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>l</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mi>p</mi> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(16)</label> </div> </div></div><div class='html-p'><a href="#mathematics-10-00408-t001" class="html-table">Table 1</a> presents Hill’s elastic moduli for (10, 10) single-walled carbon nanotubes for various chiral indices.</div></section><section id='sec2dot2-mathematics-10-00408' type=''><h4 class='html-italic' data-nested='2'> 2.2. Volume Fraction of the Reinforcement Fibers</h4><div class='html-p'>The volume fraction of the fibers for various distribution patterns is expressed as the following relations:</div><div class='html-p'>Uniform distribution of the fibers <math display='inline'><semantics> <mrow> <mi>U</mi> <msub> <mi>D</mi> <mi>f</mi> </msub> </mrow> </semantics></math>:<div class='html-disp-formula-info' id='FD17-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>V</mi> <mi>f</mi> </msub> <mo>=</mo> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>+</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>−</mo> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </semantics></math> </div> <div class='l'> <label >(17)</label> </div> </div></div><div class='html-p'>Functionally graded fiber distribution <math display='inline'><semantics> <mrow> <mi>F</mi> <msub> <mi>G</mi> <mi>f</mi> </msub> <mo>−</mo> <mi>X</mi> </mrow> </semantics></math>:<div class='html-disp-formula-info' id='FD18-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>V</mi> <mi>f</mi> </msub> <mo>=</mo> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>−</mo> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </semantics></math> </div> <div class='l'> <label >(18)</label> </div> </div></div><div class='html-p'>Functionally graded fiber distribution <math display='inline'><semantics> <mrow> <mi>F</mi> <msub> <mi>G</mi> <mi>f</mi> </msub> <mo>−</mo> <mi>O</mi> <mtext> </mtext> </mrow> </semantics></math>:<div class='html-disp-formula-info' id='FD19-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>V</mi> <mi>f</mi> </msub> <mo>=</mo> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>−</mo> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(19)</label> </div> </div></div><div class='html-p'>Functionally graded fiber distribution <math display='inline'><semantics> <mrow> <mi>F</mi> <msub> <mi>G</mi> <mi>f</mi> </msub> <mo>−</mo> <mi>A</mi> <mtext> </mtext> </mrow> </semantics></math>:<div class='html-disp-formula-info' id='FD20-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>V</mi> <mi>f</mi> </msub> <mo>=</mo> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>−</mo> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(20)</label> </div> </div></div><div class='html-p'><math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> </mrow> </semantics></math> are the maximum and minimum volume fractions of the fibers. <span class='html-italic'>N</span> is the number of layers (odd number). The volume fractions along the thickness of the shell for various distribution patterns are plotted in <a href="#mathematics-10-00408-f003" class="html-fig">Figure 3</a>.</div></section></section><section id='sec3-mathematics-10-00408' type=''><h2 data-nested='1'> 3. Mathematical Formulations</h2><div class='html-p'>In the present work, we proposed a hyperbolic sine function shear deformation theory to define the governing equations for the free vibration problem of functionally graded CNTs/fiber-reinforced composite laminated shell. The displacement field can be expressed as:<div class='html-disp-formula-info' id='FD21-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mi>z</mi> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>−</mo> <mi>z</mi> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mi>z</mi> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>−</mo> <mi>z</mi> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>w</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </semantics></math> </div> <div class='l'> <label >(21)</label> </div> </div></div><div class='html-p'>The displacements of the midplane of the composite plate are <math display='inline'><semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> </semantics></math>, whereas <math display='inline'><semantics> <mrow> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> </semantics></math> are the rotations of the transverse normal at the middle surface <span class='html-italic'>z</span> = 0. The shape function <math display='inline'><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> is proposed as hyperbolic sine, and it is given as:<div class='html-disp-formula-info' id='FD22-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>h</mi> <mi>sinh</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>z</mi> <mi>h</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mfrac> <mrow> <mn>3</mn> <msup> <mi>z</mi> <mn>3</mn> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </semantics></math> </div> <div class='l'> <label >(22)</label> </div> </div> where <div class='html-disp-formula-info' id='FD23-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> </div> <div class='l'> </div> </div></div><div class='html-p'>The nonzero strains can be determined from the displacement as:<div class='html-disp-formula-info' id='FD24-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>ε</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>+</mo> <mi>z</mi> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>ε</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>γ</mi> <mrow> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>γ</mi> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>z</mi> </mrow> </mfrac> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>y</mi> <mi>z</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>z</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </semantics></math> </div> <div class='l'> <label >(23)</label> </div> </div> where:<div class='html-disp-formula-info' id='FD25-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow/> </mtd> </mtr> </mtable> </mrow> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mtext> </mtext> </mrow> <mspace linebreak="newline"/> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>φ</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>φ</mi> <mn>2</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>φ</mi> <mn>2</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>φ</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mrow> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>y</mi> <mi>z</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>z</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>φ</mi> <mi>x</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>φ</mi> <mi>y</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(24)</label> </div> </div></div><div class='html-p'>The stresses relations associated with the strains can be written as:<div class='html-disp-formula-info' id='FD26-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>σ</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>σ</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>τ</mi> <mrow> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>τ</mi> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>τ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msup> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>11</mn> </mrow> <mi>k</mi> </msubsup> </mrow> </mtd> <mtd> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>12</mn> </mrow> <mi>k</mi> </msubsup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>12</mn> </mrow> <mi>k</mi> </msubsup> </mrow> </mtd> <mtd> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>22</mn> </mrow> <mi>k</mi> </msubsup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>44</mn> </mrow> <mi>k</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>55</mn> </mrow> <mi>k</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>66</mn> </mrow> <mi>k</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <msup> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>ε</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>γ</mi> <mrow> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>γ</mi> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msup> </mrow> </semantics></math> </div> <div class='l'> <label >(25)</label> </div> </div></div><div class='html-p'>The transformed material constants <math display='inline'><semantics> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mi>k</mi> </msubsup> </mrow> </semantics></math> are expressed as:<div class='html-disp-formula-info' id='FD27-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>11</mn> </mrow> <mi>k</mi> </msubsup> <mo>=</mo> <msub> <mi>Q</mi> <mrow> <mn>11</mn> </mrow> </msub> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mn>4</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>Q</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>Q</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mn>22</mn> </mrow> </msub> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mn>4</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> </mrow> <mspace linebreak="newline"/> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>12</mn> </mrow> <mi>k</mi> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>Q</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mn>22</mn> </mrow> </msub> <mo>−</mo> <mn>4</mn> <msub> <mi>Q</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mn>12</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mn>4</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mo>+</mo> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mn>4</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mspace linebreak="newline"/> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>22</mn> </mrow> <mi>k</mi> </msubsup> <mo>=</mo> <msub> <mi>Q</mi> <mrow> <mn>11</mn> </mrow> </msub> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mn>4</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>Q</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>Q</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mn>22</mn> </mrow> </msub> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mn>4</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> </mrow> <mspace linebreak="newline"/> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>66</mn> </mrow> <mi>k</mi> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>Q</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mn>22</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mi>Q</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mi>Q</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mn>66</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mn>4</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mo>+</mo> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mn>4</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mspace linebreak="newline"/> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>44</mn> </mrow> <mi>k</mi> </msubsup> <mo>=</mo> <msub> <mi>Q</mi> <mrow> <mn>44</mn> </mrow> </msub> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mn>55</mn> </mrow> </msub> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> </mrow> <mspace linebreak="newline"/> <mrow> <msubsup> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mrow> <mn>55</mn> </mrow> <mi>k</mi> </msubsup> <mo>=</mo> <msub> <mi>Q</mi> <mrow> <mn>55</mn> </mrow> </msub> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mn>44</mn> </mrow> </msub> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mi>k</mi> </msub> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(26)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mi>k</mi> </msub> </mrow> </semantics></math> is the lamination angle of the <span class='html-italic'>k</span>th layer:<div class='html-disp-formula-info' id='FD28-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>Q</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>E</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow> <mn>12</mn> </mrow> </msub> <msub> <mi>ν</mi> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </mfrac> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <msub> <mi>Q</mi> <mrow> <mn>22</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>E</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow> <mn>12</mn> </mrow> </msub> <msub> <mi>ν</mi> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </mfrac> <mo>,</mo> <msub> <mi>Q</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>ν</mi> <mrow> <mn>12</mn> </mrow> </msub> <msub> <mi>E</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow> <mn>12</mn> </mrow> </msub> <msub> <mi>ν</mi> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>Q</mi> <mrow> <mn>44</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>G</mi> <mrow> <mn>23</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>Q</mi> <mrow> <mn>55</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>G</mi> <mrow> <mn>13</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>Q</mi> <mrow> <mn>66</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>G</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </semantics></math> </div> <div class='l'> </div> </div></div><div class='html-p'>By integrating Equation (25), the stress relations, moment, and additional moment resultants can be obtained as:<div class='html-disp-formula-info' id='FD29-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mi>N</mi> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mi>M</mi> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mi>P</mi> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo>[</mo> <mi>A</mi> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mrow> <mo>[</mo> <mi>B</mi> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mrow> <mo>[</mo> <mi>C</mi> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>[</mo> <mi>B</mi> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mrow> <mo>[</mo> <mi>D</mi> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mrow> <mo>[</mo> <mi>F</mi> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>[</mo> <mi>C</mi> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mrow> <mo>[</mo> <mi>F</mi> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mrow> <mo>[</mo> <mi>H</mi> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mrow> <msup> <mi>ε</mi> <mn>0</mn> </msup> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mrow> <msup> <mi>ε</mi> <mn>1</mn> </msup> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mrow> <msup> <mi>ε</mi> <mn>2</mn> </msup> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(27)</label> </div> </div><div class='html-disp-formula-info' id='FD30-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>R</mi> <mrow> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>R</mi> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>J</mi> <mrow> <mn>44</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>J</mi> <mrow> <mn>45</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>J</mi> <mrow> <mn>45</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>J</mi> <mrow> <mn>55</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>y</mi> <mi>z</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>z</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(28)</label> </div> </div> where:<div class='html-disp-formula-info' id='FD31-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mi>N</mi> <mo>}</mo> </mrow> <mo>=</mo> <msup> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>N</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>N</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>N</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mrow> <mi>T</mi> </msup> <mo>,</mo> <mrow> <mo>{</mo> <mi>M</mi> <mo>}</mo> </mrow> <mo>=</mo> <msup> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mrow> <mi>T</mi> </msup> <mo>,</mo> <mrow> <mo>{</mo> <mi>P</mi> <mo>}</mo> </mrow> <mo>=</mo> <msup> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>P</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>P</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>P</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mrow> <msup> <mi>ε</mi> <mn>0</mn> </msup> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <msup> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>0</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mrow> <mi>T</mi> </msup> <mo>,</mo> <mrow> <mo>{</mo> <mrow> <msup> <mi>ε</mi> <mn>1</mn> </msup> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <msup> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mrow> <mi>T</mi> </msup> <mo>,</mo> <mrow> <mo>{</mo> <mrow> <msup> <mi>ε</mi> <mn>2</mn> </msup> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <msup> <mrow> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>ε</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </semantics></math> </div> <div class='l'> <label >(29)</label> </div> </div></div><div class='html-p'>The coefficients <math display='inline'><semantics> <mrow> <msub> <mi>A</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math>,<math display='inline'><semantics> <mrow> <mtext> </mtext> <msub> <mi>B</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>D</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>H</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math> are defined as:<div class='html-disp-formula-info' id='FD32-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mrow> <msub> <mi>A</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo> </mo> <msub> <mi>B</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo> </mo> <msub> <mi>D</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo> </mo> <msub> <mi>C</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>F</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>H</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∑</mo> </mstyle> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> </mstyle> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mrow> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>h</mi> <mi>n</mi> </msub> </mrow> </munderover> <msub> <mi>Q</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mrow/> <mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </msup> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>z</mi> <mo>,</mo> <msup> <mi>z</mi> <mn>2</mn> </msup> <mo>,</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>z</mi> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>f</mi> <msup> <mrow> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>}</mo> </mrow> <mi mathvariant="normal">d</mi> <mi>z</mi> <mo>,</mo> <mo> </mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>J</mi> <mrow> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∑</mo> </mstyle> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> </mstyle> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mrow> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>h</mi> <mi>n</mi> </msub> </mrow> </munderover> <msub> <mi>Q</mi> <mrow> <mi>i</mi> <mi>i</mi> </mrow> </msub> <msup> <mrow/> <mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mo>[</mo> <mrow> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mrow> <mn>2</mn> </msup> <mi mathvariant="normal">d</mi> <mi>z</mi> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </semantics></math> </div> <div class='l'> <label >(30)</label> </div> </div></div></section><section id='sec4-mathematics-10-00408' type=''><h2 data-nested='1'> 4. Variational Statements</h2><div class='html-p'>Hamilton’s principle is employed in this analysis to derive the equations of motion of the CNTs/fiber composite shell:<div class='html-disp-formula-info' id='FD33-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mi>δ</mi> <msubsup> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> </msubsup> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mo>+</mo> <mi>V</mi> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo> </mo> <mo>=</mo> <mo> </mo> <mn>0</mn> </mrow> </semantics></math> </div> <div class='l'> <label >(31)</label> </div> </div></div><div class='html-p'>The variation of the strain energy of the composite shell can be expressed as:<div class='html-disp-formula-info' id='FD34-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mi>δ</mi> <msub> <mi>U</mi> <mi>p</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mi>V</mi> <mrow/> </msubsup> <mrow> <mo>[</mo> <mrow> <msubsup> <mi>σ</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> <mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <mi>δ</mi> <msub> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> <mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <mi>δ</mi> <msub> <mi>ε</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <mi>δ</mi> <msub> <mi>γ</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow> <mi>x</mi> <mi>z</mi> </mrow> <mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <mi>δ</mi> <msub> <mi>γ</mi> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow> <mi>y</mi> <mi>z</mi> </mrow> <mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <mi>δ</mi> <msub> <mi>γ</mi> <mrow> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mi mathvariant="normal">d</mi> <mi>V</mi> <mo>.</mo> <mtext> </mtext> </mrow> </semantics></math> </div> <div class='l'> <label >(32)</label> </div> </div></div><div class='html-p'>The variation of the kinetic energy of the composite shell at any moment is stated as:<div class='html-disp-formula-info' id='FD35-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mi>δ</mi> <mi>T</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>L</mi> </msubsup> <msubsup> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mi>A</mi> <mrow/> </msubsup> <mi>ρ</mi> <mrow> <mo>(</mo> <mrow> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mi>δ</mi> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>v</mi> <mo>˙</mo> </mover> <mi>δ</mi> <mover accent="true"> <mi>v</mi> <mo>˙</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mi>δ</mi> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>A</mi> <mi>d</mi> <mi>x</mi> </mrow> </semantics></math> </div> <div class='l'> <label >(33)</label> </div> </div><div class='html-disp-formula-info' id='FD36-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mi>δ</mi> <mi>T</mi> <mo>=</mo> <msub> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mi>V</mi> </msub> <mrow> <mo>{</mo> <mrow> <msub> <mi>I</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mi>δ</mi> <msub> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mo>+</mo> <msub> <mover accent="true"> <mi>v</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mi>δ</mi> <msub> <mover accent="true"> <mi>v</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mo>+</mo> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mi>δ</mi> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>I</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mi>δ</mi> <msub> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mo>+</mo> <mfrac> <mrow> <mo>∂</mo> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi>δ</mi> <msub> <mover accent="true"> <mi>v</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mo>+</mo> <msub> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mfrac> <mrow> <mo>∂</mo> <mi>δ</mi> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mrow> <mover> <mrow> <mo>+</mo> <mi>v</mi> </mrow> <mo>˙</mo> </mover> </mrow> <mn>0</mn> </msub> <mfrac> <mrow> <mo>∂</mo> <mi>δ</mi> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </mrow> <mspace linebreak="newline"/> <mrow> <mo>+</mo> <msub> <mi>I</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <mi>δ</mi> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mo>∂</mo> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <mi>δ</mi> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>I</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>x</mi> </msub> <mi>δ</mi> <msub> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mo>+</mo> <msub> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mi>δ</mi> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>x</mi> </msub> <mo>+</mo> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>y</mi> </msub> <mi>δ</mi> <msub> <mover accent="true"> <mi>v</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mo>+</mo> <msub> <mover accent="true"> <mi>v</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> <mi>δ</mi> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>y</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mspace linebreak="newline"/> <mrow> <mrow> <mrow> <mo>+</mo> <msub> <mi>I</mi> <mn>4</mn> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>x</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <mi>δ</mi> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mo>∂</mo> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mi>δ</mi> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>x</mi> </msub> <mo>+</mo> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>y</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <mi>δ</mi> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mo>∂</mo> <msub> <mover accent="true"> <mi>w</mi> <mo>˙</mo> </mover> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi>δ</mi> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>y</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>I</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>x</mi> </msub> <mi>δ</mi> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>x</mi> </msub> <mo>+</mo> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>y</mi> </msub> <mi>δ</mi> <msub> <mover accent="true"> <mi>φ</mi> <mo>˙</mo> </mover> <mi>y</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mi mathvariant="normal">d</mi> <mi>z</mi> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(34)</label> </div> </div> where:<div class='html-disp-formula-info' id='FD37-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mo>{</mo> <mrow> <msub> <mi>I</mi> <mn>0</mn> </msub> <mo>,</mo> <mo> </mo> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>,</mo> <mo> </mo> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>,</mo> <mo> </mo> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi>I</mi> <mn>4</mn> </msub> <mo>,</mo> <msub> <mi>I</mi> <mn>5</mn> </msub> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∑</mo> </mstyle> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>k</mi> </munderover> <mi>ρ</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>z</mi> <mo>,</mo> <msup> <mi>z</mi> <mn>2</mn> </msup> <mo>,</mo> <mi>Φ</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>z</mi> <mi>Φ</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>,</mo> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>Φ</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>}</mo> </mrow> <mi>d</mi> <mi>z</mi> </mrow> </semantics></math> </div> <div class='l'> <label >(35)</label> </div> </div></div><div class='html-p'>By inserting Equations (32)–(34) into Equation (19), the equilibrium equations for a CNTs/fiber composite shell can be obtained as follows:<div class='html-disp-formula-info' id='FD38-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <msub> <mi>A</mi> <mrow> <mn>11</mn> </mrow> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mn>66</mn> </mrow> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>−</mo> <msub> <mi>B</mi> <mrow> <mn>11</mn> </mrow> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> </mrow> <mspace linebreak="newline"/> <mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace linebreak="newline"/> <mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>0</mn> </msub> <mo>+</mo> <mn>2</mn> <mfrac> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>3</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> <msup> <mrow/> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>4</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(36)</label> </div> </div><div class='html-disp-formula-info' id='FD39-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mn>22</mn> </mrow> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mn>66</mn> </mrow> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>−</mo> <msub> <mi>B</mi> <mrow> <mn>22</mn> </mrow> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> </mfrac> </mrow> <mspace linebreak="newline"/> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mspace linebreak="newline"/> <mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>0</mn> </msub> <mo>+</mo> <mn>2</mn> <mfrac> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> <msup> <mrow/> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>4</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(37)</label> </div> </div><div class='html-disp-formula-info' id='FD40-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <msub> <mi>B</mi> <mrow> <mn>11</mn> </mrow> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <msub> <mi>B</mi> <mrow> <mn>22</mn> </mrow> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace linebreak="newline"/> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <msub> <mi>D</mi> <mrow> <mn>11</mn> </mrow> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>4</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msub> <mi>D</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>D</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>4</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <msub> <mi>D</mi> <mrow> <mn>22</mn> </mrow> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>4</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>4</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mspace linebreak="newline"/> <mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> <msup> <mrow/> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mn>2</mn> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> <msup> <mrow/> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>+</mo> <msubsup> <mi>D</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>D</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace linebreak="newline"/> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>D</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>D</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>=</mo> <msub> <mi>I</mi> <mn>0</mn> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mspace linebreak="newline"/> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <msub> <mi>I</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>4</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>4</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>I</mi> <mn>4</mn> </msub> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(38)</label> </div> </div><div class='html-disp-formula-info' id='FD41-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>−</mo> <msubsup> <mi>D</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>D</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mspace linebreak="newline"/> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <msubsup> <mi>A</mi> <mrow> <mn>44</mn> </mrow> <mi>s</mi> </msubsup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>F</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace linebreak="newline"/> <mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>4</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <msub> <mi>I</mi> <mn>4</mn> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msub> <mi>I</mi> <mn>5</mn> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(39)</label> </div> </div><div class='html-disp-formula-info' id='FD42-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>D</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>−</mo> <msubsup> <mi>D</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> </mfrac> </mrow> <mspace linebreak="newline"/> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>F</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <msubsup> <mi>A</mi> <mrow> <mn>44</mn> </mrow> <mi>s</mi> </msubsup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mspace linebreak="newline"/> <mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>4</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>−</mo> <msub> <mi>I</mi> <mn>4</mn> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msub> <mi>I</mi> <mn>5</mn> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(40)</label> </div> </div></div></section><section id='sec5-mathematics-10-00408' type=''><h2 data-nested='1'> 5. Analytical Solution</h2><div class='html-p'>The aim of this work is to expand the use of analytical solutions to analyze the response of various structures such as shells by considering different boundary conditions, as shown in <a href="#mathematics-10-00408-t002" class="html-table">Table 2</a>. Therefore, the Galerkin approach can provide accurate solutions. The expressions of generalized displacements can be expressed as:<div class='html-disp-formula-info' id='FD43-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>ψ</mi> <mi>x</mi> </msub> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∑</mo> </mstyle> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mo>∞</mo> </munderover> </mstyle> <mstyle displaystyle="true"> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∑</mo> </mstyle> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mo>∞</mo> </munderover> </mstyle> <mrow> <mo>{</mo> <mrow> <msub> <mi>U</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>{</mo> <mrow> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>ψ</mi> <mi>y</mi> </msub> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∑</mo> </mstyle> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mo>∞</mo> </munderover> </mstyle> <mstyle displaystyle="true"> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∑</mo> </mstyle> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mo>∞</mo> </munderover> </mstyle> <mrow> <mo>{</mo> <mrow> <msub> <mi>V</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Z</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <msub> <mi>X</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∑</mo> </mstyle> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mo>∞</mo> </munderover> </mstyle> <mstyle displaystyle="true"> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∑</mo> </mstyle> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mo>∞</mo> </munderover> </mstyle> <msub> <mi>W</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </semantics></math> </div> <div class='l'> <label >(41)</label> </div> </div></div><div class='html-p'><math display='inline'><semantics> <mrow> <msub> <mi>U</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>X</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msub> <mi>Z</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math> are arbitrary parameters. The boundary conditions that can be imposed on all the four boundaries of the CNTs/F-RC shells are given in <a href="#mathematics-10-00408-t002" class="html-table">Table 2</a> as:</div><div class='html-p'>The functions <math display='inline'><semantics> <mrow> <msub> <mi>X</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> that satisfy the above boundary conditions are given in <a href="#mathematics-10-00408-t003" class="html-table">Table 3</a>.</div><div class='html-p'>By substituting Equation (41) in Equations (36)–(40), one obtains:<div class='html-disp-formula-info' id='FD44-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mrow> <mo>[</mo> <mi>K</mi> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>13</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>14</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>15</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>23</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>24</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>25</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>13</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>23</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>33</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>34</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>35</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>14</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>24</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>34</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>44</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>45</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>15</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>25</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>35</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>45</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>K</mi> <mrow> <mn>55</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>,</mo> <mo> </mo> </mrow> <mrow> <mrow> <mo>[</mo> <mi>M</mi> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>13</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>14</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>15</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>23</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>24</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>25</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>13</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>23</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>33</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>34</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>35</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>14</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>24</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>34</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>44</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>45</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>15</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>25</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>35</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>45</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mn>55</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(42)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mrow> <mo>[</mo> <mi>K</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> et [M] are the rigidity matrix and mass matrix, respectively.</div><div class='html-p'>The elements <math display='inline'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math> of the matrix <math display='inline'><semantics> <mrow> <mrow> <mo>[</mo> <mi>K</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mrow> <mo>[</mo> <mi>M</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> are given in <a href="#app1-mathematics-10-00408" class="html-app">Appendix A</a>.</div></section><section id='sec6-mathematics-10-00408' type='results'><h2 data-nested='1'> 6. Results and Discussions</h2><section id='sec6dot1-mathematics-10-00408' type=''><h4 class='html-italic' data-nested='2'> 6.1. Verification Analysis</h4><div class='html-p'>Firstly, to examine the accuracy and effectiveness of the developed model, the results for the free vibration of functionally graded shells were compared with those generated in the literature using various solution techniques (see <a href="#mathematics-10-00408-t004" class="html-table">Table 4</a>). The materials used were Alumina (AlO<sub>2</sub>) as the ceramic (<math display='inline'><semantics> <mrow> <msub> <mi>E</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>380</mn> <mo> </mo> <mi>GPa</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>ρ</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>3800</mn> <mo> </mo> <mi>kg</mi> <mo>/</mo> <msup> <mi mathvariant="normal">m</mi> <mn>3</mn> </msup> </mrow> </semantics></math>) and Aluminium (Al) as the metal (<math display='inline'><semantics> <mrow> <msub> <mi>E</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>70</mn> <mo> </mo> <mi>GPa</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>ρ</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>2707</mn> <mo> </mo> <mi>kg</mi> <mo>/</mo> <msup> <mi mathvariant="normal">m</mi> <mn>3</mn> </msup> </mrow> </semantics></math>). Poisson’s ratio was taken as <math display='inline'><semantics> <mrow> <mi>υ</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>. It can be observed that the results computed by the proposed HSDT are absolutely identical to those generated in the literature.</div></section><section id='62ParametricStud' type=''><h4 class='html-italic' data-nested='2'> 6.2. Parametric Stud</h4><div class='html-p'>The analyzed composite shell was made of a mixture of the polymer, armchair (10, 10) single-walled CNTs, and long fibers as reinforcements. The material properties of the polymer were [<a href="#B57-mathematics-10-00408" class="html-bibr">57</a>]: <math display='inline'><semantics> <mrow> <msub> <mi>E</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>2.5</mn> <mo> </mo> <mi>GPa</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>ρ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1190</mn> <mo> </mo> <mi>kg</mi> <mo>/</mo> <msup> <mi mathvariant="normal">m</mi> <mn>3</mn> </msup> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msub> <mi>υ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>0.19</mn> </mrow> </semantics></math>, while the material properties of the fiber were [<a href="#B58-mathematics-10-00408" class="html-bibr">58</a>]: <math display='inline'><semantics> <mrow> <msubsup> <mi>E</mi> <mrow> <mn>11</mn> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>233.5</mn> <mo> </mo> <mi>GPa</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>E</mi> <mrow> <mn>22</mn> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>23.1</mn> <mo> </mo> <mi>GPa</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>G</mi> <mrow> <mn>12</mn> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>8.96</mn> <mo> </mo> <mi>GPa</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>ρ</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1750</mn> <mo> </mo> <mi>kg</mi> <mo>/</mo> <msup> <mi mathvariant="normal">m</mi> <mn>3</mn> </msup> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <msub> <mi>υ</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>. The density of the CNTs was assumed to be equal to <math display='inline'><semantics> <mrow> <msub> <mi>ρ</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1400</mn> <mo> </mo> <mi>kg</mi> <mo>/</mo> <msup> <mi mathvariant="normal">m</mi> <mn>3</mn> </msup> </mrow> </semantics></math>. To standardize and simplify calculations, the normalized parameters for the vibration analyses of the CNTs/fiber shells are described using the following forms:<div class='html-disp-formula-info' id='FD45-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mover accent="true"> <mi>ω</mi> <mo>¯</mo> </mover> <mo>=</mo> <mi>ω</mi> <mi>h</mi> <msqrt> <mrow> <mfrac> <mrow> <msub> <mi>ρ</mi> <mi>m</mi> </msub> </mrow> <mrow> <msub> <mi>E</mi> <mi>m</mi> </msub> </mrow> </mfrac> </mrow> </msqrt> </mrow> </semantics></math> </div> <div class='l'> <label >(43)</label> </div> </div></div><div class='html-p'>In the following analysis, a parametric study on the vibration of CNTs/F-RC shells is carried out. Two types of laminated shells are proposed: cross-ply laminates and unidirectional laminates. For the unidirectional laminates, the angle of the orientation <math display='inline'><semantics> <mi>θ</mi> </semantics></math> was equal to 0°. For the cross-ply laminates, the angle of the orientation <math display='inline'><semantics> <mi>θ</mi> </semantics></math> of each layer changed alternately (<math display='inline'><semantics> <mi>θ</mi> </semantics></math> = 0° or 90°), for example, in the case of five layers, the shell could be described as [0°/90°/0°/90°/0°] laminate. The effect of the number of layers <span class='html-italic'>N</span> and the fiber distribution patterns on the dimensionless frequency of simply supported cross-ply F/CNT-RC shells for various curvature radii was examined as shown in <a href="#mathematics-10-00408-t005" class="html-table">Table 5</a>. The maximum volume fraction was taken as <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>30</mn> <mo>%</mo> </mrow> </semantics></math>, while the minimum is <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>10</mn> <mo>%</mo> </mrow> </semantics></math>. The volume fraction of the CNTs was taken as <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>%</mo> </mrow> </semantics></math>. The same analysis is discussed on the unidirectional F/CNT-RC shells in <a href="#mathematics-10-00408-t006" class="html-table">Table 6</a>.</div><div class='html-p'><a href="#mathematics-10-00408-t007" class="html-table">Table 7</a> shows the impact of the volume fraction and the fiber distribution patterns on the dimensionless frequency of simply supported cross-ply F/CNT-RC laminated shells. The number of layers was fixed at nine (9) layers. It is clear from this table that the change in the volume fraction of the CNTs had a significant influence on the free vibration behaviour of the shells.</div><div class='html-p'>The effect of the volume fraction and the distribution patterns of the fibers on the dimensionless frequency of simply supported cross-ply F/CNT-RC shells is demonstrated in <a href="#mathematics-10-00408-t008" class="html-table">Table 8</a>. The volume fraction of the CNTs was proposed as <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>%</mo> </mrow> </semantics></math>, whereas the minimal volume fraction of the fibers was <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>. The increase in the fiber volume fraction further reinforced the composite plate; therefore, the dimensionless frequencies increased.</div><div class='html-p'><a href="#mathematics-10-00408-t009" class="html-table">Table 9</a> presents the influence of the fiber distribution patterns on the dimensionless frequency of cross-ply F/CNT-RC shells for various boundary conditions. The fully clamped plate (CCCC) had the highest frequencies, while the lowest frequencies were for the simply supported shells, wherever the fiber distribution pattern was.</div><div class='html-p'>The action of the geometric parameters <span class='html-italic'>a</span>/<span class='html-italic'>h</span> and <span class='html-italic'>b</span>/<span class='html-italic'>a</span> on the dimensionless frequency of simply supported cross-ply F/CNT-RC shells for various curvature radii is tabulated in <a href="#mathematics-10-00408-t010" class="html-table">Table 10</a>. It is observed that the frequencies decreased when the thickness ratio <span class='html-italic'>a</span>/<span class='html-italic'>h</span> was decreased and the aspect ratio <span class='html-italic'>b</span>/<span class='html-italic'>a</span> was increased.</div><div class='html-p'><a href="#mathematics-10-00408-f004" class="html-fig">Figure 4</a> shows the the impact of the number of layers “<span class='html-italic'>N</span>” on the vibration frequencies of simply supported CNTs/F-RC shells using different fiber distribution patterns. Two types of shells were analyzed: cross-ply fibers and unidirectional fibers. The number of layers “<span class='html-italic'>N</span>” changed from 3 to 19. In the case of cross-ply fibers, the increase in the number of layers “<span class='html-italic'>N</span>” led to an increment in the dimensionless frequencies. The FG-X fiber distribution shells had the highest frequencies because of their excellent rigidity. In the case of unidirectional fibers, contradictory results were obtained. Precisely, the dimensionless frequencies increased with the increase in the number of layers for the patterns FG-O and FG-B, while the opposite effect was seen for the FG-X patterns. As is known, to increase the rigidity of the structure, it is obligatory to increase the reinforcement materials at the superior and the inferior sheets. On the other hand, the increase in the number of layers meant that the thickness of each layer decreased; therefore, the thickness of the strongest layer (superior and inferior layers) decreased, and this can explain the reduction in the rigidity of the plate. For the case of unidirectional fibers with uniform distribution UD, the number of layers did not have any influence.</div><div class='html-p'><a href="#mathematics-10-00408-f005" class="html-fig">Figure 5</a> shows the variation of dimensionless frequencies in the function of the radii of curvature <math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> <mo>/</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. The radii of curvature <math display='inline'><semantics> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math> is fixed at inf, 5, and −5. In general, the increase in the radii of curvature <math display='inline'><semantics> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> <mo>/</mo> <mi>a</mi> </mrow> </semantics></math> led to a critical decrement in frequencies for the values <math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> <mo>/</mo> <mi>a</mi> <mo>≤</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, and then, the frequencies continued to decrease in the cylindrical and the elliptical–paraboloid shells (<span class='html-italic'>R</span>/<span class='html-italic'>b</span> = inf, 5) slightly, and they increased in the case of the hyperbolic–paraboloid shell (<span class='html-italic'>R</span>/<span class='html-italic'>b</span> = −5).</div><div class='html-p'>In <a href="#mathematics-10-00408-f006" class="html-fig">Figure 6</a>, the influences of the volume fraction of fibers and CNTs on the vibration frequencies of simply supported FG-X CNTs/F-RC plate are plotted. The minimal volume fraction of the fibers was <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>. It is evident that the increase in the volume fraction of the reinforcement led to an increment in plate stiffness; therefore, the natural frequencies increased. Comparing the two reinforcement materials, the increase in the volume fraction of the CNTs had a more important effect than the volume fraction of the fibers.</div><div class='html-p'>The effect of both the aspect ratio <span class='html-italic'>b</span>/<span class='html-italic'>a</span> and the thickness ratio <span class='html-italic'>a</span>/<span class='html-italic'>h</span> on the dimensionless frequency of a spherical shell <math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> <mo>/</mo> <mi>a</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>y</mi> </msub> <mo>/</mo> <mi>b</mi> <mo>=</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> for various boundary conditions is demonstrated in <a href="#mathematics-10-00408-f007" class="html-fig">Figure 7</a>. The increase in the aspect ratio led to a decrement in the dimensionless frequencies regardless of the boundary condition type. For the thickness ratio <span class='html-italic'>a</span>/<span class='html-italic'>h</span> effect, the frequencies decreased critically for values <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>≤</mo> <mn>10</mn> </mrow> </semantics></math>. The fully clamped shells had the highest values of frequency, while the lowest values were for the simply supported shells.</div><div class='html-p'>Finally, in <a href="#mathematics-10-00408-f008" class="html-fig">Figure 8</a>, the effect of the modes of vibration “<span class='html-italic'>m</span>” and “<span class='html-italic'>n</span>” on the dimensionless frequency of a spherical shell is plotted. We can see that the dimensionless frequency is influenced by mode shapes <span class='html-italic'>m</span> and <span class='html-italic'>n</span>, where the frequency increased with the increase in the vibrational mode shapes.</div></section></section><section id='sec7-mathematics-10-00408' type='conclusions'><h2 data-nested='1'> 7. Conclusions</h2><div class='html-p'>The free vibration behavior of composite laminated plates and shells reinforced by both randomly oriented (10, 10) single-walled carbon nanotubes (SWCNTs) and functionally graded fibers is presented in this work. Four distribution patterns of fiber reinforcement including UD-distribution, V-distribution, O-distribution, and X-distribution were analyzed. The problem was tackled theoretically based on the Galerkin technique and accounting for different boundary conditions. A parametric study was performed systematically to check for the effect of some significant factors on the free vibration response of CNTs/F-RC laminated shells, namely the fiber reinforcement patterns and the volume fraction of CNTs, together with the geometric parameters of the shells. The results showed that the present solution is in close agreement with the other available solutions in the literature. Based on the parametric investigation, the following concluding remarks are revealed:</div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'>It seems that the dimensionless frequencies increased for an increased volume fraction of CNTs/fibers because of a global augmentation in the stiffness of the CNTs/F-RC laminated shell;</div></li><li><div class='html-p'>Because of the considerable reinforcement in the superior and inferior layers, the distribution pattern of FG-X fibers produced the highest values of frequencies;</div></li><li><div class='html-p'>The variation of the geometric parameters such as aspect ratio, thickness ratio, and radii of curvature had a significant effect on the vibration response of the proposed structure;</div></li><li><div class='html-p'>The analytical solution proposed in this work could represent valid benchmarks for engineers and researchers for the purposes of the practical design of shell structures.</div></li></ul></div></section> </div> <div class="html-back"> <section class='html-notes'><h2 >Author Contributions</h2><div class='html-p'>A.M. (project administration, funding acquisition, data curation); A.A.D. (software, validation, formal analysis, investigation); M.B. (formal analysis, investigation); A.W. (software, visualization, data curation); R.O. (Conceptualization, methodology, formal analysis); K.H.A. (software, investigation, resources); M.A.H. (methodology, review and editing); A.A. (methodology, software, validation); M.A.E. (Conceptualization, methodology, review and editing). All authors have read and agreed to the published version of the manuscript.</div></section><section class='html-notes'><h2 >Funding</h2><div class='html-p'>This research was funded by the Institutional Fund Projects under Grant no. (IFPRC-012-135-2020). Therefore, the authors gratefully acknowledge technical and financial support from the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia. </div></section><section class='html-notes'><h2 >Institutional Review Board Statement</h2><div class='html-p'>Not applicable.</div></section><section class='html-notes'><h2 >Informed Consent Statement</h2><div class='html-p'>Not applicable.</div></section><section class='html-notes'><h2 >Data Availability Statement</h2><div class='html-p'>Not applicable.</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>The authors declare no conflict of interest.</div></section><section><section id='app1-mathematics-10-00408' type=''><h2 data-nested='1'> Appendix A</h2><div class='html-p'>Rigidity matrix elements; <div class='html-disp-formula-info' id='FD46-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mn>11</mn> </mrow> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mn>66</mn> </mrow> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD47-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD48-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <msub> <mi>K</mi> <mrow> <mn>13</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <msub> <mi>B</mi> <mrow> <mn>11</mn> </mrow> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> </mrow> <mspace linebreak="newline"/> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD49-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <msub> <mi>K</mi> <mrow> <mn>14</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>B</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mspace linebreak="newline"/> <mrow> <msub> <mi>K</mi> <mrow> <mn>14</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>B</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD50-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>15</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD51-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>21</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>1</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD52-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>22</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mn>22</mn> </mrow> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mn>66</mn> </mrow> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD53-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <msub> <mi>K</mi> <mrow> <mn>23</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <msub> <mi>B</mi> <mrow> <mn>22</mn> </mrow> </msub> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mspace linebreak="newline"/> <mrow> <msub> <mi>K</mi> <mrow> <mn>23</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <msub> <mi>B</mi> <mrow> <mn>22</mn> </mrow> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mspace linebreak="newline"/> <mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD54-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>24</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD55-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>25</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>B</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD56-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>31</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msub> <mi>B</mi> <mrow> <mn>11</mn> </mrow> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>4</mn> </msup> <msub> <mi>X</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD57-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>32</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msub> <mi>B</mi> <mrow> <mn>22</mn> </mrow> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>4</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>4</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>B</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD157-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <msub> <mi>K</mi> <mrow> <mn>33</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>B</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>B</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>B</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mspace linebreak="newline"/> <mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> <msup> <mrow/> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mn>2</mn> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>A</mi> <mrow> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> <msup> <mrow/> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <msub> <mi>D</mi> <mrow> <mn>11</mn> </mrow> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>4</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mspace linebreak="newline"/> <mrow> <mo>−</mo> <msub> <mi>D</mi> <mrow> <mn>22</mn> </mrow> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>4</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>4</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>D</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>D</mi> <mrow> <mn>66</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD59-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>34</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msubsup> <mi>D</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>4</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>D</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD60-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>35</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msubsup> <mi>D</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>4</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>4</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>D</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD61-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>41</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>B</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD62-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>42</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD63-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>43</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <msubsup> <mi>D</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>D</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD64-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>44</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>F</mi> <mrow> <mn>11</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <msubsup> <mi>A</mi> <mrow> <mn>44</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD65-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>45</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>F</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD66-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>51</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD67-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>52</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>B</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msubsup> <mi>B</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD68-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>53</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <msubsup> <mi>D</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>D</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD69-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>54</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>F</mi> <mrow> <mn>12</mn> </mrow> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD70-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mn>55</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>F</mi> <mrow> <mn>22</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>3</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mn>66</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <msubsup> <mi>A</mi> <mrow> <mn>55</mn> </mrow> <mi>s</mi> </msubsup> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div></div><div class='html-p'>Mass matrix elements; <div class='html-disp-formula-info' id='FD71-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>11</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>0</mn> </msub> <mo>+</mo> <mn>2</mn> <mfrac> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>3</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> <msup> <mrow/> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD72-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>13</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD73-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>14</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>4</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD74-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>22</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>0</mn> </msub> <mo>+</mo> <mn>2</mn> <mfrac> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> <msup> <mrow/> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD75-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>23</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD76-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>25</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>4</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD77-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>31</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD78-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>32</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD79-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>33</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mn>0</mn> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>−</mo> <msub> <mi>I</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> <mo>+</mo> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD80-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>34</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>I</mi> <mn>4</mn> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD81-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>34</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>I</mi> <mn>4</mn> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD82-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>41</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>4</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD83-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>43</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>I</mi> <mn>4</mn> </msub> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD84-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>44</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mn>5</mn> </msub> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>b</mi> </msubsup> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>x</mi> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD85-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>52</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mn>4</mn> </msub> </mrow> <mrow> <msub> <mi>R</mi> <mi>y</mi> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD86-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>53</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>I</mi> <mn>4</mn> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD87-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>55</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>I</mi> <mn>5</mn> </msub> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>a</mi> </munderover> <munderover> <mstyle mathsize="140%" displaystyle="true"> <mo>∫</mo> </mstyle> <mn>0</mn> <mi>b</mi> </munderover> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>y</mi> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> <mi mathvariant="normal">d</mi> <mi>y</mi> </mrow> </semantics></math> </div> <div class='l'> </div> </div><div class='html-disp-formula-info' id='FD88-mathematics-10-00408'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>12</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mn>15</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mn>21</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mn>24</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mn>42</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mn>45</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mn>51</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mn>54</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> </div> <div 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data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g001.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g001-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f001"></a> </div> </div> <div class="html-fig_description"> <b>Figure 1.</b> The material properties, geometry, and coordinate system of the shell, (<b>a</b>) Geometry and coordinates, (<b>b</b>) Gradation types of material.  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_mathematics-10-00408-f001" > <div class="html-caption" > <b>Figure 1.</b> The material properties, geometry, and coordinate system of the shell, (<b>a</b>) Geometry and coordinates, (<b>b</b>) Gradation types of material.</div> <div class="html-img"><img alt="Mathematics 10 00408 g001" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g001.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g001.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g001.png" /></div> </div><div class="html-fig-wrap" id="mathematics-10-00408-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f002"> <img alt="Mathematics 10 00408 g002 550" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g002.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g002.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g002-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> Forms of various plate/shells.  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_mathematics-10-00408-f002" > <div class="html-caption" > <b>Figure 2.</b> Forms of various plate/shells.</div> <div class="html-img"><img alt="Mathematics 10 00408 g002" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g002.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g002.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g002.png" /></div> </div><div class="html-fig-wrap" id="mathematics-10-00408-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f003"> <img alt="Mathematics 10 00408 g003 550" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g003.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g003.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> The volume-fractions of the fibers along the thickness of the shell (<math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_mathematics-10-00408-f003" > <div class="html-caption" > <b>Figure 3.</b> The volume-fractions of the fibers along the thickness of the shell (<math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</div> <div class="html-img"><img alt="Mathematics 10 00408 g003" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g003.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g003.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g003.png" /></div> </div><div class="html-fig-wrap" id="mathematics-10-00408-f004"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f004"> <img alt="Mathematics 10 00408 g004 550" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g004.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g004.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g004-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f004"></a> </div> </div> <div class="html-fig_description"> <b>Figure 4.</b> The effect of the number of layers “<span class='html-italic'>N</span>”: (<b>a</b>) Cross-ply fibers (<b>b</b>) Unidirectional fibers (Plate, SSSS, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_mathematics-10-00408-f004" > <div class="html-caption" > <b>Figure 4.</b> The effect of the number of layers “<span class='html-italic'>N</span>”: (<b>a</b>) Cross-ply fibers (<b>b</b>) Unidirectional fibers (Plate, SSSS, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</div> <div class="html-img"><img alt="Mathematics 10 00408 g004" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g004.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g004.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g004.png" /></div> </div><div class="html-fig-wrap" id="mathematics-10-00408-f005"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f005"> <img alt="Mathematics 10 00408 g005 550" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g005.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g005.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g005-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f005"></a> </div> </div> <div class="html-fig_description"> <b>Figure 5.</b> The effect of the radius of curvature <span class='html-italic'>R</span>/<span class='html-italic'>a</span> on the dimensionless frequency of various shell types (Cross-ply, SSSS, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_mathematics-10-00408-f005" > <div class="html-caption" > <b>Figure 5.</b> The effect of the radius of curvature <span class='html-italic'>R</span>/<span class='html-italic'>a</span> on the dimensionless frequency of various shell types (Cross-ply, SSSS, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</div> <div class="html-img"><img alt="Mathematics 10 00408 g005" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g005.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g005.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g005.png" /></div> </div><div class="html-fig-wrap" id="mathematics-10-00408-f006"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f006"> <img alt="Mathematics 10 00408 g006 550" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g006.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g006.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g006-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f006"></a> </div> </div> <div class="html-fig_description"> <b>Figure 6.</b> The effect of the volume fraction of fibers and CNTs (Plate, Cross-ply, SSSS, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>).  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_mathematics-10-00408-f006" > <div class="html-caption" > <b>Figure 6.</b> The effect of the volume fraction of fibers and CNTs (Plate, Cross-ply, SSSS, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>).</div> <div class="html-img"><img alt="Mathematics 10 00408 g006" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g006.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g006.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g006.png" /></div> </div><div class="html-fig-wrap" id="mathematics-10-00408-f007"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f007"> <img alt="Mathematics 10 00408 g007 550" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g007.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g007.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g007-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f007"></a> </div> </div> <div class="html-fig_description"> <b>Figure 7.</b> The effect of the geometry parameters <span class='html-italic'>b</span>/<span class='html-italic'>a</span> and <span class='html-italic'>a</span>/<span class='html-italic'>h</span> on the dimensionless frequency of a spherical shell for various boundary conditions (<math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>). (<b>a</b>) inplane ratio <span class='html-italic'>b</span>/<span class='html-italic'>a</span>, (<b>b</b>) slenderness ratio <span class='html-italic'>a</span>/<span class='html-italic'>h</span>.  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_mathematics-10-00408-f007" > <div class="html-caption" > <b>Figure 7.</b> The effect of the geometry parameters <span class='html-italic'>b</span>/<span class='html-italic'>a</span> and <span class='html-italic'>a</span>/<span class='html-italic'>h</span> on the dimensionless frequency of a spherical shell for various boundary conditions (<math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>). (<b>a</b>) inplane ratio <span class='html-italic'>b</span>/<span class='html-italic'>a</span>, (<b>b</b>) slenderness ratio <span class='html-italic'>a</span>/<span class='html-italic'>h</span>.</div> <div class="html-img"><img alt="Mathematics 10 00408 g007" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g007.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g007.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g007.png" /></div> </div><div class="html-fig-wrap" id="mathematics-10-00408-f008"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f008"> <img alt="Mathematics 10 00408 g008 550" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g008.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g008.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g008-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#fig_body_display_mathematics-10-00408-f008"></a> </div> </div> <div class="html-fig_description"> <b>Figure 8.</b> The effect of the modes of vibration “<span class='html-italic'>m</span>” and “<span class='html-italic'>n</span>” on the dimensionless frequency of a spherical shell (<math display='inline'><semantics> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>5</mn> <mo> </mo> </mrow> </semantics></math>, SSSS, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_mathematics-10-00408-f008" > <div class="html-caption" > <b>Figure 8.</b> The effect of the modes of vibration “<span class='html-italic'>m</span>” and “<span class='html-italic'>n</span>” on the dimensionless frequency of a spherical shell (<math display='inline'><semantics> <mrow> <msub> <mi>R</mi> <mi>x</mi> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>5</mn> <mo> </mo> </mrow> </semantics></math>, SSSS, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</div> <div class="html-img"><img alt="Mathematics 10 00408 g008" data-large="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g008.png" data-original="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g008.png" data-lsrc="/mathematics/mathematics-10-00408/article_deploy/html/images/mathematics-10-00408-g008.png" /></div> </div><div class="html-table-wrap" id="mathematics-10-00408-t001"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href='#table_body_display_mathematics-10-00408-t001'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#table_body_display_mathematics-10-00408-t001"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 1.</b> Hill’s elastic moduli for (10, 10) single-walled carbon nanotubes [<a href="#B52-mathematics-10-00408" class="html-bibr">52</a>]. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_mathematics-10-00408-t001" > <div class="html-caption" ><b>Table 1.</b> Hill’s elastic moduli for (10, 10) single-walled carbon nanotubes [<a href="#B52-mathematics-10-00408" class="html-bibr">52</a>].</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msub> <mi mathvariant="bold-italic">k</mi> <mrow> <mi mathvariant="bold-italic">c</mi> <mi mathvariant="bold-italic">n</mi> <mi mathvariant="bold-italic">t</mi> </mrow> </msub> <mo> </mo> <mrow> <mo>[</mo> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi mathvariant="bold">G</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>]</mo> </mrow> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msub> <mi mathvariant="bold-italic">l</mi> <mrow> <mi mathvariant="bold-italic">c</mi> <mi mathvariant="bold-italic">n</mi> <mi mathvariant="bold-italic">t</mi> </mrow> </msub> <mo> </mo> <mrow> <mo>[</mo> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi mathvariant="bold">G</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>]</mo> </mrow> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msub> <mi mathvariant="bold-italic">m</mi> <mrow> <mi mathvariant="bold-italic">c</mi> <mi mathvariant="bold-italic">n</mi> <mi mathvariant="bold-italic">t</mi> </mrow> </msub> <mo> </mo> <mrow> <mo>[</mo> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi mathvariant="bold">G</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>]</mo> </mrow> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msub> <mi mathvariant="bold-italic">n</mi> <mrow> <mi mathvariant="bold-italic">c</mi> <mi mathvariant="bold-italic">n</mi> <mi mathvariant="bold-italic">t</mi> <mo> </mo> </mrow> </msub> <mrow> <mo>[</mo> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi mathvariant="bold">G</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>]</mo> </mrow> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msub> <mi mathvariant="bold-italic">p</mi> <mrow> <mi mathvariant="bold-italic">c</mi> <mi mathvariant="bold-italic">n</mi> <mi mathvariant="bold-italic">t</mi> </mrow> </msub> <mo> </mo> <mrow> <mo>[</mo> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi mathvariant="bold">G</mi> <mi mathvariant="bold">P</mi> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>]</mo> </mrow> </mrow> </mstyle> </semantics> </math></th></tr></thead><tbody ><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >271</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >88</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >17</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1089</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >442</td></tr></tbody> </table> </div><div class="html-table-wrap" id="mathematics-10-00408-t002"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href='#table_body_display_mathematics-10-00408-t002'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#table_body_display_mathematics-10-00408-t002"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 2.</b> The essential boundary conditions. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_mathematics-10-00408-t002" > <div class="html-caption" ><b>Table 2.</b> The essential boundary conditions.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >BCs.</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Boundaries Parallel to the <span class='html-italic'>x</span>-Axis</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Boundaries Parallel to the <span class='html-italic'>y</span>-Axis</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >Simply supported (S)</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>x</mi> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>N</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>y</mi> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>N</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Clamped (C)</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>x</mi> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>x</mi> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math></td></tr></tbody> </table> </div><div class="html-table-wrap" id="mathematics-10-00408-t003"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href='#table_body_display_mathematics-10-00408-t003'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#table_body_display_mathematics-10-00408-t003"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 3.</b> The admissible functions <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="bold-italic">X</mi> <mi mathvariant="bold-italic">m</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="bold-italic">Y</mi> <mi mathvariant="bold-italic">n</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_mathematics-10-00408-t003" > <div class="html-caption" ><b>Table 3.</b> The admissible functions <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="bold-italic">X</mi> <mi mathvariant="bold-italic">m</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="bold-italic">Y</mi> <mi mathvariant="bold-italic">n</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >BCs.</th><th colspan='2' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <mi mathvariant="bold">The</mi> <mtext> </mtext> <mi mathvariant="bold">Functions</mi> <mtext> </mtext> <msub> <mi mathvariant="bold-italic">X</mi> <mi mathvariant="bold-italic">m</mi> </msub> </mrow> </mstyle> </semantics> </math><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <mtext> </mtext> <mi mathvariant="bold">and</mi> <mtext> </mtext> <msub> <mi mathvariant="bold-italic">Y</mi> <mi mathvariant="bold-italic">n</mi> </msub> </mrow> </mstyle> </semantics> </math></th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' > </td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <msub> <mi mathvariant="bold-italic">X</mi> <mi mathvariant="bold-italic">m</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">x</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math></td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <msub> <mi mathvariant="bold-italic">Y</mi> <mi mathvariant="bold-italic">n</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">y</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >SSSS</td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math></td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCCC</td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math></td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSCS</td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics> </math></td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCSS</td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math></td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSSS</td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics> </math></td><td align='left' valign='middle' class='html-align-left' ><math display='inline'> <semantics> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >CCCS</td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><math display='inline'> <semantics> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math></td><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><math display='inline'> <semantics> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics> </math></td></tr></tbody> </table> <div class='html-table_foot html-p'><div class='html-p' style='text-indent:0em;'><span class='html-fn-content'>Where <math display='inline'><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mi>m</mi> <mi>π</mi> <mo>/</mo> <mi>a</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mi>n</mi> <mi>π</mi> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math>. <math display='inline'><semantics> <mi>m</mi> </semantics></math> and <math display='inline'><semantics> <mi>n</mi> </semantics></math> are mode numbers.</span></div><div style='clear:both;'></div></div> </div><div class="html-table-wrap" id="mathematics-10-00408-t004"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href='#table_body_display_mathematics-10-00408-t004'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#table_body_display_mathematics-10-00408-t004"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 4.</b> Comparison of the natural frequency parameter <math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mover accent="true"> <mi>ω</mi> <mo>˜</mo> </mover> <mo>=</mo> <mi>ω</mi> <mi>h</mi> <msqrt> <mrow> <msub> <mi>ρ</mi> <mi>c</mi> </msub> <mo>/</mo> <msub> <mi>E</mi> <mi>c</mi> </msub> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> for simply supported Al/Al<sub>2</sub>O<sub>3</sub> functionally graded square plates and doubly curved shells (<span class='html-italic'>a</span> = <span class='html-italic'>b</span> = 10<span class='html-italic'>h</span>, <span class='html-italic'>m</span> = <span class='html-italic'>n</span> = 1). </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_mathematics-10-00408-t004" > <div class="html-caption" ><b>Table 4.</b> Comparison of the natural frequency parameter <math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mover accent="true"> <mi>ω</mi> <mo>˜</mo> </mover> <mo>=</mo> <mi>ω</mi> <mi>h</mi> <msqrt> <mrow> <msub> <mi>ρ</mi> <mi>c</mi> </msub> <mo>/</mo> <msub> <mi>E</mi> <mi>c</mi> </msub> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> for simply supported Al/Al<sub>2</sub>O<sub>3</sub> functionally graded square plates and doubly curved shells (<span class='html-italic'>a</span> = <span class='html-italic'>b</span> = 10<span class='html-italic'>h</span>, <span class='html-italic'>m</span> = <span class='html-italic'>n</span> = 1).</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' > </th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <mi mathvariant="bold-italic">a</mi> <mo>/</mo> <msub> <mi mathvariant="bold-italic">R</mi> <mi mathvariant="bold-italic">x</mi> </msub> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <mi mathvariant="bold-italic">b</mi> <mo>/</mo> <msub> <mi mathvariant="bold-italic">R</mi> <mi mathvariant="bold-italic">y</mi> </msub> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mi mathvariant="bold-italic">p</mi> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Alijani<br>[<a href="#B53-mathematics-10-00408" class="html-bibr">53</a>]</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Matsunaga<br>[<a href="#B54-mathematics-10-00408" class="html-bibr">54</a>]</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Chorfi<br>[<a href="#B55-mathematics-10-00408" class="html-bibr">55</a>]</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Trinh<br>[<a href="#B56-mathematics-10-00408" class="html-bibr">56</a>]</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Present</th></tr></thead><tbody ><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Plate</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' class='html-align-center' >0</td><td align='center' valign='middle' class='html-align-center' >0.0597</td><td align='center' valign='middle' class='html-align-center' >0.0588</td><td align='center' valign='middle' class='html-align-center' >0.0577</td><td align='center' valign='middle' class='html-align-center' >0.0577</td><td align='center' valign='middle' class='html-align-center' >0.0577</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >0.0506</td><td align='center' valign='middle' class='html-align-center' >0.0492</td><td align='center' valign='middle' class='html-align-center' >0.0490</td><td align='center' valign='middle' class='html-align-center' >0.0490</td><td align='center' valign='middle' class='html-align-center' >0.0490</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >0.0456</td><td align='center' valign='middle' class='html-align-center' >0.0430</td><td align='center' valign='middle' class='html-align-center' >0.0442</td><td align='center' valign='middle' class='html-align-center' >0.0442</td><td align='center' valign='middle' class='html-align-center' >0.0442</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >0.0396</td><td align='center' valign='middle' class='html-align-center' >0.0381</td><td align='center' valign='middle' class='html-align-center' >0.0383</td><td align='center' valign='middle' class='html-align-center' >0.0381</td><td align='center' valign='middle' class='html-align-center' >0.0380</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0380</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0364</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0366</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0364</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0363</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Spherical</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.5</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >0</td><td align='center' valign='middle' class='html-align-center' >0.0779</td><td align='center' valign='middle' class='html-align-center' >0.0751</td><td align='center' valign='middle' class='html-align-center' >0.0762</td><td align='center' valign='middle' class='html-align-center' >0.0761</td><td align='center' valign='middle' class='html-align-center' >0.0753</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >0.0676</td><td align='center' valign='middle' class='html-align-center' >0.0657</td><td align='center' valign='middle' class='html-align-center' >0.0664</td><td align='center' valign='middle' class='html-align-center' >0.0662</td><td align='center' valign='middle' class='html-align-center' >0.0653</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >0.0617</td><td align='center' valign='middle' class='html-align-center' >0.0601</td><td align='center' valign='middle' class='html-align-center' >0.0607</td><td align='center' valign='middle' class='html-align-center' >0.0605</td><td align='center' valign='middle' class='html-align-center' >0.0595</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >0.0519</td><td align='center' valign='middle' class='html-align-center' >0.0503</td><td align='center' valign='middle' class='html-align-center' >0.0509</td><td align='center' valign='middle' class='html-align-center' >0.0506</td><td align='center' valign='middle' class='html-align-center' >0.0496</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0482</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0464</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0471</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0467</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0459</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Cylindrical</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.5</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0</td><td align='center' valign='middle' class='html-align-center' >0</td><td align='center' valign='middle' class='html-align-center' >0.0648</td><td align='center' valign='middle' class='html-align-center' >0.0622</td><td align='center' valign='middle' class='html-align-center' >0.0629</td><td align='center' valign='middle' class='html-align-center' >0.0628</td><td align='center' valign='middle' class='html-align-center' >0.0622</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >0.0553</td><td align='center' valign='middle' class='html-align-center' >0.0535</td><td align='center' valign='middle' class='html-align-center' >0.0540</td><td align='center' valign='middle' class='html-align-center' >0.0538</td><td align='center' valign='middle' class='html-align-center' >0.0533</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >0.0501</td><td align='center' valign='middle' class='html-align-center' >0.0485</td><td align='center' valign='middle' class='html-align-center' >0.0490</td><td align='center' valign='middle' class='html-align-center' >0.0488</td><td align='center' valign='middle' class='html-align-center' >0.0482</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >0.0430</td><td align='center' valign='middle' class='html-align-center' >0.0413</td><td align='center' valign='middle' class='html-align-center' >0.0419</td><td align='center' valign='middle' class='html-align-center' >0.0416</td><td align='center' valign='middle' class='html-align-center' >0.0410</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0408</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0390</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0395</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0392</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0387</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Hyperbolic–paraboloidal</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.5</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−0.5</td><td align='center' valign='middle' class='html-align-center' >0</td><td align='center' valign='middle' class='html-align-center' >0.0597</td><td align='center' valign='middle' class='html-align-center' >0.0563</td><td align='center' valign='middle' class='html-align-center' >0.0580</td><td align='center' valign='middle' class='html-align-center' >0.0577</td><td align='center' valign='middle' class='html-align-center' >0.0563</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >0.0506</td><td align='center' valign='middle' class='html-align-center' >0.0479</td><td align='center' valign='middle' class='html-align-center' >0.0493</td><td align='center' valign='middle' class='html-align-center' >0.0490</td><td align='center' valign='middle' class='html-align-center' >0.0478</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >0.0456</td><td align='center' valign='middle' class='html-align-center' >0.0432</td><td align='center' valign='middle' class='html-align-center' >0.0445</td><td align='center' valign='middle' class='html-align-center' >0.0442</td><td align='center' valign='middle' class='html-align-center' >0.0431</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >0.0396</td><td align='center' valign='middle' class='html-align-center' >0.0372</td><td align='center' valign='middle' class='html-align-center' >0.0385</td><td align='center' valign='middle' class='html-align-center' >0.0381</td><td align='center' valign='middle' class='html-align-center' >0.0371</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0380</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0355</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0368</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0364</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0354</td></tr></tbody> </table> </div><div class="html-table-wrap" id="mathematics-10-00408-t005"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href='#table_body_display_mathematics-10-00408-t005'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#table_body_display_mathematics-10-00408-t005"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 5.</b> The effect of the number of layers and fiber distribution patterns on the dimensionless frequency of cross-ply CNTs/F-RC shells for various curvature radii (SSSS, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>). </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_mathematics-10-00408-t005" > <div class="html-caption" ><b>Table 5.</b> The effect of the number of layers and fiber distribution patterns on the dimensionless frequency of cross-ply CNTs/F-RC shells for various curvature radii (SSSS, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' > </th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msub> <mi mathvariant="bold-italic">R</mi> <mi mathvariant="bold-italic">x</mi> </msub> <mo>/</mo> <mi mathvariant="bold-italic">a</mi> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msub> <mi mathvariant="bold-italic">R</mi> <mi mathvariant="bold-italic">y</mi> </msub> <mo>/</mo> <mi mathvariant="bold-italic">b</mi> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><span class='html-italic'>N</span></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >UD</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >FG-X</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >FG-O</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >FG-V</th></tr></thead><tbody ><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Plate</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >inf</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >inf</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >0.1896</td><td align='center' valign='middle' class='html-align-center' >0.1862</td><td align='center' valign='middle' class='html-align-center' >0.1750</td><td align='center' valign='middle' class='html-align-center' >0.1779</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >7</td><td align='center' valign='middle' class='html-align-center' >0.1900</td><td align='center' valign='middle' class='html-align-center' >0.1840</td><td align='center' valign='middle' class='html-align-center' >0.1734</td><td align='center' valign='middle' class='html-align-center' >0.1783</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >11</td><td align='center' valign='middle' class='html-align-center' >0.1900</td><td align='center' valign='middle' class='html-align-center' >0.1927</td><td align='center' valign='middle' class='html-align-center' >0.1823</td><td align='center' valign='middle' class='html-align-center' >0.1881</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >15</td><td align='center' valign='middle' class='html-align-center' >0.1901</td><td align='center' valign='middle' class='html-align-center' >0.2040</td><td align='center' valign='middle' class='html-align-center' >0.1934</td><td align='center' valign='middle' class='html-align-center' >0.1984</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1901</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1976</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1821</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1893</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Cylindrical shell</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >inf</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >0.1917</td><td align='center' valign='middle' class='html-align-center' >0.1883</td><td align='center' valign='middle' class='html-align-center' >0.1778</td><td align='center' valign='middle' class='html-align-center' >0.1789</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >7</td><td align='center' valign='middle' class='html-align-center' >0.1921</td><td align='center' valign='middle' class='html-align-center' >0.1862</td><td align='center' valign='middle' class='html-align-center' >0.1762</td><td align='center' valign='middle' class='html-align-center' >0.1792</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >11</td><td align='center' valign='middle' class='html-align-center' >0.1922</td><td align='center' valign='middle' class='html-align-center' >0.1948</td><td align='center' valign='middle' class='html-align-center' >0.1848</td><td align='center' valign='middle' class='html-align-center' >0.1888</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >15</td><td align='center' valign='middle' class='html-align-center' >0.1922</td><td align='center' valign='middle' class='html-align-center' >0.2059</td><td align='center' valign='middle' class='html-align-center' >0.1959</td><td align='center' valign='middle' class='html-align-center' >0.1988</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1922</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1997</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1844</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1899</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Spherical shell</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >0.1986</td><td align='center' valign='middle' class='html-align-center' >0.1953</td><td align='center' valign='middle' class='html-align-center' >0.1856</td><td align='center' valign='middle' class='html-align-center' >0.1855</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >7</td><td align='center' valign='middle' class='html-align-center' >0.1991</td><td align='center' valign='middle' class='html-align-center' >0.1933</td><td align='center' valign='middle' class='html-align-center' >0.1842</td><td align='center' valign='middle' class='html-align-center' >0.1854</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >11</td><td align='center' valign='middle' class='html-align-center' >0.1992</td><td align='center' valign='middle' class='html-align-center' >0.2016</td><td align='center' valign='middle' class='html-align-center' >0.1924</td><td align='center' valign='middle' class='html-align-center' >0.1947</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >15</td><td align='center' valign='middle' class='html-align-center' >0.1992</td><td align='center' valign='middle' class='html-align-center' >0.2125</td><td align='center' valign='middle' class='html-align-center' >0.2031</td><td align='center' valign='middle' class='html-align-center' >0.2041</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1992</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2064</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1917</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1955</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Elliptical–paraboloid shell</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >7.5</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >0.1959</td><td align='center' valign='middle' class='html-align-center' >0.1925</td><td align='center' valign='middle' class='html-align-center' >0.1825</td><td align='center' valign='middle' class='html-align-center' >0.1828</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >7</td><td align='center' valign='middle' class='html-align-center' >0.1964</td><td align='center' valign='middle' class='html-align-center' >0.1905</td><td align='center' valign='middle' class='html-align-center' >0.1811</td><td align='center' valign='middle' class='html-align-center' >0.1829</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >11</td><td align='center' valign='middle' class='html-align-center' >0.1964</td><td align='center' valign='middle' class='html-align-center' >0.1989</td><td align='center' valign='middle' class='html-align-center' >0.1894</td><td align='center' valign='middle' class='html-align-center' >0.1923</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >15</td><td align='center' valign='middle' class='html-align-center' >0.1964</td><td align='center' valign='middle' class='html-align-center' >0.2098</td><td align='center' valign='middle' class='html-align-center' >0.2003</td><td align='center' valign='middle' class='html-align-center' >0.2019</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1964</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2038</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1888</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1932</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Hyperbolic–paraboloidal shell</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−5</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >0.1888</td><td align='center' valign='middle' class='html-align-center' >0.1855</td><td align='center' valign='middle' class='html-align-center' >0.1744</td><td align='center' valign='middle' class='html-align-center' >0.1769</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >7</td><td align='center' valign='middle' class='html-align-center' >0.1892</td><td align='center' valign='middle' class='html-align-center' >0.1834</td><td align='center' valign='middle' class='html-align-center' >0.1728</td><td align='center' valign='middle' class='html-align-center' >0.1776</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >11</td><td align='center' valign='middle' class='html-align-center' >0.1893</td><td align='center' valign='middle' class='html-align-center' >0.1920</td><td align='center' valign='middle' class='html-align-center' >0.1816</td><td align='center' valign='middle' class='html-align-center' >0.1873</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >15</td><td align='center' valign='middle' class='html-align-center' >0.1893</td><td align='center' valign='middle' class='html-align-center' >0.2032</td><td align='center' valign='middle' class='html-align-center' >0.1927</td><td align='center' valign='middle' class='html-align-center' >0.1975</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1893</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1969</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1814</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1885</td></tr></tbody> </table> </div><div class="html-table-wrap" id="mathematics-10-00408-t006"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href='#table_body_display_mathematics-10-00408-t006'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#table_body_display_mathematics-10-00408-t006"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 6.</b> The effect of the number of layers and fiber distribution patterns on the dimensionless frequency of unidirectional F/CNT-RC shells for various curvature radii (SSSS, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>). </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_mathematics-10-00408-t006" > <div class="html-caption" ><b>Table 6.</b> The effect of the number of layers and fiber distribution patterns on the dimensionless frequency of unidirectional F/CNT-RC shells for various curvature radii (SSSS, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' > </th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msub> <mi mathvariant="bold-italic">R</mi> <mi mathvariant="bold-italic">x</mi> </msub> <mo>/</mo> <mi mathvariant="bold-italic">a</mi> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msub> <mi mathvariant="bold-italic">R</mi> <mi mathvariant="bold-italic">y</mi> </msub> <mo>/</mo> <mi mathvariant="bold-italic">b</mi> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><b><span class='html-italic'>N</span></b></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><b>UD</b></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><b>FG-X</b></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><b>FG-O</b></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><b>FG-V</b></th></tr></thead><tbody ><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Plate</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >inf</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >inf</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >0.1894</td><td align='center' valign='middle' class='html-align-center' >0.1858</td><td align='center' valign='middle' class='html-align-center' >0.1748</td><td align='center' valign='middle' class='html-align-center' >0.1776</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >7</td><td align='center' valign='middle' class='html-align-center' >0.1894</td><td align='center' valign='middle' class='html-align-center' >0.1834</td><td align='center' valign='middle' class='html-align-center' >0.1735</td><td align='center' valign='middle' class='html-align-center' >0.1773</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >11</td><td align='center' valign='middle' class='html-align-center' >0.1894</td><td align='center' valign='middle' class='html-align-center' >0.1919</td><td align='center' valign='middle' class='html-align-center' >0.1818</td><td align='center' valign='middle' class='html-align-center' >0.1868</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >15</td><td align='center' valign='middle' class='html-align-center' >0.1894</td><td align='center' valign='middle' class='html-align-center' >0.2029</td><td align='center' valign='middle' class='html-align-center' >0.1928</td><td align='center' valign='middle' class='html-align-center' >0.1968</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1894</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1967</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1817</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1879</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Cylindrical shell</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >inf</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >0.1913</td><td align='center' valign='middle' class='html-align-center' >0.1877</td><td align='center' valign='middle' class='html-align-center' >0.1771</td><td align='center' valign='middle' class='html-align-center' >0.1790</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >7</td><td align='center' valign='middle' class='html-align-center' >0.1913</td><td align='center' valign='middle' class='html-align-center' >0.1854</td><td align='center' valign='middle' class='html-align-center' >0.1754</td><td align='center' valign='middle' class='html-align-center' >0.1787</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >11</td><td align='center' valign='middle' class='html-align-center' >0.1913</td><td align='center' valign='middle' class='html-align-center' >0.1937</td><td align='center' valign='middle' class='html-align-center' >0.1840</td><td align='center' valign='middle' class='html-align-center' >0.1881</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >15</td><td align='center' valign='middle' class='html-align-center' >0.1913</td><td align='center' valign='middle' class='html-align-center' >0.2046</td><td align='center' valign='middle' class='html-align-center' >0.1948</td><td align='center' valign='middle' class='html-align-center' >0.1979</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1913</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1985</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1838</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1891</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Spherical shell</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >0.1979</td><td align='center' valign='middle' class='html-align-center' >0.1941</td><td align='center' valign='middle' class='html-align-center' >0.1841</td><td align='center' valign='middle' class='html-align-center' >0.1855</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >7</td><td align='center' valign='middle' class='html-align-center' >0.1979</td><td align='center' valign='middle' class='html-align-center' >0.1916</td><td align='center' valign='middle' class='html-align-center' >0.1822</td><td align='center' valign='middle' class='html-align-center' >0.1849</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >11</td><td align='center' valign='middle' class='html-align-center' >0.1979</td><td align='center' valign='middle' class='html-align-center' >0.1995</td><td align='center' valign='middle' class='html-align-center' >0.1901</td><td align='center' valign='middle' class='html-align-center' >0.1939</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >15</td><td align='center' valign='middle' class='html-align-center' >0.1979</td><td align='center' valign='middle' class='html-align-center' >0.2102</td><td align='center' valign='middle' class='html-align-center' >0.2007</td><td align='center' valign='middle' class='html-align-center' >0.2035</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1979</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2048</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1906</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1955</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Elliptical–paraboloid shell</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >7.5</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >0.1953</td><td align='center' valign='middle' class='html-align-center' >0.1915</td><td align='center' valign='middle' class='html-align-center' >0.1813</td><td align='center' valign='middle' class='html-align-center' >0.1829</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >7</td><td align='center' valign='middle' class='html-align-center' >0.1953</td><td align='center' valign='middle' class='html-align-center' >0.1891</td><td align='center' valign='middle' class='html-align-center' >0.1795</td><td align='center' valign='middle' class='html-align-center' >0.1824</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >11</td><td align='center' valign='middle' class='html-align-center' >0.1953</td><td align='center' valign='middle' class='html-align-center' >0.1972</td><td align='center' valign='middle' class='html-align-center' >0.1877</td><td align='center' valign='middle' class='html-align-center' >0.1916</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >15</td><td align='center' valign='middle' class='html-align-center' >0.1953</td><td align='center' valign='middle' class='html-align-center' >0.2079</td><td align='center' valign='middle' class='html-align-center' >0.1983</td><td align='center' valign='middle' class='html-align-center' >0.2012</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1953</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2023</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1879</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1929</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Hyperbolic–paraboloidal shell</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >−5</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >0.1887</td><td align='center' valign='middle' class='html-align-center' >0.1852</td><td align='center' valign='middle' class='html-align-center' >0.1743</td><td align='center' valign='middle' class='html-align-center' >0.1768</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >7</td><td align='center' valign='middle' class='html-align-center' >0.1887</td><td align='center' valign='middle' class='html-align-center' >0.1831</td><td align='center' valign='middle' class='html-align-center' >0.1728</td><td align='center' valign='middle' class='html-align-center' >0.1768</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >11</td><td align='center' valign='middle' class='html-align-center' >0.1887</td><td align='center' valign='middle' class='html-align-center' >0.1917</td><td align='center' valign='middle' class='html-align-center' >0.1817</td><td align='center' valign='middle' class='html-align-center' >0.1864</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >15</td><td align='center' valign='middle' class='html-align-center' >0.1887</td><td align='center' valign='middle' class='html-align-center' >0.2026</td><td align='center' valign='middle' class='html-align-center' >0.1926</td><td align='center' valign='middle' class='html-align-center' >0.1960</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1887</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1959</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1810</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1867</td></tr></tbody> </table> </div><div class="html-table-wrap" id="mathematics-10-00408-t007"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href='#table_body_display_mathematics-10-00408-t007'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#table_body_display_mathematics-10-00408-t007"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 7.</b> The effect of the volume fraction of CNTs and fiber distribution patterns on the dimensionless frequency of cross-ply F/CNT-RC shells (SSSS, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>). </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_mathematics-10-00408-t007" > <div class="html-caption" ><b>Table 7.</b> The effect of the volume fraction of CNTs and fiber distribution patterns on the dimensionless frequency of cross-ply F/CNT-RC shells (SSSS, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</div> <table > <thead ><tr ><th rowspan='2' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Type of Shells</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' > </th><th colspan='4' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Fiber Distribution Pattern</th></tr><tr ><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msub> <mi mathvariant="bold-italic">V</mi> <mrow> <mi mathvariant="bold-italic">c</mi> <mi mathvariant="bold-italic">n</mi> <mi mathvariant="bold-italic">t</mi> </mrow> </msub> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC UD</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC FG-X</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC FG-O</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC FG-V</th></tr></thead><tbody ><tr ><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Plate</td><td align='center' valign='middle' class='html-align-center' >0.0%</td><td align='center' valign='middle' class='html-align-center' >0.1176</td><td align='center' valign='middle' class='html-align-center' >0.1288</td><td align='center' valign='middle' class='html-align-center' >0.1033</td><td align='center' valign='middle' class='html-align-center' >0.1140</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.2%</td><td align='center' valign='middle' class='html-align-center' >0.1387</td><td align='center' valign='middle' class='html-align-center' >0.1508</td><td align='center' valign='middle' class='html-align-center' >0.1246</td><td align='center' valign='middle' class='html-align-center' >0.1358</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.4%</td><td align='center' valign='middle' class='html-align-center' >0.1543</td><td align='center' valign='middle' class='html-align-center' >0.1658</td><td align='center' valign='middle' class='html-align-center' >0.1414</td><td align='center' valign='middle' class='html-align-center' >0.1521</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.6%</td><td align='center' valign='middle' class='html-align-center' >0.1676</td><td align='center' valign='middle' class='html-align-center' >0.1781</td><td align='center' valign='middle' class='html-align-center' >0.1560</td><td align='center' valign='middle' class='html-align-center' >0.1659</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.8%</td><td align='center' valign='middle' class='html-align-center' >0.1794</td><td align='center' valign='middle' class='html-align-center' >0.1888</td><td align='center' valign='middle' class='html-align-center' >0.1691</td><td align='center' valign='middle' class='html-align-center' >0.1780</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1.0%</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1900</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1984</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1811</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1891</td></tr><tr ><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Cylindrical shell</td><td align='center' valign='middle' class='html-align-center' >0.0%</td><td align='center' valign='middle' class='html-align-center' >0.1181</td><td align='center' valign='middle' class='html-align-center' >0.1292</td><td align='center' valign='middle' class='html-align-center' >0.1039</td><td align='center' valign='middle' class='html-align-center' >0.1140</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.2%</td><td align='center' valign='middle' class='html-align-center' >0.1398</td><td align='center' valign='middle' class='html-align-center' >0.1517</td><td align='center' valign='middle' class='html-align-center' >0.1258</td><td align='center' valign='middle' class='html-align-center' >0.1358</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.4%</td><td align='center' valign='middle' class='html-align-center' >0.1557</td><td align='center' valign='middle' class='html-align-center' >0.1671</td><td align='center' valign='middle' class='html-align-center' >0.1430</td><td align='center' valign='middle' class='html-align-center' >0.1522</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.6%</td><td align='center' valign='middle' class='html-align-center' >0.1693</td><td align='center' valign='middle' class='html-align-center' >0.1797</td><td align='center' valign='middle' class='html-align-center' >0.1578</td><td align='center' valign='middle' class='html-align-center' >0.1661</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.8%</td><td align='center' valign='middle' class='html-align-center' >0.1813</td><td align='center' valign='middle' class='html-align-center' >0.1906</td><td align='center' valign='middle' class='html-align-center' >0.1712</td><td align='center' valign='middle' class='html-align-center' >0.1785</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1.0%</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1922</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2004</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1834</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1896</td></tr><tr ><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Spherical shell</td><td align='center' valign='middle' class='html-align-center' >0.0%</td><td align='center' valign='middle' class='html-align-center' >0.1200</td><td align='center' valign='middle' class='html-align-center' >0.1310</td><td align='center' valign='middle' class='html-align-center' >0.1062</td><td align='center' valign='middle' class='html-align-center' >0.1163</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.2%</td><td align='center' valign='middle' class='html-align-center' >0.1434</td><td align='center' valign='middle' class='html-align-center' >0.1551</td><td align='center' valign='middle' class='html-align-center' >0.1298</td><td align='center' valign='middle' class='html-align-center' >0.1389</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.4%</td><td align='center' valign='middle' class='html-align-center' >0.1605</td><td align='center' valign='middle' class='html-align-center' >0.1715</td><td align='center' valign='middle' class='html-align-center' >0.1481</td><td align='center' valign='middle' class='html-align-center' >0.1561</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.6%</td><td align='center' valign='middle' class='html-align-center' >0.1749</td><td align='center' valign='middle' class='html-align-center' >0.1850</td><td align='center' valign='middle' class='html-align-center' >0.1639</td><td align='center' valign='middle' class='html-align-center' >0.1707</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.8%</td><td align='center' valign='middle' class='html-align-center' >0.1877</td><td align='center' valign='middle' class='html-align-center' >0.1967</td><td align='center' valign='middle' class='html-align-center' >0.1779</td><td align='center' valign='middle' class='html-align-center' >0.1836</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1.0%</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1992</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2071</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1907</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1953</td></tr><tr ><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Elliptical–paraboloid shell</td><td align='center' valign='middle' class='html-align-center' >0.0%</td><td align='center' valign='middle' class='html-align-center' >0.1193</td><td align='center' valign='middle' class='html-align-center' >0.1303</td><td align='center' valign='middle' class='html-align-center' >0.1053</td><td align='center' valign='middle' class='html-align-center' >0.1154</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.2%</td><td align='center' valign='middle' class='html-align-center' >0.1419</td><td align='center' valign='middle' class='html-align-center' >0.1538</td><td align='center' valign='middle' class='html-align-center' >0.1283</td><td align='center' valign='middle' class='html-align-center' >0.1377</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.4%</td><td align='center' valign='middle' class='html-align-center' >0.1586</td><td align='center' valign='middle' class='html-align-center' >0.1698</td><td align='center' valign='middle' class='html-align-center' >0.1461</td><td align='center' valign='middle' class='html-align-center' >0.1545</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.6%</td><td align='center' valign='middle' class='html-align-center' >0.1727</td><td align='center' valign='middle' class='html-align-center' >0.1829</td><td align='center' valign='middle' class='html-align-center' >0.1615</td><td align='center' valign='middle' class='html-align-center' >0.1688</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.8%</td><td align='center' valign='middle' class='html-align-center' >0.1852</td><td align='center' valign='middle' class='html-align-center' >0.1943</td><td align='center' valign='middle' class='html-align-center' >0.1753</td><td align='center' valign='middle' class='html-align-center' >0.1815</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1.0%</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1964</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2045</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1878</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1929</td></tr><tr ><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Hyperbolic–paraboloidal shell</td><td align='center' valign='middle' class='html-align-center' >0.0%</td><td align='center' valign='middle' class='html-align-center' >0.1171</td><td align='center' valign='middle' class='html-align-center' >0.1283</td><td align='center' valign='middle' class='html-align-center' >0.1029</td><td align='center' valign='middle' class='html-align-center' >0.1125</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.2%</td><td align='center' valign='middle' class='html-align-center' >0.1382</td><td align='center' valign='middle' class='html-align-center' >0.1502</td><td align='center' valign='middle' class='html-align-center' >0.1241</td><td align='center' valign='middle' class='html-align-center' >0.1348</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.4%</td><td align='center' valign='middle' class='html-align-center' >0.1537</td><td align='center' valign='middle' class='html-align-center' >0.1651</td><td align='center' valign='middle' class='html-align-center' >0.1408</td><td align='center' valign='middle' class='html-align-center' >0.1511</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.6%</td><td align='center' valign='middle' class='html-align-center' >0.1669</td><td align='center' valign='middle' class='html-align-center' >0.1774</td><td align='center' valign='middle' class='html-align-center' >0.1554</td><td align='center' valign='middle' class='html-align-center' >0.1649</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.8%</td><td align='center' valign='middle' class='html-align-center' >0.1786</td><td align='center' valign='middle' class='html-align-center' >0.1888</td><td align='center' valign='middle' class='html-align-center' >0.1684</td><td align='center' valign='middle' class='html-align-center' >0.1771</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1.0%</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1893</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1976</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1804</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1881</td></tr></tbody> </table> </div><div class="html-table-wrap" id="mathematics-10-00408-t008"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href='#table_body_display_mathematics-10-00408-t008'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#table_body_display_mathematics-10-00408-t008"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 8.</b> The effect of the volume fraction and distribution patterns of the fibers on the dimensionless frequency of cross-ply F/CNT-RC shells (SSSS, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>). </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_mathematics-10-00408-t008" > <div class="html-caption" ><b>Table 8.</b> The effect of the volume fraction and distribution patterns of the fibers on the dimensionless frequency of cross-ply F/CNT-RC shells (SSSS, <math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>).</div> <table > <thead ><tr ><th rowspan='2' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Type of Shells</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' > </th><th colspan='4' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Fiber Distribution Pattern</th></tr><tr ><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <msub> <mi mathvariant="bold-italic">V</mi> <mi mathvariant="bold-italic">F</mi> </msub> </mrow> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC UD</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC FG-X</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC FG-O</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC FG-V</th></tr></thead><tbody ><tr ><td rowspan='4' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Plate</td><td align='center' valign='middle' class='html-align-center' >5%</td><td align='center' valign='middle' class='html-align-center' >0.1735</td><td align='center' valign='middle' class='html-align-center' >0.1735</td><td align='center' valign='middle' class='html-align-center' >0.1735</td><td align='center' valign='middle' class='html-align-center' >0.1735</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >10%</td><td align='center' valign='middle' class='html-align-center' >0.1765</td><td align='center' valign='middle' class='html-align-center' >0.1790</td><td align='center' valign='middle' class='html-align-center' >0.1740</td><td align='center' valign='middle' class='html-align-center' >0.1765</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >20%</td><td align='center' valign='middle' class='html-align-center' >0.1822</td><td align='center' valign='middle' class='html-align-center' >0.1891</td><td align='center' valign='middle' class='html-align-center' >0.1751</td><td align='center' valign='middle' class='html-align-center' >0.18158</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >30%</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1875</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1982</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1760</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1859</td></tr><tr ><td rowspan='4' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Cylindrical shell</td><td align='center' valign='middle' class='html-align-center' >5%</td><td align='center' valign='middle' class='html-align-center' >0.1758</td><td align='center' valign='middle' class='html-align-center' >0.1758</td><td align='center' valign='middle' class='html-align-center' >0.1758</td><td align='center' valign='middle' class='html-align-center' >0.1758</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >10%</td><td align='center' valign='middle' class='html-align-center' >0.1788</td><td align='center' valign='middle' class='html-align-center' >0.1812</td><td align='center' valign='middle' class='html-align-center' >0.1763</td><td align='center' valign='middle' class='html-align-center' >0.1782</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >20%</td><td align='center' valign='middle' class='html-align-center' >0.1844</td><td align='center' valign='middle' class='html-align-center' >0.1912</td><td align='center' valign='middle' class='html-align-center' >0.1774</td><td align='center' valign='middle' class='html-align-center' >0.1823</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >30%</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1897</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2002</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1784</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1860</td></tr><tr ><td rowspan='4' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Spherical shell</td><td align='center' valign='middle' class='html-align-center' >5%</td><td align='center' valign='middle' class='html-align-center' >0.1831</td><td align='center' valign='middle' class='html-align-center' >0.1831</td><td align='center' valign='middle' class='html-align-center' >0.1831</td><td align='center' valign='middle' class='html-align-center' >0.1831</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >10%</td><td align='center' valign='middle' class='html-align-center' >0.1861</td><td align='center' valign='middle' class='html-align-center' >0.1884</td><td align='center' valign='middle' class='html-align-center' >0.1837</td><td align='center' valign='middle' class='html-align-center' >0.1850</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >20%</td><td align='center' valign='middle' class='html-align-center' >0.1916</td><td align='center' valign='middle' class='html-align-center' >0.1982</td><td align='center' valign='middle' class='html-align-center' >0.1848</td><td align='center' valign='middle' class='html-align-center' >0.1883</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >30%</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1968</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2069</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1859</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1912</td></tr><tr ><td rowspan='4' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Elliptical–paraboloid shell</td><td align='center' valign='middle' class='html-align-center' >5%</td><td align='center' valign='middle' class='html-align-center' >0.1802</td><td align='center' valign='middle' class='html-align-center' >0.1802</td><td align='center' valign='middle' class='html-align-center' >0.1802</td><td align='center' valign='middle' class='html-align-center' >0.1802</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >10%</td><td align='center' valign='middle' class='html-align-center' >0.1832</td><td align='center' valign='middle' class='html-align-center' >0.1856</td><td align='center' valign='middle' class='html-align-center' >0.1808</td><td align='center' valign='middle' class='html-align-center' >0.1822</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >20%</td><td align='center' valign='middle' class='html-align-center' >0.1888</td><td align='center' valign='middle' class='html-align-center' >0.1954</td><td align='center' valign='middle' class='html-align-center' >0.1819</td><td align='center' valign='middle' class='html-align-center' >0.1858</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >30%</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1940</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2042</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1829</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1890</td></tr><tr ><td rowspan='4' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Hyperbolic–paraboloidal shell</td><td align='center' valign='middle' class='html-align-center' >0.0%</td><td align='center' valign='middle' class='html-align-center' >0.1728</td><td align='center' valign='middle' class='html-align-center' >0.1728</td><td align='center' valign='middle' class='html-align-center' >0.1728</td><td align='center' valign='middle' class='html-align-center' >0.1728</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.2%</td><td align='center' valign='middle' class='html-align-center' >0.1758</td><td align='center' valign='middle' class='html-align-center' >0.1783</td><td align='center' valign='middle' class='html-align-center' >0.1733</td><td align='center' valign='middle' class='html-align-center' >0.1757</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >0.8%</td><td align='center' valign='middle' class='html-align-center' >0.1815</td><td align='center' valign='middle' class='html-align-center' >0.1884</td><td align='center' valign='middle' class='html-align-center' >0.1743</td><td align='center' valign='middle' class='html-align-center' >0.1808</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1.0%</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1868</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1974</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1753</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1850</td></tr></tbody> </table> </div><div class="html-table-wrap" id="mathematics-10-00408-t009"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href='#table_body_display_mathematics-10-00408-t009'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#table_body_display_mathematics-10-00408-t009"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 9.</b> The effect of different boundary conditions and fiber distribution patterns on the dimensionless frequency of cross-ply F/CNT-RC shells (<math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>). </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_mathematics-10-00408-t009" > <div class="html-caption" ><b>Table 9.</b> The effect of different boundary conditions and fiber distribution patterns on the dimensionless frequency of cross-ply F/CNT-RC shells (<math display='inline'><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</div> <table > <thead ><tr ><th rowspan='2' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Type of Shells</th><th rowspan='2' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >BCs.</th><th colspan='4' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Fiber Distribution Pattern</th></tr><tr ><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC UD</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC FG-X</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC FG-O</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FRC FG-V</th></tr></thead><tbody ><tr ><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Plate</td><td align='center' valign='middle' class='html-align-center' >SSSS</td><td align='center' valign='middle' class='html-align-center' >0.1900</td><td align='center' valign='middle' class='html-align-center' >0.1984</td><td align='center' valign='middle' class='html-align-center' >0.1811</td><td align='center' valign='middle' class='html-align-center' >0.1891</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCCC</td><td align='center' valign='middle' class='html-align-center' >0.3599</td><td align='center' valign='middle' class='html-align-center' >0.3778</td><td align='center' valign='middle' class='html-align-center' >0.3398</td><td align='center' valign='middle' class='html-align-center' >0.3572</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCSS</td><td align='center' valign='middle' class='html-align-center' >0.2978</td><td align='center' valign='middle' class='html-align-center' >0.3162</td><td align='center' valign='middle' class='html-align-center' >0.2767</td><td align='center' valign='middle' class='html-align-center' >0.2950</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSCS</td><td align='center' valign='middle' class='html-align-center' >0.3409</td><td align='center' valign='middle' class='html-align-center' >0.3567</td><td align='center' valign='middle' class='html-align-center' >0.3237</td><td align='center' valign='middle' class='html-align-center' >0.3389</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSSS</td><td align='center' valign='middle' class='html-align-center' >0.2821</td><td align='center' valign='middle' class='html-align-center' >0.2989</td><td align='center' valign='middle' class='html-align-center' >0.2634</td><td align='center' valign='middle' class='html-align-center' >0.2797</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >CCCS</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2386</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2581</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2155</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2353</td></tr><tr ><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Cylindrical shell</td><td align='center' valign='middle' class='html-align-center' >SSSS</td><td align='center' valign='middle' class='html-align-center' >0.1922</td><td align='center' valign='middle' class='html-align-center' >0.2004</td><td align='center' valign='middle' class='html-align-center' >0.1834</td><td align='center' valign='middle' class='html-align-center' >0.1896</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCCC</td><td align='center' valign='middle' class='html-align-center' >0.3661</td><td align='center' valign='middle' class='html-align-center' >0.3839</td><td align='center' valign='middle' class='html-align-center' >0.3461</td><td align='center' valign='middle' class='html-align-center' >0.3633</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCSS</td><td align='center' valign='middle' class='html-align-center' >0.3053</td><td align='center' valign='middle' class='html-align-center' >0.3235</td><td align='center' valign='middle' class='html-align-center' >0.2844</td><td align='center' valign='middle' class='html-align-center' >0.3023</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSCS</td><td align='center' valign='middle' class='html-align-center' >0.3449</td><td align='center' valign='middle' class='html-align-center' >0.3607</td><td align='center' valign='middle' class='html-align-center' >0.3277</td><td align='center' valign='middle' class='html-align-center' >0.3421</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSSS</td><td align='center' valign='middle' class='html-align-center' >0.2866</td><td align='center' valign='middle' class='html-align-center' >0.3033</td><td align='center' valign='middle' class='html-align-center' >0.2681</td><td align='center' valign='middle' class='html-align-center' >0.2837</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >CCCS</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2484</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2676</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2258</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2452</td></tr><tr ><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Spherical shell</td><td align='center' valign='middle' class='html-align-center' >SSSS</td><td align='center' valign='middle' class='html-align-center' >0.1992</td><td align='center' valign='middle' class='html-align-center' >0.2071</td><td align='center' valign='middle' class='html-align-center' >0.1907</td><td align='center' valign='middle' class='html-align-center' >0.1953</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCCC</td><td align='center' valign='middle' class='html-align-center' >0.3729</td><td align='center' valign='middle' class='html-align-center' >0.3904</td><td align='center' valign='middle' class='html-align-center' >0.3533</td><td align='center' valign='middle' class='html-align-center' >0.3706</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCSS</td><td align='center' valign='middle' class='html-align-center' >0.3083</td><td align='center' valign='middle' class='html-align-center' >0.3264</td><td align='center' valign='middle' class='html-align-center' >0.2877</td><td align='center' valign='middle' class='html-align-center' >0.3042</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSCS</td><td align='center' valign='middle' class='html-align-center' >0.3501</td><td align='center' valign='middle' class='html-align-center' >0.3656</td><td align='center' valign='middle' class='html-align-center' >0.3332</td><td align='center' valign='middle' class='html-align-center' >0.3470</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSSS</td><td align='center' valign='middle' class='html-align-center' >0.2911</td><td align='center' valign='middle' class='html-align-center' >0.3075</td><td align='center' valign='middle' class='html-align-center' >0.2729</td><td align='center' valign='middle' class='html-align-center' >0.2868</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >CCCS</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2611</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2794</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2398</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2584</td></tr><tr ><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Elliptical–paraboloid shell</td><td align='center' valign='middle' class='html-align-center' >SSSS</td><td align='center' valign='middle' class='html-align-center' >0.1964</td><td align='center' valign='middle' class='html-align-center' >0.2045</td><td align='center' valign='middle' class='html-align-center' >0.1878</td><td align='center' valign='middle' class='html-align-center' >0.1929</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCCC</td><td align='center' valign='middle' class='html-align-center' >0.3694</td><td align='center' valign='middle' class='html-align-center' >0.3871</td><td align='center' valign='middle' class='html-align-center' >0.3496</td><td align='center' valign='middle' class='html-align-center' >0.3669</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCSS</td><td align='center' valign='middle' class='html-align-center' >0.3069</td><td align='center' valign='middle' class='html-align-center' >0.3251</td><td align='center' valign='middle' class='html-align-center' >0.2862</td><td align='center' valign='middle' class='html-align-center' >0.3032</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSCS</td><td align='center' valign='middle' class='html-align-center' >0.3476</td><td align='center' valign='middle' class='html-align-center' >0.3632</td><td align='center' valign='middle' class='html-align-center' >0.3305</td><td align='center' valign='middle' class='html-align-center' >0.3446</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSSS</td><td align='center' valign='middle' class='html-align-center' >0.2892</td><td align='center' valign='middle' class='html-align-center' >0.3057</td><td align='center' valign='middle' class='html-align-center' >0.2708</td><td align='center' valign='middle' class='html-align-center' >0.2853</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >CCCS</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2546</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2734</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2327</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2517</td></tr><tr ><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Hyperbolic–paraboloidal shell</td><td align='center' valign='middle' class='html-align-center' >SSSS</td><td align='center' valign='middle' class='html-align-center' >0.1893</td><td align='center' valign='middle' class='html-align-center' >0.1976</td><td align='center' valign='middle' class='html-align-center' >0.1804</td><td align='center' valign='middle' class='html-align-center' >0.1881</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCCC</td><td align='center' valign='middle' class='html-align-center' >0.3703</td><td align='center' valign='middle' class='html-align-center' >0.3879</td><td align='center' valign='middle' class='html-align-center' >0.3506</td><td align='center' valign='middle' class='html-align-center' >0.3671</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CCSS</td><td align='center' valign='middle' class='html-align-center' >0.3056</td><td align='center' valign='middle' class='html-align-center' >0.3238</td><td align='center' valign='middle' class='html-align-center' >0.2848</td><td align='center' valign='middle' class='html-align-center' >0.3038</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSCS</td><td align='center' valign='middle' class='html-align-center' >0.3469</td><td align='center' valign='middle' class='html-align-center' >0.3625</td><td align='center' valign='middle' class='html-align-center' >0.3298</td><td align='center' valign='middle' class='html-align-center' >0.3443</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >CSSS</td><td align='center' valign='middle' class='html-align-center' >0.2865</td><td align='center' valign='middle' class='html-align-center' >0.3031</td><td align='center' valign='middle' class='html-align-center' >0.2680</td><td align='center' valign='middle' class='html-align-center' >0.2851</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >CCCS</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2566</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2752</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2349</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.2532</td></tr></tbody> </table> </div><div class="html-table-wrap" id="mathematics-10-00408-t010"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href='#table_body_display_mathematics-10-00408-t010'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2227-7390/10/3/408/display" href="#table_body_display_mathematics-10-00408-t010"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 10.</b> The effect of geometric parameters <span class='html-italic'>a</span>/<span class='html-italic'>h</span> and <span class='html-italic'>b</span>/<span class='html-italic'>a</span> on the dimensionless frequency of cross-ply F/CNT-RC shells for various curvature radii (SSSS, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>). </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_mathematics-10-00408-t010" > <div class="html-caption" ><b>Table 10.</b> The effect of geometric parameters <span class='html-italic'>a</span>/<span class='html-italic'>h</span> and <span class='html-italic'>b</span>/<span class='html-italic'>a</span> on the dimensionless frequency of cross-ply F/CNT-RC shells for various curvature radii (SSSS, <math display='inline'><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>f</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</div> <table > <thead ><tr ><th rowspan='2' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Type of Shells</th><th rowspan='2' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <mrow> <mi mathvariant="bold-italic">b</mi> <mo>/</mo> <mi mathvariant="bold-italic">a</mi> </mrow> </mstyle> </semantics> </math></th><th colspan='4' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' ><span class='html-italic'>a</span>/<span class='html-italic'>h</span></th></tr><tr ><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >20</th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >30</th></tr></thead><tbody ><tr ><td rowspan='4' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Plate</td><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >1.4689</td><td align='center' valign='middle' class='html-align-center' >0.4559</td><td align='center' valign='middle' class='html-align-center' >0.1231</td><td align='center' valign='middle' class='html-align-center' >0.0556</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >0.7077</td><td align='center' valign='middle' class='html-align-center' >0.1984</td><td align='center' valign='middle' class='html-align-center' >0.0513</td><td align='center' valign='middle' class='html-align-center' >0.0230</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >0.5230</td><td align='center' valign='middle' class='html-align-center' >0.1437</td><td align='center' valign='middle' class='html-align-center' >0.0369</td><td align='center' valign='middle' class='html-align-center' >0.0165</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.4952</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1358</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0348</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0156</td></tr><tr ><td rowspan='4' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Cylindrical shell</td><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >1.4715</td><td align='center' valign='middle' class='html-align-center' >0.4587</td><td align='center' valign='middle' class='html-align-center' >0.1260</td><td align='center' valign='middle' class='html-align-center' >0.0584</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >0.7092</td><td align='center' valign='middle' class='html-align-center' >0.2004</td><td align='center' valign='middle' class='html-align-center' >0.0535</td><td align='center' valign='middle' class='html-align-center' >0.0251</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >0.5226</td><td align='center' valign='middle' class='html-align-center' >0.1440</td><td align='center' valign='middle' class='html-align-center' >0.0374</td><td align='center' valign='middle' class='html-align-center' >0.0170</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.4944</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1356</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0350</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0157</td></tr><tr ><td rowspan='4' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Spherical shell</td><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >1.4749</td><td align='center' valign='middle' class='html-align-center' >0.4620</td><td align='center' valign='middle' class='html-align-center' >0.1294</td><td align='center' valign='middle' class='html-align-center' >0.0617</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >0.7164</td><td align='center' valign='middle' class='html-align-center' >0.2071</td><td align='center' valign='middle' class='html-align-center' >0.0597</td><td align='center' valign='middle' class='html-align-center' >0.0307</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >0.5276</td><td align='center' valign='middle' class='html-align-center' >0.1486</td><td align='center' valign='middle' class='html-align-center' >0.0417</td><td align='center' valign='middle' class='html-align-center' >0.0210</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.4973</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1383</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0375</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0181</td></tr><tr ><td rowspan='4' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Elliptical–paraboloid shell</td><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >1.4741</td><td align='center' valign='middle' class='html-align-center' >0.4609</td><td align='center' valign='middle' class='html-align-center' >0.1282</td><td align='center' valign='middle' class='html-align-center' >0.0605</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >0.7137</td><td align='center' valign='middle' class='html-align-center' >0.2045</td><td align='center' valign='middle' class='html-align-center' >0.0572</td><td align='center' valign='middle' class='html-align-center' >0.0286</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >0.5255</td><td align='center' valign='middle' class='html-align-center' >0.1466</td><td align='center' valign='middle' class='html-align-center' >0.0399</td><td align='center' valign='middle' class='html-align-center' >0.0193</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.4960</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1371</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0363</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0170</td></tr><tr ><td rowspan='4' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Hyperbolic–paraboloidal shell</td><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >1.4652</td><td align='center' valign='middle' class='html-align-center' >0.4552</td><td align='center' valign='middle' class='html-align-center' >0.1235</td><td align='center' valign='middle' class='html-align-center' >0.0562</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >0.7049</td><td align='center' valign='middle' class='html-align-center' >0.1976</td><td align='center' valign='middle' class='html-align-center' >0.0511</td><td align='center' valign='middle' class='html-align-center' >0.0229</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >0.5219</td><td align='center' valign='middle' class='html-align-center' >0.1438</td><td align='center' valign='middle' class='html-align-center' >0.0374</td><td align='center' valign='middle' class='html-align-center' >0.0170</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.4947</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.1361</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0354</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.0162</td></tr></tbody> </table> </div></section><section class='html-fn_group'><table><tr id=''><td></td><td><div class='html-p'><b>Publisher’s Note:</b> MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.</div></td></tr></table></section> <section id="html-copyright"><br>© 2022 by the authors. 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A Dynamic Analysis of Randomly Oriented Functionally Graded Carbon Nanotubes/Fiber-Reinforced Composite Laminated Shells with Different Geometries