Document Zbl 0887.73038

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I: Justification of membrane shell equations.</strong> <i>(English)</i> <a class="label nowrap" href="/0887.73038">Zbl 0887.73038</a> </h2> <div class="source"> <a href="/serials/37" title="Journal Profile">Arch. Ration. Mech. Anal.</a> <a href="/?q=in%3A38324" title="Articles in this Issue">136, No. 2, 119-161 (1996)</a>. </div> <div class="abstract"><div class="pre">In this three-part work, the authors analyze the asymptotic behaviour of the scaled three-dimensional displacement field of a linearly elastic shell. In the first part, a family of such shells with thickness $2\varepsilon$ clamped along their entire lateral face is considered. All shells have the same middle surface $S=\varphi(\overline\omega)\subset \mathbb{R}^3$, where $\omega\subset\mathbb{R}^2$ is a bounded and connected open set with Lipschitz-continuous boundary $\gamma$ and $\varphi\in C^3(\overline\omega,\mathbb{R}^3)$. It is supposed that $\gamma$ and $\varphi$ are smooth enough, and that two principal radii of curvature are either both positive at all points of $S$, or are both negative at all points of $S$.<br class="zbmathjax-paragraph">Let the applied body force density be $O(1)$ with respect to $\varepsilon$, and $u_i(\varepsilon)$ denote three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity. Then the field $u(\varepsilon)= (u_i(\varepsilon))$, once scaled so as to be defined over the fixed domain $\Omega= \omega\times [-1,1]$, converges as $\varepsilon\to 0$ to a limit $u\in H^1(\Omega)\times H^1(\Omega)\times L^2(\Omega)$ which is independent of the transverse variable. The average $\xi= {1\over 2} \int^1_{-1} udx_3$ belongs to the space $V_m(\omega)= H^1_0(\omega)\times H^1_0(\omega)\times L^2(\omega)$ and satisfies the two-dimensional equations of a “membrane shell” $\int_\omega a^{\alpha\beta\sigma\tau} \gamma_{\sigma\tau}(\xi) \gamma_{\alpha\beta}(\eta)\sqrt{a} dy= \int_\omega \left\{\int^1_{-1} f^idx_3\right\} \eta_i\sqrt ady$ for all $\eta= (\eta_i)\in V_M(\omega)$; here $a^{\alpha\beta\sigma\tau}$ are the components of the two-dimensional elasticity tensor of the surface $S$, $\gamma_{\alpha\beta}$ are the components of the linearized change of the metric tensor of $S$, and $f^i$ are the scaled components of the applied body force.</div><div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A7722">E.Gavrilova (Sofia)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A969906">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A969906">Cited in <strong>78</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A74K15" title="MSC2020">74K15</a> </td> <td class="space"> Membranes </td> </tr><tr> <td class="mono"> 35Q72 </td> <td class="space"> Other PDE from mechanics (MSC2000) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aconvergence">convergence</a>; <a href="/?q=ut%3Amiddle+surface">middle surface</a>; <a href="/?q=ut%3ALipschitz-continuous+boundary">Lipschitz-continuous boundary</a>; <a href="/?q=ut%3Athree-dimensional+elasticity">three-dimensional elasticity</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0887.73038" data-ciurl="/ci/00969906" data-biburl="/bibtex/00969906.bib" data-amsurl="/amsrefs/00969906.bib" data-xmlurl="/xml/00969906.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00969906.pdf" title="Zbl 0887.73038 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/BF02316975" aria-label="DOI for “Asymptotic analysis of linearly elastic shells. I: Justification of membrane shell equations”" title="10.1007/BF02316975">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Acerbi, E.; Buttazzo, G.; Percivale, D., Thin inclusions in linear elasticity: a variational approach, J. reine angew. Math., 386, 99-115 (1988) · <a href="/0633.73021" class="nowrap">Zbl 0633.73021</a></td> </tr><tr> <td>[2]</td> <td class="space">Acerbi, E.; Buttazzo, G.; Percivale, D., A variational definition of the strain energy for an elastic string, J. 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