Vergleichsspannung – Wikipedia

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id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a> <a class="mw-jump-link" href="#searchInput">Zur Suche springen</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Vergleichsspannung.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Vergleichsspannung.svg/330px-Vergleichsspannung.svg.png" decoding="async" width="330" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Vergleichsspannung.svg/495px-Vergleichsspannung.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Vergleichsspannung.svg/660px-Vergleichsspannung.svg.png 2x" data-file-width="512" data-file-height="171" /></a><figcaption>Vergleichsspannung</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:ReinDeviatorischeFlie%C3%9Ffkt.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/ReinDeviatorischeFlie%C3%9Ffkt.svg/220px-ReinDeviatorischeFlie%C3%9Ffkt.svg.png" decoding="async" width="220" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/ReinDeviatorischeFlie%C3%9Ffkt.svg/330px-ReinDeviatorischeFlie%C3%9Ffkt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/ReinDeviatorischeFlie%C3%9Ffkt.svg/440px-ReinDeviatorischeFlie%C3%9Ffkt.svg.png 2x" data-file-width="512" data-file-height="417" /></a><figcaption>Tresca- und Mises-Festigkeitskriterium im Spannungsraum</figcaption></figure> <p>Die <b>Vergleichsspannung</b> ist ein Begriff aus der <a href="/wiki/Festigkeitslehre" title="Festigkeitslehre">Festigkeitslehre</a>. Dieser bezeichnet eine fiktive einachsige <a href="/wiki/Spannung_(Mechanik)" class="mw-redirect" title="Spannung (Mechanik)">Spannung</a>, die aufgrund eines bestimmten werkstoffmechanischen bzw. mathematischen <a href="/wiki/Festigkeitskriterium" title="Festigkeitskriterium">Kriteriums</a> eine hypothetisch gleichwertige Materialbeanspruchung darstellt wie ein realer, mehrachsiger <a href="/wiki/Spannungszustand" title="Spannungszustand">Spannungszustand</a>. </p><p>Anhand der Vergleichsspannung kann der wirkliche, im Allgemeinen dreidimensionale Spannungszustand im Bauteil in der <a href="/wiki/Festigkeitsnachweis" title="Festigkeitsnachweis">Festigkeits-</a> oder in der <a href="/wiki/Plastizit%C3%A4tstheorie" title="Plastizitätstheorie">Fließbedingung</a> mit den Kennwerten aus dem einachsigen <a href="/wiki/Zugversuch" title="Zugversuch">Zugversuch</a> (Material-Kennwerte, z. B. <a href="/wiki/Streckgrenze" title="Streckgrenze">Streckgrenze</a> oder <a href="/wiki/Zugfestigkeit" title="Zugfestigkeit">Zugfestigkeit</a>) verglichen werden. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Grundlagen"><span class="tocnumber">1</span> <span class="toctext">Grundlagen</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Gestaltänderungshypothese_(von_Mises)"><span class="tocnumber">2</span> <span class="toctext">Gestaltänderungshypothese (von Mises)</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Schubspannungshypothese_(Tresca,_Coulomb,_Saint-Venant,_Guest)"><span class="tocnumber">3</span> <span class="toctext">Schubspannungshypothese (Tresca, Coulomb, Saint-Venant, Guest)</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Hauptnormalspannungshypothese_(Rankine)"><span class="tocnumber">4</span> <span class="toctext">Hauptnormalspannungshypothese (Rankine)</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Quadratisches_rotationssymmetrisches_Kriterium_(Burzyński-Yagn)"><span class="tocnumber">5</span> <span class="toctext">Quadratisches rotationssymmetrisches Kriterium (Burzyński-Yagn)</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Kombiniertes_rotationssymmetrisches_Kriterium_(Huber)"><span class="tocnumber">6</span> <span class="toctext">Kombiniertes rotationssymmetrisches Kriterium (Huber)</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#Unified_Strength_Theory_(Mao-Hong_Yu)"><span class="tocnumber">7</span> <span class="toctext">Unified Strength Theory (Mao-Hong Yu)</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Cosinus-Ansatz_(Altenbach-Bolchoun-Kolupaev)"><span class="tocnumber">8</span> <span class="toctext">Cosinus-Ansatz (Altenbach-Bolchoun-Kolupaev)</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#Literatur"><span class="tocnumber">9</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Einzelnachweise"><span class="tocnumber">10</span> <span class="toctext">Einzelnachweise</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Weblinks"><span class="tocnumber">11</span> <span class="toctext">Weblinks</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Grundlagen">Grundlagen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergleichsspannung&veaction=edit&section=1" title="Abschnitt bearbeiten: Grundlagen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vergleichsspannung&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Grundlagen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zur vollständigen Beschreibung des Spannungszustandes in einem Bauteil ist im Allgemeinen die Angabe des <a href="/wiki/Spannungstensor" title="Spannungstensor">Spannungstensors</a> (symmetrischer <a href="/wiki/Tensor" title="Tensor">Tensor</a> 2. Stufe) notwendig. Dieser enthält im allgemeinen Fall (Kräfte- und Momentengleichgewicht) sechs verschiedene Spannungswerte (da einander zugeordnete <a href="/wiki/Spannung_(Mechanik)" class="mw-redirect" title="Spannung (Mechanik)">Schubspannungen</a> gleich sind). Durch die Transformation des Spannungstensors in ein ausgezeichnetes <a href="/wiki/Koordinatensystem" title="Koordinatensystem">Koordinatensystem</a> (das <a href="/wiki/Hauptkomponentenanalyse" title="Hauptkomponentenanalyse">Hauptachsensystem</a>) werden die Schubspannungen zu Null und drei ausgezeichnete (Normal)Spannungen (die <a href="/wiki/Spannung_(Mechanik)#Hauptspannung_und_Hauptspannungsrichtung" class="mw-redirect" title="Spannung (Mechanik)">Hauptspannungen</a>) beschreiben den Beanspruchungszustand des Systems <a href="/wiki/Gleichwertigkeit" class="mw-redirect" title="Gleichwertigkeit">äquivalent</a>. </p><p>Die Elemente des <a href="/wiki/Vektor" title="Vektor">Vektors</a> der Hauptspannungen bzw. des Spannungstensors können nun in einen <a href="/wiki/Skalar_(Physik)" class="mw-redirect" title="Skalar (Physik)">Skalar</a> überführt werden, der zwei Bedingungen genügen soll: </p> <ul><li>zum einen soll er den Spannungszustand möglichst umfassend beschreiben (Äquivalenz kann hier nicht mehr erreicht werden: es treten immer Informationsverluste beim Übergang vom Vektor der Hauptspannungen zur Vergleichsspannung auf)</li> <li>zum anderen soll er auf jeden Fall eine versagensrelevante Information darstellen.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Spannungshypo.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Spannungshypo.gif/440px-Spannungshypo.gif" decoding="async" width="440" height="291" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/f3/Spannungshypo.gif 1.5x" data-file-width="462" data-file-height="306" /></a><figcaption>Anwendungsbereiche von Festigkeitshypothesen. SH: Schubspannungshypothese, GEH: Gestaltänderungshypothese, NH: Normalspannungshypothese.</figcaption></figure> <p>Die Rechenvorschrift zur Bildung dieser skalaren Vergleichsspannung bezeichnet man als <b>Vergleichspannungshypothese</b> bzw. als <a href="/w/index.php?title=Versagen&action=edit&redlink=1" class="new" title="Versagen (Seite nicht vorhanden)">Versagensregel</a>. Im Rahmen einer <a href="/wiki/Tragf%C3%A4higkeit_(Technik)" title="Tragfähigkeit (Technik)">Tragfähigkeitsanalyse</a> vergleicht man die Vergleichsspannung mit zulässigen Spannungen. Durch die Wahl der Hypothese enthält sie implizit den Versagensmechanismus und ist damit ein Wert, der die Gefährdung des Bauteils unter der gegebenen Beanspruchung ausdrückt. Die Wahl der jeweiligen Vergleichspannungshypothese hängt also immer vom <a href="/wiki/Festigkeit" title="Festigkeit">Festigkeitsverhalten</a> des nachzuweisenden Materials sowie vom Lastfall (statisch, schwingend, Stoß) ab. </p><p>Es gibt eine ganze Anzahl von <a href="/wiki/Hypothese" title="Hypothese">Hypothesen</a> zur Berechnung der Vergleichsspannung. Sie werden in der <a href="/wiki/Technische_Mechanik" title="Technische Mechanik">Technischen Mechanik</a> häufig unter dem Begriff <b>Festigkeitshypothesen</b> zusammengefasst. Die Anwendung hängt vom Materialverhalten und teilweise auch vom Anwendungsgebiet (wenn etwa eine Norm die Anwendung einer bestimmten Hypothese fordert) ab. </p><p>Am häufigsten wird im Maschinenbau und im Bauwesen die Gestaltänderungsenergiehypothese nach <a href="/wiki/Richard_von_Mises" title="Richard von Mises">von Mises</a> angewendet. Außer den hier genannten gibt es noch weitere Hypothesen. </p> <div class="mw-heading mw-heading2"><h2 id="Gestaltänderungshypothese_(von_Mises)"><span id="Gestalt.C3.A4nderungshypothese_.28von_Mises.29"></span><span id="Gestalt.C3.A4nderungshypothese"></span><span id="Gestaltänderungshypothese"></span> Gestaltänderungshypothese (von Mises)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergleichsspannung&veaction=edit&section=2" title="Abschnitt bearbeiten: Gestaltänderungshypothese (von Mises)" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vergleichsspannung&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Gestaltänderungshypothese (von Mises)"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Lode%27s_parameter_in_Mohr%27s_diagram_cos.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Lode%27s_parameter_in_Mohr%27s_diagram_cos.svg/220px-Lode%27s_parameter_in_Mohr%27s_diagram_cos.svg.png" decoding="async" width="220" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Lode%27s_parameter_in_Mohr%27s_diagram_cos.svg/330px-Lode%27s_parameter_in_Mohr%27s_diagram_cos.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Lode%27s_parameter_in_Mohr%27s_diagram_cos.svg/440px-Lode%27s_parameter_in_Mohr%27s_diagram_cos.svg.png 2x" data-file-width="512" data-file-height="531" /></a><figcaption>Mithilfe der <a href="/wiki/Mohrscher_Spannungskreis#Mohr’sche_Spannungskreise_in_3D" title="Mohrscher Spannungskreis">Mohr’schen Spannungskreisen</a> kann man grafisch die Mises-Vergleichspannung bestimmen<sup id="cite_ref-hellmich2018UEskriptum_1-0" class="reference"><a href="#cite_note-hellmich2018UEskriptum-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </figcaption></figure> <p>Nach der <b>Gestaltänderungshypothese</b>, auch <b>Gestaltänderungsenergiehypothese</b> (kurz: <b>GEH</b>) oder <b>Mises-Vergleichsspannung</b> nach <a href="/wiki/Richard_von_Mises" title="Richard von Mises">Richard von Mises</a> benannt, tritt Versagen des Bauteils dann auf, wenn die Gestaltänderungsenergie einen Grenzwert überschreitet (s. auch <a href="/wiki/Kontinuumsmechanik" title="Kontinuumsmechanik">Verzerrungen</a> bzw. <a href="/wiki/Verformung" title="Verformung">Deformation</a>). Verwendet wird diese Hypothese für zähe Werkstoffe (z. B. <a href="/wiki/Stahl" title="Stahl">Stahl</a>) unter ruhender und wechselnder Beanspruchung. Die Mises-Vergleichsspannung wird im Maschinenbau und im Bauwesen am häufigsten eingesetzt - für die meisten gängigen Materialien (nicht allzu spröde) unter normaler Belastung (wechselnd, nicht stoßartig) ist die GEH einsetzbar. Wichtige Anwendungsgebiete sind die Berechnungen von <a href="/wiki/Welle_(Mechanik)" class="mw-redirect" title="Welle (Mechanik)">Wellen</a>, die sowohl auf <a href="/wiki/Biegung_(Mechanik)" title="Biegung (Mechanik)">Biegung</a> als auch auf <a href="/wiki/Torsion_(Mechanik)" title="Torsion (Mechanik)">Torsion</a> beansprucht werden, sowie der <a href="/wiki/Stahlbau" title="Stahlbau">Stahlbau</a>. Die GEH ist so konstruiert, dass sich bei <a href="/wiki/Spannungszustand#Hydrostatischer_Spannungszustand" title="Spannungszustand">hydrostatischen Spannungszuständen</a> (gleich große Spannungen in allen drei Raumrichtungen) eine Vergleichsspannung von Null ergibt, da plastisches Fließen von Metallen <a href="/wiki/Isochore_Zustands%C3%A4nderung" title="Isochore Zustandsänderung">isochor</a> ist und selbst extreme hydrostatische Drücke keinen Einfluss auf den Fließbeginn haben (Experimente von Bridgman). </p><p>Beschreibung im allgemeinen Spannungszustand: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,M}={\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+\sigma _{z}^{2}-\sigma _{x}\sigma _{y}-\sigma _{x}\sigma _{z}-\sigma _{y}\sigma _{z}+3(\tau _{xy}^{2}+\tau _{xz}^{2}+\tau _{yz}^{2})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <mn>3</mn> <mo stretchy="false">(</mo> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,M}={\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+\sigma _{z}^{2}-\sigma _{x}\sigma _{y}-\sigma _{x}\sigma _{z}-\sigma _{y}\sigma _{z}+3(\tau _{xy}^{2}+\tau _{xz}^{2}+\tau _{yz}^{2})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69a1303d10b3f7a3886c6650ffe49d4f5cf5f7ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:66.137ex; height:4.843ex;" alt="{\displaystyle \sigma _{v,M}={\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+\sigma _{z}^{2}-\sigma _{x}\sigma _{y}-\sigma _{x}\sigma _{z}-\sigma _{y}\sigma _{z}+3(\tau _{xy}^{2}+\tau _{xz}^{2}+\tau _{yz}^{2})}}}" /></span></dd></dl> <p>andere Schreibweise: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,M}={\sqrt {{\frac {1}{2}}{\big [}{\left({\sigma _{x}-\sigma _{y}}\right)^{2}+\left({\sigma _{y}-\sigma _{z}}\right)^{2}+\left({\sigma _{z}-\sigma _{x}}\right)^{2}}+6\left({\tau _{xy}^{2}+\tau _{yz}^{2}+\tau _{xz}^{2}}\right){\big ]}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,M}={\sqrt {{\frac {1}{2}}{\big [}{\left({\sigma _{x}-\sigma _{y}}\right)^{2}+\left({\sigma _{y}-\sigma _{z}}\right)^{2}+\left({\sigma _{z}-\sigma _{x}}\right)^{2}}+6\left({\tau _{xy}^{2}+\tau _{yz}^{2}+\tau _{xz}^{2}}\right){\big ]}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed3128d7ff5b29021fab7a5fe634e7a3d7145edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:72.491ex; height:6.176ex;" alt="{\displaystyle \sigma _{v,M}={\sqrt {{\frac {1}{2}}{\big [}{\left({\sigma _{x}-\sigma _{y}}\right)^{2}+\left({\sigma _{y}-\sigma _{z}}\right)^{2}+\left({\sigma _{z}-\sigma _{x}}\right)^{2}}+6\left({\tau _{xy}^{2}+\tau _{yz}^{2}+\tau _{xz}^{2}}\right){\big ]}}}}" /></span></dd></dl> <p>Beschreibung im Hauptspannungszustand: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,M}={\sqrt {{\frac {1}{2}}{\big [}(\sigma _{I}-\sigma _{II})^{2}+(\sigma _{II}-\sigma _{III})^{2}+(\sigma _{III}-\sigma _{I})^{2}{\big ]}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,M}={\sqrt {{\frac {1}{2}}{\big [}(\sigma _{I}-\sigma _{II})^{2}+(\sigma _{II}-\sigma _{III})^{2}+(\sigma _{III}-\sigma _{I})^{2}{\big ]}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233b77fcf17045ca7d190fff34a765f9d9fb25c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:55.994ex; height:6.176ex;" alt="{\displaystyle \sigma _{v,M}={\sqrt {{\frac {1}{2}}{\big [}(\sigma _{I}-\sigma _{II})^{2}+(\sigma _{II}-\sigma _{III})^{2}+(\sigma _{III}-\sigma _{I})^{2}{\big ]}}}}" /></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4ee5bcbadab1ff17234020dcbe714206458707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.388ex; height:2.009ex;" alt="{\displaystyle \sigma _{I}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{II}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{II}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2626e4ff4bf5881a6d8522607e1410fd54b11798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.217ex; height:2.009ex;" alt="{\displaystyle \sigma _{II}}" /></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{III}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{III}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e881213a7d69b36a7af62cec193acd09059aa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.045ex; height:2.009ex;" alt="{\displaystyle \sigma _{III}}" /></span> sind die Hauptspannungen. </p><p>Beschreibung im ebenen Spannungszustand: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,M}={\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}-\sigma _{x}\sigma _{y}+3\tau _{xy}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,M}={\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}-\sigma _{x}\sigma _{y}+3\tau _{xy}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/156b901c8a3f61793027682c1a2fb406d113d94d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.414ex; height:4.843ex;" alt="{\displaystyle \sigma _{v,M}={\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}-\sigma _{x}\sigma _{y}+3\tau _{xy}^{2}}}}" /></span></dd></dl> <p>Beschreibung im ebenen Verzerrungszustand mit: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{z}=\nu (\sigma _{x}+\sigma _{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi>ν</mi> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{z}=\nu (\sigma _{x}+\sigma _{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046966903fb429a039aea86bc92b3dfab7cf30b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.186ex; height:3.009ex;" alt="{\displaystyle \sigma _{z}=\nu (\sigma _{x}+\sigma _{y})}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,M}={\sqrt {(\sigma _{x}^{2}+\sigma _{y}^{2})(\nu ^{2}-\nu +1)+\sigma _{x}\sigma _{y}(2\nu ^{2}-2\nu -1)+3\tau _{xy}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>ν</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,M}={\sqrt {(\sigma _{x}^{2}+\sigma _{y}^{2})(\nu ^{2}-\nu +1)+\sigma _{x}\sigma _{y}(2\nu ^{2}-2\nu -1)+3\tau _{xy}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2137039913970f981725acfeb53a40f217597cc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:60.941ex; height:4.843ex;" alt="{\displaystyle \sigma _{v,M}={\sqrt {(\sigma _{x}^{2}+\sigma _{y}^{2})(\nu ^{2}-\nu +1)+\sigma _{x}\sigma _{y}(2\nu ^{2}-2\nu -1)+3\tau _{xy}^{2}}}}" /></span></dd></dl> <p>Beschreibung in Invariantendarstellung: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,M}={\sqrt {3I_{2}^{'}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mo>′</mo> </msup> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,M}={\sqrt {3I_{2}^{'}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded0526086396d4e986aef864cefed725890ce2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.203ex; height:4.676ex;" alt="{\displaystyle \sigma _{v,M}={\sqrt {3I_{2}^{'}}}}" /></span></dd></dl> <p>wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{2}^{'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mo>′</mo> </msup> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{2}^{'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96be292f328d5700c321f6a18f45142fda7b13c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.077ex; height:3.343ex;" alt="{\displaystyle I_{2}^{'}}" /></span> die zweite Invariante des <a href="/wiki/Spannungsdeviator" title="Spannungsdeviator">Spannungsdeviators</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f24e8bfce665d4fc56a7b863e3ea90c23f17f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.568ex; height:2.343ex;" alt="{\displaystyle s_{ij}}" /></span> ist: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{2}^{'}={\frac {1}{2}}s_{ij}s_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{2}^{'}={\frac {1}{2}}s_{ij}s_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd056688390cfff0d5418bdcd0084434d9a9a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.31ex; height:5.176ex;" alt="{\displaystyle I_{2}^{'}={\frac {1}{2}}s_{ij}s_{ij}}" /></span></dd></dl> <p>Die Gestaltänderungshypothese stellt einen Spezialfall des Drucker-Prager-Fließkriteriums dar, bei dem die Grenzspannungen für Druck <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b436b43abda74fce1a6859e03d34c914c6a240f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.272ex; height:2.009ex;" alt="{\displaystyle \sigma _{c}}" /></span> und Zug <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dd6db0238ac32f34c6feb604748e253e842356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle \sigma _{t}}" /></span> gleich groß sind.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Schubspannungshypothese_(Tresca,_Coulomb,_Saint-Venant,_Guest)"><span id="Schubspannungshypothese_.28Tresca.2C_Coulomb.2C_Saint-Venant.2C_Guest.29"></span><span id="Schubspannungshypothese"></span> Schubspannungshypothese (Tresca, Coulomb, Saint-Venant, Guest)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergleichsspannung&veaction=edit&section=3" title="Abschnitt bearbeiten: Schubspannungshypothese (Tresca, Coulomb, Saint-Venant, Guest)" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vergleichsspannung&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Schubspannungshypothese (Tresca, Coulomb, Saint-Venant, Guest)"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Es wird davon ausgegangen, dass für das Versagen des Werkstoffes die größte Hauptspannungsdifferenz verantwortlich ist (Bezeichnung in einigen <a href="/wiki/Finite-Elemente-Methode" title="Finite-Elemente-Methode">FE-Programmen</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mathrm {int} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mathrm {int} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e224477534bc8b80c017683d39cbfae83dd0f3ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.571ex; height:2.009ex;" alt="{\displaystyle \sigma _{\mathrm {int} }}" /></span> Intensität). Diese Hauptspannungsdifferenz entspricht dem doppelten Wert der maximalen Schubspannung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{\max }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\max }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f179650efdae923fc11b7c328c6f74b6060d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.307ex; height:2.009ex;" alt="{\displaystyle \tau _{\max }}" /></span> – dadurch wird sie bei <a href="/wiki/Duktilit%C3%A4t" title="Duktilität">zähem Material</a> unter statischer Belastung, welches durch Fließen (Gleitbruch) versagt, angewandt. Im <a href="/wiki/Mohrscher_Spannungskreis" title="Mohrscher Spannungskreis">Mohr’schen Spannungskreis</a> ist die kritische Größe der Durchmesser des größten Kreises. Die Schubspannungshypothese findet aber auch im <a href="/wiki/Maschinenbau" title="Maschinenbau">Maschinenbau</a> ganz allgemein Anwendung, da der <a href="/wiki/Mathematische_Formel" class="mw-redirect" title="Mathematische Formel">Formelapparat</a> im Vergleich zur GEH einfacher zu handhaben ist und man mit ihr im Vergleich zu Von Mises (GEH) auf der sicheren Seite liegt (es kommen im Zweifelsfall etwas größere Werte für die Vergleichsspannung und damit auch etwas mehr Sicherheitsreserven heraus). </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,T}=2\tau _{\max }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,T}=2\tau _{\max }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb8525086c24bd5f738887ebc12cf4d69e03ece3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.539ex; height:2.843ex;" alt="{\displaystyle \sigma _{v,T}=2\tau _{\max }}" /></span></dd></dl> <p>Räumlicher Spannungszustand: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,T}=\max(\vert \sigma _{I}-\sigma _{II}\vert ;\vert \sigma _{II}-\sigma _{III}\vert ;\vert \sigma _{III}-\sigma _{I}\vert )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">|</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> </mrow> </msub> <mo fence="false" stretchy="false">|</mo> <mo>;</mo> <mo fence="false" stretchy="false">|</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mrow> </msub> <mo fence="false" stretchy="false">|</mo> <mo>;</mo> <mo fence="false" stretchy="false">|</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo fence="false" stretchy="false">|</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,T}=\max(\vert \sigma _{I}-\sigma _{II}\vert ;\vert \sigma _{II}-\sigma _{III}\vert ;\vert \sigma _{III}-\sigma _{I}\vert )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd6688138a45f0c58b3353319432e447c417e0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.975ex; height:3.009ex;" alt="{\displaystyle \sigma _{v,T}=\max(\vert \sigma _{I}-\sigma _{II}\vert ;\vert \sigma _{II}-\sigma _{III}\vert ;\vert \sigma _{III}-\sigma _{I}\vert )}" /></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4ee5bcbadab1ff17234020dcbe714206458707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.388ex; height:2.009ex;" alt="{\displaystyle \sigma _{I}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{II}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{II}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2626e4ff4bf5881a6d8522607e1410fd54b11798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.217ex; height:2.009ex;" alt="{\displaystyle \sigma _{II}}" /></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{III}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{III}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e881213a7d69b36a7af62cec193acd09059aa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.045ex; height:2.009ex;" alt="{\displaystyle \sigma _{III}}" /></span> sind die Hauptspannungen. </p><p>Ebener Spannungszustand (vorausgesetzt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf4ab08fe163a6c495cad6f4d67653287c4044c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \sigma _{x}}" /></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22d96fceb022e70169a37383278199a26b3534a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.377ex; height:2.343ex;" alt="{\displaystyle \sigma _{y}}" /></span> haben unterschiedliche Vorzeichen<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,T}={\sqrt {(\sigma _{x}-\sigma _{y})^{2}+4\tau _{xy}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,T}={\sqrt {(\sigma _{x}-\sigma _{y})^{2}+4\tau _{xy}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f7f67c7f3c90e0172916c873109e19dc58a61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:26.983ex; height:4.843ex;" alt="{\displaystyle \sigma _{v,T}={\sqrt {(\sigma _{x}-\sigma _{y})^{2}+4\tau _{xy}^{2}}}}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Hauptnormalspannungshypothese_(Rankine)"><span id="Hauptnormalspannungshypothese_.28Rankine.29"></span>Hauptnormalspannungshypothese (Rankine)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergleichsspannung&veaction=edit&section=4" title="Abschnitt bearbeiten: Hauptnormalspannungshypothese (Rankine)" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vergleichsspannung&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Hauptnormalspannungshypothese (Rankine)"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Es wird davon ausgegangen, dass das Bauteil aufgrund der größten Normalspannung versagt. Im <a href="/wiki/Mohrscher_Spannungskreis" title="Mohrscher Spannungskreis">Mohr'schen Spannungskreis</a> ist der kritische Punkt die maximale Hauptspannung. Die Hypothese wird angewendet für Werkstoffe, welche mit <a href="/wiki/Spr%C3%B6dbruch" title="Sprödbruch">Trennbruch</a>, ohne Fließen, versagen: </p> <ul><li><a href="/wiki/Spr%C3%B6digkeit" title="Sprödigkeit">spröde Werkstoffe</a> (z. B. <a href="/wiki/Grauguss" class="mw-redirect" title="Grauguss">Grauguss</a> oder <a href="/wiki/Schwei%C3%9Fnaht" class="mw-redirect" title="Schweißnaht">Schweißnähte</a>) bei vorwiegend ruhender Zugbeanspruchung</li> <li>spröde und zähe Materialien bei <a href="/wiki/Sto%C3%9F_(Physik)" title="Stoß (Physik)">stoßartiger</a> Belastung.</li></ul> <p>Räumlicher Spannungszustand: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,R}=\sigma _{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>R</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,R}=\sigma _{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b1f8bbdebc967f63e6f98f8c176d6136393a67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.548ex; height:2.343ex;" alt="{\displaystyle \sigma _{v,R}=\sigma _{I}}" /></span></dd></dl> <p>für </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{I}\geq 0\quad \&\quad \sigma _{I}\geq \vert \sigma _{III}\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mspace width="1em"></mspace> <mi mathvariant="normal">&</mi> <mspace width="1em"></mspace> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo>≥</mo> <mo fence="false" stretchy="false">|</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mrow> </msub> <mo fence="false" stretchy="false">|</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{I}\geq 0\quad \&\quad \sigma _{I}\geq \vert \sigma _{III}\vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ec984c7d13ee1c008fa6c4907da2e939c3d848" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.928ex; height:2.843ex;" alt="{\displaystyle \sigma _{I}\geq 0\quad \&\quad \sigma _{I}\geq \vert \sigma _{III}\vert }" /></span></dd></dl> <p>ansonsten </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,R}=\vert \sigma _{III}\vert =-\sigma _{III}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">|</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mrow> </msub> <mo fence="false" stretchy="false">|</mo> <mo>=</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,R}=\vert \sigma _{III}\vert =-\sigma _{III}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e84cc02ab05f47da183c8282302f3d6031256c95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.451ex; height:3.009ex;" alt="{\displaystyle \sigma _{v,R}=\vert \sigma _{III}\vert =-\sigma _{III}}" /></span></dd></dl> <p>für </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{III}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> <mi>I</mi> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{III}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37e93c9c8d10f4adad3488ca4eae409541d4f56c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.306ex; height:2.509ex;" alt="{\displaystyle \sigma _{III}<0}" /></span></dd></dl> <p>Ebener Spannungszustand: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v,R}={\frac {(\sigma _{x}+\sigma _{y})+{\sqrt {(\sigma _{x}-\sigma _{y})^{2}+4\tau _{xy}^{2}}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v,R}={\frac {(\sigma _{x}+\sigma _{y})+{\sqrt {(\sigma _{x}-\sigma _{y})^{2}+4\tau _{xy}^{2}}}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1382eb9a99f4bd5c68e2ce4b7d6ddbe1c57d5018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.276ex; height:7.676ex;" alt="{\displaystyle \sigma _{v,R}={\frac {(\sigma _{x}+\sigma _{y})+{\sqrt {(\sigma _{x}-\sigma _{y})^{2}+4\tau _{xy}^{2}}}}{2}}}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Quadratisches_rotationssymmetrisches_Kriterium_(Burzyński-Yagn)"><span id="Quadratisches_rotationssymmetrisches_Kriterium_.28Burzy.C5.84ski-Yagn.29"></span>Quadratisches rotationssymmetrisches Kriterium (Burzyński-Yagn)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergleichsspannung&veaction=edit&section=5" title="Abschnitt bearbeiten: Quadratisches rotationssymmetrisches Kriterium (Burzyński-Yagn)" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vergleichsspannung&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Quadratisches rotationssymmetrisches Kriterium (Burzyński-Yagn)"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mit dem Ansatz<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3I_{2}'={\frac {\sigma _{\mathrm {eq} }-\gamma _{1}I_{1}}{1-\gamma _{1}}}{\frac {\sigma _{\mathrm {eq} }-\gamma _{2}I_{1}}{1-\gamma _{2}}},\qquad \gamma _{1}\in {[0,1[}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3I_{2}'={\frac {\sigma _{\mathrm {eq} }-\gamma _{1}I_{1}}{1-\gamma _{1}}}{\frac {\sigma _{\mathrm {eq} }-\gamma _{2}I_{1}}{1-\gamma _{2}}},\qquad \gamma _{1}\in {[0,1[}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef77d112b054dfb047fe98f1a4d6a336ca0f00e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.132ex; height:6.009ex;" alt="{\displaystyle 3I_{2}'={\frac {\sigma _{\mathrm {eq} }-\gamma _{1}I_{1}}{1-\gamma _{1}}}{\frac {\sigma _{\mathrm {eq} }-\gamma _{2}I_{1}}{1-\gamma _{2}}},\qquad \gamma _{1}\in {[0,1[}}" /></span></dd></dl> <p>folgen die Kriterien: </p><p>- Konus von Drucker-Prager (Mirolyubov) mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}=\gamma _{2}\in {]0,1[}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}=\gamma _{2}\in {]0,1[}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376b616e8aa88d847918667f393444de65f61b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.109ex; height:2.843ex;" alt="{\displaystyle \gamma _{1}=\gamma _{2}\in {]0,1[}}" /></span>, </p><p>- Paraboloid von Balandin (Burzyński-Torre) mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}\in {]0,1[},\gamma _{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mrow> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}\in {]0,1[},\gamma _{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee71d5b17f22f7a4002c0c86804210f2ec94396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.305ex; height:2.843ex;" alt="{\displaystyle \gamma _{1}\in {]0,1[},\gamma _{2}=0}" /></span>, </p><p>- Ellipsoid von Beltrami mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}=-\gamma _{2}\in {]0,1[}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}=-\gamma _{2}\in {]0,1[}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c2dc6c58a77f12dd79adddb448682e19e1a35b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.917ex; height:2.843ex;" alt="{\displaystyle \gamma _{1}=-\gamma _{2}\in {]0,1[}}" /></span>, </p><p>- Ellipsoid von Schleicher mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}\in {]0,1[},\gamma _{2}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mrow> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}\in {]0,1[},\gamma _{2}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb52a6ac60acb16f6d72ff893e47d56a502a038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.305ex; height:2.843ex;" alt="{\displaystyle \gamma _{1}\in {]0,1[},\gamma _{2}<0}" /></span>, </p><p>- Hyperboloid von Burzyński-Yagn mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}\in {]0,1[},\gamma _{2}\in {]0,\gamma _{1}[}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mrow> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}\in {]0,1[},\gamma _{2}\in {]0,\gamma _{1}[}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f483fbaac805bb671506b6d551b279784085e75e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.633ex; height:2.843ex;" alt="{\displaystyle \gamma _{1}\in {]0,1[},\gamma _{2}\in {]0,\gamma _{1}[}}" /></span>, </p><p>- einschaliges Hyperboloid. </p><p>Die quadratischen Kriterien lassen sich explizit nach <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mathrm {eq} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mathrm {eq} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccdd3f83efa31532961d9e7425ac2bcb7dd88191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.169ex; height:2.343ex;" alt="{\displaystyle \sigma _{\mathrm {eq} }}" /></span> auflösen, was ihren praktischen Einsatz förderte. </p><p>Die Querkontraktionszahl bei Zug lässt sich mit </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{+}^{\mathrm {pl} }={\frac {-1+2(\gamma _{1}+\gamma _{2})-3\gamma _{1}\gamma _{2}}{-2+\gamma _{1}+\gamma _{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3</mn> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mo>−</mo> <mn>2</mn> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{+}^{\mathrm {pl} }={\frac {-1+2(\gamma _{1}+\gamma _{2})-3\gamma _{1}\gamma _{2}}{-2+\gamma _{1}+\gamma _{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda0746ab8a883275375be408e880bdc4eb44614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.456ex; height:6.176ex;" alt="{\displaystyle \nu _{+}^{\mathrm {pl} }={\frac {-1+2(\gamma _{1}+\gamma _{2})-3\gamma _{1}\gamma _{2}}{-2+\gamma _{1}+\gamma _{2}}}}" /></span></dd></dl> <p>berechnen. Die Anwendung von rotationssymmetrischen Kriterien für sprödes Versagen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{+}^{\mathrm {pl} }\in {]-1,\nu _{+}^{\mathrm {el} }]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msubsup> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <msubsup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msubsup> <mo stretchy="false">]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{+}^{\mathrm {pl} }\in {]-1,\nu _{+}^{\mathrm {el} }]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c9d782d8756f01ec59df0e475d36576b5992413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.677ex; height:3.509ex;" alt="{\displaystyle \nu _{+}^{\mathrm {pl} }\in {]-1,\nu _{+}^{\mathrm {el} }]}}" /></span></dd></dl> <p>wurde nicht genügend untersucht.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Kombiniertes_rotationssymmetrisches_Kriterium_(Huber)"><span id="Kombiniertes_rotationssymmetrisches_Kriterium_.28Huber.29"></span>Kombiniertes rotationssymmetrisches Kriterium (Huber)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergleichsspannung&veaction=edit&section=6" title="Abschnitt bearbeiten: Kombiniertes rotationssymmetrisches Kriterium (Huber)" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vergleichsspannung&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Kombiniertes rotationssymmetrisches Kriterium (Huber)"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Das Kriterium von Huber<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> besteht aus dem Ellipsoid von Beltrami </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3I_{2}'={\frac {\sigma _{eq}-\gamma _{1}I_{1}}{1-\gamma _{1}}}{\frac {\sigma _{eq}+\gamma _{1}I_{1}}{1+\gamma _{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>q</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>q</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3I_{2}'={\frac {\sigma _{eq}-\gamma _{1}I_{1}}{1-\gamma _{1}}}{\frac {\sigma _{eq}+\gamma _{1}I_{1}}{1+\gamma _{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d722ae89770700cf1e21c9881566222e3427d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.527ex; height:6.009ex;" alt="{\displaystyle 3I_{2}'={\frac {\sigma _{eq}-\gamma _{1}I_{1}}{1-\gamma _{1}}}{\frac {\sigma _{eq}+\gamma _{1}I_{1}}{1+\gamma _{1}}}}" /></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0688dbf720280412428f70b9a7dc21f21562d10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.338ex; height:2.509ex;" alt="{\displaystyle I_{1}>0}" /></span></dd></dl> <p>und einem zu ihm im Schnitt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baf3e3cf555bed4f7cce11a56a8096c1dd02be6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.338ex; height:2.509ex;" alt="{\displaystyle I_{1}=0}" /></span> gekoppelten Zylinder </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3I_{2}'={\frac {\sigma _{eq}}{1-\gamma _{1}}}{\frac {\sigma _{eq}}{1+\gamma _{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>q</mi> </mrow> </msub> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>q</mi> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3I_{2}'={\frac {\sigma _{eq}}{1-\gamma _{1}}}{\frac {\sigma _{eq}}{1+\gamma _{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa712f47b34f06858611675f17cb00dcbcde376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.533ex; height:5.509ex;" alt="{\displaystyle 3I_{2}'={\frac {\sigma _{eq}}{1-\gamma _{1}}}{\frac {\sigma _{eq}}{1+\gamma _{1}}}}" /></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1}\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1}\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f525514fd3603e3bdf40c8bbaba3d5952b96bc5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.338ex; height:2.509ex;" alt="{\displaystyle I_{1}\leq 0}" /></span></dd></dl> <p>mit dem Parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}\in {[0,1[}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}\in {[0,1[}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db9d4f9891ac028827d940e3479a9e75de9cee57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.752ex; height:2.843ex;" alt="{\displaystyle \gamma _{1}\in {[0,1[}}" /></span>. </p><p>Der Übergang im Schnitt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baf3e3cf555bed4f7cce11a56a8096c1dd02be6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.338ex; height:2.509ex;" alt="{\displaystyle I_{1}=0}" /></span> ist stetig-differenzierbar. Die Querkontraktionszahlen bei Zug und Druck ergeben sich zu </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{+}^{pl}={\frac {1}{2}}(1-3\gamma _{1}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>l</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{+}^{pl}={\frac {1}{2}}(1-3\gamma _{1}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24f82b911ea748c3f6679d4d4bd834e8c7002d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.212ex; height:5.176ex;" alt="{\displaystyle \nu _{+}^{pl}={\frac {1}{2}}(1-3\gamma _{1}^{2})}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{-}^{pl}={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>l</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{-}^{pl}={\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fa7503258a238a8deb0d20a68e34b8c93166b24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.904ex; height:5.176ex;" alt="{\displaystyle \nu _{-}^{pl}={\frac {1}{2}}}" /></span></dd></dl> <p>Das Kriterium wurde 1904 entwickelt. Es setzte sich jedoch zunächst nicht durch, da es von mehreren Wissenschaftler<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> als unstetiges Modell verstanden wurde. </p> <div class="mw-heading mw-heading2"><h2 id="Unified_Strength_Theory_(Mao-Hong_Yu)"><span id="Unified_Strength_Theory_.28Mao-Hong_Yu.29"></span>Unified Strength Theory (Mao-Hong Yu)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergleichsspannung&veaction=edit&section=7" title="Abschnitt bearbeiten: Unified Strength Theory (Mao-Hong Yu)" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vergleichsspannung&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Unified Strength Theory (Mao-Hong Yu)"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Unified Strength Theory (UST)<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> besteht aus zwei sechseckigen Pyramiden von Sayir,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> die um 60° gegeneinander gedreht sind: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mathrm {I} }-{\frac {\alpha }{1+b}}(b\sigma _{\mathrm {II} }+\sigma _{\mathrm {III} })-\sigma _{\mathrm {eq} }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>b</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mathrm {I} }-{\frac {\alpha }{1+b}}(b\sigma _{\mathrm {II} }+\sigma _{\mathrm {III} })-\sigma _{\mathrm {eq} }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db16f413e7cf61d80734174a0aea69c3119f3824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.836ex; height:5.009ex;" alt="{\displaystyle \sigma _{\mathrm {I} }-{\frac {\alpha }{1+b}}(b\sigma _{\mathrm {II} }+\sigma _{\mathrm {III} })-\sigma _{\mathrm {eq} }=0}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \sigma _{\mathrm {I} }-{\frac {1}{1+b}}(b\sigma _{\mathrm {II} }+\sigma _{\mathrm {III} })+\sigma _{\mathrm {eq} }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>b</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \sigma _{\mathrm {I} }-{\frac {1}{1+b}}(b\sigma _{\mathrm {II} }+\sigma _{\mathrm {III} })+\sigma _{\mathrm {eq} }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ad0f90f1fc9874ead833fcf761e3af85baf84b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:34.323ex; height:5.509ex;" alt="{\displaystyle \alpha \sigma _{\mathrm {I} }-{\frac {1}{1+b}}(b\sigma _{\mathrm {II} }+\sigma _{\mathrm {III} })+\sigma _{\mathrm {eq} }=0}" /></span></dd></dl> <p>mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {\sigma _{+}}{\sigma _{-}}}\in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mfrac> </mrow> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {\sigma _{+}}{\sigma _{-}}}\in [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd75d3451c22d9f3e894b4efa5cfbfc70f5ad56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.754ex; height:5.343ex;" alt="{\displaystyle \alpha ={\frac {\sigma _{+}}{\sigma _{-}}}\in [0,1]}" /></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b={\frac {\tau \,(\sigma _{+}+\sigma _{-})-\sigma _{+}\sigma _{-}}{\sigma _{-}\,(\sigma _{+}-\tau )}}\in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>τ</mi> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mrow> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b={\frac {\tau \,(\sigma _{+}+\sigma _{-})-\sigma _{+}\sigma _{-}}{\sigma _{-}\,(\sigma _{+}-\tau )}}\in [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de3481d4aa4480832dac715c71c80876c39f083b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.857ex; height:6.509ex;" alt="{\displaystyle b={\frac {\tau \,(\sigma _{+}+\sigma _{-})-\sigma _{+}\sigma _{-}}{\sigma _{-}\,(\sigma _{+}-\tau )}}\in [0,1]}" /></span>. </p><p>Mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19206e7d4dab695ccb34c502eff0741e98dbdfc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=0}" /></span> ergibt sich das Kriterium von Mohr-Coulomb (Single-Shear Theorie von Yu), mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=({\sqrt {3}}-1)/2\approx 0{,}366}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>≈</mo> <mn>0,366</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=({\sqrt {3}}-1)/2\approx 0{,}366}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/368a4eccdd1d41985332302178459e7122d64e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.727ex; height:3.009ex;" alt="{\displaystyle b=({\sqrt {3}}-1)/2\approx 0{,}366}" /></span> das Pisarenko-Lebedev Kriterium und mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f55bc77dec8088791b5c1ed51e634cc1b431fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=1}" /></span> folgt die Twin-Shear Theorie von Yu (vgl. Pyramide von Haythornthwaite). </p><p>Die Querkontraktionszahlen beim Zug und beim Druck folgen als </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{+}^{\mathrm {pl} }={\frac {\alpha }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{+}^{\mathrm {pl} }={\frac {\alpha }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37fb93f119cff4117315e32b44f15a28c200f3a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.283ex; height:4.676ex;" alt="{\displaystyle \nu _{+}^{\mathrm {pl} }={\frac {\alpha }{2}}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{-}^{\mathrm {pl} }={\frac {1}{2\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{-}^{\mathrm {pl} }={\frac {1}{2\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348f4f471cc1f3ab594d9528289dcbf525c9d035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.446ex; height:5.176ex;" alt="{\displaystyle \nu _{-}^{\mathrm {pl} }={\frac {1}{2\alpha }}}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Cosinus-Ansatz_(Altenbach-Bolchoun-Kolupaev)"><span id="Cosinus-Ansatz_.28Altenbach-Bolchoun-Kolupaev.29"></span>Cosinus-Ansatz (Altenbach-Bolchoun-Kolupaev)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergleichsspannung&veaction=edit&section=8" title="Abschnitt bearbeiten: Cosinus-Ansatz (Altenbach-Bolchoun-Kolupaev)" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vergleichsspannung&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Cosinus-Ansatz (Altenbach-Bolchoun-Kolupaev)"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Oft werden die Festigkeitshypothesen unter Verwendung des Spannungswinkels </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos 3\theta ={\frac {3{\sqrt {3}}}{2}}{\frac {I_{3}'}{I_{2}'^{\frac {3}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mn>3</mn> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos 3\theta ={\frac {3{\sqrt {3}}}{2}}{\frac {I_{3}'}{I_{2}'^{\frac {3}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23edbc58bfc1d6afa0b4d67f8202f189b5e21fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:18.187ex; height:8.509ex;" alt="{\displaystyle \cos 3\theta ={\frac {3{\sqrt {3}}}{2}}{\frac {I_{3}'}{I_{2}'^{\frac {3}{2}}}}}" /></span></dd></dl> <p>formuliert. Mehrere Kriterien isotropen Materialverhaltens werden im Ansatz </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3I_{2}')^{3}{\frac {1+c_{3}\cos 3\theta +c_{6}\cos ^{2}3\theta }{1+c_{3}+c_{6}}}=\displaystyle \left({\frac {\sigma _{\mathrm {eq} }-\gamma _{1}\,I_{1}}{1-\gamma _{1}}}\right)^{6-l-m}\,\left({\frac {\sigma _{\mathrm {eq} }-\gamma _{2}\,I_{1}}{1-\gamma _{2}}}\right)^{l}\,\sigma _{\mathrm {eq} }^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mn>3</mn> <mi>θ</mi> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mn>3</mn> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>−</mo> <mi>l</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3I_{2}')^{3}{\frac {1+c_{3}\cos 3\theta +c_{6}\cos ^{2}3\theta }{1+c_{3}+c_{6}}}=\displaystyle \left({\frac {\sigma _{\mathrm {eq} }-\gamma _{1}\,I_{1}}{1-\gamma _{1}}}\right)^{6-l-m}\,\left({\frac {\sigma _{\mathrm {eq} }-\gamma _{2}\,I_{1}}{1-\gamma _{2}}}\right)^{l}\,\sigma _{\mathrm {eq} }^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d138c4099b06c159b811537bbe46de13bda66e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:74.524ex; height:6.676ex;" alt="{\displaystyle (3I_{2}')^{3}{\frac {1+c_{3}\cos 3\theta +c_{6}\cos ^{2}3\theta }{1+c_{3}+c_{6}}}=\displaystyle \left({\frac {\sigma _{\mathrm {eq} }-\gamma _{1}\,I_{1}}{1-\gamma _{1}}}\right)^{6-l-m}\,\left({\frac {\sigma _{\mathrm {eq} }-\gamma _{2}\,I_{1}}{1-\gamma _{2}}}\right)^{l}\,\sigma _{\mathrm {eq} }^{m}}" /></span></dd></dl> <p>zusammengefasst. </p><p>Die Parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1dc52bfbaf6e577fbed72a716068f4533700bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{3}}" /></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87670df150c1e4444386652b709c7a72106b8bcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{6}}" /></span> beschreiben die Geometrie der Fläche in der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span>-Ebene. Sie müssen die Bedingungen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{6}={\frac {1}{4}}(2+c_{3}),\qquad c_{6}={\frac {1}{4}}(2-c_{3}),\qquad c_{6}\geq {\frac {5}{12}}\,c_{3}^{2}-{\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em"></mspace> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em"></mspace> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>12</mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{6}={\frac {1}{4}}(2+c_{3}),\qquad c_{6}={\frac {1}{4}}(2-c_{3}),\qquad c_{6}\geq {\frac {5}{12}}\,c_{3}^{2}-{\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0a56baa6ea1ca51c636bbc0ab77739b83e4149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:57.029ex; height:5.176ex;" alt="{\displaystyle c_{6}={\frac {1}{4}}(2+c_{3}),\qquad c_{6}={\frac {1}{4}}(2-c_{3}),\qquad c_{6}\geq {\frac {5}{12}}\,c_{3}^{2}-{\frac {1}{3}}}" /></span></dd></dl> <p>erfüllen, welche sich aus der Konvexitätsanforderung ergeben. In<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> wird eine Verbesserung der dritten Bedingung vorgeschlagen. </p><p>Die Parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}\in [0,1[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}\in [0,1[}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9876adb9ebd8939878a523b6875d4b9ae5e217f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.752ex; height:2.843ex;" alt="{\displaystyle \gamma _{1}\in [0,1[}" /></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482832093b568cdc09c3aeaa2585c5fc49100b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{2}}" /></span> beschreiben die Lage der Schnittpunkte der <a href="/w/index.php?title=Flie%C3%9Ffl%C3%A4che&action=edit&redlink=1" class="new" title="Fließfläche (Seite nicht vorhanden)">Fließfläche</a> mit der hydrostatischen Achse (Raumdiagonale im Hauptspannungsraum). Diese Schnittpunkte werden hydrostatische Knoten genannt. Für die Materialien, die unter der gleichmäßigen 3D-Druckbelastung nicht versagen (Stahl, Messing usw.), ergibt sich <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{2}\in [0,\gamma _{1}[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">[</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{2}\in [0,\gamma _{1}[}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af754823cc31698a2108082759a34456c149c42c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.848ex; height:2.843ex;" alt="{\displaystyle \gamma _{2}\in [0,\gamma _{1}[}" /></span>. Für die Materialien, die unter dem gleichmäßigen 3D-Druck versagen (harte Schäume, Keramiken, gesinterte Materialien), gilt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{2}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{2}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9675e594d06aca28a220e94e9d2c2a3f404eccb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.519ex; height:2.676ex;" alt="{\displaystyle \gamma _{2}<0}" /></span>. </p><p>Die ganzzahligen Potenzen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8d7debe3dbe2599a0b1df61cd66e8b1a896fa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.954ex; height:2.343ex;" alt="{\displaystyle l\geq 0}" /></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d2d765e4cfd7adfbca9ae0e37e75a2811c0333" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.301ex; height:2.343ex;" alt="{\displaystyle m\geq 0}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l+m<6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>+</mo> <mi>m</mi> <mo><</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l+m<6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96fd721a12b1beb3e6d3245973ba11f9c16eb643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.835ex; height:2.343ex;" alt="{\displaystyle l+m<6}" /></span> beschreiben die Krümmung des Meridians. Der Meridian ist mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=m=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=m=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e29d58a8338cb372b2ffce11fd78502744ce82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.093ex; height:2.176ex;" alt="{\displaystyle l=m=0}" /></span> eine Gerade und mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66485a3e3da13d226eb36a131bf1fc7e16403a5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.954ex; height:2.176ex;" alt="{\displaystyle l=0}" /></span> eine Parabel. </p> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergleichsspannung&veaction=edit&section=9" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vergleichsspannung&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>J. Sauter, N. Wingerter: <i>Neue und alte Festigkeitshypothesen</i>. (= <i>VDI-Fortschrittsberichte. Reihe 1.</i> Band 191). VDI-Verlag, Düsseldorf 1990, <a href="/wiki/Spezial:ISBN-Suche/3181491012" class="internal mw-magiclink-isbn">ISBN 3-18-149101-2</a>.</li></ul> <ul><li>S. Sähn, H. Göldner: <i>Bruch- und Beurteilungskriterien in der Festigkeitslehre.</i> 2. Auflage. Fachbuchverlag, Leipzig 1993, <a href="/wiki/Spezial:ISBN-Suche/3343008540" class="internal mw-magiclink-isbn">ISBN 3-343-00854-0</a>.</li></ul> <ul><li>H. Mertens: <i>Zur Formulierung von Festigkeitshypothesen für mehrachsige phasenverschobene Schwingbeanspruchungen.</i> In: <i>Z. angew. Math. und Mech.</i> Band 70, Nr. 4, 1990, S. T327–T329.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergleichsspannung&veaction=edit&section=10" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vergleichsspannung&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-hellmich2018UEskriptum-1"><span class="mw-cite-backlink"><a href="#cite_ref-hellmich2018UEskriptum_1-0">↑</a></span> <span class="reference-text">Christian Hellmich und weitere: <cite style="font-style:italic">Skriptum zur Übung aus Festigkeitslehre</cite>. In: <cite style="font-style:italic">Festigkeitslehreskriptum für Bauingenieure</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>2017/18</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>202.665</span>. Institut der Mechanik für Werkstoffe und Strukturen, TU Wien; Wien, Januar 2018 (<a rel="nofollow" class="external text" href="https://tuwel.tuwien.ac.at/course/view.php?id=11461">tuwien.ac.at</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Vergleichsspannung&rft.atitle=Skriptum+zur+%C3%9Cbung+aus+Festigkeitslehre&rft.au=Christian+Hellmich+und+weitere&rft.date=2018-01&rft.genre=journal&rft.issue=202.665&rft.jtitle=Festigkeitslehreskriptum+f%C3%BCr+Bauingenieure&rft.pub=Institut+der+Mechanik+f%C3%BCr+Werkstoffe+und+Strukturen%2C+TU+Wien%3B+Wien&rft.volume=2017%2F18" style="display:none"> </span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span class="cite">Frank Faulstich: <a rel="nofollow" class="external text" href="https://vergleichsspannung.de/vergleichsspannungen/fliesskriterium-nach-drucker-prager/"><i>Drucker-Prager-Vergleichsspannung.</i></a><span class="Abrufdatum"> Abgerufen am 2. April 2020</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AVergleichsspannung&rft.title=Drucker-Prager-Vergleichsspannung&rft.description=Drucker-Prager-Vergleichsspannung&rft.identifier=https%3A%2F%2Fvergleichsspannung.de%2Fvergleichsspannungen%2Ffliesskriterium-nach-drucker-prager%2F&rft.creator=Frank+Faulstich&rft.date=&rft.language=de"> </span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20201026122218/https://www.bau.uni-siegen.de/subdomains/baustatik/lehre/bst/unterlagen_vertieft/fliessgelenkverfahren/bs3_plastizitaetstheorie_ws201213.pdf"><i>Vorlesung zu Plasizität der Uni Siegen.</i></a> Universität Siegen, archiviert vom <style data-mw-deduplicate="TemplateStyles:r250917974">.mw-parser-output .dewiki-iconexternal>a{background-position:center right!important;background-repeat:no-repeat!important}body.skin-minerva .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/a/a4/OOjs_UI_icon_external-link-ltr-progressive.svg")!important;background-size:10px!important;padding-right:13px!important}body.skin-timeless .mw-parser-output .dewiki-iconexternal>a,body.skin-monobook .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/3/30/MediaWiki_external_link_icon.svg")!important;padding-right:13px!important}body.skin-vector .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/9/96/Link-external-small-ltr-progressive.svg")!important;background-size:0.857em!important;padding-right:1em!important}</style><span class="dewiki-iconexternal"><a class="external text" href="https://redirecter.toolforge.org/?url=https%3A%2F%2Fwww.bau.uni-siegen.de%2Fsubdomains%2Fbaustatik%2Flehre%2Fbst%2Funterlagen_vertieft%2Ffliessgelenkverfahren%2Fbs3_plastizitaetstheorie_ws201213.pdf">Original</a></span> (nicht mehr online verfügbar) am <span style="white-space:nowrap;">26. Oktober 2020</span><span>;</span><span class="Abrufdatum"> abgerufen am 2. April 2020</span>.</span> <small class="archiv-bot"><span class="wp_boppel noviewer" aria-hidden="true" role="presentation"><span typeof="mw:File"><span title="i"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Pictogram_voting_info.svg/15px-Pictogram_voting_info.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Pictogram_voting_info.svg/23px-Pictogram_voting_info.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Pictogram_voting_info.svg/30px-Pictogram_voting_info.svg.png 2x" data-file-width="250" data-file-height="250" /></span></span></span> <b>Info:</b> Der Archivlink wurde automatisch eingesetzt und noch nicht geprüft. Bitte prüfe Original- und Archivlink gemäß <a href="/wiki/Benutzer:InternetArchiveBot/Anleitung/Archivlink" title="Benutzer:InternetArchiveBot/Anleitung/Archivlink">Anleitung</a> und entferne dann diesen Hinweis.</small><span style="display:none"><a rel="nofollow" class="external text" href="http://IABotmemento.invalid/https://www.bau.uni-siegen.de/subdomains/baustatik/lehre/bst/unterlagen_vertieft/fliessgelenkverfahren/bs3_plastizitaetstheorie_ws201213.pdf">@1</a></span><span style="display:none"><a rel="nofollow" class="external text" href="https://www.bau.uni-siegen.de/subdomains/baustatik/lehre/bst/unterlagen_vertieft/fliessgelenkverfahren/bs3_plastizitaetstheorie_ws201213.pdf">@2</a></span><span style="display:none"><a href="/w/index.php?title=Vorlage:Webachiv/IABot/www.bau.uni-siegen.de&action=edit&redlink=1" class="new" title="Vorlage:Webachiv/IABot/www.bau.uni-siegen.de (Seite nicht vorhanden)">Vorlage:Webachiv/IABot/www.bau.uni-siegen.de</a></span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AVergleichsspannung&rft.title=Vorlesung+zu+Plasizit%C3%A4t+der+Uni+Siegen&rft.description=Vorlesung+zu+Plasizit%C3%A4t+der+Uni+Siegen&rft.identifier=https%3A%2F%2Fweb.archive.org%2Fweb%2F20201026122218%2Fhttps%3A%2F%2Fwww.bau.uni-siegen.de%2Fsubdomains%2Fbaustatik%2Flehre%2Fbst%2Funterlagen_vertieft%2Ffliessgelenkverfahren%2Fbs3_plastizitaetstheorie_ws201213.pdf&rft.publisher=Universit%C3%A4t+Siegen&rft.date=&rft.source=https://www.bau.uni-siegen.de/subdomains/baustatik/lehre/bst/unterlagen_vertieft/fliessgelenkverfahren/bs3_plastizitaetstheorie_ws201213.pdf&rft.language=de"> </span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">L. Issler, H. Ruoß, P. Häfele: <cite style="font-style:italic">Festigkeitslehre - Grundlagen</cite>. Springer, Berlin/ Heidelberg 2003, <a href="/wiki/Spezial:ISBN-Suche/3540407057" class="internal mw-magiclink-isbn">ISBN 3-540-40705-7</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>178</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Vergleichsspannung&rft.au=L.+Issler%2C+H.+Ruo%C3%9F%2C+P.+H%C3%A4fele&rft.btitle=Festigkeitslehre+-+Grundlagen&rft.date=2003&rft.genre=book&rft.isbn=3540407057&rft.pages=178&rft.place=Berlin%2F+Heidelberg&rft.pub=Springer" style="display:none"> </span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">W. Burzyński: <i> Über die Anstrengungshypothesen.</i> In: <i>Schweizerische Bauzeitung.</i> Band 94, Nr. 21, 1929, S. 259–262.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">N. M. Beljaev: <i>Strength of materials.</i> Mir Publ., Moscow 1979.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">M. T. Huber: <i>Die spezifische Formänderungsarbeit als Maß der Anstrengung.</i> Czasopismo Techniczne, Lwow 1904.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">H. Ismar, O. Mahrenholz: <i>Technische Plastomechanik.</i> Vieweg, Braunschweig 1979.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text">M.-H. Yu: <i>Unified Strength Theory and its Applications.</i> Springer, Berlin 2004.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text">M. Sayir: <i>Zur Fließbedingung der Plastizitätstheorie.</i> In: <i>Ing. Arch.</i> 39, 1970, S. 414–432.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text">H. Altenbach, A. Bolchoun, V. A. Kolupaev: <i>Phenomenological Yield and Failure Criteria.</i> In: H. Altenbach, A. Öchsner (Hrsg.): <i>Plasticity of Pressure-Sensitive Materials.</i> (= <i>Serie ASM</i>). 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