GermanWikipedia - /page/entry.mu

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<title>Schubmodul</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Schubmodul</h1>

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<th colspan="2" style="background:#ABCDEF; color:inherit;"><a href="Physikalische_Gr%C3%B6%C3%9Fe" title="Physikalische Größe">Physikalische Größe</a>

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<td style="width:130px;">Name

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<td>Schubmodul

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<td><a href="Formelzeichen" title="Formelzeichen">Formelzeichen</a>

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<td>G

</td></tr>

<tr>

<th><a href="Gr%C3%B6%C3%9Fensystem" title="Größensystem">Größen-</a> und <a href="Einheitensystem" title="Einheitensystem">Einheitensystem</a>

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<th><a href="Ma%C3%9Feinheit" title="Maßeinheit">Einheit</a>

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<th><a href="Dimension_(Gr%C3%B6%C3%9Fensystem)" title="Dimension (Größensystem)">Dimension</a>

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<tr>

</td>

<td><a href="Pascal_(Einheit)" title="Pascal (Einheit)">Pa</a> = <a href="Newton_(Einheit)" title="Newton (Einheit)">N</a>/<a href="Quadratmeter" title="Quadratmeter">m</a>2 = <a href="Kilogramm" title="Kilogramm">kg</a>·<a href="Meter" title="Meter">m</a>−1·<a href="Sekunde" title="Sekunde">s</a>−2

</td>

<td><a href="Masse_(Physik)" title="Masse (Physik)">M</a>·<a href="L%C3%A4nge_(Physik)" title="Länge (Physik)">L</a>−1·<a href="Zeit" title="Zeit">T</a>−2

</td></tr>

<tr>

</td>

<td><a href="Barye_(Einheit)" title="Barye (Einheit)">Ba</a> = <a href="Dyn_(Einheit)" title="Dyn (Einheit)">dyn</a>/<a href="Meter#cm" title="Meter">cm</a>2 = cm−1·g·s−2

</td>

<td>

</td></tr>

</tbody></table>

</td></tr>

<tr>

<td colspan="2">Siehe auch: <a href="Elastizit%C3%A4tsmodul" title="Elastizitätsmodul">Elastizitätsmodul</a> E <a href="Spannung_(Mechanik)" class="mw-redirect" title="Spannung (Mechanik)">Spannung (Mechanik)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">

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<th>Material

</th>

<th>Typische Werte für den Schubmodul in <a href="Vors%C3%A4tze_f%C3%BCr_Ma%C3%9Feinheiten" title="Vorsätze für Maßeinheiten">G</a><a href="Pascal_(Einheit)" title="Pascal (Einheit)">Pa</a> (bei Raumtemperatur)<a href="#cite_note-CDL-1">[1]</a>

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<tr>

<td><a href="Stahl" title="Stahl">Stahl</a>

</td>

</td></tr>

<tr>

<td><a href="Silicium" title="Silicium">Silicium</a> (<a href="Polykristall" title="Polykristall">polykristallin</a>)

</td>

</td></tr>

<tr>

<td><a href="Kupfer" title="Kupfer">Kupfer</a>

</td>

<td>47

</td></tr>

<tr>

<td><a href="Titan_(Element)" title="Titan (Element)">Titan</a>

</td>

<td>41,4

</td></tr>

<tr>

</td>

<td>26,2

</td></tr>

<tr>

<td><a href="Aluminium" title="Aluminium">Aluminium</a>

</td>

<td>25,5

</td></tr>

<tr>

<td><a href="Magnesium" title="Magnesium">Magnesium</a>

</td>

<td>17

</td></tr>

<tr>

<td><a href="Polyethylen" title="Polyethylen">Polyethylen</a>

</td>

<td>00,117

</td></tr>

<tr>

<td><a href="Gummi" title="Gummi">Gummi</a>

</td>

<td>00,0003

</td></tr></tbody></table>

Der Schubmodul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> (auch Gleitmodul, G-Modul, Schermodul oder Torsionsmodul) ist eine <a href="Materialkonstante" class="mw-redirect" title="Materialkonstante">Materialkonstante</a>, die Auskunft gibt über die <a href="Elastizit%C3%A4t_(Physik)" title="Elastizität (Physik)">linear-elastische</a> <a href="Verformung" title="Verformung">Verformung</a> eines Bauteils infolge einer <a href="Scherung_(Mechanik)" title="Scherung (Mechanik)">Scherkraft</a> oder <a href="Schubspannung" class="mw-redirect" title="Schubspannung">Schubspannung</a>. Die <a href="SI-Einheit" class="mw-redirect" title="SI-Einheit">SI-Einheit</a> ist Newton pro Quadratmeter (1 N/m² = 1 <a href="Pascal_(Einheit)" title="Pascal (Einheit)">Pa</a>), also die Einheit einer <a href="Spannung_(Mechanik)" class="mw-redirect" title="Spannung (Mechanik)">mechanischen Spannung</a>. In <a href="Materialdatenbank" class="mw-redirect" title="Materialdatenbank">Materialdatenbanken</a> wird der Schubmodul üblicherweise in N/mm² (=MPa) oder kN/mm² (=GPa) angegeben.

Im Rahmen der <a href="Elastizit%C3%A4tstheorie" title="Elastizitätstheorie">Elastizitätstheorie</a> entspricht der Schubmodul der <a href="Lam%C3%A9-Konstante" class="mw-redirect" title="Lamé-Konstante">zweiten Lamé-Konstanten</a> und trägt dort das Symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Der Schubmodul beschreibt das Verhältnis zwischen der Schubspannung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> und dem Tangens des Schubwinkels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> (Gleitung):

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<annotation encoding="application/x-tex">{\displaystyle \tau =G\cdot \tan \gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6faa211fd818da8a7eaba2a950bc8f807613d047.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.815ex; height:2.676ex;" alt="{\displaystyle \tau =G\cdot \tan \gamma }" loading="lazy"></dd></dl>

Für kleine Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> kann in erster Näherung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \gamma \approx \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \tan \gamma \approx \gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d9c9a0964a4ca565c811a1d3cb6b41eef42be4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.37ex; height:2.509ex;" alt="{\displaystyle \tan \gamma \approx \gamma }" loading="lazy"> gesetzt werden (<a href="Kleinwinkeln%C3%A4herung" title="Kleinwinkelnäherung">Kleinwinkelnäherung</a>).

Diese Formel ist analog zum <a href="Hookesches_Gesetz" title="Hookesches Gesetz">Hooke’schen Gesetz</a> für den 1-achsigen <a href="Spannungszustand" title="Spannungszustand">Spannungszustand</a>:

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<annotation encoding="application/x-tex">{\displaystyle \sigma =E\cdot \varepsilon }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d703f44b1381590e63929e5042074df101cc6e3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.966ex; height:2.176ex;" alt="{\displaystyle \sigma =E\cdot \varepsilon }" loading="lazy"></dd></dl>

Die <a href="Schubsteifigkeit" class="mw-redirect" title="Schubsteifigkeit">Schubsteifigkeit</a> ist das Produkt aus dem Schubmodul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> des Werkstoffs und der Querschnittsfläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">:

<mtext>Schubsteifigkeit</mtext>

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<mrow>

<mrow>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Schubsteifigkeit}}=G\cdot A\cdot \kappa \left(=G\cdot A_{S}\right),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb0437ba92fc7285208eb70979246c1b72c9f520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.626ex; height:2.843ex;" alt="{\displaystyle {\text{Schubsteifigkeit}}=G\cdot A\cdot \kappa \left(=G\cdot A_{S}\right),}" loading="lazy"> zum Beispiel in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {N} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a91302c621d1e18627cb635f8bd86852ab4b800b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {N} }" loading="lazy"></dd></dl>

Der querschnittsabhängige Korrekturfaktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

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<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> berücksichtigt dabei die über den Querschnitt ungleichförmige Verteilung der Schubspannung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">. Oft wird die Schubsteifigkeit auch mithilfe der <a href="Schubfl%C3%A4che" title="Schubfläche">Schubfläche</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{S}}">

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle A_{S}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62b524b6b6ff2f8e3f78d17a0635dc3e05b9e4fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.035ex; height:2.509ex;" alt="{\displaystyle A_{S}}" loading="lazy"> ausgedrückt.

Bei <a href="Torsion_(Mechanik)" title="Torsion (Mechanik)">Torsionsbelastung</a> eines Bauteils berechnet sich seine <a href="Torsionssteifigkeit" class="mw-redirect" title="Torsionssteifigkeit">Torsionssteifigkeit</a> aus dem Schubmodul und dem <a href="Torsionstr%C3%A4gheitsmoment" class="mw-redirect" title="Torsionsträgheitsmoment">Torsionsträgheitsmoment</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathrm {T} }}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle I_{\mathrm {T} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e8dd326890bdbf376a586df479864f7e67fd4f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.442ex; height:2.509ex;" alt="{\displaystyle I_{\mathrm {T} }}" loading="lazy">, das auf die Achse bezogen ist, um die der Körper tordiert wird:

<msub>

</mrow>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Torsionssteifigkeit_{T}} =G\cdot I_{\mathrm {T} },}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d0554da4f52470a8459490f1d6ff0c1d3ccbc36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.784ex; height:2.509ex;" alt="{\displaystyle \mathrm {Torsionssteifigkeit_{T}} =G\cdot I_{\mathrm {T} },}" loading="lazy"></dd></dl>

analog zur Ermittlung der <a href="Steifigkeit#Dehnsteifigkeit" title="Steifigkeit">Dehnsteifigkeit</a> (aus dem Produkt von <a href="Elastizit%C3%A4tsmodul" title="Elastizitätsmodul">Elastizitätsmodul</a> und Querschnittsfläche).

<div class="mw-heading mw-heading2"><h2 id="Zusammenhang_mit_anderen_Materialkonstanten">Zusammenhang mit anderen Materialkonstanten</h2></div>

Bei einem <a href="Isotrop" class="mw-redirect" title="Isotrop">isotropen</a> Material steht der Schubmodul mit dem <a href="Elastizit%C3%A4tsmodul" title="Elastizitätsmodul">Elastizitätsmodul</a> E, der <a href="Poissonzahl" title="Poissonzahl">Querkontraktionszahl</a> ν (Poissonzahl) und dem <a href="Kompressionsmodul" title="Kompressionsmodul">Kompressionsmodul</a> K in folgender Beziehung:

<mfrac>

<mrow>

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</mfrac>

</mrow>

<mfrac>

<mrow>

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<mrow>

</mrow>

</mfrac>

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<mfrac>

<mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G={\frac {1}{2(1+\nu )}}\cdot E={\frac {3KE}{9K-E}}={\frac {3(1-2\nu )}{2(1+\nu )}}\cdot K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202547501c1923bf105a1c7f524cefe397854061.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.251ex; height:6.509ex;" alt="{\displaystyle G={\frac {1}{2(1+\nu )}}\cdot E={\frac {3KE}{9K-E}}={\frac {3(1-2\nu )}{2(1+\nu )}}\cdot K}" loading="lazy"></dd></dl>

Für linear-elastisches, nicht-<a href="Auxetik" class="mw-redirect" title="Auxetik">auxetisches</a> Material ist die Poissonzahl größer-gleich null.

Aus der Energieerhaltung ergibt sich die positive <a href="Definitheit" title="Definitheit">Definitheit</a> von <a href="Kompressionsmodul" title="Kompressionsmodul">Kompressionsmodul</a> und <a href="E-Modul" class="mw-redirect" title="E-Modul">E-Modul</a>. Daraus folgt, dass die Poissonzahl unter 0,5 liegt. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(0\leq \nu <0{,}5\right)}">

<mrow>

<mrow>

</mrow>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \left(0\leq \nu <0{,}5\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48dd40e985fe512364083d299a8a3f275082359f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.373ex; height:2.843ex;" alt="{\displaystyle \left(0\leq \nu <0{,}5\right)}" loading="lazy">

Somit ergibt sich für den Schubmodul der meisten Materialien im linear-elastischen Bereich:

<mfrac>

</mfrac>

</mrow>

<mfrac>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}E<G\leq {\frac {1}{2}}E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df01baa06036c4caac5710cd9aa409f9cce522cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.572ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}E<G\leq {\frac {1}{2}}E}" loading="lazy"></dd></dl>

<a href="Auxetik" class="mw-redirect" title="Auxetik">Auxetische Materialien</a> sind so definiert, dass sie eine negative Poissonzahl haben, was nur bei wenigen Materialien der Fall ist.

Da der Schubmodul aufgrund der Energieerhaltung eine positiv definite Größe hat, gilt für auxetische Materialien im linear-elastischen Bereich:

<mfrac>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}E<G_{\mathrm {aux} }<+\infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d12c7f038112b0ec4128f2c0d665bfcfeeb9d9d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.766ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}E<G_{\mathrm {aux} }<+\infty }" loading="lazy"></dd></dl>

Da auch der <a href="E-Modul" class="mw-redirect" title="E-Modul">E-Modul</a> positiv definit ist, ergibt sich für die Poissonzahl der Gültigkeitsbereich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1<\nu _{\mathrm {aux} }<0.}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle -1<\nu _{\mathrm {aux} }<0.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47891f9c282a8ceccd83d0ccba9d587d9f9ecc79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.961ex; height:2.509ex;" alt="{\displaystyle -1<\nu _{\mathrm {aux} }<0.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Umrechnung_zwischen_den_elastischen_Konstanten_isotroper_Festkörper">Umrechnung zwischen den elastischen Konstanten isotroper Festkörper</h2></div>

</caption>

<th style="vertical-align:bottom" rowspan="2">Der Modul…

</th>

<th colspan="11">…ergibt sich aus:<a href="#cite_note-5">[5]</a>

</th></tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (K,\,E)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/996a015d617843aa00212a26839fbae210c834a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.072ex; height:2.843ex;" alt="{\displaystyle (K,\,E)}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (K,\,\lambda )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/423e5f4d1c8ee39ba6de7bc5e48df6837fff232a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.652ex; height:2.843ex;" alt="{\displaystyle (K,\,\lambda )}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (K,\,G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bae48568e7a818eb72621f7cd602d0effc7e3648.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.123ex; height:2.843ex;" alt="{\displaystyle (K,\,G)}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (K,\,\nu )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2f6df6fb9936ea5a0e15358e66802cafe9b747d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.528ex; height:2.843ex;" alt="{\displaystyle (K,\,\nu )}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,\,\lambda )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5c948c9a642e3318131878edf4a3d45bfb90f8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.361ex; height:2.843ex;" alt="{\displaystyle (E,\,\lambda )}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,\,G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee52d776f2cb07c88fbac6567029158dff718e01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.833ex; height:2.843ex;" alt="{\displaystyle (E,\,G)}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,\,\nu )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d858acfba62fc12da4ae6fa0759b9df87c51d96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.238ex; height:2.843ex;" alt="{\displaystyle (E,\,\nu )}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\lambda ,\,G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bffd6d22cc67fd1a35a585b13d7b32a74ce6b1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.412ex; height:2.843ex;" alt="{\displaystyle (\lambda ,\,G)}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\lambda ,\,\nu )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbafda646cd2c8251ec73dbe5efd428aee76544e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.818ex; height:2.843ex;" alt="{\displaystyle (\lambda ,\,\nu )}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,\,\nu )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03de031e73561f10f24638feaf69d90abe32aea5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.289ex; height:2.843ex;" alt="{\displaystyle (G,\,\nu )}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G,\,M)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40060aa6964ead07257199a60095ec47599c9455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.499ex; height:2.843ex;" alt="{\displaystyle (G,\,M)}" loading="lazy">

</th></tr>

<tr>

<td style="text-align:left"><a href="Kompressionsmodul" title="Kompressionsmodul">Kompressionsmodul</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/492094dd0f8aec54c51b011cf9a1ae0aeaf89e38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.453ex; height:2.176ex;" alt="{\displaystyle K\,}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E+3\lambda )/6+}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a677df319f754cb29aa51ce205fcb5e51a408b71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.076ex; height:2.843ex;" alt="{\displaystyle (E+3\lambda )/6+}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\sqrt {(E+3\lambda )^{2}-4\lambda E}}{6}}}">

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\sqrt {(E+3\lambda )^{2}-4\lambda E}}{6}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/100c2b34cf8dddb904b7b3a5b56da79294f65900.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.219ex; height:5.676ex;" alt="{\displaystyle {\tfrac {\sqrt {(E+3\lambda )^{2}-4\lambda E}}{6}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {EG}{3(3G-E)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94fef5491daefd3b9667bfd32eb6e4b3f31a67aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.585ex; height:4.343ex;" alt="{\displaystyle {\tfrac {EG}{3(3G-E)}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {E}{3(1-2\nu )}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49073e20128a9ecf62b06e95a6e611808b4a1015.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.731ex; height:4.176ex;" alt="{\displaystyle {\tfrac {E}{3(1-2\nu )}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda +}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c08fed708218dfe53f1c9ed3e60b5381dc5321e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.163ex; height:2.343ex;" alt="{\displaystyle \lambda +}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G}{3}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G}{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbe6eaf3b831f42c3953bc9fce83f851f229b7fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.95ex; height:3.843ex;" alt="{\displaystyle {\tfrac {2G}{3}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8668d7ae1d4ba0bea79bc24b22969de2e0020b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.046ex; height:4.343ex;" alt="{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7f8f5046df3cc6a7152f9fa56845d5dbce78033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.201ex; height:4.843ex;" alt="{\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M-}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/491f170e0c60c5650dee4d2e0f9e94206e2755b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.25ex; height:2.343ex;" alt="{\displaystyle M-}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4G}{3}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {4G}{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/349005a0ea0771949c65f6b25ff9b44c7b8f0b69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.95ex; height:3.843ex;" alt="{\displaystyle {\tfrac {4G}{3}}}" loading="lazy">

</td></tr>

<tr>

<td style="text-align:left"><a href="Elastizit%C3%A4tsmodul" title="Elastizitätsmodul">Elastizitätsmodul</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9123abddc2ec35f72035ec59f443c79ee052c9ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.163ex; height:2.176ex;" alt="{\displaystyle E\,}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3233ebb13df867ba5551fefa7e092733fea80c42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.096ex; height:4.343ex;" alt="{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {9KG}{3K+G}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5440c8c124cded6c55e978cbf920e76e4883a51b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.689ex; height:4.009ex;" alt="{\displaystyle {\tfrac {9KG}{3K+G}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3K(1-2\nu )\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd567543f973005ad7031789e6219f93463e6d87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.822ex; height:2.843ex;" alt="{\displaystyle 3K(1-2\nu )\,}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0994ccfa74411f35a8f0762379eddc3660bfec4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.58ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16ba478690ce90bc61e86d5b21ef89e8d97c0223.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.119ex; height:4.009ex;" alt="{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2G(1+\nu )\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ecc39d8499aa0848351a6510b116bd2dfea06d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.421ex; height:2.843ex;" alt="{\displaystyle 2G(1+\nu )\,}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b53b71a131246c16879e071ac283f0cf16bc255.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.348ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}" loading="lazy">

</td></tr>

<tr>

<td style="text-align:left"><a href="Lam%C3%A9-Konstanten" title="Lamé-Konstanten">1. Lamé-Konstante</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda \,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/988b7b8a22b11081bc97378c30391f573535c21c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.742ex; height:2.176ex;" alt="{\displaystyle \lambda \,}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cad8348883df915b9b03fb2820ec6b096640f4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.215ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K-}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bbddd69a002990563c7f6aa72e066204fb94f24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.874ex; height:2.343ex;" alt="{\displaystyle K-}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G}{3}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G}{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbe6eaf3b831f42c3953bc9fce83f851f229b7fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.95ex; height:3.843ex;" alt="{\displaystyle {\tfrac {2G}{3}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K\nu }{1+\nu }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1194366850f7d5421ab50b6c2706e58498f7ea96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.99ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3K\nu }{1+\nu }}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b592ec0480b2ca3a99ea29d9cbd0b631d777d4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.055ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7aa443ecbbd2c92a028d26ee25863bbc1592b4b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.16ex; height:4.176ex;" alt="{\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d011b18df127233873f4af03b50318509583006e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.63ex; height:3.843ex;" alt="{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M-2G\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4777994467bbd835a560ddf3d9b7a9bb88f6be15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.659ex; height:2.343ex;" alt="{\displaystyle M-2G\,}" loading="lazy">

</td></tr>

<tr>

<td style="text-align:left"><a class="mw-selflink selflink">Schubmodul</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> (2. Lamé-Konstante)

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3KE}{9K-E}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdd679253fa436a6900d1cf949d909d3b09cf3c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.653ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3KE}{9K-E}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3(K-\lambda )}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d4ffacd8fb2de88a240a663a13067a7c3e921f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.635ex; height:4.176ex;" alt="{\displaystyle {\tfrac {3(K-\lambda )}{2}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31c8588d1514e15c703af716b1d98c94ef2ad4c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.192ex; height:4.843ex;" alt="{\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E-3\lambda )+}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95f8b0e46d945e0bca51d36787ca9bf399951a39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.751ex; height:2.843ex;" alt="{\displaystyle (E-3\lambda )+}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\sqrt {(E-3\lambda )^{2}+8\lambda E}}{4}}}">

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\sqrt {(E-3\lambda )^{2}+8\lambda E}}{4}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4021c82c55b79608f6405f145bb8a950ff3dcfa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.219ex; height:5.509ex;" alt="{\displaystyle {\tfrac {\sqrt {(E-3\lambda )^{2}+8\lambda E}}{4}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {E}{2(1+\nu )}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea91afdc4bb2364d6deb22ee5653d33451c8d7b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.909ex; height:4.176ex;" alt="{\displaystyle {\tfrac {E}{2(1+\nu )}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0ae2f9d9801984c3a421791f68ed6d206906ce4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.868ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">

</td></tr>

<tr>

<td style="text-align:left"><a href="Poissonzahl" title="Poissonzahl">Poissonzahl</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu \,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32a13ab64f1a72c7827e70aed42c8fcb323bd7bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.619ex; height:1.676ex;" alt="{\displaystyle \nu \,}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K-E}{6K}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52e22f238cec78b55c3b21bb03bd0f6bc122006b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.653ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3K-E}{6K}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1c70a68ad578ac446b83ae2af368a85554e2f4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.356ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b11dec1ab89c26fc5ad5719fd2932e1dc68f4cb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.791ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -(E+\lambda )+}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b40dc9a8ad1072a3137d454e5aa059c685db2c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.397ex; height:2.843ex;" alt="{\displaystyle -(E+\lambda )+}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\sqrt {(E+\lambda )^{2}+8\lambda ^{2}}}{4\lambda }}}">

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\sqrt {(E+\lambda )^{2}+8\lambda ^{2}}}{4\lambda }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1f53e6f856cb9edc51f219929716b151c0398dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.973ex; height:5.676ex;" alt="{\displaystyle {\tfrac {\sqrt {(E+\lambda )^{2}+8\lambda ^{2}}}{4\lambda }}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {E}{2G}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4418d64c78abbd933103b2cf55cab0ccd145e8e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.95ex; height:3.676ex;" alt="{\displaystyle {\tfrac {E}{2G}}}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/569977b3b51798797b8ed7d287e828ef07cdb54a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.466ex; height:4.343ex;" alt="{\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {M-2G}{2M-2G}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d95ab5446ac8be62e025bbc4461d19ad03ddb0ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:6.777ex; height:4.009ex;" alt="{\displaystyle {\tfrac {M-2G}{2M-2G}}}" loading="lazy">

</td></tr>

<tr>

<td style="text-align:left"><a href="Longitudinalmodul" title="Longitudinalmodul">Longitudinalmodul</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa66ea8156203e93f2dca12aca24538c2bdce761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.829ex; height:2.176ex;" alt="{\displaystyle M\,}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c924e463a20a0f63bd2f876283e3fb1a965db68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.215ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3K-2\lambda \,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dccc234a5195e4cc32a106e7280ce2e082a9da43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.974ex; height:2.343ex;" alt="{\displaystyle 3K-2\lambda \,}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K+}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/707d7e6fce435495b9592608fd9372f1f15a728e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.874ex; height:2.343ex;" alt="{\displaystyle K+}" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4G}{3}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {4G}{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/349005a0ea0771949c65f6b25ff9b44c7b8f0b69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.95ex; height:3.843ex;" alt="{\displaystyle {\tfrac {4G}{3}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c09e8d1337a766b61ba2b9948fa4af1503d2ede0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.37ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {E-\lambda +{\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d1445f09c01236e46d0e21aa00dff9589fc7085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.555ex; height:5.509ex;" alt="{\displaystyle {\tfrac {E-\lambda +{\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}{2}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06fef6489910b32a86944f5fe2f5f0c1473e87c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.055ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7abf15539e84aef12ea7f6aed6ac61b939e0897a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.16ex; height:4.843ex;" alt="{\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda +2G\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f04bd3c70df3389cd672961e8ba5636aa8d5fb48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.572ex; height:2.343ex;" alt="{\displaystyle \lambda +2G\,}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4b0e21d0522f123eb9043c4e4352bbfd0944365.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.046ex; height:4.009ex;" alt="{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/036c5b70861f3145642c2d797f6cbf666ea26dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.201ex; height:4.343ex;" alt="{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Komplexer_Schubmodul" title="Komplexer Schubmodul">Komplexer Schubmodul</a></li>

<li><a href="Schubfestigkeit" class="mw-redirect" title="Schubfestigkeit">Schubfestigkeit</a></li>

<li><a href="Schubfluss" title="Schubfluss">Schubfluss</a></li>

<li><a href="Kontinuumsmechanik" title="Kontinuumsmechanik">Kontinuumsmechanik</a></li>

<li><a href="Festigkeitslehre" title="Festigkeitslehre">Festigkeitslehre</a></li>

<li><a href="Rheometrie" class="mw-redirect" title="Rheometrie">Rheometer</a></li>

<li><a href="Momenten-Magnituden-Skala" title="Momenten-Magnituden-Skala">Momenten-Magnituden-Skala</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.ieap.uni-kiel.de/surface/ag-berndt/lehre/aprakt/teil-2/torsion.pdf">Physikalisches Praktikum für Anfänger.</a> (PDF; 146 kB) Uni Kiel.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-CDL-1"><a href="#cite_ref-CDL_1-0">↑</a> Crandall, Dahl, Lardner: <cite class="lang" lang="en" dir="auto" style="font-style:italic">An Introduction to the Mechanics of Solids</cite>. McGraw-Hill, Boston 1959 (englisch).<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:Schubmodul&rft.au=Crandall%2C+Dahl%2C+Lardner&rft.btitle=An+Introduction+to+the+Mechanics+of+Solids&rft.date=1959&rft.genre=book&rft.place=Boston&rft.pub=McGraw-Hill" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a rel="nofollow" class="external text" href="https://www.eurocode-online.de/de/eurocode-inhalte/eurocode-3">Eurocode 3: Stahlbau.</a> Abgerufen am 7. Mai 2020.<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%3ASchubmodul&rft.title=Eurocode+3%3A+Stahlbau&rft.description=Eurocode+3%3A+Stahlbau&rft.identifier=https%3A%2F%2Fwww.eurocode-online.de%2Fde%2Feurocode-inhalte%2Feurocode-3">

</li>

<li id="cite_note-Hopcroft-3"><a href="#cite_ref-Hopcroft_3-0">↑</a> Matthew A. Hopcroft, <a href="William_D._Nix" title="William D. Nix">William D. Nix</a>, Thomas W. Kenny: <cite style="font-style:italic">What is the Young’s Modulus of Silicon?</cite> In: <cite style="font-style:italic">Journal of Microelectromechanical Systems</cite>. Band 19, Nr. 2, 2010, S. 229–238, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109/JMEMS.2009.2039697">10.1109/JMEMS.2009.2039697</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:Schubmodul&rft.atitle=What+is+the+Young%E2%80%99s+Modulus+of+Silicon%3F&rft.au=Matthew+A.+Hopcroft%2C+William+D.+Nix%2C+Thomas+W.+Kenny&rft.date=2010&rft.doi=10.1109%2FJMEMS.2009.2039697&rft.genre=journal&rft.issue=2&rft.jtitle=Journal+of+Microelectromechanical+Systems&rft.pages=229-238&rft.volume=19" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> <a rel="nofollow" class="external text" href="http://glassproperties.com/shear_modulus/">Berechnung des Schubmoduls von Gläsern</a> (englisch).

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> G. Mavko, T. Mukerji, J. Dvorkin: The Rock Physics Handbook. Cambridge University Press, 2003, ISBN 0-521-54344-4 (paperback).

</li>

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