GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Maxwell-Körper</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Maxwell-Körper</span></h1>

</header>

</div>

<p>Als <b>Maxwell-Körper</b> wird in der <a href="Rheologisches_Modell" title="Rheologisches Modell">rheologischen Modellierung</a> die <a href="Federkonstante#Kombination_von_Federn" title="Federkonstante">Reihenschaltung</a> einer (linearen) <a href="Hooke-Element" class="mw-redirect" title="Hooke-Element">Hookeschen</a> <a href="Feder_(Technik)" title="Feder (Technik)">Feder</a> und eines <a href="Schwingungsd%C3%A4mpfer" title="Schwingungsdämpfer">Dämpfers</a> bezeichnet.

</p><p>Die Modelleigenschaften dieser Elemente sind:

</p>

<ul><li>das <a href="Hookesches_Gesetz" title="Hookesches Gesetz">Hookesche Gesetz</a> <span class="mwe-math-element mwe-math-element-inline"><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 =E\cdot \varepsilon }">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> für die Feder

<ul><li>der <a href="Elastizit%C3%A4tsmodul" title="Elastizitätsmodul">Elastizitätsmodul</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span></li>

<li>die <a href="Dehnung" title="Dehnung">Dehnung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"></span></li></ul></li>

<li>die Geschwindigkeits- oder <a href="%C3%84nderungsrate" title="Änderungsrate">Raten</a>abhängigkeit <span class="mwe-math-element mwe-math-element-inline"><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 =\eta \cdot {\dot {\varepsilon }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75d342f19a77e054fa9f6295fb83ea07e93366fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.633ex; height:2.676ex;" alt="{\displaystyle \sigma =\eta \cdot {\dot {\varepsilon }}}" loading="lazy"></span> des Dämpfers

<ul><li>die <a href="Mechanische_Spannung" title="Mechanische Spannung">Spannung</a> <span class="mwe-math-element mwe-math-element-inline"><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 }">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span></li>

<li>die <a href="Viskosit%C3%A4t" title="Viskosität">Viskosität</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"></span>.</li></ul></li></ul>

<p>In Kombination mit der Annahme, dass die Dehnungen von Feder und Dämpfer zur Gesamtdehnung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =\varepsilon _{F}+\varepsilon _{D}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon =\varepsilon _{F}+\varepsilon _{D}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95d87a1a3bdf4a4105aba7b19452f823c5dcf19b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.245ex; height:2.343ex;" alt="{\displaystyle \varepsilon =\varepsilon _{F}+\varepsilon _{D}}" loading="lazy"></span> zu addieren sind, ebenso ihre Raten zur Gesamtdehnungsrate <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon }}={\dot {\varepsilon }}_{F}+{\dot {\varepsilon }}_{D}}">

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon }}={\dot {\varepsilon }}_{F}+{\dot {\varepsilon }}_{D}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54bcaef2529a1f27042ebed4a2b64e14dc9e5609.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.063ex; height:2.509ex;" alt="{\displaystyle {\dot {\varepsilon }}={\dot {\varepsilon }}_{F}+{\dot {\varepsilon }}_{D}}" loading="lazy"></span>, ergibt sich die beschreibende <a href="Differentialgleichung" title="Differentialgleichung">Differentialgleichung</a>

</p>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Rightarrow {\dot {\varepsilon }}={\frac {{\dot {\sigma }}(t)}{E}}+{\frac {\sigma (t)}{\eta }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5389cf4ac5cc78e439e5e42186d2b1fa569675d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.893ex; height:6.176ex;" alt="{\displaystyle \Rightarrow {\dot {\varepsilon }}={\frac {{\dot {\sigma }}(t)}{E}}+{\frac {\sigma (t)}{\eta }}}" loading="lazy"></span>.</dd></dl>

<p>Die Eigenschaften dieses Systems lassen sich am besten diskutieren, wenn man seine Reaktionen auf ein Kriech- bzw. Retardationsexperiment sowie ein Relaxationsexperiment betrachtet.

</p>

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

<p>Ein <a href="Kriechen_(Werkstoffe)" title="Kriechen (Werkstoffe)">Kriech- oder Retardations</a>experiment bedeutet die Beaufschlagung des Systems mit einem Spannungssprung <span class="mwe-math-element mwe-math-element-inline"><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)=\sigma _{0}\,H(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (t)=\sigma _{0}\,H(t)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74c8176f59d2a2bf090b6060c06be9fc978f7ab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.558ex; height:2.843ex;" alt="{\displaystyle \sigma (t)=\sigma _{0}\,H(t)}" loading="lazy"></span>, wobei wir mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(t)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d1b6c8837aed2794e7b52afa88ad371f1d275fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.713ex; height:2.843ex;" alt="{\displaystyle H(t)}" loading="lazy"></span> die <a href="Heaviside-Funktion" title="Heaviside-Funktion">Heaviside-Funktion</a> bezeichnen, also einen Sprung von Null auf Eins zur Zeit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{0}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=t_{0}=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4791afb60e0dc918a010f8adfb3613c19bb639f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.093ex; height:2.509ex;" alt="{\displaystyle t=t_{0}=0}" loading="lazy"></span>.

</p><p>Aus der beschreibenden Differentialgleichung erhält man durch <a href="Integralrechnung" title="Integralrechnung">Integration</a> nach der Zeit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> für die Dehnungsantwort dieses Körpers unter einem Spannungssprung auf <span class="mwe-math-element mwe-math-element-inline"><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 _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2f330c7d4a4ee92e5d43dfe3e23f0de3406ec78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.009ex;" alt="{\displaystyle \sigma _{0}}" loading="lazy"></span>:

</p>

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

<mover>

</mover>

</mrow>

</mrow>

</mtd>

<mtd>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon (t)=\int \limits _{0}^{t}{\dot {\varepsilon }}{\rm {d}}t&={\frac {\sigma _{0}}{E}}+\int \limits _{0}^{t}{\frac {\sigma }{\eta }}{\rm {d}}t\\&={\frac {\sigma _{0}}{E}}+{\frac {\sigma _{0}}{\eta }}\,t\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6eeb99865774604866fb044a797fa43addddac42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:29.687ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\varepsilon (t)=\int \limits _{0}^{t}{\dot {\varepsilon }}{\rm {d}}t&={\frac {\sigma _{0}}{E}}+\int \limits _{0}^{t}{\frac {\sigma }{\eta }}{\rm {d}}t\\&={\frac {\sigma _{0}}{E}}+{\frac {\sigma _{0}}{\eta }}\,t\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Dies zeigt das bekannte Verhalten einer konstanten Antwort <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sigma _{0}}{E}}}">

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\sigma _{0}}{E}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4216041781e07376ac15c409df7493bc4ef0fb2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.218ex; height:4.676ex;" alt="{\displaystyle {\frac {\sigma _{0}}{E}}}" loading="lazy"></span> aufgrund des Spannungssprungs am Federelement, aber auch eine (lineare) Zeitabhängigkeit. Gerade dies beschreibt das unbegrenzte "Weiter-Dehnen" ("<i>Kriechen</i>") dieses Systems bei der hier angelegten (konstanten) Spannung <span class="mwe-math-element mwe-math-element-inline"><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 _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2f330c7d4a4ee92e5d43dfe3e23f0de3406ec78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.009ex;" alt="{\displaystyle \sigma _{0}}" loading="lazy"></span>.

</p>

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

<p>Das <a href="Relaxation_(Naturwissenschaft)" title="Relaxation (Naturwissenschaft)">Relaxations</a>experiment zeigt die Antwort des Systems auf einen Dehnungssprung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (t)=\varepsilon _{0}\,H(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon (t)=\varepsilon _{0}\,H(t)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/907ca7a405a0c90bd2f9e3c0ab0c621db58d3972.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.068ex; height:2.843ex;" alt="{\displaystyle \varepsilon (t)=\varepsilon _{0}\,H(t)}" loading="lazy"></span>.

Hierbei sehen wir aus der obigen, beschreibenden Differentialgleichung, dass nur die <a href="Homogene_Gleichung" class="mw-redirect" title="Homogene Gleichung">homogene</a> DGL gelöst werden muss:

</p>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\dot {\varepsilon }}(t)&=0\\\Leftrightarrow {\frac {{\dot {\sigma }}(t)}{E}}+{\frac {\sigma (t)}{\eta }}&=0\\\Leftrightarrow \tau ^{\star }{\dot {\sigma }}(t)+\sigma (t)&=0\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27bb24cc7c328369fe22f196b3b8ca24563622d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:21.091ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}{\dot {\varepsilon }}(t)&=0\\\Leftrightarrow {\frac {{\dot {\sigma }}(t)}{E}}+{\frac {\sigma (t)}{\eta }}&=0\\\Leftrightarrow \tau ^{\star }{\dot {\sigma }}(t)+\sigma (t)&=0\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>mit der typischen <a href="Relaxationszeit" class="mw-redirect" title="Relaxationszeit">Relaxationszeit</a> <span class="mwe-math-element mwe-math-element-inline"><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 ^{\star }={\frac {\eta }{E}}}">

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau ^{\star }={\frac {\eta }{E}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bf78a6c4a6d79f6114a752350aef9992ba15de9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.022ex; height:4.843ex;" alt="{\displaystyle \tau ^{\star }={\frac {\eta }{E}}}" loading="lazy"></span>.

</p><p>Die Lösung dieser DGL ist eine <a href="Exponentialfunktion" title="Exponentialfunktion">e-Funktion</a> der Form <span class="mwe-math-element mwe-math-element-inline"><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)=c\,e^{-t/\tau ^{\star }}}">

<msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (t)=c\,e^{-t/\tau ^{\star }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e41cb2f19982984e9d6472a271efd48770b9242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.202ex; height:3.343ex;" alt="{\displaystyle \sigma (t)=c\,e^{-t/\tau ^{\star }}}" loading="lazy"></span>,

wobei sich die <a href="Integrationskonstante" class="mw-redirect" title="Integrationskonstante">Integrationskonstante</a> c aus der <a href="Anfangsbedingung" title="Anfangsbedingung">Anfangsbedingung</a> <span class="mwe-math-element mwe-math-element-inline"><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=0)=E\,\varepsilon _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (t=0)=E\,\varepsilon _{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12dc312c18e3b3338f246d87c21d2f8b2c0bd581.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.638ex; height:2.843ex;" alt="{\displaystyle \sigma (t=0)=E\,\varepsilon _{0}}" loading="lazy"></span> ergibt.

</p><p>Damit ist die Lösung:

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (t)=E\,\varepsilon _{0}\,e^{-t/\tau ^{\star }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b570d196c5391c909d0058caac1ece5a1d6eb945.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.496ex; height:3.343ex;" alt="{\displaystyle \sigma (t)=E\,\varepsilon _{0}\,e^{-t/\tau ^{\star }}}" loading="lazy"></span>.</dd></dl>

<p>Der Dehnungssprung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acb0a8377db20e42274444cb181d51b5532b5844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}" loading="lazy"></span> an <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"></span> bewirkt also einen Spannungssprung <span class="mwe-math-element mwe-math-element-inline"><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=0)=E\varepsilon _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (t=0)=E\varepsilon _{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/517c7ca31e266b684d721d39129812a8b98fb673.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.251ex; height:2.843ex;" alt="{\displaystyle \sigma (t=0)=E\varepsilon _{0}}" loading="lazy"></span>.

Dann zieht sich die Feder zusammen, und die Dehnung geht in den Dämpfer über. Damit entspannt sich das System bei vorgegebener Gesamtdehnung immer weiter. Dies nennt man "Relaxation."

</p><p>Für exemplarische Kennwerte <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=4}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a9e5e23da29196057a970eea761e48a2d2e7d03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.037ex; height:2.176ex;" alt="{\displaystyle E=4}" loading="lazy"></span> <a href="Pascal_(Einheit)#Megapascal" title="Pascal (Einheit)">MPa</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =3}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85351f618f4f05e9b46217015506dcd98fae8070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.43ex; height:2.676ex;" alt="{\displaystyle \eta =3}" loading="lazy"></span> M<a href="Pascalsekunde" class="mw-redirect" title="Pascalsekunde">Pa·s</a> ist oben der Relaxationsverlauf zu sehen. Die Relaxationszeit ist damit <span class="mwe-math-element mwe-math-element-inline"><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 ^{\star }=0{,}75\,\mathrm {s} }">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau ^{\star }=0{,}75\,\mathrm {s} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b619bc215e93c34cbde85171717dccc1721f7209.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.848ex; height:2.676ex;" alt="{\displaystyle \tau ^{\star }=0{,}75\,\mathrm {s} }" loading="lazy"></span>.

</p>

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

<ul><li>H. Altenbach: <cite style="font-style:italic">Kontinuumsmechanik</cite>. Springer, 2012, ISBN 978-3-642-24118-5.<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:Maxwell-K%C3%B6rper&rft.au=H.+Altenbach&rft.btitle=Kontinuumsmechanik&rft.date=2012&rft.genre=book&rft.isbn=9783642241185&rft.pub=Springer" style="display:none"> </span></li>

<li>P. Haupt: <cite style="font-style:italic">Continuum Mechanics and Theory of Materials</cite>. Springer, 2000, ISBN 3-540-66114-X.<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:Maxwell-K%C3%B6rper&rft.au=P.+Haupt&rft.btitle=Continuum+Mechanics+and+Theory+of+Materials&rft.date=2000&rft.genre=book&rft.isbn=354066114X&rft.pub=Springer" style="display:none"> </span></li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-08-05" href="https://de.wikipedia.org/wiki/?title=Maxwell-K%C3%B6rper&oldid=247415274">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>