iMechanica - hypoelastic
https://imechanica.org/taxonomy/term/1102
enIncrementally linear constitutive model. Nonlinear solution procedure
https://imechanica.org/node/20642
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Hello everyone,</p>
<p>My doubt is related with the obtenion of the true stress when using incrementally linear constitutive models (hypoelastic models). These models, alternatively to total stress strain models, related increment of strain and increment of stress. The predicted stress is obtained by adding to the previous stress the stress increment obtained by using the tangent matrix. By using total stress-strain models it is clear that the true stress is obtained by substituting the current strain into the constitutive equation. How do we do this for hypoelastic models?</p>
<p>Thank you!</p>
<p>Carlos</p>
</div></div></div><div class="field field-name-taxonomy-forums field-type-taxonomy-term-reference field-label-above"><div class="field-label">Forums: </div><div class="field-items"><div class="field-item even"><a href="/forum/109">Ask iMechanica</a></div></div></div><div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Free Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/371">nonlinear</a></div><div class="field-item odd"><a href="/taxonomy/term/1102">hypoelastic</a></div><div class="field-item even"><a href="/taxonomy/term/4460">Constitutive model</a></div><div class="field-item odd"><a href="/taxonomy/term/11448">residual force</a></div></div></div>Fri, 02 Dec 2016 14:30:59 +0000carmegi20642 at https://imechanica.orghttps://imechanica.org/node/20642#commentshttps://imechanica.org/crss/node/20642Hypoelastic-plasticity with logarithmic spin
https://imechanica.org/node/1652
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1102">hypoelastic</a></div><div class="field-item odd"><a href="/taxonomy/term/1103">plastic</a></div><div class="field-item even"><a href="/taxonomy/term/1104">logarithmic spin</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In small strain elastoplasticity we start off with an additive decomposition of the total strain into elastic and plastic parts. In terms of strain rates we write</p>
<p></p>
<p><img src="http://www.eng.utah.edu/~banerjee/iMechanica/Hypoelastic/img1.png" border="0" alt="$\displaystyle \dot{\ensuremath{\boldsymbol{\varepsilon}}} = \dot{\ensuremath{\boldsymbol{\varepsilon}}_e} + \dot{\ensuremath{\boldsymbol{\varepsilon}}_p}.<br /> $" width="96" height="32" align="middle" /></p>
<p>Prior to 1990 most large deformation plasticity algorithms extended this idea by postulating an additive decomposition of the Eulerian stretching tensor (rate of deformation):</p>
<p></p>
<p><img src="http://www.eng.utah.edu/~banerjee/iMechanica/Hypoelastic/img2.png" border="0" alt="$\displaystyle \ensuremath{\boldsymbol{d}}= \ensuremath{\boldsymbol{d}}_e + \ensuremath{\boldsymbol{d}}_p .<br /> $" width="105" height="34" align="middle" /> </p>
<p>Typically, a hypoelastic constitutive model was used to compute the Kirchhoff stress. For isotropic materials such a model is of the form</p>
<p></p>
<p><img src="http://www.eng.utah.edu/~banerjee/iMechanica/Hypoelastic/img3.png" border="0" alt="$\displaystyle \ensuremath{\overset{\circ}{\ensuremath{\boldsymbol{\tau}}}} = \l...<br /> ...ht)} \ensuremath{\boldsymbol{\mathit{1}}}+ 2 \mu \ensuremath{\boldsymbol{d}}_e<br /> $" width="185" height="46" align="middle" /> </p>
<p>A question that occupied many stalwarts of mechanics for a while was how to choose the correct objective rate of the Kirchhoff stress from an infinite number of possibilities. For various reasons, the Zaremba-Jaumann rate became popular. However, it was soon discovered that there was a problem with this rate. An oscillating shear stress response was predicted for a monotonically applied shear deformation when the Zaremba-Jaumann rate was used. The shear oscillation problem was ameliorated to some extent by using the corotational Green-Naghdi rate but other issues remained.</p>
<p>It was found that not only did the hypoelastic model deviate considerably from Hooke's law for large deformations, the model was dissipative and path-dependent, and could not be derived from a potential. Therefore the hypoelastic material considered above was not really elastic in the usual (hyperelastic) sense of the word. Over time people forgot that purely elastic materials could be obtained as a subclass of hypoelastic materials under some circumstances. The use of hypoelastic material models became taboo in the finite plasticity community. </p>
<p>As the plasticity community moved to problems that involved initial elastic anisotropy they also discovered that classical hypoelastic material models were only applicable to initially isotropic materials. This put the final nail in the hypoelastic/additive decomposition coffin. Some researchers (including me) continue to pursue the classical path saying that most metals undergo only small elastic strains. Also simulations show that for the errors introduced by the hypoelastic assumption are smaller than the error in the experimental data that are used to fit various models used in elastoplasticity. However, these justifications are not really satisfactory.</p>
<p>Researchers tried to find a way out of the mess and decided to change tracks. They started using a multiplicative decomposition of the deformation gradient and hyperelastic material models to represent the elastic response. This approach has been quite successful though some fundamental questions regarding uniqueness of the decomposition remain. The literature is huge and often hard to square with experimental data.</p>
<p>New life was breathed into the additive decomposition approach by the discovery by a number of researchers in the late 1990s that an Eulerian corotational logarithmic strain rate was power conjugate to the Kirchhoff stress [<a href="Hypoelastic.html#Xiao06">1</a>].</p>
<p>The Eulerian Hencky (logarithmic) strain is defined as</p>
<p></p>
<p><img src="http://www.eng.utah.edu/~banerjee/iMechanica/Hypoelastic/img4.png" border="0" alt="$\displaystyle \ensuremath{\boldsymbol{h}}= \ln\ensuremath{\boldsymbol{V}}= \sum...<br /> ...ath{\ensuremath{\mathbf{n}}_i\boldsymbol{\otimes}\ensuremath{\mathbf{n}}_i} .<br /> $" width="228" height="72" align="middle" /> </p>
<p>An isotropic elastic model in terms of the Hencky strain can be written as</p>
<p></p>
<p><img src="http://www.eng.utah.edu/~banerjee/iMechanica/Hypoelastic/img5.png" border="0" alt="$\displaystyle \ensuremath{\boldsymbol{\tau}}= \lambda \ensuremath{\text{tr}\lef...<br /> ...ht)} \ensuremath{\boldsymbol{\mathit{1}}}+ 2 \mu \ensuremath{\boldsymbol{h}} .<br /> $" width="183" height="36" align="middle" /> </p>
<p>The newly found relationship between the Hencky strain and the rate of deformation was</p>
<p></p>
<p><img src="http://www.eng.utah.edu/~banerjee/iMechanica/Hypoelastic/img6.png" border="0" alt="$\displaystyle \ensuremath{\boldsymbol{d}}= \ensuremath{\overset{\circ}{\ensurem...<br /> ...<br /> \ensuremath{\boldsymbol{\Omega}}^{\text{log}}\cdot\ensuremath{\boldsymbol{h}}<br /> $" width="239" height="56" align="middle" /> </p>
<p>where the logarithmic spin is given by</p>
<p></p>
<p><img src="http://www.eng.utah.edu/~banerjee/iMechanica/Hypoelastic/img7.png" border="0" alt="$\displaystyle \ensuremath{\boldsymbol{\Omega}}^{\text{log}} = \ensuremath{\bold...<br /> ...suremath{\boldsymbol{m}}_i\boldsymbol{\otimes}\ensuremath{\boldsymbol{m}}_i})<br /> $" width="555" height="71" align="middle" /></p>
<p> <br />
and</p>
<p></p>
<p><img src="http://www.eng.utah.edu/~banerjee/iMechanica/Hypoelastic/img8.png" border="0" alt="$\displaystyle \ensuremath{\boldsymbol{b}}= \ensuremath{\boldsymbol{F}}\cdot\ens...<br /> ...uremath{\boldsymbol{m}}_i\boldsymbol{\otimes}\ensuremath{\boldsymbol{m}}_i} .<br /> $" width="241" height="68" align="middle" /></p>
<p> <br />
It was shown that purely elastic behavior without path dependence or dissipation could be obtained if the objective rate in the relation</p>
<p></p>
<p><img src="http://www.eng.utah.edu/~banerjee/iMechanica/Hypoelastic/img3.png" border="0" alt="$\displaystyle \ensuremath{\overset{\circ}{\ensuremath{\boldsymbol{\tau}}}} = \l...<br /> ...ht)} \ensuremath{\boldsymbol{\mathit{1}}}+ 2 \mu \ensuremath{\boldsymbol{d}}_e<br /> $" width="185" height="46" align="middle" /></p>
<p> <br />
was given by</p>
<p></p>
<p><img src="http://www.eng.utah.edu/~banerjee/iMechanica/Hypoelastic/img9.png" border="0" alt="$\displaystyle \ensuremath{\overset{\circ}{\ensuremath{\boldsymbol{\tau}}}} = \d...<br /> ...uremath{\boldsymbol{\Omega}}^{\text{log}}\cdot\ensuremath{\boldsymbol{\tau}} .<br /> $" width="214" height="46" align="middle" /></p>
<p>It was also shown that this objective rate is the only one that is allowable in a consistent Eulerian rate theory of elastoplasticity.</p>
<p>Thus some of the major objections to using a hypoelastic model and an additive decomposition of the rate of deformation tensor have been removed, at least for moderate strains. It remains to be seen what modifications are needed to the theory to allow for initially anisotropic materials. I plan to try out the model in the next few days. If you have already done so please comment on what you found.</p>
<p> </p>
<p><a name="SECTION00010000000000000000" title="SECTION00010000000000000000" id="SECTION00010000000000000000"></a></p>
<p><strong>Bibliography</strong></p>
<dl><dt><a name="Xiao06" title="Xiao06" id="Xiao06"></a>1 H. Xiao, O. T. Bruhns, and A. Meyers.<br />
Elastoplasticity beyond small deformations.<br /><em>Acta Mechanica</em>, 182:31-111, 2006.<br /></dt>
</dl></div></div></div>Tue, 03 Jul 2007 19:42:45 +0000Biswajit Banerjee1652 at https://imechanica.orghttps://imechanica.org/node/1652#commentshttps://imechanica.org/crss/node/1652