Loss of ellipticity for non-coaxial plastic deformations in additive logarithmic finite strain plasticity

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• 作者：Patrizio Neff^a ; ^{patrizio.neff@uni-due.de" class="auth_mail" title="E-mail the corresponding author} ; Ionel-Dumitrel Ghiba^b ; ^c ; ^{dumitrel.ghiba@uni-due.de" class="auth_mail" title="E-mail the corresponding author} ; ^{dumitrel.ghiba@uaic.ro" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Hencky energy ; Ellipticity domain ; Isotropic formulation ; Additive plasticity
• 刊名：International Journal of Non-Linear Mechanics
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：81
• 期：Complete
• 页码：122-128
• 全文大小：623 K

文摘

In this paper we consider the additive logarithmic finite strain plasticity formulation from the view point of loss of ellipticity in elastic unloading. We prove that even if an elastic energy class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0020746216000044&_mathId=si0004.gif&_user=111111111&_pii=S0020746216000044&_rdoc=1&_issn=00207462&md5=c5332ae1d4679f7a7eaf257e767a1615">class="imgLazyJSB inlineImage" height="17" width="121" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0020746216000044-si0004.gif">class="mathContainer hidden">class="mathCode">$F\mapsto W\left(F\right)=\widehat{W}\left(\log U\right)$ defined in terms of logarithmic strain class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0020746216000044&_mathId=si0005.gif&_user=111111111&_pii=S0020746216000044&_rdoc=1&_issn=00207462&md5=07385d0e23a98c7867eaa0416001a887" title="Click to view the MathML source">logUclass="mathContainer hidden">class="mathCode">$\log U$, where class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0020746216000044&_mathId=si0006.gif&_user=111111111&_pii=S0020746216000044&_rdoc=1&_issn=00207462&md5=ed5a4dc07a40bb7ece26719df462759f">class="imgLazyJSB inlineImage" height="17" width="64" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0020746216000044-si0006.gif">class="mathContainer hidden">class="mathCode">$U=\sqrt{F^{T}F}$, happens to be everywhere rank-one convex as a function of F  , the new function class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0020746216000044&_mathId=si0007.gif&_user=111111111&_pii=S0020746216000044&_rdoc=1&_issn=00207462&md5=443799401604e5a3194bafe55a764888">class="imgLazyJSB inlineImage" height="18" width="172" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0020746216000044-si0007.gif">class="mathContainer hidden">class="mathCode">$F\mapsto \tilde{W}\left(F\right)=\widehat{W}(\log U-\log U_p)$ need not remain rank-one convex at some given plastic stretch U_p (viz. class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0020746216000044&_mathId=si0008.gif&_user=111111111&_pii=S0020746216000044&_rdoc=1&_issn=00207462&md5=19efd1e7da71a884d1385793689efa1d">class="imgLazyJSB inlineImage" height="20" width="71" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0020746216000044-si0008.gif">class="mathContainer hidden">class="mathCode">$E_p^{\log }≔\log U_p$). This is in complete contrast to multiplicative plasticity (and infinitesimal plasticity) in which class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0020746216000044&_mathId=si0009.gif&_user=111111111&_pii=S0020746216000044&_rdoc=1&_issn=00207462&md5=f5ad41545e49829539cbcafb8770cce2">class="imgLazyJSB inlineImage" height="19" width="78" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0020746216000044-si0009.gif">class="mathContainer hidden">class="mathCode">$F\mapsto W\left(F{F}_p^{-1}\right)$ remains rank-one convex at every plastic distortion F_p if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0020746216000044&_mathId=si0010.gif&_user=111111111&_pii=S0020746216000044&_rdoc=1&_issn=00207462&md5=a6c67312b0625176bd8160010b06b995" title="Click to view the MathML source">F↦W(F)class="mathContainer hidden">class="mathCode">$F\mapsto W\left(F\right)$ is rank-one convex (class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0020746216000044&_mathId=si0011.gif&_user=111111111&_pii=S0020746216000044&_rdoc=1&_issn=00207462&md5=ab3fb552887b8271a13467901062561c">class="imgLazyJSB inlineImage" height="17" width="126" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0020746216000044-si0011.gif">class="mathContainer hidden">class="mathCode">$\nabla u\mapsto \parallel \mathrm{sym}\nabla u-{\epsilon }_p{\parallel }^2$ remains convex). We show this disturbing feature of the additive logarithmic plasticity model with the help of a recently introduced family of exponentiated Hencky energies.