Document Zbl 1425.74070

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Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci.</a> <a href="/?q=in%3A432614" title="Articles in this Issue">377, No. 2144, Article ID 20180074, 18 p. (2019)</a>. </div> <div class="abstract">Summary: A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors satisfying the boundary conditions in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed according to the symmetry group of the material at fixed initial stresses. These results allow one to extend Ball’s existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7131126">Cited in <strong>8</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A74B10" title="MSC2020">74B10</a> </td> <td class="space"> Linear elasticity with initial stresses </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74B20" title="MSC2020">74B20</a> </td> <td class="space"> Nonlinear elasticity </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aconstitutive+theory">constitutive theory</a>; <a href="/?q=ut%3Ainitial+stress">initial stress</a>; <a href="/?q=ut%3Ametric+distortion">metric distortion</a>; <a href="/?q=ut%3Aelastic+minimizers">elastic minimizers</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1425.74070" data-ciurl="/ci/07131126" data-biburl="/bibtex/07131126.bib" data-amsurl="/amsrefs/07131126.bib" data-xmlurl="/xml/07131126.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07131126.pdf" title="Zbl 1425.74070 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1098/rsta.2018.0074" aria-label="DOI for “On the existence of elastic minimizers for initially stressed materials”" title="10.1098/rsta.2018.0074">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1812.05579"title="Note: arXiv document may differ from published version">arXiv</a> <a class="btn btn-default btn-xs" type="button" href="http://europepmc.org/articles/pmc6452035" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Barenblatt GI, Joseph DD (eds). 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