Document Zbl 0604.73013

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A.</a>; <a href="/authors/murat.francois" title="Author Profile">Murat, F.</a></div> <h2 class="title"> <strong>Homogenization and optimal bounds in linear elasticity.</strong> <i>(English)</i> <a class="label nowrap" href="/0604.73013">Zbl 0604.73013</a> </h2> <div class="source"> <a href="/serials/37" title="Journal Profile">Arch. Ration. Mech. Anal.</a> <a href="/?q=in%3A152153" title="Articles in this Issue">94, 307-334 (1986)</a>. </div> <div class="abstract"><span class="zbmathjax-textit">Z. Hashin</span> and <span class="zbmathjax-textit">S. Shtrikman</span>’s derivation [J. Mech. Phys. Solids 11, 127-140 (1963; Zbl 0108.369)] of bounds on the bulk and shear moduli of a two phase isotropic elastic composite, and their optimality, are reconsidered in the context of homogenization using the techniques of ”H-convergence” and ”compensated compactness”. The bounds on the bulk modulus obtained coincide with those of Hashin and Shtrikman however they are not restricted to the case of well-ordered phases. The bounds on the shear modulus are claimed to be better than those obtained by <span class="zbmathjax-textit">B. Paul</span> [Trans. ASME 218, 36-41 (1960)]. Also, it is shown that Hashin and Shtrikman bounds on the bulk and shear moduli are simultaneously achieved through a finite number of layering processes.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A3601">J.Ignaczak</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A3976469">Cited in <strong>3</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A3976469">Cited in <strong>120</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A74E05" title="MSC2020">74E05</a> </td> <td class="space"> Inhomogeneity in solid mechanics </td> </tr><tr> <td class="mono"> 74S30 </td> <td class="space"> Other numerical methods in solid mechanics (MSC2010) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aoptimal+bounds">optimal bounds</a>; <a href="/?q=ut%3Abounds+on+the+bulk+and+shear+moduli">bounds on the bulk and shear moduli</a>; <a href="/?q=ut%3Atwo+phase+isotropic+elastic+composite">two phase isotropic elastic composite</a>; <a href="/?q=ut%3Ahomogenization">homogenization</a>; <a href="/?q=ut%3AH-convergence">H-convergence</a>; <a href="/?q=ut%3Acompensated+compactness">compensated compactness</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0108.369">Zbl 0108.369</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 0604.73013" data-ciurl="/ci/03976469" data-biburl="/bibtex/03976469.bib" data-amsurl="/amsrefs/03976469.bib" data-xmlurl="/xml/03976469.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/03976469.pdf" title="Zbl 0604.73013 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/10.1007/BF00280908" aria-label="DOI for “Homogenization and optimal bounds in linear elasticity”" title="10.1007/BF00280908">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">G. E. Backus [1] ?Long-Wave Elastic Anisotropy Produced by Horizontal Layering?, J. Geophys. Res., 1962, V. 67, p. 4427-4440. · <a href="/1369.86005" class="nowrap">Zbl 1369.86005</a> · <a href="https://doi.org/10.1029/JZ067i011p04427" class="nowrap">doi:10.1029/JZ067i011p04427</a></td> </tr><tr> <td>[2]</td> <td class="space">A. Bensoussan, J. L. Lions, & G. Papanicolaou [1] Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. · <a href="/0404.35001" class="nowrap">Zbl 0404.35001</a></td> </tr><tr> <td>[3]</td> <td class="space">R. M. Christensen [1] Mechanics of Composite Materials, Wiley Interscience, New York, 1979.</td> </tr><tr> <td>[4]</td> <td class="space">P. Germain [1] Mécanique des Milieux Continus, Masson, Paris, 1973.</td> </tr><tr> <td>[5]</td> <td class="space">K. Golden & G. Papanicolaou [1 ] ?Bounds for Effective Parameters of Heterogeneous Media by Analytic Continuation?, Commun. Math. 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