iMechanica - Comments for "Seeking a logarithmic operator for a 4th order tensor"
https://imechanica.org/node/5589
Comments for "Seeking a logarithmic operator for a 4th order tensor"enisotropic elasticity tensor eigensystem
https://imechanica.org/comment/11163#comment-11163
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<p><em>In reply to <a href="https://imechanica.org/comment/11158#comment-11158">Cijkm is elasticity tensor</a></em></p>
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If your elasticity tensor is isotropic, there are six 2nd order eigentensors for it. The first is the second order identity tensor, which you might want to scale by 1/SQRT(3) to normalize it. The corresponding eigenvalue is 3 x the bulk modulus. The remaining eigentensors are five orthogonal deviatoric tensors, which you can construct using Gram-Schmidt orthonormalization. Their associated eigenvalue is 2 x the shear modulus.
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Another way of looking at this is to consider using projection operators. In that case, we represent the elasticity tensor as 3K Psp + 2G Pd where Psp is the spherical projection operator, Pd is the deviatoric projection operator, K is the bulk modulus, and G is the shear modulus. ln(C) is then ln(3K) Psp + ln(2G) Pd. The spherical projection operator is 1/3 i x i (the dyadic product of two second order identity tensors). The deviatoric projection operator is simply the 4th order identity tensor for symmetric second order tensors minus the spherical projection operator.
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Hope this helps,
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Matt
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Matt Lewis<br />
Los Alamos, New Mexico
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</ul>Tue, 09 Jun 2009 13:53:19 +0000Matt Lewiscomment 11163 at https://imechanica.orgspectral decomposition
https://imechanica.org/comment/11159#comment-11159
<a id="comment-11159"></a>
<p><em>In reply to <a href="https://imechanica.org/node/5589">Seeking a logarithmic operator for a 4th order tensor</a></em></p>
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Perhaps this might be related:
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<a href="http://www.imechanica.org/node/1091">http://www.imechanica.org/node/1091</a>
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</ul>Mon, 08 Jun 2009 22:56:03 +0000Nachiket Gokhalecomment 11159 at https://imechanica.orgCijkm is elasticity tensor
https://imechanica.org/comment/11158#comment-11158
<a id="comment-11158"></a>
<p><em>In reply to <a href="https://imechanica.org/node/5589">Seeking a logarithmic operator for a 4th order tensor</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
In my application, C<em>ijkm</em> is the isotropic linear elasticity tensor, which comes with an associated inverse - the compliance tensor.
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Any recommended readings that detail how to approach spectral decomposition for order 4 tensors?
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None of my textbooks seem to touch on this topic.
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I did find today an <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V29-4123CDP-2&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=8173461300c42c8f12635f617a007cfb" target="_blank">article</a> by M. Itskov, but haven't digested it yet. (<a href="http://www.sciencedirect.com/science/journal/00457825"><strong>Computer Methods in Applied Mechanics and Engineering</strong></a>,<a href="http://www.sciencedirect.com/science?_ob=PublicationURL&_tockey=%23TOC%235697%232000%23998109997%23207740%23FLA%23&_cdi=5697&_pubType=J&view=c&_auth=y&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=64e6b7f250129b7861fcb768bac30d76">Volume 189, Issue 2</a>,<br />
1 September 2000,<br />
Pages 419-438)
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</ul>Mon, 08 Jun 2009 22:46:36 +0000wvmarscomment 11158 at https://imechanica.orgLog of a 4th order tensor
https://imechanica.org/comment/11154#comment-11154
<a id="comment-11154"></a>
<p><em>In reply to <a href="https://imechanica.org/node/5589">Seeking a logarithmic operator for a 4th order tensor</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Interesting question. If Cijkm is not singular, then it should have an eigensystem. If you can find the eigenvectors (2nd order tensors) and normalize them appropriately, you can certainy take the ln of each corresponding eigenvalue. The math for all of this is made easier if C has major and minor symmetries.
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Matt Lewis<br />
Los Alamos, New Mexico
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</ul>Mon, 08 Jun 2009 18:25:53 +0000Matt Lewiscomment 11154 at https://imechanica.orgError | iMechanica