iMechanica - Comments for "constitutive relations (stress-strain) for non-homogeneous materials?"
https://imechanica.org/node/9967
Comments for "constitutive relations (stress-strain) for non-homogeneous materials?"enConsider tensor notation of
https://imechanica.org/comment/16456#comment-16456
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<p><em>In reply to <a href="https://imechanica.org/node/9967">constitutive relations (stress-strain) for non-homogeneous materials?</a></em></p>
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Consider tensor notation of Hook`s law:
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Sigma_ij = C_ijkl*Epsilon_ij
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Generally 4th rank tensor has 81 independent components. However due to symmetry of Sigma and Epsilon, in our case C has up to 36 independent components (so called minor symmetry of C, i.e. Cijkl=C_jikl=Cijlk=...). There is also major symmetry which comes from the fact that stress and strain are work conjugate. Major symmetry means that C_ijkl = C_klij and therefore reduces number of independent components to 21.
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So in general case, properties of elastic body are described with 21 constants. These constants describe anisotropy of material, which is the dependence of stiffness from orientation in our case. Different cases of symmetry require fewer constants (up to 2 in isotropic case). This has nothing to do with coordinate dependence as we deal with so called first-order constitutive models, i.e. the models which account only for values of stress and strain in a one point but not their gradients with respect to coordinates. So classical constitutive models give a relation between stresses and strains in a one point. An the difference of elastic moduli in x and x+dx relates only to heterogeniety which means that different parts of a body have different mechanical properties. This fact is important, for example, when modelling a body with inclusions or voids.
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</ul>Wed, 23 Mar 2011 07:01:44 +0000Alexander Polishchukcomment 16456 at https://imechanica.orgThank you for the
https://imechanica.org/comment/16454#comment-16454
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<p><em>In reply to <a href="https://imechanica.org/comment/16451#comment-16451">The linear elastic</a></em></p>
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Thank you for the reply.
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So these "constants" (wheter for the anisotropic or isotropic case) are realy just constants with respect to time?
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What is a little confusing when thinking about these constants in this way is for example in the case of an isotropic material, where say in point (x1,y1,z1) the constants
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are the same regardles of the direction. But if you move to an adjacent point (x1_dx,y1+dy,z1+dz) the set of constants is different, then at the interface between these two points
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because the constants are different in one side than in the other isn't this akin to anisotropy?
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I still haven't found a book that explains this clearly for the elastic or viscoelastic cases.
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Thanks again, I really appreciate your help.
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Regards,
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</ul>Tue, 22 Mar 2011 15:26:53 +0000pvagcomment 16454 at https://imechanica.orgThe linear elastic
https://imechanica.org/comment/16451#comment-16451
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<p><em>In reply to <a href="https://imechanica.org/node/9967">constitutive relations (stress-strain) for non-homogeneous materials?</a></em></p>
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The linear elastic constitutive relations should be understood as hold at each point: that is Stress(x,y,z)=C(x,y,z)strain(x,y,z). It don't know matter whether it is isotropic or anisotropic or not. You need to differentiate between directional dependence (isotropic/anisotropic) versus pisitional dependence (heterogeneous/homogeneous). If you want a way to avoid the positional variance of the material properties, you need to use homogenization.
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</ul>Mon, 21 Mar 2011 16:29:10 +0000Wenbin Yucomment 16451 at https://imechanica.org