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constitutive relations (stress-strain) for non-homogeneous materials?

Dear All,

I am student new to this field and I recently found this amazing forum. I have a question that relates to the basic understanding of deformable materials.

I would like to understand better or find references that talk about the constitutive relations (stress-strain) for materials with non homogenous material properties, for

example materials that might have modulus of elasticity of the form E(x), E(x,y), or even E(x,y,z).

First, is a material having E(x,y,z) an anisotropic material?

The books on elasticity or viscoelasticiy  I have read so far talk about elasticity "constants" [C] such that Stress=[C] strain, even for anisotropic materials.

Is it infered that for anisotropic materials these "constants" can change in space (x,y,z) and that they are only constants with respect to time?

If not, how does one model the stress-strain relation for materials with spatially varying material properties (E(x), E(x,y), etc..)?

I would really appreciate being pointed to good references, papers or books that cover this question.

Thanks in advance for you help!

Best regards,




Wenbin Yu's picture

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.

Thank you for the reply.

So these "constants" (wheter for the anisotropic or isotropic case) are realy just constants with respect to time?

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

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

because the constants are different in one side than in the other isn't this akin to anisotropy? 

I still haven't found a book that explains this clearly for the elastic or viscoelastic cases.

Thanks again, I really appreciate your help.



Alexander Polishchuk's picture

Consider tensor notation of Hook`s law:

Sigma_ij = C_ijkl*Epsilon_ij

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.

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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