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The arguments below are about Cauchy-elastic/elastic material. And the source of my information is Gerhard A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering.

When defining the constitutive equation of stress, we express the stress state as a function of the response function that depends on the deformation gradient tensor. If the material considered undergoes homogeneous deformation, then the response function is in terms of the deformation gradient that is considered to be the same at each point of the continuum body/ material region and so stress is constant at all points.

As for an isotropic elastic material, the assumption is that the response function depends on the left Cauchy-elastic tensor.

My question is that, can we consider the constitutive equation of a homogeneous elastic material the same as that of an isotropic elastic material? I think homogeneity is when the material has the same response everywhere, and for isotropy the response is the same in all directions. So they are the same, right?

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Homogeneous means that it is the same in all points in space. Isotropic means that it is the same in all directions.

E.g., when we speak that the space is homogeneous and isotropic (in the context of Newtonian mechanics and relativity), we speak about symmetries in respect to translation and in respect to rotation in space.

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I was able to answer my question when I watched this video: https://www.youtube.com/watch?v=WgNd6E2odxA

If a material is isotropic then at a given point, the stress state is the same in all directions, but might differ from one point to another. So in general we have different stress states at each point, but at each point same magnitude in all directions.

If a material is homogeneous then the stress state is the same at each point except that the stress is not equal in different directions. So, throughout the material, we have everywhere the same stress state but when we vary the point, we will still have the same stress state but not same magnitude in different directions.

Now that, the two concepts are not the same, we can't have the same constitutive equation for both materials except if both properties are satisfied.

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