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?