On inconstant Poisson's ratios in non-homogeneous elastic media

For non-homogeneous linear elastic materials, it is demonstrated that even for the simplest loading case, i.e. quasi-static uniaxial, the Poisson's ratio (PR) is space dependent and not a constant. Furthermore, the assumption of constant PR values or of separable temporal and spatial PR functions leads to ill-posed overdetermined problems. Additionally, elastic PRs become space and time dependent under time dependent stresses in non-homogeneous elastic media. Under these and more general circumstances, PRs cannot be considered material property descriptors since they now become functions of the spatially changing moduli and stresses, and vary accordingly. The same conclusions are also drawn for nonlinear elastic media with small or large deformations.