On pseudo-strain energies for elastic damaging and nonlocal materials

We discuss the problem of finding strain function whose gradient coincides with a given stress-strain relationship. To this purpose, classic theorems for the existence of potentials of nonlinear operator are applied. Two types of stress-strain relationships are addressed. First, a stress-strain relationship of gradient type for elastic materials is studied, where the stress depends on the second gradient of the strain. Furthermore, a stress-strain relationship for isotropic elasto-damaging materials is discussed, where the damage parameter depends explicitly on the strain.