Uniqueness studies in boundary value problems involving some second gradient models

Abstract

Enhanced models such as second gradient models have been mainly theoretically developed during the sixties and the seventies. New attention has been paid to these models recently when it has been recognized that classical inelastic constitutive equations, especially the ones not obeying a normality rule, induce ill-posed boundary value problems. However little work has been done to study the well-posedness properties of boundary value problems involving these enhanced models. The aim of this paper is a partial study of this point. Only uniqueness of solution of numerical discretization of boundary value problems involving second gradient models is considered. We extend to second gradient models a numerical search method to find non-uniqueness, already established for classical media. In the case of loss of uniqueness for the underlying model, our numerical results show clearly that in some cases uniqueness is lost for second gradient model as well. It seems that second gradient extensions of classical models do not restore the uniqueness properties, but only the objectivity of the numerical computation. Finally our numerical experiments show once more that post-peak behavior is a structural effect. They also suggest that some constitutive equations often used in geomaterials modeling may be questionable, and may not be complete because even adding second gradient enhancement to these models yields non-unique solutions.