Some theoretical results about second-order work, uniqueness, existence and controllability independent of the constitutive equation

Abstract

When they are studied as continuum media, granular materials and other soils and rocks exhibit a complex behavior. Contrary to metals, their isotropic and deviatoric behavior are coupled. This implies some mathematical difficulties concerning boundary-value problems solved with constitutive equations modelling the salient features of such geomaterials. One of the well-known consequences is that the so-called second-order work can be negative long before theoretical failure occurs. Keeping this in mind, the starting point of this work is the pioneering and illuminating work of Nova (1994), who proved that using an isotropic hardening elasto-plastic model not obeying the normality rule, it is possible to exhibit either loss of uniqueness or loss of existence of the solution of a boundary-value problem as soon as the second-order work is negative. Because the geomaterial behavior is quite difficult to model, in practice many different constitutive equations are used. It is then important to study the point raised by Nova for other constitutive equations. In this paper, his result is generalized for any inelastic rate-independent constitutive equation. Similarly the link between localization and controllability proved by Nova (1989) is extended to some extent to a general inelastic model.