Stability of equilibrium in elastic-plastic solids

Abstract

A suitable definition of a general standard material has allowed us to consider the evolution problem in its general non-linearized form, in order to formulate a criterion of stability valid for finite perturbations of the equilibrium under dead loading. It should be relatively easy to complete this study by taking into account thermal effects in the now well defined thermodynamical frame of standard materials which exhibit normal dissipativity.

For practical use it is worthwhile to give a mechanical interpretation of the estimation obtained. The function B is in fact defined in such a way that plastically admissible stress $\sum = \sum ^1 + \frac{{\partial B}}{{\partial E}}$ corresponds to a given final state E, which is a well known property of a Hencky material. It follows that the lower bound of the work supplied is given by a kind of deformation theory, i.e. essentially as if the behaviour were “elastic”. Therefore, in the applications it is sufficient to consider, as a first approximation, the stability of corresponding elastic structures with a convex strain potential appropriately defined on the basis of experimental loading curves.