GENERALIZED FORCES AND DISSIPATION FUNCTIONS IN THE CONTEXT OF AN INTERNAL VARIABLE APPROACH APPLIED TO THE SOLUTION OF ELASTIC-PLASTIC PROBLEMS

Abstract

A non-traditional approach to the numerical analysis of elastic-plastic systems is discussed by focusing on a formulation that makes use of internal variables and dissipation functions. These functions are used in order to enforce the constitutive law, so that they play the role of the yield functions in the framework of the classical theory of plasticity.

With reference to finite element discrete models, it is shown that the solution of an elastic-plastic problem corresponds to the minimum point of a convex function (when the material is stable in Drucker’s sense) and that convergence is guaranteed when a convenient time integration method (usually known as backward-difference scheme) is applied. As a matter of fact, it can be proved that the value of that function progressively decreases (iteration by iteration) when a proper time integration strategy is implemented.

Elastic-plastic systems will be considered, which are subjected to uniaxial and multiaxial stress states (by assuming Mises’ yield condition for two-dimensional and three-dimensional finite elements). In all cases, it will be easily noticed that the dissipation functions depend on convenient generalized forces, whose features are obvious in the presence of uniaxial stress states. Instead, when the structural system is subjected to multiaxial stress states, the actual meaning of the generalized forces must be properly understood in order to define convenient dissipation functions and/or yield functions: this is the main issue of the present paper and represents a topic which, to the authors’ knowledge, has not been adequately investigated, yet.