A generalization to nonlinear hardening of the first shakedown theorem for discrete elastic-plastic structural models

Maier, Giulio

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 81 (1987) no. 2, pp. 161-174

Résumé

In the plastic constitutive laws the yield functions are assumed to be linear in the stresses, but generally non-linear in the internal variables which are non-decreasing measures of the contribution to plastic strains by each face of the yield surface. The structural models referred to for simplicity are aggregates of constant-strain finite elements. Influence of geometry changes on equilibrium are allowed for in a linearized way (the equilibrium equation contains a bilinear term in the displacements and pre-existing stresses). It is shown that shakedown (which means plastic work bounded in time) is guaranteed under variable-repeated quasi-static external actions, when the hardening behaviour exhibits reciprocal interaction, a suitably defined energy function of the internal variables is convex and the yield conditions can be satisfied at any time by some constant internal variable vector and by the linear elastic stress response. Some interpretations and extensions of this result are envisaged. By specialization to linear hardening, earlier results are recovered, which reduce to Melan's classical theorem for non-hardening (perfectly plastic) cases.