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A generalization to nonlinear hardening of the first shakedown theorem or discrete elastic-plastic structural models. (English) Zbl 0667.73024
In the plastic contitutive laws the yield functions are assumed to be linear in the stresses, but generally nonlinear 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.

74R20 Anelastic fracture and damage
74C99 Plastic materials, materials of stress-rate and internal-variable type
74S30 Other numerical methods in solid mechanics (MSC2010)