A quasistatic evolution model for perfectly plastic plates derived by Γ-convergence

Abstract

The subject of this paper is the rigorous derivation of a quasistatic evolution model for a linearly elastic-perfectly plastic thin plate. As the thickness of the plate tends to zero, we prove via Γ -convergence techniques that solutions to the three-dimensional quasistatic evolution problem of Prandtl-Reuss elastoplasticity converge to a quasistatic evolution of a suitable reduced model. In this limiting model the admissible displacements are of Kirchhoff-Love type and the stretching and bending components of the stress are coupled through a plastic flow rule. Some equivalent formulations of the limiting problem in rate form are derived, together with some two-dimensional characterizations for suitable choices of the data.

Stress regularity for a new quasistatic evolution model of perfectly plastic plates

“…For this reason we obtain a slightly better regularity of the stress with respect to the in-plane variables: the regularity estimate (1.4) is indeed global in the out-of-plane direction x 3 , whereas (1.5) is local with respect to both in-plane and out-of-plane variables. In particular, we observe that (1.4) cannot be deduced from the regularity estimates in [1] for the fully three-dimensional Prandtl-Reuss problem, using the convergence result of [5]; indeed, the estimates of [1] (whose dependence on the domain should be explicited if one wished to pass to the limit as the thickness of the plate tends to zero) are local in all directions.…”

“…In other words, the set of admissible stresses for the fully three-dimensional Prandtl-Reuss problem is a cylinder B α 0 + RI 3×3 , where B α 0 is a ball of radius α 0 in the space of trace-free M 3×3 sym matrices and I 3×3 is the identity matrix in M 3×3 sym . By the characterization in [5] this implies that the set K r is an ellipsoid of the form…”

“…In this paper we continue the study of the quasistatic evolution model for perfectly plastic plates, that has been derived in [5] starting from three-dimensional Prandtl-Reuss plasticity. Under suitable regularity assumptions for the applied body forces, we prove W 1,2 loc regularity of the stress with respect to the space variables.…”

“…where the product at the right-hand side is meant in the sense of the stress-strain duality introduced in [5] (see also Sect. 3).…”

“…In [5] this model has been rigorously justified via Γ -convergence techniques, starting from the three-dimensional Prandtl-Reuss quasistatic evolution model. In other words the system (sf1)-(sf5) describes (up to a suitable scaling) the asymptotic behaviour of the quasistatic evolutions in a three-dimensional plate, when the plate thickness approaches zero.…”