Quasistatic evolution in the theory of perfect elasto-plastic plates. Part II: Regularity of bending moments

Abstract

We study differentiability of solutions of quasistatic problems for perfect elasto-plastic plates. We prove that in the isotropic case bending moments has locally square-integrable first derivatives:The result is based on discretization of time and uniform estimates of solutions of the incremental problems, which generalize the estimates in the static case of perfect elasto-plastic plates.Remark 2.1. We refer to [21, Chapter III] for the main properties of BH(Ω) and for the definition of weak * convergence in BH(Ω). Remark, that for u ∈ BH(Ω) we have the following embedding:Let us introduce the notationsym : m(x) ∈ K for a.e. x ∈ Ω .

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

“…Thus, the problem has in general a genuinely three-dimensional nature and differs from the classical two-dimensional plastic plate model that has been extensively studied in the literature [2,6,9,10]. This comparison is discussed in the last section of the paper, where an explicit solution to (sf1)-(sf5) is shown for a specific choice of data.…”

“…Local regularity of stresses in the fully three-dimensional Prandtl-Reuss plasticity has been proved in [1,8], see also [11] for a recent global regularity result for the stress velocity. For the classical two-dimensional plastic plate model local regularity has been established in [10], using different techniques and assuming K r to be a ball, C r to be the identity tensor, and γ d = ∂ω.…”

Regularity of stresses in Prandtl-Reuss perfect plasticity

“…The method proposed in this paper will be used for proving the differentiability of stresses for other models occurring in plasticity (see [4,5]). …”

Existence and regularity of solutions for an evolution model of perfectly plastic plates

“…Indeed, this paper may be seen as the dynamic counterpart of [9], where the authors analyse the stress regularity for the reduced model derived in [8], describing the quasistatic evolution of a plastic plate. The regularity of the elastic stress is a classical issue in perfect plasticity, investigated for instance also in [4], [11], and [13] (see the related results in [2]). For general treatment of quasistatic evolutions of rate-independent systems we refer to [17,19].…”