Existence and Regularity of the Plastic Multiplier in Static and Quasistatic Plasticity

We consider rate-independent models of static and quasistatic plasticity with hardening. These models are usually stated in terms of variational inequalities of the first or second kind, for the primal and dual formulations, respectively. In the static case, the dual variational inequality can be interpreted as the necessary and sufficient optimality conditions for an underlying optimization problem which aims to minimize the energy associated with the generalized stresses Σ. In this context, the plastic multiplier can be viewed as the Lagrange multiplier associated with the admissibility constraint for the generalized stresses, Φ(Σ) ≤ 0, where Φ denotes the yield function. This interpretation of the plastic multiplier is valid in case of smooth yield functions. The existence of the plastic multiplier in a pointwise sense, even in the case of nonsmooth convex yield functions, was shown, for instance, in [1, Lemma 4.7]. Indeed, relations analogous to ours but in a pointwise setting can already be found in [1, Section 4.2]. The purpose of this paper is to extend these results into function space. Using standard assumptions, we show that the plastic multiplier λ belongs to L(0, T ;L(Ω)) in the quasistatic case. The proof uses arguments from convex analysis, which can be found, for instance, in [3, 5]. Indeed, for practical models and yield functions, λ ∈ L(0, T ;L(Ω)) holds. For the important special cases of linear kinematic hardening (including perfect plasticity), or combined linear kinematic and isotropic hardening (including pure isotropic hardening), a proof is included in Section 3. In the case of static, or incremental, plasticity, λ ∈ L(Ω) respectively λ ∈ L(Ω) holds. The latter result was also shown in [2] using optimization methods.