Globally stable quasistatic evolution for a coupled elastoplastic–damage model

Abstract

Abstract. We show the existence of globally stable quasistatic evolutions for a rateindependent material model with elastoplasticity and incomplete damage, in small strain assumptions. The main feature of our model is that the scalar internal variable which describes the damage affects both the elastic tensor and the plastic yield surface. It is also possible to require that the history of plastic strain up to the current state influences the future evolution of damage.

Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model

“…The existence of quasistatic evolutions for a coupled elastoplastic-damage model was proved in [4] using the ansatz of global stability; more precisely, at any time instant the configuration is assumed to be a global minimizer of the sum of the stored elastic energy and of the plastic dissipation. This approach, based on the so-called energetic formulation of rate-independent processes (see [22] and references therein), was followed e.g.…”

“…The formulation of the model is essentially the same as in [4] and follows [12] for modeling damage and [5] for plasticity.…”

“…Another possible choice for the regularizing term would be ∇α γ γ with γ > n, used e.g. in [4]; however, in the setting of vanishing viscosity this choice does not allow to get an energy equality, as shown in [16]. The total mechanical energy is then E(α, e) := Q(α, e) + D(α) + for every α 1 ≤ α 2 , where 0 < r < R are constant.…”

“…We use the fact, proved in [4], that V H (α, p; s, t) = Finally we introduce the "generalized energy" E λ (α, e; p, t) := E(α, e) + λ V H (α, p; 0, t) , with λ ∈ [0, 1]. Its meaning will be discussed in the final part of this introduction.…”

“…More precisely, for every k ∈ N we set τ k := T /k and define inductively (α k , so the problem is well posed.) We observe that, if one sets ε = 0, then (0.1) reduces to the incremental global minimum problem used in [4], so one obtains the energetic formulation in the limit as k → ∞. Instead, passing to the limit first as k → ∞ and then for ε → 0 allows one to select stationary points that are not necessarily global minimizers.…”