Stress-driven solution to rate-independent elasto-plasticity with damage at small strains and its computer implementation

Abstract

Quasistatic rate-independent damage combined with linearized plasticity with hardening at small strains is investigated. Fractional-step time discretization is devised with the purpose of obtaining a numerically efficient scheme, possibly converging to a physically relevant stress-driven solution, which however is to be verified a posteriori using a suitable integrated variant of the maximum-dissipation principle. Gradient theories both for damage and for plasticity are considered to make the scheme numerically stable with guaranteed convergence within the class of weak solutions. After finite-element approximation, this scheme is computationally implemented and illustrative 2-dimensional simulations are performed.

Convergence of alternate minimization schemes for phase-field fracture and damage

“…Here, M = 1, which means that at each time step only one minimization with respect to u and one minimization with respect to z is done. This algorithm corresponds to those investigated for instance in [27,25]. For n = 600, the numerical effort of this algorithm is comparable to the one described in Ib.…”

“…Proof. We first estimate γ n k from (25). By Lemma 3.1 there exists a constant C > 0 such that for every p ∈ [2, p) and for all n ≥ 1, 0 ≤ k ≤ n, i ≥ 0 it holds…”

“…Exploiting the convexity of I with respect to the variable z, from (76) we obtain in the same way as for instance in [17] that the solution (t, u, z) satisfies the following semistability inequality for t (s) > 0: This shows that the solution according to Theorem 4.8 is a local solution (see [31], where the notion of local solutions was introduced in the context of crack propagation) that satisfies an additional semi-stability property in the sense of [26] (so-called semi energetic solutions). In the papers [28,25], for rate independent problems local solutions were obtained as the limit of time-discrete solutions that are constructed by an alternate minimization procedure at each time step but only with one minimization step for each variable (fractional step methods or semi-implicit Rothe methods). The paper [15] treats the same damage model but in the context of visco-elasticity with inertia terms.…”

Damage model for plastic materials at finite strains

“…Take a sequence ⇀ in  where = ( , , ) and assume without loss of generality that sup ( , ) ≤ . We can use estimate (48) and choose a (not relabeled) subsequence such that ( , ) converges to lim inf →∞ ( , ) and ∇ ⇀ ∇ weakly in (Ω),…”

Fatigue and phase transition in an oscillating elastoplastic beam

“…We remark that our viewpoint is different from the contributions already present in the literature: on one hand we focus on the dynamics (compared, for instance, with [18,26,30] that go into the direction of rateindependent damage processes in nonlinear elasticity and/or thermo-elasto-plasticity, see also [8,28,29]) and also because the approach used in other papers, that interpret damage processes as a kind of phase transition in the material (see for instance [1][2][3]27]), is based on the idea that damage processes are driven by large deformations. Moreover, the multiyield character emerging from our bending problem does not seem to have been taken into consideration before, neither in the multidimensional setting: see for instance [5,7,21,24].…”