A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

Abstract

This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps.Therefore we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling, and approximate the model by its viscous regularization. In fact, the analysis of the latter PDE system presents remarkable difficulties, due to its highly nonlinear character. We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with careful regularity estimates tailored to this specific system, relying on a q-Laplacian type gradient regularization of the damage variable. Hence for the viscous problem we conclude the existence of weak solutions, satisfying a suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis. The latter leads to the notion of (weak) parameterized solution to our rate-independent system, which encompasses the influence of viscosity in the description of the jump regime.

Analysis of Staggered Evolutions for Nonlinear Energies in Phase Field Fracture

“…Roughly speaking, the equilibrium condition (i) says that at continuity times, i.e., when t ′ (s) > 0, the pair (u(s), z(s)) is an equilibrium configuration for F , while the energy-dissipation balance (ii) gives us a complete description of the behavior of a solution at discontinuity times. As it was already noticed in [24], the characterization (i)-(ii) is very similar to the one obtained in [25,26,27] with a vanishing viscosity approach. The main advantage of the iterative minimization (1.8)-(1.9) is that we do not have to add a fictitious viscosity term.…”

“…On the contrary, in [24] the authors provide a global description of the evolution by introducing an arc-length reparametrization of time, that is, a reparametrization based on the distance between the steps of the scheme (1.3)- (1.4). This reminds of the usual approach to viscous approximation (see, e.g., [25,26,27] in the context of phase field). The crucial point in [24] is the choice of the norms used to compute the arc-length of the algorithm: while in the viscous setting it is natural to employ the viscosity norm, in (1.3)-(1.4) it is not clear whether there are preferable norms.…”

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

“…Moreover, we characterize the jumps in time of the limit evolution by means of a suitable time-reparametrization; here we follow a technique first proposed in [9], then refined in [23,24], and used e.g. in [17,16] for damage, in [7,8] for plasticity, and in [15,19] for brittle fracture.…”

“…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.…”

Phase‐Field Fracture at Finite Strains Based on Modified Invariants: A Note on its Analysis and Simulations

“…in the works [HK11, BB08, RR15, HKRR17], also in combination with dynamics, heat transport, and phase separation. Vanishing-viscosity limits from viscous damage models at small strains to rate-independent ones are investigated in the series of works [KRZ13, KRZ15,KRZ17]. Therein, the H 1 -gradient regularization for the damage variable is replaced by stronger gradient terms (of Sobolev-Slobodeckij-type, or W 1,p with p > d) to ensure an embedding into the space of continuous functions.…”