BV solutions and viscosity approximations of rate-independent systems

Abstract

Abstract. In the nonconvex case, solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate-independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential that is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of "BV solutions" involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play crucial roles in the description of the associated jump trajectories. We shall prove general convergence results for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.Mathematics Subject Classification. 49Q20, 58E99.

Quadratic and rate-independent limits for a large-deviations functional

“…In Sect. 4, we show how generalized gradient flows for finite α and β, with strictly convex ψ, converge to a specific rigorous rate-independent solution concept, the so-called BV solutions [31,32]. We define this concept in Definition 3.…”

“…The so-called energetic solutions have been introduced and analysed in [35][36][37] and are based on the combination of a pointwise global minimality property and an energy balance. Here we concentrate on BV solutions, as defined in [32,34]. Our limiting system will be of this type.…”

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

“…To avoid the drawbacks of global minimality, in this paper we adopt a vanishing viscosity approach, i.e., we obtain a quasistatic evolution as a limit of solutions to some rate-dependent systems containing a viscous dissipation that tends to zero. 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.…”

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

“…Starting from the seminal paper [EM06], this technique has by now been thoroughly developed both for abstract rate-independent systems [MRS09, MRS12,MZ13], and in the applications to fracture [TZ09,Cag08,KMZ08,KZM10,LT11], and to plasticity [DDMM08,BFM12,DDS11,DDS12,FS13].…”