Generation of balanced viscosity solutions to rate-independent systems via variational convergence

Abstract

In this paper we investigate the origin of the Balanced Viscosity solution concept for rate-independent evolution in the setting of a finite-dimensional space. Namely, given a family of dissipation potentials (Ψn)n with superlinear growth at infinity and a smooth energy functional E, we enucleate sufficient conditions on them ensuring that the associated gradient systems (Ψn, E) Evolutionary cf. [Mie16], to a limiting rate-independent system, understood in the sense of Balanced Viscosity solutions. In particular, our analysis encompasses both the vanishing-viscosity approximation of rate-independent systems from [MRS12a, MRS16], and their stochastic derivation developed in [BP16].where ∂Ψ 0 : R d ⇒ R d is the subdifferential of Ψ 0 in the sense of convex analysis, whereas DE is the differential of the map u → E(t, u). Due to the 0-homogeneity of ∂Ψ 0 , (1.1) is invariant for time rescalings, i.e. it is rateindependent. Now, it is well known that, even in the case of a smooth energy E, if u → E(t, u) fails to be strictly convex, then absolutely continuous solutions to (1.1) need not exist. In the last two decades, this has motivated the development of various weak solvability concepts for (1.1) and, in general, for rate-independent systems in infinite-dimensional Banach spaces, or even topological spaces. The analysis of these solution notions has posed several challenges. Date: October 09, 2017. G.A.B. kindly acknowledges support from the Nederlandse Oxrganisatie voor Wetenschappelijk Onderzoek (NWO), VICI Grant 639.033.008. R.R. has been partially supported by GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). 1 2 GIOVANNI A. BONASCHI AND RICCARDA ROSSI Energetic and Balanced Viscosity solutions to rate-independent systems. While referring to [Mie11] and [MR15] for a survey of all weak notions of rate-independent evolution, we may recall here the concept of Energetic solution, first proposed in [MT99] (cf. also [DMT02] for the concept of quasistatic evolution in fracture), and fully analyzed in [MT04]. It consists of the global stability conditionand of the (Ψ 0 , E)-energy balanceinvolving the dissipated energy Var Ψ0 (u; [0, t]) (where Var Ψ0 denotes the notion of total variation induced by Ψ 0 ), the stored energy E(t, u(t)) at the process time t, the initial energy E(0, u(0)), and the work of the external forces. Since the energetic formulation (S)-(E Ψ0,E ) only features the (assumedly smooth) power of the external forces ∂ t E, and no other derivatives, it is particularly suited to solutions with discontinuities in time. It is also considerably flexible and can be indeed given for rate-independent processes in general topological spaces, cf.[MM05]. That is why, it has been exploited in a great variety of applicative contexts, cf. [Mie05, MR15]. Nonetheless, over the years it has become apparent that, in the very case of a nonconvex dependence u → E(t, u), the global stability (S) fails provide a truthful description o...

Rate-independent stochastic evolution equations: parametrized solutions

“…In the deterministic case, this approach is indeed classical and has been applied to a variety of different rate-independent settings. Without claiming completeness, we refer the reader to [16] as well as to [7,17,37,38,40] for a collection of abstract results. Applications to plasticity [5,12,13], fracture [1,10,21,28,42], damage [11,22,29], adhesive contact [8,47], delamination [18,46,50], optimal control [58], and topology optimization [2,3] are also available.…”