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...ly Lipschitz with respect to s, such that the rescaled functions p◦ ε(s, x) := pε(tε(s), x) are uniformly Lipschitz with respect to s, in a suitable function space. This idea has already been used in =-=[8, 17, 18]-=- for rate independent dissipative problems in finiteQUASISTATIC EVOLUTION FOR CAM-CLAY PLASTICITY 3 dimension. The authors of the last two papers have used the same idea to study a similar problem in...

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...deed the gradient system ∂Ψε(u ′(t)) + Ft(u(t)) ∋ 0 in V ∗ for a.a. t ∈ (0, T ) with Ψε(v) = |v|+ ε 2 ∥v∥2. (1.20) We mention that the vanishing viscosity limit of (1.20) as ε ↘ 0 has been studied in =-=[34]-=- in the case of a finite-dimensional ambient space V . Moving from the existence results for viscous doubly nonlinear equations of the present paper, we are going to address the vanishing viscosity an...

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...but nontrivial and well motivated case of inelasticity on the surface, namely delamination of viscoelastic bodies at small strains. In contrast with bulk inelastic processes like damage or fatigue, this surface inelasticity benefits analytically from compactness of the trace operator, which simplifies (anyhow still complicated) some technicalities and is more lucid for demonstrating the desired effects. It should be also emphasized that the inelastic process itself (i.e. here the delamination) is considered fully rate-independent, in contrast to a usual vanishing-viscosity approach as e.g. in [5, 8, 10, 14, 17, 22, 23, 40]. Linear visco-elastic materials for vanishing viscosity naturally approximate in some sense the elastic materials. We consider the Kelvin-Voigt rheology, which is the simplest rheology which makes the desired effect of natural prevention of too early delamination, cf. Sect. 4. In principle, a combination of Maxwell and Kelvin-Voigt’s rheologies (called Jeffrey’s rheology) would serve equally but, e.g., mere Maxwell rheology is not “enough dissipative” to yield the asymptotical effects investigated below. We further consider a general so-called 2nd-grade nonsimple material, cf. e.g. [27, 41], ...

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...scription of the process is meaningful at least up to the first jump time. In order to overcome such a drawback, one may adopt the so-called vanishing viscosity technique (we refer the reader to e.g. =-=[26, 27]-=- for an abstract treatment and to [9], [10], [21], [22] for some applications): rate-independent evolutions are seen as limits of solutions to some rate-dependent systems containing a viscous dissipat...

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...tional arguments supporting the energetic-solution concept, there are also some other concepts of solutions that are sometimes applicable and successfully competing with energetic solutions, cf. also =-=[16,47,48,53]-=- for a comparison with other notions in general, and, e.g., [10, 31–33, 36, 37, 54, 55] in the context of crack propagation, or, e.g., [4, 13, 14] in the context of plasticity. Moreover, we refer to e...

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...eously in a very drastic way, jumping into very far-apart energetic configurations (see, for instance, [Mie03, Ex. 6.1], [KMZ08, Ex. 6.3], and [MRS09, Ex. 1]), while a local force balance criterion would predict a slow evolution. In this paper, we discuss the vanishing viscosity approach as an alternative for the derivation of a local rate-independent damage model. The philosophy that rate-independence should be considered as limit of systems with smaller and smaller viscosity has by now been widely adopted in the applications, see, e.g., [TZ09, DDMM08, Cag08]. In the mainstream of the papers [EM06, MRS09, MRS10, MZ10] on general rate-independent systems, and [KMZ08, KZM10, LT11] for rate-independent models of crack propagation, we exploit this vanishing viscosity approach to obtain a more precise description of the system behavior at jumps. Hence, we approximate (1.6) with the doubly nonlinear equation ∂R(z′(t)) + DzI(t, z(t)) 3 0 in Z∗ for a.a. t ∈ (0, T ), (1.8) where the dissipation functional R features an additional L2-viscosity term, viz. R(η) = R1(η) +R2,(η) with R2,(η) = 2 ‖η‖ 2 L2(Ω) . = R2(η) for η ∈ Z. (1.9) Let us mention that damage models with viscous dissipation (possibly with viscosi...

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...E(x(t), t). (1) Here E is the energy that drives the system, and the convex function ψ is a dissipation potential, with subdifferental ∂ψ. For gradient flows, ψ typically is quadratic, and ∂ψ single-valued and linear; for rate-independent systems, ψ is 1-homogeneous, and ∂ψ is a degenerate monotone graph. Rate-independent systems have some unusual properties. Solutions are expected to be discontinuous, and therefore the concept of smooth solutions is meaningless. Currently two rigorous definitions of weak solutions are used, which we refer to as ‘energetic solutions’ [MM05] and ‘BV solutions’ [MRS12a]. Heuristically, the first corresponds to the principle ‘jump whenever it lowers the energy’, while the second can be characterized as ‘don’t jump until you become unstable’. For time-dependent convex energies the two definitions coincide, but in the non-convex case they need not be. Various rigorous justifications of rate-independent evolutions have been constructed, which underpin the rate-independent nature by obtaining it through upscaling from a ‘microscopic’ underlying system (e.g. [ACJ96, Cag08, DMDM06, DMDMM08, Fia09, Mie12, PT05, Sul09]). The general approach in these results is to ch...