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Knees, Dorothee; Rossi, Riccarda; Zanini, Chiara

A vanishing viscosity approach to a rate-independent damage model. (English) Zbl 1262.74030

Math. Models Methods Appl. Sci. 23, No. 4, 565-616 (2013).
Summary: We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps.{ }Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arclength reparametrization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.

Cited in 1 Review
Cited in 68 Documents

MSC:

74R05 Brittle damage
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
35D40 Viscosity solutions to PDEs
35K86 Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators
49J40 Variational inequalities

Keywords:

rate-independent damage evolution; vanishing viscosity method; arclength reparametrization; time discretization
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\textit{D. Knees} et al., Math. Models Methods Appl. Sci. 23, No. 4, 565--616 (2013; Zbl 1262.74030)
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