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This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals E and the dissipation distance D. For sequences (E k ) k∈N and (D k ) k∈N we address the question under which conditions the limits q ∞ of solutions q k : [0, T ] → Q satisfy a suitable limit problem with limit functionals E ∞ and D ∞ , which are the corresponding -limits. We derive a sufficient condition, called conditional upper semi-continuity of the stable sets, which is essential to guarantee that q ∞ solves the limit problem. In particular, this condition holds if certain joint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate 123 388 A. Mielke et al. the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator converge if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model. Mathematics Subject Classification (2007)

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Damage of an elastic body undergoing large deformations by a "hard-device" loading possibly combined with an impact (modeled by a unilateral frictionless contact) of another, ideally rigid body is formulated as an activated, rate-independent process. The damage is assumed to absorb a specific and prescribed amount of energy. A solution is defined by energetic principles of stability and balance of stored and dissipated energies with the work of external loading, realized here through displacement on a part of the boundary. Rigorous analysis by time discretization is performed.

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SUMMARYThe so-called generalized standard solids (of Halphen-Nguyen type) involving also activated typically rate-independent processes such as plasticity, damage, or phase transformations are described as a system of a momentum equilibrium equation and a variational inequality for inelastic evolution of internalparameter variables. Various definitions of solutions are examined, especially from the viewpoint of the ability to combine rate-independent processes and other rate-dependent phenomena, as viscosity or also inertia. If those rate-dependent phenomena are suppressed, then the system becomes fully rate-independent. Illustrative examples are presented as well.

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