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- ItemQuasistatic small-strain plasticity in the limit of vanishing hardening and its numerical approximation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Bartels, Sören; Mielke, Alexander; Roubíček, ThomášThe quasistatic rate-independent evolution of the Prager--Ziegler-type model of linearized plasticity with hardening is shown to converge to the rate-independent evolution of the Prandtl--Reuss elastic/perfectly plastic model. Based on the concept of energetic solutions we study the convergence of the solutions in the limit for hardening coefficients converging to 0 by using the abstract method of Gamma-convergence for rate-independent systems. An unconditionally convergent numerical scheme is devised and 2D and 3D numerical experiments are presented. A two-sided energy inequality is a posteriori verified to document experimental convergence rates.
- ItemWeighted energy-dissipation functionals for gradient flows(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Mielke, Alexander; Stefanelli, UlisseWe investigate a global-in-time variational approach to abstract evolution by means of the weighted energy-dissipation functionals proposed by Mielke & Ortiz in ``A class of minimum principles for characterizing the trajectories of dissipative systems''. In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with time-discretization. Applications of the theory to various classes of parabolic PDE problems are presented. In particular, we focus on two examples of microstructure evolution from S. Conti and M. Ortiz ``Minimum principles for the trajectories of systems governed by rate problems'