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- Item[Gamma]-limits and relaxations for rate-independent evolutionary problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mielke, Alexander; Toubíček, Tomáš; Stefanelli, UlisseThis work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals ε and the dissipation distance D. For sequences (ε k)k ∈ ℕ and (D k)k ∈ ℕ we address the question under which conditions the limits q∞ of solutions qk: [0,T] → Q satisfy a suitable limit problem with limit functionals ε∞ and D∞, which are the corresponding Γ-limits. We derive a sufficient condition, called emphconditional 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 emphjoint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate 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 convergece 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.
- ItemA class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mielke, Alexander; Ortiz, MichaelThis work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently formally passing to the limit of continuous time. The resulting functionals may be regarded as elliptic regularizations of the original evolutionary problem. We find that the $Gamma$-limits of interest are highly degenerate and provide limited information regarding the limiting trajectories of the system. Instead we seek to characterize the minimizing trajectories directly. The special class of problems characterized by a rate-independent dissipation functional is amenable to a particularly illuminating analysis. For these systems it is possible to derive a priori bounds that are independent of the regularizing parameter, whence it is possible to extract convergent subsequences and find the limiting trajectories. Under general assumptions on the functionals, we show that all such limits satisfy the energetic formulation (S) & (E) for rate-independent systems. Moreover, we show that the accumulation points of the regularized solutions solve the associated limiting energetic formulation.
- ItemAnalytical and numerical methods for finite-strain elastoplasticity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Gürses, Ercan; Mainik, Andreas; Miehe, Christian; Mielke, AlexanderAn important class of finite-strain elastoplasticity is based on the multiplicative decomposition of the strain tensor $F=F_el F_pl$ and hence leads to complex geometric nonlinearities. This survey describes recent advances on the analytical treatment of time-incremental minimization problems with or without regularizing terms involving strain gradients. For a regularization controlling all of $nabla F_pl$ we provide an existence theory for the time-continuous rate-independent evolution problem, which is based on a recently developed energetic formulation for rate-independent systems in abstract topological spaces. In systems without gradient regularization one encounters the formation of microstructures, which can be described by sequential laminates or more general gradient Young measures. We provide a mathematical framework for the evolution of such microstructure and discuss algorithms for solving the associated space-time discretizations. We outline in a finite-step-sized incremental setting of standard dissipative materials details of relaxation-induced microstructure development for strain softening von Mises plasticity and single-slip crystal plasticity. The numerical implementations are based on simplified assumptions concerning the complexity of the microstructures.
- ItemA mathematical framework for standard generalized materials in the rate-independent case(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mielke, AlexanderStandard generalized materials are described by an elastic energy density and a dissipation potential. The latter gives rise to the evolution equation (flow law) for the internal variables. The energetic formulation provides a very weak, derivative-free form of this flow law. It is based on a global stability condition and an energy balance. Using time-incremental minimization problems, which allow for the usage of the rich theory in the direct method of the calculus of variations, it is possible to establish general, abstract existence results as well as convergence for numerical approximations. Applications to shape-memory materials and to magnetostrictive or piezoelectric materials are surveyed.
- ItemCrack growth in polyconvex materials(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Knees, Dorothee; Zanini, Chiara; Mielke, AlexanderWe discuss a model for crack propagation in an elastic body, where the crack path is described a-priori. In particular, we develop in the framework of finite-strain elasticity a rate-independent model for crack evolution which is based on the Griffith fracture criterion. Due to the nonuniqueness of minimizing deformations, the energy-release rate is no longer continuous with respect to time and the position of the crack tip. Thus, the model is formulated in terms of the Clarke differential of the energy, generalizing the classical crack evolution models for elasticity with strictly convex energies. We prove the existence of solutions for our model and also the existence of special solutions, where only certain extremal points of the Clarke differential are allowed.