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Author: Mielke, Alexander × End date: 2009 × Start date: 2006 × Type: Text × Subject: microstructure ×

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    A 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, Michael
    This 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.
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    Analytical and numerical methods for finite-strain elastoplasticity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Gürses, Ercan; Mainik, Andreas; Miehe, Christian; Mielke, Alexander
    An 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.