Time-Continuous Evolution of Microstructures in Finite Plasticity
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Plastic deformation of crystalline solids very often gives rise to the initation ofmaterialmicrostructures experimentally visible as dislocation patterns. These microstructures are not inherent to the material but occur as a result of deformation. Modeling a physically deformed crystal in finite plasticity by means of the displacement field and in terms of a set of internal variables which capture the microstructural characteristics, we employ energy principles to analyze the microstructure formation and evolution as a result of energy minimization. In particular, for non-quasiconvex energy potentials the minimizers are no longer continuous deformation fields but small-scale fluctuations related to probability distributions of deformation gradients to be calculated via energy relaxation. We briefly review the variational concept of the underlying energy principles for inelastic materials. As a first approximation of the relaxed energy density, we assume first-order laminate microstructures, thus approximating the relaxed energy by the rank-one convex envelope. Based on this approach, we present explicit time-evolution equations for the volume fractions and the internal variables, then outline a numerical scheme by means of which the microstructure evolution can be computed and we show numerical results for particular examples in single and double-slip plasticity. In contrast to many approaches before we do not globally minimize a condensed energy functional to determine themicrostructure but instead incrementally solve the evolution equations at each time step, in particular accounting for the dissipation required to rearrange the microstructure during a finite time increment with already existing mictrostructure at the beginning of the time step.
KeywordsSlip System Internal Variable Deformation Gradient Young Measure Active Slip System
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