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MiS Preprint
10/2005
Lower semi-continuity and existence of minimizers in incremental finite-strain elastoplasticity
Alexander Mielke and Stefan Müller
Abstract
We study incremental problems in geometrically nonlinear elastoplasticity. Using the multiplicative decomposition $D\varphi= F_{el} F_{pl}$ we consider general energy functionals of the form \[ \mathcal{I}(\varphi,F_{pl})= \int_\Omega U(x,D\varphi \;\! F_{pl}^{-1}, F_{pl},\mathcal{G}(F_{pl})) d x -\langle \ell,\varphi\rangle, \] which occur as the sum of the stored energy and the dissipation in one time step. Here $\mathcal{G}(F_{pl})$ is the dislocation tensor which takes the form $\frac1{\det F_{pl}}curl_3(F_{pl})\,F_{pl}^T$ in dimension $d=3$.
Imposing the usual constraint $\det F_{pl} \equiv 1$ and suitable growth and polyconvexity conditions on $U$ we show that the minimum of $\mathcal{I}$ is attained in the natural Sobolev spaces. Moreover, we are able to treat multiple time steps by controlling the stored and dissipated energies. We also address the relation of the incremental problem to the time-continuous energetic formulation of elastoplasticity.
