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Seminario Matematico e Fisico di Milano
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Direttore: Paolo Stellari
Vice Direttore: Gabriele Grillo
Segretario: Daniele Cassani

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Seminario Matematico e Fisico di Milano
Patrick Dondl
Albert Ludwigs University of Freiburg
Relaxation, Existence, and Numerical Implementation for Models of Single-Crystal Elastoplasticity

Martedì 28 Marzo 2023, ore 17:00
Sala Consiglio 7 piano, Edificio La Nave
We will consider non-convex models for single-crystal elasto-plasticity, where the non-convexity arises through so-called cross-hardening, i.e., deformation in one slip system is prohibiting deformation in a different slip system at the same position. Such models are typically ill-posed and require relaxation. A different avenue to pursue is the addition of a strain-gradient term which regularizes the model by introducing a length scale on which the plastic strain may oscillate. We will see that physically reasonable strain gradient penalizations, which only assign an energy to geometrically necessary dislocations - penalizing only the curl of the plastic strain - still require some relaxation. Further issues arise regarding numerical implementation of the strict conditions of infinite cross-hardening. We will thus regularize the side-condition by introducing a large cross-hardening penalty into the plastic energy. The regularized model is then amenable to implementation with finite-element methods, and, with the aid of div-curl arguments, one can show that it Gamma-converges to the single-plane model for large penalization. Finally we show some microstructures arising in the numerical implementation of this model.