ALEX 2018 Workshop: Abstracts

The effect of forest dislocations on the evolution of a phase-field model for plastic slip

Patrick Dondl${}^{(1)}$, Matthias Kurzke${}^{\mathrm{(}\mathrm{2}\mathrm{)}}$, and Stephan Wojtowytsch${}^{\mathrm{(}\mathrm{3}\mathrm{)}}$

(1) Albert-Ludwigs-Universität Freiburg, Abteilung für Angewandte Mathematik (Germany)

(2) University of Nottingham, School of Mathematical Sciences (UK)

(3) Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh (USA)

We consider the gradient flow evolution of a phase-field model for crystal dislocations in a single slip system in the presence of forest dislocations. The model is based on a Peierls-Nabarro type energy penalizing non-integer slip and elastic stress. Forest dislocations are introduced as a perforation of the domain by small disks where slip is prohibited.

The $\mathrm{\Gamma }$-limit of this energy was deduced by Garroni and Müller [1, 2]. Our main result shows that the gradient flows of these $\mathrm{\Gamma }$-convergent energy functionals do not approach the gradient flow of the limiting energy. Indeed, the gradient flow dynamics remains a physically reasonable model in the case of non-monotone loading.

Our proofs rely on the construction of explicit sub- and super-solutions to a fractional Allen-Cahn equation on a flat torus or in the plane, with Dirichlet data on a union of small discs. The presence of these obstacles leads to an additional friction in the viscous evolution which appears as a stored energy in the $\mathrm{\Gamma }$-limit, but it does not act as a driving force. In terms of physics, our results explain how in this phase field model the presence of forest dislocations still allows for plastic as opposed to only elastic deformation.