### Abstract

We consider systems of $n$ parallel edge dislocations in a single slip system, represented by points in a two-dimensional domain; the elastic medium is modeled as a continuum. We formulate the energy of this system in terms of the empirical measure of the dislocations and prove several convergence results in the limit $n\to\infty$. The main aim of the paper is to study the convergence of the evolution of the empirical measure as $n\to\infty$. We consider rate-independent, quasi-static evolutions, in which the motion of the dislocations is restricted to the same slip plane. This leads to a formulation of the quasi-static evolution problem in terms of a modified Wasserstein distance, which is only finite when the transport plan is slip-plane-confined. Since the focus is on interaction between dislocations, we renormalize the elastic energy to remove the potentially large self- or core energy. We prove Gamma-convergence of this renormalized energy, and we construct joint recovery sequences for which both the energies and the modified distances converge. With this augmented Gamma-convergence we prove the convergence of the quasi-static evolutions as $n\to\infty$.

Original language | English |
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Pages (from-to) | 4149–4205 |

Number of pages | 57 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 49 |

Issue number | 5 |

Early online date | 24 Oct 2017 |

DOIs | |

Publication status | Published - 2017 |

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## Cite this

Mora, M. G., Peletier, M. A., & Scardia, L. (2017). Convergence of interaction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane confinement.

*SIAM Journal on Mathematical Analysis*,*49*(5), 4149–4205. https://doi.org/10.1137/16M1096098