In this paper we introduce Peierls–Nabarro type models for edge dislocations at semi-coherent interfaces between two heterogeneous crystals, and prove the optimality of uniformly distributed edge dislocations.
This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the core radius approach and the Ginzburg-Landau energy. Denoting by the length scale parameter in such models, we focus on the |log | energy regime. It is well known that, for configurations whose energy is bounded by c|log |, the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying I|log | energy, plus a measure supported on small zero-average sets.
We address the question of the convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals. Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time.
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