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Abstract
We consider systems of n parallel edge dislocations in a single slip system, represented by points in a twodimensional domain; the elastic medium is modelled 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→∞. The main aim of the paper is to study the convergence of the evolution of the empirical measure as n→∞. We consider rateindependent, quasistatic evolutions, in which the motion of the dislocations is restricted to the same slip plane. This leads to a formulation of the quasistatic evolution problem in terms of a modified Wasserstein distance, which is only finite when the transport plan is slipplaneconfined. Since the focus is on interaction between dislocations, we renormalize the elastic energy to remove the potentially large self or core energy. We prove Gammaconvergence of this renormalized energy, and we construct joint recovery sequences for which both the energies and the modified distances converge. With this augmented Gammaconvergence we prove the convergence of the quasistatic evolutions as n→∞.
Original language  English 

Pages (fromto)  4149–4205 
Number of pages  57 
Journal  SIAM Journal on Mathematical Analysis (SIMA) 
Volume  49 
Issue number  5 
Early online date  24 Oct 2017 
DOIs  
Publication status  Published  2017 
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Dive into the research topics of 'Convergence of interactiondriven evolutions of dislocations with Wasserstein dissipation and slipplane confinement'. Together they form a unique fingerprint.Projects
 1 Finished

Dislocation Patterns Beyond Optimality
Scardia, L. (PI)
Engineering and Physical Sciences Research Council
1/10/16 → 30/09/18
Project: Research council