考虑Wasserstein耗散和滑移面约束的位错相互作用驱动演化的收敛性
Convergence of interaction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane confinement
摘要：
我们认为系统的$n$平行刃位错在一个单一的滑移系统，在一个二维域的点表示的弹性介质被建模为一个连续体。我们用位错的经验测度来表示该系统的能量，并证明了在极限$n\to\infty$内的几个收敛结果。本文的主要目的是研究经验测度$n\to\infty$演化的收敛性。我们考虑速率无关的，准静态的演变，其中的位错的运动被限制在同一个滑移面。这导致了准静态演化问题的一个修改后的Wasserstein距离，这是唯一有限的运输计划时，滑面限制的制定。由于重点是位错之间的相互作用，我们重新归一化的弹性能量，以消除潜在的大的自我或核心的能量。我们证明伽玛收敛的重整化能量，我们构造联合恢复序列的能量和修改后的距离收敛。利用这种增广的Gamma收敛性，我们证明了准静态演化的收敛性为$n\to\infty$ .
英文摘要：
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 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 $n\to\infty$. 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 $n\to\infty$.
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