从 Peierls-Nabarro 模型推导一维 Groma-Balogh 方程
Derivation of the 1-D Groma-Balogh equations from the Peierls-Nabarro model
AI总结：
本文证明一维演化 Peierls-Nabarro 模型在大量异号位错并经重标度后收敛于宏观完全非线性积分微分方程，从而形式推导出考虑碰撞的一维 Groma-Balogh 方程。
摘要：
我们考虑一维中与半 Laplace 算子相关的半线性积分微分方程，其解表示晶体中的原子位错。该方程包含经典 Peierls-Nabarro 模型的演化形式。我们证明，对于大量位错，经过适当重标度的解收敛于一个完全非线性积分微分方程的解，后者是具有位错密度的宏观晶体塑性模型。这给出了一维 Groma-Balogh 方程的形式推导；该方程是描述正、负取向平行直位错线密度演化的常用模型。本文完善了文献中的既有工作。主要创新在于我们允许位错具有不同取向，因此必须处理位错之间的碰撞。
英文摘要：
We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls-Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a fully nonlinear integro-differential equation which is a model for the macroscopic crystal plasticity with density of dislocations. This leads to the formal derivation of the 1-D Groma-Balogh equations \cite{groma}, a popular model describing the evolution of the density of positive and negative oriented parallel straight dislocation lines. This paper completes the work of \cite{patsan}. The main novelty here is that we allow dislocations to have different orientation and so we have to deal with collisions of them.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
