论文标题
离散的脱位动力学，an灭是在一个维度中的PEIERLS-NABARRO模型的极限
Discrete Dislocations Dynamics with annihilation as the limit of the Peierls-Nabarro model in one dimension
论文作者
论文摘要
金属的可塑性是许多晶体缺陷（位错）的新兴现象，它们在显微镜时间和长度尺度上相互作用并移动。描述此类脱位动力学的两个常用模型是PEIERLS-NABARRO模型和所谓的离散脱位动力学模型。 但是，这两个模型之间的一致性仅在几个位错数量或两个脱位碰撞的第一次。在本文中，我们解决了这些限制，并确定任何位错数量的一致性，并且在其初始位置或方向上没有任何限制。 更详细的是，我们考虑的Evoltution peierls-nabarro模型描述了相位函数$ v_ \ e（t，x）$的演变，该模型代表晶体中原子变形。该模型是艾伦·卡纳（Allen-Cahn）类型的反应扩散方程，与半拉普拉斯（Laplacian）。小参数$ \ ep $是原子距离与$ v_ \ e $之间的相位过渡之间的典型距离之间的比率。相变的位置决定了位错的位置，并且过渡（向上或向下）的符号决定了方向。 本文的目的是推导功能的渐近行为$ v_ \ e $作为$ \ ep \ to to $ to $ to $ to $ to to to nutary entery time $ t $;特别是碰撞之外。我们证明，$ v_ \ e $收敛到分段常数函数$ v $，其空间变量中的跳跃点满足了代表带有歼灭的离散脱位动力学的ODE系统。我们的证明方法是将$ v_ \ e $的几个子和超填充物明确构建和修补，并表明它们会收敛到相同的限制$ v $。
Plasticity of metals is the emergent phenomenon of many crystal defects (dislocations) which interact and move on microscopic time and length scales. Two of the commonly used models to describe such dislocation dynamics are the Peierls-Nabarro model and the so-called discrete dislocation dynamics model. However, the consistency between these two models is known only for a few number of dislocations or up to the first time at which two dislocations collide. In this paper we resolve these restrictions, and establish the consistency for any number of dislocations and without any restriction on their initial position or orientation. In more detail, the evolutive Peierls-Nabarro model which we consider describes the evolution of a phase-field function $v_\e(t,x)$ which represents the atom deformation in a crystal. The model is a reaction-diffusion equation of Allen-Cahn type with the half Laplacian. The small parameter $\ep$ is the ratio between the atomic distance and the typical distance between phase transitions in $v_\e$. The position of a phase transition determines the position of a dislocation, and the sign of the transition (up or down) determines the orientation. The goal of this paper is to derive the asymptotic behavior of the function $v_\e$ as $\ep\to0$ up to arbitrary end time $T$; in particular beyond collisions. We prove that $v_\e$ converges to a piecewise constant function $v$, whose jump points in the spatial variable satisfy the ODE system which represents discrete dislocation dynamics with annihilation. Our proof method is to explicitly construct and patch together several sub- and supersolutions of $v_\e$, and to show that they converge to the same limit $v$.
