螺位错动力学的定性和定量性质
Qualitative and quantitative properties of the dynamics of screw dislocations
摘要：
本文收集了关于弹性介质中螺位错行为的一些结果。通过使用半离散模型，我们能够研究动力学的两个具体方面，即（i）与自由边界和碰撞事件的相互作用和（ii）当施加适当的Dirichlet型边界条件时，域内的限制。在第一种情况下，我们分析证明，自由边界吸引位错，我们提供了一个表达式的桃-克勒力的边界附近的位错。此外，我们用它来证明一个上界的碰撞时间的位错与边界，提供一定的几何条件得到满足。得到了两个Burgers矢量相反的位错碰撞时间的上界。在第二种情况下，我们转向其边界受到外部应力的域。在这种情况下，我们证明，位错找到它积极有利的留在材料内，而不是越来越接近边界。首先证明了一个单一的位错在材料中的结果被扩展到一个系统的许多位错，其中的分析需要仔细处理的相互作用项。
英文摘要：
This note collects some results on the behaviour of screw dislocation in an elastic medium. By using a semi-discrete model, we are able to investigate two specific aspects of the dynamics, namely (i) the interaction with free boundaries and collision events and (ii) the confinement inside the domain when a suitable Dirichlet-type boundary condition is imposed. In the first case, we analytically prove that free boundaries attract dislocations and we provide an expression for the Peach--Koehler force on a dislocation near the boundary. Moreover, we use this to prove an upper bound on the collision time of a dislocation with the boundary, provided certain geometric conditions are satisfied. An upper bound on the collision time for two dislocations with opposite Burgers vectors hitting each other is also obtained. In the second case, we turn to domains whose boundaries are subject to an external stress. In this situation, we prove that dislocations find it energetically favourable to stay confined inside the material instead of getting closer to the boundary. The result first proved for a single dislocation in the material is extended to a system of many dislocations, for which the analysis requires the careful treatments of the interaction terms.
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