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A continuum theory of edge dislocations

机译：边缘位错的连续理论

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Continuum theory of dislocation aims to describe the behavior of large ensembles of dislocations. This task is far from completion, and, most likely, does not have a "universal solution", which is applicable to any dislocation ensemble. In this regards it is important to have guiding lines set by benchmark cases, where the transition from a discrete set of dislocations to a continuum description is made rigorously. Two such cases have been considered recently: equilibrium of dislocation walls and screw dislocations in beams. In this paper one more case is studied, equilibrium of a large set of 2D edge dislocations placed randomly in a 2D bounded region. The major characteristic of interest is energy of dislocation ensemble, because it determines the structure of continuum equations. The homogenized energy functional is obtained for the periodic dislocation ensembles with a random contents of the periodic cell. Parameters of the periodic structure can change slowly on distances of order of the size of periodic cells. The energy functional is obtained by the variational-asymptotic method. Equilibrium positions are local minima of energy. It is confirmed the earlier assertion that energy density of the system is the sum of elastic energy of averaged elastic strains and microstructure energy, which is elastic energy of the neutralized dislocation system, i.e. the dislocation system placed in a constant dislocation density field making the averaged dislocation density zero. The computation of energy is reduced to solution of a variational cell problem. This problem is solved analytically. The solution is used to investigate stability of simple dislocation arrays, i.e. arrays with one dislocation in the periodic cell. The relations obtained yield two outcomes: First, there is a state parameter of the system, dislocation polarization; averaged stresses affect only dislocation polarization and cannot change other characteristics of the system. Second, the structure of dislocation phase space is strikingly simple. Dislocation phase space is split in a family of subspaces corresponding to constant values of dislocation polarizations; in each equipolarization subspace there are many local minima of energy; for zero external stresses the system is stuck in a local minimum of energy; for non-zero slowly changing external stress, dislocation polarization evolves, while the system moves over local energy minima of equipolarization subspaces. Such a simple picture of dislocation dynamics is due to the presence of two time scales, slow evolution of dislocation polarization and fast motion of the system over local minima of energy. The existence of two time scales is justified for a neutral system of edge dislocations.
机译：连续性位错理论旨在描述大位错集合体的行为。该任务远未完成，很可能没有适用于任何脱臼装置的“通用解决方案”。在这方面，重要的是要通过基准案例设置指导方针，其中严格地进行从离散位错集到连续体描述的过渡。最近已经考虑了两种这样的情况：位错壁的平衡和梁中的螺钉位错。在本文中，将研究另一种情况，即随机放置在2D有界区域中的大量2D边缘位错的平衡。感兴趣的主要特征是位错集合的能量，因为它确定了连续方程的结构。对于具有周期性细胞随机含量的周期性位错集合，获得了均质的能量函数。周期性结构的参数可以在周期性单元格大小的数量级距离上缓慢变化。能量函数是通过变分渐近方法获得的。平衡位置是能量的局部最小值。可以肯定的是，较早的论断是系统的能量密度是平均弹性应变和微结构能的和，即中和的位错系统的弹性能，即位错系统位于恒定位错密度场中，使之平均。位错密度为零。能量的计算被简化为变分单元问题的解决方案。这个问题可以通过解析来解决。该解决方案用于研究简单位错阵列（即在周期单元中具有一个位错的阵列）的稳定性。得到的关系产生两个结果：第一，系统的状态参数是位错极化。平均应力仅影响位错极化，而不能改变系统的其他特性。第二，位错相空间的结构非常简单。位错相空间被分成一系列子空间，这些子空间对应于位错极化的常数。在每个等极化子空间中，有许多局部的能量最小值。当外部应力为零时，系统陷入局部能量最低状态；对于非零且缓慢变化的外部应力，会产生位错极化，而系统会在等极化子空间的局部能量最小值上移动。如此简单的位错动力学图景是由于存在两个时间尺度，位错极化的缓慢演化以及系统在局部能量最小值上的快速运动。对于中性的边缘错位系统，存在两个时间尺度是合理的。

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