A Self-Organization Model of Collective Dislocations Based on Geometry of Crystal Defects and Thermodynamics of Complexity

建立了疲劳晶体中持久滑移带（PSBs）的数学模型，并利用该模型对位错的空间分布进行了数值模拟，提出了具有颗粒图像的移动和固定位错密度张量，使这些密度与几何晶体缺陷精确对应，并同时具有边缘位错和螺旋位错分量。从热力学的复杂性出发，推导了集合位错自组织的同时反应扩散方程。在方程组中引入了新的反应项，分别表示偶极子形成引起的可动边组分的固定化和十字滑移引起的不可动螺组分的固定化。在方程中加入了交叉滑移后螺位错对湮灭和基于热活化的偶极子对湮灭的产生项。将得到的反应差分方程应用于二维问题，然后将尺寸效应参数定义为位错结构（如脉状或阶梯状）的自发本征长度尺度。给出了结构的产生条件，并通过线性稳定性分析预测了结构的理论波长。用有限差分法进行了计算分析，得到以下结果：脉状和阶梯状结构的波长可以固定在1 μm的尺寸上，由于尺寸效应参数的存在，相似定律不成立m这一理论。该模型能定性地表达实验观察到的位错模式，并能通过图灵分岔模拟各向同性脉状结构产生各向异性阶梯结构的形成过程。螺旋位错被交叉滑移单调地湮灭，其可移动分量在psb的通道中略有保留。少

A mathematical model of persistent slip bands (PSBs) generated in fatigued f.c.c. crystals is rigorously developed, and a numerical simulation for spatial patterning of dislocations is carried out by use of this model Mobile and immobile dislocation density tensors with particulate image are proposed so that these densities have a precise correspondence to geometrical crystal defects and have both components of edge and screw dislocations.Simultaneous reaction-diffusion equations for self-organization of collect dislocations are derived on the basis of thermodynamics for complexity. New reaction terms are introduced into the equation system, which represent an immobilization of mobile edge component caused by dipole formation and a mobilization of immobile screw component by cross slip. Production terms concerning pair annihilation of screw dislocations after cross slip and pair annihilation of a dipole based on thermal activations are added to the equations.The obtained reaction -diff … More usion equations are applied to a two-dimensional problem and then a size effect parameter is defined as a spontaneous intrinsic length scale of dislocation structure such as vein or ladder one. Generation conditions of the structures are given and theoretical wavelengths of the structures are predicted through the linear stability analysis. A computational analysis using the finite difference method is performed and it leads to the following results. The wavelengths of vein and ladder structures can be fixed in 1 μm size, since the similarity law does not hold m this theory because of existence of the size effect parameter. The present model of PSBs can qualitatively express the dislocation patterns observed experimentally and can simulate the formation process of anisotropic ladder structure generated from the isotropic vein-like structure by the Turing bifurcation. The screw dislocations monotonously annihilate by the cross slip and their mobile components slightly remain in the channels of PSBs. Less