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

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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