Simulation of Dislocation Annihilation by Cross-Slip

Abstract

This contribution deals with the numerical simulation of dislocation dynamics, their interaction, merging and changes in the dislocation topology. The glide dislocations are represented by parametrically described curves moving in slip planes. The simulation model is based on the numerical solution of the dislocation motion law belonging to the class of curvature driven curve dynamics. We focus on the simulation of the cross-slip of two dislocation curves where each curve evolves in a dierent slip plane. The dislocations evolve, under their mutual interaction and under some external force, towards each other and at a certain time their evolution continues outside slip planes. During this evolution the dislocations merge by the cross-slip occurs. As a result, there will be two dislocations evolving in three planes, two planes, and one plane where cross-slip occurred. The goal of our work is to simulate the motion of the dislocations and to determine the conditions under which the cross-slip occurs. The simulation of the dislocation evolution and merging is performed by improved parametric approach and numerical stability is enhanced by the tangential redistribution of the discretization points. where B is a drag coecient, and v(x, t) is the normal velocity of a dislocation at x ∈ Γ and time t. The term Lκ represents self-force expressed in the line tension approximation as the product of the line tension L and local curvature κ(x, t). The term τ app represents the local shear stress acting on the dislocation segment produced by the bulk elastic eld. The term τ int represents interaction force between dislocations. In our simulations, we consider the stress controlled regime where the applied stress in the channel is kept uniform. In the slip plane, the applied stress τ app is the same at each point of the line and for numerical computations we use τ app = const.The strain controlled regime analyzed in [3] could be an alternative. Parametric descriptionThe motion law (1.1) in the case of dislocation dynamics is treated by parameterization where the planar curve Γ (t) is described by a smooth time-dependent vector function X : S × I → R 2 , where S = [0, 1] is a xed interval for the curve parameter and I = [0, T ] is the (509)

An Energy-Consistent Model of Dislocation Dynamics in an Elastic Body

“…We remark that X is a space of finite energy displacements with a gap across S. In the following lines, we suppose that V (0,Ψ ) = / 0. If u| Ω ± ∈ H 2 (Ω ± ; R 3 ), and u satisfies the equations of (10), u is called a strong solution to (10), where the boundary conditions are considered in the sense of the trace operator. We also define a weak solution as follows.…”

“…A solution of Problem 2.1 is called a weak solution to (10). In particular, it is not difficult to show that a strong solution to ( 10) is a weak solution, by a standard computation with integration by parts.…”

“…Then, u is a strong solution to (10) if and only if u is a weak solution and u| Ω + and u| Ω − belong to H 2 (Ω + ; R 3 ) and H 2 (Ω − ; R 3 ), respectively.…”

“…It is also known that the dislocation curve sometimes can change its slip plane. For example, some numerical simulations of dislocations with end points which change their slip planes are shown in [10]. The model presented in this paper can also be applied in the case where the end points are fixed on the boundary of the slip plane, if we impose the Dirichlet boundary condition of the phase field variable ϕ as illustrated in Figure 3.…”

Exact solution for dislocation bowing and a posteriori numerical technique for dislocation touching-splitting

“…for a positive constant C. According to this evolution law, the dislocation curves move toward circular obstacles (strong precipitates), then they touch the obstacles and bend in the direction, say ⃗ v, of a line perpendicular to the line connecting two obstacles [2]. See Fig.…”