Discrete models of dislocations and their motion in cubic crystals

A discrete model describing defects in crystal lattices and having the standard linear anisotropic elasticity as its continuum limit is proposed. The main ingredients entering the model are the elastic stiffness constants of the material and a dimensionless periodic function that restores the translation invariance of the crystal and influences the Peierls stress. Explicit expressions are given for crystals with cubic symmetry: sc (simple cubic), fcc, and bcc. Numerical simulations of this model with conservative or damped dynamics illustrate static and moving-edge and screw dislocations, and describe their cores and profiles. Dislocation loops and dipoles are also numerically observed. Cracks can be created and propagated by applying a sufficient load to a dipole formed by two edge dislocations.

Figure 1

(Color online) Deformed cubic lattice in the presence of an edge dislocation for the piecewise linear $g\left(x\right)$ of Eq. (2) with $\alpha =0.24$.

Figure 2

(Color online) Peierls stress in dimensionless units for a 2D edge dislocation in a sc crystal with the stiffnesses of tungsten as a function of the parameter $\alpha$ in the periodic function $g\left(x\right)$.

Figure 3

(Color online) Screw dislocation for the piecewise linear $g\left(x\right)$ of Eq. (2) with $\alpha =0.24$.

Figure 4

(Color online) Superimposed zooms of the moving core of a screw dislocation.

Figure 5

(Color online) Edge dislocation for the piecewise linear $g\left(x\right)$ of Eq. (2) with (a) $\alpha =0.27$, (b) $\alpha =0.29$, and (c) $\alpha =0.32$.

Figure 6

(Color online) Profile of $u_1(l,m)$ for an edge dislocation with the piecewise linear $g\left(x\right)$ of Eq. (2) when (a) $\alpha =0.24$ and (b) $\alpha =0.32$.

Figure 7

(Color online) Profile of $u_2(l,m)$ for an edge dislocation with the piecewise linear $g\left(x\right)$ of Eq. (2) when (a) $\alpha =0.24$, (b) $\alpha =0.27$, (c) $\alpha =0.29$, and (d) $\alpha =0.32$.

Figure 8

(Color online) Repulsion of like-sign edge dislocations.

Figure 9

(Color online) Attraction of opposite-sign edge dislocations leading to formation of a dislocation loop.

Figure 10

(Color online) Attraction of opposite-sign edge dislocations leading to formation of a dislocation dipole.

Figure 11

(Color online) Snapshots showing crack generation and growth induced by applying a tension in the direction $y$ to a dislocation dipole.

Figure 12

(Color online) Perfect edge dislocation in a gold lattice displaying a twofold stacking sequence of planes. Lines locating the dislocation core are a guide for the eye.

Figure 13

(Color online) Screw dislocation in a gold lattice. Lines locating the dislocation core are a guide for the eye.

Figure 14

(Color online) Edge dislocation in an iron lattice. Lines locating the dislocation core are a guide for the eye.

Figure 15

(Color online) Screw dislocation in an iron lattice. Lines locating the dislocation core are a guide for the eye.