Impact of lattice rotation on dislocation motion

We introduce a phenomenological theory of dislocation motion appropriate for two-dimensional lattices. A coarse grained description is proposed that involves as primitive variables local lattice rotation and Burgers vector densities along distinguished slip systems of the lattice. We then use symmetry considerations to propose phenomenological equations for both defect energies and their dissipative motion. As a consequence, the model includes explicit dependencies on the local state of lattice orientation, and allows for differential defect mobilities along distinguished directions. Defect densities and lattice rotation need to be determined self-consistently and we show specific results for both square and hexagonal lattices. Within linear response, dissipative equations of motion for the defect densities are derived which contain defect mobilities that depend nonlocally on defect distribution.

Figure 1

An illustration of the two decompositions used. The Burger's vector is shown in blue and its projections onto the nearest glide planes are in red. The green vectors represent the triplet density. The dislocation densities are the sum of the corresponding projection (red) and the triplet density (green). The case shown is when the dislocation density $b^{\left(2\right)}$ vanishes and $b^{\left(1\right)}$ does not receive a projection. ${\widehat{\theta }}^{\left(0\right)}$ is shown vertical for convenience, but its orientation with respect to $\widehat{x}$ is given by $\theta$.

Figure 2

(a) The dislocation position on the $y=0$ line $x\left(t\right)$ as a function of time $t$ for two opposite edge dislocations that are initially separated by a distance of 10, in units of $b$. For each case there is no climb mobility. The blue and red curves are the positions in the absence of lattice rotations, while the gold and green curves pertain to a system with lattice rotations. Note that, in the latter case, motion is arrested after some time. (b) The corresponding dislocation position $y\left(t\right)$ as a function of time $t$ for the two dislocations.

Figure 3

(a) The dislocation position $x\left(t\right)$ as a function of time $t$ for the case in which both lattice rotations and climb are operative. In this case $D_c/D_g=0.02$. The dipolar configuration seen in the previous figure is unstable and annihilation results. (b) The corresponding dislocation position $y\left(t\right)$ as a function of time $t$.