Homogeneous nucleation of dislocations as bifurcations in a periodized discrete elasticity model

A novel analysis of homogeneous nucleation of dislocations in sheared two-dimensional crystals described by periodized discrete elasticity models is presented. When the crystal is sheared beyond a critical strain $F=F_{c}$, the strained dislocation-free state becomes unstable via a subcritical pitchfork bifurcation. Selecting a fixed final applied strain $F_{f}>F_{c}$, different simultaneously stable stationary configurations containing two or four edge dislocations may be reached by setting $F=F_{f}t/t_{r}$ during different time intervals $t_{r}$. At a characteristic time after $t_{r}$, one or two dipoles are nucleated, split, and the resulting two edge dislocations move in opposite directions to the sample boundary. Numerical continuation shows how configurations with different numbers of edge dislocation pairs emerge as bifurcations from the dislocation-free state.