The landscape of instabilities versus shear angle in simulations

Abstract

We argue that, in order to appreciate fully how disordered solids fail under sufficiently large strains or thermal noise, a material should be regarded as a network of potentially interacting instabilities occurring along different directions of forcing. We develop a tool for simulations with periodic boundary conditions that allows strains to be applied at a continuously variable angle, $\theta$. This reveals an underlying landscape which consists of lines of instabilities in two dimensions. A modified polar plot provides a visualization of this strain landscape while also allowing a graphic measure of an instability's quadrupolar structure. Many lines persist over a broad angular range of applied strain; some pass through one another; others change continuously with angle or end by smoothly decreasing their magnitudes to zero. Appropriate paths through phase space allow instabilities to be circumvented and hence avoided entirely. We explain this by examining hysterons, i.e., instabilities that undo themselves upon reversing shear direction, and show that as the end of an instability line is approached, the separation between the forward and backward instabilities vanishes.