On the stability of amorphous solid

The plasticity of amorphous material occurs via local plastic rearrangements, shear transformation zones(STZ). The elastic coupling between STZs can generate large-scale avalanches of plastic events. We study the stability condition of amorphous solid toward extensive avalanches. We argue that stability is controlled by the distribution $P(x)$ of the local stress increase $x$ that would lead to an instability. In particular stability requres that $P(x)\sim x^{\theta}$ where $\theta$ satisfies a lower bound. To investigate this, we use a elasto-plastic model based on two ingredients: local plastic events above microscopic stress, and the non-local elastic stress release generated by these plastic events. For a class of models of lond range interaction, $\theta$ is found to lie near saturation. For quadrupole interaction, the model yields $\theta\approx 0.6$ in 2D, and $\theta\approx 0.4$ in 3D.