Shearbanding in Amorphous Solids and Interacting Eshelby Singularities

We will describe recent work in which it was found that the fundamental shear-localizing instability of amorphous solids under external strain, which eventually results in a shear band and failure, consists of a highly correlated line of Eshelby quadrupoles all having the same orientation and some density $\rho$. We describe how the energy $E(\rho,\gamma)$ associated with such highly correlated structures as a function of the density $\rho$ and the external strain $\gamma$ can be calculated. We then show that when the strain $\gamma$ is smaller than some characteristic yield stress $\gamma_y$ the minimum energy solution is attained for $\rho=0$ (i.e. isolated localized plastic events). While for $\gamma \ge \gamma_y$ there is a bifurcation allowing a finite density of quadrupoles. We finally suggest how the universal Johnson-Samwer $T^{2/3}$ temperature reduction of the yield stress in metallic glasses can be accounted for by such ideas.