Shear banding instabilities in amorphous solids: Predicting the yield strain

We present a short review of the theory of shear localization which results in shear bands in amorphous solids. As this is the main mechanism for the failure of metallic glasses, understanding the instability is invaluable in finding how to stabilize such materials against the tendency to shear localize. We explain the mechanism for shear localization under external shear-strain, which in 2-dimensions is the appearance of highly correlated lines of Eshelby-like quadrupolar singularities which organize the non-affine plastic flow of the amorphous solid into a shear band. We prove analytically that such highly correlated solutions in which 𝒩 equi-distant quadrupoles are aligned with equal orientations are minimum energy states when the strain is high enough. The line lies at 45 degrees to the compressive stress. We use the theory to first predict the yield strain at zero temperature and quasi-static conditions, but later generalize to the case of finite temperature and finite shear rates, deriving the Johnson-Samwer T^2/3 law.