Yield strain in shear banding amorphous solids

In recent research 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 array of Eshelby quadrupoles all having the same orientation and some density ρ. In this paper we calculate analytically the energy E(ρ,γ) associated with such highly correlated structures as a function of the density ρ and the external strain γ. We show that for strains smaller than a characteristic strain γ(Y) the total strain energy initially increases as the quadrupole density increases, but that for strains larger than γ(Y) the energy monotonically decreases with quadrupole density. We identify γ(Y) as the yield strain. Its value, derived from values of the qudrupole strength based on the atomistic model, agrees with that from the computed stress-strain curves and broadly with experimental results.