Stochastic Approach to Plasticity and Yield in Amorphous Solids

We focus on the probability distribution function (pdf) $P(\Delta \gamma; \gamma)$ where $\Delta \gamma$ are the {\em measured} strain intervals between plastic events in an athermal strained amorphous solids, and $\gamma$ measures the accumulated strain. The tail of this distribution as $\Delta \gamma\to 0$ (in the thermodynamic limit) scales like $\Delta \gamma^\eta$. The exponent $\eta$ is related via scaling relations to the tail of the pdf of the eigenvalues of the {\em plastic modes} of the Hessian matrix $P(\lambda)$ which scales like $\lambda^\theta$, $\eta=(\theta-1)/2$. The numerical values of $\eta$ or $\theta$ can be determined easily in the unstrained material and in the yielded state of plastic flow. Special care is called for in the determination of these exponents between these states as $\gamma$ increases. Determining the $\gamma$ dependence of the pdf $P(\Delta \gamma; \gamma)$ can shed important light on plasticity and yield. We conclude that the pdf's of both $\Delta \gamma$ and $\lambda$ are not continuous functions of $\gamma$. In slowly quenched amorphous solids they undergo two discontinuous transitions, first at $\gamma=0^+$ and then at the yield point $\gamma=\gamma_{_{\rm Y}}$ to plastic flow. In quickly quenched amorphous solids the second transition is smeared out due to the non existing stress peak before yield. The nature of these transitions and scaling relations with the system size dependence of $\langle \Delta \gamma\rangle$ are discussed.