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 measured strain intervals between plastic events in a athermal strained amorphous solids, and. measures the accumulated strain. The tail of this distribution as Delta gamma -> 0 (in the thermodynamic limit) scales like Delta gamma(eta). The exponent. is related via scaling relations to the tail of the PDF of the eigenvalues of the 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(Y) to plastic flow. In quickly quenched amorphous solids the second transition is smeared out due to the nonexisting stress peak before yield. The nature of these transitions and scaling relations with the system size dependence of <Delta gamma > are discussed.