A Green’s function approach to third-order elastic constants of disordered solids

Abstract

Starting from a Lippmann–Schwinger-type equation, which is very similar to that of quantum mechanical multiple scattering theory, Zeller and Dederichs [Phys. Status Solidi B 55, 831 (1973)] have developed the effective medium theory. This theory has found wide application in understanding the mechanical behavior of disordered solids. However, unlike the problem in quantum mechanics, this equation of the random elasticity is only approximate since this is a linear response theory. So, it is proposed in this work for the first time to go beyond this approximation to treat nonlinear properties of such solids of which the third-order elastic constant is a generic. Again, so far as the nonlinear elastic behavior of these solids is concerned, no work has been done except the simple Voigt- and Reuss-type averaging. Both are extreme approximations and are, moreover, known to lead to violation of the equilibrium condition. The salient feature of the present calculation is to get an exact formal solution of the problem in terms of an appropriate Green’s function in a closed form. The result obtained is quite general and may be adopted to treat nonlinearity in any tensor property of disordered materials. Finally several approximations, including a self-consistent solution, have been discussed for obtaining the effective nonlinear static mechanical susceptibility.