THEORETICAL MULTIPLE-SCATTERING CALCULATION OF NONLINEAR ELASTIC-CONSTANTS OF DISORDERED SOLIDS

The multiple-scattering theory developed in various ways has significantly contributed to our understanding of the dielectric, mechanical, thermal, and other physical properties of disordered solids mainly in the linear-response regime. Its application has, however, been limited when both nonlinearity and disorder are present. In particular, very little work has been done to treat nonlinear elastic properties of disordered materials apart from the simple constant stress or strain averaging--an extreme approximation that is well known to lead to violation of equilibrium condition. Some of our earlier work goes beyond this simple approximation and provides a formal solution of the strain distribution including nonlinearity for an arbitrary disordered solid within the framework of the multiple-scattering theory. Following a similar approach, we propose in this work to develop a general theoretical framework and deduce explicit analytical expression for the third-order elastic constant, which is a generic quantity. As a specific application we have calculated the three independent third-order elastic constants for a particular type of disordered solid, namely, cubic polycrystals following two methods: a perturbative and an approximate self-consistent method developed in the present investigation. These methods have been applied to evaluate third-order elastic constants of eight different materials. The results seem to indicate that for low-anisotropy cases, the two methods give values that closely agree whereas for large anisotropy, they differ. In view of the scarcity of data on the third-order elastic constant of polycrystals, we have also calculated another important nonlinear parameter, namely, the pressure derivative of the second-order shear modulus, for which accurate measurements are available. In all cases, the calculated values are found to compare favorably with experiment. Lastly it may be mentioned that the method developed is quite general and may be adopted to treat nonlinearity in any tensor property of disordered materials.