Linear and nonlinear elasticity of granular media: Stress-induced anisotropy of a random sphere pack

We develop an effective medium theory of the nonlinear elasticity of a random sphere pack based upon the underlying Hertz-Mindlin theory of grain-grain contacts. We compare our predictions for the stress-dependent sound speeds against new experimental data taken on samples with stress-induced uniaxial anistropy. We show that the second-order elastic moduli, C_ijkl, and therefore the sound speeds, can be calculated as unique path-independent functions of an arbitrary strain environment, (ε_kl), thus generalizing earlier results due to Walton. However, the elements of the stress tensor, σ_y, are not unique functions of (ε_kl) and their values depend on the strain path. Consequently, the sound speeds, considered as functions of the applied stresses, are path dependent. Illustrative calculations for three cases of combined hydrostatic and uniaxial strain are presented. We show further, that, even when the additional applied uniaxial strain is small, these equations are not consistent with the usual equations of third-order hyperelasticity. Nor should they be, for the simple reason that there does not exist an underlying energy function which is simply a function of the current state of the strain. Our theory provides a good understanding of our new data on sound speeds as a function of uniaxial stress.