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-depend ent sound speeds against new experimental data taken on samples with s tress-induced uniaxial anistropy. We show that the second-order elasti c moduli, C-ijkl, and therefore the sound speeds, can be calculated as unique path-independent functions of an arbitrary strain environment, {epsilon(kl)}, thus generalizing earlier results due to Walton. Howev er, the elements of the stress tensor, oil, are not unique functions o f {epsilon(kl)} and their values depend on the strain path. Consequent ly, the sound speeds, considered as functions of the applied stresses, are path dependent. Illustrative calculations for three cases of comb ined hydrostatic and uniaxial strain are presented. We show further, t hat, even when the additional applied uniaxial strain is small, these equations are not consistent with the usual equations of third-order h yperelasticity. Nor should they be, for the simple reason that there d oes not exist an underlying energy function which is simply a function of the current state of the strain. Our theory provides a good unders tanding of our new data an sound speeds as a function of uniaxial stre ss.