Nonlinear Elasticity of Granular Media

The finite and incremental elasticity of a random packing of identical spheres is derived using energy methods. We consider different models for the contact forces between spheres, all of which are based upon or related to the fundamental Hertz theory; we consider only the special cases of perfect friction (no tangential slip) or no tangential friction. The existence of a strain energy function for the medium depends critically upon the type of contact. If the tangential contact stiffness is independent of the normal force, then the energy is well defined for all values of the macroscopic strain. Otherwise, the strain energy of the system is path dependent, in general. However, the concept of a quadratic strain energy function is always well defined for incremental motion superimposed on large confining stress and strain. For all models considered, we derive the changes in wave speeds due to incremental strains. For the models based upon an energy function we derive expressions for the third-order elastic constants as a function of confining pressure.