Small-on-large theory with applications to granular materials and fluid/solid systems

Some mechanics involved in calculating wave speeds in pre-stressed materials of geophysical interest are discussed. Two topics are explored: granular materials, and fluid-solid composite systems. Elasticity in the granular material is dominated by intergranular contact which produces strong nonlinearity. The contact also introduces possible hysteresis and loss of strict macroscopic hyperelasticity. Starting with the Hertz solution for two spheres, these notes develop the theory to the stage where macroscopic elastic properties can be estimated, and the stress dependence of bulk waves calculated. Fluid-solid composite systems, such as a fluid-filled borehole in an elastic solid, are complicated by the ability of the fluid to slip under pre-stress. The interface conditions in the different material descriptions are examined and a procedure is described to determine the modified wave speed using a perturbation integral approach. The example of the fundamental guided mode of the borehole, the tube wave, is described in detail.