Nonlinear plane waves in saturated porous media with incompressible constituents

We consider the propagation of nonlinear plane waves in porous media within the framework of the Biot-Coussy biphasic mixture theory. The tortuosity effect is included in the model, and both constituents are assumed incompressible (Yeoh-type elastic skeleton, and saturating fluid). In this case, the linear dispersive waves governed by Biot's theory are either of compression or shear-wave type, and nonlinear waves can be classified in a similar way. In the special case of a neo-Hookean skeleton, we derive the explicit expressions for the characteristic wave speeds, leading to the hyperbolicity condition. The sound speeds for a Yeoh skeleton are estimated using a perturbation approach. Then we arrive at the evolution equation for the amplitude of acceleration waves. In general, it is governed by a Bernoulli equation. With the present constitutive assumptions, we find that longitudinal jump amplitudes follow a nonlinear evolution, while transverse jump amplitudes evolve in an almost linearly degenerate fashion.