Nonlinear plane waves in saturated porous media with incompressible constituents

TL;DR

This paper investigates nonlinear wave propagation in saturated porous media with incompressible constituents, deriving explicit wave speeds and evolution equations within the Biot-Coussy framework, highlighting differences between longitudinal and transverse wave behaviors.

Contribution

It introduces a model incorporating tortuosity and incompressibility, deriving explicit wave speeds and evolution equations for nonlinear waves in saturated porous media.

Findings

Explicit wave speed expressions for Yeoh skeletons

Nonlinear evolution of longitudinal wave amplitudes

Almost linear evolution of transverse wave amplitudes

Abstract

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.