Nonlinear plane waves in saturated porous media with incompressible constituents

Harold Berjamin

doi:10.1098/rspa.2021.0086

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.

Highlights

Originating in the field of geophysics, the theory of porous media has a long history that goes back to the eighteenth century
A second approach known as mixture theory derives from the ground principles of continuum mechanics, namely, balance principles and thermodynamic restrictions
The main features are the existence of shear waves and slow compression waves, which linear dispersive properties follow from the Biot theory

Summary

Introduction

Originating in the field of geophysics, the theory of porous media has a long history that goes back to the eighteenth century (see the historical review by De Boer [1]). Motivated by the above-mentioned observations, the present study aims at gaining insight into the wave physics of nonlinear porous materials by estimating wave speeds and amplitudes analytically Such results are quite rare in the nonlinear mixture theory literature, where most analytical results have been obtained in the linear limit [15,16,17]. A second approach known as mixture theory derives from the ground principles of continuum mechanics, namely, balance principles and thermodynamic restrictions This ‘rational’ approach has been used in various biomechanical applications [2], among which some of the most recent quasi-static brain mechanics studies [4,5,6].

Biphasic mixture theory

Nonlinear plane waves

Conclusion