Wave propagation and localisation in a softening two-phase medium

Within the framework of the generalised theory of heterogeneous media, the complete set of equations is derived for a three-dimensional fluid-saturated porous medium. Subsequently, stability and dispersion analyses are carried out for an infinite one-dimensional continuum that has been deforming homogeneously prior to the application of the perturbation. For common values of the stiffness, the porosity and the mass densities, the two-phase medium is found to remain stable when the solid is elastic. When an elastoplastic or an elasticity-based damage model is adopted for the solid constituent, loss of stability occurs at the peak in the uniaxial stress–strain relation for the solid constituent, which is identical to a single-phase medium. From the dispersion analysis it appears that a dispersive wave is obtained, but that the internal length scale associated with it vanishes in the short wavelength limit, at least for the assumptions made regarding the constitutive behaviour of the solid and of the fluid. This result leads to the conclusion that, upon the introduction of softening, localisation in a zero width will occur and no regularisation will be present. This conclusion is corroborated by the results of numerical analyses of wave propagation in a finite one-dimensional bar.