Marie-Angèle Abellan

A Precis of Two-Scale Approaches for Fracture in Porous Media

A derivation is given of two-scale models that are able to describe deformation and flow in a fluid-saturated and progressively fracturing porous medium. From the micromechanics of the flow in the cavity, identities are derived that couple the local momentum and the mass balances to the governing equations for a fluid-saturated porous medium, which are assumed to hold on the macroscopic scale. By exploiting the partition-of-unity property of the finite element shape functions, the position and direction of the fracture is independent from the underlying discretisation. The finite element equations are derived for this two-scale approach and integrated over time. The resulting discrete equations are nonlinear due to the cohesive crack model and the nonlinearity of the coupling terms. A consistent linearisation is given for use within a Newton—Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach.

A discrete model for the propagation of discontinuities in a fluid-saturated medium

The first part of this manuscript discusses a finite element method that captures arbitrary discontinuities in a two-phase medium by exploiting the partition-of-unity property of finite element shape functions. The fluid flow away from the discontinuity is modelled in a standard fashion using Darcys relation, and at the discontinuity a discrete analogy of Darcys relation is used. Subsequently, dynamic shear banding is studied numerically for a biaxial, plane-strain specimen. A Tresca-like as well as a Coulomb criterion are used as nucleation criterion. Decohesion is controlled by a mode-II fracture energy, while for the Coulomb criterion, frictional forces are transmitted across the interface in addition to the cohesive shear tractions. The effect of the different interface relations on the onset of cavitation is studied.

Dispersion and Localisation in a Strain–Softening Two–Phase Medium

In fluid–saturated media wave propagation is dispersive, but the associated internal length scale vanishes in the short wave-length limit. Accordingly, upon the introduction of softening, localisation in a zero width will occur and no regularisation is present. This observation is corroborated by numerical analyses of wave propagation in a finite one-dimensional bar.

Instabilities and Discontinuities in Two-Phase Media

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, dispersion analyses are carried out for an infinite one-dimensional continuum, that has been deforming homogeneously prior to the application of the perturbation. A dispersive wave is obtained, but the internal length scale associated with it vanishes in the short wave--length 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 [1]. The result has severe implications for finite element analyses of damaging multiphase media, since they will be mesh-dependent.