Farid Laouafa

Three-Dimensional Multiscale Bifurcation Analysis of Granular Media

This paper deals with Hills bifurcation criterion, which is very well suited to describe various failure modes in granular media. The first part of this paper is dedicated to the analytical and numerical investigation of this criterion by considering phenomenological constitutive relations: the incrementally piece-wise linear and nonlinear relations proposed by Darve. The 3D bifurcation domain and 3D cones of unstable directions are given for these two relations. A similar analysis is performed with a micromechanical model in the second part of the article. Finally, a qualitative comparison of the results obtained with these two different approaches leads us to some key conclusions about this material instability criterion, which is studied for the first time for general 3D conditions, by considering these nonconventional constitutive relations.

3D bifurcation analysis in geomaterials. Investigation of the second order work criterion

ABSTRACT In this paper, a study of failure in geomaterials is proposed using the second order work criterion and a phenomenological approach. In a first part, an analytical investigation of this criterion is proposed. General 3D equation of instability cones as well as of the 3D bifurcation domain limit are given for every incrementally piece-wise linear constitutive model. In the second part, these instability cones and bifurcation domain are displayed for constitutive models of Darve. Then a physical interpretation of these results is proposed. Noticeably, it is proved that, when the second order work vanishes along a given loading path, an extension a generalized flow rule can be defined for proper conjugated variables. Finally we conclude that this approach constitutes a good starting point when investigating bifurcation in geomaterials, and opens new horizons in experimental testing.

Constitutive Equations and Instabilities of Granular Materials

Constitutive equations for geomaterials constitute a very intricate field. In the first part of this chapter, a synthetic view of constitutive formalism is presented. An intrinsic classification of all existing constitutive relations is deduced. Then examples of incrementally non-linear relations are given and some applications follow. A numerical study of the so-called “yield surfaces” is presented, and is followed by a discussion on the validity of the principle of superposition for incremental loading. Finally the question of bifurcations and instabilities in geomaterials is investigated. Essentially because of the non-associative character of geomaterial plastic strains, a large domain of failure with various modes of ruptures is exhibited.

Dispersion in Porous Media with Heterogeneous Nonlinear Reactions

The upscaling of mass transport in porous media with a heterogeneous reaction at the fluid–solid interface, typical of dissolution problems, is carried out with the method of volume averaging, starting from a pore-scale transport problem involving thermodynamic equilibrium or nonlinear reactive boundary conditions. A general expression to describe the macro-scale mass transport is obtained involving several effective parameters which are given by specific closure problems. For representative unit cell with a simple stratified geometry, the effective parameters are obtained analytically and numerically, while for those with complicated geometries, the effective parameters are only obtained numerically by solving the corresponding closure problems. The impact on the effective parameters of the fluid properties, in terms of pore-scale Péclet number Pe, and the process chemical properties, in terms of pore-scale Damköhler number Da and reaction order (n), is studied for periodic stratified and 3D unit cells. It is found that the tortuosity effects play an important role on the longitudinal dispersion coefficient in the 3D case, while it is negligible for the stratified geometry. When Da is very small, the effective reaction rate coefficient is nearly identical to the pore-scale one, while when Da is very large, the reactive condition turns out to be equivalent to pore-scale thermodynamic equilibrium, and the macro-scale mass exchange term is consequently given in a different form from the reactive case. An example of the application of the macro-scale model is presented with the emphasis on the potential impact of additional, non-traditional effective parameters appearing in the theoretical development on the improvement of the accuracy of the macro-scale model.