Introduction

I.1. Background

Since Coulomb’s historical publication of 1773 [COU 73], many investigations have been carried out on the role of physical friction in the mechanical behavior of granular materials. These investigations, supported by pioneering works published by Rankine [RAN 57], Prandtl [PRA 20], Caquot [CAQ 34], Terzaghi [TER 43], and many others, have been progressively incorporated into the body of knowledge of Soil Mechanics, a pillar of civil engineering sciences. However, a direct link between the initial cause – friction at the grain contacts – and the elements of practical interest concerning the behavior at a macroscopic scale, such as the failure criterion or the 3D stress–strain relationship, has not been clearly established. Significant advances in this direction have been made, such as Rowe’s stress– dilatancy theory [ROW 62], which was enriched later by Horne in 1965–1969 [HOR 65, HOR 69], or more recent statistical mechanics approaches. Their conditions of validity, however, limited to axisymmetric stress conditions, or 2D granular assemblies made of disks, are more restrictive to apply them in a general case.

The approach presented here has a larger scope and finds a solution to more general 3D quasi-static problems for granular media with grains of random irregular shapes (Figure I.1). It provides us with the access to an explicit expression of a wide set of macroscopic properties such as stress–dilatancy laws, failure criterion, strain localization with internal structure of the shear bands, orientation and development of failure lines, the intricate relations between friction, shear strength, and volume changes, and the cyclic compaction under alternate shear motion.

This specific multi-scale approach was developed from the following observations:

• – Granular materials, even considered as pseudo-continuum at large scale, remain densely discontinuous at small scale; therefore, the large-scale pseudo-continuous behavior is likely to be highly conditional upon the small-scale behavior of elementary discontinuities: the inter-granular contacts.
• – Within a granular material in motion, internal mechanical processes are highly irreversible, and the main source of this irreversibility is at small-scale dissipation of energy by sliding friction at inter-particle contacts.

Figure I.1. Typical rockfill (basalt) used in civil engineering. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

• – This small-scale energy dissipation by contact friction can be simply formulated with relevant local elementary quantities, such as inter-granular contact forces and contact sliding movements, by direct application of classical friction laws.
• – By a multi-scale analysis, the transposition to the macroscopic scale should lead to a macroscopic energy dissipation relation, linking macroscopic relevant quantities, such as stress and strain rates, and connecting to the thermodynamics of dissipative processes.

In classical standard mechanical behaviors, such as basic fluid mechanics in hydraulics or standard elastoplasticity, the energy dissipation may often be conceptually regarded as a perturbation or a complement within the main framework provided by a regular non-dissipative behavior (e.g. the “perfect incompressible fluid” mechanics in hydraulics, or elasticity in elastoplasticity).

In most of the chapters in this book, energy dissipation by contact friction will stand “alone on stage”; therefore, all of the properties developed are its direct consequences: the whole set of behavioral characteristics displayed appear as a mechanical dissipative structure, hence the name dissipative mechanics.

I.2. Main assumptions

To achieve a clear formulation, this energy dissipation approach requires a set of material and mechanical assumptions, selected to preserve the core of the mechanical behavior. The granular media considered are under slow motion, slow enough to neglect macroscopic dynamical effects or variations in kinetic energy (quasi-static conditions). These media are material sets constituted by rigid, cohesionless mineral particles, with random irregular convex shapes, resulting in no resistance to macroscopic tensile stresses. The inter-particle contacts are unilateral and purely frictional with a uniform friction coefficient.

Relevant internal movements considered in the granular media in motion are the relative sliding movements at contacts. Particle rotations do exist in the granular mass in movements, but remain limited to kinematic shear rotations on average (i.e. with random irregular shapes, there are no macroscopic significant “ball-bearinglike” movements within the granular mass in motion, as described in Chapters 1 and 6). Therefore, the incidence of macroscopic strains of rolling and spinning relative movements is considered here on average as relatively negligible to sliding movements.

Relevant internal forces considered in the granular media in motion are locally the resultant vectors of contact forces exerted on very small contact areas, which are considered as point contacts; the energy effects of contact moments (rolling and spinning) are considered here on average as relatively negligible to the effects of resultant vectors. In this condition, the internal work is made only by contact forces against the relative contact displacements, and the mechanical energy dissipated in the contacts is due to contact sliding motions.

If the granular material is saturated by a fluid filling the inter-granular voids, the fluid pressure is taken as the origin of pressures: the reasoning is conducted on inter-granular forces or macroscopic effective stresses.

With the Eulerian description of the equivalent pseudo-continuum, compressive stress and contraction strain will be denoted as positive, according to the usual conventions in geomechanics. The local values of these stress and strain rates will be considered as the sum of:

• – an average component, on which the large-scale gradients are exerted due to external actions (such as gravity);
• – a component of local random fluctuations, due to the inherent heterogeneity of the medium.

Under regular boundary conditions, the correlations between these fluctuations will be considered to decay sufficiently with the distance beyond a certain scale, so that they have a negligible effect on the macroscopic work rate of internal forces and on the norm of internal actions.

The granular mass in slow dissipative motion close to static equilibrium may be considered resulting from a statistical population of dissipative moving contacts with greater degrees of freedom. Therefore, we assume that it satisfies a “minimum dissipation rule” stated as follows: under regular, monotonic, quasi-equilibrium boundary conditions, the moving medium tends toward a regime of minimum energy dissipation compatible with the imposed boundary conditions; this regime is independent of the initial particular conditions. This rule, strongly suggested by a set of theoretical and experimental results, may be shown [FRO 04] to be a corollary of the Prigogine minimum entropy production theorem based on the thermodynamics of dissipative systems near equilibrium [PRI 68], see Appendix A.I.1.

I.3. Key of the multi-scale approach: the internal actions, a new tensor concept

Deriving constitutive relations from a local discontinuous granular media toward its equivalent pseudo-continuum representation raises numerous basic questions of mechanics, which bring up the need for some new “tool”, both conceptually relevant and clearly formalized, involving the following six key properties regarding the mechanics:

• – to be a simple function of internal movements and internal forces, including a built-in orientation referential objectively linked to the material set in motion;
• – to be an additive physical quantity: the quantity over a whole material set shall be the sum of the quantities related to parts of the whole set (eventually with the addition of boundary terms), which is not the case for internal movements or internal forces considered separately;
• – to have a physical meaning in the discontinuous media, both at local elementary scale (the particle) and the global scale (set of particles in contact), in order to derive relations between local properties (local scale) and average properties (global scale);
• – to also have physical meaning in the equivalent pseudo-continuum, in order to allow the transposition of properties derived in the discontinuous media toward its equivalent continuum representation;
• – to be compatible with the mechanical heterogeneity, inherent to granular media (strongly heterogeneous distributions of internal movements and internal forces);
• – to have a direct link with strain energy, or more precisely, the work rate of internal forces, in order to provide a simple formulation of energy balance, interchanges within the material involved in the energy dissipation.

Such a tool with these six properties has been found in the second-order symmetric tensors resulting from the symmetric product of internal forces and internal movements, holding the work rate of internal forces as the first invariant.

This tool revealed the tensor structures induced by contact friction (Chapter 1) and made possible the general multi-scale approach from an elementary contact to the macroscopic behavior presented in the following chapters. It turned out to be particularly relevant for our specific approach of contact friction dissipative structure, as the resulting key behavior equations operates on its eigenvalues.

From the author’s point of view, the above considerations justify paying particular attention to this new tool and proposing a specific name: the internal actions.