Nearly all solids are compised of grains. However most studies treat materials as a continious solid. The book applies analysis used on loose granular materials to dense grainular materials. This title's main focus is devoted to static or dynamic loadings applied to dense materials, although rapid flows and widely dispersed media are also mentioned briefly. Three essential areas are covered: Local variable analysis: Contact forces, displacements and rotations, orientation of contacting particles and fabric tensors are all examples of local variables. Their statistical distributions, such as spatial distribution and possible localization, are analyzed, taking into account experimental results or numerical simulations. Change of scales procedures: Also known as "homogenization techniques", these procedures make it possible to construct continuum laws to be used in a continuum mechanics approach or performing smaller scale analyses. Numerical modeling: Several methods designed to calculate approximate solutions of dynamical equations together with unilateral contact and frictional laws are presented, including molecular dynamics, the distinct element method and non-smooth contact dynamics. Numerical examples are given and the quality of numerical approximations is discussed.
Experimental and Numerical Analysis of Local Variables in Granular Materials1
1.1. Introduction
Granular materials consist of densely packed solid particles and a pore-filling material which can be a fluid or a solid matrix. The particles interact via elastic repulsion, friction, adhesion and other surface forces. By nature, the length scales involved in these contact interactions are much smaller than the particle size. External loading leads to particle deformations as well as cooperative particle rearrangements. The particle deformations are of particular importance in powder metallurgy, for example, but the particles may be considered as quasi-rigid beyond the elastic response times.
The contact network and pore space are the two facets of the microstructure of granular materials to which we will refer, in this chapter, as granular texture. At the particle scale, the granular texture involves three basic vectors from which other local geometrical variables can be defined: (1) the branch vector
joining the centers of contacting particles; (2) the contact orientation vector (contact normal)
defined as the unit vector normal to the particle boundary at the contact zone α; and (3) the contact vectors
joining the particle centers to the contact point; see Figure 1.1. The reaction forces
and â
acting on two particles at their contact zone have a unique application point. This point may be considered as their contact point in the case of extended contacts between two polyhedral particles.
Figure 1.1.Local vectors at the contact α between two particles 1α and 2α: branch vector
contact normal
contact force
and contact vectors
Two different local frames can be associated with a pair of contacting particles: (1) the frame defined by the contact normal
and two orthogonal unit vectors
in the contact plane (tangential to the two particles at the contact point); and (2) the frame defined by the âradialâ unit vector
and two orthogonal unit vectors
in a orthoradial plane (orthogonal to the branch vector). These two frames coincide in the case of spherical particles. In two-dimensions (2D), the local frame is uniquely defined by a single tangent unit vector t or tâČ.
The granular texture is disordered with many different variants depending on the composition (particle shapes and sizes), interactions and assembling procedure. The granular disorder is essentially characterized by the fact that, as a result of geometrical exclusions among particles, the local vectors vary discontinuously from one contact to another. In other words, the local environments fluctuate in space. The contact network evolves with loading so that the local environments also fluctuate in time. The highly inhomogenous distribution of contact forces reflects granular disorder in static equilibrium. In particular, the force chains reveal long-range correlations whereas the presence of a broad population of very weak forces results from the arching effect. The force and fabric anisotropies are two complimentary aspects of stress transmission. They can be employed in a local (particle-scale) description of granular media in the quasi-static state.
The geometrical changes of granular texture are at the origin of the complex rheology of granular materials. These changes are highly nonlinear, involving creation and loss of contacts, rotation frustration and frictional sliding. They depend on the dissipative nature of contact interactions and steric exclusions among particles. In quasi-static deformation, various features of the plastic behavior such as shear strength and dilatancy can be traced back to the evolution of granular texture. Two issues are of primary interest in microscopic modeling of granular plasticity: (1) what is the lowest level of textural information, and to what extent does it control the effective properties of the material? and (2) how do the effective properties depend on higher order textural information?
In this chapter, we introduce several concepts and tools for the description of granular texture, kinematics and force transmission with examples and illustrations from discrete element simulations (molecular dynamics and contact dynamics, see Chapter 4) and experiments. We first consider the description of granular texture in terms of particle positions and contact orientations. The kinematics and mechanisms of plastic deformation are then analyzed. Finally, we focus on stress transmission and its link with granular texture.
1.2. Description of granular texture
The granular texture is generally described in terms of the distributions of the vector...
