Foundations and applications of non-equilibrium statistical mechanics

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Statistical mechanics approaches to granular media: between micromechanics and macromechanics

The mechanical behavior of granular materials has been traditionally approached through two theoretical and computational frameworks: macromechanics and micromechanics. Macromechanics focuses on continuum based models. In consequence it is assumed that the matter in the granular material is homogeneous and continuously distributed over its volume so that the smallest element cut from the body possesses the same physical properties as the body. In particular, it has some equivalent mechanical properties, represented by complex and non-linear constitutive relationships. Engineering problems are usually solved using computational methods such as FEM or FDM. On the other hand, micromechanics is the analysis of heterogeneous materials on the level of their individual constituents. In granular materials, if the properties of particles are known, a micromechanical approach can lead to a predictive response of the whole heterogeneous material. Two classes of numerical techniques can be differentiated: computational micromechanics, which consists on applying continuum mechanics on each of the phases of a representative volume element and then solving numerically the equations, and atomistic methods (DEM), which consist on applying rigid body dynamics together with interaction potentials to the particles. Statistical mechanics approaches arise between micro and macromechanics. It tries to state which the expected macroscopic properties of a granular system are, by starting from a micromechanical analysis of the features of the particles and the interactions. The main objective of this paper is to introduce this approach.

Statistical distributions obtained from the compression of monodisperse, soft and frictionless particles

The cyclic compression of several granular systems has been simulated with a molecular dynamics code. All the samples consisted of bidimensional, soft, frictionless and equal-sized particles that were initially arranged according to a squared lattice and were compressed by randomly generated irregular walls. The compression protocols can be described by some control variables (volume or external force acting on the walls) and by some dimensionless factors, that relate stiffness, density, diameter, damping ratio and water surface tension to the external forces, displacements and periods. Each protocol, that is associated to a dynamic process, results in an arrangement with its own macroscopic features: volume (or packing ratio), coordination number, and stress; and the differences between packings can be highly significant. The statistical distribution of the force-moment state of the particles (i.e. the equivalent average stress multiplied by the volume) is analyzed. In spite of the lack of a theoretical framework based on statistical mechanics specific for these protocols, it is shown how the obtained distributions of mean and relative deviatoric force-moment are. Then it is discussed on the nature of these distributions and on their relation to specific protocols.