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.

Representing microstates of static granular matter in a stress phase space

A stress phase space is proposed to compare the static packings of a granular system (microstates) that are compatible to a macrostate described by external stresses. The equivalent stress of each particle of a static packing can be obtained from the mechanical interaction forces, and the associated volume is given by the respective Voronoi cell. Therefore, particles can be located at different stress levels and grouped into categories or configurations, which are defined in base of the geometrical features of the local arrangement (in particular, of the number of forces that keep them force-balanced). They can be represented as points in a stress phase space. The nature of this space is analyzed in detail. The integration limits of the stress variables that avoid or limit tensile states and the capability of each configuration to represent specific stress states establish its main features. Furthermore, if some stress variables are used, instead of the usual components of the Cauchy stress tensor, then some symmetries can be found. Results obtained from molecular dynamics simulations are used to check this nature. Finally, some statistical ensembles are written in terms of the coordinates of this phase space. These require some assumptions that are made in base on continuum mechanics principles.

Permanent deformation estimates of dynamic equipment foundations: Application to a gas turbine in granular soils

Permanent displacements of a gas turbine founded on a fine, poorly graded, and medium density sand are studied. The amplitudes and modes of vibration are computed using Barkan´s formulation, and the “High-Cycle Accumulation” (HCA) model is employed to account for accumulated deformations due to the high number of cycles. The methodology is simple: it can be easily incorporated into standard mathematical software, and HCA model parameters can be estimated based on granulometry and index properties. Special attention is devoted to ‘transient’ situations at equipment´s start-up, during which a range of frequencies – including frequencies that could be similar to the natural frequencies of the ground – is traversed. Results show that such transient situations could be more restrictive than stationary situations corresponding to normal operation. Therefore, checking the stationary situation only might not be enough, and studying the influence of transient situations on computed permanent displacements is needed to produce a proper foundation design