A Master equation for force distributions in polydisperse frictional particles

An incremental evolution equation, i.e. a Master equation in statistical mechanics, is introduced for force distributions in polydisperse frictional particle packings. As basic ingredients of the Master equation, the conditional probability distributions of particle overlaps are determined by molecular dynamics simulations. Interestingly, tails of the distributions become much narrower in the case of frictional particles than frictionless particles, implying that correlations of overlaps are strongly reduced by microscopic friction. Comparing different size distributions, we find that the tails are wider for the wider distribution.Comment: 12 pages, 7 figures. Conference proceedings for PARTICLES 2015, 28-30 September, 2015, Barcelona, Spai

A master equation for force distributions in soft particle packings - Irreversible mechanical responses to isotropic compression and decompression

Mechanical responses of soft particle packings to quasi-static deformations are determined by the microscopic restructuring of force-chain networks, where complex non-affine displacements of constituent particles cause the irreversible macroscopic behavior. Recently, we have proposed a master equation for the probability distribution functions of contact forces and interparticle gaps [K. Saitoh et al., Soft Matter 11, 1253 (2015)], where mutual exchanges of contacts and interparticle gaps, i.e. opening and closing contacts, are also involved in the stochastic description with the aid of Delaunay triangulations. We describe full details of the master equation and numerically investigate irreversible mechanical responses of soft particle packings to cyclic loading. The irreversibility observed in molecular dynamics simulations is well reproduced by the master equation if the system undergoes quasi-static deformations. We also confirm that the degree of irreversible responses is a decreasing function of the area fraction and the number of cycles.Comment: 17 pages, 21 figures (6 figures are not displayed

Macroscopic model with anisotropy based on micro-macro informations

Physical experiments can characterize the elastic response of granular materials in terms of macroscopic state-variables, namely volume (packing) fraction and stress, while the microstructure is not accessible and thus neglected. Here, by means of numerical simulations, we analyze dense, frictionless, granular assemblies with the final goal to relate the elastic moduli to the fabric state, i.e., to micro-structural averaged contact network features as contact number density and anisotropy. The particle samples are first isotropically compressed and later quasi-statically sheared under constant volume (undrained conditions). From various static, relaxed configurations at different shear strains, now infinitesimal strain steps are applied to "measure" the effective elastic response; we quantify the strain needed so that plasticity in the sample develops as soon as contact and structure rearrangements happen. Because of the anisotropy induced by shear, volumetric and deviatoric stresses and strains are cross-coupled via a single anisotropy modulus, which is proportional to the product of deviatoric fabric and bulk modulus (i.e. the isotropic fabric). Interestingly, the shear modulus of the material depends also on the actual stress state, along with the contact configuration anisotropy. Finally, a constitutive model based on incremental evolution equations for stress and fabric is introduced. By using the previously measured dependence of the stiffness tensor (elastic moduli) on the microstructure, the theory is able to predict with good agreement the evolution of pressure, shear stress and deviatoric fabric (anisotropy) for an independent undrained cyclic shear test, including the response to reversal of strain

Master equation for the probability distribution functions of forces in soft particle packings

Employing molecular dynamics simulations of jammed soft particles, we study microscopic responses of force-chain networks to quasi-static isotropic (de)compressions. We show that not only contacts but also interparticle gaps between the nearest neighbors must be considered for the stochastic evolution of the probability distribution functions (PDFs) of forces, where the mutual exchange of contacts and interparticle gaps, i.e. opening and closing contacts, are also crucial to the incremental system behaviors. By numerically determining the transition rates for all changes of contacts and gaps, we formulate a Master equation for the PDFs of forces, where the insight one gets from the transition rates is striking: The mean change of forces reflects non-affine system response, while their fluctuations obey uncorrelated Gaussian statistics. In contrast, interparticle gaps are reacting mostly affine in average, but imply multi-scale correlations according to a wider stable distribution function.Comment: 5 pages, 4 figures, submitted to Soft Matte

Effect of cohesion on shear banding in quasi-static granular material

It is widely recognized in particle technology that adhesive powders show a wide range of different bulk behavior due to the peculiarity of the particle interaction. We use Discrete Element simulations to investigate the effect of contact cohesion on the steady state of dense powders in a slowly sheared split-bottom Couette cell, which imposes a wide stable shear band. The intensity of cohesive forces can be quantified by the {\em granular Bond number} ($Bo$), namely the ratio between maximum attractive force and average force due to external compression. We find that the shear banding phenomenon is almost independent of cohesion for Bond numbers $Bo<1$, but for $Bo \ge 1$ cohesive forces start to play an important role, as both width and center position of the band increase for $Bo > 1$. Inside the shear band, the mean normal contact force is always independent of cohesion and depends only on the confining stress. In contrast, when the behavior is analyzed focusing on the eigen-directions of the local strain rate tensor, a dependence on cohesion shows up. Forces carried by contacts along the compressive and tensile directions are symmetric about the mean force (larger and smaller respectively), while the force along the third, neutral direction follows the mean force. This anisotropy of the force network increases with cohesion, just like the heterogeneity in all (compressive, tensile and neutral) directions.Comment: 12 pages, 20 figures, accepted in Physical Review

Tuning the bulk properties of bidisperse granular mixtures by small amount of fines

We study the bulk properties of isotropic bidisperse granular mixtures using discrete element simulations. The focus is on the influence of the size (radius) ratio of the two constituents and volume fraction on the mixture properties. We show that the effective bulk modulus of a dense granular (base) assembly can be enhanced by up to 20% by substituting as little as 5% of its volume with smaller sized particles. Particles of similar sizes barely affect the macroscopic properties of the mixture. On the other extreme, when a huge number of fine particles are included, most of them lie in the voids of the base material, acting as rattlers, leading to an overall weakening effect. In between the limits, an optimum size ratio that maximizes the bulk modulus of the mixture is found. For loose systems, the bulk modulus decreases monotonically with addition of fines regardless of the size ratio. Finally, we relate the mixture properties to the 'typical' pore size in a disordered structure as induced by the combined effect of operating volume fraction (consolidation) and size ratio

Contact anisotropy and coordination number for a granular assembly:a comparison between DEM simulation and theory

We study an ideal granular aggregate consisting of elastic spherical particles, isotropic in stress and anisotropic in the contact network. Because of the contact anisotropy, a confining pressure applied at zero deviatoric stress, produces shear strain as well as volume strain. Our goal is to predict the coordination number k, the average number of contacts per particle, and the magnitude of the contact anisotropy ɛ, from knowledge of the elastic moduli of the aggregate. We do this through a theoretical model based upon the well known effective medium theory. However, rather than focusing on the moduli, we consider their ratios over the moduli of an equivalent isotropic state. We observe good agreement between numerical simulation and theory

Hydro-micromechanical modeling of wave propagation in saturated granular media

Biot's theory predicts the wave velocities of a saturated poroelastic granular medium from the elastic properties, density and geometry of its dry solid matrix and the pore fluid, neglecting the interaction between constituent particles and local flow. However, when the frequencies become high and the wavelengths comparable with particle size, the details of the microstructure start to play an important role. Here, a novel hydro-micromechanical numerical model is proposed by coupling the lattice Boltzmann method (LBM) with the discrete element method (DEM. The model allows to investigate the details of the particle-fluid interaction during propagation of elastic waves While the DEM is tracking the translational and rotational motion of each solid particle, the LBM can resolve the pore-scale hydrodynamics. Solid and fluid phases are two-way coupled through momentum exchange. The coupling scheme is benchmarked with the terminal velocity of a single sphere settling in a fluid. To mimic a pressure wave entering a saturated granular medium, an oscillating pressure boundary condition on the fluid is implemented and benchmarked with one-dimensional wave equations. Using a face centered cubic structure, the effects of input waveforms and frequencies on the dispersion relations are investigated. Finally, the wave velocities at various effective confining pressures predicted by the numerical model are compared with with Biot's analytical solution, and a very good agreement is found. In addition to the pressure and shear waves, slow compressional waves are observed in the simulations, as predicted by Biot's theory.Comment: Manuscript submitted to International Journal for Numerical and Analytical Methods in Geomechanic