Purpose -To present a discrete particle model for granular materials. Design/methodology/approach -Starting with kinematical analysis of relative movements of two typical circular grains with different radii in contact, both the relative rolling and the relative sliding motion measurements at contact, including translational and angular velocities (displacements) are defined. Both the rolling and sliding friction tangential forces, and the rolling friction resistance moment, which are constitutively related to corresponding relative motion measurements defined, are formulated and integrated into the framework of dynamic model of the discrete element method. Findings -Numerical results demonstrate that the importance of rolling friction resistance, including both rolling friction tangential force and rolling friction resistance moment, in correct simulations of physical behavior in particulate systems; and the capability of the proposed model in simulating the different types of failure modes, such as the landslide (shear bands), the compression cracking and the mud avalanching, in granular materials. Research limitations/implications -Each grain in the particulate system under consideration is assumed to be rigid and circular. Do not account for the effects of plastic deformation at the contact points. Practical implications -To model the failure phenomena of granular materials in geo-mechanics and geo-technical engineering problems; and to be a component model in a combined discrete-continuum macroscopic approach or a two-scale discrete-continuum micro-macro-scopic approach to granular media. Originality/value -This paper develops a new discrete particle model to describe granular media in several branches of engineering such as soil mechanics, power technologies or sintering processes.

One of the purposes in this study is to develop a micromorphic continuum model for granular materials based on a micromechanics approach. A symmetric curvature tensor is proposed in this model, and a symmetric couple stress tensor conjugated with the symmetric curvature tensor is derived. In addition, a symmetric stress tensor is obtained conjugating a symmetric strain tensor. The presented model provides a complete deformation pattern for granular materials by considering the decomposition for motions (displacement and rotation) of particles. Consequently, the macroscopic elastic constitutive relationships and constitutive moduli are derived in expressions of the microstructural information. Furthermore, the balance equations and boundary conditions are obtained for the presented micromorphic model. The other purpose in this study is to predict the dispersion behaviors of granular materials using the micromechanics-based micromorphic model. Five wave modes are predicted based on the presented model, including coupled transverse-rotational transverse, longitudinal, rotational longitudinal, transverse shear and rotational transverse waves. Investigating the propagations of these waves in the elastic granular media, the dispersion behaviors are predicted for coupled transverse-rotational transverse, longitudinal, rotational longitudinal waves, and the corresponding frequency band gaps are obtained.

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SUMMARYBased on the discrete particle model for solid-phase deformation of granular materials consisting of dry particulate assemblages, a discrete particle-continuum model for modelling the coupled hydro-mechanical behaviour in saturated granular materials is developed. The motion of the interstitial fluid is described by two parallel continuum schemes governed by the averaged incompressible N-S equations and Darcy's law, respectively, where the latter one can be regarded as a degraded case of the former.Owing to the merits in both Lagrangian and mesh-free characters, the characteristic-based smoothed particle hydrodynamics (SPH) method is proposed in this paper for modelling pore fluid flows relative to the deformed solid phase that is modelled as packed assemblages of interacting discrete particles. It is assumed that the formulation is Lagrangian with the co-ordinate system transferring with the movement of the solid particles. The assumed continuous fluid field is discretized into a finite set of Lagrangian (material) points with their number equal to that of solid particles situated in the computational domain. An explicit meshless scheme for granular materials with interstitial water is formulated. Numerical results illustrate the capability and performance of the present model in modelling the fluid-solid interaction and deformation in granular materials saturated with water.

Advancement of a fine slit along a planar grain boundary in an electric field E0, applied parallel to the slit, is investigated by considering electromigration along both the grain boundary and the slit surface. Electrically induced flux in the grain boundary Igb (+ toward the slit tip) and both electrically and curvature-induced fluxes on the slit surfaces are considered assuming 2Is>Igb, where Is is the flux (+ away from the slit tip) on each of the parallel slit surfaces far removed from the tip. Steady-state solutions of the transport equations are classified according to the value of a parameter β=tan−1 (2Is/Igb) which, under reasonable assumptions, depends on material parameters only. For 5π/4≥β≥β2, unique steady-state solutions exist; for β2>β>β1, multiple steady-state solutions occur; below β1≥π/4, no steady-state solution is possible. Since β1<π/2, Igb>0 (flux exiting the grain boundary into the slit) for all cases in which no steady-state solution is possible. In the case of multiple solutions, those corresponding to smallest width (and hence largest velocity) are determined. For all steady-state solutions, slit width and tip velocity scale as E−1/20 and E3/20, respectively. Results also apply to the propagation of a slit within a grain or along a passivation layer. Generally, tip velocities can approach 1 nm/s (3.6 μm/h), thereby representing a likely failure mechanism in fine-line (near bamboo structure) interconnects.