From particle simulations to macroscopic constitutive relations

The goal is to determine the constitutive behavior of granular packings under various deformations (isotropic and anisotropic) from particle simulations. For this we consider deformations, stress, structure and the contact forces as the basis. In a previous study [6,7] we investigated using DEM, the evolution of the coordination number (and the packing structure) and pressure as functions of the volume fraction for a polydisperse granular packing of spheres under isotropic compression. Here we focus on anisotropic deformation by implementing the triaxial test setup in a similar way. We study the effect of polydispersity changing the width of the particle size distribution. We find that an increase in polydispersity leads to a decrease in pressure at constant volume fraction whereas the macroscopic friction angle seems to increase with polydispersity. Furthermore, we performed triaxial test simulations with soft friction which is characterized by a small tangential contact stiffness. Our main observation is that using the same initial packing configuration with different friction coefficients does not lead to an obvious trend in simulation results. Introduction Granular matter is widely considered as a model material to understand more complex behaviour [1]. For example the concept of jamming applicable to a broad class of materials like glasses, molecular liquids or colloids is often analyzed numerically, using model systems of hard and soft spheres [2, 3]. When the size of the systems under consideration is relatively small individual particles can be tracked explicitly. However, this “microscopic” method becomes unmanageable for large scale real life applications where billions of grains are involved. Hence, macroscopic constitutive models [4] are required to relate basic mechanical properties such as stress and strain. The main drawback of the macroscopic approach is its empirical nature, which neglects microscopic particle properties and lacks physical ground, e.g., on the contact level [5]. The goal of this study is to understand the effect of polydispersity and soft interparticle friction on the macroscopic behaviour of granular materials under mechanical loading. Using the Discrete Element Method, in a previous study, isotropic deformations were studied in detail [6, 7]. Here we simulate the deformation of assemblies of particles in a triaxial test. Simulation Setup The assemblies consist of 9261 spherical particles initially enclosed in a cubic volume with periodic boundary conditions. The particle radius distribution function varies uniformly between rmin and rmax from which we define the polydispersity w=rmax/rmin. The initial packing is obtained from a dilute random granular gas (with volume fraction ν =0.3) via isotropic compression up to ν =0.7 and relaxation at this constant volume fraction. This configuration is then used as the starting point for further deformations. Particles interact through the simple repulsive linear spring dashpot normal contact force law and Coulomb type friction involving a (rather soft) tangential spring. In addition to the viscous damping at the contacts, (artificial) background dissipation is introduced to accelerate relaxation. For details on the contact models see [5]. Numerical values of the parameters used in the simulations are as follows: density ρ = 2000 kg/m, average particle radius rav = 1 mm, spring constant kn = 10 kg/s, particle damping γp= 1 kg/s, background dissipation γb= 0.1 kg/s, total time of simulation T = 5 ms for one cycle of loading. These material parameters lead to the contact duration tc = 0.64 μs and a coefficient of restitution e = 0.92. The first quantity, when related to T indicates that the simulations are moderately † Email: [email protected]