Micromechanical study of elastic moduli of three-dimensional granular assemblies

In micromechanics of granular materials, relationships are investigated between micro-scale characteristics of particles and contacts and macro-scale continuum characteristics. For three-dimensional isotropic assemblies the macro-scale elastic characteristics are described by the bulk and the shear modulus, which depend on the micro-scale characteristics of the coordination number (i.e. the average number of contacts per particle) and the interparticle contact stiffnesses in directions normal and tangential to the contact. It is well-known that the uniform-strain theory (or mean-field theory) overpredicts the elastic moduli. To find improved predictions, approximations of the particle displacement and rotations fields are obtained here by solving the equilibrium equations for small subassemblies that are centred around particles. At the boundary of these subassemblies, the particle displacements and rotations are prescribed such that they conform to the mean field. Employing this approach, improved predictions of bulk and shear moduli are obtained, in comparison with those according to the uniform-strain assumption, especially when the size is increased of the subassemblies for which equilibrium equations are solved. The elastic moduli are evaluated from the particle displacement and rotations fields by two methods. In the first, stress-based method the micromechanical expression for the average stress tensor, in terms of the forces at contacts and the branch vectors that connect particles in contacts, is employed. In the second, energy-based method the minimum potential-energy principle is used to obtain rigorous upper bounds to the moduli. It is generally observed that the moduli obtained from the stress-based method give closer agreement with the results from Discrete Element Method simulations than those from the energy-based method. These improvements in the predictions of the elastic moduli are observed over the range of coordination numbers and interparticle stiffnesses considered here