Estimates of elastic moduli for granular material with anisotropic random packing structure

For randomly packed spheres, the packing structure can be characterized by a fabric tensor representing the orientation distribution of inter-particle contacts. The densities of inter-particle contacts are generally not the same for all inter-particle orientations, exhibiting the anisotropy of a packing structure. Utilizing the fabric tensor, elastic moduli of granular material with anisotropic packing structures are derived in explicit terms of inter-particle properties. The derivation is based on two different methods, namely the kinematic method and the static method. Results derived from kinematic and static methods provide, respectively, the upper and lower estimates of elastic moduli. Ranges of elastic moduli are discussed for granular materials with different inter-particle properties and fabric parameters. The close relationship between fabric tensor and material symmetry is demonstrated.