Anisotropic elastic theory of preloaded granular media

A macroscopic elastic description of stresses in static, preloaded granular media is derived systematically from the microscopic elasticity of individual inter-grain contacts. The assumed preloaded state and friction at contacts ensure that the network of inter-grain contacts is not altered by small perturbations. The texture of this network, set by the preparation of the system, is encoded in second and fourth order fabric tensors. A small perturbation generates both normal and tangential inter-grain forces, the latter causing grains to reorient. This reorientation response and the incremental stress are expressed in terms of the macroscopic strain. The transmission of stress in granular media has a rich phenomenology [1, 2, 3], as illustrated by the emblematic sand pile problem. In a conical pile obtained by pouring grains from a point source (hopper outlet), the pressure profile at the base of the pile is not proportional to the height of the pile, as it would if the weight of each grain were transmitted strictly vertically; nor does the pressure vary monotonically from edge to center, as predicted by traditional isotropic elastic or elasto-plastic approaches. Rather, the pressure profile develops a local minimum (often termed ‘pressure dip’) at the center of the pile base, below the apex of the pile [4, 5, 6, 7]. By contrast, if a pile of the same shape is prepared layer by layer, e.g., by sprinkling from an extended source, the pressure profile does acquire a maximum at the center of the base (‘pressure hump’) [4, 5, 6, 7]. These and similar experiments indicate that the local structure of the pile (often called ‘texture’), which governs stress transmission, depends