Self Consistent Estimate of the Elastic Constants of a Random Array of Equal Spheres with Application to Granular Soil under Isotropic Conditions

摘要

The need for a micromechanical approach to modeling the stress-strain response of granular soil is discussed and justified. The report focuses on the small shear strain (gamma < or = 0.01%) behavior, and investigates the validity of analytically modeling uniform, rounded-grained quartz sands by arrays of identical elastic quartz spheres. First the stress-strain properties of 6 regular arrays of spheres are studied, focusing on isotropic and transversely isotropic boundary loading. An analytical procedure is established for determining the elastic moduli of a random assemblage of equal elastic spheres of arbitrary mean porosity, subjected to isotropic confining pressure. The procedure uses the properties of the regular arrays already described, accounts for the spatial distribution of porosity, and calculates the macroscopic moduli through the self consistent method. The procedure was applied to compute the shear and bulk moduli of assemblages of quartz spheres which were then compared with static and dynamic measurements on quartz sands from the literature. The theoretical sands are significantly stiffer than actual soils due to the lower number of effective contacts in actual sands. However, excellent agreement was found with resonant column shear modulus measurements on Ottawa sand, after subjecting it to a large number of cycles of shear prestraining which increased the number of contacts toward the theoretical value.