Numerical study of one-dimensional compression of granular materials. Ⅱ. Elastic moduli, stresses, and microstructure

摘要

The elastic moduli of a transversely isotropic model granular material, made of slightly polydisperse elasticfrictionalspherical beads, in equilibrium along a one-dimensional (oedometric) compression path, as describedin the companion paper [M. H. Khalili et al., Phys. Rev. E 95, 032907 (2017)], are investigated by numericalsimulations. The relations of the five independent moduli to stresses, density, coordination number, fabric andforce anisotropies are studied for different internal material states along the oedometric loading path. It isobserved that elastic moduli, as in isotropic packs, are primarily determined by the coordination number, withanomalously small shear moduli in poorly coordinated systems, whatever their density. Such states also exhibitfaster increasing moduli in compression, and larger off-diagonal moduli and Poisson ratios. Anisotropy affectsthe longitudinal moduli C_(11) in the axial direction and C_(22) in the transverse directions, and the shear modulus inthe transverse plane C_(44), more than the shear modulus in a plane containing the axial direction C_(55_. The resultsare compared to available experiments on anisotropic bead packs, revealing, despite likely differences in internalstates, a very similar range of stiffness level (linked to coordination), and semiquantitative agreement as regardsthe influence of anisotropy. Effectivemedium theory (the Voigt approach) provides quite inaccurate predictions ofthe moduli. It also significantly underestimates ratios C_(11)/C_(22) (varying between 1 and 2.2) and C_(55)/C_(44) (varyingfrom 1 to 1.6), which characterize elastic anisotropy, except in relatively weakly anisotropic states. The bulkmodulus for isotropic compression and the compliance corresponding to stress increments proportional to theprevious stress values are the only elastic coefficients to be correctly estimated by available predictive relations.We discuss the influences of fabric and force anisotropies onto elastic anisotropy, showing in particular that theformer dominates in sample series that are directly assembled in anisotropic configurations and keep a roughlyconstant lateral to axial stress ratio under compression.