Using Mindlin theory to model friction-dependent shear modulus in granular media

An explicit expression for the effective shear modulus of a random packing of identical spheres is derived as a function of Mindlin's tangential stiffness with interparticle contact friction. The motivation behind the approach is to incorporate the effect of intergrain friction to predict velocities in unconsolidated sands. The Mindlin friction term, allowing partial slip across the contact area between pairs of spheres, can be viewed as a parameter accounting for the growing macroscopic intergrain friction in sands as burial progresses. Hence, both moduli and velocities will gradually increase as the compressional- to shear-wave velocity ratio (V-P/V-S) or Poisson's ratio (nu) decreases. An estimate of effective elastic constants in particular shear modulus can be obtained for a spherical grain pack with an arbitrary frictional behavior ranging between two special contact boundary conditions representing infinite friction and zero friction. The proposed model predicts anonlinear transition between the two special grain-contact conditions when compared to previously published linear relationships. Comparison of elastic properties, i.e., dynamic shear-modulus predictions assuming zero contact friction with experimental data on loose glass bead and sand samples undergoing hydrostatic compression, appears to match reasonably well at low confining stress (less than 5 MPa) but deviates gradually as stress increases. It is advocated that the increasing effective internal frictional resistance of the experimental core samples control both the frictional attenuation mechanism in loose grain packs under low confining stress for strain amplitudes typical of seismic waves (less than 10(-6)) and the higher stress-velocity sensitivity. Circumstantial evidence of this is found in publications describing both laboratory attenuation analysis and consolidation experiments on granular materials with different degrees of competence or static shear strength.