  Despite the industrial importance of dense suspensions of hard particles, few
constitutive models for them have been proposed or tested. Most of these are
effectively "fabric evolution models" (FEMs) based on a stress rule connecting
the bulk stress to a rank-2 microstructural fabric tensor Q and a closed
time-evolution equation for Q. In dense suspensions most of the stress comes
from short-ranged pairwise steric or lubrication interactions at near-contacts,
so a natural choice for Q is the deviatoric 2nd moment of the distribution P(p)
of the near-contact orientations p. Here we test directly whether a closed
time-evolution equation for such a Q can exist for inertialess non-Brownian
hard spheres in a Newtonian solvent. We perform extensive numerical simulations
for the evolution of P(p) under shear reversal, providing a stringent test for
FEMs. We consider a generic class of these as defined by Hand (1962)
constrained only by frame indifference. Motivated by the small microstructural
anisotropy in the dense regime, we start with linear models and successively
consider increasingly nonlinear ones. Based on these results we suggest that no
closed FEM properly describes the dynamics of Q under reversal. We attribute
this to the fact that Q gives a poor description of the microstructure during
large parts of the microstructural evolution following shear reversal.
Specifically, the truncation of P(p) at 2nd spherical harmonic level describes
two-lobed distributions of near-contact orientations, whereas on reversal we
observe distributions that are markedly four-lobed; moreover dP/dt (p) has
oblique axes, not collinear with those of Q in the shear plane. This structure
likely precludes any adequate closure at second-rank level. Instead, our
numerical data suggest that closures involving the coupled evolution of both a
fabric tensor and a fourth-rank tensor might be reasonably accurate.

Soft Condensed Matter;Statistical Mechanics;Fluid Dynamics