Singular behavior of the stresses in the limit of random close packing in collisional, simple shearing flows of frictionless spheres

We consider a steady, homogeneous, shearing flow of identical, rigid, frictionless, inelastic spheres. We adopt nonlinear stress relations that include anisotropy in the velocity fluctuations and, in the balance equations for the second moment of the velocity fluctuations, we incorporate velocity correlations in the expressions for the collisional production. We use the resulting solutions of the balance equations in existing expressions for the dimensionless pressure, shear stress, and the normal stress difference to determine their behavior with volume fraction. The first normal stress difference remains near zero, the pressure, shear stress and second normal stress difference increase monotonically, becoming singular at random close packing with powers 5/2, 5/2, and 7/4, respectively.

Figure 1

The variation of ${\lambda }^2$ with $\nu$ in discrete numerical simulations of frictionless spheres with $e=0.70$ and a linear stiffness, $k$, which, when made dimensionless by ${\rho }_s{\dot{\gamma }}^2\sigma$, has a value of ${10}^7$ (Vescovi, personal communication).

Figure 2

Dimensionless pressure as a function of distance to random close packing: theoretical (lines) and numerical (measured in the discrete numerical simulations of Mitarai and Nakanishi (Ref. [7], circles); Vescovi and Luding (Ref. [12], squares); Chialvo and Sundaresan (Ref. [13], diamonds); Shen (personal communication, triangles). Colors indicate different coefficient of restitution: $e=0.70$ (black); $e=0.80$ (red); $e=0.90$ (blue); $e=0.92$ (cyan); $e=0.95$ (green); $e=0.98$ (magenta); $e=0.99$ (purple).

Figure 3

Same as in Fig. 2, but for the dimensionless shear stress.

Figure 4

Same as in Fig. 2, but for the dimensionless second normal stress difference.

Figure 5

Stress ratio (a) and second normal stress difference (b) as functions of volume fraction. Same legend as in Fig. 2.