^{1,a)}, C. Wassgren

^{2}, B. Hancock

^{3}, W. Ketterhagen

^{3}and J. Curtis

^{1}

### Abstract

Granular shear flows of flat disks and elongated rods are simulated using the Discrete Element Method. The effects of particle shape, interparticle friction, coefficient of restitution, and Young's modulus on the flow behavior and solid phase stresses have been investigated. Without friction, the stresses decrease as the particles become flatter or more elongated due to the effect of particle shape on the motion and interaction of particles. In dense flows, the particles tend to have their largest dimension aligned in the flow direction and their smallest dimension aligned in the velocity gradient direction, such that the contacts between the particles are reduced. The particle alignment is more significant for flatter disks and more elongated rods. The interparticle friction has a crucial impact on the flow pattern, particle alignment, and stress. Unlike in the smooth layer flows with frictionless particles, frictional particles are entangled into large masses which rotate like solid bodies under shear. In dense flows with friction, a sharp stress increase is observed with a small increase in the solid volume fraction, and a space-spanning network of force chains is rapidly formed with the increase in stress. The stress surge can occur at a lower solid volume fraction for the flatter and more elongated particles. The particle Young's modulus has a negligible effect on dilute and moderately dense flows. However, in dense flows where the space-spanning network of force chains is formed, the stress depends strongly on the particle Young's modulus. In shear flows of non-spherical particles, the stress tensor is found to be symmetric, but anisotropic with the normal component in the flow direction greater than the other two normal components. The granular temperature for the non-spherical particle systems consists of translational and rotational temperatures. The translational temperature is not equally partitioned in the three directions with the component in the flow direction greater than the other two. The rotational temperature is less than the translational temperature at low solid volume fractions, but may become greater than the translational temperature at high solid volume fractions.

This research is based on the funding from NSF-CBET Grant No. 0854005, NASA-STTR Phase II Program, and the State of Florida Space Research Initiative. The authors also acknowledge the University of Florida High-Performance Computing Center for providing computational resources.

I. INTRODUCTION

II. METHODOLOGY

A. Discrete element method

B. Numerical model of shear flow

III. RESULTS AND DISCUSSION

A. Stresses in shear flows of disks and rods

B. Particle alignment

C. Contact types

D. Effect of coefficient of restitution

E. Effect of interparticle friction

F. Effect of particle Young's modulus

G. Anisotropy and symmetry of the stress tensors

H. Non-equipartition of granular temperature

IV. CONCLUSIONS

### Key Topics

- Friction
- 81.0
- Granular flow
- 50.0
- Shear flows
- 43.0
- Elastic moduli
- 28.0
- Granular solids
- 20.0

## Figures

Numerical models of granular shear flows with (a) flat disks and (b) elongated rods at the solid volume fraction of 0.1. The dimensions of cylindrical particles are illustrated in (c).

Numerical models of granular shear flows with (a) flat disks and (b) elongated rods at the solid volume fraction of 0.1. The dimensions of cylindrical particles are illustrated in (c).

Normalized shear stress as a function of solid volume fraction for various particle aspect ratios (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1). The kinetic theory curve is for frictionless spheres.

Normalized shear stress as a function of solid volume fraction for various particle aspect ratios (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1). The kinetic theory curve is for frictionless spheres.

Normalized shear stresses as a function of the maximum value of L/d f and d f/L at different solid volume fractions (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Normalized shear stresses as a function of the maximum value of L/d f and d f/L at different solid volume fractions (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Apparent friction coefficient as a function of solid volume fraction for various aspect ratios (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1). The kinetic theory curve is for frictionless spheres.

Apparent friction coefficient as a function of solid volume fraction for various aspect ratios (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1). The kinetic theory curve is for frictionless spheres.

Particle alignment during shear flows with (a) disks and (b) elongated rods at a solid volume fraction of 0.5 (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Particle alignment during shear flows with (a) disks and (b) elongated rods at a solid volume fraction of 0.5 (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

(a) Description of particle orientation using the angles α and β, (b) probability density distributions of the particle inclination angle α, and (c) probability density distributions of the particle azimuthal angle β for the particles of various aspect ratios at the solid volume fraction of 0.5 (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

(a) Description of particle orientation using the angles α and β, (b) probability density distributions of the particle inclination angle α, and (c) probability density distributions of the particle azimuthal angle β for the particles of various aspect ratios at the solid volume fraction of 0.5 (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Variation of order parameter S with the maximum dimensional ratio of L/d f and d f /L for dense flows at the solid volume fraction of ν = 0.5 (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Variation of order parameter S with the maximum dimensional ratio of L/d f and d f /L for dense flows at the solid volume fraction of ν = 0.5 (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Six types of cylinder-cylinder contacts.

Six types of cylinder-cylinder contacts.

Number percentage of contacts of a specific type as a function of solid volume fraction for (a) AR = 0.1, (b) AR = 1, and (c) AR = 6 (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Number percentage of contacts of a specific type as a function of solid volume fraction for (a) AR = 0.1, (b) AR = 1, and (c) AR = 6 (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Number percentage of contacts of a specific type as a function of particle aspect ratio (AR). The solid volume fraction is fixed at 0.6 for various aspect ratio particles (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Number percentage of contacts of a specific type as a function of particle aspect ratio (AR). The solid volume fraction is fixed at 0.6 for various aspect ratio particles (μ = 0, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Normalized shear stress as a function of solid volume fraction for the cylindrical particles with various coefficients of restitution (AR = 4, μ = 0.0, E = 8.7 × 109 Pa, γ = 100 s−1).

Normalized shear stress as a function of solid volume fraction for the cylindrical particles with various coefficients of restitution (AR = 4, μ = 0.0, E = 8.7 × 109 Pa, γ = 100 s−1).

Apparent friction coefficient as a function of solid volume fraction for cylindrical particles with various coefficients of restitution (AR = 4, μ = 0.0, E = 8.7 × 109 Pa, γ = 100 s−1).

Apparent friction coefficient as a function of solid volume fraction for cylindrical particles with various coefficients of restitution (AR = 4, μ = 0.0, E = 8.7 × 109 Pa, γ = 100 s−1).

Variation of normalized shear stress with solid volume fraction for particles with and without friction. The surrounding images show snapshots of force chains at the specified solid volume fractions. The force chains are plotted by connecting the centers of two particles in contact. The thicknesses of the lines, scaled by the current maximum contact force, indicate the magnitudes of the contact forces. The inserted diagram shows the variation of time-averaged coordination number with solid volume fraction for particles with and without friction (AR = 4, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Variation of normalized shear stress with solid volume fraction for particles with and without friction. The surrounding images show snapshots of force chains at the specified solid volume fractions. The force chains are plotted by connecting the centers of two particles in contact. The thicknesses of the lines, scaled by the current maximum contact force, indicate the magnitudes of the contact forces. The inserted diagram shows the variation of time-averaged coordination number with solid volume fraction for particles with and without friction (AR = 4, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Snapshot of particle velocity vectors for a dense shear flow of frictionless particles at the solid volume fraction of ν = 0.5 (μ = 0.0, AR = 4, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Snapshot of particle velocity vectors for a dense shear flow of frictionless particles at the solid volume fraction of ν = 0.5 (μ = 0.0, AR = 4, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Three snapshots of above-average-force chains at different dimensionless time instants (γ · t) are shown in the top row (a)-(c) and the corresponding snapshots of particle velocity vectors are shown in the bottom row (d)-(f) for the dense shear flow with frictional particles at the solid volume fraction of ν = 0.5 (μ = 0.5, AR = 4, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Three snapshots of above-average-force chains at different dimensionless time instants (γ · t) are shown in the top row (a)-(c) and the corresponding snapshots of particle velocity vectors are shown in the bottom row (d)-(f) for the dense shear flow with frictional particles at the solid volume fraction of ν = 0.5 (μ = 0.5, AR = 4, e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Normalized shear stress as a function of solid volume fraction for (a) flat disks (AR < 1) and (b) rods (AR ≥ 1) with different interparticle friction coefficients (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Normalized shear stress as a function of solid volume fraction for (a) flat disks (AR < 1) and (b) rods (AR ≥ 1) with different interparticle friction coefficients (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Apparent friction coefficient as a function of solid volume fraction for (a) flat disks (AR < 1) and (b) rods (AR ≥ 1) with different interparticle friction coefficients (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Apparent friction coefficient as a function of solid volume fraction for (a) flat disks (AR < 1) and (b) rods (AR ≥ 1) with different interparticle friction coefficients (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Order parameter S as a function of solid volume fraction for the flows with and without interparticle friction (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Order parameter S as a function of solid volume fraction for the flows with and without interparticle friction (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Normalized shear stress as a function of solid volume fraction for rods with various normalized Young's moduli (e = 0.95, γ = 100 s−1).

Normalized shear stress as a function of solid volume fraction for rods with various normalized Young's moduli (e = 0.95, γ = 100 s−1).

Variation of apparent friction coefficient with the solid volume fraction for rods with various normalized Young's moduli (e = 0.95, γ = 100 s−1).

Variation of apparent friction coefficient with the solid volume fraction for rods with various normalized Young's moduli (e = 0.95, γ = 100 s−1).

Comparison of the average scaled overlaps and average dimensionless normal forces for particles with two different Young's moduli (e = 0.95, γ = 100 s−1).

Comparison of the average scaled overlaps and average dimensionless normal forces for particles with two different Young's moduli (e = 0.95, γ = 100 s−1).

Normalized shear stress as a function of normalized Young's modulus for dense, frictional flows with various particle aspect ratios (μ = 0.5, e = 0.95). Quasi-static flows at a low shear rate of γ = 1 s−1 are also considered for the rods of AR = 4 at the solid volume fraction of ν = 0.6.

Normalized shear stress as a function of normalized Young's modulus for dense, frictional flows with various particle aspect ratios (μ = 0.5, e = 0.95). Quasi-static flows at a low shear rate of γ = 1 s−1 are also considered for the rods of AR = 4 at the solid volume fraction of ν = 0.6.

Variation of the apparent friction coefficient with the normalized Young's modulus for dense frictional flows (μ = 0.5, e = 0.95, γ = 100 s−1).

Variation of the apparent friction coefficient with the normalized Young's modulus for dense frictional flows (μ = 0.5, e = 0.95, γ = 100 s−1).

Three normal stress components as a function of solid volume fraction for shear flows of flat disks (AR = 0.1) and elongated rods (AR = 6) (a) without and (b) with friction of μ = 0.5 (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Three normal stress components as a function of solid volume fraction for shear flows of flat disks (AR = 0.1) and elongated rods (AR = 6) (a) without and (b) with friction of μ = 0.5 (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Comparison of three temperature components, normalized by their average value, for shear flows of flat disks (AR = 0.1) and elongated rods (AR = 6) (a) without and (b) with friction of μ = 0.5 (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Comparison of three temperature components, normalized by their average value, for shear flows of flat disks (AR = 0.1) and elongated rods (AR = 6) (a) without and (b) with friction of μ = 0.5 (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Normalized rotational and translational temperatures as a function of solid volume fraction for shear flows of flat disks (AR = 0.1) and elongated rods (AR = 6) with μ = 0.5 (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

Normalized rotational and translational temperatures as a function of solid volume fraction for shear flows of flat disks (AR = 0.1) and elongated rods (AR = 6) with μ = 0.5 (e = 0.95, E = 8.7 × 109 Pa, γ = 100 s−1).

## Tables

Particle properties and simulation parameters.

Particle properties and simulation parameters.

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