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Review Articles

Coulomb–Mohr Granular Materials: Quasi-static Flows and the Highly Frictional Limit

[+] Author and Article Information
Grant M. Cox, Ngamta Thamwattana, James M. Hill

School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia

Scott W. McCue

School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia

Appl. Mech. Rev 61(6), 060802 (Oct 07, 2008) (23 pages) doi:10.1115/1.2987874 History: Received February 07, 2008; Revised February 17, 2008; Published October 07, 2008

One approach to modeling fully developed shear flow of frictional granular materials is to use a yield condition and a flow rule, in an analogous way to that commonly employed in the fields of metal plasticity and soil mechanics. Typically, the yield condition of choice for granular materials is the Coulomb–Mohr criterion, as this constraint is relatively simple to apply but at the same time is also known to predict stresses that are in good agreement with experimental observations. On the other hand, there is no strong agreement within the engineering and applied mechanics community as to which flow rule is most appropriate, and this subject is still very much open to debate. This paper provides a review of the governing equations used to describe the flow of granular materials subject to the Coulomb–Mohr yield condition, concentrating on the coaxial and double-shearing flow rules in both plane strain and axially symmetric geometries. Emphasis is given to highly frictional materials, which are defined as those granular materials that possess angles of internal friction whose trigonometric sine is close in value to unity. Furthermore, a discussion is provided on the practical problems of determining the stress and velocity distributions in a gravity flow hopper, as well as the stress fields beneath a standing stockpile and within a stable rat-hole.

Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of a converging hopper

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Figure 2

Schematic of (a) mass-flow and (b) funnel-flow in a silo

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Figure 3

Numerical solutions for ψ and U/U¯ of the double-shearing theory with respect to θ, where ϕ=π/3 (—), π/4 (– – –), and π/6(⋯), for gravity flow of a granular material through a wedge hopper

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Figure 4

Numerical solutions for ψ and U/U¯ of the double-shearing theory with respect to θ, where ϕ=π/3 (—), π/4 (– – –), and π/6(⋯), for gravity flow of a granular material through a conical hopper

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Figure 23

Variation of stresses with y/x2 for a two-dimensional plane strain parabolic curve shaped rat-hole

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Figure 24

Variation of stresses with z/r3 for a axially symmetric cubic curve shaped rat-hole

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Figure 11

Coordinates for an axially symmetric hopper with a cone-in-cone insert

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Figure 10

Comparison between the numerical (- - -), the leading order (—), and the perturbation (⋯) solutions for (a) ψ, (b) F=q/ρgr, and (c) U/U¯, where ϕ=π/4, for gravity flow of a granular material through a wedge hopper with a wedge-in-wedge insert

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Figure 9

Coordinates for (a) a two-dimensional asymmetrical wedge hopper and (b) a two-dimensional hopper with a wedge-in-wedge insert

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Figure 8

Comparison between the numerical (⋯) and the perturbation (- - -) solutions for (a) ψ, (b) G=q/ρgr, and (c) U/U¯ where ϕ=π/4, for gravity flow of a granular material through a conical hopper

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Figure 7

Variation of ψ and U/U¯ with respect to θ according to the exact solution for ϕ=π/2 (—) and the numerical solution, where ϕ=π/3(⋯), for gravity flow of a granular material through a conical hopper

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Figure 6

Comparison between the numerical (⋯) and the perturbation (- - -) solutions for (a) ψ, (b) F=q/ρgr, and (c) U/U¯, where ϕ=π/4, for gravity flow of a granular material through a wedge hopper

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Figure 5

Variation of ψ and U/U¯ with respect to θ according to the exact solution for ϕ=π/2 (—) and the numerical solution, where ϕ=π/3(⋯), for gravity flow of a granular material through a wedge hopper

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Figure 16

The theoretical normal and vertical stress profiles at the base of a stockpile for two-dimensional plane strain and axially symmetric geometries (124): ((a) and (b)) plane strain and ((c) and (d)) axially symmetric

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Figure 15

Theoretical predictions showing a dip in the normal pressure beneath a stockpile according to Didwania (123)

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Figure 14

The normal force at the base of a numerically simulated stockpile for two different distributions, see Baxter (112). A dip in the pressure is found for one of the distributions.

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Figure 13

The experimental data of Smid and Novosad (101) showing the existence of a dip in the pressure at the base of a stockpile: (a) vertical and (b) horizontal pressure distributions

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Figure 12

Comparison between the numerical (- - -), the leading order (—), and the perturbation (⋯) solutions for (a) ψ, (b) G=q/ρgr, and (c) U/U¯, where ϕ=π/4, for gravity flow of a granular material through a conical hopper with a cone-in-cone insert

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Figure 22

(a) Coordinates for an axially symmetric sloping rat-hole resting on a sloping rigid base and (b) coordinates for an axially symmetric cubic curve shaped rat-hole

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Figure 21

(a) Coordinates for a two-dimensional sloping rat-hole resting on a sloping rigid plane and (b) coordinates for a two-dimensional parabolic curve shaped rat-hole

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Figure 20

Variation of stresses along the base of an axially symmetric cubic curve shaped standing stockpile: (a) the horizontal stress σr=σrz and (b) the vertical stress σz=σzz

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Figure 19

Variation of stresses along the base of a two-dimensional wedge-shaped stockpile: (a) the horizontal stress σx=σxy and (b) the vertical stress σy=σyy

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Figure 18

Schematic of axially symmetric standing stockpiles, comprising of an inner rigid region and an outer yield region: (a) conical and (b) cubic curve shaped stockpiles

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Figure 17

Schematic of axially symmetric standing stockpiles, comprising of an inner rigid region and an outer yield region: (a) wedge-shaped and (b) parabolic curve shaped stockpiles

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