Chapter
1.5. Balance Laws for Continuum Models
1.5.2. Integral and Differential Balances
1.5.4. The Stress Vector and the Stress Tensor
1.5.5. The Linear Momentum Balance
1.5.6. The Angular Momentum Balance
1.5.7. The Energy Balance
1.7. Fluid--Particle Interaction
1.1. An estimate of the height of a heap
1.2. Fisher’s toroidal approximation for a liquid bridge
1.3. The variation of the drained angle of repose with the Bond number
1.4. The electrostatic force between two particles
1.5. The angular momentum balance for a rigid body
1.6. The divergence theorem
1.7. The generalized transport theorem and the transport theorem
1.8. Symmetry of the stress tensor for a classical continuum
1.9. The angle of stability of a heap
1.10. The translation of a sphere in a fluid, in the limit of creeping flow
1.11. A discrete model for the stress dip at the base of a heap
1.12. The dynamics of a spring–mass–dashpot system
1.13. The force exerted on a load cell during the emptying of a hopper
1.14. Derivation of the differential mass and momentum balances from the integral balances
1.16. An application of the local volume-averaged equations
2 Theory for Slow Plane Flow
2.1. Qualitative Observations
2.2. The Wall Yield Condition
2.3. The Janssen Solution for the Static Stress Field in a Bin
2.4. The Coulomb Yield Condition
2.5. Generalization of the Coulomb Yield Condition
2.5.1. Invariants and Principal Stresses
2.6. The Mohr–Coulomb Yield Condition
2.7. The Mohr's Circle for the Two-Dimensional Stress Tensor
2.8. The Relation Between the Coulomb and Mohr–Coulomb Yield Conditions
2.9. Active and Passive States of Stress and the Value of the Janssen K-Factor
2.10.1. The Critical State
2.10.2. The Hvorslev Surface
2.10.3. The Roscoe Surface
2.10.4. The Yield Surface
2.11. Yield Surfaces in sigma1-sigma2-nu space
2.12. Yield Loci in the sigma1-sigma2 and sigma-tau Planes
2.13.1. The Lévy–Mises and the Prandtl–Reuss Equations
2.13.2. The Coaxiality Condition
2.13.3. The Plastic Potential
2.13.4. Positive Dissipation
2.13.5. Associated and Nonassociated Flow Rules
2.14. Equations for Plane Flow
2.14.1. The Mohr's Circle for the Rate of Deformation Tensor
2.14.2. The Coaxiality Condition
2.14.4. Implications of the Associated Flow Rule
2.14.5. Rowe's Stress–Dilatancy Relation
2.14.6. Summary of the Governing Equations for Plane Flow
2.15. The Relation Between Yield Loci in the N-T and sigma-tau Planes
2.16. The Double-Shearing Model
2.1. Behavior of wet beach sand
2.2. Forces acting on a block
2.3. Forces acting on an hourglass immersed in water
2.4. Stresses exerted on the wall of a hopper
2.5. A block on an inclined plane
2.6. An application of the Coulomb yield condition to the stability of a slope
2.7. Eigenvalues and eigenvectors of a symmetric second-order tensor in two dimensions
2.8. Invariance of the eigenvalues and the principal invariants under rotation of coordinate axes
2.9. Determination of the principal stresses and the principal stress axes
2.10. The orientation of the major principal stress axis at the wall of a hopper
2.12. Interpretation of the components of the rate of deformation tensor
2.13. Value of the angle of dilation νd for incompressible plane flow
2.14. A yield condition and the associated flow rule
2.15. Relation between yield loci in the N–T and sigma–tau planes
2.17. Arching in a conical hopper
2.18. Rotation of a cylinder of sand
2.20. An application of the double-shearing model
3.1. Experimental Observations
3.1.3. Solids Fraction Profiles
3.2. Theory for Steady, Plane Flow
3.2.1. The Critical State Approximation
3.3. The Smooth Wall, Radial Gravity (SWRG) Problem
3.4. The Effect of Wall Roughness
3.5. Solutions with Allowance for Rough Walls and Vertical Gravity
3.5.1. The Brennen–Pearce Solution
3.5.2. The Radial Stress and Velocity Fields
3.5.3. Linearized Stability Analysis
3.5.4. Downward Integration from the Radial Fields
3.5.5. The Successive Approximation Procedure
3.6. A Re-Examination of the Exit Condition
3.7. An Alternative Exit Condition
3.8. The Smooth Wall, Radial Gravity Problem for Compressible Flow
3.1. Stresses exerted on the wall of a hopper
3.2. Profile of a slip plane in a hopper
3.3. Use of correlations to predict the mass flow rate
3.4. Discharge of sand from a hopper
3.5. Stability of the radial stress field: smooth wall analysis
3.6. Yield locus for Leighton Buzzard sand
3.7. Jump conditions across the exit shock
3.8. Solids fraction profile for vertical free-fall below the hopper
3.9. Asymptotic fields for compressible smooth wall, radial gravity problem
4 Flow through Wedge-Shaped Bunkers
4.1. Experimental Observations
4.2. Models for Bunker Flow
4.2.2. The Transition Region
4.2.3. The Hopper Section
4.1. The Janssen–Walker solution for the stress field in a bin
4.2. The Janssen solution for the stress field in a bin of arbitrary cross section
4.3. Alternative derivation of the Rankine solution
5 Theory for Slow Three-Dimensional Flow
5.1. Constitutive Equations Involving a Yield Condition
5.1.1. The Yield Condition
5.1.2. Symmetry Considerations
5.1.3. Conventional Triaxial Tests
5.1.4. Isotropic Compression Tests
5.1.6. Cubical Triaxial Tests
5.1.7. Comparison of Yield Conditions with Data
5.1.9. Data Related to Flow Rules
5.1.10. Steady, Fully Developed Flow of a Rigid-Plastic Material
5.1.11. One-Dimensional Deformation of a Rigid-Plastic Material
5.2. Constitutive Equations That Do Not Involve a Yield Condition
5.2.1. Hypoelastic and Hypoplastic Models
5.2.2. Some Features of (5.66)
5.2.3. Steady, Fully Developed Flow of a Hypoelastic Material
5.2.4. One-Dimensional Deformation of a Hypoelastic Material
5.1. Expressions for the distances measured along and perpendicular to the space diagonal
5.2. A feature of the yield locus for Granta-gravel
5.3. Cross sections corresponding to the Tresca and Mohr–Coulomb yield surfaces
5.4. An alternative formulation of the associated flow rule for the extended von Mises yield condition
5.5. Form of the Drucker–Prager yield condition for plane flow
5.6. Relation between the local angular velocity of the material and the vorticity tensor
5.7. Derivation of the plastic potential flow rule for the deformation of metals Consider a flow rule of the form
5.8. Representation of the constitutive equation for an isotropic tensor valued function of a second order tensor and a scalar
5.9. The critical state line for modified Cam Clay
5.10. Expressions for the components of the stress tensor in terms of ν for onedimensional deformation
5.11. Determination of the yield locus for plane flow from the three-dimensional yield surface
5.12. Elastic solution for a spherical shell
5.13. Estimate of the coefficient of earth pressure using elasticity theory
5.14. Shearing of a granular material in a cylindrical Couette cell
5.15. Forms for the functions in the hypoelastic model of Davis and Mullenger (1978)
6 Flow through Axisymmetric Hoppers and Bunkers
6.1. Experimental Observations
6.2. Theory for Steady, Axisymmetric Flow Through a Hopper
6.2.1. The Haar–von Karman Hypothesis
6.2.2. The Radial Stress and Velocity Fields for the Mohr–Coulomb Yield Condition and the Haar–von Karman Hypothesis
6.2.3. The Drucker–Prager Yield Condition and Levy's Flow Rule
6.2.4. Comparison of Predicted and Measured Velocity Profiles
6.2.5. Criteria for Mass Flow
6.3. A Hybrid Hypoplastic-Viscous Model
6.4. The Kinematic Model for Batch Discharge from a Bin
6.1. Profile of the free surface for the batch discharge of material from a bin
6.2. A feature of the incompressible radial velocity field
6.3. Derivation of the solution to the diffusion equation
7 Theory for Rapid Flow of Smooth, Inelastic Particles
7.1. Preliminaries and Scaling
7.1.1. Model for Inelastic Collisions
7.1.2. Hydrodynamic Description of Rapid Granular Flows
7.2. Heuristic Hydrodynamic Theory for High-Density Flows
7.2.1. Application to Uniform Plane Shear
7.3. Kinetic Theory for a Granular Gas of SmoothInelastic Particles
7.3.1. Statistical Preliminaries
7.3.2. The Evolution of f(1)
7.3.3. The Equilibrium Distribution Function
7.3.4. The Departure from Equilibrium
7.3.5. Maxwell Transport Equation
7.3.6. The Equations of Motion
7.3.7. The Chapman–Enskog Expansion
7.3.8. Constitutive Relations at Leading Order
7.3.9. Distribution Function at Leeps (K)
7.3.11. Constitutive Relations at Leeps (K)
7.3.12. Distribution Function and Constitutive Relations at Leeps ()
7.3.13. Constitutive Relations to First Order in K and
7.4. Anisotropy of the Microstructure
7.5. Extension to Granular Mixtures
7.6. Summary and Discussion
8 Analysis of Rapid Flow in Simple Geometries
8.1. Boundary Conditions at Solid Walls
8.2.1. Predictions of the High-Density Theory
8.2.2. Some Features of the High-Density Solutions
8.2.3. Predictions of the Kinetic Theory
8.3. Flow in Inclined Chutes
8.3.1. Some Experimental Observations of Chute Flow
8.3.2. Analysis of Steady, Fully Developed Flow
8.3.3. High-Density Theory
8.3.4. Some Features of the High-Density Solutions
8.3.5. Predictions of the Kinetic Theory
8.4. Stability of Rapid Shear Flows
8.4.1. Stability of Unbounded Plane Shear Flow
8.4.2. Stability of Plane Couette Flow
9 Theory for Rapid Flow of Rough,Inelastic Particles
9.1. Collision Models for Rough Particles
9.2. Equations of Motion for a Granular Gas of Rough, Inelastic Spheres
9.3. The Velocity Distribution Function
9.3.1. Nearly Elastic, Nearly Perfectly Rough Particles
9.3.2. Nearly Elastic, Nearly Smooth Particles
9.4. Constitutive Relations up to First Order in K, , and
9.4.1. Nearly Elastic, Nearly Perfectly Rough Particles
9.4.2. Nearly Elastic, Nearly Smooth Particles
10.1. The Frictional-Kinetic Model
10.2. Application to Flow in Chutes
10.3. Other Hybrid Models
Appendix A Operations with Vectors and Tensors
A.2. The Summation Convention
A.3. The Scalar Product of Two Vectors
A.4. Second-Order Tensors
A.4.2. The Trace of a Second-Order Tensor
A.5. Cartesian Tensor Notation
A.6. Third and Higher Order Tensors
A.6.1. The Alternating Tensor
A.7. Operations with Vectors and Tensors
A.7.1. The Vector Product of Two Vectors
A.7.2. The Product of Two Second-Order Tensors
A.7.3. The Scalar Product of Two Second-Order Tensors
A.7.4. The Transpose of a Tensor
A.7.5. The Inverse of a Second-Order Tensor
A.7.6. The Determinant of a Second-Order Tensor
A.7.7. Orthogonal Second-Order Tensors
A.7.8. The Gradient Operator
A.7.9. The Gradient of Scalars and Vectors
A.7.10. The Divergence of a Second-Order Tensor
A.7.11. The Curl of a Vector
A.8. Equations in Orthogonal Curvilinear Coordinate Systems
A.8.1. Cylindrical Coordinates
A.8.2. Spherical Coordinates
A.1. Determinant of the product of two second-order tensors
A.2. Identities involving the vector triple product
A.3. An identity involving the premutation symbol
A.4. The inverse of a second-order tensor
Appendix B The Stress Tensor
Appendix C Hyperbolic Partial Differential Equations of First Order
C.1. Solution by the Method of Characteristics
C.1. The variation of γ along a traction-free curve
C.2. Stress characteristics for a material with curved yield loci
C.3. Characteristics of the velocity equations for plane flow
D.1 The Jump Mass Balance
D.1. The jump momentum balance
Appendix E Discontinuous Solutions of Hyperbolic Equations
E.3. Jump Conditions for Linear Equations
E.1. The inclination of the velocity jump across a velocity characteristic
Appendix F Proof of the Coaxiality Condition
F.1. Relations to be satisfied by the {Qij}
F.2. Examples of coordinate transformations
Appendix G Material Frame Indifference
G.2. Frame Indifferent Scalars, Vectors, and Tensors
G.2.3. Second-Order Tensors
G.3. The Principle of Material Frame Indifference
G.4. An Alternative Interpretation of a Change of Frame
G.1. Transformation of some tensors under a change of frame
G.2. Transformation of the acceleration under a change of frame
G.3. The angular velocity of the unprimed frame relative to the primed frame
G.4. Transformation of the body force under a change of frame
G.5. Transformation of the components of the rate of deformation and vorticity tensors under a particular change of frame
G.6. Proof that the density is a frame-indifferent scalar
Appendix H The Evaluation of Some Integrals
H.2. Integration Over k for Boundary Conditions
Appenidx I A Brief Introduction to Linear Stability Theory
Appendix J Pseudo Scalars, Vectors, and Tensors
Appendix K Answers to Selected Problems
An Introduction to Granular Flow
( Cambridge Series in Chemical Engineering )
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