The Physics of the Deformation of Densely Packed Granular Materials

The Physics of the Deformation of Densely Packed Granular Materials
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This book is of interest for those that are concerned professionally with granular materials: civil engineers, geologists and geophysicists, chemical engineers, pharmacists, food technologists, agriculturalists, biologists and astronomers. Granular materials play a role in nearly all human activities. For example, users of sand, from children in sandpits to sophisticated geotechnical engineers, know that it is a fascinating — and to some extent, unpredictable — material. In addition to sand, which itself may be of many compositions, there are various types of materials including gravel, fine-particle aggregates as employed in cosmetics, pharmaceuticals, dust, crushed rock and granules that occur in a domestic environment, such as breakfast cereals, sugar, salt and (instant or ground) coffee granules. The aim of the book is to present a theory that explains the physics behind the phenomena during the deformation of densely packed granular media. The physics that describes such features is rather subtle and is developed from the micro to macro level (the latter is the continuum mechanics level that is used in practical applications). It requires the analysis of anisotropy and the heterogeneity of the packing evaluated against the background of a frictional inter-particle interaction. Contents: PrefaceAbout the AuthorGeneral ConceptsContinuum Mechanics and Cartesian Tensor CalculusThe Bounds of Static EquilibriumHeterogeneityFabric DescriptionStress-Strain Relations of Granular Assemblies: A Frictionless AssemblyStress-Strain Relations of Granular Assemblies: Normal and Tangential InteractionsFrictional Granular MaterialsAppendix A: Mathematical AppendixAppendix B: List of Symbols and NotationsIndex Readership: Mathematical physicists, civil engineers (especially soil mechanics), chemical engineers with a special interest in cake formation, geologists; applied mathematicians.Granular Mechanics;Dilatancy;Rupture Layer Formation;Continuum Mechanics;Connected Media;Heterogeneity;Static Friction;Contact Mechanics;Civil Engineering;Soil Mechanics;Geology;Chemical Engineering;Cake Formation;Stress-Strain;Mechanical Properties of Materials;Deformation0 Key Features: Targeted as postgraduate textThere is currently no other volume in which the various concepts that are needed in a consistent analytical form are drawn togetherThe range of specialisms that may apply the ideas in the book is rather wide: physics, mathematics, civil and chemical engineering and geology may all benefit, while the student who is devoted to the mechanics of granular or connected media will also find this a very useful text

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M A C Koenders. The Physics of the Deformation of Densely Packed Granular Materials

Preface

About the Author

Contents

Chapter 1. General Concepts. 1.1Introduction

1.2The isostatic state and jamming

1.3The statically indeterminate case and computer simulations

1.4Contact laws

1.5The frictional interaction

1.5.1Friction in three dimensions

1.6Contact laws in terms of material parameters

1.7Interaction for small particles in a fluid environment

References

Chapter 2. Continuum Mechanics and Cartesian Tensor Calculus. 2.1Initial considerations

2.2Rotations

2.3The strain tensor

2.4The stress tensor

2.5Tensors

2.6Material response

2.7Isotropic materials

2.8Elastic behaviour

2.9Anisotropic materials

2.10Coaxiality

2.11Objectivity and pre-stressed materials, the Jaumann derivative

2.12Frictional materials

References

Chapter3. The Bounds of Static Equilibrium. 3.1Criteria for rupture

3.2The context of second order, partial differential equations

3.3Wave speeds and strong ellipticity

References

Chapter 4. Heterogeneity

Chapter 5. Fabric Description

Chapter 6. Stress-Strain Relations of Granular Assemblies: A Frictionless Assembly. 6.1General considerations

6.2Kinematics

6.3Mean-field approximation

6.4Perturbations to the mean-field theory

6.5Solution in two dimensions

6.6Connectivity in a granular medium

6.7Estimates of the correction to the moduli due to heterogeneity

References

Chapter 7. Stress-Strain Relations of Granular Assemblies: Normal and Tangential Interactions. 7.1Particle spin

7.2The interaction and the quasi-static equilibrium equations

7.3Mean-field stiffness estimate

7.4Heterogeneity with tangential interactions

7.5Solution of the strain fluctuations for a two-dimensional isotropic medium

7.6Considerations for an isotropic contact distribution in two special cases

7.7Anisotropic calculation

7.8A few remarks on the theory

References

Chapter 8. Frictional Granular Materials. 8.1The frictional interaction

8.2Mean-field estimate

8.3Mean-field estimate with randomly distributed slip angles

8.4Mean-field estimate with concentrated slip angles

8.4.1A slip angle associated with the deviatoric stress

8.4.2Further investigation of the slip angle and fabric heterogeneity

8.4.3Verification of the friction criterion

8.5Strain fluctuations

8.5.1Determining the spins

8.5.2The displacement fluctuations associated with a sliding neighbouring particle derived from the force equilibrium equations

8.5.3The stress contribution associated with the fluctuations due to sliding; strain fluctuations and spin fluctuations

8.6Fabric heterogeneity and other refinements

8.7The evolution of an assembly in a biaxial cell test

References

Appendix A. Mathematical Appendix. A.1Isotropic tensors

A.1.1Isotropic 2-tensor

A.1.2Isotropic 4-tensor

A.2Integrals of strings of unit vectors

A.3Elastic constants

A.4Fourier transforms and harmonic density

A.5Bessel functions

A.6Various integrals. A.6.1Integrals involving Bessel functions

A.6.2Integrals with confluent hypergeometric functions

A.6.3Multiple integrals

References

Appendix B. List of Symbols and Notations. B.1List of symbols

B.2List of notations

B.2.1 Structural sums

B.2.2 Other notations

Index

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In the early 1980s, the idea first took hold that the mechanical response of a dense granular medium can be understood from a basis of the inter-particle contact properties. The initial efforts, a mean-field theory, had very poor results and papers on ‘micro-mechanics’ were usually relegated to the last section of conference proceedings. Gradually, the insight came about that a granular medium cannot be captured in a mean-field theory and that some form of non-homogeneity in the fabric properties has to be accounted for. The beginnings of this concept were implemented using the available continuum theories on heterogeneity and a few papers came out in the early 90s showing that in certain special cases the mechanical response was captured, but — irritatingly — not all cases. Highly anisotropic packed beds, for example, could not be accounted for and the failure of a granular medium at high stress ratio remained a mystery.

While progress since then has been slow, it is now clear that a proper theory of granular deformation must include a method that deals with heterogeneity that is particularly applicable to a system of particles. This turns out to be the theory of ‘connected media’, which captures the physics of contacting particulates in an appropriate manner. It has also been extended to anisotropic cases. A rigorous approach to Coulomb friction as an inter-particle interaction is required as well. Together these developments can now be implemented with great success.

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8.6Fabric heterogeneity and other refinements

8.7The evolution of an assembly in a biaxial cell test

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