Microstructures in Elastic Media
Principles and Computational Methods
1st Edition
Published on 25. August 1994
257 pages
978-0-19-535807-0 (ISBN)
Description
This monograph describes various methods for solving deformation problems of particulate solids, taking the reader from analytical to computational methods. The book is the first to present the topic of linear elasticity in mathematical terms that will be familiar to anyone with a grounding in fluid mechanics. It incorporates the latest advances in computational algorithms for elliptic partial differential equations, and provides the groundwork for simulations on high performance parallel computers. Numerous exercises complement the theoretical discussions, and a related set of self-documented programs is available to readers with Internet access. The work will be of interest to advanced students and practicing researchers in mechanical engineering, chemical engineering, applied physics, computational methods, and developers of numerical modeling software.
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11/1994
Oxford University Press Inc
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Content
- Intro
- Contents
- 1 Fundamental Equations
- 1.1 Introduction and Motivation
- 1.2 Stress and Strain
- 1.3 Equations of Equilibrium
- 1.4 Strain Energy
- 1.4.1 Uniqueness
- 1.4.2 Extremum Principles
- 1.5 Betti's Reciprocal Theorem
- 1.6 Integral Representation
- 1.6.1 Classification of Integral Equations
- 1.6.2 Kelvin State
- 1.6.3 Integral Representation
- 1.6.4 Rigid Inclusion
- 1.6.5 Eliminating Single or Double Layer
- 1.7 Single and Double Layer Potentials
- 1.7.1 Single Layer
- 1.7.2 Double Layer
- 1.7.3 Liapunov-Tauber Theorem
- 1.8 Boundary Integral Equations
- 1.8.1 Direct BEM
- 1.8.2 Indirect BEM
- 1.9 Spectral Properties
- 1.9.1 Banach's Theorem
- 1.9.2 ? = -1
- 1.9.3 ? = +1
- 1.9.4 Type II Problems
- 1.9.5 Spectral Radius of ?
- 1.10 Exercises
- 1.10.1 Rigid-Body Displacement
- 1.10.2 Stretching
- 1.10.3 Simple Shearing
- 1.10.4 Moduli of Elasticity
- 1.10.5 Integral Representation
- 1.10.6 Transmission of Force and Torque
- 1.10.7 Reciprocal Relation
- 1.10.8 Translating Rigid Sphere 1
- 1.10.9 Translating Rigid Sphere 2
- 1.10.10 Kelvin's Solution
- 1.10.11 On Green's Equations
- 1.10.12 Papkovich-Neuber Representation
- 1.10.13 Galerkin Vector
- 1.10.14 Self-Adjoint Property of G
- 1.10.15 Elastic Inclusion
- 1.10.16 Constant c[sub(ij)]
- 1.10.17 Thin, Rigid Inclusion
- 1.10.18 Liapunov-Tauber Theorem
- 2 Multipole Expansion and Rigid Inclusions
- 2.1 Singularity Solutions
- 2.1.1 Papkovich-Neuber Representation
- 2.1.2 Potential Deformation
- 2.1.3 Rotlet Deformation
- 2.1.4 Kelvinlet Deformation
- 2.1.5 Half-Space Solutions
- 2.1.6 Interior Deformation
- 2.2 Multipole Expansion
- 2.2.1 Stresslet
- 2.3 Spherical Rigid Inclusion
- 2.3.1 Translating a Rigid Sphere
- 2.3.2 Rotating a Rigid Sphere
- 2.3.3 Rigid Sphere in a Linear Deformation
- 2.3.4 Rigid Sphere in a Quadratic Ambient Field
- 2.3.5 Translating an Elastic Spherical Inclusion
- 2.4 Exercises
- 2.4.1 Navier Solutions
- 2.4.2 Navier Solutions
- 2.4.3 Navier Solutions
- 2.4.4 Galerkin Vector
- 2.4.5 Force and Torque on a Rigid Spherical Inclusion
- 2.4.6 Rigid Spherical Inclusion in High-Order Field
- 3 Faxén Relations and Ellipsoidal Inclusions
- 3.1 Faxén Relations
- 3.2 Rigid Spherical Inclusion
- 3.3 Rigid Ellipsoidal Inclusion
- 3.3.1 Singularity Solution for Translation
- 3.3.2 Singularity Solution for Linear Ambient Field
- 3.3.3 Degenerate Cases
- 3.3.4 Faxén Relations for the Rigid Ellipsoid
- 3.3.5 Interactions between Two Ellipsoids
- 3.4 Exercises
- 3.4.1 Traction Functionals
- 3.4.2 Faxén Relations for Torque and Stresslet
- 3.4.3 Multipole Expansion for Ellipsoids
- 3.4.4 Tractions for the Translating Ellipsoid
- 4 Load Transfer Problem and Boundary Collocation
- 4.1 The Method of Reflection
- 4.2 Load Transfer between Two Spheres
- 4.2.1 Far Field by Reflection
- 4.2.2 Near Touching
- 4.3 Kelvin Solutions
- 4.3.1 Spherical Harmonics
- 4.3.2 Kelvin's General Solutions
- 4.4 Boundary Collocation
- 4.4.1 Twin Multipole Expansions
- 4.4.2 Collocation Equations for Translation Problems
- 4.5 Comparison
- 4.6 Constitutive Relation
- 4.6.1 Constitutive Theory
- 4.6.2 Cubic Lattices
- 4.7 Kelvinlet near a Rigid Sphere
- 4.7.1 The Axisymmetric Kelvinlet
- 4.7.2 The Transverse Kelvinlet
- 4.8 Exercises
- 4.8.1 Solid Spherical Harmonics
- 4.8.2 Lurié Solution
- 4.8.3 Type I Problems
- 5 Completed Double Layer Boundary Element Method
- 5.1 Introduction
- 5.2 Direct Formulation
- 5.3 Completed Double Layer Boundary Element Method
- 5.3.1 Range Completer
- 5.3.2 Null Functions of (1+?)
- 5.3.3 Completion Process
- 5.3.4 Container Surface
- 5.3.5 A Summary
- 5.4 Rigid Inclusion
- 5.4.1 Translational Displacement
- 5.4.2 On Picard Iteration
- 5.4.3 Rotational Displacement
- 5.4.4 Homogeneous Deformation
- 5.5 Stresslet
- 5.6 Spectrum for a Sphere
- 5.6.1 Type I Problems - Ill-posed
- 5.7 Completed Double Layer Traction Problem
- 5.8 Exercises
- 5.8.1 Symmetry Relations
- 5.8.2 On Eigenfunctions
- 5.8.3 Incompressible Case
- 5.8.4 Gram-Schmidt Orthonormalization
- 5.8.5 Hadamard Ill-posed Problem
- 6 Numerical Implementation
- 6.1 Numerical Quadrature
- 6.2 Boundary Discretization
- 6.2.1 Constant Element
- 6.2.2 Higher Order Element
- 6.3 Evaluation of Boundary Integrals
- 6.3.1 Multivalued Traction
- 6.3.2 Regular Integrals
- 6.3.3 Singular Integrals
- 6.3.4 Rigid-Body Displacement
- 6.3.5 Adaptive Integration Schemes
- 6.3.6 Far-Field Approximation
- 6.4 Solution Methods
- 6.4.1 Direct Solver
- 6.4.2 Iterative Methods
- 6.4.3 Domain Decomposition
- 6.5 Distributed Computing under PVM
- 6.5.1 Some Concepts in Distributed Computing
- 6.5.2 Master/Slave Implementation
- 6.6 Exercises
- 6.6.1 Newton-Cotes rules
- 6.6.2 Quadrature
- 6.6.3 Galerkin Expansion
- 6.6.4 Jacobian
- 6.6.5 Evaluation of ?[sub(?)] G[sub(ij)]dS and ?[sub(?)] K[sub(ij)]dS
- 7 Some Applications of CDL-BIEM
- 7.1 Translating Sphere
- 7.1.1 Direct Formulation
- 7.1.2 CDL-BIEM
- 7.2 Sphere in Homogeneous Deformation
- 7.3 Two Spheroids
- 7.4 CDL in Half-Space
- 7.5 Container Surface
- 7.6 Deformation of a Cluster
- 7.7 Distributed Computing under PVM
- 7.7.1 Arrays of Spheres
- 7.7.2 Epilogue: Sedimentation through an Array of Spheres
- References
- Index
- A
- B
- C
- D
- E
- F
- G
- H
- I
- J
- K
- L
- M
- N
- O
- P
- Q
- R
- S
- T
- U
- V
- W
- Y
- Z
