Elasticity

Preface
Contents
Part I General Considerations
1 Introduction
	1.1 Notation for Stress and Displacement
		1.1.1 Stress
		1.1.2 Index and vector notation and the summation convention
		1.1.3 Vector operators in index notation
		1.1.4 Vectors, tensors and transformation rules
		1.1.5 Principal stresses and von Mises stress
		1.1.6 Displacement
	1.2 Strains and their Relation to Displacements
		1.2.1 Tensile strain
		1.2.2 Rotation and shear strain
		1.2.3 Transformation of coördinates
		1.2.4 Definition of shear strain
	1.3 Stress-strain Relations
		1.3.1 Isotropic constitutive law
		1.3.2 Lamé's constants
		1.3.3 Dilatation and bulk modulus
		1.3.4 Deviatoric stress
	Problems
2 Equilibrium and Compatibility
	2.1 Equilibrium Equations
	2.2 Compatibility Equations
		2.2.1 The significance of the compatibility equations
	2.3 Equilibrium Equations in terms of Displacements
	Problems
Part II Two-dimensional Problems
3 Plane Strain and Plane Stress
	3.1 Plane Strain
		3.1.1 The corrective solution
		3.1.2 Saint-Venant's principle
	3.2 Plane Stress
		3.2.1 Generalized plane stress
		3.2.2 Relationship between plane stress and plane strain
	Problems
4 Stress Function Formulation
	4.1 Choice of a Suitable Form
	4.2 The Airy Stress Function
		4.2.1 Transformation of coördinates
		4.2.2 Non-zero body forces
	4.3 The Governing Equation
		4.3.1 The compatibility condition
		4.3.2 Method of solution
		4.3.3 Reduced dependence on elastic constants
	Problems
5 Problems in Rectangular Coördinates
	5.1 Biharmonic Polynomial Functions
		5.1.1 Second and third degree polynomials
	5.2 Rectangular Beam Problems
		5.2.1 Bending of a beam by an end load
		5.2.2 Higher order polynomials — a general strategy
		5.2.3 Manual solutions — symmetry considerations
	5.3 Fourier Series and Transform Solutions
		5.3.1 Choice of form
		5.3.2 Fourier transforms
	Problems
6 End Effects
	6.1 Decaying Solutions
	6.2 The Corrective Solution
		6.2.1 Separated-variable solutions
		6.2.2 The eigenvalue problem
	6.3 Other Saint-Venant Problems
	6.4 Mathieu's Solution
	Problems
7 Body Forces
	7.1 Stress Function Formulation
		7.1.1 Conservative vector fields
		7.1.2 The compatibility condition
	7.2 Particular Cases
		7.2.1 Gravitational loading
		7.2.2 Inertia forces
		7.2.3 Quasi-static problems
		7.2.4 Rigid-body kinematics
	7.3 Solution for the Stress Function
		7.3.1 The rotating rectangular bar
		7.3.2 Solution of the governing equation
	7.4 Rotational Acceleration
		7.4.1 The circular disk
		7.4.2 The rectangular bar
		7.4.3 Weak boundary conditions and the equation of motion
	Problems
8 Problems in Polar Coördinates
	8.1 Expressions for Stress Components
	8.2 Strain Components
	8.3 Fourier Series Expansion
		8.3.1 Satisfaction of boundary conditions
		8.3.2 Circular hole in a shear field
		8.3.3 Degenerate cases
	8.4 The Michell Solution
		8.4.1 Hole in a tensile field
	Problems
9 Calculation of Displacements
	9.1 The Cantilever with an End Load
		9.1.1 Rigid-body displacements and end conditions
		9.1.2 Deflection of the free end
	9.2 The Circular Hole
	9.3 Displacements for the Michell Solution
		9.3.1 Equilibrium considerations
		9.3.2 The cylindrical pressure vessel
	Problems
10 Curved Beam Problems
	10.1 Loading at the Ends
		10.1.1 Pure bending
		10.1.2 Force transmission
	10.2 Eigenvalues and Eigenfunctions
	10.3 The Inhomogeneous Problem
		10.3.1 Beam with sinusoidal loading
		10.3.2 The near-singular problem
	10.4 Some General Considerations
		10.4.1 Conclusions
	Problems
11 Wedge Problems
	11.1 Power-law Tractions
		11.1.1 Uniform tractions
		11.1.2 The rectangular body revisited
		11.1.3 More general uniform loading
		11.1.4 Eigenvalues for the wedge angle
	11.2 Williams' Asymptotic Method
		11.2.1 Acceptable singularities
		11.2.2 Eigenfunction expansion
		11.2.3 Nature of the eigenvalues
		11.2.4 The singular stress fields
		11.2.5 Other geometries
	11.3 General Loading of the Faces
	Problems
12 Plane Contact Problems
	12.1 Self-Similarity
	12.2 The Flamant Solution
	12.3 The Half-Plane
		12.3.1 The normal force Fy
		12.3.2 The tangential force Fx
		12.3.3 Summary
	12.4 Distributed Normal Tractions
	12.5 Frictionless Contact Problems
		12.5.1 Method of solution
		12.5.2 The flat punch
		12.5.3 The cylindrical punch (Hertz problem)
	12.6 Problems with Two Deformable Bodies
	12.7 Uncoupled Problems
		12.7.1 Contact of cylinders
	12.8 Combined Normal and Tangential Loading
		12.8.1 Cattaneo and Mindlin's problem
		12.8.2 Steady rolling: Carter's solution
	Problems
13 Forces, Dislocations and Cracks
	13.1 The Kelvin Solution
		13.1.1 Body force problems
	13.2 Dislocations
		13.2.1 Dislocations in Materials Science
		13.2.2 Similarities and differences
		13.2.3 Dislocations as Green's functions
		13.2.4 Stress concentrations
	13.3 Crack Problems
		13.3.1 Linear Elastic Fracture Mechanics
		13.3.2 Plane crack in a tensile field
		13.3.3 Energy release rate
	13.4 Disclinations
		13.4.1 Disclinations in a crystal structure
	13.5 Method of Images
	Problems
14 Thermoelasticity
	14.1 The Governing Equation
	14.2 Heat Conduction
	14.3 Steady-state Problems
		14.3.1 Dundurs' Theorem
	Problems
15 Antiplane Shear
	15.1 Transformation of Coördinates
	15.2 Boundary Conditions
	15.3 The Rectangular Bar
	15.4 The Concentrated Line Force
	15.5 The Screw Dislocation
	Problems
16 Moderately Thick Plates
	16.1 Boundary Conditions
	16.2 Edge Effects
	16.3 Body Force Problems
	16.4 Normal Loading of the Faces
		16.4.1 Steady-state thermoelasticity
	Problems
Part III End Loading of the Prismatic Bar
17 Torsion of a Prismatic Bar
	17.1 Prandtl's Stress Function
		17.1.1 Solution of the governing equation
		17.1.2 The warping function
	17.2 The Membrane Analogy
	17.3 Thin-walled Open Sections
	17.4 The Rectangular Bar
	17.5 Multiply-connected (Closed) Sections
		17.5.1 Thin-walled closed sections
	Problems
18 Shear of a Prismatic Bar
	18.1 The Semi-inverse Method
	18.2 Stress Function Formulation
	18.3 The Boundary Condition
		18.3.1 Integrability
		18.3.2 Relation to the torsion problem
	18.4 Methods of Solution
		18.4.1 The circular bar
		18.4.2 The rectangular bar
	Problems
Part IV Complex-Variable Formulation
19 Prelinary Mathematical Results
	19.1 Holomorphic Functions
	19.2 Harmonic Functions
	19.3 Biharmonic Functions
	19.4 Expressing Real Harmonic and Biharmonic Functions in Complex Form
		19.4.1 Biharmonic functions
	19.5 Line Integrals
		19.5.1 The residue theorem
		19.5.2 The Cauchy integral theorem
	19.6 Solution of Harmonic Boundary-value Problems
		19.6.1 Direct method for the interior problem for a circle
		19.6.2 Direct method for the exterior problem for a circle
		19.6.3 The half-plane
	19.7 Conformal Mapping
	Problems
20 Application to Elasticity Problems
	20.1 Representation of Vectors
		20.1.1 Transformation of coördinates
	20.2 The Antiplane Problem
		20.2.1 Solution of antiplane boundary-value problems
	20.3 In-plane Deformations
		20.3.1 Expressions for stresses
		20.3.2 Rigid-body displacement
	20.4 Relation between the Airy Stress Function and the Complex Potentials
	20.5 Boundary Tractions
		20.5.1 Equilibrium considerations
	20.6 Boundary-value Problems
		20.6.1 Solution of the interior problem for the circle
		20.6.2 Solution of the exterior problem for the circle
	20.7 Conformal Mapping for In-plane Problems
		20.7.1 The elliptical hole
	Problems
Part V Three-Dimensional Problems
21 Displacement Function Solutions
	21.1 The Strain Potential
	21.2 The Galerkin Vector
	21.3 The Papkovich-Neuber Solution
		21.3.1 Change of coördinate system
	21.4 Completeness and Uniqueness
		21.4.1 Methods of partial integration
	21.5 Body Forces
		21.5.1 Conservative body force fields
		21.5.2 Non-conservative body force fields
	Problems
22 The Boussinesq Potentials
	22.1 Solution A: The Strain Potential
	22.2 Solution B
	22.3 Solution E: Rotational Deformation
	22.4 Other Coördinate Systems
		22.4.1 Cylindrical polar coördinates
		22.4.2 Spherical polar coördinates
	22.5 Solutions Obtained by Superposition
		22.5.1 Solution F: Frictionless isothermal contact problems
		22.5.2 Solution G: The surface free of normal traction
		22.5.3 A plane strain solution
	22.6 A Three-dimensional Complex-Variable Solution
	Problems
23 Thermoelastic Displacement Potentials
	23.1 The Method of Strain Suppression
	23.2 Boundary-value Problems
		23.2.1 Spherically-symmetric Stresses
		23.2.2 More general geometries
	23.3 Plane Problems
		23.3.1 Axisymmetric problems for the cylinder
		23.3.2 Steady-state plane problems
		23.3.3 Heat flow perturbed by a circular hole
		23.3.4 Plane stress
	23.4 Steady-state Temperature: Solution T
		23.4.1 Thermoelastic plane stress
	Problems
24 Singular Solutions
	24.1 The Source Solution
		24.1.1 The centre of dilatation
		24.1.2 The Kelvin solution
	24.2 Dimensional Considerations
		24.2.1 The Boussinesq solution
	24.3 Other Singular Solutions
	24.4 Image Methods
		24.4.1 The traction-free half-space
	Problems
25 Spherical Harmonics
	25.1 Fourier Series Solution
	25.2 Reduction to Legendre's Equation
	25.3 Axisymmetric Potentials and Legendre Polynomials
		25.3.1 Singular spherical harmonics
		25.3.2 Special cases
	25.4 Non-axisymmetric Harmonics
	25.5 Cartesian and Cylindrical Polar Coördinates
	25.6 Harmonic Potentials with Logarithmic Terms
		25.6.1 Logarithmic functions for cylinder problems
	25.7 Non-axisymmetric Cylindrical Potentials
	25.8 Spherical Harmonics in Complex-variable Notation
		25.8.1 Bounded cylindrical harmonics
		25.8.2 Singular cylindrical harmonics
	Problems
26 Cylinders and Circular Plates
	26.1 Axisymmetric Problems for Cylinders
		26.1.1 The solid cylinder
		26.1.2 The hollow cylinder
	26.2 Axisymmetric Circular Plates
		26.2.1 Uniformly loaded plate on a simple support
	26.3 Non-axisymmetric Problems
		26.3.1 Cylindrical cantilever with an end load
	Problems
27 Problems in Spherical Coördinates
	27.1 Solid and Hollow Spheres
		27.1.1 The solid sphere in torsion
		27.1.2 Spherical hole in a tensile field
	27.2 Conical Bars
		27.2.1 Conical bar transmitting an axial force
		27.2.2 Inhomogeneous problems
		27.2.3 Non-axisymmetric problems
	Problems
28 Eigenstrains and Inclusions
	28.1 Governing Equations
	28.2 Galerkin Vector Formulation
		28.2.1 Non-differentiable eigenstrains
		28.2.2 The stress field
	28.3 Uniform Eigenstrains in a Spherical Inclusion
		28.3.1 Stresses outside the inclusion
	28.4 Green's Function Solutions
		28.4.1 Nuclei of strain
	28.5 The Ellipsoidal Inclusion
		28.5.1 The stress field
		28.5.2 Anisotropic material
	28.6 The Ellipsoidal Inhomogeneity
		28.6.1 Equal Poisson's ratios
		28.6.2 The ellipsoidal hole
	28.7 Energetic Considerations
		28.7.1 Evaluating the integral
		28.7.2 Strain energy in the inclusion
	Problems
29 Axisymmetric Torsion
	29.1 The Transmitted Torque
	29.2 The Governing Equation
	29.3 Solution of the Governing Equation
	29.4 The Displacement Field
	29.5 Cylindrical and Conical Bars
		29.5.1 The centre of rotation
	29.6 The Saint Venant Problem
	Problems
30 The Prismatic Bar
	30.1 Power-series Solutions
		30.1.1 Superposition by differentiation
		30.1.2 The problems mathcalP0 and mathcalP1
		30.1.3 Properties of the solution to mathcalPm
	30.2 Solution of mathcalPm by Integration
	30.3 The Integration Process
	30.4 The Two-dimensional Problem mathcalP0
	30.5 Problem mathcalP1
		30.5.1 The corrective antiplane solution
		30.5.2 The circular bar
	30.6 The Corrective In-plane Solution
	30.7 Corrective Solutions using Real Stress Functions
		30.7.1 Airy function
		30.7.2 Prandtl function
	30.8 Solution Procedure
	30.9 Example
		30.9.1 Problem mathcalP1
		30.9.2 Problem mathcalP2
		30.9.3 End conditions
	Problems
31 Frictionless Contact
	31.1 Boundary Conditions
		31.1.1 Mixed boundary-value problems
	31.2 Determining the Contact Area
	31.3 Contact Problems Involving Adhesive Forces
	Problems
32 The Boundary-value Problem
	32.1 Hankel Transform Methods
	32.2 Collins' Method
		32.2.1 Indentation by a flat punch
		32.2.2 Integral representation
		32.2.3 Basic forms and surface values
		32.2.4 Reduction to an Abel equation
		32.2.5 Smooth contact problems
		32.2.6 Choice of form
	32.3 Non-axisymmetric Problems
		32.3.1 The full stress field
	Problems
33 The Penny-shaped Crack
	33.1 The Penny-shaped Crack in Tension
	33.2 Thermoelastic Problems
	Problems
34 Hertzian Contact
	34.1 Elastic Deformation
		34.1.1 Field-point integration
	34.2 Solution Procedure
		34.2.1 Axisymmetric bodies
	Problems
35 The Interface Crack
	35.1 The Uncracked Interface
	35.2 The Corrective Solution
		35.2.1 Global conditions
		35.2.2 Mixed conditions
	35.3 The Penny-shaped Crack in Tension
		35.3.1 Reduction to a single equation
		35.3.2 Oscillatory singularities
	35.4 The Contact Solution
	35.5 Implications for Fracture Mechanics
	Problems
36 Anisotropic Elasticity
	36.1 The Constitutive Law
	36.2 Two-dimensional Solutions
	36.3 Orthotropic Material
		36.3.1 Normal loading of the half-plane
		36.3.2 Degenerate cases
	36.4 Lekhnitskii's Formalism
		36.4.1 Polynomial solutions
		36.4.2 Solutions in linearly transformed space
	36.5 Stroh's Formalism
		36.5.1 The eigenvalue problem
		36.5.2 Solution of boundary-value problems
		36.5.3 The line force solution
		36.5.4 Internal forces and dislocations
		36.5.5 Planar crack problems
		36.5.6 The Barnett-Lothe tensors
	36.6 End Loading of the Prismatic Bar
		36.6.1 Bending and axial force
		36.6.2 Torsion
	36.7 Three-dimensional Problems
		36.7.1 Concentrated force on a half-space
	36.8 Transverse Isotropy
	Problems
37 Variational Methods
	37.1 Strain Energy
		37.1.1 Strain energy density
	37.2 Conservation of Energy
	37.3 Potential Energy of the External Forces
	37.4 Theorem of Minimum Total Potential Energy
	37.5 Approximate Solutions — the Rayleigh-Ritz Method
	37.6 Castigliano's Second Theorem
	37.7 Approximations using Castigliano's Second Theorem
		37.7.1 The torsion problem
		37.7.2 The in-plane problem
	37.8 Uniqueness and Existence of Solution
		37.8.1 Singularities
	Problems
38 The Reciprocal Theorem
	38.1 Maxwell's Theorem
		38.1.1 Example: Mindlin's problem
	38.2 Betti's Theorem
		38.2.1 Change of volume
		38.2.2 A tilted punch problem
		38.2.3 Indentation of a half-space
	38.3 Eigenstrain Problems
		38.3.1 Deformation of a traction-free body
		38.3.2 Displacement constraints
	38.4 Thermoelastic Problems
	Problems
Appendix A Using Maple and Mathematica
Index