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