This instructive volume offers a comprehensive survey of the methods and theories of linear elasticity. Three major sections cover general theory, two-dimensional elasticity, and three-dimensional elasticity. An ideal text for a two-course sequence on elasticity, this volume can also introduce the subject in a more general math course.
A comprehensive survey of the methods and theories of linear elasticity, this three-part introductory treatment covers general theory, two-dimensional elasticity, and three-dimensional elasticity. Ideal text for a two-course sequence on elasticity. 1984 edition.
A founding member of the American Society of Biomechanics, Robert William Soutas-Little is Professor Emeritus in the Department of Mechanical Engineering and the Department of Materials Science and Mechanics at Michigan State University.
Preface
Part I
Review of Mathematical Notation and Continuum Mechanics
Basic Equations of the Linear Theory of Elasticity
1 Vectors and Tensors
1. Notation
2. Vectors
3. Transformation Relations
3.1 Scalars
3.2 Vectors
3.3 Properties of the transformation matrix
4. Second Order Tensors
5. Higher Order Tensors
6. Dual Vector of an Antisymetric Tensor
7. Eigenvalue Problem
8. Isotropic Tensors
9. Tensor Fields
10. Integral Theorems
11. Classification of Vector Fields
2. Kinematics
1. Introduction
2. Spatial and Material Coordinates
3. Velocity and Material Time Derivative
4. Volume Elements
5. Reynold's Transport Theorem
6. Displacement Vector
7. Compatibility Equation
8. Infinitesimal Strain Tensor in Curvilinear Coordinates
9. Spherical and Deviatroic Strain Tensors
3. Stress
1. Introduction
2. Stress Tractions
3. Stress Tensor in the Material Sense
4. Properties of the Stress Tensor
4. Basic Equations of Continuum Mechanics
1. Introduction
2. Conservation of Mass
3. Cauchy's Equations of Motion
4. Considerations of Angular Momentum
5. Energy Conservation Equation
5. Linear Elasticity
1. Introduction
2. Geranlized Hooke's Law
3. Summary of the Equations of Isotropic Elasticity
4. Boundary Conditions
5. Uniqueness and Superposition
6. Saint-Venant's Principle
7. Displacement Formulation
8. Thermoelasticity
Part II
Two-Dimensional Elasticity
6 General Theory of Plane Elasticity
1. Introduction
2. Plane Deformation or Plane Strain
3. Plane Stress
4. Biharmonic Solutions
7 Problems in Cartesian Coordinates
1. Introduction
2. Mathematical Preliminaries
3. Polynomial Solutions
3.1 Uniaxial tension
3.2 Simply supported beam under pure moments
3.3 Beam bent by its own weight
4. Fourier Series Solutions
4.1 Beam subjected to sinusoidal load
5. Fourier Analysis
5.1 Fourier trigonometric series
6. General Fourier Solution of Elasticity Problem
6.1 Case 4-odd in x and even in y
6.2 Displacement solution using Marguerre function
7. Multiple Fourier Analysis
8. Problems Involving Infinite or Semi-Infinite Dimensions
8.1 Infinite strip loaded by uniform pressure
8.2 Fourier transform solutions
8.3 Solution of the infinite strip problem using integral transforms
8.4 Semi-infinite strip problems
8.5 Solution for the half-plane
9. Saint-Venant Boundary Region in Elastic Strips
10. Nonorthogonal Boundary Function Expansions
10.1 Point-matching
10.2 Least squares
10.3 Iterative improvements to point-matching techniques
11. Plane Elasticity Problems Using Nonorthogonal Functions
11.1 Examples requring functions nonorthogonal on the boundaries
8 Problems in Polor Coordinates
1. Introduction
2. Axially Symmetric Problems
2.1 Lamé problem
2.2 Pure bending of a curved beam
2.3 Rotational dislocation
3. Solution of Axisymmetric Problems Using the Navier Equation
4. Michell Solution
5. Examples Using the Michell Solution
5.1 Interior problem-stresses distributed around the edge of a disk
5.2 Exterior problem-infinite plane with circular hole
5.3 Annulus problem
5.4 Symmetry conditions
6. General Solutions Not Involving Orthogonal Functions
7. Wedge Problem
7.1 Wedge under uniform side load
7.2 Stress singularities at the tip of a wedge (M.L. williams solution)
7.3 Truncated semi-infinite wedge
8. Special Problems Using the Flamant Solution
8.1 Concentrated load in hole in infinite plate
9 Complex Variable Solutions
1. Introduction
2. Complex Variables
3. Complex Stress Formulation
4. Polar Coordinates
5. Interior Problem
6. Conformal Transformations
10 Finite Difference and Finite Element Methods
1. Introduction
2. Finite Element Method
3. Displacement Functions
4. Stresses and Strains
5. Nodal Force-Displacement Relations
6. Analysis of a Structure
7. Facet Stiffness Matrix
8. Local and Global Coordinates
9. Finite Element Example
10. Finite Difference Methods
11 Energy Theorems and Variational Techniques
1. Introduction
2. Calculus of Variations
3. Strain Energy Methods
4. Theorem of Stationary Potential Energy
5. Rayleigh-Ritz Method
6. Energy Method for Problems Involving Multiply Connected Domains
Part III
Three-Dimensional Elasticity
12 Saint-Venant Tension and Bending Theory
1. Torsion of Circular Cylinder
2. Non-Circular Section
3. Uniqueness of Saint-Venant Torsion Problem
4. Semi-Inverse Approach-Problems of Saint-Venant
5. Torsion of Rectangular Bars Using Fourier Anaylsis
6. Prandtl Stress Functions
7. Solution of a Hollow Cylinder
8. Solution by Use of Orthogonal Series
9. Polar Coordinates
10. Nonorthogonal Functions on Boundary
11. Complex Variable Solutions of Torsion Problems
12. Membrane Anaogy for Torsion
13. Torsion of Circular Shafts of Variable Diameter
14. Saint-Venant Approximation for Circular Cylinders
15. Flexure of Beams by Transverse End Loads
16. Flexure of a Circular Beam Under a Load Py Through the Centroid
13 Navier Equation and the Galerkin Vector
1. Solution of the Navier Equation in Three-Dimensional Elastostatics
2. Stress Functions
3. The Galerkin Vector
4. Equivalent Galerkin Vector
5. Mathematical Notes on the Galerkin Vector
6. Love's Strain Function
7. Long Solid Cylinders Axisymmetrically Loaded
8. Infinite Cylinder or Hole in an Elastic Body
9. Thick Axisymmetrically Loaded Plate
10. Axisymmetrically Loaded Plate with a Hole
11. Hankel Transform Methods
12. Axisymmetric Problem of a Half Space
13. Boussinesq Problem
14. Contact Problems
15. Short Cylinders-Multiple Fourier-Bessel Series Analysis
16. End-Loading on a Semi-Infinite Cylinder
17. Axisymmetric Torsion
18. Asymmetric Loadings
14 Papkovich-Neuber Solution
1. Introduction
2. Concentrated Force in the Infinite Solid
3. Concentrated Load Not at the Origin
4. Other Special Singular Solutions
5. Concentrated Load Tangential to the Surface of the Half Space-Cerruti's Problem
6. Concentrated Loads at the Tip of an Elastic Cone
7. Symmetrically Loaded Spheres
8. Spherical Harmonics
9. The Internal Problem
10. Interior Problem with Body Forces
11. Gravitating Sphere
12. Rotating Sphere
13. The External Problem
14. Stress Concentration Due to a Spherical Cavity
15. Hollow Sphere
16. End Effects in a Truncated Semi-Infinited Cone
17. General Solution of Non-Axismmetric Problems by
6. Complex Variable Formulae
7. Bessel Functions
8. Hankel Transform
9. Legendre Functions