# Elasticity in Engineering Mechanics, 3rd Edition

# Elasticity in Engineering Mechanics, 3rd Edition

ISBN: 978-0-470-95000-5 November 2010 656 Pages

## Description

**Comprehensive, accessible, and logical—an outstanding treatment of elasticity in engineering mechanics**

*Elasticity in Engineering Mechanics* has been prized by many aspiring and practicing engineers as an easy-to-navigate guide to an area of engineering science that is fundamental to aeronautical, civil, and mechanical engineering, and to other branches of engineering. With its focus not only on elasticity theory, including nano- and biomechanics, but also on concrete applications in real engineering situations, this acclaimed work is a core text in a spectrum of courses at both the undergraduate and graduate levels, and a superior reference for engineering professionals. With more than 200 graphs, charts, and tables, this Third Edition includes:

- Clear explorations of such topics as deformation and stress, stress-strain-temperature relations, plane elasticity, thermal stresses, and end loads
- Discussions of deformation and stress treated separately for clarity, with emphasis on both their independence and mathematical similarities
- An overview of the mathematical preliminaries to all aspects of elasticity, from stress analysis to vector fields, from the divergence theorem to tensor algebra
- Real-world examples and problem sets illustrating the most common elasticity solutions—such as equilibrium equations, the Galerkin vector, and Kelvin's problem
- Highlights of the similarities and differences between molecular dynamics and continuum theory
- Presentations of molecular dynamics, including the subjects of definition of temperature at atomistic scale, and interatomic potentials, forces, and stiffness matrices
- Discussions and real-world examples of biomechanics, including the subjects of finite strain elasticity, constitutive equations of soft biological tissues, incompressibility, aneurysm, plaque on artery wall, and active stresses
- A series of appendixes covering advanced topics such as complex variables, couple-stress theory, micromorphic theory, and concurrent atomistic/continuum theory

**Preface.**

**CHAPTER 1 INTRODUCTORY CONCEPTS AND MATHEMATICS.**

**Part I Introduction.**

1-1 Trends and Scopes.

1-2 Theory of Elasticity.

1-3 Numerical Stress Analysis.

1-4 General Solution of the Elasticity.

1-5 Experimental Stress Analysis.

1-6 Boundary Value Problems of Elasticity.

**Part II Preliminary Concepts.**

1-7 Brief Summary of Vector Algebra.

1-8 Scalar Point Functions.

1-9 Vector Fields.

1-10 Differentiation of Vectors.

1-11 Differentiation of a Scalar Field.

1-12 Differentiation of a Vector Field.

1-13 Curl of a Vector Field.

1-14 Eulerian Continuity Equation for Fluids.

1-15 Divergence Theorem.

1-16 Divergence Theorem in Two Dimensions.

1-17 Line and Surface Integrals (Application of Scalar Product).

1-18 Stokes's Theorem.

1-19 Exact Differential.

1-20 Orthogonal Curvilinear Coordiantes in Three-Dimensional Space.

1-21 Expression for Differential Length in Orthogonal Curvilinear Coordinates.

1-22 Gradient and Laplacian in Orthogonal Curvilinear Coordinates.

**Part III Elements of Tensor Algebra.**

1-23 Index Notation: Summation Convention.

1-24 Transformation of Tensors under Rotation of Rectangular Cartesian Coordinate System.

1-25 Symmetric and Antisymmetric Parts of a Tensor.

1-26 Symbols *dij* and *ijk* (the Kronecker Delta and the Alternating Tensor).

1-27 Homogeneous Quadratic Forms.

1-28 Elementary Matrix Algebra.

1-29 Some Topics in the Calculus of Variations.

**CHAPTER 2 THEORY OF DEFORMATION.**

2-1 Deformable, Continuous Media.

2-2 Rigid-Body Displacements.

2-3 Deformation of a Continuous Region. Material Variables. Spatial Variables.

2-4 Restrictions on Continuous Deformation of a Deformable Medium.

2-5 Gradient of the Displacement Vector. Tensor Quantity.

2-6 Extension of an Infinitesimal Line Element.

2-7 Physical Significance of *ii*. Strain Definitions.

2-8 Final Direction of Line Element. Definition of Shearing Strain. Physical Significance of *ij(i* ? *j)*.

2-9 Tensor Character of *aß*. Strain Tensor.

2-10 Reciprocal Ellipsoid. Principal Strains. Strain Invariants.

2-11 Determination of Principal Strains. Principal Axes.

2-12 Determination of Strain Invariants. Volumetric Strain.

2-13 Rotation of a Volume Element. Relation to Displacement Gradients.

2-14 Homogeneous Deformation.

2-15 Theory of Small Strains and Small Angles of Rotation.

2-16 Compatibility Conditions of the Classical Theory of Small Displacements.

2-17 Additional Conditions Imposed by Continuity.

2-18 Kinematics of Deformable Media.

Appendix 2A Strain–Displacement Relations in Orthogonal Curvilinear Coordinates.

Appendix 2B Derivation of Strain–Displacement Relations for Special Coordinates by Cartesian Methods.

Appendix 2C Strain–Displacement Relations in General Coordinates.

**CHAPTER 3 THEORY OF STRESS.**

3-1 Definition of Stress.

3-2 Stress Notation.

3-3 Summation of Moments. Stress at a Point. Stress on an Oblique Plane.

3-4 Tensor Character of Stress. Transformation of Stress Components under Rotation of Coordinate Axes.

3-5 Principal Stresses. Stress Invariants. Extreme Values.

3-6 Mean and Deviator Stress Tensors. Octahedral Stress.

3-7 Approximations of Plane Stress. Mohr's Circles in Two and Three Dimensions.

3-8 Differential Equations of Motion of a Deformable Body Relative to Spatial Coordinates.

Appendix 3A Differential Equations of Equilibrium in Curvilinear Spatial Coordinates.

Appendix 3B Equations of Equilibrium Including Couple Stress and Body Couple.

Appendix 3C Reduction of Differential Equations of Motion for Small-Displacement Theory.

**CHAPTER 4 THREE-DIMENSIONAL EQUATIONS OF ELASTICITY.**

4-1 Elastic and Nonelastic Response of a Solid.

4-2 Intrinsic Energy Density Function (Adiabatic Process).

4-3 Relation of Stress Components to Strain Energy Density Function.

4-4 Generalized Hooke's Law.

4-5 Isotropic Media. Homogeneous Media.

4-6 Strain Energy Density for Elastic Isotropic Medium.

4-7 Special States of Stress.

4-8 Equations of Thermoelasticity.

4-9 Differential Equation of Heat Conduction.

4-10 Elementary Approach to Thermal-Stress Problem in One and Two Variables.

4-11 Stress–Strain–Temperature Relations.

4-12 Thermoelastic Equations in Terms of Displacement.

4-13 Spherically Symmetrical Stress Distribution (The Sphere).

4-14 Thermoelastic Compatibility Equations in Terms of Components of Stress and Temperature. Beltrami–Michell Relations.

4-15 Boundary Conditions.

4-16 Uniqueness Theorem for Equilibrium Problem of Elasticity.

4-17 Equations of Elasticity in Terms of Displacement Components.

4-18 Elementary Three-Dimensional Problems of Elasticity. Semi-Inverse Method.

4-19 Torsion of Shaft with Constant Circular Cross Section.

4-20 Energy Principles in Elasticity.

4-21 Principle of Virtual Work.

4-22 Principle of Virtual Stress (Castigliano's Theorem).

4-23 Mixed Virtual Stress–Virtual Strain Principles (Reissner’s Theorem).

Appendix 4A Application of the Principle of Virtual Work to a Deformable Medium (Navier–Stokes Equations).

Appendix 4B Nonlinear Constitutive Relationships.

Appendix 4C Micromorphic Theory.

Appendix 4D Atomistic Field Theory.

**CHAPTER 5 PLANE THEORY OF ELASTICITY IN RECTANGULAR CARTESIAN COORDINATES.**

5-1 Plane Strain.

5-2 Generalized Plane Stress.

5-3 Compatibility Equation in Terms of Stress Components.

5-4 Airy Stress Function.

5-5 Airy Stress Function in Terms of Harmonic Functions.

5-6 Displacement Components for Plane Elasticity.

5-7 Polynomial Solutions of Two-Dimensional Problems in Rectangular Cartesian Coordinates.

5-8 Plane Elasticity in Terms of Displacement Components.

5-9 Plane Elasticity Relative to Oblique Coordinate Axes.

Appendix 5A Plane Elasticity with Couple Stresses.

Appendix 5B Plane Theory of Elasticity in Terms of Complex Variables.

**CHAPTER 6 PLANE ELASTICITY IN POLAR COORDINATES.**

6-1 Equilibrium Equations in Polar Coordinates.

6-2 Stress Components in Terms of Airy Stress Function *F* = *F(r,0 )*.

6-3 Strain–Displacement Relations in Polar Coordinates.

6-4 Stress–Strain–Temperature Relations.

6-5 Compatibility Equation for Plane Elasticity in Terms of Polar Coordinates.

6-6 Axially Symmetric Problems.

6-7 Plane Elasticity Equations in Terms of Displacement Components.

6-8 Plane Theory of Thermoelasticity.

6-9 Disk of Variable Thickness and Nonhomogeneous Anisotropic Material.

6-10 Stress Concentration Problem of Circular Hole in Plate.

6-11 Examples.

Appendix 6A Stress–Couple Theory of Stress Concentration Resulting from Circular Hole in Plate.

Appendix 6B Stress Distribution of a Diametrically Compressed Plane Disk.

**CHAPTER 7 PRISMATIC BAR SUBJECTED TO END LOAD.**

7-1 General Problem of Three-Dimensional Elastic Bars Subjected to Transverse End Loads.

7-2 Torsion of Prismatic Bars. Saint-Venant's Solution. Warping Function.

7-3 Prandtl Torsion Function.

7-4 A Method of Solution of the Torsion Problem: Elliptic Cross Section.

7-5 Remarks on Solutions of the Laplace Equation, v2*F* = 0.

7-6 Torsion of Bars with Tubular Cavities.

7-7 Transfer of Axis of Twist.

7-8 Shearing–Stress Component in Any Direction.

7-9 Solution of Torsion Problem by the Prandtl Membrane Analogy.

7-10 Solution by Method of Series. Rectangular Section.

7-11 Bending of a Bar Subjected to Transverse End Force.

7-12 Displacement of a Cantilever Beam Subjected to Transverse End Force.

7-13 Center of Shear.

7-14 Bending of a Bar with Elliptic Cross Section.

7-15 Bending of a Bar with Rectangular Cross Section.

Appendix 7A Analysis of Tapered Beams.

**CHAPTER 8 GENERAL SOLUTIONS OF ELASTICITY.**

8-1 Introduction.

8-2 Equilibrium Equations.

8-3 The Helmholtz Transformation.

8-4 The Galerkin (Papkovich) Vector.

8-5 Stress in Terms of the Galerkin Vector **F**.

8-6 The Galerkin Vector: A Solution of the Equilibrium Equations of Elasticity.

8-7 The Galerkin Vector **k***Z* and Love's Strain Function for Solids of Revolution.

8-8 Kelvin's Problem: Single Force Applied in the Interior of an Infinitely Extended Solid.

8-9 The Twinned Gradient and Its Application to Determine the Effects of a Change of Poisson's Ratio.

8-10 Solutions of the Boussinesq and Cerruti Problems by the Twinned Gradient Method.

8-11 Additional Remarks on Three-Dimensional Stress Functions.

References.

Bibliography.

**INDEX.**