# Elasticity

## 1st Edition

### Theory, Applications, and Numerics

**Authors:**Martin Sadd

**eBook ISBN:**9780080477473

**Imprint:**Academic Press

**Published Date:**20th August 2004

**Page Count:**480

## Description

Although there are several books in print dealing with elasticity, many focus on specialized topics such as mathematical foundations, anisotropic materials, two-dimensional problems, thermoelasticity, non-linear theory, etc. As such they are not appropriate candidates for a general textbook. This book provides a concise and organized presentation and development of general theory of elasticity. This text is an excellent book teaching guide.

## Key Features

- Contains exercises for student engagement as well as the integration and use of MATLAB Software
- Provides development of common solution methodologies and a systematic review of analytical solutions useful in applications of

## Readership

graduate students, engineering industries and governmental research centers

## Table of Contents

Part I: Foundations and Elementary Applications (Part I - the first half of the text, provides a standard first semester course in beginning elasticity theory and applications. Particular topics from Part II could also be used to supplement such a first course.)

1. Mathematical Preliminaries (A self-contained review is provided of mathematical principles and notation needed in the text. This material can be covered at the beginning of the course to bring the entire class to a common point or various sections can be referred to as later course material is presented. Vector and index notation is introduced and Cartesian tensor notation will become the primary notational scheme for the formulation part of the course. MATLAB is first introduced here and is used to conduct rotational transformations and solve eigenvalue problems.)

1.1 Introduction

1.2 Scalar, Vector and Matrix Notation

1.3 Index Notational Properties

1.4 Kronecker Delta and Alternating Symbol

1.5 Determinants

1.6 Coordinate Rotation Transformations (This section is the first MATLAB application,

and a specific code is to be introduced for coordinate rotations. This will reinforce the theoretical material presented and allow students to explicitly calculate particular transformations and evaluate the nature of this concept.)

1.7 Cartesian Tensors

1.8 Principal Values, Axes and Invariants of Symmetric Second Order Tensors

(Another MATLAB application is covered, which will solve the general eigenvalue problem. This will again reinforce the theoretical material covered in this section.)

1.9 Algebra of Cartesian Tensors

1.10 Tensor Fields, Derivatives and Integral Theorems

1.11 Orthogonal Curvilinear Coordinates

2. Deformation: Displacements and Strains (Starting with a brief presentation of general deformation theory, the chapter moves into small deformation kinematics, and develops basic definitions and relations for linear elasticity.)

2.1 Introduction

2.2 General Deformations

2.3 Small Deformation Theory - Strain-Displacement Relations

2.4 Principal Strains

2.5 Deviatoric Strains

2.6 Strain Compatibility

2.7 Strain Transformation (MATLAB applications from sections 1.6 and 1.8 are applied to specific strain transformation problems. Applications related to strain measurement systems will be included.)

3. Stress and Equilibrium (This chapter will cover the appropriate theory for force and stress distribution in solids undergoing small deformation.)

3.1 Introduction

3.2 Body and Surface Forces

3.3 The Traction Vector

3.4 The Stress Tensor

3.5 Principal Stresses

3.6 Deviatoric Stresses

3.7 Stress Transformation (Similar to section 2.7, MATLAB applications will be introduced for specific stress transformation problems)

3.8 Equilibrium Equations

4. Material Behavior-Constitutive Equations (The chapter introduces linear elastic material behavior as formalized by Hooke's Law. The formulation starts with the general case, pointing out the complexities for heterogeneous anisotropic materials. Further more detailed study of anisotropic elasticity is made in chapter 11. Reduction for homogeneous and isotropic materials is then made. Thermoelastic relations are initially presented, and further applications of this case are made in chapter 12. Discussion of bounds on elastic constants is saved for chapter 6 where strain energy is introduced.)

4.1 Introduction

4.2 Generalized Hooke's Law

4.3 Isotropic Case

4.4 Thermoelastic Relations

4.5 Material Characterization

4.6 Physical Data (In order to provide students with some experience of real materials, this section will present tables of elastic constants for common materials used in engineering applications.)5. Formulation and Solution Strategies (This chapter is very important for student understanding of theory and application for problem solution. I feel that I have provided more educational innovation than found in competitor texts. Collecting and understanding all of the previous theory and equations is not an easy task for students. Developing the final formulations will be done concisely and organized around solution strategies. Material in section 5.7 will provide useful material that will link with later chapters in the text.)

5.1 Introduction

5.2 Boundary Conditions and Fundamental Problem Classifications (Some additional material is presented here to provide students with more experience on proper formulation of boundary conditions, since this has been a perceived area of student difficulty)

5.3 Stress Formulation (Stress function solution strategy is discussed)

5.4 Displacement Formulation (Displacement potential solution strategy is mentioned)

5.5 Principal of Superposition

5.6 Saint-Venant's Principle

5.7 General Solution Strategies (This particular section will contain special extended discussions on solution strategies in order to provide students with a more broad and general understanding of the variety of such techniques. Many of these solution methods will then be explicitly demonstrated in appropriate problems in later chapters. Students wishing to apply elasticity theory to their own future work will need such information to make a successful solution method choice.)

- Direct Method

Example 5.1 Stretching of Prismatic Bar Under Its Own Weight

- Inverse Method

Example 5.2 Pure Beam Bending

- Semi_Inverse Method

Example 5.3 Torsion of Prismatic Bars

- Analytical Solution Procedures

Power Series, Fourier Methods, Integral Transform and Complex Variable

Techniques

- Approximate Solution Procedures

Ritz Technique

- Numerical Solution Procedures

Finite Difference, Finite and Boundary Element Techniques

6. Strain Energy and Related Principles (Final elasticity formulation is completed in this chapter through the introduction of strain energy and related topics. Standard uniqueness and bounds on elastic moduli are discussed. The reciprocal theorem is presented and this will lead into the integral formulation that is later used in developing the numerical boundary element method in chapter 15.)

6.1 Introduction

6.2 Strain Energy

6.3 Uniqueness of the Elasticity Boundary-Value Problem

6.4 Bounds on the Elastic Constants

6.5 Reciprocal Theorem

6.6 Integral Formulation of Elasticity: Somigliana's Identity

6.7 Principle of Virtual Work

6.8 Principles of Minimum Potential and Complimentary Energy

6.9 Ritz Method

7. Two-Dimensional Formulations (This chapter establishes the standard two-dimensional theories of plane strain, plane stress and generalized plane stress. The Airy stress function solution methodology is covered, and is latter used to develop several analytical solutions in chapter 8.)

7.1 Introduction

7.2 Plane Strain

7.3 Plane Stress

7.4 Generalized Plane Stress

7.5 Airy Stress Function

7.6 Polar Coordinate Formulation

8. Two-Dimensional Problem Solution (Numerous analytical solutions are developed in this chapter. In order to provide comparisons with student's previous knowledge and experience, particular stress and displacement solutions are compared with elementary mechanics of materials solutions. Continued use of MATLAB software is made in many examples to evaluate and plot the analytical stress and displacement solution fields. This is accomplished through the use of X-Y, polar and contour plots as illustrated in Figures 8.4, 8.5, 8.13, 8.22, and 8.36. A unique feature of this activity is the plotting of maximum shear stress contours and the subsequent comparison with the corresponding photoelastic fringe patterns. This is done in Figure 8.36 and in exercises 8-18 and 8-20. It is felt that students will appreciate and understand more clearly such solutions when they are able to view such graphical representations. This concept also fits nicely with the numerical solution procedures presented in chapter 15.)

8.1 Introduction

8.2 Cartesian Coordinate Solutions Using Polynomials

Example 8.1 Uniaxial Tension of a Beam

Example 8.2 Pure Bending of a Beam

Example 8.3 Bending of a Beam by Uniform Transverse Loading

8.3 Cartesian Coordinate Solutions Using Fourier Methods

Example 8.4 Beam Subjected to Transverse Sinusoidal Loading

Applications Involving Fourier Series

Example 8.5 Rectangular Domain with Arbitrary Boundary Loading

8.4 General Solutions in Polar Coordinates

- Axisymmetric Solution

- Mitchell Solution

8.5 Polar Coordinate Solutions

Example 8.6 Thick-Walled Cylinder Under Uniform Boundary Pressure

Pressurized Hole in an Infinite Medium

Stress Free Hole in an Infinite Medium Under Uniform Tension at Infinity

Example 8.7 Infinite Medium with a Stress Free Hole Under Uniform Far Field

Loading

Biaxial and Shear Loading Cases

Example 8.8 Wedge and Semi-Infinite Domain Problems

Quarter Plane Example

Half Space Examples

Half Space Under Uniform Normal Stress Over x [0

Half Space Under Concentrated Force at the Origin (Flamant Solution)

Half Space Under a Surface Concentrated Moment

Half Space Under Uniform Normal Loading Over -a / x / a

Notch and Crack Problems

Example 8.9 Curved Beam Problems

Pure Bending Example

Curved Cantilever Under End Loading

Example 8.10 Disk Under Diametrical Compression

Example 8.11 Rotating Disk Problem

9. Saint - Venant Extension, Torsion and Flexure (Some basic three dimensional analytical solutions are presented in this chapter. Similar to the previous chapter, MATLAB is again used to evaluate particular stress and displacement solutions of examples 9.1, 9.2 and 9.3.

9.1 Introduction

9.2 Extension Formulation

Example 9.1 Longitudinal Loading Example

9.3 Torsion Formulation

Stress Function Formulation

Displacement Formulation

Membrane Analogy

9.4 Torsion Solutions Derived from Boundary Equation

Example 9.2 Elliptical Section

Example 9.3 Equilateral Triangular Section

9.5 Torsion Solution Using Fourier Methods

Example 9.4 Rectangular Section

9.6 Torsion of Multiply Connected Cross Sections

Example 9.5 Hollow Elliptical Section

9.7 Torsion of Circular Shafts of Variable Diameter

Example 9.6 Conical Shaft

9.8 Flexure Formulation

9.9 Flexure Problems without Twist

Example 9.7 Circular Section

Example 9.8 Rectangular Section

Part II: Advanced Applications (This second part of the text includes more advanced topics which could form the basis of a second elasticity course or selected topics could be used to supplement a first course.)

10. Complex Variable Methods (This chapter provides an overview of complex variable methods for solution of two-dimensional elasticity problems. A brief mathematical review of complex variable theory is provided within the chapter. Formulation and solution schemes are based on using Kolosov-Muskhelishvili complex potentials. Applications include examples of stress concentration and fracture mechanics (previously introduced in example 8.8). MATLAB will again be used to evaluate and plot particular solutions in the chapter examples.)

10.1 Introduction

10.2 Review of Complex Variable Theory

10.3 Complex Formulation of the Plane Problem

Example 10.1 Constant Stress State Example

10.4 Resultant Boundary Conditions

10.5 General Structure of the Complex Potentials

- Finite Simply-Connected Domains

- Finite Multiply-Connected Domains

- Infinite Domains

10.6 Circular Domain Examples

Example 10.2 Disk Under Uniform Compression

Example 10.3 Circular Plate with Concentrated Edge Loading

10.7 Plane and Half-Plane Problems

Example 10.4 Concentrated Force-Moment System in an Infinite Plane

Example 10.5 Concentrated Force System on the Surface of a Half-Plane

Example 10.6 Stressed Infinite Plate with a Circular Hole

10.8 Applications Using the Method of Conformal Mapping

Example 10.7 Stressed Infinite Plate with an Elliptical Hole

10.9 Applications to Fracture Mechanics

Example 10.8 Infinite Plate with a Central Crack

10.10 Westergaard Method for Crack Analysis

11. Anisotropic Elasticity (This chapter picks up from the material in section 4.2 and presents a concise development of elasticity applied to anisotropic materials. After a brief description of a torsion example, the chapter focuses primarily on two-dimensional applications using the complex potentials as introduced in the preceding chapter. Similar to chapter 10, applications to stress concentration and fracture mechanics are developed.)

11.1 Introduction

11.2 Material Symmetry

- Plane of Symmetry (Monoclinic Material)

- Three Perpendicular Planes of Symmetry (Orthotropic Material)

- Axis of Symmetry (Transversely Isotropic Material)

- Complete Symmetry (Isotropic Material)

Example 11.1 Hydrostatic Compression of a Monoclinic Cube

11.3 Restrictions on the Elastic Moduli

11.4 Torsion of a Solid Possessing a Plane of Material Symmetry

- Stress Formulation

- Displacement Formulation

- General Solution to Governing Equation

Example 11.2 Torsion of an Elliptical Orthotropic Bar

11.5 Plane Deformation Problems

Example 11.3 Simple Tension of an Anisotropic Sheet

Example 11.4 Concentrated Force in an Infinite Plane

Example 11.5 Concentrated Force on the Surface of a Half-Plane

Example 11.6 Infinite Plate with an Elliptical Hole

Example 11.7 Stressed Infinite Plate with an Elliptical Hole

11.6 Applications to Fracture Mechanics

12. Thermoelasticity (Starting from the introductory material in section 4.4, this chapter provides a brief formulation of thermoelasticity and solution of problems of engineering interest. Most of the application problems are to be two-dimensional, and examples will be solved using methods previously discussed in chapters 8, 10 and 11. As in previous chapters, examples of stress concentration and fracture mechanics will be presented.)

12.1 Introduction

12.2 Heat Conduction and the Energy Equation

12.3 General Uncoupled Formulation

12.4 Two-Dimensional Formulation

- Plane Strain

- Plane Stress

12.5 Displacement Potential Solution

12.6 Stress Function Formulation

Example 12.1 Thermal Stresses in an Elastic Strip

12.7 Polar Coordinate Formulation

12.8 Radially Symmetric Problems

Example 12.2 Circular Plate Problems

12.9 Complex Variable Methods for Plane Problems

Example 12.3 Annular Plate Problem

Example 12.4 Stresses Around a Circular Hole in an Infinite Plane Under

Uniform Heat Flow

Example 12.5 Stresses Around an Elliptical Hole in an Infinite Plane Under

Uniform Heat Flow

13. Displacement Potentials and Stress Functions (This chapter will present the use of displacement potentials and stress functions for the solution of three-dimensional problems. These important techniques provide convenient solution methods for a variety of classical problems. Further applications of this technique (the Papkovich method) appear in chapter 14 for development of singular stress states.)

13.1 Introduction

13.2 Helmoholtz Displacement Vector Representation

13.3 LamJ's Strain Potential

13.4 The Galerkin Vector Representation

Example 13.1 Kelvin's Problem: A Concentrated Force Acting in the Interior of an Infinite Solid

Example 13.2 Boussinesq's Problem: A Concentrated Force Acting Normal to the Free Surface of a Semi-Infinite Solid

Example 13.3 Cerruti's Problem: A Concentrated Force Acting Parallel to the Free Surface of a Semi-Infinite Solid

13.5 The Papkovich-Neuber Representation

Example 13.4 Boussinesq's Problem

13.6 Spherical Coordinate Formulations

Example 13.5 Spherical Cavity in an Infinite Medium Subject to Uniform Far-Field

Tension

13.7 Stress Functions

- Maxwell Stress Function Representation

- Morera Stress Function Representation

14. Micromechanics Applications (This unique chapter provides an introduction to the use of elasticity theory in micromechanical modeling of materials. A series of topics are covered to provide a background on several of the more common and popular theories that have been proposed in the literature. No other elasticity book provides such a presentation, and this material should prove to be useful in connecting the course with current material modeling.)

14.1 Introduction

14.2 Dislocation Modeling

Example 14.1 Edge Dislocation in x-Direction

Example 14.2 Screw Dislocation in z-Direction

14.3 Singular Stress States

Example 14.3 Concentrated Force in an Infinite Medium (Kelvin Problem)

Example 14.4 Kelvin State with Unit Loads in Coordinate Directions

Example 14.5 Force Doublet

Example 14.6 Force Doublet with a Moment (About (-Axis)

Example 14.7 Center of Compression/Dilatation

Example 14.8 Center of Rotation

Example 14.9 Half-Line of Dilatation

14.4 Elasticity Theory with Distributed Cracks

Example 14.10 Isotropic Dilute Crack Distribution

Example 14.11 Planar Transverse Isotropy with Dilute Crack Distribution

Example 14.12 Isotropic Self-Consistent Crack Distribution

14.5 Micropolar/Couple-Stress Elasticity

- Two-Dimensional Couple-Stress Theory

Example 14.13 Stress Concentration Around a Circular Hole: Micropolar Elasticity

14.5 Elasticity Theory with Voids

Example 14.14 Stress Concentration Around a Circular Hole: Elasticity with Voids

15. Numerical Methods: Finite and Boundary Element Methods (A brief introduction is given to the important numerical finite and boundary elements methods. Two-dimensional formulations are developed for each method, and some example applications are provided. Use of the MATLAB PDE Toolbox allows easy finite element solutions to be developed, and comparisons are made with analytical solutions from chapter 8.)

15.1 Introduction

15.2 Virtual Work Formulation

15.3 Weighted Residual Formulation

15.4 Displacement Interpolation and Element Development

15.5 Model Assembly and Boundary Conditions

Example 15.1 Rectangular Plate

15.6 MATLAB FEM CODE: PDE Toolbox

Example 15.2 Circular Hole in a Stressed Plate

Example 15.3 Circular Disk Under Diametrical Compression

15.7 Boundary Integral Formulation

15.8 Boundary Element Formulation

15.9 Calculation of Internal Stresses and Displacements

15.10 Boundary Element Example

Example 15.4 Circular Hole in a Stressed Plate

Appendix A: Basic Field Equations in Cartesian, Cylindrical and Spherical Coordinates

Appendix B: Stress Transformation Relations Between Cartesian, Cylindrical and Spherical

Appendix C: Theory of Photoelasticity

Appendix D: MATLAB Primer

## Details

- No. of pages:
- 480

- Language:
- English

- Copyright:
- © Academic Press 2005

- Published:
- 20th August 2004

- Imprint:
- Academic Press

- eBook ISBN:
- 9780080477473

## About the Author

### Martin Sadd

Martin H. Sadd is Emeritus Professor of Mechanical Engineering and Applied Mechanics at the University of Rhode Island. He received his Ph.D. in mechanics from the Illinois Institute of Technology and began his academic career at Mississippi State University. In 1979 he joined the faculty at Rhode Island and served as department chair from 1991 to 2000. Professor Sadd’s teaching background is in the area of solid mechanics with emphasis in elasticity, continuum mechanics, wave propagation, and computational methods. He has taught elasticity at two academic institutions, in several industries, and at a government laboratory. Professor Sadd’s research has been in the area of computational modeling of materials under static and dynamic loading conditions using finite, boundary, and discrete element methods. Much of his work has involved micromechanical modeling of geomaterials including granular soil, rock, and concretes. He has authored more than 70 publications and has given numerous presentations at national and international meetings.

### Affiliations and Expertise

Mechanical Engineering & Applied Mechanics Department, University of Rhode Island, USA

## Reviews

This book is a welcome addition to the set of textbooks available to beginning graduate students and advanced undergraduates in mechanical engineering. I have previously taught the subject of elasticity from textbooks written by Barber, Salughter, and Shames with frequent references to the classics by Timoshenko, Love, Sokolnikoff, and Green and Zerna. However, students have found these books either too difficult to understand or too dated in their notation. When I received the new textbook by Professor Sadd, I read it briefly and then handed it over to one of my graduate students who was preparing for his qualifying examinations. His response was unqualified admiration of the ease with which he was able to navigate the book and grasp its contents. - Biswajit Banerjee - Department of Mechanical Engineering, University of Utah