Theory, Applications, and Numerics To order this title, and for more information, click here
By Martin Sadd, Professor, Mechanical Engineering & Applied Mechanics Department,
University of Rhode Island, USA
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. Complemented
by a Solutions Manual and including MatLab codes and coding, this text is an excellent book teaching guide.
Audience
graduate students, engineering industries and governmental research centers
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
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Theory, Applications, and Numerics
