Elasticity

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 Introduction1.2 Scalar, Vector and Matrix Notation1.3 Index Notational Properties1.4 Kronecker Delta and Alternating Symbol1.5 Determinants1.6 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 Tensors1.8 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 Tensors1.10 Tensor Fields, Derivatives and Integral Theorems1.11 Orthogonal Curvilinear Coordinates2. 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 Introduction2.2 General Deformations2.3 Small Deformation Theory - Strain-Displacement Relations2.4 Principal Strains2.5 Deviatoric Strains2.6 Strain Compatibility2.7 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 Introduction3.2 Body and Surface Forces3.3 The Traction Vector3.4 The Stress Tensor3.5 Principal Stresses3.6 Deviatoric Stresses3.7 Stress Transformation (Similar to section 2.7, MATLAB applications will be introduced for specific stress transformation problems)3.8 Equilibrium Equations4. 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 Introduction4.2 Generalized Hooke's Law4.3 Isotropic Case4.4 Thermoelastic Relations4.5 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 Introduction5.2 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 Superposition5.6 Saint-Venant's Principle5.7 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 MethodExample 5.1 Stretching of Prismatic Bar Under Its Own Weight- Inverse MethodExample 5.2 Pure Beam Bending - Semi_Inverse MethodExample 5.3 Torsion of Prismatic Bars - Analytical Solution ProceduresPower Series, Fourier Methods, Integral Transform and Complex VariableTechniques - Approximate Solution ProceduresRitz Technique - Numerical Solution ProceduresFinite Difference, Finite and Boundary Element Techniques6. 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 Introduction6.2 Strain Energy6.3 Uniqueness of the Elasticity Boundary-Value Problem6.4 Bounds on the Elastic Constants6.5 Reciprocal Theorem6.6 Integral Formulation of Elasticity: Somigliana's Identity6.7 Principle of Virtual Work 6.8 Principles of Minimum Potential and Complimentary Energy6.9 Ritz Method7. 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 Introduction7.2 Plane Strain7.3 Plane Stress7.4 Generalized Plane Stress7.5 Airy Stress Function7.6 Polar Coordinate Formulation8. 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 Introduction8.2 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 Loading8.3 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 Loading8.4 General Solutions in Polar Coordinates - Axisymmetric Solution - Mitchell Solution8.5 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 Problem9. 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 Introduction9.2 Extension Formulation Example 9.1 Longitudinal Loading Example9.3 Torsion Formulation Stress Function Formulation Displacement Formulation Membrane Analogy9.4 Torsion Solutions Derived from Boundary Equation Example 9.2 Elliptical Section Example 9.3 Equilateral Triangular Section9.5 Torsion Solution Using Fourier Methods Example 9.4 Rectangular Section9.6 Torsion of Multiply Connected Cross Sections Example 9.5 Hollow Elliptical Section9.7 Torsion of Circular Shafts of Variable Diameter Example 9.6 Conical Shaft9.8 Flexure Formulation9.9 Flexure Problems without Twist Example 9.7 Circular Section Example 9.8 Rectangular SectionPart 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 Introduction10.2 Review of Complex Variable Theory10.3 Complex Formulation of the Plane Problem Example 10.1 Constant Stress State Example10.4 Resultant Boundary Conditions10.5 General Structure of the Complex Potentials- Finite Simply-Connected Domains- Finite Multiply-Connected Domains- Infinite Domains10.6 Circular Domain Examples Example 10.2 Disk Under Uniform Compression Example 10.3 Circular Plate with Concentrated Edge Loading10.7 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 Hole10.8 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 Analysis11. 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 Introduction11.2 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 Cube11.3 Restrictions on the Elastic Moduli11.4 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 Bar11.5 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 Hole11.6 Applications to Fracture Mechanics12. 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 Introduction12.2 Heat Conduction and the Energy Equation12.3 General Uncoupled Formulation12.4 Two-Dimensional Formulation- Plane Strain- Plane Stress12.5 Displacement Potential Solution12.6 Stress Function Formulation Example 12.1 Thermal Stresses in an Elastic Strip12.7 Polar Coordinate Formulation12.8 Radially Symmetric Problems Example 12.2 Circular Plate Problems12.9 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 Flow13. 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 Introduction13.2 Helmoholtz Displacement Vector Representation13.3 LamJ's Strain Potential13.4 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 Solid13.5 The Papkovich-Neuber Representation Example 13.4 Boussinesq's Problem13.6 Spherical Coordinate Formulations Example 13.5 Spherical Cavity in an Infinite Medium Subject to Uniform Far-Field Tension13.7 Stress Functions - Maxwell Stress Function Representation- Morera Stress Function Representation14. 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 Introduction14.2 Dislocation ModelingExample 14.1 Edge Dislocation in x-DirectionExample 14.2 Screw Dislocation in z-Direction14.3 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 Dilatation14.4 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 Distribution14.5 Micropolar/Couple-Stress Elasticity - Two-Dimensional Couple-Stress TheoryExample 14.13 Stress Concentration Around a Circular Hole: Micropolar Elasticity14.5 Elasticity Theory with Voids Example 14.14 Stress Concentration Around a Circular Hole: Elasticity with Voids15. 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 Introduction15.2 Virtual Work Formulation15.3 Weighted Residual Formulation15.4 Displacement Interpolation and Element Development15.5 Model Assembly and Boundary Conditions Example 15.1 Rectangular Plate15.6 MATLAB FEM CODE: PDE Toolbox Example 15.2 Circular Hole in a Stressed Plate Example 15.3 Circular Disk Under Diametrical Compression15.7 Boundary Integral Formulation15.8 Boundary Element Formulation15.9 Calculation of Internal Stresses and Displacements15.10 Boundary Element Example Example 15.4 Circular Hole in a Stressed PlateAppendix A: Basic Field Equations in Cartesian, Cylindrical and Spherical CoordinatesAppendix B: Stress Transformation Relations Between Cartesian, Cylindrical and SphericalAppendix C: Theory of PhotoelasticityAppendix D: MATLAB Primer