This book examines the issues across the breadth of elasticity theory. Firstly, the underpinning mathematics of vectors and matrices is covered. Thereafter, the equivalence between the inidicial, symbolic and matrix notations used for tensors is illustrated in the preparation for specific types of material behaviour to be expressed, usually as a response function from which a constitutive stress-strain relation follow.
Mechanics of Elastic Solids shows that the elastic response of solid materials has many forms. Metals and their alloys confirm dutifully to Hooke's law. Non-metals do not when the law connecting stress to strain is expressed in polynomial, exponential and various empirical, material specific forms. Hyper- and hypo- elasticity theories differ in that the former is restricted to its thermodynamic basis while the latter pervades many an observed response with its release from thermal restriction, but only at the risk of contravening the laws of thermodynamics.
This unique compendium is suitable for a degree or diploma course in engineering and applied mathematics, as well as postgraduate and professional researchers.
Author(s): David W. A. Rees
Publisher: World Scientific Publishing Europe Ltd.
Year: 2019
Language: English
Pages: xvi+737
CONTENTS
Symbols and Units
Preface
CHAPTER 1 VECTORS
1.1 Introduction
1.2 Properties
1.3 Addition and Subtraction
1.4 The Unit Vector
1.5 Direction Cosines
1.6 Position Vectors
1.6.1 Line
1.6.2 Parametric Equation of a Line
1.6.3 Plane
1.6.4 Parallel and Perpendicular Planes
1.6.5 Parametric Equation for a Plane
1.7 Vector Algebra
1.8 Scalar (Dot) Product
1.9 Vector (Cross) Product
1.10 Triple Products
References
Exercises
CHAPTER 2 MATRICES AND DETERMINANTS
2.1 Introduction
2.2 Matrix Formation
2.3 Matrix Types
2.4 Transpose of a Matrix
2.5 Addition and Subtraction
2.6 Multiplication
2.7 Pre- and Post-Multiplication
2.8 Matrix Operations
2.9 The Inverse of a Matrix
2.9.1 Co-Factors and Determinant
2.9.2 Partition Method
2.9.3 Upper and Lower Triangular Matrices
2.9.4 Gaussian Elimination
2.9.5 Canonical Inversion
2.10 Matrix Function Transformation
2.11 The Rotation Matrix
2.12 Vector and Tensor Transformations
2.13 Powers and Functions of a Matrix
2.14 The Characteristic Equation
2.15 Determinants
2.15.1 Properties and Laws
2.15.2 Applications Through Example
Reference
Exercises
Answers to Selected Exercises
CHAPTER 3 THE INDEX NOTATION
3.1 Introduction
3.2 Summation Convention
3.3 Kronecker Delta
3.4 Alternating Unit Tensor
3.5 Vector Transformation Equations
3.5.1 Three Dimensions
3.5.2 Two Dimensions
3.6 Tensor Transformation Equations
3.6.1 Second Order
3.6.2 Higher Orders
3.6.3 Two Dimensions
3.7 Dual Vector and Scalar
3.8 Vector Products
3.9 Vector Identities
3.10 Principal Co-ordinates
3.10.1 Symmetric Matrix
3.10.2 Non-Symmetric Matrix
3.11 Eigen Vectors as Co-ordinates
3.12 Canonical Form of a Matrix
3.12.1 Symmetry
3.12.2 Non-Symmetry
3.13 Powers of a Matrix
3.13.1 Positive Integer Powers
3.13.2 Negative and Fractional Powers
3.14 Functions of a Matrix
3.14.1 Exponential
3.14.2 Arbitrary Forms
3.14.3 Simultaneous Differential Equations
3.14.4 Higher-Order Differential Equations
3.14.5 Constitutive Relation
3.15 Skew Symmetric Matrix
Bibliography
Exercises
CHAPTER 4 STRESS ANALYSIS
4.1 Introduction
4.1.1 Direct Stress and Strain
4.1.2 Shear Stress and Strain
4.2 Cauchy Stress Tensor
4.2.1 Stress Intensity
4.2.2 General Stress State
4.2.3 Stress Tensor
4.3 Stress Analysis in 3D
4.3.1 Direction Cosines
4.3.2 Force Resolution
4.3.3 Stress Transformations in Tensor and Matrix Notations
4.4 Invariants and Principal Stress
4.4.1 Magnitudes of Principal Stresses
4.4.2 Principal Stress Directions
4.4.3 Reductions to Plane Stress
4.5 Principal Stress Co-ordinates
4.5.1 Maximum Shear Stress
4.5.2 Octahedral Plane
4.5.3 Plane Stress Reductions
4.6 Piola and Kirchoff Stress Tensors
4.7 Relationships Between Stress Tensors and F
Bibliography
Exercises
CHAPTER 5 STRAIN ANALYSIS
5.1 Introduction
5.1.1 Contraction
5.1.2 Inner Product
5.1.3 Outer Product
5.1.4 Co-ordinate Transformation
5.1.5 Inverse Co-ordinate Transformation
5.2 Infinitesimal Strain Tensor
5.2.1 Distortion, Strain and Rotation
5.2.2 Strain Transformation
5.2.3 Reduction to Plane Strain
5.3 Large Strain Measures
5.3.1 True Strain
5.3.2 Extension Ratio
5.3.3 Finite Homogenous Strain
5.4 Finite Strain Tensors
5.4.1 Lagrangian
5.4.2 Eulerian
5.4.3 Extension (Stretch) Ratio
5.4.4 Relationship Between Extension Ratios
5.4.5 Finite Principal Strains
5.4.6 Velocity and Acceleration
5.4.7 Angular Velocity Vector and Tensor
5.4.8 Deformation Rate and Spin
5.4.9 Finite Strain Rates
5.5 Stretch, Rotation and Translation
5.5.1 Line Element in Spacial Co-ordinates
5.5.2 Rigid Body Motion
5.5.3 The Rotation Matrix
5.5.4 Rigid Body Angular Velocity
5.5.5 Plane Stretch and Rotation
5.6 Polar Decomposition
5.6.1 Geometrical Interpretation
5.6.2 Relationship Between Stretch Tensors
5.6.3 Reduction to Infinitesimal Strain
5.6.4 Right and Left Stretch Tensors
5.6.5 Principal Stretch Tensors
5.7 Strain Compatibility
5.8 Concluding Remarks
References
Exercises
CHAPTER 6 PLANE ELASTICITY THEORY
6.1 Introduction
6.2 Elastic Constants
6.2.1 Tension and Compression
6.2.2 Shear and Torsion
6.2.3 Hydrostatic Pressure
6.2.4 Relationships
6.3 Stress and Strain Tensors
6.3.1 Cauchy Stress
6.3.2 Eulerian Strain and Rotation
6.3.3 Alternative Notation
6.4 Linear Constitutive Relations
6.4.1 Principal Triaxial Form
6.4.2 Principal Biaxial Form
6.4.3 General Isotropic Forms
6.5 Plane Cartesian Elasticity
6.5.1 Constitutive Relations for Plane Stress
6.5.2 Constitutive Relations for Plane Strain
6.5.3 Force Equilibrium
6.5.4 Compatibility Conditions
6.5.5 Bi-Harmonic Equation and Stress Functions
6.5.6 Body Forces
6.5.7 Polynomial Series for Plates and Bars
6.5.8 Fourier Series for Edge Loading
6.6 Plane Cylindrical Elasticity
6.6.1 Strain-Displacement Relations
6.6.2 Strain Compatibility
6.6.3 Equilibrium Equations
6.6.4 Bi-Harmonic Equation without Body Forces
6.6.5 Stress Functions without Body Forces
6.6.6 Body Forces
6.6.7 Axial Symmetry
6.7 Summary of Plane Elasticity Theory
6.8 Polar Stress Functions
6.8.1 Stress Concentration Around a Hole
6.8.2 Displacements Around a Hole Under Tension
6.8.3 Point Loading
6.8.4 Superposition and Integral Equations
6.8.5 Maximum Shear Stress and Principal Stress
6.8.6 Stress Functions Independent of r
6.8.7 Semi-Infinite Plate
6.8.8 Disc Under Compression
6.9 Infinite Sheet with Loaded Stiffener
6.9.1 Remote Uniform Tension
6.9.2 Non-Symmetrical Loading
6.10 Stress Concentration for an Elliptical Hole
6.10.1 Uniaxial Loading
6.10.2 Biaxial Loading
6.11 Concluding Remarks
References
Exercises
CHAPTER 7 EXPERIMENTAL ELASTICITY
7.1 Introduction
7.2 The Photoelastic Bench
7.2.1 Bi-Refringence
7.2.2 Material and Model Calibrations
7.3 Beam Stress Analysis
7.4 Photoelastic Coating
7.5 Principal Stress Separation along a Stress Trajectory
7.5.1 Lamê-Maxwell Equations
7.5.2 Filon’s Transformation
7.5.3 Axes of Symmetry
7.6 Slope-Equilibrium Method of Principal Stress Separation
7.6.1 Tie Bar With Central Hole
7.6.2 Notched Beam
7.6.3 Ring Under Compression
7.7 Shear Difference Method of Component Stress Separation
7.8 Strain Gauges and the Wheatstone Bridge
7.8.1 Bridge in Balance
7.8.2 Strain Gauge Bridge
7.8.3 Voltage from an Unbalanced Bridge
7.8.4 AC Bridge
7.9 Strain Conversion
7.9.1 Gauge Factor
7.9.2 Cross Sensitivity
7.10 Strain Gauge Rosette
7.10.1 Rosette Analysis
7.10.2 Cross Sensitivity in Rosettes
7.11 Hole in a Stressed Plate
7.11.1 Photoelasticity
7.11.2 Strain Gauges
7.12 Central Slot and Edge-Notch in a Stressed Plate
7.12.1 Strain Gauges
7.12.2 Photoelasticity
7.12.3 Finite Elements
References
Exercises
CHAPTER 8 ANISOTROPIC ELASTICITY
8.1 Introduction
8.2 Constitutive Relations
8.3 Crystal Symmetries
8.3.1 Monoclinic
8.3.2 Orthotropic
8.3.3 Transverse Isotropy
8.3.4 Cubic
8.3.5 Isotropic
8.4 Compliance and Stiffness Matrix Symmetry
8.4.1 Isotropic Symmetry
8.4.2 Cubic Symmetry
8.4.3 Cylindrical and Hexagonal Symmetries
8.4.4 Orthotropic Symmetry
8.4.5 Special Orthotropy
8.4.6 Transverse Isotropy
8.4.7 Orthotropic to Isotropic Reduction
8.5 Global Transverse Isotropy
8.5.1 Off-Axis tension
8.5.2 Off-Axis Shear and Torsion
8.5.3 Predictions and Experimental Results
8.6 Orthotropic Material Under Plane Stress
8.6.1 Matrix Transformation
8.6.2 Vector Transformation
8.6.3 Coefficients of Mutual Influence
8.6.4 Poisson’s Ratios
8.6.5 Cross-Plane Shear Coefficients
8.6.6 Chentsov’s Coefficients
8.6.7 Cross-Plane Thickness Coefficients
8.6.8 Symmetry in Off-Axis Coefficients
8.6.9 Transformed Compliance Matrix
8.6.10 Principal Axes of Stress and Strain
8.7 Loading Woven and Cross-Ply Composites
8.7.1 Off-Axis Tension and Shear
8.7.2 Off-Axis Bending
8.7.3 Off-Axis Torsion
8.7.4 Average Modulus Prediction
8.7.5 Weighted Stress
8.7.6 Equilibrium and Compatibility
8.7.7 Interfacial Shear
8.8 Laminated Composite under Plane Stress
8.8.1 Global Elastic Constants for an Inclined Laminate
8.8.2 Inversion of the Global Compliance Matrix
8.8.3 Non-Dimensional Stiffness Matrix
8.9 Concluding Remarks
References
Exercises
CHAPTER 9 NON-LINEAR ELASTICITY
9.1 Introduction
9.2 Generalised Elastic Relations
9.2.1 Stress Invariant Function
9.2.2 Secant Moduli Gs and Ks
9.2.3 Tangent Moduli Et, Gt and Kt
9.2.4 Combined Secant and Tangent Moduli
9.2.5 Variable Moduli
9.3 Elastic-Plastic Theory
9.3.1 Yield Function
9.3.2 Variable Moduli
9.3.3 Flow Rule
9.3.4 Octahedral Equivalence
9.3.5 Pre-Strain and Hysteresis Effects
9.3.6 Hencky Theory
9.4 Hyper-Elastic Theory
9.4.1 Strain Energy
9.4.2 Potential Energy
9.4.3 Complementary Energy
9.4.4 Complementary Work
9.5 Hypo-Elastic Theory
9.5.1 Total Strain Theory
9.5.2 Strain Invariant Basis
9.5.3 Stress Invariant Basis
9.6 Incremental Hypo-Elastic Theory
9.6.1 Stiffness and Compliance
9.6.2 First-Order Reductions
9.6.3 General First-Order Forms
9.6.4 Higher-Order Forms
9.7 Thermodynamic State Parameters
9.7.1 Laws of Thermodynamics
9.7.2 Elastic Deformation
9.7.3 Elastic Cycles with Full Recovery
9.7.4 Cyclic Elastic-Plastic Deformation
9.7.5 Non-Cyclic Elastic-Plastic Deformation
9.8 Rheological Models
9.8.1 Standard Linear Solids
9.8.2 Maxwell-Weichert Elements in Parallel
9.8.3 Voigt–Kelvin Elements in Series
9.9 Curve Fitting Matrices
9.9.1 Polynomial Function σ = σ(e) with Common Strain Cut-Off
9.9.2 Polynomial Function e = e(σ) with Common Stress Cut-Off
9.10 Uniaxial Rate and Temperature Effects
9.10.1 Rate Independence
9.10.2 Strain (Displacement) Rate
9.10.3 Stress (Load) Rate
9.10.4 Temperature Influence at Constant Strain Rate
9.10.5 Temperature Influence at Constant Stress Rate
9.10.6 Stiffness and Compliance
9.10.7 Varying Temperature
References
Exercises
CHAPTER 10 FINITE ELASTICITY
10.1 Introduction
10.2 Finite Non-Linear Elasticity
10.2.1 Applications to Incompressible Material
10.2.2 Applications to Compressible Material
10.2.3 Uniaxial Tension
10.2.4 Homogenous Stretching
10.2.5 Simple Shear
10.2.6 Plane Strain Compression
10.2.7 Bulge Test
10.2.8 Channel Die Compression
10.3 Rubber Elasticity
10.3.1 Free Energy
10.3.2 Entropy Change
10.3.3 Uniaxial Stress
10.3.4 Pure Shear
10.4 Elastic Hysteresis
10.4.1 Empirical Description
10.4.2 Points of Inflection
10.4.3 Master Curves
10.5 Hysteresis in Polymers
10.5.1 Thermodynamics
10.5.2 Multi-Axial Loading
10.5.3 Back Stress Model
10.6 Compression of Foams
10.6.1 Exponential Inflection
10.6.2 Hysteresis with Inverse Reflection
10.6.3 Hysteresis Loop Fitting
10.7 Standard Linear Solids
10.7.1 SLS Type I
10.7.2 SLS Type II
10.8 Alternative Spring-Dashpot Connections
10.8.1 Maxwell-Weichert Elements in Parallel
10.8.2 Voigt-Kelvin Elements in Series
10.8.3 Non-Linear Maxwell Elements in Series
10.8.4 Non-Linear Voigt Elements in Parallel
References
Bibliography
Exercises
INDEX
