Introduction to Linear Elasticity

1. Introduction and Mathematical Preliminaries.- 1.1 Scope.- 1.2 Vector Algebra.- 1.3 Scalar and Vector Fields.- 1.3.1 Definitions.- 1.3.2 Gradient.- 1.3.3 Operators.- 1.3.4 Divergence.- 1.3.5 Curl.- 1.3.6 Integral Theorems.- 1.4 Indicial Notation.- 1.5 Coordinate Rotation.- 1.6 Cartesian Tensors.- 1.7 Algebra of Cartesian Tensors.- 1.8 Operational Tensors.- 1.9 Computational Examples.- Exercises.- References.- 2. Traction, Stress and Equilibrium.- 2.1 Introduction.- 2.2 State of Stress.- 2.2.1 Traction and Couple-Stress Vectors.- 2.2.2 Components of Stress.- 2.2.3 Stress at a Point.- 2.2.4 Stress on a Normal Plane.- 2.2.5 Dyadic Representation of Stress.- 2.2.6 Computational Example.- 2.3 Equilibrium.- 2.3.1 Physical and Mathematical Principles.- 2.3.2 Linear Momentum.- 2.3.3 Angular Momentum.- 2.3.4 Computational Example.- 2.4 Principal Stress.- 2.4.1 Definition and Derivation.- 2.4.2 Computational Format, Stress Invariants and Principal Coordinates.- 2.4.3 Computational Example.- 2.5 Stresses in Principal Coordinates.- 2.5.1 Stresses on an Oblique Plane.- 2.5.2 Stresses on Octahedral Planes.- 2.5.3 Absolute Maximum Shearing Stress.- 2.5.4 Computational Example.- 2.6 Properties and Special States of Stress.- 2.6.1 Projection Theorem.- 2.6.2 Plane Stress.- 2.6.3 Linear Stress.- 2.6.4 Pure Shear.- 2.6.5 Hydrostatic Stress.- Exercises.- References.- 3. Deformations.- 3.1 Introduction.- 3.2 Strain.- 3.3 Physical Interpretation of Strain Tensor.- 3.4 Principal Strains.- 3.5 Volume and Shape Changes.- 3.6 Compatibility.- 3.7 Computational Example.- Exercises.- References.- 4. Material Behavior.- 4.1 Introduction.- 4.2 Uniaxial Behavior.- 4.3 Generalized Hooke's Law.- 4.4 Thermal Strains.- 4.5 Physical Data.- Exercises.- References.- 5. Formulation, Uniqueness and Solution Strategies.- 5.1 Introduction.- 5.2 Displacement Formulation.- 5.3 Force Formulation.- 5.4 Other Formulations.- 5.5 Uniqueness.- 5.6 Membrane Equation.- 5.7 Solution Strategies.- Exercises.- References.- 6. Extension, Bending and Torsion.- 6.1 Introduction.- 6.2 Prismatic Bar under Axial Loading.- 6.3 Cantilever Beam under End Loading.- 6.3.1 Elementary Beam Theory.- 6.3.2 Elasticity Theory.- 6.4 Torsion.- 6.4.1 Torsion of Circular Shaft.- 6.4.2 Torsion of Solid Prismatic Shafts.- 6.4.3 Torsion of Elliptical Shaft.- 6.4.4 Membrane Analogy.- Exercises.- References.- 7. Two-Dimensional Elasticity.- 7.1 Introduction.- 7.2 Plane Stress Equations.- 7.3 Plane Strain Equations.- 7.4 Cylindrical Coordinates.- 7.4.1 Geometric Relations.- 7.4.2 Transformation of Stress Tensor and Compatibility Equation.- 7.4.3 Axisymmetric Stresses and Displacements.- 7.5 Thick-Walled Cylinder or Disk.- 7.6 Sheet with Small Circular Hole.- 7.7 Curved Beam.- 7.8 Rotational Dislocation.- 7.9 Narrow, Simply Supported Beam.- 7.10 Semi-Infinite Plate with a Concentrated Load.- Exercises.- References.- 8. Bending of Thin Plates.- 8.1 Introduction.- 8.2 Assumptions.- 8.3 Formulation.- 8.3.1 Geometric Relationships.- 8.3.2 Strains and Stresses.- 8.3.3 Plate Equation.- 8.3.4 Polar Coordinates.- 8.4 Solutions.- 8.4.1 Rectangular Plate.- 8.4.2 Circular Plate.- 8.5 Commentary.- Exercises.- References.- 9. Time-Dependent Effects.- 9.1 Introduction.- 9.2 Vibrations in an Infinite Elastic Medium.- 9.2.1 Equilibrium Equations.- 9.2.2 Longitudinal Vibrations.- 9.2.3 Transverse Vibrations.- 9.2.4 Harmonic Vibrations.- 9.3 Free Vibration.- 9.3.1 Equations of Motion.- 9.3.2 Orthogonality Conditions.- 9.3.3 Rayleigh's Quotient.- 9.3.4 Axial Vibration of a Bar.- 9.4 Uniform Rotation of a Beam.- 9.4.1 Equilibrium Equations.- 9.4.2 Boundary Conditions.- 9.4.3 Semi-Inverse Solution.- 9.4.4 Two-Dimensional Problem.- 9.4.5 Circular Cross Section.- Exercises.- References.- 10. Energy Principles.- 10.1 Introduction.- 10.2 Conservation of Energy.- 10.3 Strain Energy.- 10.3.1 Strain Energy Density.- 10.3.2 Strain Energy Density of Distortion.- 10.4 Work of External Loading.- 10.5 Principle of Virtual Work.- 10.5.1 Definitions.- 10.5.2 Principle of Virtual Displacements.- 10.5.3 Principle of Virtual Forces.- 10.5.4 Reciprocal Theorems.- 10.6 Variational Principles.- 10.6.1 Definitions.- 10.6.2 Principle of Minimum Total Potential Energy.- 10.6.3 Principle of Minimum Complementary Energy.- 10.7 Direction Variational Methods.- 10.7.1 Motivation.- 10.7.2 Rayleigh-Ritz Method.- 10.7.3 Torsion of Rectangular Cross Section.- 10.7.4 Commentary.- Exercises.- References.- 11. Strength and Failure Criteria.- 11.1 Introduction.- 11.2 Isotropic Materials.- 11.2.1 Classical Tests.- 11.2.2 Failure Theories.- 11.2.3 Invariants.- 11.3 Yield Surfaces.- 11.3.1 General.- 11.3.2 Tresca Yield Condition.- 11.3.3 von Mises Yield Condition.- 11.3.4 General Criterion for Isotropic Media.- 11.4 Anisotropic Materials.- 11.4.1 Objectives.- 11.4.2 Failure Surface.- 11.4.3 Specializations.- 11.4.4 Evaluation of Components.- 11.5 Failure of Structures.- Exercises.- References.- 12. Something New.