Elasticity is a well-established subject, and many books have been written on diverse aspects of elasticity theory. In this course, we will concentrate on the application of small-strain linear elasticity to the exact or approximate solution of many engineering problems. This course is divided into two main parts; the first emphasizes formulation details and elementary applications.

Introduction to Elasticity: What is Elasticity? A Brief History of Elasticity; Tools of the Trade; Engineering Applications of Elasticity; Fundamental Concepts in Elasticity; Assumptions of Elasticity Theory; Geometry of Elastic Solids; Topics That Will Be Covered; Greek Alphabet.

Mathematical Preliminary: Scalar, Vector and Matrix; Indicial Notation and Summation Convention; Kronecker Delta; Alternating Symbol; Coordinate Transformation; Tensor; Principal Values and Directions; Tensor Algebra; Tensor Calculus; Integral Theorems; Curvilinear Coordinates.

Deformation – Displacement & Strain: Generalized Displacement; Small Deformation Theory; Continuum Motion & Deformation; Strain & Rotation; Principal Strains; Spherical and Deviatoric Strain; Strain Compatibility; Domain Connectivity; Cylindrical Strain and Rotation; Spherical Strain and Rotation.

Stress and Equilibrium: Body and Surface Forces; Traction/Stress Vector; Stress Tensor; Traction on Oblique Planes; Principal Stresses and Directions; Mohr’s Circles of Stresses; Octahedral Stresses; Spherical and Deviatoric Stresses; Conservation of Linear Momentum; Conservation of Angular Momentum; Equilibrium Equations; Equilibrium Equations in Curvilinear Coordinates.

Constitutive Laws: Strain Energy; Linear Constitutive Relations; Anisotropy and Stiffness Tensor; Isotropic Hooke’s Law; Physical Meaning of Elastic Moduli; Simple Engineering Tests; Relationships among Elastic Constants; Hooke’s Law in Curvilinear Coordinates; Thermoelastic Constitutive Relations; Typical Values of Elastic Constants; Inhomogeneity.

Formulation and Solution Strategies: Review of Field Equations; Types of BCs; BCs on Coordinate Surfaces; BCs on Oblique Surface; Line of Symmetry BCs; Interface BCs; Problem Classification; Stress Formulation; Displacement Formulation; Principle of Superposition; Uniqueness of Elastic Solution; Saint-Venant’s Principle; Solution Strategies; Mathematical Techniques.

Two-Dimensional Formulation: Introduction; Two vs. Three-Dimensional Problems; Plane Strain; Plain Stress; Boundary Conditions; Correspondence between Plane Strain and Plain Stress; Combined Plane Formulations; Anti-Plane Strain; Airy Stress Function; Polar Coordinate Formulation.

Two-Dimensional Problems in Cartesian Coordinates: Introduction; Polynomial Solutions; Uniaxial Tension of a Beam; Pure Bending of a Beam; Beam under Uniform Transverse Loading; River Dam; Fourier Methods; Beam under Sinusoidal Loading; Rectangular Domain with Arbitrary Boundary Loading.

Two-Dimensional Problems in Polar Coordinates: Polar Coordinate Formulation; Axisymmetric Solutions to Biharmonic Equations; Cylinders under Boundary Pressures; Hole in Infinite Media; Pure Bending of Curved Beams; General Solutions to Biharmonic equation; Stress Concentration around a Hole; Wedge Problems; Quarter-Plane Problems; Half-Plane Problems; Disk under Diametrical Compression; Transverse Bending of Curved Beams; Notch/Crack Problem; Rotating Disk/Cylinder Problem.

Part II of the course continues the study into more advanced topics:

Elastic Cylinders Subjected to End Loadings: Introduction; Extension of Cylinders; Torsion of Cylinders; Stress Function Formulation; Displacement Formulation; Multiply Connected Cross-Sections; Membrane Analogy; Boundary Equation Scheme; Fourier Method – Rectangular Section; Hollow Sections; Bending of Cylinders; Bending of Circular, Rectangular, and Elliptic Cylinders without Twisting.

Energy Method and Variational Principle:Work Done by External Load; Strain Energy; Complimentary Strain Energy; The Delta Operator; Principle of Virtual Work; Principle of Minimum Potential Energy; Castigliano’s First Theorem; Ritz Method; Galerkin Method; Principle of Complimentary Virtual Work; Principle of Minimum Complimentary Potential Energy; Castigliano’s Second Theorem; Variational Approach in terms of Stress; Application in Plane Elasticity.

Three-Dimensional Problems: Displacement Formulation Review; Half-Space under Uniform Pressure and Gravity; Spherical Shell; General Solution – Helmholtz Representation; Particular Case – Lamé Strain Potential; Galerkin Vector Potential; Love Strain Potential – Axi-symmetry; Completeness of Displacement Potentials; Harmonic and Bi-harmonic Functions; Kelvin’s Problem；Boussinesq’s Problem；Cerruti’s Problem；Distributed Pressure on Half-Space; Hertz Contact Problem; Stress Formulation Review; Pure Bending of Straight Beams.

Thermoelasticity (2-D): Heat Conduction Equation; General 3-D Formulation; Combined Plane Hooke’s Law; Stress Compatibility and Airy Stress Function; Displacement Equilibrium and Displacement Potentials; Thermal Stresses in Thin-Plates; Summary of Solution Strategy; Polar Coordinate: Airy Stress Function; Polar Coordinate: Displacement Potentials; Axi-Symmetric Problems – Direct Solution; Thermal Stresses in Circular Plates.

Bending of Thin Plates: Introduction；Elementary Beam Theory; Assumptions; Formulation in terms of Deflection; Internal Force per Unit Length; Relations between Internal Force and Stress; Differential Element Equilibrium – Alternative Approach; Boundary Conditions; Boundary Element Scheme; Fourier Method; Summary.

Numerical Method: Finite Element Method; Finite Difference Method; Boundary Element Method; Method of Weighted Residuals.