## Theory of Elasticity

(Department)  Biomedical Engineering         (Division)

(Level and Major)

Course Title                Theory of Elasticity

Number of Credits       3             Prerequisite

Course Topics:
• Mathematical Preliminaries, Kronecker Delta and Alternating Symbol
• Coordinate Transformations & Cartesian Tensors
• Vector, Matrix, and Tensor Algebra, Calculus of Cartesian Tensors
• Orthogonal Curvilinear Coordinates
• Deformation: Displacements and Strains
• Geometric Construction of Small Deformation Theory
• Strain Transformation, Principal Strains, Spherical and Deviatoric Strains
• Strain Compatibility
• Curvilinear Cylindrical and Spherical Coordinates
• Stress and Equilibrium
• Body and Surface Forces
• Traction Vector and Stress Tensor
• Stress Transformation, Principal Stresses, Spherical and Deviatoric Stresses
• Equilibrium Equations,
• Relations in Curvilinear Cylindrical and Spherical Coordinates
• Material Behavior—Linear Elastic Solids, Material Characterization
• Linear Elastic Materials—Hooke’s Law, Physical Meaning of Elastic Moduli
• Thermoelastic Constitutive Relations,
• Formulation and Solution Strategies,
• Review of Field Equations,
• Boundary Conditions and Fundamental Problem Classifications,
• Stress & Displacement Formulations
• Principle of Superposition,
• Saint-Venant’s Principle,
• General Solution Strategies,
• Strain Energy and Related Principles,
• Uniqueness of the Elasticity Boundary-Value Problem,
• Bounds on the Elastic Constants,
• Related Integral Theorems,
• Principle of Virtual Work,
• Principles of Minimum Potential and Complementary Energy,
• Rayleigh-Ritz Method,
• Two-Dimensional Formulation,
• Plane Strain & Plane Stress,
• Generalized Plane Stress,
• Antiplane Strain,
• Airy Stress Function,
• Polar Coordinate Formulation,
• Two-Dimensional Problem Solution,
• Cartesian Coordinate Solutions Using Polynomials,
• Cartesian Coordinate Solutions Using Fourier Methods,
• General Solutions in Polar Coordinates,
• Polar Coordinate Solutions,
• Extension, Torsion, and Flexure of Elastic Cylinders, General Formulation,
• Torsion Formulation & Torsion Solutions Derived from Boundary Equation,
• Torsion Solutions Using Fourier Methods,
• Torsion of Cylinders With Hollow Sections,
• Flexure Formulation & Flexure Problems Without Twist,
• Some Aspects of Objectivity, Change of Observer, and Objective Tensor Fields
• Objective Rates
• Invariance of Elastic Material Response
• Isotropic Hyperelastic Materials,
• Incompressible & Compressible Hyperelastic Materials,
• Some Forms of Strain-energy Functions,
• Elasticity Tensors,
• Transversely Isotropic Materials
• Hyperelastic Composite Materials with Two Families of Fibers