Applied Elasticity

Week 1:  Introduction and Mathematical Preliminaries: Tensor algebra and indicial notation, Transformation of tensors, Tensor calculus.

Week 2: Deformation and Strain: Lagrangian and Eulerian descriptions of deformation, Deformation gradient tensor, Green-Lagrange and Eulerian strain tensors, Infinitesimal strain tensor, Dilation, Strain compatibility equations.

Week 3: Stress Measures and Balance Laws: Surface traction vector, Cauchy stress tensor at a point, Cauchy stress formula, First and second Piola Kirchhoff stress tensors, Mass, momentum, angular momentum, and energy balance, Stress power.

 Week 4: Constitutive Relations: Generalized Hooke’s law for linear elastic solids, Concept of material symmetry, Elastic stiffness tensors for monoclinic, orthotropic, transversely isotropic and isotropic materials, Strain energy density function, Hyperelasticity.

 Week 5: Field Equations of Elasticity: Problem formulation using field equations of elasticity, Stress and displacement based solution techniques, Lame-Navier equations and Beltrami-Michell equations, Different elasticity theorems.

 Week 6: Stress Functions and Planar Elastic Problems: Stress functions, Plane stress formulation, Plane strain formulation, Anti-plane strain problems, Effect of body forces, Different forms of stress functions.

 Week 7: Planar Bending Problems: Stress functions for solving bending of straight beams, Pure bending problem, Beams subjected to concentrated and distributed transverse loading.

 Week 8: Torsional Problems: Torsion of prismatic shaft using Prandtl’s stress function, Solution for elliptical, triangular, and rectangular shafts, Torsion of hollow sections.

 Week 9: Problems in Polar Coordinates I: Field equations in polar coordinates, Stress functions in polar coordinate, General Michell solution, Axisymmetric problem solution, Bending of curved beams.

 Week 10: Problems in Polar Coordinates II: Semi-infinite domain problems involving elastic half space, quarter plane, and plane elastic wedge under different loading conditions.

 Week 11: Problems in Polar Coordinates III: Notch and crack problems, Rotating disc problems, Stress concentration around a hole within a plate subjected to tension.

 Week 12: Advanced Topics: Disk under diametrical compression, Plane elastic contact problem solution, Thermo-elastic deformation problems.