ABOUT THE COURSE:This course focuses on the application of elasticity theory in various structural deformation problems. With a brief revisit to the theory of elasticity, the course explores the solution techniques for different solid mechanics problems using the field equations of elasticity for both stress based and displacement based approaches. Special emphasis is given for solving the structures with complex geometries described by both Cartesian and polar coordinates. The course is expected to be useful for the postgraduate course work as well as academic research.INTENDED AUDIENCE: First year postgraduate students of Mechanical Systems Design/Machine Design/ Computational Solid Mechanics/Applied Mechanics specialization, and final year undergraduate students of Mechanical EngineeringPREREQUISITES: Solid Mechanics/Mechanics of Materials (UG level)

Syllabus

Week 1: Introduction and Mathematical Preliminaries: Review of strength of materials and its limitations, Tensor algebra and indicial notation, Tensor calculus.
Week 2:Deformation and Strain: Lagrangian and Eulerian descriptions of deformation, Deformation gradient tensor, Green-Lagrange and Eulerian strain tensors, Strain displacement relations, Infinitesimal strain tensor and rotation tensor, 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 coefficient tensors for monoclinic, orthotropic, transversely isotropic and isotropic materials, Strain energy density function.
Week 5:Formulation and Solution Strategies: Problem formulation using field equations of elasticity, Stress and displacement based solution techniques, Navier equations and Beltrami-Michell equations, Direct and inverse method based solution strategies.
Week 6:Planar Elastic Problems: Plane stress and plane strain problem formulation, Solution using Airy’s stress function, Effect of body forces.
Week 7:Planar Bending Problems: Stress functions for solving bending of straight beams, Pure bending, Beams subjected to concentrated and distributed transverse loading.
Week 8:Torsional Problems: Torsion of prismatic shaft using Prandtl’s stress function, Solution for circular, elliptical, triangular, and rectangular shafts, Torsion of hollow sections.
Week 9:Formulation in Polar Coordinates: Field equations in polar coordinates, Polar coordinate formulation for plane stress and plane strain problems, Stress functions in polar coordinate.
Week 10:Problems in Polar Coordinates I: Bending of curved beams, Axisymmetric problem solution, Stress concentration around a hole within a plate subjected to tension.
Week 11:Problems in Polar Coordinates II: Flamant’s solution for semi-infinite elastic half space under concentrated surface force system, Wedge problem, Notch and crack problems.
Week 12:Advanced Topics: Plane elastic contact problem solution, Disk under diametrical compression, Thermo-elastic deformation problems.