Course Summary

This course will introduce and develop the subject of mechanics in a continuum, specially elasticity, applicable to the circumstances when the material of a body experiences large (or finite) deformation including large displacement or large rotation. The main objective is rigorous introduction to continuum mechanics emphasizing solid mechanics and linear elasticity.

1. (i)Introduction: Review of strength of materials, Solids Mechanics, fluid mechanics, and its limitations.
   

2. (ii)Mathematical Preliminaries: Vector and tensor calculus; Tensor analysis, derivatives of functions with respect to tensors; Fields, div, grad, curl; Divergence theorem, transport theorem. Introduction to the notion of a tensor and preparation of some background in tensor algebra, tensor analysis and tensor calculus. This part mainly deals with vectors and tensors with direct notation.
   

3. (iii)Kinematics: Configurations of a body, displacement, velocity, motion; Deformation gradient, rotation, stretch, strain, strain rate, spin tensor; Assumption of small deformation and small strain; Definition of motion of a body, in particular a deformable body. Definition of a reference configuration and some ways of quantifying `strain' or `strain rate'.
   

4. (iv)Balance laws: Balances of mass, linear momentum and angular momentum; Contact forces and the concept of stress; Balance of energy and Clausius-Duhem inequality; Discussion of the concept of force. Definition of Cauchy stress.  Definition of several other notions of stress. The parts (iii) and (iv) discuss kinematics and forces: deformations, rotations, strains and strain rates, force and stress, balance principles.
   

5. (v)Constitutive relations: Definition of frame indifference (symmetry of physical space) and material symmetry (symmetry of material space). Kinematic constraints (incompressibility, etc); Thermodynamical restrictions; Highlights on the role of a reference configuration in this subject.
   

6. (vi)Viscous fluids: Examples of constitutive relations, non-Newtonian fluid, boundary value problem
   

7. (vii)Finite elasticity: Hyperelasticity, isotropy, simple constitutive relations, boundary value problem. Definition of an elastic material. Discussion on the special case of an isotropic elastic material. Introduction to the three dimensional  theory of hyperelasticity. Exposure to some useful results in the case of an isotropic hyperelastic material.
   

8. (viii)Derivation of linear elasticity as a result of linearization of fully nonlinear theory discussed so far. Discussion on certain limitations of such linearization procedure. Discussion on anisotropic and isotropic cases. The topics may include stability; basic theorems of elastostatics and boundary-value problems.
   

9. (ix)Conclusion of the course with a discussion of some open problems.