Course Content:

Mathematical Preliminaries: Vector and tensors calculus, Indicial notation. Strain: Definition of small strain, Strain-Displacement relations in 3D, Physical interpretation of strain components, Principal Strains. Stress and equilibrium: Stress components in 3D, Principal Stresses, Cauchy’s principle, stress equilibrium. Constitutive law, Navier’s equations, compatibility equations. Formulation of boundary value problems and solution methods: Plane Problems - plane stress, plane strain, anti-plane shear. Fourier transform methods. Superposition principle. Additional topics from: Examples - Torsion of prismatic shaft, Contact problems, Wedge problems, Dislocations and inclusions, Cracks, Think-film problems; Advanced transform methods - Complex variable techniques, Potential methods; Advanced ideas - Energy method, Numerical approaches, Finite elements, Eigenstrains, Micromechanics.

Lecturewise Breakup (based on 75min per lecture)

I. Introduction: (1 Lectures)

• Review of strength of Materials and its limitations.

II. Mathematical Preliminaries: (4 Lectures)

• Vector and tensor calculus.
• Indicial notation.

III. Strains: (1 Lectures)

• Definition of small strain, Strain-Displacement relations in 3D, Physical interpretation of strain components, Principal Strains.

IV. Stress and equilibrium: (4 Lectures)

• Stress components in 3D and their physical interpretations. [1 Lecture]
• Principal Stresses. [1 Lecture]
• Cauchy’s principle and derivations of stress equilibrium equations in stress components. [2 Lectures]

V. Constitutive law, Navier’s equations, compatibility: (4 Lectures)

• Constitutive law for general linear elastic solid, Discussions on isotropic, orthotropic and transversely isotropic solid.
• Navier’s equations.
• Stress and displacement approaches.
• Compatibility equations.

VI. Formulation of boundary value problems and solution methods: (15 Lectures)

• Formulation of boundary value problems. [1 Lecture]
• Plane Problems - plane stress, plane strain, anti-plane shear (also in axisymmetric coordinates). [2 Lectures]
• Examples of plane problems: Stress function approach, Series solutions. [6 Lectures]
• Fourier transform methods with examples. [3 Lectures]
• Superposition principle: Flamant’s solutions; Kelvin’s solution; Boussinesq’s solution. [3 Lectures]

VII. Additional topics - a few topics to be selected from below: (11-13 Lectures)

• Further examples: Torsion of prismatic shaft; Contact problems; Wedge problems; Dislocations and inclusions; Cracks; Thin-film problems.
• Further methods: Advanced transform methods; Complex variable techniques; Potential methods.
• Further ideas: Energy methods; Numerical approaches; Finite elements; Eigenstrains; Micromechanics.

References:

1. Elasticity, J. R. Barber
2. The Linearized Theory of Elasticity, W. L. Slaughter
3. Continuum Mechanics for Engineers, G. T. Mase and G. E. Mase
4. Theory of Elasticity, S. Timoshenko and J. N. Goodier
5. Elasticity: Theory, Applications and Numerics, M. H. Sadd
6. Applied Mechanics of Solids, A. Bower