Syllabus and Lecture Notes

1. Introduction

2. Mathematical Preliminaries: Vectors and Tensors

    1. Vectors: (self-study review);
    2. Index notation (self study)
    3. Tensors and tensor operations
    4. Vector and tensor operations in polar coordinates

3. Kinematics – mathematical description of motion and deformation

    1. The continuum; Inertial reference frames; the reference configuration and current configuration of a deformed solid
    2. The displacement and velocity field, examples of deformations and motions, Eulerian and Lagrangian descriptions of motion
    3. The deformation gradient tensor; deformation of line, volume and area elements
    4. Strain tensors – Lagrange strain and Eulerian strain, Cauchy Green strain, infinitesimal strains, compatibility.
    5. Polar decomposition of the deformation gradient; rotation tensor; left and right stretch tensors
    6. Principal stretches and strains
    7. Time derivatives of motion: the velocity gradient, stretch rate, spin and vorticity.
    8. Spatial description of acceleration
    9. Reynolds transport relation
    10. Circulation-vorticity relations

4. Kinetics – mathematical description of internal forces

    1. External loading – surface tractions, body forces
    2. Internal forces – Cauchy Stress
    3. Principal stresses, stress invariants
    4. Stresses near a surface
    5. Piola-Kirchhoff stresses (Nominal and material stress)

5. Field Equations and Conservation Laws

    1. Mass Conservation
    2. Linear and angular momentum; static equilibrium
    3. Work done by stresses
    4. The principle of virtual work
    5. The first and second laws of thermodynamics for continua
    6. Conservation laws for a control volume
    7. Transformation of field quantities under changes of reference frame

6. Constitutive models – general considerations

    1. Thermodynamics – the dissipation inequality
    2. Frame indifference

7. Mechanics of elastic and compressible, viscous fluids

    1. Summary of field equations
    2. Constitutive models for fluids
    3. Solutions to simple problems

8. Mechanics of elastic solids

    1. Field equations
    2. Constitutive models for hyperelastic materials
    3. Solutions to simple boundary value problems for hyperelastic materials
    4. Linearized field equations, and examples of linear elastic solutions