## Syllabus and Lecture Notes

### Course Outcomes

After completing ENGN2210 you should

1. Be familiar with linear vector spaces relevant to continuum mechanics and able to perform vector and tensor manipulations in Cartesian and curvilinear coordinate systems
2. Be able to describe motion, deformation and forces in a continuum;
3. Be able to derive equations of motion and conservation laws for a continuum ;
4. Understand constitutive models for fluids and viscoelastic solids;
5. Be able to solve simple boundary value problems for fluids and solids.

1. Lectures: 39 hours
2. Review lecture notes, background reading: 30 mins per scheduled class (20 hrs)
3. Homework assignments: 8 at 8 hours each
4. Projects: 2 at 15 hours each
5. Midterm exam: 1 hour, in class, plus 10 hours preparation
6. Final exam, 3 hours (as scheduled by Registrar) plus 15 hrs preparation

TOTAL: 182 hours.

### Class Lecture Notes

37. L37 12/12/2016 Final Review Slides

### Detailed Reference 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

External websites on curvilinear coordinates:

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

Ancient but nicely done movies on Lagrangean/Eulerean descriptions of motion Part 1 Part 2 Part 3

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

Links to some publications discussing 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