## Syllabus and Lecture Notes

## Course Outcomes

After completing ENGN2210 you should

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

## Workload Expectation

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

TOTAL:182 hours.

## Class Lecture Notes

Lecture notes from class will be posted here

## Detailed Reference Notes

1. Introduction

2. Mathematical Preliminaries: Vectors and Tensors

- Vectors: (self-study review);
- Index notation (self study)
- Tensors and tensor operations
- Vector and tensor operations in polar coordinates

3. Kinematics– mathematical description of motion and deformation

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

4. Kinetics– mathematical description of internal forces

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

5. Field Equations and Conservation Laws

- Mass Conservation
- Linear and angular momentum; static equilibrium
- Work done by stresses
- The principle of virtual work
- The first and second laws of thermodynamics for continua
- Conservation laws for a control volume
- Transformation of field quantities under changes of reference frame

6. Constitutive models – general considerations

- Thermodynamics – the dissipation inequality
- Frame indifference
Links to some publications discussing frame indifference

http://szofi.elte.hu/~szaboa/MatolcsiKonyvek/pdf/cikk/claritymfi.pdf

http://repository.cmu.edu/cgi/viewcontent.cgi?article=1579&context=math

http://web.mit.edu/abeyaratne/Volumes/RCA_Vol_II.pdf

https://www.jstor.org/stable/2397311?seq=1#page_scan_tab_contents

7. Mechanics of elastic and compressible, viscous fluids

- Summary of field equations
- Constitutive models for fluids
- Solutions to simple problems

8. Mechanics of elastic solids

- Field equations
- Constitutive models for hyperelastic materials
- Solutions to simple boundary value problems for hyperelastic materials
- Linearized field equations, and examples of linear elastic solutions