## 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

- L1 9/7/2016 Intro Slides. Vectors and Index Notation 1 Euclidean space, vectors, basis
- L2 9/9/2016 Vectors and Index Notation 2: Basis change formulas, vector calculus
- L3 9/12/2016 Index notation exercises; Tensor basics
- L4 9/14/2016 Tensors: determinant; inverse; invariants; eigenvalues/vectors Cayley Hamilton theorem
- L5 9/16/2016 Special Tensors, Decomposition of tensors, tensor calculus, Polar Coords
- L6 9/19/2016 Calculus in polar coords; Curvilinear coords: covariant/contravariant bases and components
- L7 9/21/2016 Curvilinear coordinates: metric tensor; tensor/vector operations; covariant derivative
- L8 9/23/2016 Kinematics:: Lagrangean/Eulerean motion, deformation gradient, volume changes
- L9 9/26/2016 Kinematics: Area elements, line elements, strain measures, polar decomposition of F
- L10 9/28/2016 Infinitesimal deformation: small strain/rotation tensor, compatibility conditions
- L11 9/30/2016 Compatibility conditions; deformation rates (velocity gradient, stretch rate, vorticity)
- L12 10/3/2016 Transport relations; Kinetics: External forces; internal force - traction, Cauchy stress
- L13 10/5/2016 Nominal and Material stresses Project 1 slides
- L14 10/7/2016 Stresses (continued); conservation of mass, linear and angular momentum
- L15 10/12/2016 BLM/BAM in terms of other stress measures; Mechanical Work; simple example BVP
- L16 10/14/2016 BVP continued, proofs of principle of virtual work (power)
- L17 10/17/2016 Example using PVW, Thermodynamics for continua I
- L18 10/19/2016 Proofs of second law; dissipation inequality. Conservation laws for control volume
- L19 10/21/2016 Transformations of vectors and tensors in continuum mechanics under observer changes
- L20 10/24/2016 Constitutive Equations
- L21 10/28/2016 Constitutive equations; constitutive model for fluids part I
- L22 10/31/2016 Constitutive model for fluids part II
- L23 11/2/2016 Fluid Mechanics: Navier Stokes; simplified equations for ideal fluids; vorticity transport
- L24 11/4/2016 Bernoulli; potential flows; Stokes flows; control volume methods
- L25 11/7/2016 Fluids: control volume methods, potential flow example
- L26 11/9/2016 Fluids: stokes flow examples; acoustics
- L27 11/11/2016 Fluids: acoustic waves from a vibrating sphere; Elastic material behavior and models
- L28 11/14/2016 General structure of elastic material models
- L29 11/16/2016 Examples of strain energy potentials for hyperelastic materials
- L30 11/18/2016 Solutions for hyperelastic solids
- L31 11/21/2016 Solutions for hyperelastic solids, linearized equations of elasticity
- L32 11/28/2016 Cauchy-Navier equations for linear elasticity; solutions to static boundary value problems
- L33 11/30/2016 Solutions to dynamic problems in linear elasticity
- L34 12/2/2016 Mass Transport

## 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
External websites on curvilinear 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
External Links

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

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