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.

     

Workload Expectation

    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

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

 

 

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:

    Rebecca Brannon, Utah
    P.A. Kelly (University of Aukland)

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

    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

    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

    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

    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