The course
This page describes my notes from the winter 2012 session of the University of Toronto Continuum Mechanics course (PHY454H1S), taught by Prof. Kausik S. Das, which I took as a nondegree student.
The official course description at the time was:
The theory of continuous matter, including solid and fluid mechanics. Topics include the continuum approximation, dimensional analysis, stress, strain, the Euler and NavierStokes equations, vorticity, waves, instabilities, convection and turbulence.
That said, this was, at least at the time, a course that was really only about fluid dynamics. Anything related to solids was really just to introduce the stress and strain tensors as lead up to expressing the NavierStokes equation. There was nothing in this course about beam deformation, Euler stability, or similar topics that were touched on back when I took CIV102. Given the title and description, I’d expected some of those topics to be covered in more depth in this course.
My notes for the class are available
 in paperback or hardcover (black and white), using Amazon’s printondemand service (Kindle Direct Publishing.)
 as an ebook on leanpub for $5+ (set your own price.) Purchasers of the book also have a forum available to them.
 or as a PDF, for $0,
 or as latex (see below.)
The paper version is softcover, published using amazon’s kindle direct publishing (aka createspace) and has a 6×9″ format (358 pages.)
Mathematica notebooks associated with the class are also available.
If you download the free PDF, compile the latex version, or buy a copy and feel undercharged, feel free to send some bitcoin my way.
Contributing.
Should you wish to actively contribute typo fixes (or additions, editing, …) to these notes, you can do so by contacting me, or by forking your own copy of the associated git repositories and building the book pdf from source, and submitting a subsequent merge request.
git clone [email protected]:peeterjoot/latexnotescompilations.git peeterjoot cd peeterjoot submods="figures/phy454continuumechanics phy454continuumechanics mathematica latex" for i in $submods ; do git submodule update init $i (cd $i && git checkout master) done export PATH=`pwd`/latex/bin:$PATH cd phy454continuumechanics make
I reserve the right to impose dictatorial control over any editing and content decisions, and may not accept merge requests asis, or at all. That said, I’ll probably not refuse reasonable suggestions or merge requests.
Contents:
 Copyright
 Document Version
 Dedication
 Preface
 Contents
 List of Figures
 1 Introduction to continuum mechanics
 1.1 Continuum Mechanics
 1.2 Nomenclature and basic definitions
 1.3 Texts
 2 Strain Tensor
 2.1 Deformations
 2.2 Matrix representation, diagonalization, and deformed volume element
 2.3 Strain in cylindrical coordinates
 2.4 Compatibility condition compatibility condition for 2D strain
 2.5 Compatibility condition for 3D strain
 2.6 On the factor of two in the tensor definition
 2.7 Summary
 2.7.1 Strain Tensor
 2.7.2 Diagonal strain representation
 2.7.3 Strain in cylindrical coordinates
 2.7.4 Compatibility condition
 2.8 Problems
 3 Stress tensor
 3.1 Force per unit volume
 3.2 Stress tensor in 2D
 3.3 Stress tensor in 3D
 3.4 Cauchy tetrahedron
 3.5 Constitutive relation
 3.6 Constitutive relation for Hydrostatic compression
 3.7 Constitutive relation for uniaxial stress
 3.8 Summary
 3.8.1 Stress tensor
 3.8.2 Constitutive relation
 3.8.3 Uniform hydrostatic compression
 3.8.4 Uniaxial stress. Young’s modulus. Poisson’s ratio
 3.9 Problems
 4 Elastodynamics
 4.1 Elastic waves
 4.2 Pwaves
 4.3 Swaves
 4.4 Relative speeds of the pwaves and swaves
 4.5 Assuming a gradient plus curl representation
 4.6 A couple summarizing statements
 4.7 Phasor description of elastic waves
 4.8 Some wave types described
 4.9 Summary
 4.9.1 Elastic displacement equation
 4.9.2 Equilibrium
 4.9.3 Pwaves
 4.9.4 Swaves
 4.9.5 Scalar and vector potential representation
 4.9.6 Phasor description
 4.9.7 Some wave types
 4.10 Problems
 5 NavierStokes equation
 5.1 Time dependent displacements
 5.2 Comparing to elastostatics
 5.3 Antisymmetric term, the vorticity
 5.4 Symmetric term, the strain tensor
 5.5 Newtonian Fluids
 5.6 Dimensions of viscosity
 5.7 Conservation of mass in fluid
 5.8 Incompressible fluid
 5.9 Conservation of momentum (NavierStokes equation)
 5.10 Incompressible fluids
 5.11 Boundary value conditions
 5.12 Normals and tangents at fluid interfaces
 5.13 Solutions by intuition
 5.14 Summary
 5.14.1 Vector displacements
 5.14.2 Relative change in volume
 5.14.3 Conservation of mass
 5.14.4 Constitutive relation
 5.14.5 Conservation of momentum (NavierStokes)
 5.14.6 Observe the first order time derivative here
 5.14.7 No slip condition
 5.14.8 Traction vector matching at an interface
 5.14.9 Flux
 5.15 Problems
 6 Hydrostatics
 6.1 Steady state and static fluids
 6.2 Height matching in odd geometries
 6.3 Summary
 6.3.1 Hydrostatics
 6.3.2 Mass conservation through apertures
 7 Bernoulli’s theorem
 7.1 Derivation
 7.2 Summary
 7.2.1 Bernoulli equation
 7.3 Problems
 8 Surface tension
 8.1 Traction vector at the interface
 8.2 Surface tension gradients
 8.3 Summary
 8.3.1 Laplace pressure
 8.3.2 Surface tension gradients
 8.3.3 Surface tension for a spherical bubble
 8.4 Problems
 9 Nondimensionalisation
 9.1 Scaling
 9.2 Rescaling by characteristic length and velocity
 9.3 Reynold’s number
 9.4 Summary
 9.4.1 Nondimensionality and scaling
 9.5 Problems
 10 Boundary Layers
 10.1 Time dependent flow
 10.2 Unsteady rectilinear flow
 10.3 Review. Impulsively started flow
 10.4 Boundary layers
 10.5 Universal behavior
 10.6 Fluid flow over a solid body
 10.6.1 Scaling arguments
 10.7 Summary
 10.7.1 Impulsive flow
 10.7.2 Oscillatory flow
 10.7.3 Blassius problem (boundary layer thickness in flow over plate)
 10.8 Problems
 11 Singular perturbation theory
 11.1 Magnitude of the viscosity and inertial terms
 11.2 Asymptotic solutions of ill conditioned equations
 11.3 Summary
 11.3.1 Singular perturbation theory
 12 Thermal effects and stability
 12.1 Stability
 12.1.1 Stability. Some graphical illustrations
 12.2 Characterizing stability
 12.2.1 Case I. Oscillatory unstability
 12.2.2 Case II. Marginal unstability
 12.2.3 Case III. Neutral stability
 12.3 A mathematical description
 12.4 Thermal stability review. Rayleigh Benard Problem
 12.5 Application of the perturbation to the energy equation
 12.6 Nondimensionalisation of the thermal velocity equation
 12.7 Nondimensionalization of the energy equation
 12.8 Normal mode analysis
 12.9 Back to our coupled equations
 12.10 Multimedia presentations
 12.11 Summary
 12.11.1 Stability
 12.11.2 Thermal stability: RayleighBenard problem
 12.12 Problems
 Appendixes
 A Strain Tensor in cylindrical and spherical coordinates
 A.1 Cylindrical coordinates
 A.2 For general coordinate representation
 A.3 Cartesian tensor
 A.4 Cylindrical tensor
 A.5 Spherical tensor
 A.6 Spherical tensor. Manual derivation
 B Non coordinate strain and traction vector representation
 B.1 Motivation
 B.2 Verifying the relationship
 B.3 Cylindrical strain tensor
 B.3.1 Outwards radial normal ncap equals rcap
 B.3.2 Azimuthal normal ncap equals phicap
 B.3.3 Longitudinal normal ncap equals zcap
 B.3.4 Summary
 B.4 Spherical strain tensor
 B.4.1 Outwards radial normal ncap equals rcap
 B.4.2 Polar normal ncap equals thetacap
 B.4.3 Azimuthal normal ncap equals phicap
 B.4.4 Summary
 C Poisson’s ratio and shear modulus relations
 D Surfaces
 D.1 Normals and tangents
 D.2 Review. Surfaces
 E Identities and proofs
 E.1 Error function properties
 E.2 A Fourier series refresher
 E.3 Vector identities
 F Attempt at general inclined flow problem
 F.1 Motivation
 F.2 Equations of motion
 F.3 Boundary value constraints
 F.4 Laplacian of Pressure and Vorticity
 F.4.1 Separation of variables?
 F.4.2 In terms of vorticity?
 F.4.3 Pressure and vorticity equations with the nonlinear term retained
 F.4.4 Reworking slightly
 F.5 Now what?
 G Steady state velocity profile of stirred cup of nonbottomless coffee
 G.1 Motivation
 G.2 NavierStokes for the problem
 G.3 Working around the noslip troubles at the base of the cup
 G.4 Spin down below the stir point
 H Mathematica notebooks
 Bibliography
Changelog:
V0.1.129, Aug 22, 2023 (commit 4627d27790ff38ce78d92479adef39661cd25176)

Regenerate some of the figures (in black and white, thick attributes, MaTeX markup.)

Update figures for 3d flow between pipes to black and white (hatch shading.)
V0.1.122, Mar 12, 2022 (commit 9ff5a1527594e9a25b26225a1dc8bc042cb4e1a7)
 eqnref > cref.
 Add a GA analysis of e_i e_{ij} n_j tensor > vector expansion.
 Fix some gutter issues.
 Prune chapter 1 summary (totally redundant).
 Prune some PLACEHOLDER markers.
 Fix error in first instance of the navier stokes.
 Spelling fixes.
 Parameterize printversion.
V0.1.110, Sept 12, 2020 (commit 5688bbb641ecbbcd952649f5fa2d7d405632a781)
 Too big figure.
 Remove dmath+aligned.
 Dup sequential word fixes.
 Add spell check rules and do it.
 URL fixes.