###### Continuum Mechanics

### Continuum Mechanics

###### Notes and problems from UofT PHY454H1S 2012

# About the Book

This book is based on my lecture notes for the Winter 2012, University of Toronto Continuum Mechanics course (PHY454H1S), taught by Prof. Kausik S. Das.

#### Table of 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 P-waves
- 4.3 S-waves
- 4.4 Relative speeds of the p-waves and s-waves
- 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 P-waves
- 4.9.4 S-waves
- 4.9.5 Scalar and vector potential representation
- 4.9.6 Phasor description
- 4.9.7 Some wave types
- 4.10 Problems
- 5 Navier-Stokes 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 (Navier-Stokes 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 (Navier-Stokes)
- 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 Non-dimensionality 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 Non-dimensionalisation of the thermal velocity equation
- 12.7 Non-dimensionalization 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: Rayleigh-Benard 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 non-linear term retained
- F.4.4 Reworking slightly
- F.5 Now what?
- G Steady state velocity profile of stirred cup of non-bottomless coffee
- G.1 Motivation
- G.2 Navier-Stokes for the problem
- G.3 Working around the no-slip troubles at the base of the cup
- G.4 Spin down below the stir point
- H Mathematica notebooks
- Bibliography

#### Other books by this author

### The Leanpub 60-day 100% Happiness Guarantee

Within **60 days of purchase** you can get a **100% refund** on any Leanpub purchase, in **two clicks**.

See full terms

### 80% Royalties. Earn $16 on a $20 book.

#### We pay **80% royalties**. That's not a typo: **you earn $16 on a $20 sale**. If we sell **5000** non-refunded copies of your book or course for **$20**, you'll earn **$80,000**.

*(Yes, some authors have already earned much more than that on Leanpub.)*

In fact, authors have earnedover $12 millionwriting, publishing and selling on Leanpub.

**Learn more about writing on Leanpub**

### Free Updates. DRM Free.

If you buy a Leanpub book, you get free updates for as long as the author updates the book! Many authors use Leanpub to publish their books in-progress, while they are writing them. All readers get free updates, regardless of when they bought the book or how much they paid (including free).

Most Leanpub books are available in PDF (for computers) and EPUB (for phones, tablets and Kindle). The formats that a book includes are shown at the top right corner of this page.

Finally, Leanpub books don't have any DRM copy-protection nonsense, so you can easily read them on any supported device.

Learn more about Leanpub's ebook formats and where to read them