###### 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

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