EMA 700/Math 707: Theory of Elasticity
Homework and lecture pointers (Fall 2009)

Textbook NONE, but we will use "Intro to Continuum Mechanics," THIRD edition by Michael Lai, David Rubin and Erhard Krempl [LRK] published by Butterworth-Heineman (http://www.bh.com). Available online through the UW library. We will also use Timoshenko and Goodier, and Landau and Lifschitz.

• Read LRK Chapter 1
• Review LRK Chapter 2A: index notation, summation convention, vector notation (Gibbs notation). We will use Gibbs notation, index notation and also matrix notation.
  • Exercises : 2A 6, 7, 8, 12
• Review LRK Chapter 2B: tensors. Note that I use dyadic notation and write T.a instead of LRK's Ta.
  • In index notation T.a ≡ T_ij a_j ≠ a_i T_ij ≡ a.T = T^T.a
  • sections 2B17-20 are especially important.
  • Exercises : 2B 1, 2, 4, 7, 8, 9, 22, 24, 26, 30, 32, 34, 37, 40
• Study your class notes and LRK sections: 3.1, 3.5, 3.7, 3.18, 3.23, 3.24, 3.8, 3.16, 3.9
  • Notational differences: my J = F^T. It's the Jacobian of x(X), with J_ij=∂_i x_j = ∂x_j/∂X_i .
  • Note the inconsistency of LRK's (3.7.1.b) definition of ∇u and the dyadic product in (2B7.2a), (2B7.2b) page 21! confusing!
  • What is C? Where does it come from? What is the meaning of its components?
  • What are E* and E? What are the meanings of their components?
  • How do line, volume and area elements change during deformation? i.e. what are the formula connecting the original elements to their deformed values? Do you understand those formulas? can you derive them?
  • What are principal values and principal axes?
  • Why do C, E* and E have real principal values and orthogonal principal directions? Do they all have positive principal values? Are the principal directions the same for those 3 tensors?
  • What are the compatibility equations for E? Where do they come from?
  • Exercises: 3.17--23, 27, 29, 30, 31, 34, 61a,b,e,g,h, 62, 63, 64, 68, 69
• Chapter 4: Stress...
  • Given T at a point, find the surfaces on which there is (1) maximum normal stress, (2) minimum normal stress, (3) maximum shear stress, (4) minimum shear stress. Solve this problem for general T.
  • Exercises: 4.2 + solve previous question for this T, 4.10, 14, 15, 16, 17, 18, 30
• Chapter 5: Elastic Solid
  • Derive the form of an isotropic T_ij.
  • What are the restrictions ont values of the Lame constants λ and μ and why?
  • How are the Lame constants λ and μ related to Young's modulus, the shear modulus, the bulk modulus and Poisson's ratio?
  • What is the range of acceptable values for the Poisson ratio and why?
  • Elastic waves: what and how?
  • Elastic wave reflections at a free plane boundary (general approach as begun in class).
  • Exercises: 5.1, 2, 3
  • LRK 5:12: simple extension
  • LRK 5.16: Plane strain: Airy stress function, biharmonic equation
  • Half space problems, e.g. x < 0. Boundary conditions: stress finite as |x|, |y|, |z| → ∞ together with :
    1. Not-so-simple-extension: T₁₁ = σ exp(i k y), T₁₂=0 at x=0, where σ, k are real constants and i²=-1. (Done in class, we illustrated/discussed the validity of Saint Venant's principle (LRK p. 256) from this solution.)
    2. Not-so-simple-shear: T₁₁ = 0 , T₁₂= τ exp(i k y), at x=0, where τ, k are real constants and i²=-1. Volunteer: Mariana Kersh
    3. Not-so-simple-extension: T₁₁ = σ exp(i (k₂ x₂ +k₃ x₃ )), T₁₂= T₁₃=0 at x=0, where σ, k₂, k₃ are real constants and i²=-1. This is just a rotation of (1) above, not too hard but not trivial. Volunteer?
    4. Fundamental solution, Line-load: T₁₁ = F δ(y), T₁₂=0 at x=0, where F is a constant (force/unit length) and δ(y) is the Dirac delta function. Done in class by superposition of Fourier modes (#1 above).
    5. Fundamental solution, Line-shear: T₁₁=0, T₁₂ = T δ(y), at x=0, where T is a constant (force/unit length) and δ(y) is the Dirac delta function. By superposition of Fourier modes (#2 above). Volunteer?
  • 2D (plane strain, plane stress) problems in polar coordinates: we sketched how to derive the equation, talked about how to take care of ∇ . v =0 directly: with a curl in 3D, which translates to a perpendicular gradient ∇_⊥= in 2D. Studied LRK 5.17-5.20, and expanded upon those problems, discussing in particular the general solution to the biharmonic equation in polar coordinates.
    1. Find the stress distribution in a cylindrical annulus (shell) subjected to axial torques: i.e. σ_r = 0 at r=a and r=b, τ_rθ = τ₀ (constant) at r=a, with similar constant shear stress at r=b, as required for equilibrium, to be determined.
    2. Find the stress distribution in a cylindrical annulus with inner radius a, and σ_r = σ₀ cos θ, τ_rθ =0 at r=a (i) with whatever simplest boundary conditions at r=b to have equilibrium, (ii) with b = ∞ and bounded stresses as r → ∞. In case (ii), study the stress distribution in the limit a → 0 also.
    3. Find the stress distribution in a cylindrical tube (or a disk) with outer radius b, and σ_r = σ₀ cos n θ, τ_rθ =0 at r=b, for n=0,1,2,...
    4. Find the stress distribution in a cylindrical tube (or disk) with outer radius b, subjected to two localized and opposite forces normal to the boundary. Discuss how to use the general solution in polar coordinates to solve this problem (that is pose the problem and show how to solve it). Verify that the solution correspond to the superposition of two half-space solutions (i.e. show that the equations and the boundary conditions are satisfied).
    5. Earlier, we found the stress distribution in an semi-infinite domain (half-space) subjected to a localized compressive or shear force. Verify that those solutions satisfy the equilibrium equations in polar coordinates and obtain the Airy stress function for both cases. Find the displacements in polar coordinates for both problems.
    6. Find the stress distribution for a localized torque by combining 2 nearby localized compressive forces and taking a suitable limit (i.e. force dipole). You must define the suitable limit.
    7. Find the stress distribution in a disk rotating at constant angular velocity.