1. In linear elasticity, the stress tensor and the strain tensor are related by the constitutive relation

σ = 2µe + λtr(e)I,

where µ and λ are constants and I is the identity tensor. Show that

e = 1/2µσ - λ/6µκtr(σ)I,

where 3κ = 3λ + 2µ.

2. A cylindrical bar of length l and with generators parallel to the x₃-axis has arbitrary cross-section. The generating surface is free of applied traction, and a uniform normal stress S is applied to the end sections. Show that the boundary condition on the generating surface is satisfied if

σ₁₁n₁ + σ₁₂n₂ = 0,

σ₁₂n₁ + σ₂₂n₂ = 0,

σ₁₃n₁ + σ₂₃n₂ = 0,

where σ is the infinitesimal stress and N = n₁e₁ +n₂e₂ is the unit outward normal to the generating surface. Hence, show that the displacement field u with components

u₁ = -νSx₁/E, u₂ = -νSx₂/E, u₃ = Sx₃/E,

E and ν being the Young's modulus and the Poisson ratio, respectively, satisfies the boundary conditions of this problem.

3. Calculate the increase in length and the decrease in diameter when a cylindrical bar of length l and diameter 2a is subjected to a uniaxial tension along its length by the application of force F distributed uniformly over the ends.

4. In an isotropic linear elastic solid the displacement field is given by

u₁ = -τx₂x₃, u₂ = τ x₁x₃, u₃ = τ Φ(x₁, x₂)

where τ is a real constant, show that the equilibrium equations in the absence of body forces are satisfied if Φ is a harmonic function, i.e., if ?Φ = 0.

5. Isotropic elastic material extending to infinity in all directions contains a spherical inclusion of radius a. Assuming that body force is absent and denoting by R the radial distance from the centre of the sphere, show that, at equilibrium, the displacement throughout the material is given by

u = - p/3κ (R + 3κa³/4µR²)e_R

if a uniform pressure p is applied at infinity and the surface R = a is traction-free, and by

u = pa³/4µR²eR

if a uniform pressure p is applied at the surface R = a and the material is unloaded at infinity.

6. A rigid sphere of radius a is surrounded by a concentric spherical shell of internal radius a and external radius b, which is subject to a uniform pressure p on its outer boundary. Show that the radial displacement in the shell is

- P/(4µa³ + 3κb³).(R - a³/R²)

Hence calculate the radial and hoop stress components.

7. Investigate the problem of a self-gravitating spherical shell of internal radius a and external radius b. Assume that both boundaries R = a and R = b are traction-free.

[Hint: Note that the volume of material within radius R > a is

4/3π(R³ - a³)

and the radial component of the gravitational body force is

-4/3 πρG(R - a³/R²)

where ρ is the mass density and G is the gravitational constant.]

8. Elastic material with Lam´e constants λ₁, µ₁ occupies the spherical region R < a, and elastic material with Lam´e constants λ₂, µ₂ occupies the region a ≤ R ≤ b. When the surface R = b is subjected to a uniform pressure p, and body force is absent, explain why the stress and displacement vectors are continuous across R = a and find the equilibrium displacement in both region of the body.

9. When u = u(R)e_R and b = b(R)e_R, where R denotes the first cylindrical polar coordinate (R = x₁²+x₁²), show that the equilibrium equations reduce

(λ + 2µ) d/dr [1/R d/dR(Ru) + ρb(R) = 0.

When body force is absent, show that u has the form u = AR + B/R, where A and B are constants, and that

σe_R = [2(λ + µ)A - 2µ.B/R²]e^r.

Elastic material occupies the region a ≤ R ≤ b. The surface R = a is fixed and the surface R = b is subjected to a uniform pressure p. Show that the displacement in the material is given by

u = -pb²/2(λ + µ)b² + 2µa²(R - a²/R),

and then find the radial and hoop stresses.

10. Consider an infinite isotropic elastic medium in which the displacement depends only on x1 and t and on which no body force acts. Show that the equations of motion reduce to

∂²u₁/∂t² = (λ+ 2µ)/ρ.∂²u₁/∂x₁², ∂²u₂/∂t² = µ/ρ.∂²u²/∂x², ∂²u3/∂t² = µ/ρ∂²u3/∂x²

Prove that

u₁ = f₁(t - x₁/vp) + g₁(t + x₁/vp),

u₂ = f₂(t - x₁/vs) + g₂(t + x₁/vs),

u₃ = f₃(t - x₁/vs) + g₃(t + x₁/vs),

where v = √(λ + 2µ)/ρ, vs = √µ/ρ, fi, gi ∈ C²(R) (i = 1, 2, 3). Finally,

prove that

vp = √(2(1 - ν))/(1 - 2v)

ν being the Poisson's ratio, and deduce that

v_p > √2v_s.

11. Consider a motion in which u depends only on x₁, x₂ and t (independent of x3): u₁ = u₂ = 0, u₃ = u₃(x₁, x₂, t). Show that this is a S-wave.