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)eR
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)eR and b = b(R)eR, 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
σeR = [2(λ + µ)A - 2µ.B/R2]er.
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
vp > √2vs.
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.