Problem 1 -

Playing with component notation: For items which are manifestly 3-dimensional (i.e. ∈_ijk), don't bother with raised/lowered indices (since the metric in Euclidean space is just the identity matrix), but sum over repeated (lower) indices.

Show that A_μνB^μν = 0 if A_μν is an anti-symmetric tensor and B_μν is symmetric.

Problem 2 - Spherical stress and strain

1. Detailing your calculations using indicial notation, and starting from Hooke's law and the law of static equilibrium linking external volume forces (weight, EM forces. . . ) and the stress tensor, derive the following equation:

f_i + μ∂_j²u_i + (λ + μ)∂_i∂_ju_j = 0,

where λ and μ are the Lamé parameters, μ being the shear modulus. Note that the index in ∂_j² is considered repeated in the sense of Einstein's convention.

2. Rewrite the equation above in vector, or dyadic, form, i.e. with f^→, ∇^→. . . Even though the derivation using indicial notation was done with Cartesian coordinates in mind, the vector form of Navier's equation in valid in any coordinate system.

3. We now consider a spherical problem, in which the external volume forces are all radial (f^→ = fr^{^}), the only dependence is on the radial coordinate r and no other, and in which the only displacement is also in the radial direction (u^→ = u(r)r^{^}).

Show that: f + (2μ + λ) d/dr((1/r²)(d(r²u)/dr)) = 0.

4. We now compute the stresses and strains of a spherical shell under pressure. We assume that the deformations due to the external volume forces are negligible when compared to those induced by these pressure forces. Show that:

u = Ar + B/r², with A and B two constants to be determined later.

5. The diagonal terms of the strain tensor in spherical coordinates (which are the only non-zero ones) are, under our symmetry assumptions:

U_rr = du/dr and U_tt = u/r,

where the index 't' refers to the two tangential directions. The strain is the same in each direction in an isotropic medium under isotropic contraints such as ours.

Compute U_rr and U_tt (retain A and B for now), and briefly explain why U_tt ≠ 0: what does it term mean in terms of the deformation of a solid element?

6. Hooke's law is still valid in spherical coordinates:

τ_rr = 2μU_rr + λ(U_rr + 2U_tt) and τ_tt = 2μU_tt + λ(U_rr + 2U_tt).

Compute τ_rr and τ_tt in terms of (μ, K), with K the bulk modulus, instead of (μ, λ).

7. Let a and b the inner and outer radii, respectively, of our spherical shell, and P_a and P_b the corresponding pressure. Write down the two boundary conditions these two pressures correspond to, and use them to show that:

u = (a³b³/b³ - a³)[((P_a/b³ - P_b/a³)/3K)r - (P_b- P_a/4μr²)]

8. Numerical applications: a spherical submarine of inner radius a = 75 cm, with a cast steel wall (K = 200 GPa, μ = 75 GPa) that is 10 cm thick, dives to depths d = 10 m, 1 km and 11 km. By how much does the inner radius shrink at each depth? Use P_a = 1.0 x 10⁵ Pa for both the inner pressure and the atmospheric pressure, g = 9.8 m²s^-1 and ρ_ocean ≈ 1030 kgm^-3.