A submarine hull is treated as an elastic solid whose outer surface is in contact…
PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids
Bending, Ringing, Breaking and Flowing, Ch Strain and stress in a deformed solid
Problem
A submarine hull is treated as an elastic solid whose outer surface is in contact with seawater. The water, at rest, exerts on the hull a normal pressure p0 (no shear, since a fluid at rest carries no shear stress) over a small patch of the outer surface whose outward unit normal, pointing from the hull into the water, is n.
((a)) Consider a thin pillbox of face area A straddling the water-hull interface at this patch: its outer face, of area A, lies just outside in the water; its inner face, also of area A, lies just inside the hull; the side faces have area of order h, where h is the (vanishingly small) thickness of the pillbox. Balance the forces on this pillbox in the static limit h→0, in which the body-force term (volume Ah→0) and the side-face tractions (area →0) do not survive. Hence derive the boundary condition satisfied by the Cauchy stress σij just inside the hull at this surface, in terms of p0 and nj. State what this condition becomes for an unloaded (vacuum-exposed) free surface, p0=0.
((b)) The submarine is at a depth of 250m in seawater of density 1025kgm−3; take the pressure at the sea surface as negligible next to the hydrostatic pressure at this depth, and use g=9.8ms−2. Evaluate p0 and give the traction ti=σijnj on the hull's outer surface, as a multiple of n.