Past exam of the solid mechanics course of the University of Oxford / 2023 / 2 / b / iii / Solution

Take the inner surface to be free of traction, as appropriate to an empty cavity with no surface tension. At the outer surface the imposed pressure is compressive when positive. The outward normal vectors are $-{\mathbf{e}}_r$ at $r=a$ and $+{\mathbf{e}}_r$ at $r=b$, givingIntegrating the ordinary differential equation from part (ii) between the two surfaces givesThe outer stretch follows from incompressibility:Hence an expression entirely in terms of the inner stretch and the reference radii isEquivalently $P=-{\int }_{{\lambda }_b}^{{\lambda }_a}{h}^{\prime }(\lambda )/({\lambda }^3-1)d\lambda$. For an expanding shell, $1<{\lambda }_b<{\lambda }_a$; when $h^{\prime }>0$ on this range, $P<0$, correctly describing an outward tensile loading. If an inner pressure $P_i$ were present, the same derivation would give $P-P_i$ on the left. Thus the empty-cavity condition specifies the pressure reference needed for the requested answer.