Past exam of the solid mechanics course of the University of Oxford / 2023 / 3 / b / ii / Solution

The pressure adjustment from part (i) can be writtenIndeed, since $\mathrm{div}(q1)=\nabla q$, the difference of the two Cauchy momentum equations isThus $q+\rho \xi$ is constant in space on each connected component. The function $C(t)$ is fixed by any prescribed pressure or surface traction datum. In terms of the usual compressive pressure multiplier, $p=p_0-q=p_0+\rho \xi -C(t)$. The plus sign in $q1$ therefore does not identify $q$ with the compressive multiplier $p$ itself.