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Original solutions to every part of the elasticity and plasticity course of the University of Oxford Part C examination, held on 30 May 2023. Source: released examination PDF, bundle pages 317–320; paper pages 1–4. Numbering follows the original paper. In Question 1(c), the printed count needs a strict-cutoff qualification: at $ak=(n-1/2)\pi$ the newly counted state has no exterior exponential decay and must be excluded. The solution includes that correction and the exceptional small-parameter balances. In Question 3(d), purely elastic unloading is valid only until reverse yielding begins.

LetAt maximum pressure, the stress is the field from part (c) with $P=P_m$ and $s=s_m$. Under the assumed elastic unloading, previously accumulated plastic strain is frozen. Therefore it is the stress increment, rather than the total stress, that satisfies homogeneous linear elasticity and radial elastic compatibility. Applying the superposition principle, a pressure change $-\Delta$ produces the elastic incrementsThey have zero far-field stress and change the wall traction from $-P_m$ to $-P$. These increments are the same in the formerly plastic and formerly elastic regions because their elastic Lamé parameters agree. Adding them to the maximum-load field givesandThese formulae retain $s_m$: it records the plastic loading history and is not the instantaneous loading-front formula with $P$ substituted for $P_m$. If the entire return is elastic, setting $\Delta =P_m$ gives the residual stress after removal of the applied pressure.

To check admissibility, examine the signed difference of the principal stresses in the formerly plastic shell:It immediately moves into the interior of the Tresca yield criterion when unloading starts. It can next reach the opposite limit $D=+2{\tau }_Y$. For fixed $\Delta >0$, $D$ is largest at $r=a$ and stays above $-2{\tau }_Y$. At the wall,There is no earlier violation in the outer region. Indeed,so its magnitude is no larger than $|D(s_m)|$, which has already been controlled by the shell calculation. Hence the first reverse yielding occurs at the cavity wall, withIf $P_c<P_m<2P_c$, the stress stays within the yield surface throughout unloading to zero. If $P_m=2P_c$, it reaches reverse yield exactly at $P=0$; further unloading would produce reverse plastic deformation. If $P_m>2P_c$, reverse plastic deformation starts while the applied pressure is still positive. ThusIf “yield again while unloaded” includes first touching the yield surface at the final endpoint, the corresponding condition is $P_m\ge 8{\tau }_Y/3$. The displayed elastic unloading fields are physically valid only for $0\le \Delta \le \min (P_m,2P_c)$; beyond that interval a new reverse-plastic region must be solved for. Continuing the elastic formula to $P=0$ when $P_m>2P_c$ would violate the given Tresca yield criterion.

In a homogeneous isotropic solid without body force, a radial displacement field satisfies $(\lambda +2\mu )(u^{\prime }+2u/r{)}^{\prime }=0$. For nonzero $\lambda +2\mu$, its volumetric strain is constant. Since Hooke's law gives $\mathrm{tr}\tau =(3\lambda +2\mu )(u^{\prime }+2u/r)$, its stress tensor has constant trace. After plasticity in continuum mechanics has occurred, this conclusion applies to purely elastic increments during elastic unloading, and need not apply to the total residual stress.

Renewed plastic deformation during unloading or reversed loading when the stress reaches the opposite part of the yield surface. For the spherical elastoplastic cavity with maximum pressure $P_m>4{\tau }_Y/3$, elastic unloading first reaches the opposite Tresca yield criterion at the cavity wall after a pressure decrease $8{\tau }_Y/3$. Purely elastic return to zero pressure is possible when $P_m\le 8{\tau }_Y/3$, with equality reaching the yield surface exactly at the endpoint.