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In the completely elastic exterior, radial elastic compatibility makes $q+2t$ constant. The far-field stress vanishes, so $q+2t=0$ and $t=-q/2$. Radial force balance then gives $q^{\prime }+3q/r=0$, whence $q=C/r^3$. The pressure boundary condition is $q(a)=-P$: positive applied pressure is compressive. ThereforeFor completeness, the corresponding radial displacement field is $u=P{a}^3/(4\mu r^2)$, which gives these components through Hooke's law and decays at infinity.

The Tresca yield criterion becomesFor $P\ge 0$ this is largest at the cavity wall and decreases strictly with radius when $P>0$. The first yield point is therefore at $r=a$, where $3P/4={\tau }_Y$. HenceThe normalization matters: ${\tau }_Y$ in this paper is a shear yield stress, rather than the uniaxial yield strength.

During increasing pressure, the elastic solution reaches the Tresca yield criterion with $t-q>0$. In the plastic shell of a material with perfect plasticity, this sign is retained andRadial force balance becomes $q^{\prime }-4{\tau }_Y/r=0$. Integrating from the cavity wall, where $q(a)=-P$, givesOutside the plastic shell, radial elastic compatibility and vanishing far-field stress again give $q=-C/r^3$ and $t=C/(2r^3)$. At the advancing plastic front $r=s$, the Tresca yield criterion gives $3C/(4s^3)={\tau }_Y$, so $C=P_c{s}^3$. Continuity of radial traction now givesThusThe complete stress field of this spherical elastoplastic cavity isAt $s$, the tangential stress also matches because $-P_c+2{\tau }_Y=P_c/2$. Outside $s$, the function in the Tresca yield criterion is ${\tau }_Y(s/r{)}^3<{\tau }_Y$, which verifies that the remaining material is elastic. As $P\downarrow P_c$, $s\downarrow a$ and the solution joins the entirely elastic regime. The radius $a$ is the fixed reference radius of the small-strain model used by the paper.

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.

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.

A spherical cavity of fixed reference radius $a$ in an infinite solid with perfect plasticity, the Tresca yield criterion, and zero far-field stress first yields under internal pressure $P_c$. In the plastic region $a<r<s$, radial force balance and ${\tau }_{\theta \theta }-{\tau }_{rr}=2{\tau }_Y$ give ${\tau }_{rr}=-P+4{\tau }_Y\log (r/a)$. Matching to the outer linear elasticity solution, where ${\tau }_{rr}(s)=-P_c$, gives the displayed plastic-front radius. These are small-strain reference-geometry formulae.

The Tresca yield criterion bounds the largest resolved shear stress, expressed through the principal stresses. In spherical symmetry, the two tangential principal stresses coincide, so $f=|{\tau }_{rr}-{\tau }_{\theta \theta }|/2$. Here ${\tau }_Y$ is a shear yield stress; the corresponding uniaxial yield strength is $2{\tau }_Y$.