= Solution
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 and
$$
t-q=2\tau_Y.
$$
Radial <force balance> becomes $q'-4\tau_Y/r=0$. Integrating from the cavity wall, where $q(a)=-P$, gives
$$
q=-P+4\tau_Y\log(r/a),\qquad t=q+2\tau_Y,
\qquad a<r<s.
$$
Outside 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 $C=P_cs^3$. <Continuity> of radial <traction> now gives
$$
-P+4\tau_Y\log(s/a)=-P_c.
$$
Thus
$$
\boxed{s=a\exp\left(\frac{P-P_c}{4\tau_Y}\right).}
$$
The complete <stress> field of this <spherical elastoplastic cavity> is
$$
\boxed{\tau_{rr}=
\begin{cases}
-P+4\tau_Y\log(r/a),&a\leq r\leq s,\\
-P_c(s/r)^3,&r\geq s,
\end{cases}
\qquad
\tau_{\theta\theta}=\tau_{\phi\phi}=
\begin{cases}
-P+4\tau_Y\log(r/a)+2\tau_Y,&a\leq r\leq s,\\
\tfrac12P_c(s/r)^3,&r\geq s.
\end{cases}}
$$
At $s$, 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.
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