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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.

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.