= Spherical elastoplastic cavity
{title2=$P_c=4\tau_Y/3,\quad s=a e^{(P-P_c)/(4\tau_Y)}$}
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.
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