 Spherical elastoplastic cavity

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.