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For strain energy density $W=\mu [({\lambda }_r^2+{\lambda }_{\theta }^2+{\lambda }_{\varphi }^2{)}^{\alpha }-3^{\alpha }]/2$ with $\mu >0$ and $\alpha >0$, the critical loading for elastic cavitation is finite exactly in the displayed range. To prove this, put $S={\lambda }^{-4}+2{\lambda }^2$ in the reduced strain energy density. The critical-pressure integrand is $2\mu \alpha S^{\alpha -1}({\lambda }^{-2}+{\lambda }^{-5})$. It is finite at $\lambda =1$ and asymptotic to $\mu \alpha 2^{\alpha }{\lambda }^{2\alpha -4}$ at infinity. The comparison test for the resulting improper integral gives $2\alpha -4<-1$.

First let $A\downarrow 0$ at fixed positive cavity radius $a$ and fixed reference outer radius $B$. Thenand the pressure sustaining a finite cavity isThe onset of cavitation is the additional limit $a\downarrow 0$. The outer stretch then tends to one, and the signed critical pressure isWhen this improper integral is positive and finite before the minus sign, the tensile loading magnitude is $-P_{\mathrm{crit}}>0$. Thus the cavity opens under tension, even though the load variable is named pressure. The critical value is independent of the original sphere radius $B$.

For $A=0$ and $a>0$, the deformation map is smooth away from $R=0$ but singular at the original centre: $r=(R^3+a^3{)}^{1/3}$ approaches $a$, rather than zero, as $R\downarrow 0$. This is the elastic cavitation mechanism; it cannot be represented by a smooth deformation map through the centre.