Elastic cavitation ($P_{\mathrm{crit}}=-{\int }_1^{\infty }{h}^{\prime }(\lambda )/({\lambda }^3-1)d\lambda$)

The opening of an internal cavity under tensile loading in a solid. For spherically symmetric deformation of an isotropic hyperelastic material with incompressibility, and an empty cavity without surface tension, radial static elastic equilibrium gives the displayed critical signed external pressure, where $h(\lambda )=W({\lambda }^{-2},\lambda ,\lambda )$ and positive $P$ denotes compression. Indeed, the principal stress formula for incompressible hyperelasticity gives $d{t}_r/d\lambda =h^{\prime }(\lambda )/(1-{\lambda }^3)$. Integration between the free inner surface and the outer surface, followed by the vanishing-initial-hole and vanishing-current-cavity limits, gives the formula. For a neo-Hookean solid with shear modulus $\mu >0$, the tensile magnitude is $-P_{\mathrm{crit}}=5\mu /2$.

For the unit ball and energy $W(F)=|F{|}^2+(\det F-1{)}^2$, boundary dilation $y=tx$ admits a singular incompressible radial competitor $r(R)=(R^3+t^3-1{)}^{1/3}$. Its radial stretch is $R^2/r^2$ and its double tangential stretch is $r/R$. Although the latter diverges at zero, the three-dimensional volume element makes its squared gradient integrable. Its energy is $4\pi {\int }_0^1[R^6/r^4+2r^2],dR\sim 8\pi t^2$.

The smooth homogeneous equilibrium $y=tx$ has constant stress and energy $(4\pi /3)[3t^2+(t^3-1{)}^2]$, of sixth-order growth. It is therefore beaten for large $t$ in the finite-energy Sobolev space class admitting the cavity. This conclusion depends on that class: the singular map is not continuous at the center and cannot simply be substituted as a smooth cavity-free competitor.

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 $2\alpha -4<-1$.