= Solution
{c}
The <neo-Hookean solid> corresponds to $\alpha=1$, with positive <shear modulus> $\mu$. Substitution of the <principal stretches> gives
$$
h(\lambda)=\frac\mu2(\lambda^{-4}+2\lambda^2-3),\qquad
h'(\lambda)=2\mu(\lambda-\lambda^{-5}).
$$
The identity $\lambda^6-1=(\lambda^3-1)(\lambda^3+1)$ simplifies the integrand:
$$
\frac{h'(\lambda)}{\lambda^3-1}
=2\mu\left(\lambda^{-2}+\lambda^{-5}\right).
$$
The two <improper integrals> are elementary:
$$
\boxed{P_{\mathrm{crit}}
=-2\mu\left(\int_1^\infty\lambda^{-2}\,d\lambda
+\int_1^\infty\lambda^{-5}\,d\lambda\right)
=-2\mu\left(1+\frac14\right)=-\frac52\mu.}
$$
Thus the critical tensile <stress> magnitude is $5\mu/2$. As a check on the finite-sphere limit, part (c) gives
$$
P(a)=-2\mu\left(\lambda_b^{-1}+\frac14\lambda_b^{-4}\right),\qquad
\lambda_b=(1+a^3/B^3)^{1/3},
$$
which tends to $-5\mu/2$ as the cavity radius tends to zero.
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