= Equilibrium contraction of a self-gravitating elastic sphere
{title2=$\delta a=\rho ga^2/(15\kappa)$}
For a uniform traction-free sphere, first-order self-gravity is $b=-gx/a$. Static averaging gives $\overline\sigma=-\rho gaI/5$. Homogeneous <linear elasticity> then gives $\overline e=-\rho gaI/(15\kappa)$. The boundary strain identity identifies $\overline e=u(a)I/a$, proving the displayed positive radius decrease. The local strain is not uniform, and the cancellation of the <shear modulus> concerns only the mean contraction.
Back to article page
