Energy and Stress Distributions in Ellipsoids

Abstract

A closed form solution for a uniform, isotropic, homogenous, rotating, and self-gravitating solid ellipsoid is analyzed as a function of its material properties and angular momentum. This solution recovers the classical results for an incompressible material, namely that Maclaurin and Jacobi ellipsoids minimize the total strain energy of an ellipsoid. As the solid becomes compressible we find that Maclaurin-type spheroids are energetically preferred over all relevant values of angular momentum, although the gradient toward these spheroids is weak, at best. Compressibility allows the weak minima to encompass most observed asteroid shapes. Using this solution, a simple failure criterion of maximum principal stress is examined. Under this failure criterion, an ellipsoid rotating at a sufficiently rapid rate can have positive stress on the periphery and in the polar regions. Contours for this failure theory are compared with estimated asteroid rotation rates and ellipsoidal shape estimates. For compressible materials there is surprisingly good correspondence for such a simple theory and criterion, with the failure curve delimiting a demarcation that few observed asteroids cross. This research has implications for the shapes and distortions of rubble pile asteroids. In particular, the theory gives insight into the internal stress field and the effects of energy dissipation in an asteroid after it has been boosted into a nonminimum energy configuration due to an impact event or a close flyby with a planet. Internally there are double-lobed compressive stress regions reminiscent of the external shape of Kleopatra.