Stresses in accreted planetary bodies

An explicit analytical solution is obtained for the stress field in an accreted triaxial ellipsoid under the influence of self-gravitation and rotation. Material is assumed to attach to the surface of the accreting body in a stress-free state, after which it behaves elastically. The results differ significantly from the classical elasticity solutions that are based on the assumption that the body is fully formed before the loading is applied. These results are relevant to the strengths of accreted planetary bodies such as comets and asteroids.The solution allows both the magnitude and direction of the angular velocity to be a general function of the time-like parameter defining the progress of accretion. Simple closed-form expressions are given for two special cases—the ellipsoid accreting at constant angular velocity and the sphere accreting with an angular velocity vector that precesses through 90° during the accretion process. A Mathematica notebook permitting the solution of other problems can be downloaded from the website http://www-personal.umich.edu/jbarber/ellipsoid.nb.