Gravitational stresses in accreted bodies

Abstract

The conventional treatment of body forces in continuum mechanics implies that these forces are applied to a structure having the dimensions of the final body. This treatment is quite satisfactory when the body forces arise from the presence of an acceleration field or of an electromagnetic field. It fails, however, when we turn to the determination of the state of stress in a body which grows to its final shape by the gradual accretion of layers of material. Then the weight of each new layer loads and deforms the earlier material before hardening and becoming a part of the final structure. Successive layers are applied not to an unstressed but to a partially complete structure which is in a state of initial stress and deformation. The final gravitational stresses in such an accreted body depend upon an historical element; that is, on the order and manner in which the final shape is attained. When the final state is achieved there is, in general, a non-vanishing dislocation tensor. An appropriate method of computing the final gravitational stresses due to own weight is indicated for the case in which the stresses and deformations are small enough to permit the use of the constitutive and geometric equations of the linear theory of elasticity. The method is illustrated for the case of a gravitating sphere that has grown to its final size by accretion of material.