The strain of a gravitating, compressible elastic sphere

Abstract

In the solution of the problerll of the strain of a homogeneous elastic sphere, as given by LAME and bv IVELVIN, the bodily forces are supposed expressible as known functions of the coordinates of position. Wllen self-gravitation is among the forces considered, however, the bodily forces depend in part upon the state of strain, and it is only in the case of assumed incompressibility that the usual method of solution is applicable. The present papel has for its main object the solution of the problem of the strain of an isotropic elastic sphere, initially homogeneous;, due to disturbing forces of a certaill type, takillg into account the changes in the gravitational forces whieh result from the strain. The contents of the paper fall under the following four heads: I. Strain of a gravitatillg, compressible elastic sphere under the action of tidal or centrifugal forces, with numerical consplltations for tlle case in which the surface is free from stress. II. Strain of a gravitating, compressible elastic sphere covered by a shallow ocean, under the action of tidal or centrifugal forces. III. Effect of compressibility upon estimates of the rigidity of the earth. IV. Strain of a gravitating, compressible elastic sphere under the action of small disturbing forces derivable from a potential which is any spherical harlllonic of degree not less than 2.