An Asymptotic Solution of a Rotating Disk

The solution of the problem of a thin circular disk rotating at a constant angular velocity about its axis is obtained as a formal power series of the thickness-diameter ratio. The matching of the inner and outer expansions at a circular edge is carried out in detail for the stress conditions as well as for the displacement conditions. While the matching procedure at a stress boundary is well known, the matching procedure at a displacement boundary does not seem to have been treated thoroughly before. We accomplish the matching at a displacement boundary systematically by invoking Betti’s reciprocal theorem. The method is essentially that used by Shield in determining the resultant force on a displacement boundary. The procedure can be generalized to obtain the matching conditions at a mixed boundary.