New solutions to pressuring, shearing, torsion and extension of a cylindrically anisotropic elastic circular tube or bar

The problem of a cylindrically anisotropic elastic circular tube or bar sub jected to a uniform pressure, shearing, torsion or extension has been studi ed by Ting. The Stroh formalism for two-dimensional deformations of anisotr opic elastic materials modified for a cylindrical coordinate system was emp loyed. The solutions are in terms of the elastic stiffnesses C-alpha beta. While the modified Stroh formalism is elegant in a cylindrical coordinate s ystem, the solutions presented are rather complicated kin that most express ions involve 3 x 3 (and 4 x 4 in some cases) miners of C-alpha beta. Moreov er, degenerate cases are considered separately, resulting in up to four cas es for the problem of a uniform extension. In this paper we modify the Lekh nitskii formalism for a cylindrical coordinate system. The solutions are no w in terms of the elastic compliances s(alpha beta) and the reduced elastic compliances s'(alpha beta). We also introduce the doubly reduced elastic c ompliances (s) over cap'(alpha beta). The solutions are much simpler, and a re uniformly valid for degenerate cases. The largest miners of s(alpha beta ) that appear in the solutions are no more than 2 x 2. Moreover, the soluti ons satisfy the condition for the unwanted surface tractions on the surface s of the tube. New results are presented for a solid bar that is subjected to pure pressuring, pure tension and pure torque. The physical significance of the new solutions is more transparent. Following Lekhnitskii and Ting, the stress at the axis of a solid bar can be infinite under uniform pressur ing, torsion or extension.