A revisit of a cylindrically anisotropic tube subjected to pressuring, shearing, torsion, extension and a uniform temperature change

The problem of a cylindrically anisotropic tube or bar was seemed to be ®rst examined by Lekhnitskii (1981) [Lekhnitskii, S.G., 1981. Theory of Elasticity of an Anisotropic Body. (Trans. from the revised 1977 Russian edition.) Mir, Moscow]. Recently, a thorough investigation of the subject was performed by Ting (1996) [Ting, T.C.T., 1996. Pressuring, shearing, torsion and extension of a circular tube or bar of cylindrically anisotropic material. Proc. Roy. Soc. Lond. A452, 2397±2421] in which a formulation akin to that of Strohʹs formalism is employed to resolve the boundary value problem subjected to a uniform pressure, shearing, torsion and uniform extension. In a continuing paper, Ting (1999) [Ting, T.C.T., 1999. New solutions to pressuring, shearing, torsion and extension of a cylindrically anisotropic elastic circular tube or bar. Proc. Roy. Soc. Lond, to appear.] rederived the solutions based on a modi®ed formalism of Lekhnitskii, in which the solutions are in terms of elastic compliances, reduced elastic compliances as well as doubly reduced compliance. The results are much more compact and simpler than those of the earlier one. Independently, in this work, we construct the governing system also under the Lekhnitskiiʹs framework. Nevertheless, the present work and Tingʹs formulation (1999) are not alike. Besides the loads considered in Ting (1996, 1999), we add the e€ect of a uniform temperature change in the formulation. The assumption that the stresses depend only on r makes it possible to incorporate the various loading cases considered. In addition to the explicit forms of admissible stresses, we derive the admissible displacements which are ensured to be single-valued for a multiply-connected domain. In contrast to the Tingʹs works (1996, 1999), which often require superpositions of two or more basic solutions, the present solutions o€er complete forms of solutions ready for direct calculations. We also report that, as in rectilinearly anisotropic solids, an entire analogy is observed between the ®elds of a uniform axial extension and a uniform temperature change in cylindrically anisotropic solids.