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

1999; Royal Society; Volume: 455; Issue: 1989 Linguagem: Inglês

10.1098/rspa.1999.0464

1471-2946

T. C. T. Tings,

Dynamics and Control of Mechanical Systems

The problem of a cylindrically anisotropic elastic circular tube or bar subjected to a uniform pressure, shearing, torsion or extension has been studied by Ting. The Stroh formalism for two–dimensional deformations of anisotropic elastic materials modified for a cylindrical coordinate system was employed. The solutions are in terms of the elastic stiffnesses Cαβ. While the modified Stroh formalism is elegant in a cylindrical coordinate system, the solutions presented are rather complicated in that most expressions involve 3 times 3 (and 4 times 4 in some cases) minors of Cαβ. Moreover, degenerate cases are considered separately, resulting in up to four cases for the problem of a uniform extension. In this paper we modify the Lekhnitskii formalism for a cylindrical coordinate system. The solutions are now in terms of the elastic compliances sαβ and the reduced elastic compliances sśαβ. We also introduce the doubly reduced elastic compliancessśαβ. The solutions are much simpler, and are uniformly valid for degenerate cases. The largest minors of sαβ that appear in the solutions are no more than 2 times 2. Moreover, the solutions satisfy the condition for the unwanted surface tractions on the surfaces 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 pressuring, torsion or extension.