A shear-relaxation boundary layer near the pinned ends of a buckled elastic-walled tube. (13th August 2015)

• Record Type: Journal Article
• Authors: Whittaker, Robert J.
• Is Part Of: IMA journal of applied mathematics. Volume 80:Number 6(2015:Dec.)
• Page Start: 1932
• Page End: 1967
• Publication Date: 2015-08-13
• Subjects: shell theory -- boundary layers -- in-plane shear -- elastic-walled tubes; Mathematics -- Periodicals; Mathematics; Periodicals; 519
• DOI: 10.1093/imamat/hxv022
• Languages: English
• ISSNs: 0272-4960

Abstract

Abstract : We consider small-amplitude deformations of a thin-walled elastic tube, which initially has a uniform elliptical cross-section and is subject to a large axial pre-stress. We derive a boundary-layer model for the deformations near an end of such a tube that is pinned to a rigid elliptical support. The model is appropriate in the limit in which $\mathscr {F} \equiv d^2 F/[24{\pi } aK (1-\nu ^2)] \ll 1$, where $d$ is the wall thickness, $F$ is the axial tension that gives rise to the pre-stress, $2\pi a$ is the tube circumference, $K$ is the bending stiffness of the tube wall and $\nu $ is its Poisson ratio. In particular, the model takes into account in-plane shear forces arising because of geometrical constraints. These forces are asymptotically small outside the boundary layer, and so were not present in the previous tube-law model of Whittaker et al. (2010a, A rational derivation of a tube law from shell theory. Q. J. Mech. Appl. Math. 63, 465–492). Deformation profiles from the boundary-layer model are matched to solutions for the interior arising from the tube-law model of Whittaker et al. (2010a, A rational derivation of a tube law from shell theory. Q. J. Mech. Appl. Math. 63, 465–492). The net effect is to modify the previous tube-end boundary condition on the interior solution, from zero normal displacement to a Robin-type condition. The predictions from the matched models compare favourably with full numerical simulations of the tube wall deformations. While the additional shear forces are only important in the boundary layer near the end, they can have a significant effect on the global solution when $\mathscr {F} \ll 1$ .