Bending boundary layers near the clamped end of a thin-walled elastic tube

We consider small-amplitude deformations of a long thin-walled elastic tube, caused by a pressure difference between the interior and exterior. The tube initially has a uniform elliptical cross-section and is subject to a large axial pre-stress. The tube length and wall thinness can be exploited to derive simplified models of the wall mechanics. Such models typically neglect effects such as axial bending, which are small over most of the tube but contain higher-order axial derivatives. The resulting models are unable to satisfy the full set of clamped boundary conditions where an elastic section of tube is joined to a rigid support. In this work, we examine the asymptotic boundary layers that arise near the clamped end of an elastic-walled tube, which allow a bulk solution to a simplified model in the interior to be matched to the boundary conditions at the tube ends. We consider the region of parameter space where the width of the thinnest bending boundary-layer is small compared with the tube diameter, but still much larger than the thickness of the tube wall. Within this region, we find three distinct regimes which give rise to different sets of nested boundary layers involving different physical effects. Our matched asymptotic solutions show excellent agreement with an exact solution in a case where the full problem can be solved analytically.