A new solution for the deformations of an elastic-walled tube

We investigate the small-amplitude deformations of a long, thin-walled elastic tube

having an initially axially uniform elliptical cross-section. The tube is deformed by

a (possibly non-uniform) transmural pressure. At leading-order its deformations are

shown to be governed by a single partial differential equation (PDE) for the azimuthal

displacement as a function of the axial and azimuthal co-ordinates and time. Previous

authors have obtained solutions of this PDE by making ad-hoc approximations based

on truncating an approximate Fourier representation. In the present work, we instead

write the azimuthal displacement as a sum over the azimuthal eigenfunctions of a

generalised eigenvalue problem. We show that we are able to derive an uncoupled

system of linear PDEs with constant coefficients for the amplitude of the azimuthal

modes as a function of the axial co-ordinate and time. This results in a formal solution

of the whole system being found as a sum over the azimuthal modes. We show that

the nth mode’s contribution to the tube’s relative area change is governed by a second-

order PDE, and examine the case in which the tube’s deformations are driven by a

uniform transmural pressure. Finally, we investigate how our solution method can be

adapted to investigate tubes with different initial cross-sections.