The residually stressed unloaded state of arteries: Membrane and thin cylinder approximations

A solution is obtained for incompressible non-linearly elastic membranes that describes the bending of a cylindrical sector to form a perfect cylinder for a wide class of materials that includes isotropic materials and orthotropic materials reinforced by two families of mechanically equivalent fibres that are wound helically about the axial direction. Despite the relative simplicity of the physical problem, the solution of the corresponding boundary value problem for thick cylinders requires a numerical solution for even the simplest models. It is shown, however, that the thin shell solution provides an excellent approximation to the solution of the problem for cylindrical sectors whose thicknesses are an order of magnitude greater than that assumed for membranes. The approximate stress distribution in such thin shells is obtained. In these residually stressed cylinders, the radial stress is approximately zero but the hoop and axial stresses are finite. Estimates of the residual stresses in the unloaded state are obtained. A closed-form solution for the bending moment necessary to effect the deformation is also obtained.