Solutions to the First-order Buckling Equations of a Fung Hyperelastic Cylindrical Shell Subjected to Torsion, Internal Pressure, and Axial Tension

In this study a theoretical model is proposed for the buckling of a vein subjected to torsion, internal pressure, and axial tension using a formation of elasticity theory for shells. The vein is assumed to be an anisotropic hyperelastic cylindrical shell which obeys the Fung constitutive model.

The approach uses finite deformation theory for thick-walled blood vessels to characterize the vessel dilation in the pre-buckling state. The pre-buckling state is identified by its midpoint and then perturbed by a displacement vector field dependent on the circumferential and axial directions to define the buckled state. The buckling equations of static equilibrium are derived using the nominal stress measure and traction boundary conditions are applied. A side result is shown proving the existence of a moment traction although typically taken to be zero for torsional problems. Perturbational displacements raised to the power of two or greater are assumed negligible thereby linearizing the coupled partial differential equations of equilibrium. The coupled equations are solved by supposing first-order and single Fourier term trigonometric forms for the displacement field components.

The model and the assumptions used are validated by experimental data for five human great saphenous vein (GSV) samples taken from a previous study. The theoretical model is unstable but using an eigenvalue compatibility condition as a selection method yields strong quantitative results for three out of five GSVs in the entire tested pressure range (6-100 mmHg). The other two sampless showed excessive stiffening upon loading and may indicate limitations of the model although quantitative predictions were still moderately accurate. The strongest results are in the 6-20 mmHg pressure range where all vessels matched well with predicted values. In general the model showed increased error as pressure increased hinting that effects of vessel stiffening are poorly predicted. The eigenmodes predicted were consistently inaccurate indicating the assumptions used in this solution method are inadequate to characterize the buckling modes of a nonlinear material. It may suggest that nonlinear buckling conformation is determined by nonlinear perturbation terms.