## On the Energy Conserved in a Buckling Fung Hyperelastic Cylindrical Shell Subjected to Torsion, Internal Pressure and Axial Tension

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A theoretical model is proposed for the buckling of a three-dimensional vein subjected to torsion, internal pressure, and axial tension using energy conservation methods. The vein is assumed to be an anisotropic hyperelastic cylindrical shell which obeys the Fung constitutive model. Finite deformation theory for thick-walled blood vessels is used 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 total potential energy functional of the system is extremized by minimizing the first variation with respect to the elements of the set of all continuous bounded functions on

A second solution method approximates the first variation of the total potential energy functional using a variational Taylor series expansion. The approximation is minimized and combined with equations of equilibrium derived from elasticity theory to yield a polynomial relating buckling eigenmodes, material parameters, geometric parameters, and the critical angle of twist which induces buckling. Various properties of the total potential energy functional specific to the problem are proved. Another solution method is outlined using the first variation approximation and the basis of the kernel of the linear transformation which maps buckling displacement amplitudes during static equilibrium.