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

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

Ramsey Shadfan

ABSTRACT
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 and thickness decrease 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 ℝ3. The Euler-Lagrange equations form three coupled linear
partial differential equations with Dirichlet boundary conditions characterizing the buckling
displacement field under equilibrium. 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.

DATE: August 1, 2018
TIME: 9:00 AM
LOCATION: FLN 4.01.20
CHAIR: Changfeng Gui, Ph.D.