 # Nonlinear deformations of a thick-walled hyperelastic tube under external pressure

Zhu, Yunfei (2010) Nonlinear deformations of a thick-walled hyperelastic tube under external pressure. PhD thesis, University of Glasgow.

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## Abstract

This research deals with several novel aspects of the nonlinear
behaviour of thick-walled cylindrical hyperelastic tubes under
external pressure.

Initially, we consider bifurcation from a circular cylindrical
deformed configuration of a thick-walled circular cylindrical tube
of incompressible isotropic elastic material subject to combined
axisymmetric and asymmetric modes of bifurcation. The analysis is
based on the three-dimensional incremental equilibrium equations,
which are derived and then solved numerically for a specific
material model using the Adams-Moulton method. We assess the effects
of wall-thickness and the ratio of length to (external) radius on
the bifurcation behaviour.

The problem of the finite axisymmetric deformation of a thick-walled
circular cylindrical elastic tube subject to pressure on its
external lateral boundaries and zero displacement on its ends is
formulated for an incompressible isotropic neo-Hookean material. The
formulation is fully nonlinear and can accommodate large strains and
large displacements. The governing system of nonlinear partial
differential equations is derived and then solved numerically using
the C++ based object-oriented finite element library Libmesh. The
weighted residual-Galerkin method and the Newton-Krylov nonlinear
solver are adopted for solving the governing equations. Since the
nonlinear problem is highly sensitive to small changes in the
numerical scheme, convergence was obtained only when the analytical
Jacobian matrix was used. A Lagrangian mesh is used to discretize
the governing partial differential equations. Results are presented
for different parameters, such as wall thickness and aspect ratio,
and comparison is made with the corresponding linear elasticity
formulation of the problem, the results of which agree with those of
the nonlinear formulation only for small external pressure. Not
surprisingly, the nonlinear results depart significantly from the
linear ones for larger values of the pressure and when the strains
in the tube wall become large. Typical nonlinear characteristics
exhibited are the ``corner bulging'' of short tubes, and multiple
modes of deformation for longer tubes.

Finally the general fully nonlinear governing equations in
Lagrangian form for the three dimensional large deformations of an
elastic tube under external pressure are developed.

Item Type: Thesis (PhD) Doctoral Elastic stability, Finite deformation, Bifurcation, Elastic tube, Nonlinear elasticity, Finite deformation, Large strain, Elastic tubes, Axisymmetric deformations. Q Science > QC PhysicsT Technology > TA Engineering (General). Civil engineering (General)Q Science > Q Science (General) College of Science and Engineering > School of Mathematics and Statistics > Mathematics Luo, Professor Xiaoyu and Ogden, Professor Ray 2010 Mr yunfei zhu glathesis:2010-1627 Copyright of this thesis is held by the author. 16 Mar 2010 10 Dec 2012 13:44 http://theses.gla.ac.uk/id/eprint/1627 View Item