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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 axial loading and external pressure. In particular, we examine both 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)
Qualification Level: Doctoral
Keywords: Elastic stability, Finite deformation, Bifurcation, Elastic tube, Nonlinear elasticity, Finite deformation, Large strain, Elastic tubes, Axisymmetric deformations.
Subjects: Q Science > QC Physics
T Technology > TA Engineering (General). Civil engineering (General)
Q Science > Q Science (General)
Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Supervisor's Name: Luo, Professor Xiaoyu and Ogden, Professor Ray
Date of Award: 2010
Depositing User: Mr yunfei zhu
Unique ID: glathesis:2010-1627
Copyright: Copyright of this thesis is held by the author.
Date Deposited: 16 Mar 2010
Last Modified: 10 Dec 2012 13:44
URI: http://theses.gla.ac.uk/id/eprint/1627

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