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Zhu, Yunfei (2010) Nonlinear Deformations of a Thick-Walled Hyperelastic Tube Under External Pressure

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Zhu, Yunfei (2010) 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. http://theses.gla.ac.uk/1627/ Copyright and moral rights for this thesis are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given. Glasgow Theses Service http://theses.gla.ac.uk/ [email protected] Nonlinear deformations of a thick-walled hyperelastic tube under external pressure by Yunfei Zhu A thesis submitted to the Faculty of Information and Mathematical Sciences at the University of Glasgow for the degree of Doctor of Philosophy December 2009 °c Yunfei Zhu 2009 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 con¯guration 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 axisymmet- ric 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 speci¯c material model using the Adams-Moulton method. We assess the e®ects of wall- thickness and the ratio of length to (external) radius on the bifurcation behaviour. The problem of the ¯nite axisymmetric deformation of a thick-walled circular cylindri- cal 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 for- mulation is fully nonlinear and can accommodate large strains and large displacements. The governing system of nonlinear partial di®erential equations is derived and then solved numerically using the C++ based object-oriented ¯nite 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 analyt- ical Jacobian matrix was used. A Lagrangian mesh is used to discretize the governing partial di®erential equations. Results are presented for di®erent parameters, such as wall thickness and aspect ratio, and comparison is made with the corresponding linear elas- ticity 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 signi¯cantly 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 i ii 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. Acknowledgements I would like to thank my supervisors, Professor Xiaoyu Luo and Professor Ray W. Ogden, for their continued support and advice through out the period of this study. I feel extremely fortunate to have been given the chance to work under them as a postgraduate and to continue working with them as a research assistant. I am also grateful to the Overseas Research Students Awards Scheme, and the Faculty of Information and Mathematical Science, and the Department of Mathematics at Glasgow University for the generous scholarships. I also wish to thank Dr. David Haughton, University of Glasgow, and Professor Chris Berttram, University of New South Wales, for their valuable advice on Chapter 4. I also want to express my appreciation to Libmesh developers at the University of Texas and Dr. Boyce Gri±th at New York University for their help with using Libmesh. My parents and elder sister deserve the deepest thanks for their never-ending support during my education. Finally, I would like to thank the cool kids in my o±ce, such as Robert, Stephen, Ehsan, Susan, Monica and Marjory for their help in the last three years. iii Contents Abstract i Acknowledgements iii 1 Introduction 1 2 Basic equations 11 2.1 Deformation . 11 2.2 Stress theory and equilibrium . 13 2.3 Constitutive law for a Cauchy elastic material . 14 2.4 Green elastic material . 16 2.4.1 Stress-deformation relations in terms of invariants . 17 2.5 Internal constraints . 18 2.6 Example strain-energy functions for isotropic elastic material . 19 2.7 Incremental deformations . 20 3 Finite element method, libmesh library 22 3.1 Finite element nonlinear analysis in solid mechanics . 22 3.1.1 Introduction . 22 3.1.2 Procedure for ¯nite element solution with libmesh . 23 3.1.3 Stress recovery . 25 3.2 Libmesh library . 27 3.2.1 Object-Oriented Scienti¯c Computing . 28 3.2.2 libmesh: Adaptive mesh re¯nement scheme . 29 3.2.3 Data structure of libmesh . 30 iv CONTENTS v 4 Bifurcation 34 4.1 The elastic constitutive law and strain-energy function . 34 4.2 The circular cylindrical con¯guration . 35 4.3 Incremental equations . 37 4.4 Asymmetric bifurcations and numerical methods . 40 4.5 Numerical results and discussion . 44 4.5.1 Equilibrium pressure curves . 44 4.5.2 Axisymmetric bifurcation . 45 4.5.3 Asymmetric bifurcation . 46 4.5.4 Discussion . 56 4.6 Conclusion . 58 5 Nonlinear axisymmetric deformations 60 5.1 Basic equations . 60 5.1.1 Deformation . 60 5.1.2 Material properties and equilibrium . 61 5.2 Linear and nonlinear equations . 63 5.2.1 Radially symmetric case . 63 5.2.2 The linear case . 65 5.2.3 The nonlinear case . 66 5.3 Finite element algorithm . 67 5.3.1 Discretization . 67 5.3.2 Newton's method . 68 5.3.3 Detailed discretizing integrations . 68 5.4 Numerical algorithms . 70 5.5 Numerical results . 71 5.5.1 Thick-walled short tubes: A=B = 0:5 and L=B =1.......... 71 5.5.2 Thick-walled longer tubes: A=B = 0:5 and L=B =5 ......... 87 5.5.3 Thinner and longer tubes: A=B = 0:8 and L=B =5 ......... 90 5.6 Discussion and conclusions . 91 6 Three-dimensional large deformations 94 6.1 Nonlinear case: cylindrical polar coordinates . 94 6.1.1 Deformation gradient . 94 CONTENTS vi 6.1.2 Nominal stress and Cauchy stress . 96 6.1.3 Equilibrium equations . 97 6.2 Nonlinear case: cartesian coordinates . 97 6.2.1 Deformation gradient . 97 6.2.2 Nominal stress and Cauchy stress . 98 6.2.3 Governing equations . 98 6.2.4 Unit outward normal to the outer surface . 99 6.3 Linear case . 99 6.3.1 Deformation gradient and Cauchy stress . 100 6.3.2 Governing equations . 100 7 Conclusions 102 A Appendix A 105 A.1 Integration of the equilibrium equations by parts . 105 B Appendix B 109 B.1 Mesh ¯les . 109 References 113 Chapter 1 Introduction In this thesis, we focus on modelling and simulating the collapse of a cylindrical tube under external pressure. The collapse of a circular tube is of interest not only in many engineering applications, such as in the design of undersea or underground pipelines and pressure vessels, particularly submarine structures, but it is also of great interest in the biomechanics context. The main motivation for this research comes from the investigation of the behaviour of physiological conduits within our body subject to external pressure, such as veins and arteries conveying blood flow and large bronchi conducting air into the lungs. Due to the high flexibility of the soft biological tissue, these conduits may collapse under certain conditions of external and internal fluid pressure. For example, intramyocardial arteries collapse during systole [29]. More generally, the collapse of cardiovascular vessels plays an important role in the delivery of blood to other organs [30], [63]. A long cylindrical elas- tic tube when subjected to a transmural (internal minus external) pressure may collapse into a two-lobed con¯guration, as illustrated in Fig.1.1. In elastic buckling the collapse is usually sudden and catastrophic, which is a process involving some nonlinear dynamical deformations of great complexity. Depending on geometry, material properties, the pres- sure and boundary conditions, the tubes may collapse di®erently. In essence, these kinds of problems involve two physical systems interacting with each other, which are the elastic wall of the conduit and the biological fluid inside or around the conduit. Such systems are also known as coupled and such coupling may be weak or strong depending on the degree of interaction. A general de¯nition of coupled systems may be given as [80] 1 CHAPTER 1. INTRODUCTION 2 Figure 1.1: Collapsed tube (from paper by Marzo et al. [52]). De¯nition 1.0.1 Coupled systems and formulations are those applicable to multiple do- mains and dependent variables which usually but not always describe the di®erent physical phenomena and in which (a) neither domain can be solved while separated from the other; (b) neither set of dependent variables can be explicitly eliminated at the di®erential equation level. The topic of flow through collapsible tubes, an obvious coupled problem, has been studied for several decades. This subject has been reviewed briefly by Kamm and Pedley [42]. Experiments [14], [12] on a Starling resistor prototype of the system have presented a rich dynamic behaviour, with various types of self-excited oscillations.
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