Nonlinear deformations of a thick-walled hyperelastic tubeunder external pressure

QC Physics;TA Engineering (General). Civil engineering (General);Q Science (General)

Zhu, Yunfei ; Luo, Xiaoyu

University:University of Glasgow

Department:School of Mathematics and Statistics

关键词: Elastic stability, Finite deformation, Bifurcation, Elastic tube, Nonlinear elasticity, Finite deformation, Large strain, Elastic tubes, Axisymmetricdeformations.;   

Others  :  http://theses.gla.ac.uk/1627/1/2010zhuphd.pdf

来源: University of Glasgow

【 摘 要 】

This research deals with several novel aspects of the nonlinearbehaviour of thick-walled cylindrical hyperelastic tubes underexternal pressure.Initially, we consider bifurcation from a circular cylindricaldeformed configuration of a thick-walled circular cylindrical tubeof incompressible isotropic elastic material subject to combinedaxial loading and external pressure. In particular, we examine bothaxisymmetric and asymmetric modes of bifurcation. The analysis isbased on the three-dimensional incremental equilibrium equations,which are derived and then solved numerically for a specificmaterial model using the Adams-Moulton method. We assess the effectsof wall-thickness and the ratio of length to (external) radius onthe bifurcation behaviour.The problem of the finite axisymmetric deformation of a thick-walledcircular cylindrical elastic tube subject to pressure on itsexternal lateral boundaries and zero displacement on its ends isformulated for an incompressible isotropic neo-Hookean material. Theformulation is fully nonlinear and can accommodate large strains andlarge displacements. The governing system of nonlinear partialdifferential equations is derived and then solved numerically usingthe C++ based object-oriented finite element library Libmesh. Theweighted residual-Galerkin method and the Newton-Krylov nonlinearsolver are adopted for solving the governing equations. Since thenonlinear problem is highly sensitive to small changes in thenumerical scheme, convergence was obtained only when the analyticalJacobian matrix was used. A Lagrangian mesh is used to discretizethe governing partial differential equations. Results are presentedfor different parameters, such as wall thickness and aspect ratio,and comparison is made with the corresponding linear elasticityformulation of the problem, the results of which agree with those ofthe nonlinear formulation only for small external pressure.Notsurprisingly, the nonlinear results depart significantly from thelinear ones for larger values of the pressure and when the strainsin the tube wall become large. Typical nonlinear characteristicsexhibited are the ``corner bulging'' of short tubes, and multiplemodes of deformation for longer tubes.Finally the general fully nonlinear governing equations inLagrangian form for the three dimensional large deformations of anelastic tube under external pressure are developed.