Nonlinear axisymmetric deformations of an elastic tube under external pressure

Abstract

International audienceThe 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 Cþþ 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

Geometrically nonlinear dynamic analysis of functionally graded thick hollow cylinders using total Lagrangian MLPG method

“…Relations (42) to (45) represent exponential, triangular, rectangular and step loading, respectively. The obtained time histories of radial displacement and stress fields base on the presented loadings in Eqs.…”

Stress Analysis of Thick Pressure Vessel Composed of Incompressible Hyperelastic Materials

“…The same authors also investigated nonlinear axisymmetric deformations of an elastic tube under external pressure. [19] In another study behavior of a hyperelastic gas-filled spherical shell in a viscous fluid was studied by Allen and Rashid [20]. The behavior of an inflating spherical membrane investigated to provide information on the pressure-deformation response by Needleman [21], Chen and Healey [22], Müller and Struchtrup [23], Goriely et al [24], Beatty [25] and Rudykh et al [26].…”

Three-dimensional flows in a hyperelastic vessel under external pressure

“…1. The same authors (Zhu et al 2010) also carried out numerical simulations for large nonlinear deformation of thick-walled tubes with the initial configuration constrained to a circular cylindrical shape. The work was later extended to a fully 3D and nonlinear thick-walled tube model, which showed that the axisymmetric analysis provides a good Fig.…”

“…For example, only a quarter of the tube was modelled, and the focus was mostly on mode-2 collapse (Heil 1997;Heil and Pedley 1996;Marzo et al 2005), the solid mechanics was represented using a thin-shell theory or even a tube-law (Whittaker et al 2010d;Whittaker 2015). On the other extreme, Zhu et al (Zhu et al 2008(Zhu et al , 2010(Zhu et al , 2012) adopted a purely solid mechanics approach to study a three-dimensional nonlinear thick-walled tube under external pressure and axial loading. Despite the absence of fluid, their model offered some important insights on the collapsible behaviour of the tube.…”