Nonlinear axisymmetric deformations of an elastic tube under external pressure

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.