Fourier series based finite element analysis of tube hydroforming—Generalized plane strain model

Abstract

Abstract In previous paper [Fourier series based finite element analysis of tube hydroforming—An axisymmetric model, Eng. Comput. 23 (7) (2006) 697–728], an axisymmetric analysis of tube hydroforming process was discussed. In the present paper, a generalized plane strain implicit formulation of the cross-sectional expansion of an extruded aluminum tube with pressurized fluid to fill a hydroforming die is presented. The cross-section of the tube is modeled with thin straight and circular segments with constant thickness, and Fourier series are used to approximate nodal displacements. The material of the tube is assumed to obey a rate-independent, elastoplastic model that takes into account work hardening and normal anisotropy. At the tube-die interface, frictional stress is assumed, based on Coulomb friction, to be proportional to the contact pressure whenever relative sliding occurs. The kinematics relationships are derived based on thin shell theory, and the equilibrium equation is derived based on virtual work principle. The axial feed is implemented by imposing either a compressive force or strain in the tube length direction. Frictional boundary condition is introduced into the formulation in the form of a penalty function, which imposes the constraints directly into the tangent stiffness matrix. The Newton-Raphson iterative method is used to incrementally solve the resulting nonlinear equations. Two examples of tube hydroforming problems are solved and numerical predictions of the deformed shape, hydroforming pressure, and deformation strains are compared with experimental and ABAQUS results.