Approximate solution of finite deformation for the combined bending and torsion of circular tubes

In this paper, the equilibrium problem of incompressible hyperelastic circular tubes under combined bending and torsional deformation is studied. By using polar coordinates on the axis, a three-dimensional kinematic model of the longitudinal bending of a circular tube with wall thickness variation is established. Due to the adoption of the semi-inverse method, the displacement field specified in the model contains three unknown functions. Lagrangian and Eulerian analyses are performed on the model to determine the (first) Piola-Kirchhoff stress and Cauchy stress, clarify the equilibrium equations and boundary conditions, and thus solve for the unknown parameters in the kinematic model. In addition, the established model is validated by comparing it with the finite element (FE) results. The results show that the established model is effective and exhibits high accuracy. It describes the combined deformation of bending and torsion quite well and obtains the axial normal stress and torsional shear stress with relatively high precision. It can characterize the distribution of the wall thickness and the strain-neutral layer (SNL) after deformation with quite high precision. According to the deformation angle, the corresponding bending moment and torque are obtained, and the relative errors are all at a low level. Finally, the axial elongation rates of tubes with different specifications are analyzed, and it is found that they increase with the increase of the outer diameter or the inner diameter.