Analytical solutions of strain of axisymmetric curved part in sheet metal forming process using direct integral method

Highlight•A direct integral method is presented to analyse axisymmetric curved part forming.•Strain distributions based on the new method are validated using the finite element method and experiments.•Equilibrium equations of axisymmetric forming in total differential form are presented.

This paper presents equilibrium equations for axisymmetric sheet forming problems in the total differential form. The strain differential equations in the parameter equation form are obtained on the basis of equilibrium equations, compatibility equations, plane stress assumption, proportional loading, deformation theory and Hollomon power-hardening rule. The strain distributions of axisymmetric curved parts in sheet forming processes, such as deep drawing, bulging and flanging, can be obtained on the basis of the foregoing equations and corresponding boundary conditions. Given that the first-order partial derivative of strain exists in the aforementioned strain equations, these equations should generally be solved using iterative methods. However, the solving process is time consuming. The main contributions of this study are to extend the direct integral method previously used in analysing the in-plane sheet forming process to solve strain equations of axisymmetric curved parts and explain the convergence of the solutions. The newly developed method is faster than the iterative methods. The strain distributions in the flange and die arc regions of cylindrical cups in deep drawing process are obtained using the direct integral method, finite element (FE) simulations and experiments. The strain distributions obtained using the direct integral method under the plane stress condition are closer to the FE and experimental results than those calculated analytically under the plane strain condition. The radial strain distributions predicted by the direct integral method that considers the bending effects are consistent with the experimental results.