Blank design and strain estimates for sheet metal forming processes by a finite element inverse approach with initial guess of linear deformation

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Abstract

A new finite element approach is introduced for direct prediction of blank shapes and strain distributions from desired final shapes in sheet metal forming. The approach deals with the geometric compatibility of finite elements, plastic deformation theory, minimization of plastic work with constraints and a proper initial guess. The algorithm developed is applied to cylindrical cup drawing, square cup drawing and oil pan drawing in order to confirm its validity by demonstrating reasonably accurate numerical results for each problem. Rapid calculation with this algorithm enables easy determination of various process variables for design of a sheet metal forming process.

Introduction

Sheet metal forming processes experiences very complicated deformation effected by process parameters such as the die geometry, the blank shape, the sheet thickness, the blank holding force, friction, lubrication and so on. Although these process parameters have an influence on the deformation mechanism and the quality of deformed parts, the optimum condition for process parameters is determined by intuition and experience, with trial-and-error. More systematic methods for the determination of the optimum conditions have been developed with the aid of computers and numerical analysis. Numerical analysis including finite-element methods, can simulate complicated sheet metal parts and affords useful information on forming processes, thus reducing trial-and-error. However, numerical simulation is generally carried out with given process parameters and so requires numerical trial-and-error with enormous time and cost, to determine the optimum process parameters. For this reason, there arises the necessity for some approaches to find one, or some, of the optimum process parameters directly.

One of the important process parameters is the blank shape, which has a direct relationship with the quality of the deformed parts. Design methods for the blank shape have been widely studied by many researchers. Jimma [1], Hazek and Lange [2]and Karima [3]made use of the slip-line method to design the initial blank shape. Vogel and Lee [4]and Chen and Sowerby [5]used the characteristic of plane stress, while Duncan et al. [6]and Blount and Stevens [7]used geometric mapping to design the initial blank shape. These methods provide good guidance to design the initial blank shape, even though they neglect the height of the deformed parts, have geometric restrictions, or do not consider the deformation behavior of the materials. On the other hand, there have been several attempts to design the blank shape and estimate the distribution of the strain in a deformed part with deformation theory. Majlessi and Lee 8, 9, 10extended the theory of Levy et al. [11]and applied it to axisymmetric problems and axisymmetric multi-stage problems, obtaining good results. Guo and Batoz 12, 13, 14, 15derived a formulation for field problems as an inverse method to obtain the initial blank shape and the thickness distribution in a deformed part. Chung and Richmond 16, 17, 18, 19suggested ideal forming with optimum deformation to design the initial blank shape and the intermediate deformed shapes.

In this paper, the strain tensor at the deformed state is calculated from the initial state as a function of the coordinates by approximating the deformed shape to a system of discretized triangular membrane elements. The plastic work is then calculated element-wise and minimized satisfying the given constraints with the conjugate gradient method and the Newton–Raphson method. The plastic work is assumed to obey Hencky’s deformation theory and Hill’s anisotropic yield criterion. The calculation and minimization adopts an initial guess using inverse mapping with linear deformation, which is of great importance for good convergence. Minimization provides the optimum shape of the initial blank and the strain distribution in a deformed part under given friction conditions and blank-holding forces. The pre-approximated inverse method enables the determination of process parameters within a small range of error with low computing time, affording the determination of the strain distribution in a deformed part before the process design.

Section snippets

Formulation

Predicting an initial blank shape from a final deformed shape in a one-step calculation, the finite element approach using deformation theory has some different features from the conventional finite element method using the incremental theory. The problem reduces to minimizing the plastic potential energy, relating the initial state to the final state. In the problem, the given variables are the geometry of the final state and the thickness of the initial state, while the unknowns are the

Numerical results

The algorithm described above has been implemented in a finite-element code and applied to several sheet metal forming examples for validation. The blank shapes and thickness distributions of a cylindrical and a square cup have been obtained as a bench-mark test. In order to check the validity of the present algorithm, experiments have been carried out for cylindrical and square cup drawing with blank specimens prepared for the calculated blank shape. As a demonstration of its versatility, the

Conclusions

An algorithm has been developed for the prediction of the initial blank shapes and strain distributions from desired final shapes in the sheet metal forming process. Analysis with the developed algorithm can help in the design of sheet metal parts and their manufacturing processes at the initial stage of product development and in order to reduce trial-and-error as well as cost. In this approach, the material is described by Hencky’s deformation theory and Hill’s anisotropic yield criterion. A

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