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Sheet metal forming analysis with a modified membrane finite element formulation considering bending effect (굽힘 효과를 고려한 박막 요소 수식화에 의한 박판 성형 가공 해석)
Computer simulation of sheet metal forming is generally carried out by the finite element method with a membrane element or a shell element. In the real process modelling, a membrane element is regarded as more preferable rather than a shell element because of the computing efficiency and the contact treatment. Nevertheless, a membrane element has a disadvantage of disregarding the bending effect during the deformation, which leads to inaccuracy in computing the forming load and the deformed shape. To overcome such deficit in using a membrane element, some modification is required in the finite element formulation procedure to take the bending effect into account. One of them may be to make a membrane element modified with the advantage of a shell element by considering the bending effect. The remedy correctly enhances the flexural rigidity not only within an element but also among elements, which is almost zero especially in the ordinary 4 - node bilinear quadrilateral membrane element. In this paper, this idea is adopted in the finite element formulation procedure. A variational formulation is derived for the incremental analysis of the non-steady large deformation of sheet metal. Then, the strain energy term, or the plastic dissipation term is decomposed into the term due to the mean stretching throughout the thickness and the term due to the bending deformation. The two kinds of strain energy are calculated separately and added together within a membrane element. For sheet metal forming simulation, the sheet material is assumed to be rigid - plastic, possessing normal anisotropic and obeying Hill's quadratic yield criterion and its associated flow rule. The formulation uses the convected coordinate system for the sakes of convenience with geometrical nonlinearity and is combined with an effective contact algorithm to deal with the contact between the material and the punch or the die. The algorithm developed is implemented into a computer program and applied to several sheet metal forming processes. The first numerical simulation investigates bending effect on a stretching process and a deep drawing process. The numerical results are compared with the experimental results and other numerical results based on the membrane theory. The comparison shows the computed results are in better agreement with the experimental results than other numerical results. The second numerical simulation investigates the thickness effect on a square cup drawing and the size effect on an initial wrinkling phenomenon in a square cup drawing without the blank holder. The investigation of the thickness effect demonstrates the importance of bending effect by showing the difference between the corresponding drawing loads. The investigation of the size effect demonstrates the versatility of the present algorithm which can calculate the wrinkling behavior of sheet metals. The third numerical simulation investigates the draw - bead effect on a sheet metal forming process. The numerical results are compared with the experimental results and the comparison shows the computed results are in good agreement with the experimental results. The numerical results show the drawing load and the strain distribution are significantly affected by the draw - bead. Consequently, the algorithm developed is proved to improve the precision of the numerical solutions with the efficient computing time and can be effectively applied to the analysis of the sheet metal forming process which is significantly affected by bending and unattainable with the conventional membrane finite element approach.