本文採用Prandtl-Reuss塑流法則和von Mises的降伏條件，結合有限變形理論及updated Lagrangian formulation (ULF)的觀念建立一增量型彈塑性大變形三維有限元素分析程式，並應用四邊形四節點退化殼元素(degenerated shell element)所推導之形狀函數偶合入剛性矩陣中，組成三維有限元素之分析模式，以及使用廣義 法則處理板材成形時，元素降伏之判斷、最大容許應變增量、最大容許旋轉增量、料片與模具間節點之接觸或分離等問題。
為驗證此有限元素程式之可信度及準確度，本研究以材料拉伸試驗所得之試片破斷面厚度為數值分析之破裂判斷準則，探討橢圓杯引伸成形製程之沖頭負荷與衝程之關係、工件厚度變化、主應力分佈、等價應變分佈、變形歷程及成形極限等。
由數值分析與實驗結果比較得知，沖頭負荷隨著沖頭衝程之增加而遞增，直到成形為止負荷才趨於平緩。成形工件在主應力與等價應變分佈的變化可區分為三階段，最小厚度值集中在工件與沖頭長軸接觸之區域，而工件短軸區域則因為曲率半徑較大，所承受之周向拉伸應力較小，故厚度變化不大。經由橢圓柱沖頭周長與初始料片周長所定義之極限引伸比得知，其橢圓杯成形之極限引伸比為2.136。隨著料片尺寸之加大，其破裂發生之部位對應於相對位置可發現有逐漸下降之趨勢。

A methodology of formulating an elasto-plastic three-dimensional finite element model, which is based on Prandtl-Reuss flow rule and von Mises’s yield criterion respectively, associated with an updated Lagrangian formulation, is developed to simulation sheet metal flanging process. The shape function derived from a four-node quadrilateral degenerated shell element associated into the stiffness matrix to constitute the finite element model. An extended rmin algorithm is proposed to formulate the boundary condition, such as nodal penetration and separation, strain increment and rotation increment and altered elasto-plastic state of material.
In order to verify the reliability and accuracy of the FEM code, the fractured thickness of a specimen in the simple tension test is adopted as the fracture criterion of forming limit in simulation. The numerical simulation results include relationship between punch load and punch stroke, distribution of the thickness, distribution of the maximum principal stress, distribution of the normal strain, deformation history and the forming limit etc. in elliptical cup drawing process.
According to the simulation and experiment results, the maximum punch load decrease as the punch stroke, until final drawing punch load becomes to smooth and steady; drawing piece in distribution of the maximum principal stress and distribution of the normal strain can be divided into three stages, the minimum thickness are concentrated on the contact regions between workpiece and punch major axis, because camber radius is relatively large in minor axis, the ones that bear are relatively small to the circular tensile stress, so the thickness does not change much. The limiting drawing ratio (LDR) is defined between the circumference of elliptical punch profile and the circumference of the initial of the blanks. The LDR amounts to about 2.136 for penetration in elliptical cup drawing process of this study. With initial blank of increasing of size, the position taken place to break it correspondent to relative position can is it have trend to of the decline gradually.

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