本文採用Hill的異向性降伏理論及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，以模擬方杯深引伸成形製程。由於流線法在預測初始料片輪廓曲線時，並未考慮沖頭底部圓弧角，因此在方杯引伸貫穿成形後，方杯的實際杯高將會高於所給定的設定杯高。因此本文利用流線法及其補正公式預測固定杯高之初始料片外形，根據不同杯高的假設來預測各料片之初始輪廓，再進行方杯深引伸成形分析，進而判斷出方杯深引伸成形極限之最佳化初始料片之輪廓曲線。
本文並設計一組方形模具，進行方杯深引伸成形實驗，以探討方杯深引伸之杯高之分佈、厚度變化情形與破裂情況，進而得到方杯深引伸成形最佳化料片之成形極限，此數值分析所得之結果均可合理的預測實驗成果。
經由數值分析與實驗結果比較得知，固定杯高47.5mm的初始料片外形是方杯深引伸貫穿達成形極限之最佳化料片，並且與圓形料片經直接貫穿成形後之方杯在外形上做比較，經流線法及補正公式計算之方杯在平行邊處較為平整，而圓形料片其方杯平行邊之中點附近仍為曲線外形，可知由流線法及補正公式所預測之料片經深引伸成形後之工件具有較佳的外形。最後利用極限引伸比(LDR)之定義可知，其極限引伸比為2.56。

A methodology for formulating an elasto-plastic finite element model was developed to analyze the deep drawing process of the square cup.  The model was based on the Updated Lagrangian Formulation and Hill’s yield criterion.  Because the stream line didn’t consider the effect of punch radius when predicting the initial contour of optimum blank, hence the actual height of square cup would be higher than the fixed height of square cup after deep drawing.  Therefore, stream line and its modified formula were used in this study.  According to the assumption of the different cup height, the prediction of the optimum contour profile would be achieved.  Also, the simulation of the deep drawing of square cup was proceeded to determine the initial contour profile of optimal blank to the forming limit. 
A set of tools was designed to perform the deep drawing experiment of the square cup.  A good agreement was obtained from the comparison of the results between the simulation and experiment, such as the distribution of cup height and the distribution of the workpiece thickness. In addition, the forming limit could be gained from the deep drawing experiment of the optimum blank of the square cup.  Therefore, the elasto-plastic finite element program simulation result could predict the experiment result reasonably.
From the comparison of the results between the simulation and experiment, the square cup height was 47.5mm of the initial contour profile of optimal blank to the forming limit.  The appearance of the square cup of the optimum blank after deep drawing was better than the circular blank.  Finally, the limiting drawing ratio of the optimal blank was 2.56 in the deep drawing process of square cup.

目      錄
中文摘要 Ⅰ
英文摘要 Ⅱ
目    錄 Ⅳ
圖表索引 Ⅵ
第一章   緒論 1
1.1 前言 1
1.2 研究動機與目的 2
1.3 文獻回顧 3
1.4 論文之構成 4
第二章   基本理論 6
2.1 基本假設 6
2.2 有限變形之應變與應變率 6
2.3 有限變形之應力與應力率 9
2.4 有限變形之Total Lagrangian Formulation 14
2.5 有限變形之Updated Lagrangian Formulation 17
2.6 材料之彈塑性構成關係式 18
第三章	有限元素分析 27
3.1 簡介 27
3.2 虛速度原理的離散化 29
3.3 退化殼元素（Degenerated Shell Element）之分析 31
3.4 不同積分法則推導退化殼元素之剛性矩陣 33
3.5 摩擦處理 34
3.6 廣義 法之增量步驟的計算 38
第四章	預測方杯深引伸成形極限之最佳化料片之初始外形 44
4.1 流程簡介 44
4.2 預測固定杯高之方杯深引成形之初始料片外形 44
    4.2.1 流線法 44
    4.2.2 利用修正公式修正流線法預測固定杯高之條件 48
    4.2.3 依流線法預測固定杯高為47.50mm時方杯引伸貫穿成形之初始料片外形 50
    4.2.4 有限元素分析 50
    4.2.5 邊界條件 51
    4.2.6 材料參數 51
4.3 實驗設備 52
    4.3.1 實驗原理與步驟 53
4.4 數值分析與實驗結果之比較 54
    4.4.1 數值分析結果 54
    4.4.2 實驗結果與討論 55
    4.4.3 數值模擬與實驗負荷之比較 56
    4.4.4 最佳化料片之成形歷程與變形圖 56
    4.4.5 方杯工件之主應力分佈 57
    4.4.6 方杯工件之主應變分佈 57
    4.4.7 工件之厚度分佈 57
第五章	結論 79
5.1 結論 79
5.2 未來展望 80
參考文獻 81
符號索引 84
圖  表  索  引
圖2-1  定義應力之連體座標 24
圖2-2   Lagrange應力 24
圖2-3   Kirchhoff應力 25
圖2-4   Cauchy應力 25
圖2-5  平衡時之連體體積與曲面定義 26
圖3-1  退化殼元素之座標 42
圖3-2  殼元素之自由度 42
圖3-3  接觸彈簧元素 43
圖3-4  彈塑性分析中材料降伏之判定 43
圖4-1  預測固定杯高之最佳化料片初始外形之流程圖 59
圖4-2  速度與勢能場 60
圖4-3  不規則沖頭輪廓與其材料流動的勢能速率 60
圖4-4  體積控制與材料流動 61
圖4-5  實際沖頭幾何外形與流線法在預測固定杯高之方杯深引伸貫穿成形之初始料片時的沖頭外形 61
圖4-6  方形沖頭外圍輪廓圖 62
圖4-7  經流線法及補正公式預測固定杯高47.5mm之方杯深引伸貫成形初始料片與沖頭之輪廓曲線 62
圖4-8  方杯深引伸貫穿成形極限分析之沖頭網格分割 63
圖4-9  方杯深引伸貫穿成形極限分析之沖模網格分割 64
圖4-10 方杯深引伸貫穿成形極限分析之壓料板網格分割 64
圖4-11 方杯深引伸貫穿成形初始料片之邊界條件設定65
圖4-12 方杯深引伸成形的模具配置及相關尺寸圖 67
圖4-13 固定杯高47.5mm之方杯深引伸貫穿成形之工件與沖模示意圖 69
圖4-14 方杯深引伸貫穿成形後沿方杯杯口周緣杯高分佈之比較 69
圖4-15 數值模擬與實驗負荷之比較 73
圖4-16 固定杯高47.5mm方杯深引伸貫穿之成形歷程 74
圖4-17 固定杯高47.5mm方杯深引伸成形分析之最大主應力分佈圖 74
圖4-18 固定杯高47.5mm方杯深引伸成形分析之最大主應力等值線分佈圖 75
圖4-19 固定杯高47.5mm方杯深引伸成形分析之最大主應變分佈圖 75
圖4-20 固定杯高47.5mm方杯深引伸成形分析之最大主應變等值線分佈圖 76
圖4-21 杯高47.5mm方杯深引伸成形之厚度分佈圖 76
圖4-22 杯高47.5mm方杯深引伸成形之厚度等值線分佈圖 77
圖4-23 工件沿輥軋方向厚度分佈之比較 77
圖4-24 工件沿輥軋方向夾45度角厚度分佈之比較 78
圖4-25 工件沿輥軋方向夾90度角厚度分佈之比較 78
表4-1  方杯深引伸成形模具與料片之網格分割數據 63
照片4-1  實驗用之方形沖頭 65
照片4-2  實驗用之沖模 66
照片4-3  實驗用之壓料板 66
照片4-4  實驗用之間隔環 67
照片4-5  工件厚度之量測方法 68
照片4-6  工件杯口周緣杯高之量測方法 68
照片4-7  流線法及補正公式預測固定杯高47.5mm之數值模擬與實驗工件之比較 70
照片4-8  流線法及補正公式預測固定杯高47.5mm之最佳化料片經貫穿成形之照片 70
照片4-9  直徑128mm料片貫穿成形之照片 71
照片4-10 固定杯高45mm料片實驗貫穿成形之照片 71
照片4-11 固定杯高47mm料片實驗貫穿成形之照片 72
照片4-12 固定杯高47.5mm料片實驗貫穿成形之照片 72
照片4-13 固定杯高48mm料片在沖頭衝程達到33mm時發生破裂的情形 73

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