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畢業(yè)設(shè)計
文獻翻譯
院（系）名稱
工學院機械系
專業(yè)名稱
機械設(shè)計制造及其自動化
學生姓名
周凱
指導教師
薛東彬
2012年 03 月 10 日
黃河科技學院畢業(yè)設(shè)計（文獻翻譯） 第24頁
沖壓半徑和角度對管向外卷曲過程的影響
摘要：跟據(jù)更新的拉格朗日理論的公式，本研究采用彈塑性有限元法和擴展增量的測定方法，最小半徑法，包括元素的產(chǎn)生，節(jié)點接觸或分離的工具，最大應(yīng)變和極限旋轉(zhuǎn)增量。使用改性的庫侖摩擦法建立有限元方法的計算機代碼。不同半徑和角度的錐形沖頭是用于模擬成的硬銅及銅合金管的兩端。影響的各種要素，包括沖床半頂角α和半徑R的比值，管的平均直徑，管壁厚度比，力學性能，并對潤滑管向外卷曲，進行調(diào)查。模擬結(jié)果顯示，當沖床入口的彎曲半徑滿足的條件ρ≦ρc，卷曲加工出現(xiàn)在管的末端。另一方面，如果ρ≧ρc，管端遭受過擴口?？勺儲裞稱為臨界彎曲半徑。ρc的值隨著α的值增加而增加。此外，研究結(jié)果還表明，ρc既不與管的材料，也不與潤滑劑有關(guān)。
關(guān)鍵詞：彈塑性；有限元;半頂角;向外卷曲過程。
1介紹
形成金屬管的凸邊的過程中，經(jīng)常被用于連接兩個管件，連接和鎖管的一部分，其組成部分，和流體管道連接管的末端相結(jié)合的互補與加固管道。它是一種常見的與管兩端相關(guān)的工業(yè)技術(shù)。攻絲機擴口，法蘭管端進程一般分為分為擴口，加強擴口，卷曲成型。本文所討論的管卷曲變形卷曲管的結(jié)束了一圈向外開放。管卷曲變形的模擬是一項復雜而艱巨的任務(wù)，因為變形過程是高度非線性的。由于非線性變形的特點是：
1．大位移，旋轉(zhuǎn)，變形過程中金屬變形。
2．金屬材料遇到大變形的非線性材料變形行為。
3．由摩擦產(chǎn)生的金屬和工具之間的界面，和他們的接觸條件的非線性邊界條件。
這些特點，使有限元方法，使用最廣泛的金屬工藝分析。改善工藝和提高工業(yè)生產(chǎn)率，本研究開發(fā)的彈塑性有限元計算機代碼，使用選擇性降低集成（SRI）的模擬方法，它采用一個矩形的四個節(jié)點內(nèi)四個整點元素。目標是到模擬的管卷曲的變形過程。
Nadai在1943年研究管收口。理論歸納為弧形殼體理論的延伸。Nadai假設(shè)摩擦系數(shù)為常數(shù)，而忽略了殼的有效應(yīng)力變化的存在。克魯?shù)呛蜏丈M行了一系列管減噪實驗，建立管減噪的各種限制，并評估各種參數(shù)的影響。Manabe and Nishimura兩個錐形燃燒管和管減噪過程進行了一系列實驗，調(diào)查研究的參數(shù)包括不同角度的錐形打孔，潤滑，材料成形負荷和應(yīng)力應(yīng)變分布的各種參數(shù)的影響管壁厚度。唐和小林提出了剛塑性有限元理論，并開發(fā)出一種計算機代碼來模擬105毫米AISI1018鋼冷收口。黃等。模擬冷收口過程中，通過彈塑性和剛塑性有限元法的方法，并比較模擬結(jié)果與實驗數(shù)據(jù)來驗證剛塑性有限元分析的準確性。北澤等人。錐形沖頭，與假設(shè)的銅及黃銅材料進行實驗，管材料剛性完美的塑料，他們探討了以前的沖頭圓弧半徑，角度，管壁厚，管卷曲變形過程的影響。北澤采用碳鋼，銅，黃銅的管材管卷曲變形試驗。當在管端前緣擴口的能量增量是大于卷曲邊緣的能量增量。管底料的向外卷曲，是在不屈的情況下形成。另一方面，如果管端材料附著因為不屈的錐形打孔表面和失敗卷曲或形成擴口。能源規(guī)則被用來誘導管形成的比較和驗證的目的時的變形能，從而確立了向外卷曲的標準。
在這項研究中，在其他研究中使用相同的材料常數(shù)和尺寸都采用向外卷曲過程的模擬結(jié)果相比，報告的結(jié)果驗證了彈塑性有限元計算機代碼的準確性。
2基本理論的說明
2.1剛度方程
采用更新的拉格朗日公式（超低頻）金屬成形過程中的增量變形的應(yīng)用框架（體積成形，板料成形）是描述塑性流動規(guī)則的增量特性的最切實可行的辦法。在每個變形階段的當前配置中的ULF用于評估一個小的時間間隔t變形，一階理論的精度要求是一致的，作為參考狀態(tài)。
更新的拉格朗日方程的虛擬工作速率方程寫成
是基爾霍夫應(yīng)力的Jaumann率，是柯西應(yīng)力，是變形率，這是笛卡爾坐標，是速度，的名義牽引速度，是速度梯度，是固定的空間直角坐標，V和材料的體積和表面上牽引規(guī)定。
圖1.幾何初始向外卷曲變形管和一個特定階段的邊界條件。單位毫米;沖床半徑R，半頂點沖角α，沖床入口彎曲半徑ρ。
J2流動規(guī)律的模型采用的彈塑性行為的金屬片，是應(yīng)變硬化率，是有效應(yīng)力，E是楊氏模量，是泊松比，是的偏量。取1為塑性狀態(tài)和0為彈性狀態(tài)或卸載。
據(jù)推測，速度{V}，在離散化元素中的分布
其中[N]是形函數(shù)矩陣和表示節(jié)點的速度。變形率和速度梯度寫成
其中[B]和[E]中，分別代表應(yīng)變率的速度矩陣和速度梯度速度矩陣。將式（4）和（5）代入式（1），得到元素的剛度矩陣。
由于虛功率式的原則。本構(gòu)關(guān)系是非線性方程。差餉，他們可以更換任何單調(diào)增加的措施方面定義的工具，如位移增量，增量。
Alpha有限元的標準程序，以形成整體的整體剛度矩陣，
在其中
在這些方程中，[K]是全球切線剛度矩陣，元素彈塑性本構(gòu)矩陣，表示位移增量，表示在規(guī)定的結(jié)點力增量。和被定義為應(yīng)力校正矩陣，由于在電流應(yīng)力變形的任何階段。
圖2.（a）變形的幾何形狀，（b）節(jié)點速度分布在12個不同的形成階段管向外卷曲硬銅管，N= 0.05; T0 =0.8毫米;=60擄; r =3.4毫米。
2.2斯里蘭卡方案
由于四邊形元素充分整合計劃的實現(xiàn)，導致過度的約束效應(yīng)時，材料近不可壓縮成型過程中的彈塑性情況[13]，休斯提出應(yīng)變率的速度矩陣分解為擴張矩陣和偏差矩陣，即
矩陣中和由傳統(tǒng)的四點一體化集成。當材料變形到幾乎不可彈塑性狀態(tài)，修改后的擴張矩陣是由一個點的一體化集成，即必須更換擴張矩陣，即
是修改后的應(yīng)變率速度矩陣。將式（8）代入式（7），修改后的應(yīng)變率速度矩陣
明顯的，速度梯度-速度矩陣，取而代之的是修改后的速度梯度–速度矩陣
3數(shù)值分析
在分析模型管卷曲過程的是軸對稱。因此，只有右邊一半的中心軸是考慮過的。部分的有限元是由計算機自動處理。由于是從彎曲，并在管的末端卷曲的急劇變形，這一節(jié)需要更精細的單元劃分，以獲得精確的計算結(jié)果。如圖左邊一半。1表示零件和模具開始的尺寸。表1給出了詳細的尺寸。在局部坐標系，軸1表示的切線方向，管材料和工具之間的聯(lián)系而軸n為同一聯(lián)系的正常方向。常數(shù)（X，Y）坐標和地方坐標（l，n）描述結(jié)點力，位移和元素的應(yīng)力和應(yīng)變。
圖3.正如圖2，但R= 3.8毫米。
表1 沖頭的角度和半徑
沖頭頂點
角度α（度）
沖頭半徑R（毫米
60
2.0
2.4
2.8
3.1
3.4
3.8
4.1
4.4
4.8
5.1
65
2.4
2.8
3.1
3.4
3.8
4.1
4.4
4.8
5.5
5.4
70
3.1
3.4
3.8
4.1
4.4
4.8
5.1
5.4
5.8
6.1
75
3.4
3.8
4.1
4.4
4.8
5.1
5.4
5.8
6.1
6.4
80
4.8
5.4
5.8
6.1
6.4
6.8
7.2
7.6
8.0
8.4
85
5.8
6.1
6.4
6.8
7.4
9.0
9.4
9.8
10.2
10.6
11.0
表2.所用材料的機械性能
材料
外徑×厚度
真空熱處理
n
F(Mpa)
屈服應(yīng)力(Mpa)
銅
25.4×0.8
原樣
0.09
380
220
500℃ 1 h
0.53
630
26
原樣
0.05
450
280
300℃ 1 h
0.09
450
260
400℃ 1 h
0.46
610
50
600℃ 1 h
0.50
640
42
70/3黃銅
25.4±0.8
原樣
0.18
730
280
應(yīng)力 ；應(yīng)變
表2給出了我們模擬的物質(zhì)條件。管外半徑保持在25.4毫米不變;但也有三種不同的管壁厚度值，即，0.4，0.6和0.8毫米，在實驗和計算。硬銅管和黃銅管的泊松比和楊氏模量分別是0.33和110740兆帕，0.34和96500兆帕。
3.1邊界條件
如圖右手半管。1是指管卷曲變形在某一階段的變形形狀。邊界條件包括以下三個部分：
1.FG和BC段的邊界：
這里是節(jié)點的切向摩擦力增量，是普通的力量增量。假定涉及到摩擦材料，工具的接觸面積，和不等于零。，表示工具的配置文件的正常方向的位移增量，從規(guī)定的沖頭位移增量中確定。
2.邊界上的CD，DE，EF，和GA部分：
上述情況反映了這個邊界上的節(jié)點都是自由的。
3.AB段的邊界：
AB部分是固定在管底的邊界。在此位置沿Y軸方向的節(jié)點位移增量設(shè)置為零，而節(jié)點B是完全固定的沒有任何動靜。
在管卷曲過程中，邊界將會改變。因此，有必要檢查FG和BC沿邊界部分的變形階段，在每個接觸節(jié)點的法向力。如果達到零，則節(jié)點將成為自由節(jié)點和邊界條件（1）（2）轉(zhuǎn)移。同時，GA和EF沿管段的自由節(jié)點還檢查了計算。如果該節(jié)點接觸到?jīng)_頭，自由邊界條件的約束條件（1）改變。
3.2 彈塑性和接觸問題的處理
當解釋以前明確暗示的邊界條件時，接觸條件應(yīng)維持在一個增量變形過程。為了滿足這個要求，山田等人提出了γ最小值方法。[ 14]采用擴展處理了彈塑性和接觸問題[ 15]。每個加載增量的最小值均被控制，有以下六個標準。
彈塑性狀態(tài)。元素的應(yīng)力大于屈服應(yīng)力時，R1的計算方法[15]因此，以確定達到一樣的表面屈服應(yīng)力。
最大應(yīng)變增量。獲得R2默認的最大應(yīng)變增量ψ主應(yīng)變增量比dε，即，限制增長到這樣的規(guī)模，一階理論是有效的措施。
最大旋轉(zhuǎn)增量。默認的最大旋轉(zhuǎn)增量β的旋轉(zhuǎn)增量，即，限制增長到這樣的規(guī)模，一階理論是有效的措施。
滲透條件。形成的收益時，管的自由節(jié)點可能滲透的工具。比[16]的計算方法等，剛剛接觸到的工具的自由節(jié)點。
分離條件。當形成的收益，接觸節(jié)點可以脫離接觸面。 [ 16 ]計算每個接觸節(jié)點，這種正常的組成部分，節(jié)點力為零。
滑粘摩擦條件。修正的庫侖摩擦定律提供了選擇的接觸狀態(tài)，即滑動或粘狀態(tài)。這種狀態(tài)檢查每個接觸節(jié)點由以下條件：
（a）如果，還有節(jié)點處于滑動狀態(tài)
(b)如果還有則節(jié)點處于固定狀態(tài)
一個方向滑動節(jié)點是相反的方向前漸進的步驟，使接觸節(jié)點粘貼節(jié)點在下一個增量步。然后在這里得到比R，，產(chǎn)生變化的摩擦狀態(tài)的基體，在是一個很小的公差。
最大應(yīng)變增量Ψ，最大的旋轉(zhuǎn)增量β這里使用的常數(shù)是0.002o和0.5o，分別。這些常量都被證明是有效的一階理論。。此外，一個小的檢查程序的滲透和分離條件是允許的。
3.3 卸載過程
卸載后的回彈現(xiàn)象是管形成過程中具有重要意義。假設(shè)是固定的，對管底的節(jié)點卸載程序執(zhí)行。所有的元素都將復位彈性。節(jié)點來接觸到的工具的力量在扭轉(zhuǎn)成為管規(guī)定力的邊界條件，即
同時，核查的滲透，摩擦，分離條件被排除在仿真程序。
四．結(jié)果與討論
圖2顯示沖角α=60°，圓弧半徑R=3.4毫米銅管卷曲成形仿真結(jié)果。圖2（a）表示的幾何形狀的變形，在最后的形狀是卸載后的最終形狀的一部分。圖2（b）顯示了變形過程中的節(jié)點的速度分布。最后圖顯示卸貨后節(jié)點的速度分布。這些數(shù)字表明，管端材料處于起步階段順利進入沿打孔表面。管底料后，逐漸彎曲向外卷曲，造成所謂向外卷曲成形。圖3顯示的R=3.8毫米的情況下的模擬結(jié)果。管端材料還顯示所謂的燃燒形成合格后，電弧感應(yīng)部分和圓錐面形成。
圖4顯示了α=75°，R=6.1毫米卷曲銅管的模擬的結(jié)果。相比之下，圖。5顯示了擴口中的R= 6.4毫米的情況下形成的模擬結(jié)果。
圖4 如圖2條件，而α=75o；R=6.1毫米
圖5 如圖2條件，而α=75o；R=6.4毫米
圖6顯示了應(yīng)變分布相應(yīng)的那些圖2和圖3在沖床行程12毫米。圖7顯示了應(yīng)變分布相應(yīng)的那些圖4和圖5在沖床進步14毫米。變形主要是由于拉在圓周方向彎曲，在正北方向和厚度方向的剪切。最大直徑的管端擴口成形。因此，其應(yīng)變值在圓周方向遠高于向外卷曲。減少管壁厚度也更重要的擴口成形。因此，在相同的條件下，在管端擴口比在其他情況下更容易斷裂。
圖6計算硬銅管應(yīng)變分布下的卷曲和擴口過程比較
n=0.05;to=0.8毫米；α=60o
圖7 條件如圖6，而α=75o
圖8和圖9顯示仿真結(jié)果硬銅和黃銅，分別。同管壁厚，管半徑，潤滑條件，和半頂沖床是用來在模擬。然而，不同的價值觀的沖頭圓弧半徑的使用，從而產(chǎn)生不同類型的形成，即，擴口和卷曲。就是說，如果圓弧半徑，，在沖頭入口大于臨界彎曲半徑，ρ，然后管端的材料形成沿電弧燃燒沖壓部分和圓錐面。反之，如果，管底材料葉片穿孔表面形成卷曲。這一數(shù)字表明臨界彎曲半徑隨半頂沖床增加。比較結(jié)果表明，我們的數(shù)值模擬是一致的實驗數(shù)據(jù)報告[ 12]。所謂臨界彎曲半徑，ρ可表示如下：
圖8 影響壁厚硬銅管彎曲半徑和沖角之間至關(guān)重要的關(guān)系 n=0.05
圖9 條件如圖8 黃銅管n=0.05
這里
R c=沖頭半徑相應(yīng)的ρ丙
R fmin=最小值所需的耀斑沖壓半徑
R cmax=最大值所需的卷曲沖壓半徑
t0=壁厚管
圖10和圖11表明之間的相關(guān)性無量綱臨界彎曲半徑（ρ丙）和半頂角的2種不同壁厚徑比的情況下，努力按銅管，分別。價值ρ可以定義如下：
如果ρ是無關(guān)的壁厚徑比，那么幾何相似定律的存在臨界彎曲半徑。事實判斷出來的曲線顯示在圖10和11是幾乎相同的，我們一定會存在的幾何相似的臨界彎曲半徑管材料。
圖10
圖11
圖12和圖13表明之間的關(guān)系，最大半徑和半頂角的時候，在數(shù)值模擬的壁厚度值為0.4，0.6，和0.8毫米的情況下硬銅管和銅管分別發(fā)生卷曲。當雙方的半徑和相應(yīng)的半頂角的沖落在曲線的條件，將導致擴口。另一方面，當兩個值低于曲線的條件，將導致冰壺。這些數(shù)字表明，最大半徑增加的半頂角的擴大。
圖12
圖13
圖14顯示了相關(guān)材料的加工硬化指數(shù)（n）和臨界彎曲半徑。無論變化值之間的0.03和0.53，由于不同的管材料，數(shù)值模擬的結(jié)果表明，值的臨界彎曲半徑保持不變，從而形成一條直線。這些調(diào)查結(jié)果是幾乎相同的實驗數(shù)據(jù)表明[ 12 ]，這意味著，臨界彎曲半徑的時候，卷曲是無關(guān)的加工硬化指數(shù)。
圖14
圖15顯示了影響潤滑的關(guān)系臨界彎曲半徑和半頂角的沖頭。臨界彎曲半徑的增加而增加半頂角的沖頭的情況μ=0.05和μ=0.20。此外，結(jié)果顯示，有一個微不足道的差異程度臨界彎曲半徑之間的潤滑條件。
圖15黃銅管依賴的臨界彎曲半徑對潤滑條件。
五．總結(jié)
從更新的拉格朗日公式，帶有一個圓錐形的沖壓工藝模擬管卷曲的彈塑性有限元計算機代碼開發(fā)。高非線性的過程是一個漸進的方式考慮與一個最小半徑技術(shù)是通過限制每個增量步的線性關(guān)系的大小。
摩擦改性庫侖法開發(fā)是一個連續(xù)函數(shù)。如前所述，此功能可以處理工具和金屬界面的滑動和粘性很難與普通連續(xù)摩擦模型描述的現(xiàn)象。斯里蘭卡有限元是四節(jié)點四邊形單元中的4個集成點。然后再加上與大變形有限元分析的SRI進一步成功地應(yīng)用于錐形管卷曲形成工藝的分析。根據(jù)上述分析和討論，可以得出以下結(jié)論：
1、一個關(guān)鍵的彎曲半徑，存在。如果拱半徑ρ在沖頭入口大于臨界彎曲半徑，，然后管端的材料在沖壓拱感應(yīng)部分和錐形面形成擴口。另一方面，如果，管底材料從表面和結(jié)果卷曲。彎曲半徑的臨界值隨沖頭的半頂角增加而增加。
2、臨界彎曲半徑，ρ，遵循幾何相似定律。ρ值隨沖床的半頂角的增加而增加。我們也知道，ρ值是不相關(guān)的管材料，潤滑條件或壁厚直徑半徑比。
3、擴口圓周應(yīng)變大于卷邊的。擴口厚度也有比卷曲減少。因此，擴口更容易發(fā)生管端斷裂。相比之下，卷曲的產(chǎn)品可以保證處理的安全，并加強管材料的結(jié)束部分。
4、模具的形狀表示為一個數(shù)值函數(shù)。因此，在這項研究中開發(fā)的有限元模型，可用于在連續(xù)形狀的任何工具，來模擬擴口或卷曲過程中的所有類型。
參考文獻
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Int J Adv Manuf Technol (2002) 19:587596 Ownership and Copyright Springer-Verlag London Ltd 2002 Influence of Punch Radius and Angle on the Outward Curling Process of Tubes You-Min Huang and Yuung-Ming Huang Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan Using the theory of updated Lagrangian formulation, this study adopted the elasto-plastic finite-element method and extended the increment determination method, the r min method, to include the elements yielding, nodal contact with or separation from the tool, maximum strain and limit of rotation increment. The computer code for a finite-element method is established using the modified Coulombs friction law. Conical punches with different radii and angles are used in the forming simulation of hard copper and brass tube ends. The effects of various elements including the half-apex angle of punch (H9251) and its radius (R), the ratio of the thickness of the tube wall to the mean diameter of tube, mechanical properties, and lubrication on the tubes outward curling, are investigated. Simulation findings indicate that when the bending radius at the punch inlet (H9267) satisfies the condition of H9267 H11017 H9267 c , curling is present at the tube end. On the other hand, if H9267 H11084 H9267 c , the tube end experiences flaring. The variable H9267 c is called the critical bend- ing radius. The value of H9267 c increases as the value of H9251 increases. Furthermore, the findings also show that H9267 c is neither correlated with tube material nor lubrication. Keywords: Elasto-plastic; Finite elements; Half-apex angle; Outward-curling process 1. Introduction The process of forming convex edges of a metal tube is often employed for connecting two tube parts, linking and locking a tube part and its components, and connecting fluid pipelines or combining complementary pipelines with reinforcement at the tubes end. It is a common industrial technology related to tube ends. The tube end processes are generally divided into tapper flaring, flange flaring, step flaring, and curl forming. The tube-curling deformation discussed in this paper curls the opening of the tubes end outward in a circle. The simulation Correspondence and offprint requests to: Dr You-Min Huang, Depart- ment of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan. E-mail: ymhuangL50560mail.ntus- t.edu.tw of the tube-curling deformation is a complex and difficult task because the deformation process is highly nonlinear. The nonlinear deformation characteristic is due to: 1. The large displacement, rotation, and deformation during metal deformation. 2. The nonlinear material deformation behaviour when metal material experiences large deformation. 3. The nonlinear boundary condition generated by the friction between the metal and tool interface, and their contact con- ditions. These characteristics made the finite-element method the most widely used of the metal process analyses. To improve the process and increase industrial productivity, this study developed an elasto-plastic finite-element computer code using the selective reduced integration (SRI) simulation method, which employs four integral point elements within the four nodes of a rectangle. The objective is to simulate the tube- curling deformation process. Nadai 1 studied tube-nosing in 1943. The theoretical induc- tion was an extension of the curved shell theory. Nadai assumed the friction coefficient to be constant, and ignored the presence of effective stress variation in the shell. Cruden and Tompson 2 conducted a series of tube-nosing experiments to establish the various limitations of tube-nosing and evaluate the effects of various parameters. Manabe and Nishimura 37 also con- ducted a series of experiments on both conical tube-flaring and tube-nosing processes to investigate the effects of various parameters on the forming load and stressstrain distribution. The parameters studied include the conical punch with different angles, lubrication, material and tube wall thickness. Tang and Kobayashi 8 proposed a rigid-plastic finite-element theory and developed a computer code to simulate the cold-nosing of 105 mm AISI 1018 steel. Huang et al. 9 simulated the cold- nosing process through elasto-plastic and rigid-plastic finite- element methods and compared the simulation results with experimental data to verify the accuracy of the rigid-plastic finite element. Kitazawa et al. 10,11 conducted experiments with conical punches, and copper and brass materials with the assumption that the tube materials were rigid-perfectly plastic. They had explored previously the effect of the arc radius of 588 You-Min Huang and Yuung-Ming Huang the punch, angle, and tube wall thickness on the tube-curling deformation process. Kitazawa 12 used carbon steel, copper, and brass as the tube materials in his experiment of tube- curling deformation. When the flaring energy increment at the front edge of the tube end (H9254W f ) is greater than the energy increment at the curl edge (H9254W c ), the tube end material undergoes outward curling forming because of the absence of unbending. On the other hand, if H9254W c H11084H9254W f , the tube end material is attached to the conical punch surface because of unbending and fails to curl or form flaring. The energy rule was used to induce the deformation energy at the time of tube forming for the purpose of comparison and verification, thus establishing the criterion of outward curling. In this study, the same material constants and dimensions used in other studies 12 are adopted in the simulation of the outward curling process. The findings are compared with the results reported in 12 to verify the accuracy of the elasto- plastic finite-element computer code developed. 2. Description of the Basic Theory 2.1 Stiffness Equation Adopting the updated Lagrangian formulation (ULF) in the framework of the application of incremental deformation for the metal forming process (bulk forming and sheet forming) is the most practical approach for describing the incremental characteristics of the plastic flow rule. The current configuration in ULF at each deformation stage is used as the reference state for evaluating the deformation for a small time interval H9004t such that first-order theory is consistent with the accuracy requirement. The virtual work rate equation of the updated Lagrangian equation is written as Fig. 1. Boundary conditions for the deformation tubes geometry of outward curling at an initial and a particular stage. Units, mm; R, punch radius; H9251, half-apex punch angle; H9267, bending radius at punch inlet. H20885 V (H9270 * ij 2H9268 ik H9280 kj ) H9254H9280 ij -V + H20885 V H9268 jk L ik H9254L ij -V = H20885 S f tH6032 i H9254v i -S (1) where H9270 * ij is the Jaumann rate of Kirchhoff stress, H9268 ij is the Cauchy stress, H9280 ij is the rate of deformation which is the Cartesian coordinate, v i is the velocity, tH6032 i is the rate of the nominal traction, L ij (= H11128v i /H11128X j ) is the velocity gradient, X j is the spatial fixed Cartesian coordinate, V and S f are the material volume and the surface on which the traction is prescribed. The J 2 flow rule H9270 * = E 1+v H9254 ik H9254 jl + v 1 2v H9254 ij H9254 kl 3H9261(E/(1+v) H9268 H11032 ij H9268 H11032 kl 2H9268 2 (0H H11032 + E/(1 + v) H9280 kl (2) is employed to model the elasto-plastic behaviour of sheet metal, where HH11032 is the strain hardening rate, H9268 is the effective stress, E is the Youngs modulus, v is the Poissons ratio, H9268 H11032 ij is the deviatoric part of H9268 ij . H9261 takes 1 for the plastic state and 0 for the elastic state or the unloading. It is assumed that the distribution of the velocity vina discretised element is v = Nd (3) where N is the shape function matrix and d denotes the nodal velocity. The rate of deformation and the velocity gradient are written as H9280 = Bd (4) L = Ed (5) where B and E represent the strain ratevelocity matrix and the velocity gradientvelocity matrix, respectively. Substituting Eqs (4) and (5) into Eq. (1), the elemental stiffness matrix is obtained. As the principle of virtual work rate Eq. and the constitutive relationship are linear Eq. of rates, they can be replaced by increments defined with respect to any monotonously increasing measure, such as the tool-displacement increment. Following the standard procedure of finite elements to form the whole global stiffness matrix, KH9004u = H9004F (6) in which K = H20888 H20855eH20856 H20885 VH20855eH20856 B T (D ep Q) B-V + H20888 H20855eH20856 H20885 VH20855eH20856 E T GE-V H9004F = H20873H20888 H20855eH20856 H20885 SH20855eH20856 N T tH6032-S H20874 H9004t In these equations, K is the global tangent stiffness matrix, D ep is the elemental elasto-plastic constitutive matrix, H9004u denotes the nodal displacement increment, and H9004F denotes the prescribed nodal force increment. Q and G are defined as stress-correction matrices due to the current stress at any stage of deformation. Influence of Punch Radius and Angle on Tube Curling 589 Fig. 2. (a) Deformed geometries, and (b) nodal velocity distributions in the outward curling of tubes at 12 different forming stages. Hard copper tube, n = 0.05; t 0 = 0.8 mm; H9251 = 60; R = 3.4 mm. 2.2 SRI Scheme As the implementation of the full integration (FI) scheme for the quadrilateral element leads to an excessively constraining effect when the material is in the nearly incompressible elasto- plastic situation in the forming process 13, Hughes proposed that the strain ratevelocity matrix be decomposed to the dilation matrix B dil and the deviation matrix B dev , i.e. B = B dil +B dev (7) in which the matrices B dil and B dev are integrated by conven- tional four-point integration. When the material is deformed to the nearly incompressible elasto-plastic state, the dilation matrix B dil must be replaced by the modified dilation matrix B dil that is integrated by the one point integration, i.e. B = B dil +B dev (8) in which B is the modified strain ratevelocity matrix. Substi- tuting Eq. (8) into Eq. (7), the modified strain ratevelocity matrix is B = B+(B dil B dil ) (9) Explicitly, the velocity gradientvelocity matrix E is replaced by the modified velocity gradientvelocity matrix E E = E+(E dil E dil ) (10) 3. Numerical Analysis The analytical model of the tube-curling process is axially symmetric. Thus, only the righthand half of the centre axis is considered. Division of the parts finite elements is automati- cally processed by the computer. Since there is drastic defor- mation from the bending and curling at the tubes end, a finer element division is required for this section in order to derive precise computation results. The lefthand half shown in Fig. 1 denotes the sizes of the part and die at the beginning. The detailed dimensions are given in Table 1. In the local coordi- nates, axis 1 denotes the tangential direction of the contact between tube material and the tool, while axis n denotes the normal direction of the same contact. Constant coordinates (X,Y) and local coordinates (l, n) describe the nodal force, displacement and elements stress and strain. 590 You-Min Huang and Yuung-Ming Huang Fig. 3. As Fig. 2, but R = 3.8 mm. Table 1. Punch angle and radius. Punch apex Punch radius R (mm) angle H9251 (deg.) 60 2.0, 2.4, 2.8, 3.1, 3.4, 3.8, 4.1, 4.4, 4.8, 5.1 65 2.4, 2.8, 3.1, 3.4, 3.8, 4.1, 4.4, 4.8, 5.5, 5.4 70 3.1, 3.4, 3.8, 4.1, 4.4, 4.8, 5.1, 5.4, 5.8, 6.1 75 3.4, 3.8, 4.1, 4.4, 4.8, 5.1, 5.4, 5.8, 6.1, 6.4 80 4.8, 5.4, 5.8, 6.1, 6.4, 6.8, 7.2, 7.6, 8.0, 8.4 85 5.8, 6.1, 6.4, 6.8, 7.4, 9.0, 9.4, 9.8, 10.2, 10.6, 11.0 Table 2 gives the material conditions of our simulation. The exterior radius of the tube remains unchanged at 25.4 mm; but there are three different tube wall thickness values, namely, 0.4, 0.6 and 0.8 mm, used in the experiment and computation. The Poissons ratio and Youngs modulus of the hard copper tube are 0.33 and 110740 MPa, respectively. The Poissons ratio and Youngs modulus of brass are 0.34 and 96500 MPa, respectively. 3.1 Boundary Condition The righthand half of the tube shown in Fig. 1 denotes the deformation shape at a certain stage during the tube-curling Table 2. Mechanical properties of materials used. Materials Outer Heat treatment nF Yield stress diameter in vacuum (Mpa) H9268 0.2 (Mpa) thickness Copper 25.4 0.8 As received 0.09 380 220 500C 1 h 0.53 630 26 As received 0.05 450 280 300C 1 h 0.09 450 260 400C 1 h 0.46 610 50 600C 1 h 0.50 640 42 70/30 Brass 25.4 0.8 As received 0.18 730 280 H9268 = FH9280 n ; H9268 = stress; H9280 = strain. deformation. The boundary conditions include the following three sections: 1. The boundary on the FG and BC sections: H9004f l HS33527 0, H9004f n HS33527 0, H9004v n = H9004v n where H9004f l is the nodal tangential friction forces increment, and H9004f n is the normal force increment. As the materialtool contact area is assumed to involve friction, H9004f l and H9004f n are not equal to zero. H9004v n , which denotes the nodal displacement Influence of Punch Radius and Angle on Tube Curling 591 increment in the normal direction of the profile of the tools, is determined from the prescribed displacement increment of the punch H9004v n . 2. The boundary on the CD, DE, EF, and GA sections: H9004F x = 0, H9004F y = 0 The above condition reflects that the nodes on this boundary are free. 3. The boundary on the AB section: H9004v x = 0, H9004v y = 0 The AB section is the fixed boundary at the bottom end of the tube. The displacement increment of the node at this location along the y-axis direction is set at zero, whereas node B is completely fixed without any movement. As the tube-curling process proceeds, the boundary will be changed. It is thus necessary to examine the normal force H9004f n of the contact nodes along boundary sections FG and BC in each deformation stage. If H9004f n reaches zero, then the nodes will become free and the boundary condition is shifted from (1) to (2). Meanwhile, the free nodes along the GA and EF sections of the tube are also checked in the computation. If the node comes into contact with the punch, the free-boundary condition is changed to the constraint condition (1). 3.2 Treatment of the Elasto-Plastic and Contact Problems The contact condition should remain unchanged within one incremental deformation process, as clearly implied from the interpretation in the former boundary condition. In order to satisfy this requirement, the r-minimum method proposed by Yamada et al. 14 is adopted and extended towards treating the elasto-plastic and contact problems 15. The increment of each loading step is controlled by the smallest value of the following six values. 1. Elasto-plastic state. When the stress of an element is greater than the yielding stress, r 1 is computed by 15 so as to ascertain the stress just as the yielding surface is reached. 2. The maximum strain increment. The r 2 term is obtained by the ratio of the defaulted maximum strain increment H9274 to the principal strain increment dH9280, i.e. r 2 = H9274/dH9280, to limit the incremental step to such a size that the first-order theory is valid within the step. 3. The maximum rotation increment. The r 3 term is calculated by the defaulted maximum rotation increment H9252 to the rotation increment dH9275, i.e. r 3 = H9252/dH9275, to limit the incremen- tal step to such a size that the first-order theory is valid within the step. 4. Penetration condition. When forming proceeds, the free nodes of the tube may penetrate the tools. The ratio r 4 16 is calculated such that the free nodes just come into contact with tools. 5. Separation condition. When forming proceeds, the contact nodes may be separated from the contact surface. The r 5 term 16 is calculated for each contact node, such that the normal component of nodal force becomes zero. 6. Sliding-sticking friction condition. The modified Coulombs friction law provides two alternative contact states, i.e. sliding or sticking states. Such states are checked for each contacting node by the following conditions: v rel(i) l v rel(Il) l H11084 0 (a) if H20841v rel(i) l H20841 H11022 VCRI, then f 1 = H9262f n , the node is in sliding state (b) if H20841v rel(i) l H20841 H11021 VCRI, then f 1 = H9262f n (v rel l /VCRI), the node is in quasi-sticking state. v rel(i) l v rel(I1) l H11021 0. The direction of a sliding node is opposite to the direction of the previous incremental step, making the contact node a sticking node at the next incremental step. The ratio r 6 is then obtained here, r 6 = Tolf/H20841v rel(i) l H20841, which produces the change of friction state from sliding to sticking, where Tolf (= 0.0001) is a small tolerance. The constants of the maximum strain increment H9274 and maximum rotation increment H9252 used here are 0.002 and 0.5, respectively. These constants are proved to be valid in the first- order theory. Furthermore, a small tolerance in the check pro- cedure of the penetration and separation condition is permitted. 3.3 Unloading Process The phenomenon of spring-back after unloading is significant in the tube-forming process. The unloading procedure is executed by assuming that the nodes on the bottom end of the tube are fixed. All of the elements are reset to be elastic. The force of the nodes which come into contact with the tools is reversed in becoming the prescribed force boundary condition on the tube, i.e. F =F. Meanwhile, the verification of the penetration, friction, and separation condition is excluded in the simulation program. 4. Results and Discussion Figure 2 shows the simulation results of curling forming of copper tubes with the punch semi-angle, H9251 = 60, and the arc radius, R = 3.4 mm. Figure 2(a) denotes the geometric shape of the deformation, in which the last shape is the final shape of the part after unloading. Figure 2(b) shows the nodal velocity distribution during deformation. The last diagram shows the nodal velocity distribution after unloading. These figures show that the tube end material enters smoothly along the punch surface at the initial stage. After that, the tube end material gradually bends outward and curls up, resulting in the so- called outward curling forming. Figure 3 shows the simulation result in the case of R = 3.8 mm. The tube end material still shows the so-called flaring forming after passing the arc induc- tion section and conical face forming. Figure 4 shows the result of curling simulation of copper tubes in the case of H9251 = 75 and R = 6.1 mm. In contrast, Fig. 5 shows the simulation result of flaring forming in the case of R = 6.4 mm. Figure 6 shows the strain distributions corresponding to those of Figs 2 and 3 at a punch travel of 12 mm. Figure 7 592 You-Min Huang and Yuung-Ming Huang Fig. 4. As Fig. 2, but H9251 = 75; R = 6.1 mm. shows the strain distributions corresponding to those of Figs 4 and 5 at a punch progress of 14 mm. The deformation is mostly the result of pulling in the circumference direction, bending in the meridian direction and shearing in the thickness direction. The maximum diameter of the tube end is the flaring forming. Thus, its strain value in the circumference direction is far greater than that of outward curling. The reduction in tube wall thickness is also more significant in flaring forming. Thus, under the same progress, fracture of the tube end occurs more easily in flaring forming than in other situations. Figures 8 and 9 show the simulation results for hard copper and brass, respectively. The same tube wall thickness, tube radius, lubrication condition, and half-apex of the punch are used in both simulations. However, different values of the punch arc radius are used, which generate different types of forming, namely, flaring and curling. That is, if the arc radius, H9267, at the punch ent
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