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拉中拉深壁起皺的分深模設(shè)計(jì)析 ——F．—K．Chen and Y．—C．Liao 臺(tái)灣大學(xué)機(jī)械設(shè)計(jì)研究所 在帶有斜度的方形盒和帶有階梯的方形盒的拉深中發(fā)生的起皺現(xiàn)象一直在被研究。這兩中類型的起皺現(xiàn)象有一個(gè)共同的特征：全都發(fā)生在相對(duì)無(wú)支撐、無(wú)壓邊的拉深壁處。在帶有斜度的方形盒的拉深中，常受到工序參數(shù)的影響，例如：模具的間隙值和壓邊力等，所以常用有限元模擬的方法來(lái)研究分析起皺的發(fā)生。模擬的結(jié)果表明模具的間隙值越大，起皺現(xiàn)象就越嚴(yán)重，而且增加壓邊力也不能抑制和消除起皺現(xiàn)象的發(fā)生。在帶有階梯的方形盒拉深的起皺現(xiàn)象分析中，常通過實(shí)際生產(chǎn)中一種近似的幾何結(jié)構(gòu)來(lái)研究、試驗(yàn)。當(dāng)凸模與階梯邊緣之間的金屬板料在拉深時(shí)分布并不均衡，就會(huì)在側(cè)壁發(fā)生起皺現(xiàn)象。為了消除起皺現(xiàn)象的發(fā)生，一個(gè)最優(yōu)的模具設(shè)計(jì)常采用有限元的方法進(jìn)行分析。模擬的結(jié)果和起皺試驗(yàn)論證了有限元分析的準(zhǔn)確性，并且表明了在拉深模具設(shè)計(jì)中使用有限元方法分析的優(yōu)越性。 關(guān)鍵詞:側(cè)壁起皺；拉深模；帶有階梯的方形盒；帶有斜度的方形盒 一、介紹: 起皺是金屬板料成形中常見的失效形式之一。由于功能和視覺效果的原因，起皺通常是不能為零件制品所能接受的。在金屬板料成形加工中通常存在三種類型的起皺現(xiàn)象：法蘭起皺；側(cè)壁起皺和由于殘余壓應(yīng)力在未變形區(qū)產(chǎn)生的彈性變形。在沖壓復(fù)雜形狀的時(shí)候，拉深壁起皺就是在模具型腔中形成的褶皺。由于金屬板料在拉深壁區(qū)域內(nèi)相對(duì)無(wú)支撐，因此，消除拉深壁起皺比抑制法蘭起皺要難得多。我們知道在不被支撐的拉深壁區(qū)域中材料的外力拉深可以防止起皺，這可以在實(shí)踐中通過增加壓邊力而實(shí)現(xiàn)，但是運(yùn)用過大的拉深力會(huì)引起破裂失效。因此，壓邊力必須控制在一定的范圍內(nèi)，一方面可以抑制起皺，另一方面也可以防止破裂失效。合適的壓邊力范圍是很難確定的，因?yàn)槠鸢櫾诶盍慵闹行膮^(qū)域以一個(gè)復(fù)雜的形狀形成，甚至根本不存在一個(gè)合適的壓邊力范圍。 為了研究起皺的原因，Yoshida et al.發(fā)明了一個(gè)試驗(yàn)，即：一張薄板延著對(duì)角的一個(gè)方向進(jìn)行不均勻拉深。他們還提出了一個(gè)近似的理論模型，起皺的初始是由于彈性變形導(dǎo)致橫向壓力發(fā)展成為不均勻的壓力場(chǎng)。Yu et al.用試驗(yàn)和理論分析的方法來(lái)研究起皺問題。他們發(fā)現(xiàn)根據(jù)他們的理論分析，起皺發(fā)生在兩個(gè)環(huán)形的起伏處，而且試驗(yàn)結(jié)果指出了4—6處起皺。Narayanasamy和Sowerby通過圓錐形凸模和半球形凸模的拉深來(lái)研究金屬板料的起皺。同時(shí)，他們也試圖整理防止發(fā)生起皺的特性參數(shù)。 這些試驗(yàn)都僅僅圍繞在與簡(jiǎn)單形狀成形有關(guān)的起皺問題上，例如：一個(gè)圓形的盒件等等。在20世紀(jì)90年代初期，3D動(dòng)態(tài)有限元方法的應(yīng)用成功，使得解決金屬板料成形復(fù)雜形狀的起皺現(xiàn)象的分析變成了可能。目前，研究人員都使用3D有限元方法來(lái)分析帶有斜度的方形盒和帶有階梯的方形盒零件拉深時(shí)在拉深壁處由于金屬板料流動(dòng)引起的褶皺以及在成形過程中的參數(shù)的影響因素。 一個(gè)有斜度的方形盒，如圖1（a）所示，盒形件的每一個(gè)傾斜的拉深壁都與圓錐盒形件相似。拉深成形過程中，在拉深壁處的金屬板料是相對(duì)無(wú)支撐的，因此，褶皺是傾斜的。在目前的研究中，各種關(guān)于起皺的成型過程參數(shù)都被研究。在帶有階梯的方形盒件的研究中，如圖1（b）所示，觀察到了另一種類型的起皺。在當(dāng)前的研究中，為了得出分析的效果，實(shí)際生產(chǎn)用階梯形結(jié)構(gòu)的零件來(lái)研究。使用有限元方法可以分析出起皺的原因，并且可以使一個(gè)最優(yōu)的模具設(shè)計(jì)消除起皺現(xiàn)象。有限元分析使得模具設(shè)計(jì)在實(shí)際生產(chǎn)中更為合理化。 （a）帶有斜度的方形盒件 （b）帶有階梯的方形盒件 圖1 二、有限元模型 模具的幾何結(jié)構(gòu)（包括凸模、凹模、壓邊裝置等等），通過使用CAD和PRO/ENGINEE R來(lái)設(shè)計(jì)。使用CAD將3個(gè)節(jié)點(diǎn)或4個(gè)節(jié)點(diǎn)形成殼形的單體，進(jìn)而在模型上形成網(wǎng)格體系。使用有限元模擬，模型被視為是剛性的，并且相對(duì)應(yīng)的網(wǎng)格僅僅可以定義模型的幾何形狀，不能對(duì)壓力進(jìn)行分析。使用CAD所建立的4個(gè)節(jié)點(diǎn)的殼形單體可以為板料創(chuàng)建網(wǎng)格體系。圖2給出了模型完全建立時(shí)的網(wǎng)格體系和用以成形帶有斜度的方形盒件的金屬板料。由于對(duì)稱的原因，僅僅分析了零件的1/4。在模擬過程中，金屬板料放在壓邊裝置上，凹模向下移動(dòng)，夾緊板料。凸模向上移動(dòng)，拉深板料至模具型腔。 為了精確的完成有限元分析，金屬板料的實(shí)際壓力——拉力的關(guān)系需要輸入相關(guān)的數(shù)據(jù)。從目前的研究來(lái)看，金屬板料的深拉深的特性參數(shù)已經(jīng)用于模擬。一個(gè)拉深的實(shí)驗(yàn)已經(jīng)用于樣品的生產(chǎn)，并且沿著壓延方向和與壓延方向成45°和90°的方向切斷。平均的流動(dòng)壓力σ可以通過公式σ=（σ0+2σ45+σ90）/4，計(jì)算出來(lái)，進(jìn)而準(zhǔn)確測(cè)量出實(shí)際拉力，如圖3所示，以用于帶有斜度的方形盒件和帶有階梯的方形盒件的拉深。 目前研究中的所有模擬都在SGI Indigo2工作站使用有限元可調(diào)拉深程序完成。完成了用于模擬所需數(shù)據(jù)的輸入（假定凹模速度為10m /s，并且平均摩擦系數(shù)為0.1）。 圖2 有限元模擬的網(wǎng)格體系 實(shí)際壓力（GPa） 圖3 金屬板料的實(shí)際壓力——拉力的關(guān)系 實(shí)際拉力（mm/mm） 三、帶有斜度的方形盒件的起皺 一個(gè)帶有斜度的方形盒可以給出草圖的相關(guān)尺寸，如圖1（a）所示。從圖1（a）可以看出方形凸模頂部每邊的長(zhǎng)度為2Wp，凹模口部長(zhǎng)度為2Wd以及拉深高度H——影響起皺所考慮的關(guān)鍵性尺寸。凹模的口部尺寸與凸模頂部尺寸差值的一半為凸模的間隙，即：G=Wd－Wp。拉深壁處金屬板料相對(duì)無(wú)支撐的程度可能取決于凸模的間隙，并且增加壓邊力也有可能抑制起皺現(xiàn)象的發(fā)生。在有斜度的方形盒拉深中，與發(fā)生起皺有關(guān)系的兩個(gè)參數(shù)——凸模間隙和壓邊力，他們對(duì)起皺的影響也正在研究之中。 1.凸模間隙的影響 為了研究凸模間隙對(duì)起皺的影響，現(xiàn)在分別用凸模間隙為20mm，30mm和50mm的帶有斜度的方形盒進(jìn)行拉深模擬。在每次模擬拉深中，凹?？诓砍叽鐬?00mm固定不變，并且拉深高度均為100mm。在3次模擬中，均使用尺寸為380mm×380mm的方形板料，且板料厚度均為0.7mm，凹模對(duì)板料的壓力——拉力關(guān)系，如圖3所示。 圖4 帶有斜度的方形盒件的褶皺模擬圖（G=50mm） 模擬結(jié)果表明：三個(gè)有斜度的方形盒均發(fā)生了起皺現(xiàn)象，圖4給出了凸模間隙為50mm的方形盒的形狀。從圖4可以看出，起皺分布在拉深壁處，并且拉深壁鄰近的拐角處起皺現(xiàn)象尤為嚴(yán)重。經(jīng)分析，在拉深過程中，起皺是由于拉深壁處存在過大的無(wú)支撐區(qū)域，而且凸模頂部和凹模口部長(zhǎng)度的不同是由于凸模間隙的存在。在凸模頂部與凹模之間的金屬板料的延伸變得不穩(wěn)定，是由于斷面壓力的存在。在壓力作用下，金屬板料的無(wú)約束拉深是在拉深壁處形成褶皺的主要原因。為了比較三個(gè)不同凸模間隙的試驗(yàn)結(jié)果，需要引入兩個(gè)主應(yīng)力的比值β，β為εmin/εmax, εmin/εmax是主應(yīng)力相對(duì)的最小值和最大值。Hosford和Caddell指出，β值比臨界值更重要，如果起皺發(fā)生，那么β值越大，起皺現(xiàn)象就可能越嚴(yán)重。 如圖4和圖5的曲線所示，三次不同凸模間隙的拉深模擬，沿M——N截面的相同拉深高度處的β值。從圖5可以看出，在3次模擬中位于拉深壁的拐角處起皺比較嚴(yán)重，在拉深壁的中間起皺比較弱。還可以看出，凸模間隙越大，比值β就越大。因此，增加凸模間隙將可能增加帶有斜度的方形盒件在拉深壁處起皺的可能性。 2.壓邊力的影響 眾所周知，增加壓邊力可以幫助削弱拉深過程中發(fā)生的褶皺。為了研究增加壓邊力的影響，采用凸模間隙為50mm，不同的壓邊力數(shù)值來(lái)對(duì)有斜度的方形盒進(jìn)行拉深起皺的模擬。壓邊力從100KN增加到600KN，以提供壓邊力0.33Mpa到1.98Mpa。其他模擬條件和先前的規(guī)定保持一致（在模擬當(dāng)中采用了300KN的壓邊力）。 模擬結(jié)果表明：增加壓邊力并不能消除拉深壁處起皺現(xiàn)象的發(fā)生。如圖4所示，在M——N截面處的β值，和壓邊力分別為100KN、600KN的拉深相比較，模擬結(jié)果指出，在M——N截面處的β值都是相同的。為了分析兩次不同壓邊力時(shí)出現(xiàn)起皺的不同，從拉深壁頂部到直線M——N處，對(duì)5處不同高度截面進(jìn)行了分析，如圖4所示，圖6給出了所有情況的曲線。從圖6可以看出，幾種情況截面處的波度是相似的。這就證明壓邊力與有斜度的方形盒件拉深中的起皺現(xiàn)象無(wú)關(guān)，因?yàn)轳薨櫟男纬芍饕怯捎诶畋谔幋竺娣e無(wú)支撐區(qū)域存在較大的橫斷面壓力，所以壓邊力并不影響凸模頂部與凹模肩部之間的制件形狀的不穩(wěn)定狀況。 距離（mm） 圖5 對(duì)于不同凸模間隙在M——N截面處的β值 圖6 在不同的壓邊力狀態(tài)下，拉深壁不同高度處的橫斷面線。(a)100KN.(b)600KN. 四、帶有階梯的方形盒件 在帶有階梯的方形盒件的拉深中，即使凸模間隙不是這樣重要，而在拉深壁處仍然會(huì)發(fā)生起皺。圖1（b）所示為帶有階梯的方形盒件拉深用的凸模，圖1（b）給出了拉深壁C和階梯處D、E。目前，實(shí)際生產(chǎn)中一直在研究這種類型的幾何結(jié)構(gòu)。生產(chǎn)中，板料的厚度為0.7mm，壓力——拉力關(guān)系從應(yīng)力試驗(yàn)中獲得，如圖3所示。 這種拉深件的生產(chǎn)是通過深拉深和整形兩個(gè)工序組成的。由于凸模拐角處的小圓角半徑和復(fù)雜的幾何結(jié)構(gòu)，導(dǎo)致在盒形件的頂部邊緣發(fā)生破裂，在盒形件的拉深壁處發(fā)生褶皺，如圖7所示。從圖7中可以看出，褶皺分布在拉深壁處，尤其在階梯邊緣的拐角處更為嚴(yán)重，如圖1（b）所示的A——D和B——E處。金屬板料在凸模頂部的邊緣開裂，進(jìn)而形成破裂，如圖7所示。 圖7 產(chǎn)品上的褶皺和破裂情況 圖8 模擬產(chǎn)品起皺和破裂的盒形件外形圖 為了對(duì)拉深過程中金屬板料出現(xiàn)的變形現(xiàn)象有更進(jìn)一步的了解，生產(chǎn)中仍然采用了有限元分析方法。最初的設(shè)計(jì)已經(jīng)用有限元模擬完成。模擬的盒形件外形如圖8所示。從圖8可以看出，盒形件頂部邊緣的網(wǎng)絡(luò)拉深比較嚴(yán)重，褶皺分布在拉深壁處，這與實(shí)際生產(chǎn)中的狀況是一致的。 小的凸模圓角，例如A——B邊緣的圓角和凸模拐角A處的圓角，如圖1（b）所示，是拉深壁處破裂的主要原因。然而，根據(jù)有限元分析的結(jié)果，通過加大上述兩處圓角可以避免破裂的產(chǎn)生。較大的拐角圓角這種想法通過實(shí)際生產(chǎn)加工被驗(yàn)證是可行的。 還有一些試驗(yàn)也是模擬褶皺的。最初時(shí)將壓邊力增加到初始值的2倍。然而，正如和有斜度的方形盒件拉深時(shí)獲得的結(jié)論是一樣的，壓邊力對(duì)起皺的影響并不是最主要的。相同的結(jié)論是增大摩擦或者增加坯料的尺寸。因此我們得出的結(jié)論是：通過增加壓邊力是不能抑制起皺現(xiàn)象的發(fā)生的。 起皺的形成是由于在某些區(qū)域發(fā)生多余的金屬板料流動(dòng)，所以應(yīng)在起皺的區(qū)域增加壓桿裝置來(lái)控制多余的金屬料流。壓桿應(yīng)加到平行于起皺的方向，以便能有效的控制多余的金屬料流。在這種理論分析下，兩個(gè)壓桿應(yīng)加到拉深壁的臨近處，如圖9所示以便能控制多余的金屬料流。模擬的結(jié)果表明：正如所期望的那樣，通過壓桿的作用，階梯拐角處的起皺被控制住了，但是一些褶皺還是存在于拉深壁處。這就表明：需要在拉深壁處設(shè)置更多的壓桿，以控制多余的金屬料流。但是從結(jié)構(gòu)設(shè)計(jì)的角度考慮，這種結(jié)構(gòu)是不可行的。 圖9 在拉深壁處增加的壓桿 在拉深工序中采用有限元分析的優(yōu)點(diǎn)之一就是可以通過拉深模擬來(lái)監(jiān)視、控制金屬板料的形狀變形，而這些在實(shí)際生產(chǎn)中是不可能做到的。在拉深過程中，仔細(xì)地看金屬板料的流動(dòng)，可以看出金屬板料首先由凸模拉深進(jìn)凹模腔內(nèi)，直到金屬板料到階梯邊緣D——E處時(shí)，褶皺才開始形成。褶皺的形狀如圖10所示。有限元分析還可以為模具設(shè)計(jì)的改進(jìn)提供相關(guān)的數(shù)據(jù)信息。 圖10 金屬板料接觸階梯邊緣時(shí)形成褶皺 圖11 切斷階梯拐角后的外形圖 圖12 凸模設(shè)計(jì)修改后的外形模擬圖 ? 最初推斷發(fā)生起皺的原因是由于凸模拐角圓角A處和階梯拐角圓角D處的金屬板料不均勻、不穩(wěn)定拉深形成的。因此，模具應(yīng)設(shè)計(jì)成在階梯拐角處切斷一部分，如圖11所示，以有利于改善拉深條件。通過增加階梯邊緣而使板料均勻、穩(wěn)定的拉深。然而在拉深壁處還是存在起皺現(xiàn)象。結(jié)果指出：起皺的原因是由于凸模頂部邊緣和整個(gè)階梯邊緣的板料不均勻、不穩(wěn)定的拉深，這與凸模拐角和階梯拐角不同。毫無(wú)疑問，模具的設(shè)計(jì)結(jié)構(gòu)應(yīng)有兩處需要調(diào)整，一處是切斷整個(gè)階梯；另一處是增加拉深工序，使用2次拉深可以獲得期望的形狀。如圖12所示，是這種成形方法模擬出的外形。如果較低的臺(tái)階被切斷去除，那么這種盒形件的拉深就與矩形盒件的拉深十分相似，詳見圖12。從圖12可以看出，褶皺被去除了。 在兩次拉深過程中，金屬板料首先拉深成較深的臺(tái)階，如圖13（a）所示。因此，較低的階梯是在第二次拉深工序中形成的，此時(shí)，可以獲得我們所期望的外形，如圖13（b）所示。從圖13（b）中可以清楚地看出，帶有階梯的方形盒件通過兩次拉深被制作出來(lái)，而且沒有褶皺。在兩次拉深工序中，如果假想使用相反的順序拉深，較低的階梯首先成形，然后再拉深成較高的臺(tái)階，那么在較深臺(tái)階的邊緣處，如圖1（b）A—B處，容易形成破裂現(xiàn)象，因?yàn)榘寄Ｖ性谳^低階梯處的金屬板料很難流動(dòng)。 有限元模擬分析指出要想獲得理想的帶有階梯的方形盒件，使用一次拉深幾乎是不可能成功的。然而，使用兩次拉深則增加了生產(chǎn)成本，因?yàn)槟＞叱杀竞椭圃斐杀驹黾恿恕榱司S持較低的生產(chǎn)成本，設(shè)計(jì)師對(duì)盒形件外形做了適當(dāng)?shù)男薷模⑶腋鶕?jù)有限元模擬的結(jié)果，修改了模具，切斷去除了較低的階梯，如圖12所示。修改之后，拉深模制造出來(lái)了，并且盒形件消除了褶皺問題，如圖14 所示。盒形件的外形也與用有限元模擬所獲得的外形效果一樣好。 圖13 （a）第一次拉深工序 （b）第二次拉深工序 圖14 消除褶皺后的產(chǎn)品圖 為了更進(jìn)一步驗(yàn)證有限元模擬的結(jié)論，將從模擬的結(jié)果中獲得的截面GH處的板料厚度的分布情況與實(shí)際生產(chǎn)中的情況進(jìn)行比較。比較情況如圖15所示。從圖15的比較情況可以斷定：通過有限元模擬的厚度分布與實(shí)際生產(chǎn)的情況基本上一致。這就證明了有限元分析方法的有效性。 厚度（mm） 距離（mm） 圖15 模擬與實(shí)際生產(chǎn)中，GH截面處的板料厚度分布比較圖 五、簡(jiǎn)要論點(diǎn)及結(jié)束語(yǔ) 在拉深過程中發(fā)生的兩種類型的褶皺通過有限元分析研究以及對(duì)起皺原因做的試驗(yàn)，最終發(fā)現(xiàn)了抑制起皺的方法。 第一種類型的起皺出現(xiàn)在帶有斜度的方形盒件的拉深壁處。在凹模口部的高度尺寸和凸模頂部的高度尺寸等因素中，起皺的發(fā)生歸因于較大的凸模間隙。較大的凸模間隙會(huì)導(dǎo)致拉深到凸模頂部與凹模肩部的金屬板料處產(chǎn)生較大的無(wú)支撐區(qū)域，而金屬板料較大的無(wú)支撐區(qū)域是形成起皺的最終原因。有限元模擬表明這種類型的起皺是不能通過增加壓邊力而抑制的。 另一種類型的起皺發(fā)生在實(shí)際生產(chǎn)中帶有階梯的幾何結(jié)構(gòu)的方形盒件中。研究發(fā)現(xiàn)即使凸模間隙影響不是很重要，起皺還是會(huì)發(fā)生在階梯上面的拉深壁處。根據(jù)有限元分析，起皺的原因主要是由于凸模頂部和臺(tái)階邊緣之間的不均勻拉深造成的。為了避免起皺，在模具設(shè)計(jì)中使用有限元模擬做了一些試驗(yàn)，試驗(yàn)最終確定的最優(yōu)設(shè)計(jì)就是將階梯去除。修改后的模具設(shè)計(jì)生產(chǎn)出了無(wú)缺陷的盒形零件。模具分析的結(jié)果和實(shí)際生產(chǎn)所獲得的結(jié)論證明了有限元分析的準(zhǔn)確性和使用有限元模擬的有效性。因此可以說：有限元方法可以取代傳統(tǒng)的實(shí)際生產(chǎn)試驗(yàn)的昂貴的方法。 An Analysis of Draw-Wall Wrinkling in a Stamping Die Design F.-K. Chen and Y.-C. Liao Wrinkling that occurs in the stamping of tapered square cups and stepped rectangular cups is investigated. A common characteristic of these two types of wrinkling is that the wrinkles are found at the draw wall that is relatively unsupported.In the stamping of a tapered square cup, the effect of process parameters, such as the die gap and blank-holder force, on the occurrence of wrinkling is examined using finiteelement simulations. The simulation results show that the larger the die gap, the more severe is the wrinkling, and such wrinkling cannot be suppressed by increasing the blank-holder force. In the analysis of wrinkling that occurred in the stamping of a stepped rectangular cup, an actual production part that has a similar type of geometry was examined. The wrinkles found at the draw wall are attributed to the unbalanced stretching of the sheet metal between the punch head and the step edge. An optimum die design for the purpose of eliminating the wrinkles is determined using finite-element analysis. The good agreement between the simulation results and those observed in the wrinkle-free production part validates the accuracy of the finite-element analysis, and demonstrates the advantage of using finite-element analysis for stamping die design. Keywords: Draw-wall wrinkle; Stamping die; Stepped rectangular cup; Tapered square cups 1. Introduction Wrinkling is one of the major defects that occur in the sheet metal forming process. For both functional and visual reasons,wrinkles are usually not acceptable in a finished part. There are three types of wrinkle which frequently occur in the sheet metal forming process: flange wrinkling, wall wrinkling, and elastic buckling of the undeformed area owing to residual elastic compressive stresses. In the forming operation of stamping a complex shape, draw-wall wrinkling means the occurrence Correspondence and offprint requests to: Professor F.-K. Chen, Department of Mechanical Engineering, National Taiwan University, No. 1 Roosevelt Road, Sec. 4, Taipei, Taiwan 10617. E-mail: fkchen w3.me.ntu.edu.Tw of wrinkles in the die cavity. Since the sheet metal in the wall area is relatively unsupported by the tool, the elimination of wall wrinkles is more difficult than the suppression of flange wrinkles. It is well known that additional stretching of the material in the unsupported wall area may prevent wrinkling,and this can be achieved in practice by increasing the blankholder force; but the application of excessive tensile stresses leads to failure by tearing. Hence, the blank-holder force must lie within a narrow range, above that necessary to suppress wrinkles on the one hand, and below that which produces fracture on the other. This narrow range of blank-holder force is difficult to determine. For wrinkles occurring in the central area of a stamped part with a complex shape, a workable range of blank-holder force does not even exist. In order to examine the mechanics of the formation of wrinkles, Yoshida et al. [1] developed a test in which a thin plate was non-uniformly stretched along one of its diagonals.They also proposed an approximate theoretical model in which the onset of wrinkling is due to elastic buckling resulting from the compressive lateral stresses developed in the non-uniform stress field. Yu et al. [2,3] investigated the wrinkling problem both experimentally and analytically. They found that wrinkling could occur having two circumferential waves according to their theoretical analysis, whereas the experimental results indicated four to six wrinkles. Narayanasamy and Sowerby [4]examined the wrinkling of sheet metal when drawing it through a conical die using flat-bottomed and hemispherical-ended punches. They also attempted to rank the properties that appeared to suppress wrinkling. These efforts are focused on the wrinkling problems associated with the forming operations of simple shapes only, such as a circular cup. In the early 1990s, the successful application of the 3D dynamic/explicit finite-element method to the sheetmetal forming process made it possible to analyse the wrinkling problem involved in stamping complex shapes. In the present study, the 3D finite-element method was employed to analyse the effects of the process parameters on the metal flow causing wrinkles at the draw wall in the stamping of a tapered square cup, and of a stepped rectangular part. A tapered square cup, as shown in Fig. 1(a), has an inclined draw wall on each side of the cup, similar to that existing in a conical cup. During the stamping process, the sheet metal on the draw wall is relatively unsupported, and is therefore prone to wrinkling. In the present study, the effect of various process parameters on the wrinkling was investigated. In the case of a stepped rectangular part, as shown in Fig. 1(b),another type of wrinkling is observed. In order to estimate the effectiveness of the analysis, an actual production part with stepped geometry was examined in the present study. The cause of the wrinkling was determined using finite-element analysis, and an optimum die design was proposed to eliminate the wrinkles. The die design obtained from finite-element analysis was validated by observations on an actual production part. 2. Finite-Element Model The tooling geometry, including the punch, die and blankholder,were designed using the CAD program PRO/ENGINEER. Both the 3-node and 4-node shell elements were adopted to generate the mesh systems for the above tooling using the same CAD program. For the finite-element simulation,the tooling is considered to be rigid, and the corresponding meshes are used only to define the tooling geometry and are not for stress analysis. The same CAD program using 4-node shell elements was employed to construct the mesh system for the sheet blank. Figure 2 shows the mesh system for the complete set of tooling and the sheet-blank used in the stamping of a tapered square cup. Owing to the symmetric conditions, only a quarter of the square cup is analysed. In the simulation, the sheet blank is put on the blank-holder and the die is moved down to clamp the sheet blank against the blank-holder. The punch is then moved up to draw the sheet metal into the die cavity. In order to perform an accurate finite-element analysis, the actual stress–strain relationship of the sheet metal is required as part of the input data. In the present study, sheet metal with deep-drawing quality is used in the simulations. A tensile test has been conducted for the specimens cut along planes coinciding with the rolling direction (0°) and at angles of 45°and 90°to the rolling direction. The average flow stress,calculated from the equation
 (
0  2
45 
90)/4, for each measured true strain, as shown in Fig. 3, is used for the simulations for the stampings of the tapered square cup and also for the stepped rectangular cup. All the simulations performed in the present study were run on an SGI Indigo 2 workstation using the finite-element program PAMFSTAMP. To complete the set of input data required Fig. 3. The stress–strain relationship for the sheet metal.Draw-Wall Wrinkling in a Stamping Die Design 255 for the simulations, the punch speed is set to 10 m s1 and a coefficient of Coulomb friction equal to 0.1 is assumed. 3. Wrinkling in a Tapered Square Cup A sketch indicating some relevant dimensions of the tapered square cup is shown in Fig. 1(a). As seen in Fig. 1(a), the length of each side of the square punch head (2Wp), the die cavity opening (2Wd), and the drawing height (H) are considered as the crucial dimensions that affect the wrinkling.Half of the difference between the dimensions of the die cavity opening and the punch head is termed the die gap (G) in the present study, i.e. G  Wd  Wp. The extent of the relatively unsupported sheet metal at the draw wall is presumably due to the die gap, and the wrinkles are supposed to be suppressed by increasing the blank-holder force. The effects of both the die gap and the blank-holder force in relation to the occurrence of wrinkling in the stamping of a tapered square cup are investigated in the following sections. 3.1 Effect of Die Gap In order to examine the effect of die gap on the wrinkling, the stamping of a tapered square cup with three different die gaps of 20 mm, 30 mm, and 50 mm was simulated. In each simulation, the die cavity opening is fixed at 200 mm, and the cup is drawn to the same height of 100 mm. The sheet metal used in all three simulations is a 380 mm  380 mm square sheet with thickness of 0.7 mm, the stress–strain curve for the material is shown in Fig. 3. The simulation results show that wrinkling occurred in all three tapered square cups, and the simulated shape of the drawn cup for a die gap of 50 mm is shown in Fig. 4. It is seen in Fig. 4 that the wrinkling is distributed on the draw wall and is particularly obvious at the corner between adjacent walls. It is suggested that the wrinkling is due to the large unsupported area at the draw wall during the stamping process,also, the side length of the punch head and the die cavity Fig. 4. Wrinkling in a tapered square cup (G  50 mm).opening are different owing to the die gap. The sheet metal stretched between the punch head and the die cavity shoulder becomes unstable owing to the presence of compressive transverse stresses. The unconstrained stretching of the sheet metal under compression seems to be the main cause for the wrinkling at the draw wall. In order to compare the results for the three different die gaps, the ratio  of the two principal strains is introduced,  being min/max, where max and min are the major and the minor principal strains, respectively. Hosford and Caddell [5] have shown that if the absolute value of  is greater than a critical value, wrinkling is supposed to occur,and the larger the absolute value of , the greater is the possibility of wrinkling. The  values along the cross-section M–N at the same drawing height for the three simulated shapes with different die gaps, as marked in Fig. 4, are plotted in Fig. 5. It is noted from Fig. 5 that severe wrinkles are located close to the corner and fewer wrinkles occur in the middle of the draw wall for all three different die gaps. It is also noted that the bigger the die gap, the larger is the absolute value of . Consequently,increasing the die gap will increase the possibility of wrinkling occurring at the draw wall of the tapered square cup. 3.2 Effect of the Blank-Holder Force It is well known that increasing the blank-holder force can help to eliminate wrinkling in the stamping process. In order to study the effectiveness of increased blank-holder force, the stamping of a tapered square cup with die gap of 50 mm, which is associated with severe wrinkling as stated above, was simulated with different values of blank-holder force. The blank-holder force was increased from 100 kN to 600 kN,which yielded a blank-holder pressure of 0.33 MPa and 1.98 MPa, respectively. The remaining simulation conditions are maintained the same as those specified in the previous section.An intermediate blank-holder force of 300 kN was also usedin the simulation. The simulation results show that an increase in the blankholder force does not help to eliminate the wrinkling that occurs at the draw wall. The  values along the cross-section Fig. 5. -value along the cross-section M–N for different die gaps.256 F.-K. Chen and Y.-C. Liao M–N, as marked in Fig. 4, are compared with one another for the stamping processes with blank-holder force of 100 kN and 600 kN. The simulation results indicate that the  values along the cross-section M–N are almost identical in both cases. In order to examine the difference of the wrinkle shape for the two different blank-holder forces, five cross-sections of the draw wall at different heights from the bottom to the line M–N, as marked in Fig. 4, are plotted in Fig. 6 for both cases.It is noted from Fig. 6 that the waviness of the cross-sectionsfor both cases is similar. This indicates that the blank-holder force does not affect the occurrence of wrinkling in the stamping of a tapered square cup, because the formation of wrinkles is mainly due to the large unsupported area at the draw wall where large compressive transverse stresses exist. The blankholder force has no influence on the instability mode of the material between the punch head and the die cavity shoulder. 4. Stepped Rectangular Cup In the stamping of a stepped rectangular cup, wrinkling occurs at the draw wall even though the die gaps are not so significant.Figure 1(b) shows a sketch of a punch shape used for stamping a stepped rectangular cup in which the draw wall C is followed by a step D–E. An actual production part that has this type of geometry was examined in the present study. The material used for this production part was 0.7 mm thick, and the stress–strain relation obtained from tensile tests is shown in Fig. 3. The procedure in the press shop for the production of this stamping part consists of deep drawing followed by trimming.In the deep drawing process, no draw bead is employed on the die surface to facilitate the metal flow. However, owing to the small punch corner radius and complex geometry, a split occurred at the top edge of the punch and wrinkles were found to occur at the draw wall of the actual production part,as shown in Fig. 7. It is seen from Fig. 7 that wrinkles are distributed on the draw wall, but are more severe at the corner edges of the step, as marked by A–D and B–E in Fig. 1(b).The metal is torn apart along the whole top edge of the punch,as shown in Fig. 7, to form a split. In order to provide a further understanding of the deformation of the sheet-blank during the stamping process, a finiteelement analysis was conducted. The finite-element simulation was first performed for the original design. The simulated shape of the part is shown from Fig. 8. It is noted from Fig.8 that the mesh at the top edge of the part is stretched Fig. 6. Cross-section lines at different heights of the draw wall for different blank-holder forces. (a) 100 kN. (b) 600 kN.Fig. 7. Split and wrinkles in the production part.Fig. 8. Simulated shape for the production part with split and wrinkles.significantly, and that wrinkles are distributed at the draw wall,similar to those observed in the actual part. The small punch radius, such as the radius along the edge A–B, and the radius of the punch corner A, as marked in Fig.1(b), are considered to be the major reasons for the wall breakage. However, according to the results of the finiteelement analysis, splitting can be avoided by increasing the above-mentioned radii. This concept was validated by the actual production part manufactured with larger corner radii. Several attempts were also made to eliminate the wrinkling.First, the blank-holder force was increased to twice the original value. However, just as for the results obtained in the previous section for the drawing of tapered square cup, the effect of blank-holder force on the elimination of wrinkling was not found to be significant. The same results are also obtained by increasing the friction or increasing the blank size. We conclude that this kind of wrinkling cannot be suppressed by increasing the stretching force. Since wrinkles are formed because of excessive metal flow in certain regions, where the sheet is subjected to large compressive stresses, a straightforward method of eliminating the wrinkles is to add drawbars in the wrinkled area to absorb the redundant material. The drawbars should be added parallel to the direction of the wrinkles so that the redundant metal can be absorbed effectively. Based on this concept, two drawbars are added to the adjacent walls, as shown in Fig. 9, to absorb the excessive material. The simulation results show that the Draw-Wall Wrinkling in a Stamping Die Design 257 Fig. 9. Drawbars added to the draw walls.wrinkles at the corner of the step are absorbed by the drawbars as expected, however some wrinkles still appear at the remaining wall. This indicates the need to put more drawbars at the draw wall to absorb all the excess material. This is, however,not permissible from considerations of the part design. One of the advantages of using finite-element analysis for the stamping process is that the deformed shape of the sheet blank can be monitored throughout the stamping process, which is not possible in the actual production process. A close look at the metal flow during the stamping process reveals that the sheet blank is first drawn into the die cavity by the punch head and the wrinkles are not formed until the sheet blank touches the step edge D–E marked in Fig. 1(b). The wrinkled shape is shown in Fig. 10. This provides valuable information for a possible modification of die design. An initial surmise for the cause of the occurrence of wrinkling is the uneven stretch of the sheet metal between the punch corner radius A and the step corner radius D, as indicated in Fig. 1(b). Therefore a modification of die design was carried out in which the step corner was cut off, as shown in Fig.11, so that the stretch condition is changed favourably, which allows more stretch to be applied by increasing the step edges. However, wrinkles were still found at the draw wall of the cup. This result implies that wrinkles are introduced because of the uneven stretch between the whole punch head edge and the whole step edge, not merely between the punch corner and Fig. 10. Wrinkle formed when the sheet blank touches the stepped edge.Fig. 11. Cut-off of the stepped corner.the step corner. In order to verify this idea, two modifications of the die design were suggested: one is to cut the whole step off, and the other is to add one more drawing operation, that is, to draw the desired shape using two drawing operations.The simulated shape for the former method is shown in Fig. 12. Since the lower step is cut off, the drawing process is quite similar to that of a rectangular cup drawing, as shown in Fig. 12. It is seen in Fig. 12 that the wrinkles were eliminated. In the two-operation drawing process, the sheet blank was first drawn to the deeper step, as shown in Fig. 13(a). Subsequently,the lower step was formed in the second drawing operation, and the desired shape was then obtained, as shown in Fig. 13(b). It is seen clearly in Fig. 13(b) that the stepped rectangular cup can be manufactured without wrinkling, by a two-operation drawing process. It should also be noted that in the two-operation drawing process, if an opposite sequence is applied, that is, the lower step is formed first and is followed by the drawing of the deeper step, the edge of the deeper step,as shown by A–B in Fig. 1(b), is prone to tearing because the metal cannot easily flow over the lower step into the die cavity. The finite-element simulations have indicated that the die design for stamping the desired stepped rectangular cup using one single draw operation is barely achieved. However, the manufacturing cost is expected to be much higher for the twooperation drawing process owing to the additional die cost and operation cost. In order to maintain a lower manufacturing cost, the part design engineer made suitable shape changes,and modified the die design according to the finite-element Fig. 12. Simulated shape for the modified die design.Fig. 13. (a) First operation and (b) second operation in the two-operation drawing process.simulation result to cut off the lower step, as shown in Fig.12. With the modified die design, the actual stamping die for production was manufactured and the production part was found to be free from wrinkles, as shown in Fig. 14. The part shape also agre