滅火器端蓋塑料注射模設(shè)計(jì)【高度100】
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立體光照成型的注塑模具工藝的綜合模擬
摘要 功能性零部件都需要設(shè)計(jì)驗(yàn)證測(cè)試，車間試驗(yàn)，客戶評(píng)價(jià)，以及生產(chǎn)計(jì)劃。在小批量生產(chǎn)零件的時(shí)候，通過消除多重步驟，建立了有快速成型形成的注塑模具，這種方法可以保證縮短時(shí)間和節(jié)約成本。這種潛在的一體化由快速成型形成注塑模具的方法已經(jīng)被多次證明是可行的。無論是模具設(shè)計(jì)還是注塑成型的過程中，缺少的是對(duì)如何修改這個(gè)模具材料和快速成型制造過程的影響有最根本的認(rèn)識(shí)。此外，數(shù)字模擬技術(shù)現(xiàn)在已經(jīng)成為模具設(shè)計(jì)工程師和工藝工程師開注塑模具的有用的工具。但目前所有的做常規(guī)注塑模具的模擬包已經(jīng)不再適合這種新型的注塑模具，這主要是因?yàn)槟＞卟牧系某杀咀兓艽?。在本文中，以完成特定的?shù)字模擬注塑液塑造成快速成型模具的綜合方法已經(jīng)發(fā)明出來了，而且還建立了相應(yīng)的模擬系統(tǒng)。通過實(shí)驗(yàn)結(jié)果表明，目前這個(gè)方法非常適合處理快速成型模具中的問題。
關(guān)鍵詞 注塑成型,數(shù)字模擬,快速成型
１引言
在注塑成型中，聚合物熔體在高溫和高壓下進(jìn)入模具中。因此，模具的材料需要有足夠的熱性能和機(jī)械性能來經(jīng)受高溫和高壓的塑造循環(huán)。許多研究的焦點(diǎn)都是直接有快速成型形成注塑模具的過程。在生產(chǎn)小批量零件的時(shí)候，通過消除多重步驟，直接由快速成型形成的注塑模具可以保證縮短時(shí)間和節(jié)約成本。這種潛在的有快速成型形成注塑模具的方法已經(jīng)被證明成功了?？焖俪尚湍＞咴谛阅苌鲜怯袆e與傳統(tǒng)的金屬模具。主要差異是導(dǎo)熱性能和彈性模量（剛性）。舉例來說，在立體光照成型模具中的聚合物的導(dǎo)熱率小于鋁制的工具的千分之一。在用快速成型技術(shù)來制造鑄模時(shí)，整個(gè)模具設(shè)計(jì)和注塑成型工藝參數(shù)都需要修改和優(yōu)化，傳統(tǒng)的方法是改變徹底的刀具材料．不過，目前還沒有對(duì)如何修改這個(gè)模具材料的方法有根本的了解．在當(dāng)前的模具中，僅僅改變一些材料的性能是不能得到一個(gè)合理的結(jié)果的。同樣，使用傳統(tǒng)方法的時(shí)候，實(shí)際生產(chǎn)的零件也會(huì)有出先次品。因此，研究出一個(gè)快速成型過程，材料和注塑模具之間的互動(dòng)關(guān)系是非?；鸺钡?。這樣就可以確定模具設(shè)計(jì)標(biāo)準(zhǔn)和快速模具的注塑的技術(shù)。
此外，計(jì)算機(jī)模擬是一種預(yù)測(cè)模塑件的質(zhì)量的有效的方法。目前，商用仿真軟件包已經(jīng)成為模具設(shè)計(jì)師和工藝工程師在注塑過程中例行性的工具。不幸的是，目前常規(guī)注塑成型的模擬程序已經(jīng)不再適用于這個(gè)快速成型模具，因?yàn)樗鼧O大的需要不同的刀具材料。例如，利用現(xiàn)在的仿真軟件在鋁和立體光照模具之間做個(gè)實(shí)驗(yàn)比較一下，雖然鋁模具模擬植的部分失真是合理的，但是結(jié)果是不可以接受的，因?yàn)檎`差超過了百分之五十。在注塑成型中，失真主要是由于塑料零件的收縮和翹曲，模具也是一樣的。對(duì)于通常模具，失真的主要因素是塑料件的收縮和翹曲，這個(gè)在目前的模擬中能測(cè)試準(zhǔn)確。但是對(duì)于快速成型模具，潛在的失真會(huì)更多，在當(dāng)前的測(cè)試中，其中就會(huì)有些失真會(huì)被忽視。例如，用一個(gè)簡單的三步驟模擬分析模具變形的時(shí)候，就會(huì)出現(xiàn)很多偏差。
在本文中，基于以上分析，一個(gè)新的快速成型模具的仿真系統(tǒng)已經(jīng)開發(fā)出來了。擬議制度著重于預(yù)測(cè)部分失真，主要是用與預(yù)測(cè)快速成型模具的缺陷。先進(jìn)的仿真系統(tǒng)可以用于預(yù)測(cè)快速成型模具設(shè)計(jì)和工藝是否最合理。我們的仿真系統(tǒng)已經(jīng)被我們的實(shí)驗(yàn)證明是沒有錯(cuò)誤的。
雖然有很多材料可以用于快速成型技術(shù)，但是我們還是專注于利用立體光照模具的技術(shù)來制造聚合物模具．立體光照成型的過程是利用激光能量一層一層建立零件的部分。使用立體光照則可以體現(xiàn)出雙方在快速成型工業(yè)的商業(yè)優(yōu)勢(shì)，而且在以后也可以生產(chǎn)出準(zhǔn)確的，高品質(zhì)的零部件。直到最近，立體光照主要是用于建立物理模型，為了檢查視覺效果，僅僅只利用了它的一點(diǎn)點(diǎn)功能。不過，新一代的立體光照的光改善了立體化，機(jī)械性能，熱學(xué)性能，所以它可以更好的應(yīng)用于實(shí)際的模具中。
2 綜合仿真的成型過程
2.1 方法
為了在注塑成型過程中模擬立體光照模具的功能，反復(fù)的試驗(yàn)中得到了一個(gè)方法。不同的軟件組已經(jīng)開發(fā)出來了，而且也已經(jīng)做到了這一點(diǎn)。主要的假設(shè)是，溫度和負(fù)載邊界條件造成立體光照模具的扭曲，仿真步驟如下：
１部分幾何模型則作為一個(gè)實(shí)體模型，這將通過流量分析軟件包被翻譯到一個(gè)文件中。
２模擬光聚合物模具中熔融體填充的過程，然后輸出溫度和壓力的資料。
３在前一步獲得了熱負(fù)荷和邊界條件，然后對(duì)光模具進(jìn)行結(jié)構(gòu)分析，其中失真的計(jì)算是在該注塑過程中進(jìn)行的。
?。慈绻＞叩呐で諗苛?，那么直接進(jìn)行下一步．否則，扭曲的型腔（改動(dòng)扭曲后的型腔的尺寸）返回第二個(gè)步驟，以熔體形式模擬注入扭曲的模具中。
?。等缓笞⑸涑尚土慵氖湛s和翹曲模擬就開始應(yīng)用了，算出該成型零件最終的扭曲部分．
上述的模擬流動(dòng)中，基本上是三個(gè)仿真模塊。
2.2充型模擬的熔體
2.2.1數(shù)字建模
計(jì)算機(jī)仿真技術(shù)已經(jīng)能成功的預(yù)測(cè)到在極其復(fù)雜的幾何形狀下的填充情況。然而，目前大多數(shù)字模擬是基于一種混合有限元和有限差的中性平面上的。模擬軟件包的應(yīng)用過程基于這一模型說明圖１。然而，不同與ＣＡＤ系統(tǒng)中模具設(shè)計(jì)中的表面／實(shí)體模型，這里所謂的中性平面（如圖所示，圖１Ｂ）是一個(gè)假想的在中間型腔中有距離和方向的一個(gè)平面，這個(gè)平面可能會(huì)在應(yīng)用的過程中帶來很大的不便。舉例來說，模具表面常用于目前的快速成型系統(tǒng)中（通常是ＳＴＬ格式），所以當(dāng)用模擬軟件包的時(shí)候，第二次建模是不可避免的。那是因?yàn)槟Ｐ驮诳焖俪尚拖到y(tǒng)和仿真系統(tǒng)中是不一樣的?？紤]到這些缺點(diǎn)，在模擬系統(tǒng)中，型腔的表面將以基準(zhǔn)面來引入，而不是中性平面。
根據(jù)以往的調(diào)查，流量和溫度場(chǎng)的方程式可以寫為：
X,Y是中性平面坐標(biāo)系中的兩個(gè)平面，Ｚ是高度坐標(biāo)，Ｕ，Ｖ，Ｗ是Ｘ，Ｙ，Ｚ方向上的速度．Ｕ，Ｖ是整體的平均厚度，η, ρ,CP (T), K(T)分別表示聚合物的粘性，密度，周期熱，熱導(dǎo)率。
圖１ Ａ－Ｄ是中性平面的模擬程序．Ａ是３維表面模型，Ｂ是中性平面模型，Ｃ是網(wǎng)狀的平面模型，Ｄ是最后的模擬結(jié)果
此外，在高度方向上的邊界條件的誤差可以表示為：
正如圖２中的Ａ中表示，TW 是恒壁溫度.結(jié)合方程１－４和方程５－６，表明了u, v, T, P在Ｚ坐標(biāo)上面應(yīng)該是對(duì)稱的，因此在上半個(gè)高度中的平均u, v應(yīng)該和整個(gè)高度中的平均u, v是一樣的。根據(jù)這個(gè)特點(diǎn)，我們可以把整個(gè)型腔在上下高度上分為兩個(gè)部分，正如圖２Ｂ中的第一部分和第二部分。同時(shí)，型腔（如圖２Ｂ）表面產(chǎn)生的三角有限元將替代了中性平面（如圖２Ａ）。因此，在高度方向上的有限元誤差僅僅限于型腔表面，正如圖２Ｂ所示，高度上的誤差將從０到Ｂ。這是中性平面上的單一性。此外，從圖２Ａ到圖２Ｂ，坐標(biāo)也隨之改變了。為了配合上述調(diào)整，方程仍是用方程１－４。然而，原來的邊界條件高度方向則改寫為：
與此同時(shí)，為了保持在同一坐標(biāo)（７）上的兩部分能夠流動(dòng)，那么更多的邊界條件必須滿足Ｚ＝Ｂ。
下標(biāo)I和II則分別代表第一部分和第二部分的參數(shù)．Cm-I 和Cm-II 則表示在填充階段中分開的兩個(gè)表面上的自由移動(dòng)的熔融線。
應(yīng)該指出的是，方程９與１０和方程７與８不同，９和１０在數(shù)字模擬過程中將變的更難，主要原因是以下幾點(diǎn)：
１同一個(gè)斷層的表面都已經(jīng)都已經(jīng)有著特殊的網(wǎng)格，這將導(dǎo)致同一層上的獨(dú)特的格局．因此，在比較兩個(gè)熔接口的時(shí)候，應(yīng)該計(jì)算出各自的u, v, T, P。
?。惨?yàn)閮蓚€(gè)部分都有各自的流道通向節(jié)點(diǎn)Ａ和節(jié)點(diǎn)Ｃ（如圖２Ｂ所示）．在同一段中，有可能兩個(gè)都充滿，也有可能一個(gè)滿，一個(gè)空．這兩個(gè)情況應(yīng)該分開處理，應(yīng)該平均流動(dòng)，使后者也分配到流動(dòng)。
?。尺@意味著在前線熔合處出現(xiàn)一點(diǎn)點(diǎn)小的誤差是可以允許的．通過控制時(shí)間和選擇更好的位置來控制前線熔合節(jié)點(diǎn)。
４每個(gè)流場(chǎng)的邊界都擴(kuò)張到熔線前線，所以核查方程１０是否準(zhǔn)確是相當(dāng)重要的。
?。佃b于上述分析，在同一個(gè)節(jié)點(diǎn)處的物理參數(shù)應(yīng)該加以比較和調(diào)整。所以在進(jìn)行模擬之前，描述同一節(jié)點(diǎn)有限元的信息應(yīng)該準(zhǔn)備好，也就是說，匹配的原理應(yīng)該先預(yù)備好。
圖２ Ａ－Ｂ表明表面模型中的中性平面Ｂ的高度方向Ａ上的邊界條件
2.2.2數(shù)字模擬
壓力場(chǎng)．在建模中，粘度 η是由于熔提的剪切速率，溫度和壓力引起的性能．剪切變稀后，這就代表一個(gè)跨越式的模式，例如：
其中對(duì)應(yīng)于冪律指數(shù)，τ的特點(diǎn)是在在牛頓和冪律漸近極限之間的剪應(yīng)力過渡區(qū)。無論在溫度還是壓力指數(shù)上，η0(T, P)都可以有合理的表示，詳情如下：
方程11和12構(gòu)成了一個(gè)五個(gè)常數(shù)，可以代表粘度，而且通過粘度的剪切速率的計(jì)算可以得到：
根據(jù)上述情況，通過方程1—4，我們可以推斷出一下充氣壓力方程：
其中S是由計(jì)算出來的。運(yùn)用伽遼金方法，對(duì)壓力的有限元方程推導(dǎo)為：
其中l(wèi)是所有要素的的導(dǎo)線，包括節(jié)點(diǎn)N，而且其中i和j代表此處的N節(jié)點(diǎn)的數(shù)目，的計(jì)算方法如下：
其中代表三角有限元，而代表有限元中的壓力。
溫度場(chǎng)中，為了確定高度方向上的誤差，應(yīng)該在模具表面上分為一層一層的三角有限元的網(wǎng)格。左邊的能量方程4可以表示為：
其中代表每一層N節(jié)點(diǎn)上的溫度。熱傳導(dǎo)的計(jì)算方法是：
其中l(wèi)是所有要素，包括節(jié)點(diǎn)N，而且i和j分別代表此處的N節(jié)點(diǎn)個(gè)數(shù)。
對(duì)流項(xiàng)的計(jì)算方法是：
當(dāng)是粘性熱時(shí)，計(jì)算方法是：
把方程17—20帶入方程4，溫度方程變?yōu)椋?
2.3 模具結(jié)構(gòu)分析
結(jié)構(gòu)分析的目的是預(yù)測(cè)在填充過程中，模具由于熱和機(jī)械壓力而產(chǎn)生的變形。這個(gè)模型是基于一個(gè)三維熱邊界元法。邊界元法是比較適合這個(gè)應(yīng)用的，因?yàn)橹挥凶冃蔚哪＞弑砻娌庞羞@樣的信息。此外，邊界元法有一個(gè)優(yōu)點(diǎn)，那就是在計(jì)算變形的模具的時(shí)候，它的計(jì)算是不會(huì)白費(fèi)的。
模具在所受載荷超過彈性范圍的時(shí)候會(huì)產(chǎn)生應(yīng)力。因此，在決定模具變形的時(shí)候，模具材料是一個(gè)基準(zhǔn)。模具的熱性能和力學(xué)性能是各向同性的，而且溫度也是獨(dú)立的。
盡管這個(gè)過程是循環(huán)的，但是相同時(shí)間的溫度和熱流都是可以用于計(jì)算模具變形的．通常情況下，在模具里面每個(gè)瞬間溫度都局限于型腔的表面和噴嘴的頂端。在觀察距離的時(shí)候，瞬間的衰減變化是很微笑的，小于２．５毫米．這說明在模具的噴嘴處的變形是很小的，因此，忽略這個(gè)影響也是合理的．穩(wěn)態(tài)溫度場(chǎng)滿足拉普拉斯方程?2T = 0的邊界條件。至于機(jī)械邊界條件，型腔表面受到熔體的壓力，模具的表面會(huì)連接到工作臺(tái)上的，而其他的外部表面將會(huì)假設(shè)是自由的.熱邊界的推導(dǎo)方程１０是大家都知道的，這是由于：
其中uk, pk和Ｔ分別是位移，牽引力和溫度。α, ν是代表材料的膨脹系數(shù)和泊松比。Ulk是在ＸＹ方向上基本的位移。在一個(gè)三維空間中，各向同性彈性區(qū)域中，由一個(gè)單元產(chǎn)生的負(fù)荷主要集中在xl方向上，它是以下面的形式產(chǎn)生的：
其中δlk是Kronecker三角函數(shù)，μ是該模具材料的剪切模量。Plk的基本收縮都是在模具表面的每個(gè)Ｎ節(jié)點(diǎn)處測(cè)量的，可以表示為：
整個(gè)Ｎ將分散在模具的表面上，轉(zhuǎn)變?yōu)榉匠蹋玻玻?
其中Γn是指在這個(gè)區(qū)域上的表面成分。
把恰當(dāng)?shù)木€性函數(shù)代入方程２５，得到的線性邊界方程就是模具的方程．這個(gè)方程適用于每個(gè)離散的模具表面，從而組合成線性方程組，其中Ｎ是節(jié)點(diǎn)的總數(shù)。每個(gè)節(jié)點(diǎn)有八個(gè)相關(guān)數(shù)量，三個(gè)位移組成部分，三個(gè)牽引組成部分，還有溫度和熱流量。在穩(wěn)態(tài)熱模型中，每個(gè)節(jié)點(diǎn)處的溫度和磁場(chǎng)是已知的，余下的６個(gè)量中，三個(gè)必須是已知的。此外，在若干個(gè)節(jié)點(diǎn)處的位移值的方程必須消除剛體運(yùn)動(dòng)和剛體自轉(zhuǎn)的奇異系統(tǒng)。由此產(chǎn)生的系統(tǒng)方程式是一個(gè)集合起來的綜合矩陣，它可以為有限元方法求解。
基于方程１２的注塑假設(shè)，下面將給出元件的應(yīng)力和應(yīng)變：
該偏元件的應(yīng)力和應(yīng)變分別是：
用類似的方法可以預(yù)測(cè)在回火玻璃中的殘余應(yīng)力了。以積分的形式在平面上分析粘性和彈性結(jié)構(gòu)關(guān)系時(shí)，可以表示為以下公式：
其中G1是材料的的剪切模量。擴(kuò)張的應(yīng)變的情況如下：
其中Ｋ是材料體積的彈性模量，α和θ的定義是：
如果α(t) = α0，那么方程２７到方程２９的結(jié)果則為：
同樣的，利用方程３１到方程２８消除應(yīng)變?chǔ)舩x(z, t)，得到：
利用拉普拉斯變化方程３２，輔助系數(shù)R(ξ)由下面的方程得出：
利用上述方程３３，并簡化在模具中的應(yīng)力和應(yīng)變的形式，那么注塑中殘余的應(yīng)力在冷卻階段中，由下面的方程獲得：
方程３４可以通過梯形正交被解決。由于材料的時(shí)間在快速的變化，所以需要一個(gè)準(zhǔn)數(shù)控程序來檢測(cè)。輔助模量是檢測(cè)數(shù)控梯形的規(guī)則。
關(guān)于翹曲分析，節(jié)點(diǎn)位移和曲率將以殼單元表達(dá)為：
其中[ k ]單元?jiǎng)偠染仃嚕琜Be]是衍生算子矩陣，jxnrzjj是位移，{re}是 負(fù)載單元，可以由下面的方程得出：
使用完整的三維有限元分析法的好處就是可以準(zhǔn)確知道翹曲的結(jié)果。但是，當(dāng)零件的形狀很復(fù)雜的時(shí)候，它也是相當(dāng)麻煩的。在本文中，在殼體理論基礎(chǔ)上介紹了一種二維有限元分析方法。這種方法被大量使用是因?yàn)榇蠖鄶?shù)注塑模具的零件都有一些部分幾何的厚度遠(yuǎn)遠(yuǎn)小于其他部分。因此，那些部分則可以被作為一個(gè)集會(huì)的單元來預(yù)測(cè)翹曲。每三個(gè)節(jié)點(diǎn)殼單元組合成一個(gè)恒應(yīng)變?nèi)菃卧鸵粋€(gè)離散克?；舴蛉窃?，如圖３所示，因此翹曲可以分為平面伸展變形ＣＳＴ和板彎曲變形ＤＫＴ。并相應(yīng)的以單元?jiǎng)偠染仃噥砻枋雎N曲的拉伸剛度矩陣和彎曲剛度矩陣。
圖３ a-c是殼單元在局部坐標(biāo)系統(tǒng)里的變形分解．a(chǎn)是平面伸展元素，b是平面彎曲元素，c是殼單元
三 實(shí)驗(yàn)驗(yàn)證
對(duì)提出的模型進(jìn)行了評(píng)定和發(fā)展，最后核查是非常重要的。從模型模擬中得到的扭曲數(shù)據(jù)將和文獻(xiàn)８中的立體光照模具數(shù)據(jù)比較。如圖４所示，有一個(gè)注塑尺寸36 × 36 × 6毫米和實(shí)驗(yàn)數(shù)據(jù)中是相同的。薄壁和加強(qiáng)筋的厚度都是1.5毫米，這個(gè)注塑材料是聚丙烯。注塑機(jī)的型號(hào)是ARGURY Hydronica320-210-750，它的工藝參數(shù)是，熔解溫度是２５０度，模具溫度是３０度，注塑壓力是１３.７９帕，保壓時(shí)間是３秒，冷卻時(shí)間是４８秒。立體光照模具材料使用杜邦SOMOSTM６１１０樹脂，能抵御高達(dá)３００度的高溫。如上所述，熱傳導(dǎo)是區(qū)分立體光照模具和傳統(tǒng)模具的一個(gè)重要因素。模具中的熱量轉(zhuǎn)移會(huì)產(chǎn)生溫度的不均勻分布，所以導(dǎo)致了成型零件的翹曲．立體光照成型模具的周期是可以預(yù)測(cè)的。以高的熱傳導(dǎo)率金屬為背面做的薄殼立體光照模具將會(huì)增加自身的熱傳導(dǎo)率。
圖４ 模型腔
圖５ 不同的熱傳導(dǎo)率下，在Ｘ方向上的扭曲失真比較．實(shí)驗(yàn)值，三步走和常規(guī)都是指最后的實(shí)驗(yàn)結(jié)果．常規(guī)是指實(shí)驗(yàn)中最好的結(jié)果．三步走步驟的模擬過程分別與傳統(tǒng)的注塑成型相似
圖６ 在不同的熱傳導(dǎo)率下，在Ｙ方向上的扭曲失真比較
圖７ 在不同熱傳導(dǎo)率下，在Ｚ方向上扭曲失真比較
圖８ 不同熱傳導(dǎo)率下各個(gè)捻度變量的比較
對(duì)于這個(gè)部分，扭曲包括三個(gè)方向上的位移和捻度（兩個(gè)最初的平行邊的夾角的誤差）．如圖５到圖８，實(shí)驗(yàn)結(jié)果表明，這些數(shù)值也包括通過傳統(tǒng)注塑模具模擬系統(tǒng)預(yù)測(cè)的扭曲值和報(bào)道［３］中的三步驟。
４結(jié)論
本文介紹了一個(gè)綜合模擬的快速成型模具的方法，并且建立了相應(yīng)的仿真系統(tǒng)。為了驗(yàn)證這個(gè)系統(tǒng)，實(shí)驗(yàn)還進(jìn)行了快速焊接立體光照成型模具。
很明顯，立體光照模具也會(huì)出現(xiàn)傳統(tǒng)的注塑模具模擬軟件一樣的故障．假設(shè)由于注射中的溫度和負(fù)載荷引起了扭曲．那么用三步驟完成的話，結(jié)果也會(huì)出現(xiàn)比較多的誤差。不過更先進(jìn)的模型會(huì)使結(jié)果更接近與實(shí)驗(yàn)。
立體光照模具改進(jìn)了熱傳導(dǎo)率極大的增加了零件質(zhì)量．由于溫度比壓力（負(fù)載）對(duì)模具的影響更大，所以改進(jìn)立體光照模具的熱傳導(dǎo)率可以更顯著的提高零件質(zhì)量。
無論零件多么復(fù)雜，快速成型技術(shù)可以使人們?cè)煨透?，更便捷，更便宜．在快速成型穩(wěn)步發(fā)展的基礎(chǔ)上，快速制造也將隨之而來，并且需要更多的精確工具來確定工藝過程的參數(shù)．現(xiàn)行的模擬工具不能滿足研究者研究模具相對(duì)的變化。正如本文中所述，對(duì)于一個(gè)綜合模型來說，要預(yù)測(cè)最后零件質(zhì)量是相當(dāng)重要的。在不久的將來，我們期待看到通過快速成型擴(kuò)展到快速模具制造的模擬程序。
參考文獻(xiàn)
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IntegratedIntegratedIntegratedIntegrated simulationsimulationsimulationsimulation ofofofof thethethethe injectioninjectioninjectioninjection moldingmoldingmoldingmolding processprocessprocessprocesswithwithwithwith stereolithographystereolithographystereolithographystereolithography moldsmoldsmoldsmoldsAbstractAbstractAbstractAbstractFunctional parts are needed for design verification testing,field trials,customer evaluation, and production plan ning. By eliminating multiple steps, thecreationofthe injec tion mold directly by a rapid prototyping (RP) process holds thebest promise of reducing the time and cost needed to mold low-volume quantities ofparts. The potential of this integra tion of injection molding with RP has beendemonstrated many times. Whatismissingisthe fundamental understanding of howthe modifications to the mold material and RP manufacturing process impact both themold design and the injection mold ing process. In addition, numerical simulationtechniques have now become helpful tools of mold designers and process engi neersfor traditional injection molding. Butallcurrent simulation packages for conventionalinjection molding are no longer ap plicable to this new typeofinjection molds,mainly because the propertyofthe mold material changes greatly.Inthis paper, anintegrated approach to accomplish a numerical simulation of in jection molding intorapid-prototyped moldsisestablished and a corresponding simulation systemisdeveloped. Comparisonswithexperimental results are employed for verification,which show that the present schemeiswellsuited to handle RP fabri catedstereolithography (SL) molds.KeywordsKeywordsKeywordsKeywordsInjection moldingNumerical simulationRapid prototyping1 1 1 1 IntroductionIntroductionIntroductionIntroductionIn injection molding, the polymer melt at high temperatureisinjected into themold under high pressure 1. Thus, the mold material needs to have thermal andmechanical properties capa bleofwithstanding the temperatures and pressures ofthe mold ing cycle. The focus of many studies has been to create theinjection mold directly by a rapid prototyping (RP) process. By eliminatingmultiple steps, this method of tooling holds the best promise of reducing the time andcost needed to create low-volume quantities of parts in a production material. ThepotentialofintegratinginjectionmoldingwithRPtechnologieshasbeendemonstrated many times. The properties of RP molds are very different from thoseof traditional metal molds. The key differ ences are the properties of thermalconductivity and elastic mod ulus (rigidity). For example, the polymers used inRP-fabricated stereolithography (SL) molds have a thermal conductivity thatislessthan one thousandth that of an aluminum tool. In using RP technologies to createmolds, the entire mold design and injection-molding process parameters need to bemodified and optimized from traditional methodologies due to the completelydifferent tool material. However, thereisstillnota fundamen tal understanding ofhow the modifications to the mold tooling method and material impact both the molddesign and the injec tion molding process parameters. One cannot obtain reasonableresultsbysimply changing a few material properties in current models. Also, usingtraditional approaches when making actual parts may be generating sub-optimalresults. So thereisa dire need to study the interaction between the rapid tooling (RT)pro cess and material and injection molding, so as to establish the mold designcriteria and techniques for an RT-oriented injection molding process.In addition, computer simulationisaneffective approach for predicting thequality of moldedparts. Commerciallyavailablesimulation packages of thetraditional injection molding process have now become routine toolsofthe molddesigner and pro cess engineer 2. Unfortunately, current simulation programs forconventional injection molding arenolonger applicable to RP molds, because of thedramatically dissimilar tool material. For instance, in using the existing simulationsoftware with alu minum and SL molds and comparing with experimental results,though the simulation values of part distortion are reasonable for the aluminum mold,results are unacceptable, with the error exceeding 50%. The distortion duringinjection moldingisdue to shrinkage and warpage of the plastic part, aswellas themold. For ordinarily molds, the main factoristhe shrinkage and warpage of theplastic part, whichismodeled accurately in cur rent simulations. But for RP molds,the distortion of the mold has potentially more influence, which have been neglectedin current models. For instance, 3 used a simple three-step simulation process toconsider the mold distortion, which had too much deviation.In this paper, based on the above analysis, a new simula tion system for RPmoldsisdeveloped. The proposed system focuses on predicting part distortion, whichisdominating defect in RP-molded parts. The developed simulationcanbe applied asan evaluation tool for RP mold design and process opti mization. Our simulationsystemisverifiedbyan experimental example.Although many materials are available for use in RP tech nologies, weconcentrateonusing stereolithography (SL), the original RP technology, to createpolymer molds. The SL pro cess uses photopolymer and laser energy to build a partlayerbylayer. Using SL takes advantage of both the commercial domi nanceofSLin the RP industry and the subsequent expertise base that has been developed forcreating accurate, high-quality parts.Untilrecently, SL was primarily used to createphysical models for visual inspection and form-fitstudieswithvery limitedfunc tional applications. However,thenewer generationstereolitho graphicphotopolymers have improved dimensional,mechanical and thermal propertiesmakingitpossible to use them for actual functional molds.2 2 2 2 IntegratedIntegratedIntegratedIntegrated simulationsimulationsimulationsimulation ofofofof thethethethe moldingmoldingmoldingmolding processprocessprocessprocess2.1 MethodologyIn order to simulate the use of an SL mold in the injection molding process, aniterative methodisproposed. Different soft ware modules have been developed andused to accomplish this task. The main assumptionisthat temperature and loadbound ary conditions cause significant distortions in the SL mold. The simulationsteps are as follows:1The part geometryismodeled as a solid model, whichistranslated to afilereadable by theflow analysis package.2Simulate the mold-fillingprocess of the melt into a pho topolymer mold,whichwilloutput the resulting temperature and pressure profiles.3Structural analysisisthen performed on the photopolymer mold modelusing the thermal and load boundary conditions obtained from the previous step,which calculates the distor tion that the mold undergo during the injection process.4Ifthe distortion of the mold converges, move to the next step. Otherwise,the distorted mold cavityisthen modeled (changes in the dimensions of the cavityafter distortion), and returns to the second step to simulate the melt injection into thedistorted mold.5The shrinkage and warpage simulation of the injection molded partisthenapplied, which calculates thefinaldistor tions of the molded part.In above simulationflow, there are three basic simulation mod ules.2.2Filling simulationof themelt2.2.1 Mathematical modelingIn order to simulate the use of an SL mold in the injection molding process, aniterative methodisproposed. Different software modules have been developed andused to accomplish this task. The main assumptionisthat temperature and loadboundary conditions cause significant distortionsinthe SL mold. The simulation stepsare as follows:1. The part geometryismodeled as a solid model, whichistranslated to a filereadable by the flow analysis package.2. Simulate the mold-filling process of the melt into a photopolymer mold, whichwilloutput the resulting temperature and pressure profiles.3. Structural analysisisthen performedonthe photopolymer mold model usingthe thermal and load boundary conditions obtained from the previous step, whichcalculates the distortion that the mold undergo during the injection process.4.Ifthe distortion of the mold converges, move to the next step. Otherwise, thedistorted mold cavityisthen modeled (changesinthe dimensions of the cavity afterdistortion), and returns to the second step to simulate the melt injection into thedistorted mold.5. The shrinkage and warpage simulationofthe injection molded partisthenapplied, which calculates the final distortionsofthe molded part.In above simulation flow, there are three basic simulation modules.2.2 Filling simulation ofthe melt2.2.1 Mathematical modelingComputer simulation techniques have had success in predictingfillingbehaviorin extremely complicated geometries. However, most of the current numericalimplementationisbasedona hybrid finite-element/finite-difference solution with themiddleplane model. The application processofsimulation packages basedonthismodelisillustrated in Fig. 2-1. However, unlike the surface/solidmodel inmold-design CAD systems, the so-called middle-plane (as shown in Fig. 2-1b)isanimaginary arbitrary planar geometry at the middle of the cavity in the gap-wisedirection, which should bring about great inconvenience in applications. For example,surface models are commonly used in current RP systems (generally STL file format),so secondary modelingisunavoidable when using simulation packages because themodels in the RP and simulation systems are different. Considering these defects, thesurface model of the cavityisintroduced as datum planes in the simulation, instead ofthe middle-plane.According to the previous investigations 46, fillinggoverning equations for theflow and temperature field can be written as:wherex, yare the planar coordinates in the middle-plane, andzisthe gap-wisecoordinate;u, v,ware the velocity componentsinthex, y, zdirections;u, vare theaverage whole-gap thicknesses; and, ,CP(T), K(T)represent viscosity, density,specific heat and thermal conductivity of polymer melt, respectively.Fig.2-1Fig.2-1Fig.2-1Fig.2-1 a a a a d. d. d. d. Schematic procedure of thesimulation with middle-plane model. a a a aThe3-D surfacemodelb b b bThemiddle-plane model c c c c Themeshed middle-plane modeld d d dThedisplay of thesimulation resultIn addition, boundary conditions in the gap-wise direction can be defined as:whereTWisthe constantwalltemperature (shown in Fig. 2a).Combining Eqs. 14 with Eqs. 56,itfollows that the distributions of theu, v, T,Patzcoordinates should be symmetrical, with the mirror axis beingz= 0, andconsequently theu, vaveraged in half-gap thicknessisequal to that averaged inwholegap thickness. Basedonthis characteristic, we can divide the whole cavity intotwo equal parts in the gap-wise direction, as described byPartIandPartIIin Fig. 2b.At the same time, triangular finite elements are generatedinthe surface(s) of thecavity(atz= 0 in Fig. 2b), insteadofthe middle-plane(atz= 0 in Fig. 2a).Accordingly, finite-difference increments in the gapwise direction are employed onlyin the inside of the surface(s)(wallto middle/center-line), which, in Fig. 2b, meansfromz= 0 toz=b. Thisissingle-sided instead of two-sided with respect to themiddle-plane (i.e. from the middle-line to two walls).Inaddition, the coordinatesystemischanged from Fig. 2a toFig.2b to alter the finite-element/finite-differencescheme, as shown in Fig. 2b. With the above adjustment, governing equations are stillEqs. 14. However, the original boundary conditionsinthe gapwise direction arerewritten as:Meanwhile, additional boundary conditions must be employed atz=bin orderto keep the flows at the juncture of the two parts at the same section coordinate 7:where subscripts I,IIrepresent the parametersofPartIandPartII, respectively,and Cm-I and Cm-II indicate the moving free melt-fronts of the surfaces of thedivided two parts in the filling stage.Itshould be noted that, unlike conditions Eqs. 7 and 8, ensuring conditions Eqs.9 and 10 are upheld in numerical implementations becomes more difficult due to thefollowing reasons:1. The surfaces at the same section have been meshed respectively, which leadsto a distinctive pattern of finite elements at the same section. Thus, an interpolationoperation should be employed foru, v, T, Pduring the comparison between the twoparts at the juncture.2. Because the two parts have respective flow fields with respect to the nodes atpoint A and point C (as shown in Fig. 2b) at the same section,itispossible to haveeither both filled or one filled (and one empty). These two cases should be handledseparately, averaging the operation for the former, whereas assigning operation for thelatter.3.Itfollows that a small difference between the melt-frontsispermissible. Thatallowance can be implementedbytime allowance control or preferable locationallowance control of the melt-front nodes.4. The boundaries of the flow field expandbyeach melt-front advancement, soitisnecessary to check the condition Eq. 10 after each change in the melt-front.5. In view of above-mentioned analysis, the physical parameters at the nodes ofthe same section should be compared and adjusted, so the information describingfinite elements of the same section should be prepared before simulation, that is, thematching operation among the elements should be preformed.Fig.Fig.Fig.Fig. 2a,b.2a,b.2a,b.2a,b. Illustrative of boundary conditionsinthe gap-wise direction a a a aof themiddle-planemodelb b b bof thesurfacemodel2.2.2 Numerical implementationPressure field.In modeling viscosity, whichisa functionofshear rate,temperature and pressureofmelt, the shear-thinning behavior can bewellrepresentedby a cross-type model such as:wherencorresponds to the power-law index, and*characterizes the shearstress level of the transition region between the Newtonian and power-law asymptoticlimits. In terms ofanArrhenius-type temperature sensitivity and exponential pressure dependence,0(T, P)can be represented with reasonable accuracy as follows:Equations 11 and 12 constitute a five-constant(n,* ,B,Tb,)representationfor viscosity. The shear rate for viscosity calculationisobtainedby:Based on the above, we can infer the following filling pressure equation from thegoverning Eqs. 14:whereSiscalculatedbyS=b0/(bz)2dz. Applying the Galerkin method, thepressure finite-element equationisdeduced as:wherel_ traversesallelements, including nodeN, and whereIandjrepresent thelocal node number in elementl_ corresponding to the node number N andN_ in thewhole, respectively. TheD(l_)ijiscalculated as follows:whereA(l_)represents triangular finite elements, andL(l_)iisthe pressure trialfunction in finite elements.Temperature field.To determine the temperature profile across the gap, eachtriangular finite element at the surfaceisfurther divided intoNZlayers for thefinite-difference grid.The leftitemofthe energy equation (Eq. 4)canbe expressed as:whereTN, j,trepresents the temperature of thejlayerofnodeNat timet. Theheat conductionitemiscalculatedby:whereltraversesallelements, including nodeN, andiandjrepresent the localnode number in elementlcorresponding to the node numberNandN_ in the whole,respectively.The heat convectionitemiscalculatedby:For viscous heat,itfollowsthat:Substituting Eqs. 1720 into the energy equation (Eq. 4), the temperatureequation becomes:2.3 Structural analysis ofthemoldThe purpose of structural analysisisto predict the deformation occurring in thephotopolymer mold due to the thermal and mechanical loads of the filling process.This modelisbased on a three-dimensional thermoelastic boundary element method(BEM). The BEMisideally suited for this application becauseonlythe deformationof the mold surfacesisof interest. Moreover, the BEMhasan advantage over othertechniques in that computing effortisnot wasted on calculating deformation withinthe mold.The stresses resulting from the process loads arewellwithin the elastic rangeofthe mold material. Therefore, the mold deformation modelisbasedona thermoelasticformulation. The thermal and mechanical properties of the mold are assumed to beisotropic and temperature independent.Although the processiscyclic, time-averaged values of temperature and heatflux are used for calculating the mold deformation. Typically, transient temperaturevariations within a mold have been restricted to regions local to the cavity surface andthe nozzletip8. The transients decay sharply with distance from the cavity surfaceand generally little variationisobserved beyond distances as small as 2.5 mm. Thissuggests that the contribution from the transients to the deformation at the mold blockinterfaceissmall, and thereforeitisreasonable to neglect the transient effects. Thesteadystatetemperaturefieldsatisfies Laplaces equation2T=0 andthetime-averaged boundary conditions. The boundary conditions on the mold surfacesare describedindetail by Tang et al. 9. As for the mechanical boundary conditions,the cavity surfaceissubjected to the melt pressure, the surfaces of the mold connectedto the worktable are fixed in space, and other external surfaces are assumed to bestress free.The derivation of the thermoelastic boundary integral formulationiswellknown10.Itisgivenby:whereuk,pkandTare the displacement, traction and temperature,representthe thermal expansion coefficient and Poissons ratio of the material, andr=|yx|.clk(x)isthe surface coefficient which dependsonthe local geometry atx, theorientation of the coordinate frame and Poissons ratio for the domain 11. Thefundamental displacementulkat a pointyin thexkdirection, in a three-dimensionalinfinite isotropic elastic domain, results from a unit load concentrated at a pointxacting in thexldirection andisof the form:wherelkisthe Kronecker delta function andisthe shear modulus of the moldmaterial.The fundamental tractionplk, measured at the pointyon a surface with unitnormaln n n n,is:Discretizing the surface of the mold into atotalofNelements transforms Eq. 22to:wherenrefers to thenthsurface elementonthe domain.Substituting the appropriate linear shape functions into Eq. 25, the linearboundary element formulation for the mold deformation modelisobtained. Theequationisapplied at each node on the discretized mold surface, thus giving a systemof 3Nlinear equations, whereNisthetotalnumber of nodes. Each node has eightassociated quantities: three components of displacement, three components of traction,a temperature and a heat flux. The steady state thermal model supplies temperatureand flux values as known quantities for each node, and of the remaining six quantities,three must be specified. Moreover, the displacement values specified at a certainnumber of nodes must eliminate the possibility of a rigid-body motion or rigid-bodyrotation to ensure a non-singular system of equations. The resulting system ofequationsisassembled into a integrated matrix, whichissolved withaniterativesolver.2.4 Shrinkage and warpage simulation ofthemoldedpartInternal stresses in injection-molded components are the principal cause ofshrinkage and warpage. These residual stresses are mainly frozen-in thermal stressesdue to inhomogeneous cooling, when surface layers stiffen sooner than the coreregion, as in free quenching. Based on the assumption of the linear thermo-elastic andlinearthermo-viscoelasticcompressiblebehaviorofthepolymericmaterials,shrinkage and warpage are obtained implicitly using displacement formulations, andthe governing equationscanbe solved numerically using a finite element method.With the basic assumptionsofinjection molding 12, the components of stressand strain are givenby:The deviatoric components of stress and strain, respectively, are given byUsing a similar approach developedbyLee and Rogers 13 for predicting theresidual stresses in the tempering of glass, an integral form of the viscoelasticconstitutive relationshipsisused, and the in-plane stresses can be related to the strainsby the following equation:WhereG1isthe r
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