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正文：外文資料翻譯譯文
多尺度模擬復(fù)合材料和結(jié)構(gòu)與DIGIMAT ANSYS
文件版本1.0,2009年2月,e-Xstream工程,2009年版權(quán)
info@e-Xstream.com www.e-Xstream.com
材料：工程塑料、增強(qiáng)塑料．e-Xstream
技術(shù)：DIGIMAT,Digimat-MF Digimat-FE,DIGIMAT、ANSYS,分析數(shù)據(jù)映射圖。Moldflow、Moldex3D CAE技術(shù)。
分析軟件：ANSYS。
行業(yè)：材料供應(yīng)商、汽車、航空、消費(fèi)者和工業(yè)產(chǎn)品。
法律通告：eXdigimat和e-Xstream工程是e-Xstream工程的注冊商標(biāo)。其他產(chǎn)品及公司名稱和商標(biāo)的商標(biāo)權(quán)或注冊商標(biāo)權(quán)歸他們的各自的主人所有。
概要
在這篇文章中, 簡要的介紹兩個(gè)尺度的建模方法，平均場均化處理和有限元同化方法，在進(jìn)行建模時(shí)，這些強(qiáng)大的技術(shù)用與微觀和宏觀的應(yīng)力和應(yīng)變場,可以通過影響(改變)材料內(nèi)部微觀組織來控制材料在宏觀上表現(xiàn)出來的性能(例如：纖維取向、纖維含量、纖維長度，等等) 。說明這些技巧, 我們目前的狀況是：（一） 應(yīng)用有限元分析均到納米二氧化鈦；(二)研究了注入玻璃纖維增強(qiáng)塑料霓虹燈扣使用有限元計(jì)算的宏觀尺度結(jié)合中值場均在微觀的尺度上。
多尺度建模：簡介
作為一種激勵人心的例子, 讓我們來看一個(gè)由短玻璃纖維加固的熱塑性聚合物塑料部件。作為典型的注塑生產(chǎn)過程，這種分布于成品內(nèi)部的纖維將毫無疑問的會在走向和長度上發(fā)生普遍的改變?？磮D（1），該復(fù)合材料同時(shí)呈現(xiàn)各向異性與非均質(zhì)性，這使它極難得到一個(gè)可靠準(zhǔn)確的產(chǎn)品模擬，因?yàn)樗媒?jīng)典的方法是基于宏觀的本構(gòu)模型。然而，通過多尺度的方法使預(yù)測模擬成為可能,這種預(yù)測模擬可以把這種復(fù)合材料用相當(dāng)簡單的方式進(jìn)行描述，如圖：
圖（1）：在注射玻璃纖維增強(qiáng)塑料后的離合器踏板中的纖維取向分布圖（有羅地亞公司和特瑞堡集團(tuán)提供）
此圖讓我們研究學(xué)習(xí)了異構(gòu)實(shí)體的顯微組織組成的矩陣資料并且這些所謂的“夾雜物”可以是短纖維、小片晶體、顆粒、微小孔或微裂紋。我們的目標(biāo)是根據(jù)它的顯微結(jié)構(gòu)，模擬預(yù)測產(chǎn)品在施加載荷和增加邊界條件（BCs）下所產(chǎn)生的變化和影響。我們能區(qū)分出兩種尺度,分別是微觀層次和宏觀層次。這個(gè)模型在微觀結(jié)構(gòu)尺度上與異質(zhì)性質(zhì)相符,然而從宏觀尺度上看，可以認(rèn)為是局部均勻的。如圖：
圖（2）：在實(shí)踐中,解決力學(xué)問題時(shí)的計(jì)算不可能停留在微尺度層面上。因此,我們考慮的是宏觀尺度，并且假設(shè)每個(gè)質(zhì)點(diǎn)是大量代表性的等效體積單元(RVE)的中心,這些質(zhì)點(diǎn)包含潛在的異質(zhì)性的微觀結(jié)構(gòu)。經(jīng)典的固體力學(xué)是進(jìn)行宏觀尺度分析的,只可惜在計(jì)算每個(gè)點(diǎn)后，應(yīng)力、應(yīng)變值像邊界條件傳送到潛伏的等效體積單元一樣被傳送了。換句話說:每個(gè)數(shù)值的縮放就被認(rèn)為是一個(gè)宏觀點(diǎn)。這樣等效體積單元的問題都解決了濱且每個(gè)單元都返回應(yīng)力和剛度的測試值,這個(gè)方法被用于宏觀尺度的計(jì)算中。
圖2:多尺度的材料建模的插圖，現(xiàn)在唯一的困難在于這種用二尺度的方法（和更多一般的多尺度的方法）來解決等效體積單元的問題。它可以被等價(jià)為一個(gè)在經(jīng)典邊界條件作用下的等效體積單元,此時(shí)宏觀上的應(yīng)變與應(yīng)力等于所有等效體積單元內(nèi)部未知的區(qū)域內(nèi)微應(yīng)變和應(yīng)力的體積平均值。在線彈性的條件下,運(yùn)用復(fù)合材料的宏觀尺度時(shí)，涉及到了那兩個(gè)能給出有效剛度或總體剛度的均值。為了解決這個(gè)問題,你可以使用等效體積單元有名的有限元方法算法,見圖7到10。該方法的優(yōu)點(diǎn)是既簡單又非常準(zhǔn)確。然而,它有兩個(gè)主要的缺點(diǎn)是:在計(jì)算實(shí)際的微結(jié)構(gòu)時(shí)網(wǎng)格化分非常困難和在處理非線性問題時(shí)占用大量的CPU運(yùn)算時(shí)間,比如在模擬計(jì)算非彈性材料性能的時(shí)候。另一個(gè)完全不同的方法是平均場均質(zhì)法,這種方法是基于應(yīng)力體積平均值和一個(gè)等效體積單元的每個(gè)相的應(yīng)變場之間的假設(shè)關(guān)系而形成的方法;見圖3。與絕對的有限元方法和其他所有現(xiàn)存的數(shù)值轉(zhuǎn)換方法相比,平均場均質(zhì)法（MFH）不僅是最好用的而且在占用CPU時(shí)間方面明顯是最快的。然而，平均場均質(zhì)法也有兩個(gè)缺點(diǎn),一是它無法給出每個(gè)相中的詳細(xì)應(yīng)變和應(yīng)力場數(shù)值,二是局限于夾雜物的橢球面形狀。
圖3:平均場均質(zhì)法的過程。(1)局部應(yīng)變根據(jù)宏觀應(yīng)變計(jì)算；(2)局部應(yīng)力根據(jù)局部應(yīng)變和每個(gè)相的組織模型來計(jì)算；(3)宏觀應(yīng)力根據(jù)平均局部應(yīng)力計(jì)算。
一個(gè)典型的等效體積單元的例子是Mori-Tanaka模型，已經(jīng)成功適用于具有相同和對齊尺寸的橢圓形夾雜物的兩相復(fù)合材料中。該模型假定了，如果等效體積單元是單獨(dú)存在于一個(gè)無限的由實(shí)際的基體材料組成的空間中時(shí)，每個(gè)夾雜物都包含了等效體積單元。邊界條件在解決單一的夾雜問題時(shí)相當(dāng)于實(shí)際的等效體積單元的基體相應(yīng)變場體積平均值的計(jì)算方法。 單夾雜物分析問題已經(jīng)被J.D. Eshelby在一篇標(biāo)志性論文中解決了，這是平均場均質(zhì)模型劃時(shí)代的基石。
圖四：原理的Mori-Tanaka同質(zhì)化的程序
Mori-Tanaka模型和其它平均場均質(zhì)模型已經(jīng)推廣到許多案例中了,如熱耦合、兩相非直線纖維的復(fù)合材料（使用多步驟分步處理的途徑）或多相復(fù)合材料（使用一個(gè)多層次的方法）。這個(gè)預(yù)測已經(jīng)直接廣泛地驗(yàn)證了均場均質(zhì)模型的有限元模擬和實(shí)驗(yàn)結(jié)果的校驗(yàn)。作為一種普遍的結(jié)論,人們發(fā)現(xiàn)在線(熱)彈性條件下,平均場均質(zhì)可給出有效特性的精確預(yù)測值,盡管是分布式取向,然而在終止近似值法上取得的進(jìn)步仍然是受歡迎的。另外,確定平均場均質(zhì)模型可用于UD,并可用在復(fù)合材料每個(gè)微結(jié)構(gòu)中像用在機(jī)織織物的每根紗線中一樣。一個(gè)重要并且仍持續(xù)在理論模型和計(jì)算方法上努力的在材料或幾何非線性領(lǐng)域推廣。這種擴(kuò)展包括一些主要的困難：第一個(gè)是線性化,在微觀尺度上本構(gòu)方程需要線性化,需要微線彈性-或像熱彈性一樣的格式。第二個(gè)問題是對所謂的對比資料，即定義每個(gè)相中具有均勻瞬時(shí)剛度的控制運(yùn)算符。接下來需要解決的問題是一階和二階同化，在一階均以真正的本構(gòu)模型計(jì)算作為比較材料,但不是每個(gè)相的應(yīng)變和應(yīng)力場的體積平均值。在一個(gè)二階配方,充足的統(tǒng)計(jì)信息,即每個(gè)相的應(yīng)變和應(yīng)力場的方差也考要慮進(jìn)去。最后,非常難的技術(shù)難點(diǎn)涉及Eshelby和希爾的與各向異性的瞬時(shí)剛度相關(guān)算子比較的張量計(jì)算方法。在多尺度分析耦合有限元方法用于宏觀尺度,同時(shí),確定各高斯點(diǎn)進(jìn)行了計(jì)算,無論是在線性或非線性的狀態(tài)。這是實(shí)踐中最可行的方法。見圖5。
圖5：經(jīng)典的鐵和耦合的有限元/ Digimat-MF方法對比。
廣泛的驗(yàn)證和驗(yàn)證結(jié)果表明,平均場均質(zhì)模型確定可用于實(shí)踐中存在的非線性問題,并且一般情況下可以帶來良好的非線性預(yù)測值,然而在某些情況下工作可以持續(xù)提高精度(和減少CPU時(shí)間與多尺度分析相結(jié)合）。
有限元均值處理法：一種納米復(fù)合材料的應(yīng)用方法
未來材料最有可能的是納米材料,它廣泛的為未來各種領(lǐng)域提供新的劃時(shí)代的應(yīng)用,例如如納米電子學(xué),納米生物科技和納米醫(yī)學(xué)等領(lǐng)域。這樣,越來越多的精力放在理解和模擬他們的性狀上以及得知什么是納米效應(yīng)。而目前正在開發(fā)的新工具,來解決這個(gè)工程上的挑戰(zhàn),今天有些新工具已經(jīng)提供給工程師使用。其中:有限元素均值法應(yīng)用的最多。
模型聚合物類填料,一種典型的納米效果填料。材料科學(xué)家在納米尺度上，面臨一些有關(guān)設(shè)計(jì)和加工的納米復(fù)合尺度的挑戰(zhàn),這些新的物理現(xiàn)象,從宏觀尺度上看是可以忽略不計(jì)的。舉例來說，納米填料均勻的分散在復(fù)合基體中，被認(rèn)為可以改善材料的機(jī)械性能,然而期望具有導(dǎo)電率的聚和物類和滲透類導(dǎo)熱或?qū)щ娦远夹枰黾拥幕A(chǔ)材料。
參見圖6，實(shí)現(xiàn)完成一個(gè)或其他如今是在材料加工和對其研究方面構(gòu)成的挑戰(zhàn)。
圖六：納米填料的擴(kuò)散
有限元均值法：它需要進(jìn)行幾何研究并被明確的產(chǎn)生并且是網(wǎng)狀的,可以準(zhǔn)確模擬滲流和集群效應(yīng)。如圖所示,介紹了宏觀材料質(zhì)點(diǎn)的彈性力學(xué)性能目前在塑料聚合物上的影響。
圖7給出了兩種周期性納米結(jié)構(gòu),也稱為等效體積元素(RVE),這已經(jīng)在使用Digimat-FE方法。介紹了聚合類材料參數(shù)已經(jīng)產(chǎn)生最終的幾何坐標(biāo)，聚合材料內(nèi)含物集中的聚類附近的兩個(gè)截然不同的地方。體積分?jǐn)?shù)5%的相位和夾雜物是球形的。一旦包圍,這些結(jié)構(gòu)將在等效體積單元中只受單向拉伸條件,運(yùn)用x，y，z軸的負(fù)向和有限元方法。利用ANSYS有限元求解器進(jìn)行求解之后問題就解決了。
圖7:微結(jié)構(gòu)與均勻分布夾雜物(左)與群集(右)。
結(jié)果比較：
圖8: 應(yīng)力分布在夾雜物(左)和矩陣(右)為隨機(jī)放置的雜質(zhì)
圖8到10個(gè)說明應(yīng)力分布矩陣和夾雜物的階段,在這個(gè)案例中介紹的是x軸單向拉伸試驗(yàn)測試。由于最近的聚類中心附近的包裹,應(yīng)力集中現(xiàn)象出現(xiàn)。這樣,可以提高了30%的拉應(yīng)力進(jìn)行了觀察,對聚類情況x軸向單向拉伸加載條件下進(jìn)行觀察,見圖10。
圖11是在等效體積單元中S33的應(yīng)力與應(yīng)變分布和E33的應(yīng)變分布和基質(zhì)材料的相。觀察夾雜的相時(shí)候應(yīng)用了一個(gè)明顯的更高的應(yīng)力水平。這種更高的應(yīng)力集中,不會隨機(jī)或均勻分布內(nèi)含的夾雜物,而且在施加載荷的時(shí)候可能會導(dǎo)致脫粘。
圖9:S11應(yīng)力分布在夾雜物(左)和矩陣(右)為聚物。
圖10：2D等效體積單元的部分觀點(diǎn)的群集(左)和隨機(jī)(右)。拉應(yīng)力分布。
圖11:S33(左)和E33應(yīng)力應(yīng)變(右)分布在等效體積單元的納米階段,對這兩種情況下,一個(gè)沿z軸方向的加載。
在這個(gè)低體積分?jǐn)?shù)的內(nèi)含物中,我們看到這類不明顯改變其宏觀力學(xué)性能的資料,請參閱表1。處在這樣一個(gè)位置,最好的方法是避免納米夾雜物材料的出現(xiàn)，當(dāng)試圖增加基材的剛度(基質(zhì)剛度 = 2195 MPa)結(jié)合納米填料(填料剛度 = 7000 MPa)。
有限元法和平均場均值處理法耦合計(jì)算：已經(jīng)應(yīng)用到一部分的工業(yè)中。
出于許多原因(制造成本、適應(yīng)性、加工方法、高強(qiáng)度對抗.亮度比等。) 注射部分由短的玻璃纖維增強(qiáng)塑料在我們的日常生活中已經(jīng)無處不在。但當(dāng)它用這樣的材料做成的模型,能夠模擬宏觀模型構(gòu)成的物質(zhì)模型受到捕捉效應(yīng)的影響，例如注射過程？答案是否定的,因?yàn)樗麄儾]有從中捕獲的由注塑工藝決定的對纖維的分布方向產(chǎn)生的影響。下面的例子中,它由一個(gè)霓虹燈扣受載荷的過程,介紹了耦合分析和有限元軟件ANSYS,DIGIMAT-MF Moldex3D之間的區(qū)別。這個(gè)過程在圖12中表現(xiàn)的很清楚,并且包括下列步驟：
1、注射成型工藝過程使用Moldex3D進(jìn)行了數(shù)值模擬?，F(xiàn)有結(jié)果是纖維取向張量,將作為輸入DIGIMAT結(jié)構(gòu)仿真。
2、張量計(jì)算的定位可用映射圖在準(zhǔn)備從注射網(wǎng)格映射到粗糙的結(jié)構(gòu)性沖突的映射工具(在DIGIMAT中)。
3、這個(gè)結(jié)構(gòu)仿真是利用ANSYS有限元求解加上Digimat-MF、多尺度的材料建模,模具制作的每個(gè)整合平均場均值方法進(jìn)行結(jié)構(gòu)模型。
圖12:耦合分析的過程。
DIGIMAT以獲得Moldex3D纖維取向張量作為輸入,除了材料性能之外還有作為塑料模型采用了ANSYS有限元模擬。
問題的說明：
這個(gè)輕環(huán)由四個(gè)獨(dú)立的部分組成,見圖13,也顯示了不同的零件之間的接觸結(jié)果。他們兩個(gè)都由30%的玻璃纖維增強(qiáng)尼龍和Bergamid（一種新型復(fù)合材料）注入。他們的注射過程都在Moldex3D中進(jìn)行了注塑模擬?；瑝K和支座組建假設(shè)是由鋼鐵制作的。
關(guān)閉的卡環(huán)是模擬位移滑而擋住了支持和部分的內(nèi)部。對稱性邊界條件來限制也被應(yīng)用到一半的部分研究。這個(gè)目標(biāo)是為了評價(jià)模擬零件表面的平均應(yīng)力最大值,在負(fù)載期間,比較了利用線性彈性響應(yīng)模型,利用材料的彈性模量進(jìn)行確定DIGIMAT-MF尼龍與玻璃纖維和彈塑性模型進(jìn)行平均場均值法計(jì)算的結(jié)果。
材料建模
為了模仿在DIGIMAT-MF中的PAGF模型,做了以下假設(shè):
1、 玻璃纖維仍舊在線性彈性的領(lǐng)域。
2、 聚酰胺（尼龍）具有可塑性和線彈性。
3、 纖維的縱橫（長度/直徑）比值為30。
參見圖14：拉伸反應(yīng)的材料模型。
圖13：表述的是霓虹燈扣和四個(gè)獨(dú)立部件之間觸體的關(guān)系。由Trilux和CADFEM股份有限公司提供。
圖14：Bergamid尼龍材料的模型。各向同性案例的拉伸響應(yīng),固定的纖維取向(1D)、隨機(jī)二維定位(2D)和隨機(jī)三維定位(3D)。由Trilux和CADFEM股份有限公司提供。
仿真結(jié)果
而有限元均值法具有明顯的在等效體積單元中準(zhǔn)確的應(yīng)變/應(yīng)力場優(yōu)勢，平均場均值法只能得到微觀層面的平均的應(yīng)力與應(yīng)變值。盡管如此,它給我們的信息如果我們用宏觀的本構(gòu)模型我們將不能使用這個(gè)方法。同樣地,在這個(gè)基質(zhì)的相中平均累積塑性應(yīng)變的可以直接的在塑料部件中去觀察可塑性的分布。最大的塑性變形都可以從外圍表面部分觀察到。如圖15。
圖15:平均分布于塑性應(yīng)變積累在基體相的內(nèi)部和外部的部分。范圍0.01%(藍(lán)色)0.09%(紅色)。由Trilux和CADFEM股份有限公司提供。
圖16：線性彈性各向同性反應(yīng)(經(jīng)典的有限元法)與各向異性非線性(有限元法和平均場均值法)的對比。提高到21%的不同是觀察應(yīng)力大小,用硬線彈性模型得到更高的應(yīng)力。
圖16:S11壓力分布(MPa)的各向同性彈性鉤(左)和各向異性非線性模型(右)。由Trilux和CADFEM股份有限公司提供。
參考書目：
1. Nemat-Nasser.S，Hori，M. Micromechanics：異構(gòu)體的整體性能．艾斯維爾科學(xué)出版社，1993。
2. 莫里，田中．具有彈性能量雜質(zhì)的材料的基質(zhì)的平均應(yīng)力．金屬學(xué)報(bào)，1973年,571-574,第21卷。
3. 彈性模量的確定的領(lǐng)域中的相關(guān)問題．Eschelby博士，1226年，倫敦:倫敦皇家學(xué)會．1957年,第241卷，376-396。
4. 概述聚合物基納米復(fù)合材料的工程應(yīng)用前景．Chmutin．第一卷。
5. 有納米壓痕的納米硅基的納米復(fù)合材料聚合物概要．郭等人．強(qiáng)化塑料和復(fù)合材料雜志，2004年。
附件：外文原文
Multi-Scale Modeling of Composite Materials and Structures with DIGIMAT to ANSYS
Document Version 1.0, February 2009 Copyright, e-Xstream engineering, 2009
info@e-Xstream.com www.e-Xstream.com
Materials: Engineering Plastics, Reinforced Plastics. e-Xstream Technology: DIGIMAT, Digimat-MF, Digimat-FE, Digimat to ANSYS, MAP. Complementary
CAE Technology: Moldflow, Moldex3D, SigmaSoft, ANSYS.
Industry: Material Suppliers, Automotive, Aerospace, Consumer & Industrial Products.
Legal Notice. eX, eXdigimat and e-Xstream engineering are registered trademarks of e-Xstream engineering SA. The other product and company names and logos are trademarks or registered trademarks of their respective owners.
EXECUTIVE SUMMARY
In this paper, we briefly introduce two multi-scale modeling approaches, namely the Mean-Field (MFH) and Finite Element Homogenization (FEH) methods. These powerful techniques relate the microscopic and macroscopic stress and strain fields when modeling material behaviors and hence can capture the influence of the material microstructure (i.e. fiber orientation, fiber content, fiber length, etc.) on its macroscopic response. To illustrate these techniques, we also present (i) an application of finite element homogenization to a nanostructure and (ii) the study of an injected glass fiber reinforced plastic neon light clasp using finite element computations at the macro scale coupled with MF homogenization at the micro scale.
Material Multi-Scale Modeling: an introduction
As a motivating example, let us consider a plastic part made up of a thermoplastic polymer reinforced with short glass fibers. As typical of the injection molding manufacturing process, the fiber distribution inside the final product will vary widely in terms of orientation and length, see Figure 1. The composite material will be both anisotropic and heterogeneous, which makes it extremely difficult to perform a reliable simulation of the product using a classical approach based on macroscopic constitutive models. However, a predictive simulation is possible via a multi-scale approach, which can be described in a rather general setting as follows.
Figure 1: Fiber orientation distribution in an injected glass fiber-reinforced plastic clutch pedal. Courtesy of Rhodia & Trelleborg.
Let us study a heterogeneous solid body whose microstructure consists of a matrix material and multiple phases of so-called “inclusions”, which can be short fibers, platelets, particles, micro-cavities or micro-cracks. Our objective is to predict the response of the body under given loads and boundary conditions (BCs), based on its microstructure. We can distinguish two scales, the microscopic and macroscopic levels, respectively. The former corresponds to the scale of the heterogeneities, while at the macro scale, the solid can be seen as locally homogeneous; Figure 2. In practice, it would be computationally impossible to solve the mechanical problem at the fine micro scale.
Therefore, we consider the macro scale and assume that each material point is the center of a representative volume element (RVE), which contains the underlying heterogeneous microstructure. Classical solid mechanics analysis is carried out at the macro scale, except that at each computation point, strain or stress values are transmitted as BCs to the underlying RVE. In other words, a numerical zoom is realized at each macro point. The RVE problems are solved and each of them returns stress and stiffness values, which are used at the macro scale.
Figure 2: Illustration of the multi-scale material modeling approach, after Nemat-Nasser and Hori (1). Now the only difficulty in this two-scales (and more generally multi-scale) approach is to solve the RVE problems. It can be shown that for a RVE under classical BCs, the macro strains and stresses are equal to the volume averages over the RVE of the unknown micro strain and stress fields inside the RVE. In linear elasticity, relating those two mean values gives the effective or overall stiffness of the composite at the macro scale. In order to solve the RVE problem, one can use the well-known finite element (FE) method, see Figures 7 to 10. This method offers the advantages of being very general and extremely accurate. However, it has two major drawbacks which are: serious meshing difficulties for realistic microstructures and a large CPU time for nonlinear problems, such as for inelastic material behaviour. Another completely different method is mean-field homogenization (MFH), which is based on assumed relations between volume averages of stress or strain fields in each phase of a RVE; see Figure 3. Compared to the direct FE method, and actually to all other existing scale transition methods, MFH is both the easiest to use and the fastest in terms of CPU time. However, two shortcomings of MFH are that it is unable to give detailed strain and stress fields in each phase and it is restricted to ellipsoidal inclusion shapes.
Figure 3: Mean-field homogenization process: (i) local strains are computed based on the macro strains, (ii) local stresses are computed based on the local strains and according to each phase constitutive model, and (iii) macro stresses are computed by averaging the local stresses.
A typical example of MFH is the Mori-Tanaka model (2) which is successfully applicable to two-phase composites with identical and aligned ellipsoidal inclusions. The model assumes that each inclusion of the RVE behaves as if it were alone in an infinite body made of the real matrix material. The BCs in the single inclusion problem correspond to the volume average of the strain field in the matrix phase of the real RVE. The single inclusion problem was solved analytically by J.D. Eshelby (3) in a landmark paper, which is the cornerstone of MFH models.
Figure 4: Schematic of the Mori-Tanaka homogenization procedure.
Mori-Tanaka and other MFH models were generalized to other cases, such as thermoelastic coupling, two-phase composites with misaligned fibers (using a multi-step approach) or multi-phase composites (using a multi-level method). The predictions have been extensively verified against direct FE simulation of RVEs or validated against experimental results. As a general conclusion, it was found that in linear (thermo)elasticity, MFH can give extremely accurate predictions of effective properties, although for distributed orientations, progress in closure approximation will be welcomed. Note also that MFH can be used for UD, and for each yarn in woven composites. An important and still ongoing effort both in theoretical modeling and in computational methods is the generalization of MFH to the material or geometric nonlinear realms. Such extension involves some major difficulties. The first one is linearization, where constitutive equations at microscale need to be linearized onto linear elastic- or thermoelastic-like format. The second issue is the definition of so-called comparison materials which are fictitious materials designed to possess uniform instantaneous stiffness operators in each phase. The next problem to be solved is first-order vs second-order homogenization. In first-order homogenization comparison materials are computed with real constitutive models but volume averages of strain or stress fields per phase. In a second-order formulation, richer statistical information, namely the variance of strain or stress fields per phase is also taken into account. Finally, a very technical difficulty concerns the computation of Eshelby’s or Hill’s tensors and is related to the anisotropy of the comparison instantaneous stiffness operator. Within a coupled multi-scale analysis, FE method is used at macro scale, while at each Gauss integration point, MFH computation is carried out, either in the linear or nonlinear regime. This is the most feasible approach in practice. See Figure 5.
RVE homogenization Each inclusion
Figure 5: Comparison between the classical FE and the coupled FE/Digimat-MF approaches.
Extensive verification and validation results show that MFH can be used in practice for nonlinear problems and leads to good predictions in general, while work continues on improving accuracy in some situations (and reducing CPU time for coupled multi-scale analysis).
FE Homogenization: an application to nanocomposites
Most likely will nanomaterials be the materials of tomorrow, as they offer new horizons of applications in a wide variety of fields, e.g. nanoelectronics, bio-nanotechnology and nanomedicine. As such, more and more effort is put in understanding and modeling their behavior as well as acquiring know-how about nanoeffects. While new tools are being developed to tackle this engineering challenge, some are already available to the engineer of today. Among them: Finite Element Homogenization (FEH).
Modeling Filler Clustering, a typical nanoeffect
Material scientists face several challenges related to the design and the processing of nanocomposites as, at the nano scale, new physics and phenomena that are negligible at the macro scale enter the picture. For instance, uniform dispersion of the nanofiller inside the composite matrix is sought to improve the material mechanical properties, while clustering and percolation are desired when the conductivity of a base material, thermal or electrical, needs to be increased;
see Figure 6. Achieving one or the other nowadays constitutes a challenge in terms of both material processing and study.
Figure 6: Nanofiller dispersion.
FEH, as it requires the studied geometry to be explicitly generated and meshed, allows an accurate modeling of percolation and clustering effects. As an illustration, we present the effect of clustering on the elastic mechanical properties of a macroscopic material point.
Figure 7 presents two periodic nanostructures, also referred to as Representative Volume Element (RVE), that have been generated using Digimat-FE. Clustering parameters have been introduced to generate the rightmost geometry, whose inclusions are concentrated around 2 distinct clustering points. Volume fraction of the inclusion phase is 5% and the inclusions are spherical. Once meshed, these geometries will be subjected to uniaxial tensile conditions in the RVE x-, y- and z-directions and the finite element problem will be solved using the ANSYS finite element solver
Figure 7: Microstructures with uniformly distributed inclusions (left) and clustered inclusions (right).
Result-ComparisonFigure 8: S11 stress distribution in the inclusions (left) and in the matrix (right) for randomly placed inclusions
Figure 8 to 10 illustrate the stress distribution in the matrix and inclusion phases, in the case of the x-axis uniaxial tensile test. Due to the proximity of the inclusions around the clustering centers, stress concentrations appear. As such, up to 30% higher tensile stresses are observed for the clustered case, under x-direction uniaxial tensile loading conditions, see Figure 10.
Figure 11 plots the S33 stress and E33 strain distribution in the inclusion and matrix phases, as well as in the RVE. One clearly observes the higher stress levels in the inclusion phase. Such higher stress concentrations, that are not observed for randomly or uniformly placed inclusions, could lead to debonding during loading.
Figure 9: S11 stress distribution in the inclusions (left) and in the matrix (right) for clustered inclusions.
Figure 10: 2D section view of clustered (left) and random (right) RVEs. Tensile stress distribution.
Figure 11: S33 stress (left) and E33 strain (right) distributions in the nano phases and in the RVE for both cases for a z-direction uniaxial loading.
At this low volume fraction of inclusions, we see that clustering does not significantly alter the macroscopic mechanical properties of the material, see Table 1. Such a placement of nanoinclusions is thus preferably avoided by the material scientists when trying to increase the stiffness of a base material (Ematrix = 2195 MPa) by combination with a nanofiller (Efiller = 7000 MPa).
FE/MFH Coupled Computation: an application to an industrial part
For many reasons (manufacturing costs and flexibility, processing methods, high strength vs. lightness ratio, etc.), injected parts made up of short glass fiber reinforced plastics have become omnipresent in our daily life. But when it gets to model such materials, can macroscopic constitutive material models capture effects such as the injection process? The answer is no, as they do not capture the influence of the fiber orientation which depends on the injection process. The following example, which consists of a neon light clasp subjected to loading, introduces the process of a coupled analysis between Moldex3D, DIGIMAT-MF and ANSYS. This process, which is illustrated in Figure 12, consists of the following steps:
1. The injection molding process is simulated using Moldex3D. Among the available results are the fiber orientation tensors that will serve as input to DIGIMAT in the structural simulation.
2. The orientation tensors computed in 1. are mapped from the injection mesh onto the coarser structural one using Map (the mapping tool available in DIGIMAT).
3. The structural simulation is run using the ANSYS finite element solver coupled with Digimat-MF, the multi-scale material modeler that performs MFH at each integration point of the structural mesh.
Figure 12: Coupled analysis process. DIGIMAT takes the fiber orientation tensor obtained from Moldex3D as input, in addition to the material properties and serves as material modeler for the ANSYS finite element simulation.
Problem Description
The light clasp consists of four independent parts, see Figure 13 that also illustrates the contacts between the different parts. Two of them are made up of 30% glass fiber reinforced polyamide, Bergamid, and were injected. Their injection was simulated in Moldex3D. The slide and support block are assumed to be made up of steel
Closure of the clasp is simulated by imposing a displacement to the slide while blocking the support and part of the inner part. Sym
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