外文翻譯----三維注射成型流動模擬的研究

1、<p><b>　　附 錄</b></p><p>　　Numerical Filling Simulation of Injection Molding</p><p>　　Using Three—Dimensional Model</p><p>　　Abstract： Most injection molded parts a

2、re three-dimensional, with complex geometrical configurations and thick／thin wall sections．A 3D simulation model will predict more accurately the filling process than a 2.5D mode1.This paper gives a mathematical model an

3、d numeric method based on 3D model，in which an equal-order velocity-pressure interpolation method is employed successfully．The relation between velocity and pressure is obtained from the discretized momentum equations in

4、 order to derive the </p><p>　　Key words： three dimension；equal-order interpolation；simulation；injection molding </p><p>　　1 Introduction</p><p>　　During injection molding，the

5、 theological response of polymer melts is generally non-Newtonian and no isothermal with the position of the moving flow front．Because of these inherent factors，it is difficult to analyze the filling process．Therefore，si

6、mplifications usually are used．For example，in middle-plane technique and dual domain technique[1], because the most injection molded parts have the characteristic of being thin but generally of complex shape，the Hele-Sha

7、w approximation [2] is used whil</p><p>　　However, because of the us e of the Hele-Shaw approximation，the information that 2.5D models can generate is limited and incomplete．The variation in the gapwise (thi

8、ckness) dimension of the physical quantities with the exception of the temperature，which is solved by finite difference method，is neglected．With the development of molding techniques，molded parts will have more and more

9、complex geometry and the difference in the thickness will be more and more notable，so the change in the gapwise (th</p><p>　　3D simulation model has been a research direction and hot spot in the scope of sim

10、ulation for plastic injection molding．In 3D simulation model，velocity in the gapwise (thickness) dimension is not neglected and the pressure varies in the direction of part thickness，and 3 D finite elements are used to d

11、iscretize the part geometry．After calculating，complete data are obtained(not only surface data but also internal data are obtained)．Therefore, a 3D simulation model should be able to generate comple</p><p>　

12、　This Paper presents a 3 D finite element model to deal with the three—dimensional flow, which employs an equa1-order velocity-pressure formulation method [3,4]．The relation between velocity and pressure is obtained from

13、 the discretized momentum equations, then substituted into the continuity equation to derive pressure equation．A 3D control volume scheme is employed to track the flow front．The validity of the model has been tested thro

14、ugh the analysis of the flow in cavity．</p><p>　　2 Governing Equations</p><p>　　The pressure of melt is not very big during filling the cavity, in addition，reasonable mold structure can avoid o

15、ver big pressure，so the melt is considered incompressible．Inertia and gravitation are neglected as compared to the viscous force．</p><p>　　With the above approximation，the governing equations，expressed in ca

16、rtesian coordinates，are as following：</p><p>　　Momentum equations</p><p>　　Continuity equation</p><p>　　Energy equation</p><p>　　where, x，y，z are three dimensional coor

17、dinates and u, v，w are the velocity component in the x, y, z directions．P，T，ρa(bǔ)ndη denote pressure， temperature, density and viscosty respectively．</p><p>　　Cross viscosity model has been used for the simulat

18、ions：</p><p>　　where，n,γ，r are non-Newtonian exponent，shear rate and material constant respectively．Because there is no notable change in the scope of temperature of the melt polymer during filling，Anhenius

19、model[5] for η0 is employed as following：</p><p>　　where B，Tb， β are material constants.</p><p>　　3 Numerical Simulation Method</p><p>　　3.1 Velocity-Pressure Relation </p>&

20、lt;p>　　In a 3D model，since the change of the physical quantities are not neglected in the gapwise (thickness) dimension，the momentum equations are much more complex than those in a 2.5D mode1．It is impossible to obtai

21、n the velocity—pressure relation by integrating the momentum equations in the gapwise dimension，which is done in a 2.5D model. The momentum equations must be first discretized，and then the relation between velocity and p

22、ressure is derived from it. In this paper, the momentum equations are </p><p><b>　　where</b></p><p>　　the nodal pressure coefficients are defined as</p><p>　　where repre

23、sent global velocity coefficient matrices in the direction of x, y, z coordinate respectively. denote the nodal pressure coefficients the direction of x, y, z coordinate respectively. The nodal values for are obtained

24、by assembling the element-by-element contributions in the conventional manner. N，is element interpolation and i means global node number and j , is for a node, the amount of the nodes around it.</p><p>　　3.2

25、 Pressure Equation</p><p>　　Substitution of the velocity expressions (2) into discretized continuity equation, which is discretized using Galerkin method，yields element equation for pressure:</p><

26、p>　　The element pressure equations are assembled the conventional manner to form the global pressure equations． </p><p>　　3.3 Boundary Conditions </p><p>　　In cavity wall, the no- slip bound

27、ary conditions are employed, e.g.</p><p>　　On an inlet boundary, </p><p>　　3.4 Velocity Update </p><p>　　After the pressure field has been obtained，the velocity values are updated

28、using new pressure field because the velocity field obtained by solving momentum equations does not satisfy continuity equation．The velocities are updated using the following relations</p><p>　　The overall p

29、rocedure for fluid flow calculations is relaxation iterative，as shown in Fig.l and the calculation is stable without pressure oscillation．</p><p>　　3.5 The Tracing of the Flow Fronts </p><p>　　T

30、he flow of fluid in the cavity is unsteady and the position of the flow fronts values with time．Like in 2.5D model, in this paper, the control volume method is employed to trace the position of the flow fronts after the

31、FAN(Flow Analysis Network)[6]. But 3D control volume is a special volume and more complex than the 2D control volume．</p><p>　　It is required that 3D control volumes of all nodes fill the part cavity without

32、 gap and hollow space. Two 3D control volumes are shown in Fig．2．</p><p>　　4 Results and Discussion</p><p>　　The test cavity and dimensions are shown in Fig.3(a)．The selected material is ABS780

33、 from Kumbo. The parametric constants corresponding to then, γ,B, Tb and β of the five-constant Cross-type Viscosity model are 0．2638, 4．514 ×le4 Pa, 1.3198043×le-7 Pa *S, 1．12236 ×1e4K，0．000 Pa-1．Inject

34、ion temperature is 45℃，mould temperature is 250℃, injection flow rate is 44．82 cu. cm／s. The meshed 3D model of cavity is shown in Fig. 3(b).</p><p>　　“Fountain flow” is a typical flow phenomenon during fill

35、ing．When the fluid is injected into a relatively colder mould，solid layer is formed in the cavity walls because of the diffusion cooling，so the shear near the walls takes place and is zero in the middle of cavity, and th

36、e fluid near the walls deflects to move toward the walls．The fluid near the center moves faster than the average across the thickness an d catches up with the front so the shape of the flow front is round like the founta

37、in．T</p><p>　　The flow front comparison at the filling stage is illustrated in Fig．5．It shows that the predicted results based on present 3D model agree well with that based on Moldflow 3D mode1．The gate pre

38、ssure is illustrated in Fig．6，compared with the prediction of Moldflow 3D model．It shows that the predicted gate pressure of present 3D model is mainly in agreement with that based on Moldflow 3D mode1．The major reason f

39、or this deviation is difference in dealing with the model an d material parameters．</p><p>　　5 Conclusions</p><p>　　A theoretical model and numerical scheme to simulate the filling stage based

40、on a 3D finite element model are presented．A cavity has been employed as example to test the validity. 3D numeral simulation of the filling stage in injection moulding is a development direction in the scope of simulatio

41、n for plastic injection molding in the future．The long time cost is at present a problem for 3D filling simulation，but with the development of computer hardware and improvement in simulation technique，th</p><p

42、>　　三維注射成型流動模擬的研究</p><p>　　摘要:大多數(shù)注射成型制品都是具有復(fù)雜的幾何輪廓和厚壁或薄壁的制品。這種三維仿真模型將比兩維半模型具有更精確的填充過程。本文介紹了一種基于三維模型的注射成型流動模擬的數(shù)學(xué)模型和數(shù)值實(shí)現(xiàn)，把速度和壓力同次插值方法成功地應(yīng)用到三維注塑模擬的計(jì)算中，從離散的動量方程中找出壓力和速度的關(guān)系，然后迭代到連續(xù)性方程中得到壓力方程。用三維控制體積法追蹤流動前沿，并通過算

43、例分析來說明三維模型的有效性。</p><p>　　關(guān)鍵詞：三維模型 ； 等序插值法； 模擬； 注塑成型</p><p><b>　　1引言</b></p><p>　　在注塑成型的過程中，聚合物熔化的流變反應(yīng)隨著流動前沿的方向大多是非牛頓流體和非等溫的。由于這些內(nèi)在的因素，分析它的填充過程是很困難的，因此通常進(jìn)行簡易處理。例如在

44、中面流和雙面流技術(shù)中，由于大多數(shù)注塑成型的零件都是薄壁卻有復(fù)雜的形狀的特征，當(dāng)分析流動性而厚度方向的速度和壓力變化被忽略時，通常使用Hele—Shaw流動簡化。因此這兩種技術(shù)都是兩維的填充模型，用這種方法填充一個模型的型腔就變成了流動方向的二維問題和厚度方向的一維分析。</p><p>　　但由于采用了簡化假設(shè)，它產(chǎn)生的信息是有限的、不完整的。除了用有限差分法求解溫度在壁厚方向的差異外，基本上沒有考慮物理量在厚度

45、方向上的變化 。隨著塑料成型技術(shù)的發(fā)展，注塑成型零件將具有越來越復(fù)雜的形狀，其壁的厚度的多樣性將變得越來越顯著，因此在厚度方向變化的物理量就不能被忽視。此外，熔體在型腔的表面流動模擬看起來不真實(shí)，僅當(dāng)這些流動模擬出現(xiàn)在成型型腔時它的真實(shí)性才更加明顯。</p><p>　　三維流動模型已經(jīng)是研究方向而且在塑料注塑成型模擬方面將是個熱點(diǎn)。在三維流動模型中，熔體在厚度方向的速度分量不再被忽略，熔體的壓力沿厚度方向變化，

46、并且在分解三維實(shí)體制品方面通常使用有限元分析。通過有限元計(jì)算，可以獲得完整的數(shù)據(jù)（不僅獲得實(shí)體制品表面的流動數(shù)據(jù)，還獲得實(shí)體內(nèi)部完整的流動數(shù)據(jù)。）。因此，對于薄壁制品，三維流動模擬能夠產(chǎn)生更加詳細(xì)的關(guān)于流動特征的信息和應(yīng)力分布；對于如在氣體輔助成型中遇到的有厚壁區(qū)域的制品，三維流動模擬能更加準(zhǔn)確地預(yù)測其充填行為。許多在二維模型中不能預(yù)測的充模過程中的流動行為，如熔體前沿的流動形態(tài)和推進(jìn)方式，即“噴泉”效應(yīng)在三維流動模擬技術(shù)中都可以得到很

47、好的體現(xiàn)。</p><p>　　本文提出了一種三維有限元模型來預(yù)測模擬塑料熔體的充模流動，把速度和壓力同次插值方法成功地應(yīng)用到三維注塑模擬的計(jì)算中，從離散的動量方程中找出壓力和速度的關(guān)系，然后代到連續(xù)性方程得到壓力方程。用三維控制體積法追蹤流動前沿，并通過算例來說明該三維模型的有效性。</p><p><b>　　2 控制方程</b></p><p

48、>　　充模過程中熔體壓力不是很高，且合理的模具結(jié)構(gòu)可以避免過壓現(xiàn)象，因此設(shè)熔體為未壓縮流體。由于熔體粘性較大，相對于粘度剪切應(yīng)力而言 ，慣性力和質(zhì)量力都很小，可忽略不計(jì)。 </p><p>　　經(jīng)過簡化和假設(shè)，控制方程的直角分量形式分別為： </p><p><b>　　動量方程：</b></p><p><b>　　連續(xù)性方程

49、：</b></p><p><b>　　能量方程：</b></p><p>　　式中：x, y, z—三維坐標(biāo)；u, v, w—分別表示x, y, z方向的速度；ρ—熔體密度；</p><p>　　P—壓力；T—溫度；η—熔體粘度</p><p>　　粘度模型采用 Cross模型</p><

50、;p>　　式中：n—非牛頓指數(shù)；γ—剪切速率；—材料常數(shù)；η0—零剪切粘度</p><p>　　由于在充模過程中，熔體的溫度變化范圍不大，因此η0采用 Arrhenius型表達(dá)式：</p><p>　　式中：B，Tb, β—材料常數(shù)。</p><p><b>　　3 數(shù)值模擬方法</b></p><p>　　3.

51、1 壓力 —速度關(guān)系 </p><p>　　三維有限元模型由于沒作 Hele-Shaw流動簡化，其數(shù)值處理方法和二維模型有很大不同。在三維模型中，用三維立體單元離散制品空間，采用速度和壓力同次插值和迦遼金法來離散控制方程 ，用三維控制體積法追蹤流動前沿。由于三維模型考慮了厚度方向物理量的變化，其動量方程比二維模型復(fù)雜得多，不可能像二維模型那樣直接通過在厚度方向上的積分得到速度和壓力的關(guān)系，需要首先對動量方程進(jìn)行離

52、散，從中找出壓力和速度的關(guān)系。本文采用壓力、速度雙線形插值，用 Galerkin法對動量方程離散，經(jīng)逐個單元組裝后得到節(jié)點(diǎn)速度和壓力的關(guān)系如下：</p><p>　　其中，虛擬速度定義為：</p><p>　　節(jié)點(diǎn)上的壓力系數(shù)定義為：</p><p><b>　　(3)</b></p><p>　　式中—分別表示在 x，

53、y，z 方向的總體速度系數(shù)矩陣</p><p>　　—分別表示節(jié)點(diǎn)在 x，y, z 方向的壓力系數(shù)，其值利用式(3)在整個計(jì)算域內(nèi)積分，由各單元的貢獻(xiàn)值組裝而得到</p><p>　　Ni—單元插值函數(shù)；i—總體節(jié)點(diǎn)號；j—每個節(jié)點(diǎn)所有領(lǐng)接節(jié)點(diǎn)的數(shù)量</p><p><b>　　3.2 壓力方程 </b></p><p&g

54、t;　　把連續(xù)方程式(1d)用 Galerkin法離散后，把速度方程式(2)代入，整理后得到離散的單元壓力方程：</p><p>　　把單元剛度矩陣用常規(guī)的方法在整個計(jì)算域內(nèi)組裝就得到整體壓力方程。</p><p><b>　　3.3 邊界條件</b></p><p>　　在模壁上采用無滑移邊界條件：</p><p>

55、　　在澆口處：u=v=w=給定；</p><p><b>　　3.4 速度修正 </b></p><p>　　求解壓力方程，得到壓力場。但從動量方程求解得到的速度場并不滿足連續(xù)性條件，因此，要按下式用所求得的壓力場去修正當(dāng)前得到的速度場。</p><p>　　上述壓力、速度方程采用松弛迭代求解。整個求解過程如圖1所示。</p>&

56、lt;p>　　3.5 流動前沿位置的確定 </p><p>　　熔體在模腔內(nèi)的流動是非穩(wěn)態(tài)的過程，熔體前沿位置隨時間變化。像二維模型一樣，本文沿用 FAN (Flow Analysis Network) J的思路，采用控制體積法來跟蹤熔體每一時刻的前沿位置。但三維控制體積是一個空間體積，比二維控制體積復(fù)雜得多，三維控制體積的劃分必須保證各節(jié)點(diǎn)的控制體積完全充滿制品空間，不能有空洞和縫隙。圖2是三維控制體積的

57、形態(tài)圖，箭頭處為制品表面。</p><p>　　圖1 三維計(jì)算流程框圖</p><p>　　(a)制品內(nèi)部節(jié)點(diǎn)的控制體積 (b)制品邊界節(jié)點(diǎn)的控制體積</p><p><b>　　圖2 三維控制體積</b></p><p><b>　　4 結(jié)果和討論</b></p>&l

58、t;p>　　算例的型腔如圖 3(a)所示。注射材料為 Kumbo生產(chǎn)的AKS780，對應(yīng)于五參數(shù) Cross模型中的( n，γ，B，Tb，β)粘度參數(shù)為 (0．2 638，4．515×10 Pa， 3．13 198 043×10‐7 Pa·s，1．12 236×10 K，0 Pa‐¹ )。 注射溫度為 250℃，模具溫度為 45℃，制品的三維有限元網(wǎng)格如圖3(b)所示。</p

59、><p>　　(a) 制品尺寸 (b) 立體網(wǎng)格劃分</p><p><b>　　圖3 示例制品</b></p><p>　　“噴泉”效應(yīng)也是充模流動時的一個典型現(xiàn)象。當(dāng)熔體以較快的速度注入一個相對較冷的模具中，熔體和型腔壁接觸后，由于傳導(dǎo)冷卻效應(yīng)，實(shí)際上在型腔壁處就形成固體層 ，靠近型腔壁處的熔體剪切應(yīng)力增加，

60、而中部剪切應(yīng)力為零，于是靠近型腔壁處熔體流動方向開始向模壁偏轉(zhuǎn)。又由于中部熔體流動速度比沿壁厚度方向上的平均速度快，不斷沖破熔體前沿由于降溫而形成的前沿膜并形成新的前沿膜。因此，此時流體前端呈噴泉狀，后面則以片狀流動在固體層下面通過。圖4(a)是示例制品在幾個充填時刻流動前沿的形狀，實(shí)驗(yàn)結(jié)果和這種理論相符合。相反，如圖4(b)所示兩維半模型的流動前沿形狀不會出現(xiàn)這種“噴泉”效應(yīng)。</p><p>　　(a) 三維

61、流動前沿的形狀 (b) 兩維半流動前沿的形狀</p><p>　　圖4 三維模型流動前沿形狀(a)和兩維半模型流動前沿形狀(b)的比較</p><p>　　圖5所示的幾個充填時刻流動前沿形狀的比較。它所示的當(dāng)前模型的流動前沿形狀的效果比充填模型的好。如圖6所示的是和充填模型的流動前沿形狀相比較的片門壓力圖，它所示的當(dāng)前模型的片門壓力和充填模型的相一致。產(chǎn)生

62、這種偏差的主要原因是在處理模型和材料參數(shù)的差異。</p><p>　　圖 6 當(dāng)前三維模型的片門壓力值（虛線）和充填模型的片門壓力值（實(shí)線）的比較 </p><p>　　圖 5 當(dāng)前三維模型流動前沿形狀(a)和充填模型流動前沿形狀(b)的比較</p><p><b>　　5 結(jié)論</b></p><p>　　三維有