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立体光照成型的注塑模具工艺的综合模拟摘要功能性零部件都需要设计验证测试，车间试验，客户评价，以及生产计划。在小批量生产零件的时候，通过消除多重步骤，建立了有快速成型形成的注塑模具，这种方法可以保证缩短时间和节约成本。这种潜在的一体化由快速成型形成注塑模具的方法已经被多次证明是可行的。无论是模具设计还是注塑成型的过程中，缺少的是对如何修改这个模具材料和快速成型制造过程的影响有最根本的认识。此外，数字模拟技术现在已经成为模具设计工程师和工艺工程师开注塑模具的有用的工具。但目前所有的做常规注塑模具的模拟包已经不再适合这种新型的注塑模具，这主要是因为模具材料的成本变化很大。在本文中，以完成特定的数字模拟注塑液塑造成快速成型模具的综合方法已经发明出来了，而且还建立了相应的模拟系统。通过实验结果表明，目前这个方法非常适合处理快速成型模具中的问题。关键词注塑成型,数字模拟,快速成型引言在注塑成型中，聚合物熔体在高温和高压下进入模具中。因此，模具的材料需要有足够的热性能和机械性能来经受高温和高压的塑造循环。许多研究的焦点都是直接有快速成型形成注塑模具的过程。在生产小批量零件的时候，通过消除多重步骤，直接由快速成型形成的注塑模具可以保证缩短时间和节约成本。这种潜在的有快速成型形成注塑模具的方法已经被证明成功了。快速成型模具在性能上是有别与传统的金属模具。主要差异是导热性能和弹性模量（刚性）。举例来说，在立体光照成型模具中的聚合物的导热率小于铝制的工具的千分之一。在用快速成型技术来制造铸模时，整个模具设计和注塑成型工艺参数都需要修改和优化，传统的方法是改变彻底的刀具材料不过，目前还没有对如何修改这个模具材料的方法有根本的了解在当前的模具中，仅仅改变一些材料的性能是不能得到一个合理的结果的。同样，使用传统方法的时候，实际生产的零件也会有出先次品。因此，研究出一个快速成型过程，材料和注塑模具之间的互动关系是非常火急的。这样就可以确定模具设计标准和快速模具的注塑的技术。此外，计算机模拟是一种预测模塑件的质量的有效的方法。目前，商用仿真软件包已经成为模具设计师和工艺工程师在注塑过程中例行性的工具。不幸的是，目前常规注塑成型的模拟程序已经不再适用于这个快速成型模具，因为它极大的需要不同的刀具材料。例如，利用现在的仿真软件在铝和立体光照模具之间做个实验比较一下，虽然铝模具模拟植的部分失真是合理的，但是结果是不可以接受的，因为误差超过了百分之五十。在注塑成型中，失真主要是由于塑料零件的收缩和翘曲，模具也是一样的。对于通常模具，失真的主要因素是塑料件的收缩和翘曲，这个在目前的模拟中能测试准确。但是对于快速成型模具，潜在的失真会更多，在当前的测试中，其中就会有些失真会被忽视。例如，用一个简单的三步骤模拟分析模具变形的时候，就会出现很多偏差。在本文中，基于以上分析，一个新的快速成型模具的仿真系统已经开发出来了。拟议制度着重于预测部分失真，主要是用与预测快速成型模具的缺陷。先进的仿真系统可以用于预测快速成型模具设计和工艺是否最合理。我们的仿真系统已经被我们的实验证明是没有错误的。虽然有很多材料可以用于快速成型技术，但是我们还是专注于利用立体光照模具的技术来制造聚合物模具立体光照成型的过程是利用激光能量一层一层建立零件的部分。使用立体光照则可以体现出双方在快速成型工业的商业优势，而且在以后也可以生产出准确的，高品质的零部件。直到最近，立体光照主要是用于建立物理模型，为了检查视觉效果，仅仅只利用了它的一点点功能。不过，新一代的立体光照的光改善了立体化，机械性能，热学性能，所以它可以更好的应用于实际的模具中。2 综合仿真的成型过程2.1 方法 为了在注塑成型过程中模拟立体光照模具的功能，反复的试验中得到了一个方法。不同的软件组已经开发出来了，而且也已经做到了这一点。主要的假设是，温度和负载边界条件造成立体光照模具的扭曲，仿真步骤如下：部分几何模型则作为一个实体模型，这将通过流量分析软件包被翻译到一个文件中。模拟光聚合物模具中熔融体填充的过程，然后输出温度和压力的资料。在前一步获得了热负荷和边界条件，然后对光模具进行结构分析，其中失真的计算是在该注塑过程中进行的。如果模具的扭曲收敛了，那么直接进行下一步否则，扭曲的型腔（改动扭曲后的型腔的尺寸）返回第二个步骤，以熔体形式模拟注入扭曲的模具中。然后注射成型零件的收缩和翘曲模拟就开始应用了，算出该成型零件最终的扭曲部分上述的模拟流动中，基本上是三个仿真模块。2.2充型模拟的熔体2.2.1数字建模 计算机仿真技术已经能成功的预测到在极其复杂的几何形状下的填充情况。然而，目前大多数字模拟是基于一种混合有限元和有限差的中性平面上的。模拟软件包的应用过程基于这一模型说明图。然而，不同与系统中模具设计中的表面实体模型，这里所谓的中性平面（如图所示，图）是一个假想的在中间型腔中有距离和方向的一个平面，这个平面可能会在应用的过程中带来很大的不便。举例来说，模具表面常用于目前的快速成型系统中（通常是格式），所以当用模拟软件包的时候，第二次建模是不可避免的。那是因为模型在快速成型系统和仿真系统中是不一样的。考虑到这些缺点，在模拟系统中，型腔的表面将以基准面来引入，而不是中性平面。根据以往的调查，流量和温度场的方程式可以写为：X,Y是中性平面坐标系中的两个平面，是高度坐标，是，方向上的速度，是整体的平均厚度，, ,CP (T), K(T)分别表示聚合物的粘性，密度，周期热，热导率。图 是中性平面的模拟程序是维表面模型，是中性平面模型，是网状的平面模型，是最后的模拟结果此外，在高度方向上的边界条件的误差可以表示为：正如图中的中表示，TW 是恒壁温度.结合方程和方程，表明了u, v, T, P在坐标上面应该是对称的，因此在上半个高度中的平均u, v应该和整个高度中的平均u, v是一样的。根据这个特点，我们可以把整个型腔在上下高度上分为两个部分，正如图中的第一部分和第二部分。同时，型腔（如图）表面产生的三角有限元将替代了中性平面（如图）。因此，在高度方向上的有限元误差仅仅限于型腔表面，正如图所示，高度上的误差将从到。这是中性平面上的单一性。此外，从图到图，坐标也随之改变了。为了配合上述调整，方程仍是用方程。然而，原来的边界条件高度方向则改写为：与此同时，为了保持在同一坐标（）上的两部分能够流动，那么更多的边界条件必须满足。下标I和II则分别代表第一部分和第二部分的参数Cm-I 和Cm-II 则表示在填充阶段中分开的两个表面上的自由移动的熔融线。应该指出的是，方程与和方程与不同，和在数字模拟过程中将变的更难，主要原因是以下几点：同一个断层的表面都已经都已经有着特殊的网格，这将导致同一层上的独特的格局因此，在比较两个熔接口的时候，应该计算出各自的u, v, T, P。因为两个部分都有各自的流道通向节点和节点（如图所示）在同一段中，有可能两个都充满，也有可能一个满，一个空这两个情况应该分开处理，应该平均流动，使后者也分配到流动。这意味着在前线熔合处出现一点点小的误差是可以允许的通过控制时间和选择更好的位置来控制前线熔合节点。每个流场的边界都扩张到熔线前线，所以核查方程是否准确是相当重要的。鉴于上述分析，在同一个节点处的物理参数应该加以比较和调整。所以在进行模拟之前，描述同一节点有限元的信息应该准备好，也就是说，匹配的原理应该先预备好。图 表明表面模型中的中性平面的高度方向上的边界条件2.2.2数字模拟 压力场在建模中，粘度 是由于熔提的剪切速率，温度和压力引起的性能剪切变稀后，这就代表一个跨越式的模式，例如：其中对应于幂律指数，的特点是在在牛顿和幂律渐近极限之间的剪应力过渡区。无论在温度还是压力指数上，0(T, P)都可以有合理的表示，详情如下：方程11和12构成了一个五个常数，可以代表粘度，而且通过粘度的剪切速率的计算可以得到：根据上述情况，通过方程14，我们可以推断出一下充气压力方程：其中S是由计算出来的。运用伽辽金方法，对压力的有限元方程推导为：其中l是所有要素的的导线，包括节点N，而且其中i和j代表此处的N节点的数目，的计算方法如下：其中代表三角有限元，而代表有限元中的压力。 温度场中，为了确定高度方向上的误差，应该在模具表面上分为一层一层的三角有限元的网格。左边的能量方程4可以表示为：其中代表每一层N节点上的温度。热传导的计算方法是：其中l是所有要素，包括节点N，而且i和j分别代表此处的N节点个数。对流项的计算方法是：当是粘性热时，计算方法是：把方程1720带入方程4，温度方程变为：2.3 模具结构分析 结构分析的目的是预测在填充过程中，模具由于热和机械压力而产生的变形。这个模型是基于一个三维热边界元法。边界元法是比较适合这个应用的，因为只有变形的模具表面才有这样的信息。此外，边界元法有一个优点，那就是在计算变形的模具的时候，它的计算是不会白费的。 模具在所受载荷超过弹性范围的时候会产生应力。因此，在决定模具变形的时候，模具材料是一个基准。模具的热性能和力学性能是各向同性的，而且温度也是独立的。 尽管这个过程是循环的，但是相同时间的温度和热流都是可以用于计算模具变形的通常情况下，在模具里面每个瞬间温度都局限于型腔的表面和喷嘴的顶端。在观察距离的时候，瞬间的衰减变化是很微笑的，小于毫米这说明在模具的喷嘴处的变形是很小的，因此，忽略这个影响也是合理的稳态温度场满足拉普拉斯方程2T = 0的边界条件。至于机械边界条件，型腔表面受到熔体的压力，模具的表面会连接到工作台上的，而其他的外部表面将会假设是自由的.热边界的推导方程是大家都知道的，这是由于：其中uk, pk和分别是位移，牵引力和温度。, 是代表材料的膨胀系数和泊松比。Ulk是在方向上基本的位移。在一个三维空间中，各向同性弹性区域中，由一个单元产生的负荷主要集中在xl方向上，它是以下面的形式产生的：其中lk是Kronecker三角函数，是该模具材料的剪切模量。Plk的基本收缩都是在模具表面的每个节点处测量的，可以表示为：整个将分散在模具的表面上，转变为方程：其中n是指在这个区域上的表面成分。把恰当的线性函数代入方程，得到的线性边界方程就是模具的方程这个方程适用于每个离散的模具表面，从而组合成线性方程组，其中是节点的总数。每个节点有八个相关数量，三个位移组成部分，三个牵引组成部分，还有温度和热流量。在稳态热模型中，每个节点处的温度和磁场是已知的，余下的个量中，三个必须是已知的。此外，在若干个节点处的位移值的方程必须消除刚体运动和刚体自转的奇异系统。由此产生的系统方程式是一个集合起来的综合矩阵，它可以为有限元方法求解。基于方程的注塑假设，下面将给出元件的应力和应变：该偏元件的应力和应变分别是：用类似的方法可以预测在回火玻璃中的残余应力了。以积分的形式在平面上分析粘性和弹性结构关系时，可以表示为以下公式：其中G1是材料的的剪切模量。扩张的应变的情况如下：其中是材料体积的弹性模量，和的定义是：如果(t) = 0，那么方程到方程的结果则为：同样的，利用方程到方程消除应变xx(z, t)，得到：利用拉普拉斯变化方程，辅助系数R()由下面的方程得出：利用上述方程，并简化在模具中的应力和应变的形式，那么注塑中残余的应力在冷却阶段中，由下面的方程获得：方程可以通过梯形正交被解决。由于材料的时间在快速的变化，所以需要一个准数控程序来检测。辅助模量是检测数控梯形的规则。关于翘曲分析，节点位移和曲率将以壳单元表达为：其中 k 单元刚度矩阵，Be是衍生算子矩阵，d是位移，re是负载单元，可以由下面的方程得出：使用完整的三维有限元分析法的好处就是可以准确知道翘曲的结果。但是，当零件的形状很复杂的时候，它也是相当麻烦的。在本文中，在壳体理论基础上介绍了一种二维有限元分析方法。这种方法被大量使用是因为大多数注塑模具的零件都有一些部分几何的厚度远远小于其他部分。因此，那些部分则可以被作为一个集会的单元来预测翘曲。每三个节点壳单元组合成一个恒应变三角单元和一个离散克希霍夫三角元，如图所示，因此翘曲可以分为平面伸展变形和板弯曲变形。并相应的以单元刚度矩阵来描述翘曲的拉伸刚度矩阵和弯曲刚度矩阵。图 a-c是壳单元在局部坐标系统里的变形分解a是平面伸展元素，b是平面弯曲元素，c是壳单元三 实验验证 对提出的模型进行了评定和发展，最后核查是非常重要的。从模型模拟中得到的扭曲数据将和文献中的立体光照模具数据比较。如图所示，有一个注塑尺寸36 36 6毫米和实验数据中是相同的。薄壁和加强筋的厚度都是1.5毫米，这个注塑材料是聚丙烯。注塑机的型号是ARGURYHydronica320-210-750，它的工艺参数是，熔解温度是度，模具温度是度，注塑压力是.帕，保压时间是秒，冷却时间是秒。立体光照模具材料使用杜邦SOMOSTM树脂，能抵御高达度的高温。如上所述，热传导是区分立体光照模具和传统模具的一个重要因素。模具中的热量转移会产生温度的不均匀分布，所以导致了成型零件的翘曲立体光照成型模具的周期是可以预测的。以高的热传导率金属为背面做的薄壳立体光照模具将会增加自身的热传导率。图 模型腔图 不同的热传导率下，在方向上的扭曲失真比较实验值，三步走和常规都是指最后的实验结果常规是指实验中最好的结果三步走步骤的模拟过程分别与传统的注塑成型相似图 在不同的热传导率下，在方向上的扭曲失真比较 图 在不同热传导率下，在方向上扭曲失真比较图 不同热传导率下各个捻度变量的比较对于这个部分，扭曲包括三个方向上的位移和捻度（两个最初的平行边的夹角的误差）如图到图，实验结果表明，这些数值也包括通过传统注塑模具模拟系统预测的扭曲值和报道中的三步骤。结论本文介绍了一个综合模拟的快速成型模具的方法，并且建立了相应的仿真系统。为了验证这个系统，实验还进行了快速焊接立体光照成型模具。很明显，立体光照模具也会出现传统的注塑模具模拟软件一样的故障假设由于注射中的温度和负载荷引起了扭曲那么用三步骤完成的话，结果也会出现比较多的误差。不过更先进的模型会使结果更接近与实验。立体光照模具改进了热传导率极大的增加了零件质量由于温度比压力（负载）对模具的影响更大，所以改进立体光照模具的热传导率可以更显著的提高零件质量。无论零件多么复杂，快速成型技术可以使人们造型更快，更便捷，更便宜在快速成型稳步发展的基础上，快速制造也将随之而来，并且需要更多的精确工具来确定工艺过程的参数现行的模拟工具不能满足研究者研究模具相对的变化。正如本文中所述，对于一个综合模型来说，要预测最后零件质量是相当重要的。在不久的将来，我们期待看到通过快速成型扩展到快速模具制造的模拟程序。参考文献1 Wang KK (1980) System approach to injection molding process.Polym-Plast Technol Eng 14(1):7593.2 Shelesh-Nezhad K, Siores E (1997) Intelligent system for plastic injectionmolding process design. J Mater Process Technol 63(13):458462.3 Aluru R, Keefe M, Advani S (2001) Simulation of injection moldinginto rapid-prototyped molds. Rapid Prototyping J 7(1):4251.4 Shen SF (1984) Simulation of polymeric flows in the injection moldingprocess. Int J Numer Methods Fluids 4(2):171184.5 Agassant JF, Alles H, Philipon S, Vincent M (1988) Experimental andtheoretical study of the injection molding of thermoplastic materials.Polym Eng Sci 28(7):460468.6 Chiang HH, Hieber CA, Wang KK (1991) A unified simulation of thefilling and post-filling stages in injection molding. Part I: formulation.Polym Eng Sci 31(2):116124.7 Zhou H, Li D (2001) A numerical simulation of the filling stagein injection molding based on a surface model. Adv Polym Technol20(2):125131.8 Himasekhar K, Lottey J, Wang KK (1992) CAE of mold cooling in injectionmolding using a three-dimensional numerical simulation. J EngInd Trans ASME 114(2):213221.9 Tang LQ, Pochiraju K, Chassapis C, Manoochehri S (1998) Computeraidedoptimization approach for the design of injection mold coolingsystems. J Mech Des, Trans ASME 120(2):165174.10 Rizzo FJ, Shippy DJ (1977) An advanced boundary integral equationmethod for three-dimensional thermoelasticity. Int J Numer MethodsEng 11:17531768.11 Hartmann F (1980) Computing the C-matrix in non-smooth boundarypoints. In: New developments in boundary element methods, CML Publications,Southampton, pp 367379.12 Chen X, Lama YC, Li DQ (2000) Analysis of thermal residual stress inplastic injection molding. J Mater Process Technol 101(1):275280.13 Lee EH, Rogers TG (1960) Solution of viscoelastic stress analysisproblems using measured creep or relaxation function. J Appl Mech30(1):127134.14 Li Y (1997) Studies in direct tooling using stereolithography. Dissertation,University of Delaware, Newark, DE.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 relaxation shear modulus of the material. The dilatationalstresses can be related to the strain as follows:WhereKisthe relaxation bulk modulus of the material, and the definition ofandis:If(t)=0, applying Eq. 27 to Eq. 29 resultsin:Similarly, applying Eq. 31 to Eq. 28 and eliminating strainxx(z, t)resultsin:Employing a Laplace transform to Eq. 32, the auxiliary modulusR()isgivenby:Using the above constitutive equation (Eq. 33) and simplified forms of thestresses and strains in the mold, the formulation of the residual stress of the injectionmolded part during the cooling stageisobtainby:Equation 34 can be solved through the application of trapezoidal quadrature. Dueto the rapid initial change in the material time, a quasi-numerical procedureisemployed for evaluating the integral item. The auxiliary modulusisevaluatednumerically by the trapezoidal rule.For warpage analysis, nodal displacements and curvatures for shell elements areexpressed as:wherek isthe element stiffness matrix,Be isthe derivative operator matrix,disthe displacements, andreisthe element load vector which can be evaluatedby:The use of afullthree-dimensional FEM analysiscanachieve accurate warpageresults, however,itiscumbersome when the shape of the partisvery complicated. Inthis paper, a twodimensional FEM method, basedonshell theory, was used becausemost injection-molded parts have a sheet-like geometry in which the thicknessismuch smaller than the other dimensions of the part. Therefore, the part can beregarded asanassembly of flat elements to predict warpage. Each three-nodeshellelementisa combinationofa constant strain triangular element (CST) and a discreteKirchhoff triangular element (DKT), as shown in Fig. 3. Thus, the warpage can beseparatedintoplane-stretchingdeformationoftheCSTandplate-bendingdeformation of the DKT, and correspondingly, the element stiffness matrix todescribe warpagecanalso be divided into the stretching-stiffness matrix andbending-stiffness matrix.Fig.Fig.Fig.Fig. 3a3a3a3a c. c. c. c. Deformation decomposition of shell elementinthelocal coordinate system. a a a aIn-planestretching elementb b b bPlate-bending element c c c cShell element3 3 3 3 ExperimentalExperimentalExperimentalExperimental validationvalidationvalidationvalidationTo assess the usefulnessofthe proposed model and developed program,verificationisimportant. The distortions obtained from the simulation model arecompared to the ones from SL injection molding experiments whose dataispresentedin the literature 8. A common injection molded partwiththe dimensions of 36366mmisconsidered in the experiment, as shown in Fig. 4. The thickness dimensions ofthe thin walls and rib are both 1.5 mm; and polypropylene was used as the injectionmaterial. The injection machine was a production level ARGURY Hydronica320-210-750 with the following process parameters: a melt temperatureof250 C; anambient temperatureof30 C;aninjection pressure of 13.79 MPa; an injection timeof 3s;and a cooling time of 48 s. The SL material used, Dupont SOMOSTM 6110resin,hasthe ability to resist temperatures ofupto 300 C temperatures. Asmentioned above, thermal conductivity of the moldisa major factor that differentiatesbetweenanSL and a traditional mold. Poor heat transfer in the mold would produce anon-uniform temperaturedistribution,thus causingwarpagethat distortsthecompleted parts. For an SL mold, a longer cycle time would be expected. The methodof using a thin shell SL mold backed with a higher thermal conductivity metal(aluminum) was selected to increase thermal conductivityofthe SL mold.Fig.Fig.Fig.Fig. 4. 4. 4. 4. Experimental cavity modelFig.Fig.Fig.Fig. 5. 5. 5. 5. A comparison of the distortion variationinthe X direction for different thermalconductivity; where “Experimental”, “present”, “three-step”, and “conventional” mean the resultsof the experimental, the presented simulation, the three-step simulation process and theconventional injection molding simulation, respectively.Fig.Fig.Fig.Fig. 6. 6. 6. 6. Comparison of the distortion variationinthe Y direction for different thermalconductivitiesFig.Fig.Fig.Fig. 7. 7. 7. 7. Comparison of thedistortion variationintheZdirection for different thermalconductivitiesFig.Fig.Fig.Fig. 8. 8. 8. 8. Comparison of the twist variation fordifferent thermal conductivitiesFor this part, distortion includes the displacements in three directions and thetwist (the difference in angle between two initially parallel edges). The validationresults are shown in Fig. 5 to Fig. 8. These figures also include the distortion valuespredictedbyconventional injection molding simulation and the three-step modelreported in 3.4
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