【机械类毕业论文中英文对照文献翻译】冲压半径和角度对管向外卷曲过程的影响
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【机械类毕业论文中英文对照文献翻译】冲压半径和角度对管向外卷曲过程的影响,机械类毕业论文中英文对照文献翻译,机械类,毕业论文,中英文,对照,文献,翻译,冲压,半径,角度,向外,卷曲,过程,影响
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毕业设计文献翻译院（系）名称工学院机械系专业名称机械设计制造及其自动化学生姓名周凯指导教师薛东彬2012年 03 月 10 日 黄河科技学院毕业设计（文献翻译） 第24页冲压半径和角度对管向外卷曲过程的影响摘要：跟据更新的拉格朗日理论的公式，本研究采用弹塑性有限元法和扩展增量的测定方法，最小半径法，包括元素的产生，节点接触或分离的工具，最大应变和极限旋转增量。使用改性的库仑摩擦法建立有限元方法的计算机代码。不同半径和角度的锥形冲头是用于模拟成的硬铜及铜合金管的两端。影响的各种要素，包括冲床半顶角和半径R的比值，管的平均直径，管壁厚度比，力学性能，并对润滑管向外卷曲，进行调查。模拟结果显示，当冲床入口的弯曲半径满足的条件c，卷曲加工出现在管的末端。另一方面，如果c，管端遭受过扩口。可变c称为临界弯曲半径。c的值随着的值增加而增加。此外，研究结果还表明，c既不与管的材料，也不与润滑剂有关。关键词：弹塑性；有限元;半顶角;向外卷曲过程。1介绍形成金属管的凸边的过程中，经常被用于连接两个管件，连接和锁管的一部分，其组成部分，和流体管道连接管的末端相结合的互补与加固管道。它是一种常见的与管两端相关的工业技术。攻丝机扩口，法兰管端进程一般分为分为扩口，加强扩口，卷曲成型。本文所讨论的管卷曲变形卷曲管的结束了一圈向外开放。管卷曲变形的模拟是一项复杂而艰巨的任务，因为变形过程是高度非线性的。由于非线性变形的特点是：1大位移，旋转，变形过程中金属变形。2金属材料遇到大变形的非线性材料变形行为。3由摩擦产生的金属和工具之间的界面，和他们的接触条件的非线性边界条件。这些特点，使有限元方法，使用最广泛的金属工艺分析。改善工艺和提高工业生产率，本研究开发的弹塑性有限元计算机代码，使用选择性降低集成（SRI）的模拟方法，它采用一个矩形的四个节点内四个整点元素。目标是到模拟的管卷曲的变形过程。Nadai在1943年研究管收口。理论归纳为弧形壳体理论的延伸。Nadai假设摩擦系数为常数，而忽略了壳的有效应力变化的存在。克鲁登和汤普森进行了一系列管减噪实验，建立管减噪的各种限制，并评估各种参数的影响。Manabe and Nishimura两个锥形燃烧管和管减噪过程进行了一系列实验，调查研究的参数包括不同角度的锥形打孔，润滑，材料成形负荷和应力应变分布的各种参数的影响管壁厚度。唐和小林提出了刚塑性有限元理论，并开发出一种计算机代码来模拟105毫米AISI1018钢冷收口。黄等。模拟冷收口过程中，通过弹塑性和刚塑性有限元法的方法，并比较模拟结果与实验数据来验证刚塑性有限元分析的准确性。北泽等人。锥形冲头，与假设的铜及黄铜材料进行实验，管材料刚性完美的塑料，他们探讨了以前的冲头圆弧半径，角度，管壁厚，管卷曲变形过程的影响。北泽采用碳钢，铜，黄铜的管材管卷曲变形试验。当在管端前缘扩口的能量增量是大于卷曲边缘的能量增量。管底料的向外卷曲，是在不屈的情况下形成。另一方面，如果管端材料附着因为不屈的锥形打孔表面和失败卷曲或形成扩口。能源规则被用来诱导管形成的比较和验证的目的时的变形能，从而确立了向外卷曲的标准。在这项研究中，在其他研究中使用相同的材料常数和尺寸都采用向外卷曲过程的模拟结果相比，报告的结果验证了弹塑性有限元计算机代码的准确性。2基本理论的说明2.1刚度方程采用更新的拉格朗日公式（超低频）金属成形过程中的增量变形的应用框架（体积成形，板料成形）是描述塑性流动规则的增量特性的最切实可行的办法。在每个变形阶段的当前配置中的ULF用于评估一个小的时间间隔t变形，一阶理论的精度要求是一致的，作为参考状态。更新的拉格朗日方程的虚拟工作速率方程写成是基尔霍夫应力的Jaumann率，是柯西应力，是变形率，这是笛卡尔坐标，是速度，的名义牵引速度，是速度梯度，是固定的空间直角坐标，V和材料的体积和表面上牵引规定。图1.几何初始向外卷曲变形管和一个特定阶段的边界条件。单位毫米;冲床半径R，半顶点冲角，冲床入口弯曲半径。J2流动规律的模型采用的弹塑性行为的金属片，是应变硬化率，是有效应力，E是杨氏模量，是泊松比，是的偏量。取1为塑性状态和0为弹性状态或卸载。据推测，速度V，在离散化元素中的分布其中N是形函数矩阵和表示节点的速度。变形率和速度梯度写成其中B和E中，分别代表应变率的速度矩阵和速度梯度速度矩阵。将式（4）和（5）代入式（1），得到元素的刚度矩阵。由于虚功率式的原则。本构关系是非线性方程。差饷，他们可以更换任何单调增加的措施方面定义的工具，如位移增量，增量。Alpha有限元的标准程序，以形成整体的整体刚度矩阵，在其中在这些方程中，K是全球切线刚度矩阵，元素弹塑性本构矩阵，表示位移增量，表示在规定的结点力增量。和被定义为应力校正矩阵，由于在电流应力变形的任何阶段。图2.（a）变形的几何形状，（b）节点速度分布在12个不同的形成阶段管向外卷曲硬铜管，N= 0.05; T0 =0.8毫米;=60掳; r =3.4毫米。2.2斯里兰卡方案由于四边形元素充分整合计划的实现，导致过度的约束效应时，材料近不可压缩成型过程中的弹塑性情况13，休斯提出应变率的速度矩阵分解为扩张矩阵和偏差矩阵，即矩阵中和由传统的四点一体化集成。当材料变形到几乎不可弹塑性状态，修改后的扩张矩阵是由一个点的一体化集成，即必须更换扩张矩阵，即是修改后的应变率速度矩阵。将式（8）代入式（7），修改后的应变率速度矩阵明显的，速度梯度-速度矩阵，取而代之的是修改后的速度梯度速度矩阵3数值分析在分析模型管卷曲过程的是轴对称。因此，只有右边一半的中心轴是考虑过的。部分的有限元是由计算机自动处理。由于是从弯曲，并在管的末端卷曲的急剧变形，这一节需要更精细的单元划分，以获得精确的计算结果。如图左边一半。1表示零件和模具开始的尺寸。表1给出了详细的尺寸。在局部坐标系，轴1表示的切线方向，管材料和工具之间的联系而轴n为同一联系的正常方向。常数（X，Y）坐标和地方坐标（l，n）描述结点力，位移和元素的应力和应变。图3.正如图2，但R= 3.8毫米。表1 冲头的角度和半径冲头顶点角度（度）冲头半径R（毫米602.065.4706.176.4808.489.49.810.210.611.0表2.所用材料的机械性能材料外径厚度真空热处理nF(Mpa)屈服应力(Mpa)铜25.40.8原样0.09380220500 1 h0.5363026原样0.05450280300 1 h0.09450260400 1 h0.4661050600 1 h0.506404270/3黄铜25.40.8原样0.18730280 应力 ；应变表2给出了我们模拟的物质条件。管外半径保持在25.4毫米不变;但也有三种不同的管壁厚度值，即，0.4，0.6和0.8毫米，在实验和计算。硬铜管和黄铜管的泊松比和杨氏模量分别是0.33和110740兆帕，0.34和96500兆帕。3.1边界条件如图右手半管。1是指管卷曲变形在某一阶段的变形形状。边界条件包括以下三个部分：1.FG和BC段的边界：这里是节点的切向摩擦力增量，是普通的力量增量。假定涉及到摩擦材料，工具的接触面积，和不等于零。，表示工具的配置文件的正常方向的位移增量，从规定的冲头位移增量中确定。2.边界上的CD，DE，EF，和GA部分：上述情况反映了这个边界上的节点都是自由的。3.AB段的边界：AB部分是固定在管底的边界。在此位置沿Y轴方向的节点位移增量设置为零，而节点B是完全固定的没有任何动静。在管卷曲过程中，边界将会改变。因此，有必要检查FG和BC沿边界部分的变形阶段，在每个接触节点的法向力。如果达到零，则节点将成为自由节点和边界条件（1）（2）转移。同时，GA和EF沿管段的自由节点还检查了计算。如果该节点接触到冲头，自由边界条件的约束条件（1）改变。3.2 弹塑性和接触问题的处理当解释以前明确暗示的边界条件时，接触条件应维持在一个增量变形过程。为了满足这个要求，山田等人提出了最小值方法。 14采用扩展处理了弹塑性和接触问题 15。每个加载增量的最小值均被控制，有以下六个标准。弹塑性状态。元素的应力大于屈服应力时，R1的计算方法15因此，以确定达到一样的表面屈服应力。最大应变增量。获得R2默认的最大应变增量主应变增量比d，即，限制增长到这样的规模，一阶理论是有效的措施。最大旋转增量。默认的最大旋转增量的旋转增量，即，限制增长到这样的规模，一阶理论是有效的措施。渗透条件。形成的收益时，管的自由节点可能渗透的工具。比16的计算方法等，刚刚接触到的工具的自由节点。分离条件。当形成的收益，接触节点可以脱离接触面。 16 计算每个接触节点，这种正常的组成部分，节点力为零。 滑粘摩擦条件。修正的库仑摩擦定律提供了选择的接触状态，即滑动或粘状态。这种状态检查每个接触节点由以下条件：（a）如果，还有节点处于滑动状态(b)如果还有则节点处于固定状态一个方向滑动节点是相反的方向前渐进的步骤，使接触节点粘贴节点在下一个增量步。然后在这里得到比R，产生变化的摩擦状态的基体，在是一个很小的公差。最大应变增量，最大的旋转增量这里使用的常数是0.002o和0.5o，分别。这些常量都被证明是有效的一阶理论。此外，一个小的检查程序的渗透和分离条件是允许的。3.3 卸载过程卸载后的回弹现象是管形成过程中具有重要意义。假设是固定的，对管底的节点卸载程序执行。所有的元素都将复位弹性。节点来接触到的工具的力量在扭转成为管规定力的边界条件，即同时，核查的渗透，摩擦，分离条件被排除在仿真程序。四结果与讨论图2显示冲角=60，圆弧半径R=3.4毫米铜管卷曲成形仿真结果。图2（a）表示的几何形状的变形，在最后的形状是卸载后的最终形状的一部分。图2（b）显示了变形过程中的节点的速度分布。最后图显示卸货后节点的速度分布。这些数字表明，管端材料处于起步阶段顺利进入沿打孔表面。管底料后，逐渐弯曲向外卷曲，造成所谓向外卷曲成形。图3显示的R=3.8毫米的情况下的模拟结果。管端材料还显示所谓的燃烧形成合格后，电弧感应部分和圆锥面形成。图4显示了=75，R=6.1毫米卷曲铜管的模拟的结果。相比之下，图。5显示了扩口中的R= 6.4毫米的情况下形成的模拟结果。图4 如图2条件，而=75o；R=6.1毫米图5 如图2条件，而=75o；R=6.4毫米图6显示了应变分布相应的那些图2和图3在冲床行程12毫米。图7显示了应变分布相应的那些图4和图5在冲床进步14毫米。变形主要是由于拉在圆周方向弯曲，在正北方向和厚度方向的剪切。最大直径的管端扩口成形。因此，其应变值在圆周方向远高于向外卷曲。减少管壁厚度也更重要的扩口成形。因此，在相同的条件下，在管端扩口比在其他情况下更容易断裂。图6计算硬铜管应变分布下的卷曲和扩口过程比较n=0.05;to=0.8毫米；=60o图7 条件如图6，而=75o图8和图9显示仿真结果硬铜和黄铜，分别。同管壁厚，管半径，润滑条件，和半顶冲床是用来在模拟。然而，不同的价值观的冲头圆弧半径的使用，从而产生不同类型的形成，即，扩口和卷曲。就是说，如果圆弧半径，在冲头入口大于临界弯曲半径，然后管端的材料形成沿电弧燃烧冲压部分和圆锥面。反之，如果，管底材料叶片穿孔表面形成卷曲。这一数字表明临界弯曲半径随半顶冲床增加。比较结果表明，我们的数值模拟是一致的实验数据报告 12。所谓临界弯曲半径，可表示如下：图8 影响壁厚硬铜管弯曲半径和冲角之间至关重要的关系 n=0.05图9 条件如图8 黄铜管n=0.05这里R c=冲头半径相应的丙R fmin=最小值所需的耀斑冲压半径R cmax=最大值所需的卷曲冲压半径t0=壁厚管图10和图11表明之间的相关性无量纲临界弯曲半径（丙）和半顶角的2种不同壁厚径比的情况下，努力按铜管，分别。价值可以定义如下：如果是无关的壁厚径比，那么几何相似定律的存在临界弯曲半径。事实判断出来的曲线显示在图10和11是几乎相同的，我们一定会存在的几何相似的临界弯曲半径管材料。图10图11图12和图13表明之间的关系，最大半径和半顶角的时候，在数值模拟的壁厚度值为0.4，0.6，和0.8毫米的情况下硬铜管和铜管分别发生卷曲。当双方的半径和相应的半顶角的冲落在曲线的条件，将导致扩口。另一方面，当两个值低于曲线的条件，将导致冰壶。这些数字表明，最大半径增加的半顶角的扩大。图12图13图14显示了相关材料的加工硬化指数（n）和临界弯曲半径。无论变化值之间的0.03和0.53，由于不同的管材料，数值模拟的结果表明，值的临界弯曲半径保持不变，从而形成一条直线。这些调查结果是几乎相同的实验数据表明 12 ，这意味着，临界弯曲半径的时候，卷曲是无关的加工硬化指数。图14图15显示了影响润滑的关系临界弯曲半径和半顶角的冲头。临界弯曲半径的增加而增加半顶角的冲头的情况=0.05和=0.20。此外，结果显示，有一个微不足道的差异程度临界弯曲半径之间的润滑条件。图15黄铜管依赖的临界弯曲半径对润滑条件。五总结从更新的拉格朗日公式，带有一个圆锥形的冲压工艺模拟管卷曲的弹塑性有限元计算机代码开发。高非线性的过程是一个渐进的方式考虑与一个最小半径技术是通过限制每个增量步的线性关系的大小。摩擦改性库仑法开发是一个连续函数。如前所述，此功能可以处理工具和金属界面的滑动和粘性很难与普通连续摩擦模型描述的现象。斯里兰卡有限元是四节点四边形单元中的4个集成点。然后再加上与大变形有限元分析的SRI进一步成功地应用于锥形管卷曲形成工艺的分析。根据上述分析和讨论，可以得出以下结论：1、一个关键的弯曲半径，存在。如果拱半径在冲头入口大于临界弯曲半径，然后管端的材料在冲压拱感应部分和锥形面形成扩口。另一方面，如果，管底材料从表面和结果卷曲。弯曲半径的临界值随冲头的半顶角增加而增加。2、临界弯曲半径，遵循几何相似定律。值随冲床的半顶角的增加而增加。我们也知道，值是不相关的管材料，润滑条件或壁厚直径半径比。3、扩口圆周应变大于卷边的。扩口厚度也有比卷曲减少。因此，扩口更容易发生管端断裂。相比之下，卷曲的产品可以保证处理的安全，并加强管材料的结束部分。4、模具的形状表示为一个数值函数。因此，在这项研究中开发的有限元模型，可用于在连续形状的任何工具，来模拟扩口或卷曲过程中的所有类型。参考文献1、A. Nadai，塑性应力状态在弯曲外壳：形成钢制外壳，机头高爆外壳锻造所需的力量，在美国机械工程师学会年度会议，纽约，美国， 1943.11.23-12.1。2、A. K. Cruden 和 J. F. Thompson反向挤塑罐最终的关闭者，第511卷，国家工程实验室，瑞典，格拉斯哥，1972。3、K. Manabe and H. Nishimura，成形载荷和成形极限，锥管收口收口与扩口管研究杂志，日本社会对技术的可塑性，23（255），335 -342页，19824。4、K. Manabe and H. Nishimura，实验应力应变分布在锥管收口收口与扩口管的研究杂志，日本社会对技术的可塑性，23（255），335 -342页，19824。5、K. Manabe and H. Nishimura，应力分析和载荷在锥形收口管收口与扩口管的研究，日本社会的塑性技术学报，23（258），650 -657页，19827。6、K. Manabe and H. Nishimura，锥形冲头在扩口管形成负载收口与扩口管的研究杂志五“，日本社会对技术的可塑性，24（264），47 -52页 19831。7、K. Manabe and H. Nishimura，扩口管与锥形冲头的应力和应变分布收口与扩口管研究杂志六，日本社会对技术的可塑性，24（264），pp.276282，19833。8、M. C. Tang and S. Kobayashi，收口过程的有限元方法的调查，第1部分：在室温下的收口（冷收口），交易应用工程学报，104， 305 -311页，1982。9、Y. M. Huang, Y. H. Lu and M. C. Chen，分析过程中使用的弹塑性和塑性冷去噪方法，材料加工技术杂志，30， 351 -380页，1992。10、K. Kitazawa and M. Kobayashi 在管端卷曲变形机制的实验研究 - 壳的卷曲杂志，日本社会对技术的可塑性，28（323），pp.12671274，198712。11、K. Kitazawa, M. Kobayashi and S. Yamashita，管端卷曲的基础能源分析 - 锥状壳卷曲理论分析，29（331），845 -850页，19888。12、K. Kitazawa，标准向外卷曲管，交易应用工程学报，115， 466 -471页 1993。13、J. C. Nagtegaal, D. M. Parks and J. R. Rice,，准确的数值模拟，在全塑范围内的有限元解决方案，交易中的计算方法应用于机械工程，4， 153 -177页，1974。14、Y. Yamada, N. Yoshimura and T. Sakurai，塑性应力应变矩阵及其应用的解决弹塑性问题的有限元方法，国际机械科学学报，10， 343 -354页，1968。15、A. Makinouchi，有限元建模用于加工钣金，在K. Mattiasson，A. Samuelsson, R. D. Wood and O. C. Zienkiewicz， numiform86程序会议，哥德堡， 327332页，1986。16、Y. M. Huang, Y. H. Lu and A. Makinouchi, 弹塑性有限元分析V形薄板弯曲，材料加工技术杂志，35， 129 -150页，1992Int J Adv Manuf Technol (2002) 19:587596Ownership and Copyright Springer-Verlag London Ltd 2002Influence of Punch Radius and Angle on the Outward CurlingProcess of TubesYou-Min Huang and Yuung-Ming HuangDepartment of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, TaiwanUsing the theory of updated Lagrangian formulation, this studyadopted the elasto-plastic finite-element method and extendedthe increment determination method, the rminmethod, to includethe elements yielding, nodal contact with or separation fromthe tool, maximum strain and limit of rotation increment. Thecomputer code for a finite-element method is established usingthe modified Coulombs friction law. Conical punches withdifferent radii and angles are used in the forming simulationof hard copper and brass tube ends. The effects of variouselements including the half-apex angle of punch (?) and itsradius (R), the ratio of the thickness of the tube wall to themean diameter of tube, mechanical properties, and lubricationon the tubes outward curling, are investigated. Simulationfindings indicate that when the bending radius at the punchinlet (?) satisfies the condition of?c, curling is presentat the tube end. On the other hand, if?c, the tube endexperiences flaring. The variable?cis called the critical bend-ing radius. The value of?cincreases as the value of?increases. Furthermore, the findings also show that?cis neithercorrelated with tube material nor lubrication.Keywords: Elasto-plastic; Finite elements; Half-apex angle;Outward-curling process1.IntroductionThe process of forming convex edges of a metal tube is oftenemployed for connecting two tube parts, linking and lockinga tube part and its components, and connecting fluid pipelinesor combining complementary pipelines with reinforcement atthe tubes end. It is a common industrial technology relatedto tube ends. The tube end processes are generally dividedinto tapper flaring, flange flaring, step flaring, and curl forming.The tube-curling deformation discussed in this paper curls theopening of the tubes end outward in a circle. The simulationCorrespondence and offprint requests to: Dr You-Min Huang, Depart-ment of Mechanical Engineering, National Taiwan University ofScience and Technology, Taipei, Taiwan. E-mail: ymhuang?.twof the tube-curling deformation is a complex and difficulttask because the deformation process is highly nonlinear. Thenonlinear deformation characteristic is due to:1. The large displacement, rotation, and deformation duringmetal deformation.2. The nonlinear material deformation behaviour when metalmaterial experiences large deformation.3. The nonlinear boundary condition generated by the frictionbetween the metal and tool interface, and their contact con-ditions.These characteristics made the finite-element method the mostwidely used of the metal process analyses. To improve theprocessandincreaseindustrialproductivity,thisstudydeveloped an elasto-plastic finite-element computer code usingthe selective reduced integration (SRI) simulation method,which employs four integral point elements within the fournodes of a rectangle. The objective is to simulate the tube-curling deformation process.Nadai 1 studied tube-nosing in 1943. The theoretical induc-tion was an extension of the curved shell theory. Nadai assumedthe friction coefficient to be constant, and ignored the presenceof effective stress variation in the shell. Cruden and Tompson2 conducted a series of tube-nosing experiments to establishthe various limitations of tube-nosing and evaluate the effectsof various parameters. Manabe and Nishimura 37 also con-ducted a series of experiments on both conical tube-flaring andtube-nosing processes to investigate the effects of variousparameters on the forming load and stressstrain distribution.The parameters studied include the conical punch with differentangles, lubrication, material and tube wall thickness. Tang andKobayashi 8 proposed a rigid-plastic finite-element theoryand developed a computer code to simulate the cold-nosing of105 mm AISI 1018 steel. Huang et al. 9 simulated the cold-nosing process through elasto-plastic and rigid-plastic finite-element methods and compared the simulation results withexperimental data to verify the accuracy of the rigid-plasticfinite element. Kitazawa et al. 10,11 conducted experimentswith conical punches, and copper and brass materials with theassumption that the tube materials were rigid-perfectly plastic.They had explored previously the effect of the arc radius of588You-Min Huang and Yuung-Ming Huangthe punch, angle, and tube wall thickness on the tube-curlingdeformation process. Kitazawa 12 used carbon steel, copper,and brass as the tube materials in his experiment of tube-curling deformation. When the flaring energy increment at thefront edge of the tube end (?Wf) is greater than the energyincrement at the curl edge (?Wc), the tube end materialundergoes outward curling forming because of the absence ofunbending. On the other hand, if ?Wc? ?Wf, the tube endmaterial is attached to the conical punch surface because ofunbending and fails to curl or form flaring. The energy rulewas used to induce the deformation energy at the time of tubeforming for the purpose of comparison and verification, thusestablishing the criterion of outward curling.In this study, the same material constants and dimensionsused in other studies 12 are adopted in the simulation of theoutward curling process. The findings are compared with theresults reported in 12 to verify the accuracy of the elasto-plastic finite-element computer code developed.2.Description of the Basic Theory2.1Stiffness EquationAdopting the updated Lagrangian formulation (ULF) in theframework of the application of incremental deformation forthe metal forming process (bulk forming and sheet forming)is the most practical approach for describing the incrementalcharacteristics of the plastic flow rule. The current configurationin ULF at each deformation stage is used as the referencestate for evaluating the deformation for a small time interval?t such that first-order theory is consistent with the accuracyrequirement.The virtual work rate equation of the updated Lagrangianequation is written asFig. 1. Boundary conditions for the deformation tubes geometry ofoutward curling at an initial and a particular stage. Units, mm;R, punch radius; ?, half-apex punch angle; ?, bending radius atpunch inlet.?V(?*ij 2?ik?kj) ?ij-V +?V?jkLik?Lij-V =?Sft ?i?vi-S(1)where ?*ijis the Jaumann rate of Kirchhoff stress, ?ijis theCauchy stress, ?ijis the rate of deformation which is theCartesian coordinate, viis the velocity, t ?iis the rate of thenominal traction, Lij(= ?vi/?Xj) is the velocity gradient, Xjisthe spatial fixed Cartesian coordinate, V and Sfare the materialvolume and the surface on which the traction is prescribed.The J2flow rule?*=E1 + v?ik?jl+v1 2v?ij?kl3?(E/(1+v) ?ij?kl2? 2(0H?+ E/(1 + v) ?kl(2)is employed to model the elasto-plastic behaviour of sheetmetal, where H? is the strain hardening rate, ? is the effectivestress, E is the Youngs modulus, v is the Poissons ratio, ?ijis the deviatoric part of ?ij. ? takes 1 for the plastic state and0 for the elastic state or the unloading.It is assumed that the distribution of the velocity v in adiscretised element isv = N d(3)where N is the shape function matrix and d denotes thenodal velocity. The rate of deformation and the velocitygradient are written as? = B d(4)L = E d(5)where B and E represent the strain ratevelocity matrix andthe velocity gradientvelocity matrix, respectively. SubstitutingEqs (4) and (5) into Eq. (1), the elemental stiffness matrixis obtained.As the principle of virtual work rate Eq. and the constitutiverelationship are linear Eq. of rates, they can be replaced byincrements defined with respect to any monotonously increasingmeasure, such as the tool-displacement increment.Following the standard procedure of finite elements to formthe whole global stiffness matrix,K ?u = ?F(6)in whichK =?e?V?e?BT(Dep Q) B-V +?e?V?e?ETGE-V?F =?e?S?e?NTt ?-S?tIn these equations, K is the global tangent stiffness matrix,Dep is the elemental elasto-plastic constitutive matrix, ?udenotes the nodal displacement increment, and ?F denotesthe prescribed nodal force increment. Q and G are definedas stress-correction matrices due to the current stress at anystage of deformation.Influence of Punch Radius and Angle on Tube Curling589Fig. 2. (a) Deformed geometries, and (b) nodal velocity distributions in the outward curling of tubes at 12 different forming stages. Hard coppertube, n = 0.05; t0= 0.8 mm; ? = 60; R = 3.4 mm.2.2SRI SchemeAs the implementation of the full integration (FI) scheme forthe quadrilateral element leads to an excessively constrainingeffect when the material is in the nearly incompressible elasto-plastic situation in the forming process 13, Hughes proposedthat the strain ratevelocity matrix be decomposed to thedilation matrix Bdiland the deviation matrix Bdev, i.e.B = Bdil+ Bdev(7)in which the matrices Bdiland Bdevare integrated by conven-tional four-point integration. When the material is deformed tothe nearly incompressible elasto-plastic state, the dilation matrixBdilmust be replaced by the modified dilation matrix Bdilthat is integrated by the one point integration, i.e.B = Bdil+ Bdev(8)in which B is the modified strain ratevelocity matrix. Substi-tuting Eq. (8) into Eq. (7), the modified strain ratevelocitymatrix isB = B + (Bdil Bdil)(9)Explicitly, the velocity gradientvelocity matrix E is replacedby the modified velocity gradientvelocity matrix EE = E + (Edil Edil)(10)3.Numerical AnalysisThe analytical model of the tube-curling process is axiallysymmetric. Thus, only the righthand half of the centre axis isconsidered. Division of the parts finite elements is automati-cally processed by the computer. Since there is drastic defor-mation from the bending and curling at the tubes end, a finerelement division is required for this section in order to deriveprecise computation results. The lefthand half shown in Fig. 1denotes the sizes of the part and die at the beginning. Thedetailed dimensions are given in Table 1. In the local coordi-nates, axis 1 denotes the tangential direction of the contactbetween tube material and the tool, while axis n denotes thenormal direction of the same contact. Constant coordinates(X,Y) and local coordinates (l, n) describe the nodal force,displacement and elements stress and strain.590You-Min Huang and Yuung-Ming HuangFig. 3. As Fig. 2, but R = 3.8 mm.Table 1. Punch angle and radius.Punch apexPunch radius R (mm)angle ? (deg.)602.0, 2.4, 2.8, 3.1, 3.4, 3.8, 4.1, 4.4, 4.8, 5.1652.4, 2.8, 3.1, 3.4, 3.8, 4.1, 4.4, 4.8, 5.5, 5.4703.1, 3.4, 3.8, 4.1, 4.4, 4.8, 5.1, 5.4, 5.8, 6.1753.4, 3.8, 4.1, 4.4, 4.8, 5.1, 5.4, 5.8, 6.1, 6.4804.8, 5.4, 5.8, 6.1, 6.4, 6.8, 7.2, 7.6, 8.0, 8.4855.8, 6.1, 6.4, 6.8, 7.4, 9.0, 9.4, 9.8, 10.2, 10.6, 11.0Table 2 gives the material conditions of our simulation. Theexterior radius of the tube remains unchanged at 25.4 mm; butthere are three different tube wall thickness values, namely,0.4, 0.6 and 0.8 mm, used in the experiment and computation.The Poissons ratio and Youngs modulus of the hard coppertube are 0.33 and 110740 MPa, respectively. The Poissonsratio and Youngs modulus of brass are 0.34 and 96500MPa, respectively.3.1Boundary ConditionThe righthand half of the tube shown in Fig. 1 denotes thedeformation shape at a certain stage during the tube-curlingTable 2. Mechanical properties of materials used.MaterialsOuterHeat treatmentnFYield stressdiameterin vacuum(Mpa)?0.2(Mpa) thicknessCopper25.4 0.8As received0.09380220500C 1 h0.5363026As received0.05450280300C 1 h0.09450260400C 1 h0.4661050600C 1 h0.506404270/30 Brass25.4 0.8As received0.18730280? = F?n; ? = stress; ? = strain.deformation. The boundary conditions include the followingthree sections:1. The boundary on the FG and BC sections:?fl? 0, ?fn? 0,?vn= ?v nwhere ?flis the nodal tangential friction forces increment,and ?fnis the normal force increment. As the materialtoolcontact area is assumed to involve friction, ?fland ?fnarenot equal to zero. ?vn, which denotes the nodal displacementInfluence of Punch Radius and Angle on Tube Curling591increment in the normal direction of the profile of the tools,is determined from the prescribed displacement incrementof the punch ?v n.2. The boundary on the CD, DE, EF, and GA sections:?Fx= 0,?Fy= 0The above condition reflects that the nodes on this boundaryare free.3. The boundary on the AB section:?vx= 0,?vy= 0The AB section is the fixed boundary at the bottom end ofthe tube. The displacement increment of the node at thislocation along the y-axis direction is set at zero, whereas nodeB is completely fixed without any movement.As the tube-curling process proceeds, the boundary will bechanged. It is thus necessary to examine the normal force ?fnof the contact nodes along boundary sections FG and BC ineach deformation stage. If ?fnreaches zero, then the nodeswill become free and the boundary condition is shifted from(1) to (2). Meanwhile, the free nodes along the GA and EFsections of the tube are also checked in the computation. Ifthe node comes into contact with the punch, the free-boundarycondition is changed to the constraint condition (1).3.2Treatment of the Elasto-Plastic and ContactProblemsThe contact condition should remain unchanged within oneincremental deformation process, as clearly implied from theinterpretation in the former boundary condition. In order tosatisfy this requirement, the r-minimum method proposed byYamada et al. 14 is adopted and extended towards treatingthe elasto-plastic and contact problems 15. The increment ofeach loading step is controlled by the smallest value of thefollowing six values.1. Elasto-plastic state. When the stress of an element is greaterthan the yielding stress, r1is computed by 15 so as toascertain the stress just as the yielding surface is reached.2. The maximum strain increment. The r2term is obtained bythe ratio of the defaulted maximum strain increment ? tothe principal strain increment d?, i.e. r2= ?/d?, to limit theincremental step to such a size that the first-order theory isvalid within the step.3. The maximum rotation increment. The r3term is calculatedby the defaulted maximum rotation increment ? to therotation increment d?, i.e. r3= ?/d?, to limit the incremen-tal step to such a size that the first-order theory is validwithin the step.4. Penetration condition. When forming proceeds, the freenodes of the tube may penetrate the tools. The ratio r416is calculated such that the free nodes just come into contactwith tools.5. Separation condition. When forming proceeds, the contactnodes may be separated from the contact surface. The r5term 16 is calculated for each contact node, such that thenormal component of nodal force becomes zero.6. Sliding-sticking friction condition. The modified Coulombsfriction law provides two alternative contact states, i.e.sliding or sticking states. Such states are checked for eachcontacting node by the following conditions:vrel(i)l vrel(Il)l? 0(a)if ?vrel(i)l? ? VCRI, then f1= ?fn, the node is insliding state(b)if ?vrel(i)l? ? VCRI, then f1= ?fn(vrell/VCRI), the nodeis in quasi-sticking state.vrel(i)l vrel(I1)l? 0.The direction of a sliding node is opposite to the directionof the previous incremental step, making the contact node asticking node at the next incremental step. The ratio r6is thenobtained here, r6= Tolf/?vrel(i)l?, which produces the change offriction state from sliding to sticking, where Tolf (= 0.0001)is a small tolerance.The constants of the maximum strain increment ? andmaximum rotation increment ? used here are 0.002 and 0.5,respectively. These constants are proved to be valid in the first-order theory. Furthermore, a small tolerance in the check pro-cedure of the penetration and separation condition is permitted.3.3Unloading ProcessThe phenomenon of spring-back after unloading is significantin the tube-forming process. The unloading procedure isexecuted by assuming that the nodes on the bottom end of thetube are fixed. All of the elements are reset to be elastic. Theforce of the nodes which come into contact with the tools isreversed in becoming the prescribed force boundary conditionon the tube, i.e. F= F.Meanwhile, the verification of the penetration, friction, andseparation condition is excluded in the simulation program.4.Results and DiscussionFigure 2 shows the simulation results of curling forming ofcopper tubes with the punch semi-angle, ? = 60, and the arcradius, R = 3.4 mm. Figure 2(a) denotes the geometric shapeof the deformation, in which the last shape is the final shapeof the part after unloading. Figure 2(b) shows the nodal velocitydistribution during deformation. The last diagram shows thenodal velocity distribution after unloading. These figures showthat the tube end material enters smoothly along the punchsurface at the initial stage. After that, the tube end materialgradually bends outward and curls up, resulting in the so-called outward curling forming. Figure 3 shows the simulationresult in the case of R = 3.8 mm. The tube end material stillshows the so-called flaring forming after passing the arc induc-tion section and conical face forming.Figure 4 shows the result of curling simulation of coppertubes in the case of ? = 75 and R = 6.1 mm. In contrast,Fig. 5 shows the simulation result of flaring forming in thecase of R = 6.4 mm.Figure 6 shows the strain distributions corresponding tothose of Figs 2 and 3 at a punch travel of 12 mm. Figure 7592You-Min Huang and Yuung-Ming HuangFig. 4. As Fig. 2, but ? = 75; R = 6.1 mm.shows the strain distributions corresponding to those of Figs4 and 5 at a punch progress of 14 mm. The deformation ismostly the result of pulling in the circumference direction,bending in the meridian direction and shearing in the thicknessdirection. The maximum diameter of the tube end is the flaringforming. Thus, its strain value in the circumference directionis far greater than that of outward curling. The reduction intube wall thickness is also more significant in flaring forming.Thus, under the same progress, fracture of the tube end occursmore easily in flaring forming than in other situations.Figures 8 and 9 show the simulation results for hard copperand brass, respectively. The same tube wall thickness, tuberadius, lubrication condition, and half-apex of the punch areused in both simulations. However, different values of thepunch arc radius are used, which generate different types offorming, namely, flaring and curling. That is, if the arcradius, ?, at the punch entrance is greater than the criticalbending radius, ?C, then the tube end material forms flaringalong the punch arc induction section and conical face. Onthe contrary, if ?C? ?, the tube end material leaves thepunch surface and forms curling. Both figures show that thecritical bending radius increases as the half-apex of thepunch increases. A comparison shows that the results of ournumerical simulation are identical to the experimental datareported in 12. The so-called critical bending radius, ?C,can be expressed as follows:?C= RCt02=(RCmax+ Rfmin)2t02(11)whereRC= punch radius corresponding to ?CRfmin= minimum value of punch radius required to flareRCmax= maximum value of punch radius required to curlt0= wall thickness of tubesFigures 10 and 11 show the correlation between the non-dimensional critical bending radius (? C) and the half apexangle of thepunch withtwo differentwall-thickness-to-diameter ratios in the cases of hard copper tube and brasstube, respectively. The value of ? Ccan be defined as follows:? C=?C?(t0d0)(12)If ? Cis unrelated to the wall-thickness-to-diameter ratio, thenthe geometrical similarity law of the critical bending radiusInfluence of Punch Radius and Angle on Tube Curling593Fig. 5. As Fig. 2, but ? = 75; R = 6.4 mm.Fig. 6. Comparison of calculated strain distributions under curling andflaring processes. Hard copper tube, n = 0.05; t0= 0.8 mm; ? = 60.exists. Judging from the fact that the two curves shown inFigs 10 and 11 are almost identical, we can be certain of theexistence of the geometrical similarity law of the criticalbending radius of the tube materials.Fig. 7. As Fig. 6, but ? = 75.Figures 12 and 13 show the relationship between themaximum punch radius and half-apex angle of the punch whencurling occurs in the numerical simulation of wall thicknessvalues of 0.4, 0.6, and 0.8 mm in the cases of hard coppertube and brass tube, respectively. When values of both the594You-Min Huang and Yuung-Ming HuangFig. 8. The effect of tube thickness on the relationship between criticalbending radius and punch angle. Hard copper tube, n = 0.05.Fig. 9. As Fig. 8, but brass tube, n = 0.05.Fig. 10. Non-dimensional critical bending radius half-apex angle ofpunch relationships in outward curling of tubes. Hard copper tube,n = 0.05.Fig. 11. As Fig. 10, but brass tube, n = 0.18.Fig. 12. Maximum radius of punch to curl for various half-apex anglesof punch. Hard copper tube, n = 0.05.Fig. 13. As Fig. 12, but brass tube, n = 0.18.Influence of Punch Radius and Angle on Tube Curling595Fig. 14. Dependence of the critical bending radius on the work harden-ing exponent. ? = 60; t0= 0.8 mm.radius and the corresponding half-apex angle of the punch fallon the curve, the conditions will lead to flaring. On the otherhand, when both values fall below the curve, the conditionswill lead to curling. These figures indicate that the maximumpunch radius increases as the half-apex angle of the punchenlarges.Figure 14 shows the correlation between the materials workhardening exponent (n) and the critical bending radius. Regard-less of the variation in n value between 0.03 and 0.53 owingto different tube materials, results of the numerical simulationindicate that the value of the critical bending radius remainsconstant, thus forming a straight line. These findings are almostidentical to the experimental data shown in 12, and it impliesthat the critical bending radius at the time of curling isunrelated to the work hardening exponent.Figure 15 shows the effects of lubrication on the relation-ships between the critical bending radius and half-apex angleof the punch. The critical bending radius increases with theincreasing half-apex angle of the punch in the case of ? =0.05 and ? = 0.20. Furthermore, the result revealed that thereis a negligible difference in the magnitude of the criticalbending radius between the two lubricant conditions.Fig. 15. Dependence of the critical bending radius on lubricating con-ditions. Brass tube, t0= 0.8 mm.5.ConclusionAn elasto-plastic finite-element computer code was developedfrom the updated Lagrangian formulation to simulate tube-curling with a conical punch process. The high nonlinearity ofthe process was taken into account in an incremental mannerand an rmintechnique was adopted to limit the size of eachincrement step for a linear relation.The modified Coulombs friction law developed is a continu-ous function. As discussed earlier, this function can handle thesliding and viscosity phenomena of the tool and metal interfacethat are difficult to describe with the ordinary discontinuousfrictionmodel.TheSRI finite-elementisthefour-node-quadrilateral element in four integration points. Then the SRIcoupled with the finite-element large deformation analysis isfurther applied successfully to the analysis of the conical tube-curling formation process. According to the above analysesand discussion, the following conclusions can be drawn:1. A critical bending radius, ?C, exists. If the arch radius, ?,at the punch entrance is larger than the critical bendingradius, ?C, then the tube end material travels along thepunch arch induction part and the conical face to formflaring. On the other hand, if ?C? ?, the tube end materialleaves from the punch surface and results in curling. Thevalue of the critical bending radius increases as the half-apex angle of the punch increases.2 The critical bending radius, ?C, follows the geometricalsimilarity law. The value of ?Cincreases as the half-apexangle of the punch increases. We also learned that the valueof ?Cis not correlated with the tube material, lubricationconditions or the wall-thickness-to-diameter radius ratio.3. The circumference strain of flaring is greater than that ofcurling. Flaring also has greater reduction in thickness thancurling. Thus, tube end fracture occurs more easily inflaring. In contrast, products with curling can guaranteehandling safety and reinforce the end port
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