板材多点成形中起皱和回弹的数值分析
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摘要
多点成形技术是金属板材三维自由曲面成形的一种柔性加工方法，其基
本思想是将传统的整体模具离散化，由一系列规则排列的基本体（或称冲
头）组成的“柔性多点模具”来代替，由基本体球头的包络曲面来完成板材
成形。多点成形技术作为一种新兴的技术已在飞机、船舶、汽车等诸多制造
领域有着广泛的应用前景。
无压边多点成形通常用于变形量不大的曲面成形，是板材多点成形过程
中使用频率很高的一种成形方法。多点成形在成形变形量不大的曲面时，通
常不压边，这样既可省去切边工序又可以节省材料，另一方面又能简化多点
压机结构，降低其造价。由于没有压边圈，板材面内变形力较小，主要以面
外弯曲变形为主，导致在多点成形中起皱缺陷更容易出现，特别是在薄板多
点成形中，起皱是工艺上必须克服的成形缺陷。
由于没有压边圈的作用，板材成形过程中，回弹对成形件最终形状的影
响要比有压边时大。影响回弹的因素很多，如板材厚度，板材的材质以及成
形件变形量的大小等。本文采用数值模拟技术对多点成形过程的回弹进行了
分析，另外还探讨了典型零件在多点成形时，通过控制基本体群成形面形状
消除回弹的方法。
1. 板材无压边多点成形的起皱数值模拟
采用显式算法进形起皱数值模拟时，板材必须离散成足够小的有限元单
元，使皱纹能够已相当小的波长正确的模拟出来。通过不同尺寸有限元单元
的数值结果比较，建立了板材无压边起皱分析的有限元模型。对双向曲率曲
面—球面和马鞍面进行了详细的数值模拟，研究了板材厚度、曲率半径以及
材质等对起皱的影响。板厚与变形程度是影响起皱的重要因素，增大板材厚
71
吉林大学硕士研究生学位论文
度，减小成形件曲率半径均可有效的抑制起皱的产生；文中对 08AL 和 L2Y2
铝两种不同材料的板材成形球面和马鞍面的起皱现象进行了有限元分析。结
果表明选用具有较大的弹性模量的 08AL较具有较小弹性模量的 L2Y2 不容
易产生起皱缺陷。
通过对多点成形过程中的起皱产生过程的分析，发现在无压边多点成形
中，起皱主要产生于基本体对板材的不完全约束，在成形后期起皱出现最大
值，在成形结束时，部分皱纹被压平。基于对多种不同板厚、不同曲率半径
的球面和马鞍面成形件的数值分析，得到了球面与马鞍面无压边、不起皱多
点成形极限图。在本文之前，还没有多点成形不起皱极限图方面的结果。
2. 板材无压边多点成形的回弹数值模拟。
完整的板材成形包括加载和卸载两个过程。本文建立了采用显-隐式算法
模拟回弹问题的有限元模型。即采用动态显式算法模拟板材成形过程，采用
隐式算法模拟卸载回弹过程。这种方法克服了显式算法耗时大，难以收敛的
缺点，解决了隐式算法不能处理好接触问题的缺点，极大的提高了回弹数值
的计算效率，具有实用性和高效性的特点。
主要分析了板材厚度、成形件变形量以及材质等对柱面、球面及马鞍面
的无压边多点成形的回弹影响，并且利用中心插分法，拟合出成形件回弹
前、后曲面形状，计算出回弹量的大小。由数值模拟结果可以看出，在多点
成形中，板材厚度越小，变形量越小，卸载后回弹越大，反之，厚度越大，
变形量越大则回弹越小。
3. 多点成形过程中成形面的修正与回弹控制的研究
利用数值模拟技术，在回弹分析的基础上，给出了基于回弹数值计算的
柱面、球面多点成形基本体群成形面的修正方法。对于目标曲率半径为的圆
柱面，但板厚为 1.0mm时，经过三次修正，形状总体误差只有 0.0044m；板
厚为 1.5mm时，经过五次修正，形状误差达到 0.00083m；对于目标曲率半径
为 1200mm，板厚为 1.5mm时，经过两次修正形状误差达到 0.0962m；依据
最后一次修正的成形面调整基本体群的形状进形多点成形，可以有效的补偿
板回弹影响。实现板材的一次精确成形，提高产品的成形精度。这些结果对
多点成形技术的进一步工程应用具有一定的参考价值和实际意义
Multi-point forming(MPF) is a flexible manufacturing technology for three
dimensional sheet metal forming. In multi-point forming, the conventional solid die
is replaced by “flexible Multi-point die” composed by a series of discrete elements
(or punches). The forming process of sheet metal is implemented by the envelope
surface of punches. Multi-point Forming can be applied extensively in a lot of
fields such as aircraft, stream and navel ships, vehicle, large sculpture and modern
architecture etc.
Sheet metal forming in MPF without blank holder, which is a common
process method, is usually applied in processing those kinds of surface parts which
have not big deformation amount. In this process condition, the original material
consuming and the press cost can be reduced to a low level as well as simplifying
the press structure and leaving out trimming procedure in metal sheet forming
process. without blank holder, the deformation inside the surface is distinctly
smaller than the bending outside the surface. Wrinkle is a key factor to determine
whether the part could be formed and become one of primary forming defects
especially for thin metal sheet in MPF.
On the other hand, in this condition, springback was more serious than those
forming with blank holder. Springback was affected by numerous factors, such as
thickness of metal sheet, material property and deformation amount and so on. And
the last content of this paper was to research the springback eleminating method
through controlling the forming surface of the elements group for metal sheet in
multi-point forming process.
1.Numerical simulation of wrinkling in multi-point forming for
metal sheet without blank holder
As view from mechanics, wrinkle is a kind of plastic ?in metal sheet surface.
It is more difficult with static implicit algorithm than dynamic explicit in analyzing
73
吉林大学硕士研究生学位论文
wrinkling problem. Based on the explicit algorithm, there must be offered as fine
element as to make the wrinkle to simulate with adequate small wavelength.
Obtained the numerical simulation results of wrinkle with different size of
finite element, establishing the finite element model for wrinkling numerical
analysis. The Multi-point forming process of sphere, saddle surface of different
materials with different thickness and deformations were simulated, and those
results show that metal sheet thickness, deformation and material property have
effect on wrinkle defect. With metal sheet thickness increasing, deformation
diminishing, wrinkle can be gradually weakened. Simultaneously, wrinkle was
affected by material property too. In this paper, two material, 08Al steel and L2Y2
pure aluminum are simulated, and found L2Y2 pure aluminum are more sensitive
to wrinkling than 08AL steel because the Elastic modulus of 08AL is greater than
that of L2Y2.
In this paper, the wrinkling growing up process was also analyzed, and
discovered that wrinkling in multi-point forming for metal sheet without blank
holder, derives from metal sheet under an half-baked restraining which comes from
the elements group. The maximal of wrinkle depth appear in the anaphase of the
forming processing, and some wrinkle can be planished at the end of forming. In
addition, the non-wrinkling limit of sphere and saddle surface were obtained.
2. Numerical simulation of springback in multi-point forming for
metal sheet without blank holder
There were included loading and unloading course in an intact metal sheet
forming. Based on explicit-implicit algorithm, the finite element model for
numerical simulation of springback was established. Dynamic-explicit formulation
was employed to analyze the loading process and implicit formulation to unloading
process. This method not only overcomes the time consuming and convergence
problem in explicit algorithm but also avoids the contact difficult in implicit
algorithm, with go
本思想是将传统的整体模具离散化，由一系列规则排列的基本体（或称冲
头）组成的“柔性多点模具”来代替，由基本体球头的包络曲面来完成板材
成形。多点成形技术作为一种新兴的技术已在飞机、船舶、汽车等诸多制造
领域有着广泛的应用前景。
无压边多点成形通常用于变形量不大的曲面成形，是板材多点成形过程
中使用频率很高的一种成形方法。多点成形在成形变形量不大的曲面时，通
常不压边，这样既可省去切边工序又可以节省材料，另一方面又能简化多点
压机结构，降低其造价。由于没有压边圈，板材面内变形力较小，主要以面
外弯曲变形为主，导致在多点成形中起皱缺陷更容易出现，特别是在薄板多
点成形中，起皱是工艺上必须克服的成形缺陷。
由于没有压边圈的作用，板材成形过程中，回弹对成形件最终形状的影
响要比有压边时大。影响回弹的因素很多，如板材厚度，板材的材质以及成
形件变形量的大小等。本文采用数值模拟技术对多点成形过程的回弹进行了
分析，另外还探讨了典型零件在多点成形时，通过控制基本体群成形面形状
消除回弹的方法。
1. 板材无压边多点成形的起皱数值模拟
采用显式算法进形起皱数值模拟时，板材必须离散成足够小的有限元单
元，使皱纹能够已相当小的波长正确的模拟出来。通过不同尺寸有限元单元
的数值结果比较，建立了板材无压边起皱分析的有限元模型。对双向曲率曲
面—球面和马鞍面进行了详细的数值模拟，研究了板材厚度、曲率半径以及
材质等对起皱的影响。板厚与变形程度是影响起皱的重要因素，增大板材厚
71
吉林大学硕士研究生学位论文
度，减小成形件曲率半径均可有效的抑制起皱的产生；文中对 08AL 和 L2Y2
铝两种不同材料的板材成形球面和马鞍面的起皱现象进行了有限元分析。结
果表明选用具有较大的弹性模量的 08AL较具有较小弹性模量的 L2Y2 不容
易产生起皱缺陷。
通过对多点成形过程中的起皱产生过程的分析，发现在无压边多点成形
中，起皱主要产生于基本体对板材的不完全约束，在成形后期起皱出现最大
值，在成形结束时，部分皱纹被压平。基于对多种不同板厚、不同曲率半径
的球面和马鞍面成形件的数值分析，得到了球面与马鞍面无压边、不起皱多
点成形极限图。在本文之前，还没有多点成形不起皱极限图方面的结果。
2. 板材无压边多点成形的回弹数值模拟。
完整的板材成形包括加载和卸载两个过程。本文建立了采用显-隐式算法
模拟回弹问题的有限元模型。即采用动态显式算法模拟板材成形过程，采用
隐式算法模拟卸载回弹过程。这种方法克服了显式算法耗时大，难以收敛的
缺点，解决了隐式算法不能处理好接触问题的缺点，极大的提高了回弹数值
的计算效率，具有实用性和高效性的特点。
主要分析了板材厚度、成形件变形量以及材质等对柱面、球面及马鞍面
的无压边多点成形的回弹影响，并且利用中心插分法，拟合出成形件回弹
前、后曲面形状，计算出回弹量的大小。由数值模拟结果可以看出，在多点
成形中，板材厚度越小，变形量越小，卸载后回弹越大，反之，厚度越大，
变形量越大则回弹越小。
3. 多点成形过程中成形面的修正与回弹控制的研究
利用数值模拟技术，在回弹分析的基础上，给出了基于回弹数值计算的
柱面、球面多点成形基本体群成形面的修正方法。对于目标曲率半径为的圆
柱面，但板厚为 1.0mm时，经过三次修正，形状总体误差只有 0.0044m；板
厚为 1.5mm时，经过五次修正，形状误差达到 0.00083m；对于目标曲率半径
为 1200mm，板厚为 1.5mm时，经过两次修正形状误差达到 0.0962m；依据
最后一次修正的成形面调整基本体群的形状进形多点成形，可以有效的补偿
板回弹影响。实现板材的一次精确成形，提高产品的成形精度。这些结果对
多点成形技术的进一步工程应用具有一定的参考价值和实际意义
Multi-point forming(MPF) is a flexible manufacturing technology for three
dimensional sheet metal forming. In multi-point forming, the conventional solid die
is replaced by “flexible Multi-point die” composed by a series of discrete elements
(or punches). The forming process of sheet metal is implemented by the envelope
surface of punches. Multi-point Forming can be applied extensively in a lot of
fields such as aircraft, stream and navel ships, vehicle, large sculpture and modern
architecture etc.
Sheet metal forming in MPF without blank holder, which is a common
process method, is usually applied in processing those kinds of surface parts which
have not big deformation amount. In this process condition, the original material
consuming and the press cost can be reduced to a low level as well as simplifying
the press structure and leaving out trimming procedure in metal sheet forming
process. without blank holder, the deformation inside the surface is distinctly
smaller than the bending outside the surface. Wrinkle is a key factor to determine
whether the part could be formed and become one of primary forming defects
especially for thin metal sheet in MPF.
On the other hand, in this condition, springback was more serious than those
forming with blank holder. Springback was affected by numerous factors, such as
thickness of metal sheet, material property and deformation amount and so on. And
the last content of this paper was to research the springback eleminating method
through controlling the forming surface of the elements group for metal sheet in
multi-point forming process.
1.Numerical simulation of wrinkling in multi-point forming for
metal sheet without blank holder
As view from mechanics, wrinkle is a kind of plastic ?in metal sheet surface.
It is more difficult with static implicit algorithm than dynamic explicit in analyzing
73
吉林大学硕士研究生学位论文
wrinkling problem. Based on the explicit algorithm, there must be offered as fine
element as to make the wrinkle to simulate with adequate small wavelength.
Obtained the numerical simulation results of wrinkle with different size of
finite element, establishing the finite element model for wrinkling numerical
analysis. The Multi-point forming process of sphere, saddle surface of different
materials with different thickness and deformations were simulated, and those
results show that metal sheet thickness, deformation and material property have
effect on wrinkle defect. With metal sheet thickness increasing, deformation
diminishing, wrinkle can be gradually weakened. Simultaneously, wrinkle was
affected by material property too. In this paper, two material, 08Al steel and L2Y2
pure aluminum are simulated, and found L2Y2 pure aluminum are more sensitive
to wrinkling than 08AL steel because the Elastic modulus of 08AL is greater than
that of L2Y2.
In this paper, the wrinkling growing up process was also analyzed, and
discovered that wrinkling in multi-point forming for metal sheet without blank
holder, derives from metal sheet under an half-baked restraining which comes from
the elements group. The maximal of wrinkle depth appear in the anaphase of the
forming processing, and some wrinkle can be planished at the end of forming. In
addition, the non-wrinkling limit of sphere and saddle surface were obtained.
2. Numerical simulation of springback in multi-point forming for
metal sheet without blank holder
There were included loading and unloading course in an intact metal sheet
forming. Based on explicit-implicit algorithm, the finite element model for
numerical simulation of springback was established. Dynamic-explicit formulation
was employed to analyze the loading process and implicit formulation to unloading
process. This method not only overcomes the time consuming and convergence
problem in explicit algorithm but also avoids the contact difficult in implicit
algorithm, with go
引文
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44. K.Kassem, F.Fuchs, M.Hillmann et al. Mathematical modeling of the sheet metal forming
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the static and dynamic response of three-dimensional solids. Uiversity of California,
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transverse shear deformation. Int.J.Mech.Sci., 1991, 33: 717~728
40. M.Kawka and A.Makinouchi. Finite element simulation of sheet metal forming processes
by processes. NUMIFORM’ 92, A.A.Balkemma, Rotterdam 1992, 491~496
41. M.Kawka and A.Makinouchi. Shell-element formulation in the swtatic explicit FEM code
for the simulation of sheet stamping. Joural of Materials Processing Technology, 1995,
54(1-4): 105~115
42. E.Nakamachi and A.Makinouchi. Development of process simulationm system for
autobody panel forming by FE ansysis and laser atereolithography techniques. 17th
Biennial Congress. IDDRG Congress Proc., 1992, 461~466
43. A.Makinouchi, A.B.D.Santos and E.Nakamachi. Comparison of different contact
description in finite element simulation of 3-Dsheet metal forming processes.
MECAMAT’91,Teodosiu, Raphanel & Sidoroff (eds), Balkema, Rotterdam,1993,
443~440
44. K.Kassem, F.Fuchs, M.Hillmann et al. Mathematical modeling of the sheet metal forming
process with INDEED. Num.Mech.in Industrial Forming processes, ED, Chont, Wood
&Zienkiewicz, 1992, 461~466
45. 李尧臣，金属板料冲压成形的有限元法模拟.力学学报，1995，27（3）：351~364.
46. Tang SC. Computer predictiong of the deformed sharp of a draw blank during the boinder
–wrap stge. J.Appl.Metal working.1980, (3): 695~712
47. 董相怀.轴对称其三维金属板料成形的有限元模拟. 武汉：华中理工大学，1992.
48. 柳玉起.板料成形塑性流动规律及起皱破裂回弹的数值模拟研究.长春：吉林工业大
学，1995:34~39
49. Du CO, et al.. Springback predition in sheet forming simulation. SAE Trans.Section5,
940937: 707~717
50. Kawka M.Makinouchi A. Ansysis of multroperation automotive sheet metal forming
processes. Advanceed Tech.of Plasticity. Proc.4th ITP, Beijing 1993, (ICTP’93),
1811~1816
64
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51. Toshihiko Kuwabara, Susumu Takahashi, Yohsuke Miyashita. 2-D Springback Analysis
for Stretch-Bending Processes Based on Total Strain Theory. SAE International Congress
and Exposition, Detroit, Michigan, 1995: 1~10
52. 宋黎，杨坚，黄天泽，板料弯曲成形的回弹分析与工程控制综述，锻压技术，
1996.1:18~22
53. N.He, R.HWayoner, Springback simulation in sheet metal forming.
Nmuisheet’96
54. N.Montmayeur.C.Staub. Springback prediction with Optris Numisheet’99,
France, 1999
55. Mattiasson. Kjell. Numerical results from large deflection beam and frame problems
anlysed by means of elliptic integrals. Mathematical Techematical Techniques -Numerical
Numerical Analysis, 1981, vol. 17：145~153
56. Luc Papeleux, Lean-Philippe Ponthot. Finite element simulation of springback in sheet
metal forming. In: Journal of Material Processing Technology, 2002, 125-126：785~791
57. Volakan Esat, Haluk Darendeliler, Mustafa Ilhan Gokler. Finite element analysis of
springback in bending of aluminiunm sheet. In: Materials and Design, 2002, 23: 223~229
58. K.J. Bathe, R.Slavkovic and M.Kojic. On large strain elastic–plastic and creep analysis.
Finite element methods for nonlinear problems. Europe–US Symposium, Trondheim,
Norway, 1985, 175~190
59. 蔡中义, 李明哲, 李湘吉. 板材成形回弹数值分析的静力隐式方法. 中国机械工程,
2002,13(17): 1458~1461
60. 李明哲, 李淑慧, 陈建军等. 板材多点成形不同工艺方法的数值模拟研究, 中国机械工
程, 2000, 11(9): 1042~1046
61. 张立力、戴映荣、齐恬. 数值模拟参数和工艺参数对板材成形回弹影响的研究. 锻压
技术，2002 Vol.27 No.6:65~72
62. Narkeeran Narasimhan, Michael Lovell. Predicting springback in sheet metal forming an
explicit implicit sequential solution procedure. Finite Elements in Analysis and Design
1999,33: 29~42
63. W.D. Cardena, L.M. Genga, D.K. Matlockb. Measurement of springback. International
Journal of Mechanical Sciences, 2002, 44: 79~101
64. 李广权，板材无模多点成形过程数值模拟研究，吉林工业大学博士学位论文，1997
65. 李广权，李明哲，苏世忠，李淑慧，刘纯国.板材多点成形时摩擦边界条件的处理
方法，农业机械学报，1999，30(2)：99~103
66. 李广权，李达，李东平，李明哲.板材多点成形时复杂边界条件的研究. 哈尔滨工
业大学学报，2000，32(5):75~77
67. 李明哲，李广权，苏世忠，付沛福.板材多点成形时非连续性边界条件的处理. 农
业机械学报，1997，28：85~89
68. 蔡中义. 板材多点成形过程数值模拟与最优成形路径研究. 吉林大学博士学位论
文，2000
65
吉林大学硕士研究生学位论文
69. 李明哲，蔡中义，刘纯国.板材多点反复成形的残余应力分析. 机械工程学报，
2000，36(1)：50~54
70. 蔡中义，李明哲，陈建军.板材多点成形隐式算法数值模拟专用软件及关键技术.哈
尔滨工业大学学报，2000，32（4）：82~88
71. 蔡中义，李明哲，李广权.金属成形数值模拟中的接触单元法.中国机械工程，
2000，11(8)：933~935
72. 蔡中义，李明哲，陈建军.板材多点成形隐式算法数值模拟专用软件及关键技术，
哈尔滨工业大学学报，2000，32（4）：82~88
73. Cai Zhongyi, Li Mingzhe, Feng Zhaohua. Theory and method of optimum path forming
for sheet metal. Chinese Journal of Aeronautics，2001，14(2)：118~122
74. 苏世忠.基于有限元多晶体弹塑模型模拟板材无模多点成形过程的研究.吉林大学博
士学位论文，2000
75. 苏世忠，李明哲，陈建军.有限元多晶体模型模拟铝板多点成形过程.哈尔滨工业大
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