臺灣博碩士論文加值系統

本文研究之目的，乃在以ULF(updated Lagrangian formulation)的觀念建立一彈塑性大變形有限元素法分析模式，以解析軸對稱條件下金屬管端成形過程中所需之參數對製程的影響，以及考慮非線性加工硬化之均向性與正交異向性材料之變形行為。
本文以rmin法則來精確的計算邊界條件，包含不斷在變形中發生變化的節點切入與分離及摩擦方向變換等問題，還有應變增量與旋轉增量的線性化及變形之彈塑性狀態的轉換問題之處理。本文引用修正之庫侖摩擦法則，適當的描述黏滯與滑動現象之摩擦作用。上述相關之運算與模式可結合至整體剛性矩陣中，進而構成整體的剛性統制方程式。
依上述發展之有限元素模式，首先分析軸對稱金屬管端擴口及縮口成形加工，在計算成形之過程中，可得除荷過程的管端變形形狀與衝頭負荷分佈，並配合相關文獻之實驗結果，探討不同加工製程參數對衝頭負荷與除荷回彈後之最後成形形狀的作用與影響，以期得到正確的衝頭負荷與管端成形形狀。於分析擴口和縮口之成形過程中，加工參數包含衝頭半角、材料性質、摩擦係數、擴口率(或縮口率)、最適衝頭半角、管材厚度與管材平均直徑比值等對衝頭負荷與除荷後回彈作用的影響。
此外，並將選擇減少積分法技巧偶合入所建立之有限元素法程式，並用來分析管端向外卷邊製程，探討各因素如衝頭半角及其入口圓弧半徑、管材厚度與管材平均直徑比值、機械性質、潤滑等對管端向外卷邊成形的影響。並將數值解析的結果與相關文獻之實驗結果比較，以驗證理論之正確性與程式之可信性。由結果得知，當衝頭入口彎曲半徑小於或等於臨界彎曲半徑，則管端形成卷邊成形，反之，則管端形成擴口成形。
本文所建立之分析模式，證明能正確地處理各種複雜變形與接觸之邊界條件之管端成形加工製程，且對有關管端成形現象與問題提供了進一步的了解與認識，於製程改良與模具設計上，將有相當之助益。

An elasto-plastic finite-element computer code based on an updated Lagrangian formulation is developed to analyze the influence of manufacturing parameters in axisymmetric tube end forming processes. The behavior of isotropic and normal anisotropic material is examined in a non-linear work hardening process.
An extended rmin algorithm is utilized to formulate boundary conditions, such as nodal penetration and separation, alteration frictional sliding direction, which varies throughout the entire process of tube end forming, establishing linear strain and rotational increments, and the altered state of elasto-plastic material. A modified Coulomb’s friction law is introduced to describe the alternation state of frictional sliding-sticking at the contact interface. After incorporating the modified Coulomb’s friction matrix and corresponding solution algorithms, the govering stiffness equation used in the tube end forming process is derived.
Based on the developed finite-element code, the tube-flaring and tube-nosing processes are first simulated under axisymmetric condition. Simulation results are compared with experiment data to clarify the process conditions of tube-flaring and tube-nosing operation for the correct punch load of tube end forming and precise final shape of products after unloading. The punch load and the effects of process variables, such as optimum punch-semi angle, friction coefficient, mechanical properties, tube wall thickness to mean diameter of tube, on the final shape of the product after unloading are discussed in detail.
Furthermore, introducing the reduced and selective integration method into the computer code present the deformation mechanics of tube-curling. The effects of variables, such as the half-apex angle of punch and its radius, the ratio of the thickness of the tube to its mean diameter, the mechanical properties and lubrication on the tube’s outward curling are investigated. The results of numerical simulations have exhibited a good consistency with the experimental results. Simulation findings indicate that, if the bending radius at punch inlet is smaller than the critical bending radius, curling is present at the tube end. On the contary, the tube end experiences flaring.
The simulation clearly demonstrates the efficiency of the code in conducting various tube end forming processes that proceed under complicated deformation and contact history. This study has provided a greater understanding of the tube end forming process, which will inevitably improve the manufacturing processes and the designing of tools for such processes.

中文摘要-------------------------------------------------------I
英文摘要------------------------------------------------------Ⅲ
誌 謝---------------------------------------------------------Ⅴ
目 錄---------------------------------------------------------Ⅵ
符號索引------------------------------------------------------Ⅷ
圖表索引-----------------------------------------------------XII
第一章　緒論---------------------------------------------------1
1.1 前言-------------------------------------------------------1
1.2 文獻回顧---------------------------------------------------3
1.3 論文之構成-------------------------------------------------8
第二章　基本理論----------------------------------------------10
2.1 有限變形之應變與應變率------------------------------------10
2.2 有限變形之應力與應力率------------------------------------12
2.3 有限變形之Total Lagrangian Formulation--------------------15
2.4 有限變形之Updated Lagrangian Formulation--------------16
2.5 材料之彈塑性構成關係式------------------------------------17
2.6 材料降伏之判定法則----------------------------------------21
2.7 有限元素法之公式------------------------------------------24
2.8 選擇減少積分法 (Selective Reduced Integration)------------30
2.9 穩定矩陣法 (Stabilization Matrix)-------------------------32
第三章　數值分析----------------------------------------------39
3.1 彈塑性界面與接觸節點的處理--------------------------------39
3.2 摩擦的處理------------------------------------------------42
3.3 除荷之設定------------------------------------------------44
3.4 數值分析之流程--------------------------------------------44
第四章　管端擴口與縮口成形加工分析----------------------------52
4.1 管端擴口成形加工------------------------------------------52
4.1.1 數值分析----------------------------------------------- 53
4.1.2 邊界條件----------------------------------------------- 54
4.1.3 結果與討論--------------------------------------------- 55
4.2 管端縮口成形加工------------------------------------------66
4.2.1 數值分析----------------------------------------------- 67
4.2.2 邊界條件----------------------------------------------- 68
4.2.3 結果與討論--------------------------------------------- 69
第五章　管端向外卷邊成形加工分析------------------------------78
5.1 數值分析--------------------------------------------------79
5.2 邊界條件--------------------------------------------------81
5.3 結果與討論------------------------------------------------82
第六章　結論--------------------------------------------------96
6.1 結論------------------------------------------------------96
6.2 未來研究之展望--------------------------------------------99
參考文獻-----------------------------------------------------100
作者簡介-----------------------------------------------------109
授權書-------------------------------------------------------110

參考文獻
1. R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, London (1950).
2. Y. Yamada, N. Yoshimura and T. Sakurai, “Plastic Stress Strain Matrix and its Application for the Solution of Elastic-plastic Problems by the Finite Element Method”, Int. J. Mech. Sci., Vol.10, pp.343-354(1968).
3. O. C. Ziekiewicz, S. Valliappan and I. P. King, “Elasto-Plastic Solutions of Engineering Problems, Initial-Stress, Finite Element Approch”, Int. J. Num. Meth. Enggn. Vol.1, pp.75-100(1969).
4. K. Iwata, K. Osakada and S. Fujino, “Analysis of Hydrostatic Extrusion by the Finite Element Method”, Trans. ASME, J. Engng. Ind., Vol.94, pp.697-703(1972).
5. C. H. Lee and S. Kobayashi, “Elasto-plastic Analysis of Plane Strain and Axisymmetric Flat Punch Indentation by the Finite Element Method”, Int. J. Mech. Sci., Vol.12, pp.349-370 (1970).
6. C. H. Lee, S. Masaki and S. Kobayashi, “Analysis of Ball Indentation”, Int. J. Mech. Sci., Vol.14, pp.417-426 (1972).
7. E. I. Odell, “A Study of Wall Ironing by the Finite Element Technique”, Trans. ASME, J. Engng. Ind., Vol.100, pp.31-36(1978).
8. C. H. Lee and S. Kobayashi, “New Solution to Rigid-Plastic Deformation Problems Using a Matrix Method”, Trans. ASME, J. Engng. Ind., pp.865-873 (1973).
9. J. H. Kim, S. I. Oh and S. Kobayashi, “Analysis of Stretching of Sheet Metals with Hemispherical Punch”, Int. J. Mach. Tool Des. Res., Vol.18, pp.209-226(1978).
10. C. C. Chen and S. Kobayashi, “Rigid-Plastic Finite Element Analysis of Ring Compression”, Applications of Numerical Methods to Forming Processes, ASME, AMD-28, pp.163-174 (1978).
11. I. Pillinger, P. Hartley, C. E. N. Sturgess and G. W. Rowe, “A New Linearized Expression for Strain Increment in Finite Element Analysis of Deformations Involving Finite Rotation”, Int. J. Mech. Sci., Vol.28, No.5, pp.253-262 (1986).
12. H. D. Hibbitt, P. V. Marcal and J. R. Rice, “A Finite Element Formulation for Problems of Large Strain and Large Displacement”, Int. J. Solids Struct., Vol.6, pp.1069-1086(1970).
13. R. M. McMeeking and J. R. Rice, “Finite Element Fromulations for Problems of Large Elastic-plastic Deformation”, Int. J. Solids Struct., Vol.11, pp.601-616(1975).
14. R. Hill, “Some Basic Principles in the Mechanics of Solid Without a Natural Time”, J. Mech. Phys. Solids. Vol.7, pp.208-225(1959).
15. A. Needleman, “Numerical Study of Necking in Circular Cylindrical Bars”, J. Mech. Phys. Solids. Vol.20, pp.111-127(1972).
16. E. H. Lee, R. L. Mallett and W. H. Yang, “Stress and Deformation Analysis of Metal Extrusion Process”, Comput. Meth. Appl, Mech. Engng., Vol.10, pp.339-353(1977).
17. Y. Yamada and T. Hirakawa, “Large Deformation and Instability Analysis in Metal Forming Process”, Applications of Numerical Methods to Forming Processes, ASME, AMD-28, pp.27-38 (1978).
18. Y. Yamada, T. Hirakawa and A. S. Wifi, “Analysis of Large Deformation and Bifurcation in Plasticity Problems by the Finite Element Method, Conference on Finite Elemenet in Nonlinear Solid and Structural Mechanics, Geilo, Norway, pp.393-412 (1977).
19. I. Pillinger, P. Hartley, C. E. N. Sturgess and G. W. Rowe, “An Elasti-Plastic Three Dimensional Finite Element Analysis of the Upsetting of Rectangular Blocks and Experimental Comparison”, Int. J. Mach. Tool Des. Res., Vol.25, No.5, pp.229-243(1985).
20. I. Pillinger, P. Hartley, C. E. N. Sturgess and G. W. Rowe, “A Three Dimensional Finite Element Analysis of the Cold Forging of a Model Aluminum Connecting Rod”, Proc. Inst. Mech. Engng., Vol.199, pp.319-324(1985).
21. Y. M. Huang, Y. H. Lu and J. W. Chan, “An Elasto-Plastic Finite Element and Experimental Study of The Ironing Process”, J. Mater. Process. Technol., Vol.26, pp.53-80(1991).
22. Y. M. Huang and Y. H. Lu, “An Elasto-Plastic Rate-Dependent Finite Element Analysis of the Metal Forming Process”, Computers and Structures, Vol.39, No.6, pp.615-622 (1991).
23. Y. M. Huang, Y. H. Lu and M. C. Chen, “Analyzing the Cold Nosing Process Using Elasto-Plastic and Rigid-Plastic Methods”, J. Master. Process. Technol., Vol.30, pp.351-380(1992).
24. Y. M. Huang, Y. H. Lu and A. Makinouchi, “Elasto-Plastic Finite-Element Analysis of V-shape sheet bending”, J. Mater. Process. Technol., Vol.35, pp.129-150(1992).
25. Y. M. Huang and Y. H. Lu, “An Analysis of The Axisymmetric and Nonaxisymmetric Sheet Stretching by A Hemispherical Punch”, Computers and Structures, Vol.51, No.3, pp.315-324 (1994).
26. Y. M. Huang and C. H. Liu, “The Effects of Strain Rate and Anisotropy upon the Sheet-Stretching Process”, Int. J. Mech. Sci., Vol.36, No.2, pp.105-120 (1994).
27. J. C. Nagtegaal, D. M. Parks and J. R. Rice, “On Numerically Accurate Finite Element Solutions in the Fully Plastic Range”, Comput. Meth. Appl. Mech. Engng., Vol.4, pp.153-177 (1974).
28. E. D. Pugh, E. Hinton and O. C. Zienkiewicz, “A Study of Quadrilateral Plate Bending Elements with Reduced Integration”, Int. J. Num. Meth. Engng., Vol.12, pp.1059-1079 (1978).
29. T. J. R. Hughes, “Generalization of Selective Integration Procedures to Anisotropic and Nonlinear Media”, Int. J. Num. Meth. Engng., Vol.15, pp.1413-1418(1980).
30. T. Belytschko, C. S. Tsay and W. K. Liu, “A Stabilization Matrix for the Bilinear Mindlin Plate Element”, Comput. Methods Appl. Mech. Engng., Vol.29, pp.313-327 (1981).
31. W. K. Liu, J. S.-J. Ong and R. A. Uras, “Finite Element Stabilization Matrices - A Unification Approach”, Computer Methods in Applied Mechanics and Engineering, Vol.53, pp.13-46(1985).
32. Y. H. Lu, C. L. Li and Y. M. Huang, “Strategies for Improving Efficiency on Axisymmetric Sheet Stretching Process”, J ournal of Materials Processing Technology, Vol.63, pp.111-116 (1997).
33. K. J. Bathe and A. Chaudhary, “A Solution Method for Planar and Axisymmetric Contact Problems”, Int. J. Num. Meth. Engng., Vol.21, pp. 65-88(1985).
34. N. Rebelo, J. C. Nagtegaal and H. D. Hibbitt, “Practical Aspects of Modeling Sheet Forming Processes”, Proc. NUMIFORM''89, Conf., pp.26-30 (1989).
35. Y. T. Keum, E. Nakamachi, R. H. Wagoner and J. K. Lee, “Compatible Description of Tool Surfaces and FEM Meshes for Analyzing Sheet Forming Operations”, Int. J. Num. Meth. Engng., Vol.30, pp.1471-1502(1990).
36. A. Makinouchi, Y. Shirataki, S. D. Liu and Y. Nagai, “Generalization of tool-work contact conditions for elasto-plastic analysis of forming process”, Advanced Technology of Plasticity, Vol.3, pp.1161-1166 (1990).
37. A. Makinouchi and S. D. Liu, “Finite element analysis of contact problems at finite elasto-plastic deformation”, Proc. NUMFIFORM''89, Conf., pp.26-30 (1989).
38. Y. Germain, K. Chung and R. H. Wagoner, “A rigid-viscoplastic finite element program for sheet metal forming analysis”, Int. J. Mech. Sci., Vol.31, No.1, pp.1-24 (1989).
39. P. Hartley, C. E. N. Sturgess and G. W. Rowe, “Friction in Finite Elemenet Analysis of Metalforming Process”, Int. J. Mech. Sci., Vol.21, pp.301-311 (1979).
40. M. J. Saran and R. H. Wagoner, “A consistent implicit formulation for nonlinear finite element modeling with contact and friction: part I-theory”, Trans. ASME, J. Appl. Mech., Vol.58, pp.499-506 (1991).
41. J. T. Oden and E. B. Pries, “Nonlocal and nonlinear friction law and variational principles for contact problems in elasticity”, J. Appl. Mech., Vol.50, pp.67-76 (1983).
42. J. H. Cheng and N. Kikuchi, “An Analysis of Metal Forming Processes Using Large Deformation Elastic-Plastic Formulations”, Comput. Meth. Appl. Mech. Engng., Vol.49, pp.71-108(1985).
43. B. Fredriksson, “Finite Element Solution of Surface Nonlinearities in Structural Mechanics with Special Emphasis to Contact and Fracture Mechanics Problems”, Comput. Struct., Vol.6, pp.281-290 (1976).
44. J. C. Nagtegaal and N. Rebelo, “On the Development of a General Purpose Finite Element Program for Analysis of Forming Process”, Int. J. Num. Meth. Engng., Vol.25, pp.113-131(1988).
45. F. P. T. Baaijens, F. E. Veldpans and W. A. M. Brekelmans, “On the numerical simulation of contact problems in forming processes”, Proc. NUMIFORM''86, Conf., pp.85-90 (1986).
46. Y. C. Fung, Foundation of Solid Mechanics, Prentice-Hall, Englewood Cliffs, N. J. (1965).
47. E. H. Lee, “Some Comments on Elastic-Plastic Analysis”, Int. J. Solids Struct, Vol.17, pp.859-872 (1981).
48. E. Hinton and D. R. J. Owe, Finite Element Software for Plates and Shells, Pineridge Press, Swansea, U. K. (1984).
49. O. C. Zienkiewicz, R. L. Taylor and J. M. Too, “Reduced Integration Techniques in General Analysis of Plates and Shells”, Int. J. Num. Meth. Engng., Vol.3, pp.275-290 (1971).
50. T. J. R. Hughes, M. Cohen and M. Haroun, “Reduced and Selective Integration Techniques in the Finite Element Analysis of Plate”, Nucl. Engng. Des., Vol.46, pp.203-222(1978).
51. W. K. Liu, E. S. Law, D. Lam and T. Belytschko, “Resultant-Stress Degenerated-Shell Element”, Computer Methods in Applied Mechanics and Engineering, Vol.55, pp.259-300 (1986).
52. K. Ichinose, and T. Masuda, “Inside-out Inversion Process of Aluminum Tube under Axial Loading by Conical Die”, Journal of Japan Institute of Light Metals, Vol.29, No.11, pp.483-490 (1979).
53. A. Makinouchi, Proc. of the NUMIFORM’86 Conference, Gothenburg, pp. 327-332, (1986).
54. K. Manabe, and H. Nishimura, “Stress and Strain Distributions in Tube Flaring with Conical Punch-Study on Nosing and Flaring of Tubes Ⅵ”, Journal of Japan Society for Technology of Plasticity, Vol.24, No.266, pp.276-282 (1983).
55. K. Manabe and H. Nishimura, “Contact Pressure Distributions in Nosing and Flaring of Tubes with Conical Tool”, Journal of Japan Institute of Light Metals. Vol.34 No.8, pp.439-445 (1984).
56. A. S. Chumadin and I. V. Ershov, “Investigation of the Process of Flaring Conical Blanks”, Soviet Forging and Sheet Metal Stamping Technology, No.2, pp.76-79 (1987).
57. Z. R. Wang, D. Kun and F. Yi, “The Method of the Principal Shear Stess Tracing Line and Its Application in the Flaring and expanding of a Thin-Walled Tube with a Conical Punch”, Journal of Materials Processing Technology, Vol70, pp.220-227 (1997).
58. M. Lu, “Flaring Forming Limit Determined by Instability of Aluminum Tube”, Forging & Stamping Technology, Vol.22, No.5, pp.35-37 (1997).
59. Y. M. Huang and C. L. Li, “Analysis of the Flaring Process of Metal Tube”, NSC 88-2212-E-011-009 (1999).
60. A. Nadai, “Plastic State of Stress in Curved Shells: The Forces Required for Forging of the Nose of High-Explosive Shell”, Forming of Steel Shells, Present at the annual meeting of the ASME, New York, Nov. 23-Dec. 1, (1943).
61. E. T. Onate and W. Prager, “Nosing of Shells”, Techenical Report DA 798/15, Brown University, Providence, R. I., pp.1-8 (1954).
62. A. K. Cruden and J. F. Thompson, “The End Closed of Backward Extruded Cans”, NEL Report No.511, National Engng. Lab., Glasgow, (1972).
63. G. D. Lahoti and T. Altan, “Analysis of Metal Flow in Nosing of Tubular Products”, Proc. 6th NARMC, Manufacturing Engng. Trans., (1978).
64. A. Baba, Y. Tozawa and K. Kawada, “Analysis of deformation process in forming of tube”, Journal of the JSTP. Vol.13, No.132, pp.33-41, 1972.
65. K. Kawada and Y. Tozawa, “Analysis of deformation process in forming of tube for elastic”, work-hardening material, Journal of the JSTP, Vol.16, No.179, pp.1132-1138, 1975.
66. K. Kawada and Y. Tozawa, “effects of mechanical properties of the materials on the forming of tube”, Journal of the JSTP. Vol. 20, No.219, pp.299-306, 1979.
67. K. Manabe and H. Nisimura, “Forming loads and forming limits in conical nosing of tube-study on nosing and flaring of tubes Ⅰ”, Journal of the JSTP, Vol.23, No.255, pp.342-355, 1982.
68. S. Mori, K. Manabe and H. Nisimura, “Effects on forming load of clad tubes in conical nosing and flaring”, Journal of the JSTP, Vol.37, No.420, pp.99-104, 1996.
69. K. Manabe and H. Nisimura, “Forming loads in tube-flaring with conical punch-study on nosing and flaring of tubes Ⅴ”, Journal of the JSTP, Vol.24, No.264, pp.47-52, 1983.
70. Y. M. Huang, Y. H. Lu and M. C. Chen, “Analyzing the cold nosing process using elasto-plastic and rigid-plastic methods”, Journal of Materials Processing Technology, Vol.30, pp.351-380, 1992.
71. K. Kitazawa and M. Kobayashi, “Inward curling of circular tubes by conical dies”, Journal of JSTP, Vol.28, No.316, pp.481-487, 1987-5.
72. K. Kitazawa and M. Kobayashi, “Experimental study of deformation mechanism in tube-end curling-curling of shells Ⅱ”, Journal of JSTP, Vol.28, No.323, pp.1267-1274, 1987-12.
73. K. Kitazawa, M. Kobayashi and S. Yamashita, “Elementary energy analysis of tube-end curling-theoretical analysis of tube and conical shell end curling Ⅰ”, Vol.29, No.331, pp.845-850, 1988-8.
74. K. Kitazawa, “Criteria for outward curling of tubes” , Trans. ASME, J. Engng. Ing., Vol.115, pp.466-471, 1993.
75. K. Kitazawa, “Improvement in Flaring Limit of Thin-Walled Circular Tubes Using Precurling Method”, Transactions of the Japan Socitey of Mechanical Engineers, Part C Vol.62, No.594, pp.773-778(1996).
76. A. El-Domiaty, “Curling of Thin Tubes: Analytical and Expermental Study”, Journal of Materials Engineering and Performance, Vol.6, No.4, pp.481-495(1997).