[Back to Results | New Search]

Student Number 89343001 Author Ming-Chang Wu(吳明昌) Author's Email Address s9343001@cc.ncu.edu.tw Statistics This thesis had been viewed 2053 times. Download 686 times. Department Mechanical Engineering Year 2005 Semester 2 Degree Ph.D. Type of Document Doctoral Dissertation Language zh-TW.Big5 Chinese Title Variational Upper-Bound for Upset Forging of Rings and Its Comparison with FEM Solution Date of Defense 2006-06-30 Page Count 210 Keyword natural boundary condition neutral surface upset forging of ring variational upper-bound method Abstract ABSTRACT

The objective of this thesis is to verify the validity of the variational upper-bound(VUB) method in analysis of upset forging of rings. The effect of natural boundary conditions and geometrical shape of the neutral surface on the upper-bound solution would be investigated in this project as well. As for the upper-bound method on the theoretical analysis of upset forging of rings, the most research interests concentrate on the kinematically admissible velocity field for a lower upper-bound solution of energy dissipation. The analytical result could provide some important reference for the metal forming process. However, a large gap exists between the theoretical prediction on deformation behavior and the experimental results. Therefore, in this research work, a variational upper-bound (VUB) method is used. It is the method that determines an upper-bound solution using variational calculus. Consequently, in addition to the kinematically boundary condition, a set of natural boundary conditions (NBCs) can be derived theoretically and can be applied to approximate the solution. These NBCs were found to affect the upper-bound solution of energy dissipation as well as the pattern of metal deformation significantly.

The other objective of this thesis is to investigate the effect of the geometrical shape of the neutral surface, since the results of experiment and FEM has shown that the neutral surface of the upset forging of ring is not a cylindrical shape as was assumed in the upper-bound solution. For this purpose, the radius of the neutral surface, which was assumed as a constant, will be assumed as a function of higher order in analysis. It is expected that the solution determined in this manner should be improved.

In order to verify the validity and applicability of the present method, the result obtained in this research proposal will be compared and discussed with FEM solution, and also experimentally verified by the bulged profiles of upset disks and rings, and the calibration curves which indicate the relation between the decrease in minimum internal diameter of upset rings and the reduction under different interfacial friction conditions. The result proves that while considering the neutral surface as a function form and satisfying the natural boundary conditions, not only the calibration curves and bulged profiles in upset ring will be influenced, but also the forming energy will be improved.Table of Content 目 錄

頁次

摘要I

誌謝V

目錄VI

圖表說明X

符號索引XVI

第一章 緒論1

1-1 前言1

1-2 文獻回顧2

1-3 研究動機及方法16

1-4 論文架構20

第二章 基本理論介紹21

2-1 上界限法原理21

2-2 流函數理論23

2-3 上界限流函數法27

第三章 圓環鍛粗加工之變分上界限解析30

3-1 問題定義30

3-2 基本假設31

3-3 流函數場及速度場的性質31

3-4 上界限成形功率消耗33

3-4-1 塑性變形區的功率消耗33

3-4-2 摩擦界面的功率消耗34

3-4-3 圓環鍛粗加工總功率消耗35

3-5 平衡方程式及自然邊界條件36

3-6 變分上界限解41

第四章 結果與討論43

4-1 流函數模式對解的影響及其解的驗證44

4-1-1 流函數的模式44

4-1-2 速度場的建立44

4-1-3 應變率場的建立45

4-1-4 上界限解46

4-1-5 實驗驗證47

(一) 校正曲線的實驗驗證(Calibration curve)47

(二) 圓盤桶脹輪廓的實驗驗證48

(三) 圓環桶脹輪廓的實驗驗證50

4-1-6 理論計值的討論51

(一) 速度場的比較51

(二) 等效塑性應變場的比較51

(三) 摩擦條件與半高度縮減率對變形場的影響52

(四) 摩擦條件與半高度縮減率對成形能量的影響53

4-2 自然邊界條件對圓環鍛粗的影響54

4-2-1 實驗驗證59

(一) 校正曲線的實驗驗證(Calibration curve)59

(二) 圓盤桶脹輪廓及成形負荷的實驗驗證60

(三) 圓環桶脹輪廓的實驗驗證62

4-3 非線性模式之中立面對圓環鍛粗的效應62

4-3-1 流函數的模式63

4-3-2 速度場的建立63

4-3-3 應變率場的建立64

4-3-4 上界限解66

4-3-5 實驗驗證68

(一) 校正曲線的實驗驗證(Calibration curve)68

(二) 圓盤桶脹輪廓的實驗驗證70

(三) 圓環桶脹輪廓的實驗驗證71

4-3-6 理論計值的討論72

(一) 速度場的比較72

(二) 等效塑性應變場的比較73

(三) 摩擦條件與半高度縮減率對變形場的影響73

(四) 摩擦條件與半高度縮減率對成形能量的影響74

第五章 結論與建議76

5-1 結論76

5-2 建議79

參考文獻122

附錄Ａ 變分的推導(對流函數場)138

附錄B 自然邊界條件的轉換158

附錄C 變分的推導(對速度場)163

附錄D 有限元素軟體MARC簡介176Reference 參考文獻

[1] Male, A. T. and Cockcroft, M. G., "A Method for the Determination of the Coefficient of Friction of Metals Under Conditions of Bulk Plastic Deformation", J. of the Inst. of Metals, Vol. 93, (1965), pp. 38-46.

[2] Saida, Y., Lee, C. H. and Kobayashi, S., "Some Aspects of Friction in Forging Problems", ⅡInter-American Conference on Mater. Technol., Mexico City, (1970), pp. 308.

[3] Green, A. P., "Calculation on the Theory of Sheet Drawing", British Iron and Research Association, Report MW/B/7/52, April, (1952).

[4] Avitzur, B., "Analysis of Wire Drawing and Extrusion through Conical Dies of Small Cone Angle", Journal of Engineering for Industry, Trans. ASME, Vol. 85, (1963), pp. 89-96.

[5] Stepanskii, L. G., "The Boundaries of the Area of Plastic Deformation in Extrusion", Russian Engineering Journal, Vol. 43, (1963), pp. 40-42.

[6] Kudo, H., "Upper-Bound Approach to Plane-Strain Forging and Extrusion-I", International Journal of Mechanical Sciences, Vol. 1, (1960), pp. 57-83.

[7] Kudo, H., "Upper-Bound Approach to Plane-Strain Forging and Extrusion-II", International Journal of Mechanical Sciences, Vol. 1, (1960), pp. 229-252.

[8] Kudo, H., "Upper-Bound Approach to Plane-Strain Forging and Extrusion-III", International Journal of Mechanical Sciences, Vol. 1, (1960), pp. 366-368.

[9] Kudo, H., "Some Analytical and Experimental Studies of Axisymmetric Cold Forging and Extrusion - I", International Journal of Mechanical Sciences, Vol. 2, (1960), pp. 102-127.

[10] Kudo, H., "Some Analytical and Experimental Studies of Axisymmetric Cold Forging and Extrusion - II", International Journal of Mechanical Sciences, Vol. 3, (1961), pp. 91-117.

[11] Kudo, H., "Some Analytical and Experimental Studies of Axi-Symmetric Cold Forging and Extrusion -Ⅰ", International Journal of Mechanical Science Pergamon Press, Vol. 2, (1960), pp. 102-127.

[12] Baraya, G. L. and Johnson, W., "Flat Bar Forging", Proc. of 5th Int. M.T.D.T. Conference, Birmingham, (1964), pp. 449-469.

[13] Kobayashi, S., "Upper-Bound Solution of Axisymmetry", Journal of Engineering for Industry, Trans. ASME, (1964), pp. 122-133.

[14] Kobayashi, S., "Upper-Bound Solutions of Axisymmetric Forming Problems - II", Journal of Engineering for Industry, Trans. ASME, Vol. 86, (1964), pp. 326-332.

[15] Kobayashi, S. and Thomsen, E. G., "Upper and Lower Bound Solutions to Axisymmetric Compression and Extrusion Problems", International Journal of Mechanical Sciences, Vol. 7, (1965), pp. 127.

[16] Pan, J., Pachla, W., Rosenberry, S. and Avitzur, B., "The Study of Distorted Grid Patterns for Flow through Conical Converging Dies by the Multi-triangular Velocity Field", Journal of Engineering for Industry, Trans. ASME, Vol. 106, (1984), pp. 150-160.

[17] Avitzur, B., "Strain-Hardening and Strain-Rate Effects in Plastic Flow Through Conical Converging Dies", Journal of Engineering for Industry, Trans. ASME, Vol. 89, (1967), pp. 556-562.

[18] Osakada, K., Limb, M. and Mellor, P. B., "Hydrostatic Extrusion of Composite Rods with Hard Cores", International Journal of Mechanical Sciences, Vol. 15, (1973), pp. 291-307.

[19] Avitzur, B., Wu, R., Talbert, S. and Chou, Y. T., "Criterion for the Prevention of Core Fracture during Extrusion of Bimetal Rods", Journal of Engineering for Industry, Trans. ASME, Vol. 104, (1982), pp. 293-304.

[20] Hill, R., "A General Method of Analysis for Metal Working Processes", Journal of Mechanics and Physics of Solids, Vol. 11, (1963), pp. 306-326.

[21] Avitzur, B., "Forging of Hollow Discs", Israel Journal of Technology, Vol. 2, (1964), pp. 295-304.

[22] Tarnovskii, I. Ya., Pozdeyev, A. A., and Lyashkov, V. B., "Deformation of Metals During Rolling", Pergamon Press, New-York, (1965).

[23] Avitzur, B., "Metal Forming : Process and Analysis", McGraw-Hill, New York, (1968).

[24] Male, A. T. and Depierre, V., "The Validity of Mathematical Solutions for Determining Friction from the Ring Compression Test", Journal of Lubrication Technology, (1970), pp. 389-397.

[25] Lee, C. H. and Altan, T., "Influence of Flow Stress and Friction upon Metal Flow in Upset Forging of Rings and Cylinders", Journal of Engineering for Industry, Trans. ASME, Vo1. 94, (1972), pp. 775-782.

[26] Liu, J. Y., "An Analysis of Deformation Characteristics and Interfacial Friction Conditions in Simple Upsetting of Rings", Journal of Engineering for Industry, Trans. ASME, (1972), pp. 1149-1156.

[27] Juneja, B. L., "Forging of Polygonal Discs", International Journal of Machine Tool Design and Research, Vol.13, (1973), pp. 17-28.

[28] Juneja, B. L., "Forging of Polygonal Discs with Barrelling". International Journal of Machine Tool Design and Research, Vol.13, (1973), pp. 87-93.

[29] Juneja, B. L., "Forging of Rectangular Plates", International Journal of Machine Tool Design and Research, Vol.13, (1973), pp. 141-153.

[30] Avitzur, B. and Sauerwine, F. "Limit Analysis of Hollow Disk Forging; part 1: Upper Bound," Journal of Engineering for Industry, Trans. ASME, (1978), pp. 340-346.

[31] Avitzur, B. and van Tyne, C. J., "Ring Forming : An Approach. Part 1 : Flow Pattern and Calculation of Power", Journal of Engineering for Industry, Trans. ASME, 104, (1982), pp. 231-237.

[32] Avitzur, B. and van Tyne, C. J., "Ring Forming : An Approach. Part 2 : Process Analysis and Characteristics", Journal of Engineering for Industry, Trans. ASME, 104, (1982), pp. 238-247.

[33] Avitzur, B. and van Tyne, C. J., "Ring Forming : An Approach. Part 3 : Constrained Forging and Deep Drawing Applications", Journal of Engineering for Industry, Trans. ASME, 104, (1982), pp. 248-252.

[34] Yang, D. Y. and Kin, J. H., "An Analysis for three-dimensional Upset forging of Elliptical Disk", International Journal of Machine Tool Design and Research, (1986), pp. 147-156.

[35] Kim, J. H. and Yang, D. Y., "An Analysis of Upset Forging of Square Blocks Considering The Three-Dimensional Bulging of Sides", International Journal of Machine Tool Design and Research, Vol.25, (1986), pp. 327-336.

[36] Kim, J. H., Yang, D. Y. and Kim, M. U., "An Analysis of Three-Dimensional Upset Forging of Arbitrarily-Shaped Prismatic Blocks", International Journal of Machine Tool Design and Research, Vol.27, (1987), pp. 311-323.

[37] Yang, D. Y. and Kim, J. H., "An Analysis of Three-Dimensional Upset Forging of Regular Polygonal Blocks by Using the Upper Bound Method", Journal of Engineering for Industry, Trans. ASME, Vol.109, (1987), pp. 155-160.

[38] Sagar, R. and Juneja, B. L., "Open Die Forging of a Four Sided Irregular Disc", International Journal of Machine Tools & Manufacture, Vol. 31, (1991), pp. 315-328.

[39] Hsiang, S. H. and Huang, T. F., "Analysis of Deformation Behaviours in Upsetting Processes", Proceedings of the 8th National Conference on Mechanical Engineering The Chinese Society of Mechanical Engineers, (1991), pp. 887-894.

[40]Moon, Y. H., Van Tyne, C. J. and Gordon, W. A., "An upper bound analysis of a process-induced side-surface defect in forgings Part 1: The velocity fields and power terms", Journal of Materials Processing Technology, Vol. 99, (2000), pp. 169-178.

[41]Moon, Y. H., Van Tyne, C. J. and Gordon, W. A., "An upper bound analysis of a process-induced side-surface defect in forgings Part 2: Characteristics and criteria curves", Journal of Materials Processing Technology, Vol. 99, (2000), pp. 179-184.

[42]Moon, Y. H. and Van Tyne, C. J., "Validation via FEM and Plasticine Modeling of Upper Bound Criteria of a Process-Induced Side-Surface Defect in Forgings", Journal of Materials Processing Technology, (2000), pp. 185-196.

[43] Alexandrov, S., "An Analysis of the Axisymmetric Compression of Viscous Materials", Journal of Materials Processing Technology, Vol. 105, (2000), pp. 278-283.

[44] Wang, J. P., "A New Evaluation to Friction Analysis for the Ring Test", International Journal of Machine Tools & Manufacture, (2001), pp. 311-324.

[45] Stepanskii, L. G., "The Boundaries of the Area of Plastic Deformation in Extrusion", Rassian Engr. J., issue 9, (1963), pp. 40-42.

[46] Kobayashi, S., "Upper-Bound Solutions of Axisymmetric Forming Problems - I", Journal of Engineering for Industry, Trans. ASME, Vol. 86, (1964), pp. 122-126.

[47] Zimerman, Z. and Avitzur, B., "Metal Flow through Conical Converging Dies - A Lower Upper Bound Approach Using Generlized Boundaries of the Plastic Zone", Journal of Engineering for Industry, Trans. ASME, Vol. 92, (1970), pp. 119-129.

[48] Tirosh, J., "On the Dead-Zone Formation in Plastic Axially-Symmetric Converging Flow", Journal of Mechanics and Physics of Solids, Vol. 19, (1971), pp. 39-47.

[49] Wu, S. C. and Li, M. Q., "A Study of Cup-Cup Axisymmetric Combined Extrusion by the Upper-Bound Approach, I. Upper-Bound Solutions for the Deformation Force", J. of Mech. Work Technol., Vol. 18, (1989), pp. 63-84.

[50] Monaghan, J. and Peard, M., "Closed-die cold forging: an upper-bound analysis", Journal of Materials Processing Technology, Vol. 26, (1991), pp. 181-195.

[51] Alcaraz, J. L. and Sevillano, J. G., "Safety maps in bimetallic extrusion ", Journal of Materials Processing Technology, Vol. 60, (1996), pp. 133-140.

[52] Wu, C. W. and Hsu, R. Q., "A universal velocity field for the extrusion of non-axisymmetric rods with non-uniform velocity distribution in the extrusion direction", Journal of Materials Processing Technology, Vol. 97, (2000), pp. 180-185.

[53] Wu, C. W. and Hsu, R. Q., "Theoretical analysis of extrusion of rectangular hexagonal and octagonal composite clad rods", International Journal of Mechanical Sciences, Vol. 42, (2000), pp. 473-486.

[54] Chitkara, N. R. and Aleem, A., "Axi-symmetric tube extrusion/piercing using die-mandrel combination: some experiments and a generalized upper bound analysis", International Journal of Mechanical Sciences, Vol. 43, (2001), pp. 1685-1709.

[55] Chitkara, N. R. and Aleem, A., "Extrusion of axi-symmetric bi-metallic tubes form solid circular billets: application of a generalized upper bound analysis and some experiments", International Journal of Mechanical Sciences, Vol. 43, (2001), pp. 2833-2856.

[56] Chin, T. K., "An analysis of the closed-die forging of a general non-axisymmetric shape by the upper-bound elemental technique", Journal of Materials Processing Technology, Vol. 123, (2002), pp. 197-202.

[57] Chin, T. K., "A concave circular parallelepiped element and its application to three-dimensional closed-die forging", Journal of Materials Processing Technology, Vol. 123, (2002), pp. 203-208.

[58] Manesh, H. D. and Taheri, A. K., "An investigation of deformation behavior and bonding strength of bimetal strip during rolling", Mechanics of Material, Vol. 37, (2005), pp. 531-542.

[59] Shabaik, A., Kobayashi, S. and Thomsen, E. G., "Application of Potential Flow Theory to Plane Strain Extrusion", Journal of Engineering for Industry, Trans. ASME, (1967), pp. 503-512.

[60] Nagpal, V., "General Kinematically Admissible Velocity Fields for Some Axisymmetric Metal Forming Problems", Journal of Engineering for Industry, Trans. ASME, (1974), pp. 1197-1201.

[61] Liu, J.Y., "Upper Bound Solutions of Some Axisymmetric Cold Forging Problems", Journal of Engineering for Industry, Trans. ASME, (1971), pp. 1134.

[62] Nagpal, V. and Clough, W.R., "Plane Strain Forging-A Lower Upper Bound Approach", Journal of Engineering for Industry, Trans. ASME, (1975), pp119-124.

[63] Nagpal, V., "On the Solution of Three-Dimensional Metal-Forming Process", Journal of Engineering for Industry, Trans. ASME, (1977), pp. 624-629.

[64] Nagpal, V., Lahoti, G. D. and Altan, T., "A Numerical Method For Simutaneous Prediction of Metal Flow and Temperatures in Upset Forging of Rings", Journal of Engineering for Industry, Trans. ASME, (1978), pp. 413-420.

[65] Shabaik, A. H. and Thomsen, E. G., "A Theoretical Method for Analysis of Metal-Working problems", Journal of Engineering for Industry, Trans. ASME, Vol. 90, (1968), pp. 343-352.

[66] Chen, P. C. T. and Ling, F. F., "Upper-Bound Solutions to Axisymmetric Extrusion Problems", International Journal of Mechanical Sciences, Vol. 10, (1968), pp. 863-879.

[67] Chen, P. C. T., "Upper-Bound Solution to Plane-Strain Extrusion Problems", Journal of Engineering for Industry, Trans. ASME, Vol. 92, (1970), pp. 158-164.

[68] Chang, K. T. and Choi, J. C., "Upper-Bound Solutions to Tube Extrusion Problems through Curved Die", Journal of Engineering for Industry, Trans. ASME, Vol. 94, (1972), pp. 1108-1112.

[69] Cho, N. S. and Yang, D. Y., "Hydrofilm Extrusion of Tubes through Optimized Curved Dies", Journal of Engineering for Industry, Trans. ASME, Vol. 105, (1983), pp. 243-250.

[70] Gunasekera, J. S. and Heshino, S., "Analysis of Extrusion of Polygonal Sections through Streamlined Dies", Journal of Engineering for Industry, Trans. ASME, Vol. 107, (1985), pp. 229-233.

[71] Yang, D. Y., Han, C. H. and Lee, B. C., "The Use of Generalized Deformation Boundaries for the Analysis of Axisymmetric Extrusion through Curved Dies", International Journal of Mechanical Sciences, Vol. 27, (1985), pp. 653-663.

[72] Yang, D. Y. and Han, C. H., "A New Formulation of Generalized Velocity Field for Axisymmetric Forward Extrusion through Arbitrarily Curved Dies", Journal of Engineering for Industry, Trans. ASME, Vol. 109, (1987), pp. 161-168.

[73] Yang, D. Y., Kim, Y. G. and Lee, C. M., "An Upper-Bound Solution for Axisymmetric extrusion of Composite Rods through Curved Dies", International Journal of Mechanical Sciences, Vol. 33, (1991), pp. 565-575.

[74] Tresca, H., "On the Flow of Solid Bodies Subjected to High Pressures", C. R. Acad. Sci., Paris, 59, (1864), pp. 754.; Other closely connected work by Tresca is reported in C. R. Acad. Sci., 64, (1867), pp. 809; Mem. Sav. Acad. Sci., Paris, 18, (1868), pp. 733 and 20, (1872), pp. 75.

[75] Hill, R. and Tupper, S. J., "A New Theory of Plastic Deformation in Wire Drawing", J. Iron and Steel Inst., Vol. 159, (1948), pp. 353.

[76] Yang, D. Y. and Lee, C. H., "Analysis of Three-Dimentional Extrusion of Sections through Curved Dies by Conformal Transformation", International Journal of Mechanical Sciences, Vol. 20, (1978), pp. 541-552.

[77] Yang, D. Y., Kim, M. U. and Lee, C. H., "An analysis for Extrusion of Helical Shapes from Round Billets", International Journal of Mechanical Sciences, Vol. 20, (1978), pp. 695-705.

[78] Wang, J. P. and Lin, Y. T., "The load analysis of the plane-strain forging processes using the upper-bound stream-function elemental technique", Journal of Materials Processing Technology, Vol. 47, (1995), pp. 345-359.

[79] Wang, J. P. and Lin, Y. T., "The UBST Approach to the Stress Analysis of Plane-Strain Upsetting", International Journal of Machine Tools & Manufacture, Vol. 35, (1995), pp. 607-618.

[80] Wang, J. P., "The UBST approach to the stress analysis of plane-strain upsetting with a newly constructed model of the slip-line field", Journal of Materials Processing Technology, Vol. 58, (1996), pp. 267-273.

[81] Lin, Y. T. and Wang, J. P., "A New Upper-Bound Elemental Technique Approach", Computers & Structures, Vo1. 4, (1997), pp. 601-611.

[82] Yeh, W. C. and Yang, Y. S., "A Variational Upper-Bound Method for Plane Strain Problems", Journal of Manufacturing Science and Technology, Trans. ASME, Vol. 118, (1996), pp. 301-309.

[83] Yang, Y. S. and Yeh, W. C., "A Variational Upper-Bound Method for Analysis of Plane Strain Extrusion Test", Journal of Chinese Society of Mechanical Engineering, Vol. 18, (1997), pp. 1-14.

[84] Yang, Y. S. and Yeh, W. C., "Experimental Verification of the VUB method Using Plane Strain Extrusion Tests for 6061 Aluminum Alloy", Journal of Materials Processing & Manufacturing Science, Vol. 5, (1997), pp. 267-282.

[85] Yeh, W.C. and Wu, M.C., "A Variational Upper-Bound Method for Analysis of Upset Forging of Rings", Journal of Materials Processing Technology, Vol. 170, (2005), pp. 392-402.

[86] MARC Analysis Research Corporation, (2001).

[87] Fung, Y. C., "Foundations of Solid Mechanics", Prentice-Hall, Englewood Cliffs, New Jersey, (1965), pp. 285.

[88] Prager, W. and Hodge, P. G., "Theory of Perfectly Plastic Solids", John Willey & sons, New York, (1951), pp. 237.

[89] Avitzur, B., "Metal Forming, The Application of Limit Analysis", Marcel Dekker, (1980), pp. 31-32.

[90] Thomsen, E. G., Yang, C. T. and Kobayashi, S., "Mechanics of Plastic Deformation in Metal Forming", MacMillan., New York, (1965), pp. 166-167.

[91] "User’s Manual-Fortran Subroutines for Mathematical Application", IMSL, Inc., (1991), pp.1096-1102.

[92] Hosford, W. F. and Caddell, R. M., "Metal Forming-Mechanics and Metallurgy", Prentice-Hall, Inc., (1983), pp. 150-153.

[93] "Mentat Command Refernce", MARC Analysis Research Corporation, Version 3.1, pp. 3-345~3-350.

[94] "Mentat Command Refernce", MARC Analysis Research Corporation, Version 3.1.

[95] "Theory and User Information", MARC Analysis Research Corporation, Volume A, Version K7.

[96] 張薰圭, "硬度試驗," 金屬熱處理, 第三期, (1980), pp. 54.Advisor Wei-Ching Yeh(葉維磬)

Files approve immediately

89343001.pdf Date of Submission 2006-07-12

Our service phone is (03)422-7151 Ext. 57407,E-mail is also welcomed.