在這篇研究論文裡，我們將對在液體和熱影響區域提出三維溫度場解析解，並且預測受到高能量密度移動電子束所產生的焊接凹洞附近三維熔區的形狀。探討熔區形狀是很基本且重要的，以便幫助我們去了解接合點的性質和其微結構。在本研究裡，我們利用在有限厚度工件材料上受到高斯分佈入射通量的旋轉拋物面來描述焊接凹洞的形狀。導入三維溫度場的解析解，我們可以找到熔區無因次化寬度，前緣和後緣，以及深度的數學表示式，並發現它們為控制單位深度電子束能量，工件材料表面位置和焊接凹洞形狀的無因次化參數之函數。我們可以利用在焊接凹洞底部的力平衡條件來求得控制凹洞形狀的無因次化參數。研究結果顯示了各焊接參數，如無因次電子束能量，Peclet數，凹洞開口半徑，Biot數，工件材料厚度和對流模擬參數對熔區形狀以及凹洞表面溫度分佈的影響。在線源解和本研究上對熔區形狀的預測有著顯著的差異，這是由於三維熱傳所造成的強烈影響。最後，我們可以發現本研究的理論預測和實驗數據有很好的一致性。

Analytical three-dimensional temperature field in the liquid and heat-affected zones and prediction of the three-dimensional fusion zone shape around the keyhole produced by a moving high-intensity beam are provided. Determination of the fusion zone
shapes is of fundamental and practical importance to understand properties and microstructures of joints. In this work, the keyhole is idealized by a paraboloid of revolution in a finite workpiece subject to an incident flux of a Gaussian distribution.Introducing analytical solutions of three-dimensional analytical temperature field, the dimensionless width, leading and rear
edges, and depth of the fusion zone are analytically found to be a function of the dimensionless parameters governing beam power per unit penetration, location of the workpiece surface and shape of the keyhole. The dimensionless parameters governing the keyhole shape can be evaluated from a force balance at the keyhole base. The results show the effects of welding parameters, such as the dimensionless beam power, Peclet number, cavity opening radius, Biot number, thickness of workpiece, and the parameter approximating convection, on the shape of the fusion zone and the temperature of keyhole surface. A significant difference in the fusion zone shapes predicted between the line-source solution and this work indicates the strong effects of three-dimensional heat transfer. Agreement between the prediction from this work and available
experimental data is achieved.

目錄
頁次
謝誌 Ⅰ
中文摘要 Ⅱ
英文摘要 Ⅲ
目錄 Ⅳ
圖目錄 Ⅵ
符號說明 Ⅷ

第一章 序論 1
1.1 前言與文獻回顧 1
1.2 研究動機 6
1.3 研究目的 7

第二章 數學模型之假設與分析 8
2.1 理論模型與假設 8
2.2 溫度場解析解 12
2.3 溫度場近似解 15
2.4 熔區寬度 16
2.5 熔區後緣與前緣 17
2.6熔區深度 18

第三章 結果與討論 19

第四章 結論 23

圖 24

附錄 43

參考文獻 49

1.Kou, S.,1987, Welding Metallurgy, Wiley, New York.
2.Zhao, H., and DebRoy, T. 2003. Macroporosity free
aluminum alloy weldments through numerical
simulation of keyhole mode laser welding. Journal
of Applied Physics 93:10089-10096.
3.Goldak, J. A., Burbidge, G., and Bibby, M. J. 1970.
Predicting microstructure from heat flow calculations
in electron beam welded eutectoid steels. Canadian
Metallurgical Quarterly 9:459-466.
4.Mohanty, P. S., and Mazumder, J. 1998. Solidification
behavior and microstructural evolution during laser
beam-material interaction. Metallurgical and
Materials Transactions B 29B: 1269-1279.
5.Yilbas B. S., Sami, M., Nickel, J., Coban, A., Said, S.
1998. Introduction into the electron beam welding of
austenitic 321 -type stainless steel, Journal of
Materials Processing Technology 82: 13-20.
6.Modest, M. F. 1997. Transient elastic and
viscoelastic thermal stresses during laser drilling of
ceramics, In Proceedings of the Materials
Processing Symposium ICALEO '97, San Diego, CA,
1997.
7.Du, J., Longobardi, J., Latham, W. P., and Kar, A.
2000. Weld geometry and tensile strength in laser
welded thin sheet metals. Science and Technology
of Welding and Joining 5: 304-309.
8.Tong, H., and Giedt, W. H. 1969. Radiographs of the
electron beam welding cavity. The Review of
Scientific Instruments 40: 1283-1285.
9.Schwarz, H. 1962. Electron beam processes at
different voltages. Transactions of the Second
International Vacuum Congress, Washington, D. C.,
Oct. 16-19, 1961, Pergamon Press, New York,
Vol. II, pp. 699-707.
10.Tong, H., and Giedt, W. H. 1970. A dynamic
interpretation of electron beam welding. Welding
Journal 49:259-s-266-s.
11.Chun, M. K., and Rose, K., 1970. Interaction of high-
intensity laser beams with metals. Journal of
Applied Physics 41: 614-620.
12.Schebesta, W. 1975. A contribution to explain the
deep penetration of high-power-density electron
beams in metals. Journal of Vacuum Science and
Technology 12:1218-1220.
13.Arata, Y. 1986: Plasma, Electron and Laser Beam
Technology, ASM, Ohio, 1986.
14.Taniguchi, N., Ikeda, M., Miyamoto, I., and Miyazaki,
T., 1989, Energy-Beam Processing of Materials,
Clarendon Press, Oxford.
15.Kar, A., and Mazumder, J. 1990. Two-dimensional
model for material damage due to melting and
vaporization during laser irradiation. Journal of
Applied Physics 68: 3884-3891.
16.DeBastiani, D. L., Modest, M. F., and Stubican, V. S.
1990. Mechanism of material removal from silicon
carbide by carbon dioxide laser heating. Journal of
American Ceramic Society 73:1947-1952.
17.Wei, P. S., and Ho, J. Y. 1990. Energy
considerations in high-energy beam drilling.
International Journal of Heat and Mass Transfer 33:
2207-2217.
18.Batteh, J. J., Chen, M. M., and Mazumder, J. 2000. A
stagnation flow-analysis of the heat transfer and
fluid-flow phenomena in laser drilling. Journal of
Heat Transfer 122: 801-807.
19.Wei, P. S., and Chiou, L. R. 1988. Molten metal
flow around the base of a cavity during a high-
energy beam penetrating process. Journal of Heat
Transfer 110: 918-923.
20.Ol'shanskii, N. A. 1974. Movement of molten metal
during electron-beam welding. Svarochnoe
Proizvodstvo 21: 12-14.
21.Ki, H., Mohanty, P. S., and Mazumder, J. 2002.
Modeling of laser keyhole welding: part I.
mathematical modeling, numerical methodology,
role of recoil pressure, multiple reflections, and
free surface evolution. Metallurgical and Materials
Transactions A 33A: 1817-1830.
22.Ki, H., Mohanty, P. S., and Mazumder, J. 2002.
Modeling of laser keyhole welding: part II.
simulation of keyhole evolution, velocity,
temperature profile, and experimental verification.
Metallurgical and Materials Transactions A
33A:1831-1842.
23.Wei, P. S., and Giedt, W. H. 1985. Surface tension
gradient-driven flow around an electron beam
welding cavity. Welding Journal 64: 251-s to 259-s.
24.Postacioglu, N., Kapadia, P., and Dowden, J. 1991.
A theoretical model of thermocapillary flows in
laser welding. Journal of Physics D: Applied
Physics 24: 15-20.
25.Rykalin, N. N. 1951. Calculation of Heat Flow in
Welding, translated by Zvi Paley and C. M. Adams,
Jr., Moscow.
26.Myers, P. S., Uyehara, O. A., and Borman, G. L.
1967. Fundamentals of Heat Flow in Welding.
Welding Research Council Bulletin 123.
27.Rosenthal, D. 1941. Mathematical theory of heat
distribution during welding and cutting, Welding
Journal 20: 220-s-34-s.
28.Carslaw, H. C. and Jaeger, J. C. 1959. Conduction
of Heat in Solids, Clarendon Press., Oxford.
29.Adams, C. M., Jr. 1958. Cooling rates and peak
temperatures in fusion welding. Welding Journal
37 : 210-s-215-s.
30.Hashimoto, T., and Matsuda, F. 1965. Effect of
welding variables and materials upon bead shape
in electron-beam welding. Transactions of
National Research Institute for Metals 7: 22-35.
31.Swift-Hook, D. T., and Gick, A. E. F. 1973.
Penetration welding with lasers. Welding Journal
52: 492-s-499-s.
32.Christensen, N., Davies, V. de L., and
Gjermundsen, K. 1965. Distribution of
temperatures in arc welding. British Welding
Journal 12: 54-75.
33.Lubin, B. T. 1966. A Correlation of electron beam
welding parameters. ASME Paper 66-WA/MET-18,
New York, Nov. 27-Dec. 1, 1966.
34.Stronskii, A. E., and Levin, V. V. 1982. Calculation of
the temperature field in the weld zone when
electron beam welding thick metals. Svarochnoe
Proizvodstvo 5: 3-4.
35.Lampa, C., Kaplan, A. F. H., Powell, J., and
Magnusson, C. 1997. An analytical thermodynamic
model of laser welding. Journal of Physics D:
Applied Physics 30: 1293-1299.
36.Irie, H., Tsukamoto, S., and Inagaki, M.,
1984,”Relation between Beam Properties and
Shape of Fusion Zone in Electron Beam
Welding- Abnormally Expanded Fusion Zone,”
Transactions of National Research Institute for
Metals, Vol. 26, pp. 287-296.
37.Ryzhkov, F. N., Bashkatov, A, V., and Uglov, A. A.
1972. Mechanism of formation of electron-beam
welds. Svarochnoe Proizvodstvo 5: 10-12.
38.Steen, W. M., Dowden, J., Davis, M., and Kapadia,
P. 1988. A point and line source model of laser
keyhole welding. Journal of Physics D: Applied
Physics 21: 1255-1260.
39.Mackwood, A. P., and Crafer, R. C. 2005. Thermal
modeling of laser welding and related processes:
a literature review. Optics& Laser Technology 37:
99-115.
40.Tong, H., and Giedt, W. H. 1971. Depth of
penetration during electron beam welding. Journal
of Heat Transfer 93:155-163.
41.Miyazaki, T., and Giedt, W. H. 1982. Heat transfer
from an elliptical cylinder moving through an infinite
plate applied to electron beam beam welding.
International Journal of Heat and Mass Transfer
25: 807-814.
42.Peretz, R., and Mayer, H. G. 1985. Parameter
correlations for deep penetration welding with high
energy focused beams. Optics and Lasers in
Engineering 6: 225-250.
43.Giedt, W. H., and Tallerico, L. N., 1988. Prediction
of electron beam depth of penetration. Welding
Journal 67: 299-s to 305-s.
44.Wei, P. S., Wu, T. H., and Chow, Y. T. 1990.
Investigation of high-intensity beam characteristics
on welding cavity shape and temperature
distribution. Journal of Heat Transfer 112: 163-169.
45.Schauer, D. A., and Giedt, W. H. 1978. Prediction of
electron beam welding spiking tendency. Welding
Journal 57: 189s-195s.
46.Schauer, D. A., Giedt, W. H., and Shintaku, S. M.
1978. Electron beam welding cavity temperature
distributions in pure metals and alloys. Welding
Journal 57: 127s-133s.
47.Kristensen, J. K., Hansson, L. H., and Smidth, F. L.
1986. Key-hole formation, temperature distribution
and thermal cycle in electron beam welding. in
Electron and Laser Beam Welding, Proceedings
of the International Conference, Tokyo, 14-15
July,1986, International Institute of Welding,
Pergamon Press, New York, pp.119-129.
48.Hemmer, H., and Grong, O. 1999. Prediction of
penetration depths during electron beam welding.
Science and Technology of Welding and Joining 4:
219-225.
49.Wei, P. S., and Shian, M. D. 1993. Three-
dimensional analytical temperature field around the
welding cavity produced by a moving distributed
high-intensity beam. Journal of Heat Transfer 115:
848-856.
50.Wei, P. S., Ho, C. Y., Shian, M. D., and Hu, C. L.
1997. Three-dimensional analytical temperature
field and its application to solidification
characteristics in high- or low-power-density-
beam welding. International Journal of Heat and
Mass Transfer 40:2283-2292.
51.Liu, M., Kang, J., Li, Y., and Chen, S. 2001.
Calculation of 3-D EBW temperature field in
titanium alloy plates. Acta Metallurgica Sinica 37:
301-306.
52.Burgardt, P., Knaus, S. E., and Kautz, D. D. 1987.
Faraday cup characterization of electron beam
welding parameters. Proceedings of the
International Conference on Applications of
Electron and Laser Beam Welding, Sep. 16-17,
1987, Hartford, Connecticut, American Welding
Society, pp.149-162.
53.Hicken, G. K., Giedt, W. H., and Bentley, A. E., 1991.
Correlation of joint penetration with electron beam
current distribution. Welding Journal 70: 69-s-75-s.
54.Elmer, J. W., and Teruya, A. T. 2001. An enhanced
Faraday cup for rapid determination of power
density distribution in electron beams. Welding
Journal 80: 288-s-295-s.
55.Wang, S. C., and Wei, P. S. 1992. Energy-beam
redistribution and absorption in a drilling or
welding cavity. Metallurgical Transactions B 23B:
505-511.
56.Wei, P. S., and Ho, C. Y. 1998. Beam focusing
characteristics effect on energy reflection and
absorption in a drilling or welding cavity of
paraboloid of revolution. International Journal of
Heat and Mass Transfer 41: 3299-3308.
57.Ho, C. Y., and Wei, P. S. 2001. Absorption in a
paraboloid of revolution-shaped welding or drilling
cavity irradiated by a polarized laser beam.
Metallurgical and Materials Transactions B 32B:
603-614.
58.Viskanta, R., and Anderson, E. E. 1975. Heat
transfer in semitransparent solids, in Advances in
Heat Transfer (editors: Irvine, T. F., Jr., and
Hartnett, J. P.), Vol. 11, Academic Press, New
York, pp.317-441.
59.Modest, M. F. 1993, Radiative Heat Transfer,
McGraw-Hill, New York.
60.Shui, V. H., Kivel, B., and Weyl, G. M. 1978. Effect of
vapor plasma on the coupling of laser radiation
with aluminum targets. Journal of Quantitative
Spectroscopy and Radiative Transfer 20: 627-636.
61.Collur, M. M., and DebRoy, T. 1989. Emission
spectroscopy of plasma during laser welding of
AISI 201 stainless steel. Metallurgical
Transactions 20B: 277-286.
62.Matsunawa, A., and Ohnawa, T. 1991. Beam-
plume interaction in laser materials processing.
Transactions of JWRI 20: 9-15.
63.Poueyo-Verwaerde, A., Fabbro, R., Deshors, G., de
Frutos, A. M., and Orza, J. M. 1993. Experimental
study of laser-induced plasma in welding
conditions with continuous CO2 Laser. Journal of
Applied Physics 74: 5773-5780.
64.Tu, J. F., Inoue, T., and Miyamoto, I. 2003.
Quantitative Characterization of Keyhole
Absorption Mechanisms in 20 kW-Class Laser
Welding Processes,” Journal of Physics D:
Applied Physics, Vol. 36, pp. 192-203.
65.Hoffman, J., and Szymanski, Z. 2004. Time
dependent spectroscopy of plasma plume under
laser welding conditions. Journal of Physics D:
Applied Physics 37: 1792-1799.
66.Zeng, X., Mao, X., Mao, S. S., Yoo, J. H., and Greif,
R. 2004. Laser-plasma interactions in fused silica
cavities. Journal of Applied Physics 95: 816-822.
67.Zhang, Y., Li, L., and Zhang, G. 2005.
Spectroscopic measurements of plasma inside
the keyhole in deep penetration laser welding.
Journal of Physics D: Applied Physics 38: 703-710.
68.Bejan, A., 1984, Convection Heat Transfer, Wiley,
New York.
69.Viskanta, R. 1988. Heat transfer during melting
and solidification of metals. 50th Anniversary Issue,
Journal of Heat Transfer 110: 1205-1219.
70.Gradshteyn, I. S., and Ryzhik, I. M. 1980, Table of
Integrals, Series, and Products, edited by Jeffrey,
A., 1980, translated from the Russian by Scripta
Technica, Inc., England.
71.Ion, J. C., Shercliff, H. R., and Ashby, M. F. 1992.
Diagrams for laser materials processing. Acta
Metallurgica et Materialia 40:1539-1551.
72.Fuerschbach, P. W. 1996. Measurement and
prediction of energy transfer efficiency in laser
beam welding. Welding Journal 75:24-s-34-s.
73.Wei, P. S., Kuo, Y. K., and Ku, J. S. 2000. Fusion
zone shapes in electron- beam welding dissimilar
metals. Journal of Heat Transfer 122: 626-631.
74.Incropera, F. P., and DeWitt, D. P. 1990.
Introduction to Heat Transfer, 2nd ed.,Wiley, New
York.