Nonlinear response is studied using similarity groups. The nonlinear solution is similar to the linear solution except with properties evaluated at the local temperature and is consistent with reported empirical observations that, near the heat source, high-temperature properties are needed to predict cooling times to high temperatures and that low temperature properties are needed to predict cooling time to low temperatures. For position-dependent characteristics of the solution away from the source, like magnitudes and locations of peak temperatures, nonlinear and linear solutions are similar if local properties are evaluated at current temperatures prevailing near the source. [S0022-1481(00)02701-8]

1.
Kasuya
,
D.
, and
Yurioka
,
N.
,
1993
, “
Prediction of Welding Thermal History by a Comprehensive Solution
,”
Weld. J. (Miami)
,
72
, No.
3
, pp.
107s–115s
107s–115s
.
2.
Moore, J. E., Bibby, M. J., Goldak, J. A., and Santyr, S., 1986, “A Comparison of the Point Source and Finite Element Schemes for Computing Weld Cooling,” Welding Research: The State of the Art, E. F. Nippes and D. J. Ball eds., ASM, Miami, pp. 1–9.
3.
Carslaw, H. C., and Jaeger, J. C., 1959, Conduction of Heat in Solids, 2nd Ed., Oxford University Press, London.
4.
Grosh
,
R. J.
, and
Trabant
,
E. A.
,
1956
, “
Arc Welding Temperatures
,”
Weld. J. (Miami)
,
35
, No.
8
, pp.
396
401
.
5.
Myers, P. T., Uyehara, O. A., and Borman, G. L., 1967, “Fundamentals of Heat Flow in Welding,” Welding Research Council Bulletin, No. 123, pp. 1–46.
6.
Ames, W. F., 1968, Nonlinear Ordinary Differential Equations in Transport Processes, Academic Press, New York.
7.
Logan, J. D., 1994, An Introduction to Nonlinear Partial Differential Equations, John Wiley and Sons, New York.
8.
Ozisik, M. N., 1980, Conduction Heat Transfer, John Wiley and Sons, New York.
9.
Zeldovich, Ya., and Kompaneets, A. S., 1950, “On the Theory of Propagation of Heat With Temperature-Dependent Thermal Conductivity,” Izdat. Akad. Nauk SSSR, pp. 61–71.
10.
Ames
,
W. F.
,
1964
, “
Similarity for the Nonlinear Diffusion Equation
,”
Ind. Eng. Chem. Fundam.
,
4
, pp.
72
76
.
11.
Boyer
,
R. H.
,
1961
, “
On Some Solutions of a Nonlinear Diffusion Equation
,”
J. Math. Phys.
,
40
, No.
1
, pp.
41
45
.
12.
Heaslet
,
M. A.
, and
Alksne
,
A.
,
1961
, “
Diffusion From a Fixed Surface With a Concentration Dependent Coefficient
,”
J. SIAM
,
9
, No.
4
, pp.
584
596
.
13.
Pattle
,
R. E.
,
1959
, “
Diffusion From an Instantaneous Point Source With a Concentration Dependent Coefficient
,”
Q. J. Mech. Appl. Math.
,
12
, No.
4
, pp.
407
409
.
14.
Radaj, D., 1992, Heat Effects of Welding, Springer-Verlag, New York.
15.
Ion
,
J. C.
,
Easterling
,
K. E.
, and
Ashby
,
M. F.
,
1984
, “
A Second Report on Diagrams of Microstructure and Hardness for Heat-Affected Zones in Welds
,”
Acta Metall.
,
32
, pp.
1949
1962
.
16.
Svensson, L., 1994, Control of Microstructures and Properties in Steel Arc Welds, CRC Press, Boca Raton, FL.
17.
Rosenthal
,
D.
,
1941
, “
Mathematical Theory of Heat Distribution During Welding and Cutting
,”
Weld. J. (Miami)
,
20
, No.
5
, pp.
220s–234s
220s–234s
.
18.
Rosenthal
,
D.
,
1946
, “
The Theory of Moving Sources of Heat and Its Application to Metal Treatments
,”
Trans. ASME
,
68
, pp.
849
866
.
19.
Harvey, P., 1982, Engineering Properties of Steel, ASTM, Metals Park, OH.
20.
Goldak
,
J.
,
Chakravarti
,
A.
, and
Bibby
,
M.
,
1984
, “
A New Finite Element Model for Welding Heat Sources
,”
Metall. Trans.
,
15B
, No.
6
, pp.
299
305
.
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