Temperature Fields Produced by Traveling Distributed Heat Sources

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1 Temperature Fields Produced by Traveling Distributed Heat Sources Use of a Gaussian heat distribution in dimensionless form indicates final weld pool shape can be predicted accurately for many welds and materials BY T. W. EAGAR AND N.-S. TSAI ABSTRACT. The solution of a travelling cate that the Rosenthal solution gives Symbols used when presenting the distributed heat source on a semi-infinite good agreement with the actual weld solutions discussed below are defined in plate provides information about both bead size over several orders of magni- Table 1. the size and the shape of arc weld pools. tude. However, the scatter can be as The results indicate that both welding much as a factor of three. In addition, the Formulation of the Solution process variables (current, arc length and point source solution does not provide travel speed) and material parameters any information concerning the shape of Rosenthal's solution for the tempera- (thermal diffusivity) have significant the weld pool, since all transverse iso- ture distribution produced by a steady effects on weld shape. The theoretical therms are assumed to be semicircular in state point heat source moving on the predictions are compared with experi- shape. The question of weld pool shape surface of a semi-infinite plate using the mental results on carbon steels, stainless is of considerable interest of late due to coordinate system shown in Fig. 1 is given steel, titanium and aluminum with good wide variations in welding behavior of by: agreement. heat-to-heat lots (Ref. 8). One of the purposes of the present Introduction study is to determine what shape infor- mation can be obtained from the solution It has been more than forty years since of a travelling distributed heat source. Rosenthal presented his solution of a Christensen (Ref. 2) converted this to Fortunately, several investigators have travelling point source of heat (Ref. 1) the dimensionless form: measured actual heat distributions in arcs which has been the basis for most subse- on water-cooled copper anodes (Refs. 9, quent studies of heat flow in welding. 10). Using these results, it is possible to Christensen put the results in dirnension- determine whether the presence of a less form in order to demonstrate that distributed rather than a point source of the solution applies to many materials heat can explain the range of weld shape where the operating parameter, n, over wide ranges of heat input (Ref. 2). variation measured by Christensen and includes the travel speed, the thermal Since that time, a number of refinements others. diffusivity and the net heat input to the have been offered (Refs. 3-6). Several In the following sections, a general workpiece. have attempted to use a more realistic solution of a travelling distributed heat The solution to a distributed heat distributed heat source (Refs. 3,4, 6), but source is briefly presented. Next, a source as shown in Fig. 1 can be formu- none have solved the entire temperature dimensionless solution of a travelling lated by the use of Green's functions. field for a travelling distributed source. Gaussian heat distribution is presented The steady-state heat conduction equa- More recently, a general form of the with a number of results. These are then tion in a moving coordinate of travel travelling distributed heat source has compared with experimental weld pool speed v (Ref. 1) is: been offered, but calculations of the shapes. thermal field were limited and unex- It should be emphasized at the outset plained (Ref. 7). that this solution retains all but one of the Christensen's experimental results indi- simplifying assumptions used by Rosen- thal; the assumptions include absence of convective or radiative heat flow, con- stant average thermal properties and a Paper presented at the 64th Annual A WS quasi-steady state semi-infinite medium. Convention in Philadelphia, Pennsylvania, dur- The only change is the use of a distrib- vw ing April 24-29, 1983. uted rather than a point source of heat. - 2a Despite these simplifications, it will be where T* = (T -To) e , and Q* T. W. EAGAR, Associate Professor -Materials shown that the results not only agree (w,y,z,) is the heat source moving at a Engineering, and N.-S.TSAI are with the Mas- with the Rosenthal solution in the limit, speed of v. sachusetts Institute of Technology, Cambridge, but that they are capable of explaining A derivation of the Green's function Massachusetts. most of the experimental scatter. which satisfies equation (3) and suitable 346-s I DECEMBER 1983

2 where 5Q is the amount of heat located Table 1-List of symbols at position (x',yr,z') at time t' The solution of an instantaneous a - thermal diffusivity Gaussian heat source is the superposition c - specific heat G - Green's function of a series of point heat source solutions k - thermal conductivity over the distributed region. By substitut- n - operating parameter (n = qv/ ing the Gaussian distributed heat source 4raZpc[Tc- To] for the point heat source Q, this super- q - net heat input per unit time (power) position is performed by the integration Q - power distribution as shown below and derived in Appen- Q* - heat source moving at a speed of v dix B. R - distance to the center of arc + + (R = (w2 y2 z2)'^) R* - dimensionless distance from the cen- + + ter of the arc (R* = [p rf,2 p]'72 T - temperature To - initial temperature Tc - critical temperature dt' u - dimensionless distribution parameter TO-* pc[4ra(t - t ')I3'* (u = vu/2a) v - travel speed of arc w - distance in x direction in a moving coordinate of speed v (w = x - vt) y - distance in y direction z - distance in z direction o- - distribution parameter p - density 6Q - incremental amount of heat T - dimensionless time 8 - dimensionless temperature (8 = [T - Tol/[Tc - To]) This corresponds to the rise of temper- f - dimensionless distance in the moving ature during a very short time interval ' coordinate (f = vw/2a) from time, t', to t' +dt' due to an

3 plate with no change in phase. The solu- equal. This produces a Gaussian distribu- tion may be put in dimensionless form by tion of equal total heat input as the using the following dimensionless vari- experimental distribution. This analysis ables: $ equals vw 2a; $ equals produces values of u ranging from 1.6 to vy 2a; { equals vz 2a; T equals 4 mm (0.06 to 0.16 in.); these are in v2 t " 2a; and u equals vu + 2a reasonable agreement with the results of Rykalin (Ref. 3). Equation (10) then reduces to: Qualitatively, it can be argued that u will increase with increasing current, increasing arc length (i.e., voltage) and increasing tungsten electrode tip angle. The effect of variations in shielding gas composition are less obvious (Ref. 14). A more quantitative discussion of the varia- tion in arc welding conditions will be 0 0 0.60 1.20 1.80 2.40 3.00 given in a subsequent publication (Ref. Further solution of this equation DIMENSIONLESS DISTANCE, 1/, 15). requires a numerical procedure. For the purposes of this paper, values Fig. 3 -Peak temperature distribution along It is interesting to note that the two the dimensionless transverse distance, it. as a of u between 1.6 and 4 mm (0.06 to 0.16 function of the dimensionless distribution primary dimensionless variables that in.) will be assumed for gas tungsten arc parameter, u, equal to 0, 0.4, 0.6 and 0.8, describe the heat source are u and n. The welding. For travel speeds on the order where u = vd2a value, u, represents the width of the heat of 2 mm/s (4.7 ipm), these values of u will source, while n is related to the intensity produce values of the dimensionless dis- or magnitude of the input energy. Travel tribution parameter, u, between 0.4 and speed is an integral part of both u and n. 0.8 on carbon steel. In the limit where Typical values of u and n vary markedly u = 0, the Gaussian has no width and the for different materials and different weld- solution to equation (11) reduces to the ing processes as shown schematically in Rosenthal solution. Fig. 2. It will be noted that u varies directly with the welding travel speed and Results and Discussion inversely with a material condition, i.e., the thermal diffusivity. This means that Results of the Model slow travel speeds and high thermal diffu- In order to represent the solution to sivity materials such as aluminum or cop- equation (1I), it is necessary to select per alloys are approximated better by the values of the distribution parameter, u. Rosenthal equation than are fast welds Fortunately, Nestor (Ref. 9) and Schoeck on materials such as stainless steel or (Ref. 10) have measured the heat distribu- titanium where u is larger. For conve- 0.60 1.20 1 .80 2.40 3.00 tions of arcs on water-cooled copper nience, the results presented here will DIMENSIONLESS DISTANCE, i- anodes. Although their measured distri- generally apply to values of u = 0 (Rosen- Fig. 4 -Peak temperature distribution along butions are not true Gaussians, the results thal) and 0.4, 0.6 and 0.8. the dimensionless transverse distance, t, as a can be approximated as Gaussians by a Figures 3 and 4 show the dimensionless function of the $rnensionless distribution least squares regression of their data to fit peak temperature distributions in the parameter, u, equal to 0, 0.4, 0.6 and 0.8 a Gaussian distribution, subject to the dimensionless transverse and the dimen- constraint that the total area under the sionless through thickness directions, experimental and the Gaussian curves are respectively. Note that the dimensionless temperature is divided by Christensen's operating parameter, n, in order to show all solutions on a single graph. The distrib- uted heat source solution does not pre- dict an infinite centerline temperature as the Rosenthal solution does; however, it will be noted that the center-line temper- ature may often exceed the boiling tem- perature of the metal depending on the values of u and n. This is an impossible situation. Nonetheless, for values of u and n where the evaporative power loss (Ref. 16) is not excessive, the solution given by equation (11) is valid. In other cases, it is still a somewhat better approx- imation than the Rosenthal solution. It will be noted that Figs. 3 and 4 provide shape information about the weld pool. At values of 9/17 correspond- ing to the melting temperature, Fig. 3 - predicts the weld width; this is generally, 2' but not always, greater than that pre- H a 0.01 e. t I .a 10.0 100.a dicted by the Rosenthal solution. Figure 4 OPERATING PARAMETER. n predicts the weld depth which is always Fig. 2 - Welding condition: The graph of dimensionless distribution parameter, u, and operating less than that predicted by the Rosenthal parameter, n, depicts the conditions used with different processes and materials solution. 348-s ] DECEMBER 1983

4 2 s I- 0.60 n _1 w 3 2 u 0.40 -I z 0 H If) 0.20 z H n 0.00 0.00 0.30 0.60 0.90 1 .20 1 .50 0.00 0.30 0.60 0.90 1.20 1 .50 OPERATING PARAMETER, n OPERATING PARAMETER, n Fig. 5 -Dimensionless weld width WW* (constant 0 = 1) versus oper- ating parameter, n, as a function of the dimensionless distribution Fig 6 -Dimensionless welddepth WD* vs. operatingparameter, n, as a parameter, u function of the dimensionless distribution parameter, u These points are illustrated further by ing parameters are varied (Ref. 17); how- these cases, a much more complicated Figs. 5 and 6 which show the dimension- ever, this model is not capable of predict- analysis is required. less weld width and depth as a function ing the variable penetration at constant Figure 5 also provides an explanation of the operating parameter, n. These are process parameters which is caused by of the changes in weld width with change lines of constant 6 = 1. It is easily seen convection in the weld pool (Ref. 18). As in arc length as measured by Glickstein that at low n (which corresponds to low noted previously, convection is not con- (Ref. 14) and confirmed in our laboratory. heat input and travel speed), the weld sidered in the solution presented here. At low heat inputs the width decreases width is less than the Rosenthal predic- Nonetheless, in most cases the model with increased arc length, while at higher tion. However, as n increases, the width presented here gives a good prediction heat inputs the weld width increases with rapidly increases to greater than that for of the weld pool. increasing arc length - Fig. 8. This is the Rosenthal solution, whereas the In instances when the convective pat- understood by recognizing that increases depth always remains less than the tern in the weld pool is constant but in arc length increase the distribution Rosenthal prediction. process parameters are changed, this parameter without significantly altering , Figure 7 is the depth-to-width ratio as a model should provide the proper func- the heat input.* Hence, increasing the arc function of operating parameter for dif- tional relationship between weld process length is essentially an increase in the ferent values of the distribution parame- parameters and weld pool shape. When distribution parameter at constant oper- ter. The wide variations may help to the convective pattern changes from ating parameter. explain at least some of the weld pool weld to weld, no model which neglects As shown in Fig. 5, this results in depth-to-width ratios which have been convection such as this one will give an narrower welds with increasing arc reported in the literature when the weld- accurate representationof pool shape. In length at low operating parameters but increased weld width with increased arc length at higher operating parameters. At some values of n, where the lines of constant u intersect, the weld width will be maximum at an intermediate arc length. A physical explanation for these results is obtained by considering that a very broad heat distribution at low heat inputs will not produce any melting. As the distribution narrows but the net heat 'Changes in arc length fie., voltage) do not increase the net heat input as longer arcs are less efficient than short ones (Ref. 19). This is due to the fact that most of the heat is 0.00 0.30 0.60 0.90 1.20 1.50 transferred by the electrons falling across the 0.0 2.0 4.0 6.E a 0 OPERATING PARAMETER, n ARC LENGTH, ALCmm:) anode fall space and condensing in the metal Fig. 7- Calculated depth to width ratio versus (Ref. 20). Longerplasma columns do not affect operating parameter, n, as a function of the the heat transfer processes in the anode fall dimensionless distribution parameter, u, equal Fig. 8 -Experimental weld width vs. arc length region. However, they do produce greater to 0, 0.4, 0.6, and 0.8 as a function of heat input per unit length radiative heat losses. WELDING RESEARCH SUPPLEMENT I 349-s

5 0.00 0.30 0.60 0.90 1 .20 1 .50 OPERATING PARAMETER, n OPERATING PARAMETER, n Fig. 9 -Dimensionless width of the total transformed zone (heat- affected zone plus fusion zone, constant 0 = 0.462), vs. operating Fig. 10 -Dimensionless depth of the total transformed zone, TW* vs. parameter, n, as a function of the dimensionless distribution parame- operating parameter, n, as a function of the dimensionless distribution ter, u parameter, u input remains constant, the amount of sion zone plus heat-affected zone) as a ate heat inputs. Such information may be melting increases and the weld width function of the operating parameter. important if one is either trying to mini- grows. At higher heat inputs there is always sufficient heat for melting, and a narrower distribution will result in a nar- which is equivalent to T 723OC - These graphs correspond to 9 = 0.462 (1333OF) for carbon steel. Subtracting the mize the thermal effect in the base metal or if one is trying to use the weld heat- affected zone to alter previous weld rower weld more closely approximating data of Figs. 9 and 10 from Figs. 5 and 6 passes, as is done with temper beads in the value predicted by the Rosenthal gives the predicted width and depth of heavy section alloy steels. Unfortunately, solution. the heat-affected zone as shown in Figs. the maximum in heat-affected zone size Figures 9 and 10 show the width and 11 and 12. A distinct maximum in heat- is not as pronounced in the depth direc- depth of the total transformed zone (fu- affected zone width occurs at intermedi- tion as is seen in Fig. 11. 0.00 0.30 0.60 0.90 1.20 1.50 OPERATING PARAMETER, n OPERATING PARAMETER, n Fig 1 1 -Dimensionless width of the heat affected zone, HW* vs. Fie 12-Dimensionless depth of the heat affected zone, HD* vs. operating parameter, n, as a function of the dimensionless distribution operating parameter, n, as a function of the dimensionless distribution parameter, u parameter, u 3 1 DECEMBER 1983

6 0.00 0.30 0.60 0.90 1.20 1.50 0.00 0.30 0.60 0.90 1.20 1 .S0 OPERATING PARAMETER, n OPERATING PARAMETER, n Fig. 13 -Dimensionless area of the weld metal, WA * vs. operating @. 14 -Dimensionless area of the total transformed zone, TA * vs. parameter, n, as a function of the dimensionless distribution parame- operating parameter, n, as a function of the dimensionless distribution ter, u parameter, u Figures 13, 14 and 15 give the weld tion of typical values of n for gas tungsten made on carbon steel while varying the area, total transformed area, and heat; arc welding will show that stainless steel welding process variables. In the second affected zone area, respectively. These and titanium are often welded in the test, welds were made on different met- areas were obtained by complete solu- region where the lines of constant u als to evaluate the effect of changes in tion of equation (11) in two dimensions. overlap. This may explain, in part, why thermal diffusivity. The small change in the total transformed more anomalous weld bead shape ratios For the first set of welds, the current, area with variations in u is consistent with have been reported for these materials arc length, electrode tip angle, shielding the fact that, over large distances, the than for carbon steel, aluminum or cop- gas composition and travel speed were overall heat effect in the metal is propor- per alloys. The shape of the weld bead in varied. Changes in the first four of these tional to the net heat input-or in the stainless steels and titanium is more sensi- process variables were correlated with present case- to the operating parame- tive to minor variations in the process or changes in heat distribution on the sur- ter. When smaller distances, such as the in the metal because of inherently large face of a water-cooled copper anode by fusion zone, are considered, the deviation values of u. an extensive set of experiments (Ref. 21) may be significant as seen in Fig. 13. similar to the work of Nestor (Ref. 9). Figures 16 through 18 are provided to Experimental Verification of the Predictions The resulting weld widths, depths and show the effect of larger values of u and areas are given in Figs. 19, 20 and 21, n. These values apply to materials with The accuracy of the model was tested respectively The predicted widths and low thermal diffusivity such as stainless in two ways. In one series of experi- areas as functions of u and n are in good steel and titanium. Furthermore, estima- ments, gas tungsten arc welds were agreement with the theory, although the I .60 * n I;' u I .20 a $ 0.80 H 2 3: 2 0.40 0.00 0.00 0.30 0.60 0.90 1-20 1-50 0.0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 5.0 OPERATING PARAMETER, n OPERATING PARAMETER, n OPERATING PARAMETER. n Fig. 15 -Dimensionless area of the heat-affect- Fig. 16 -Dimensionless width W* (constant ed zone, HA* vs. operating parameter, n, as a 0 = 1) vs. operatingparameter, n, as a function Fig. 17 -Dimensionless depth D* (constant function of the dimensionless distribution of the dimensionless distribution parameter, u, ' = 1) vs. operatingparameter, n, as a function 6 parameter, u equal to 0, 0.6, 1.2 and 1.8 of dimensionless distribution parameter, u WELDING RESEARCH SUPPLEMENT 1 351-s

7 O.e 1 .a 2.a 3.0 4.0 5.0 0.00 0.30 0.60 0.90 1.20 1.50 OPERATING PARAMETER, n OPERATING PARAMETER, n Fig. 18 -Dimensionless area A* (constant 6 = 1) vs. operating parame- Fig. 19-Comparison of the experimental weld width on mild steel and ter, n, as a function of dimensionless distribution parameter, u the distributed source theory, where the dimensionless distribution parameter, u, represents the heat source width. The dashed lines represent the Rosenthal prediction. The solid lines are the distributed source theory predictions. The experimental data points represent welds made at constant values of u depth predictions (Fig. 20) are in consid- sources to develop an enhancement fac- ment as shown in Fig. 22. The remaining erable error. These errors in the depth tor which estimates the temperature pro- errors in weld pool shape may be attrib- prediction are due to the assumption of a file of finite thickness plates, based upon uted either to convection in the weld semi-infinitely thick plate, whereas the the semi-infinite Rosenthal solution. This pool or to depression of the surface due true plate thickness was 0.5 in. (12.7 rnm). temperature enhancement factor was to arc forces, neither of which are consid- The prediction may be improved by use used to correct the distributed source ered in the distributed source theory. of the temperature enhancement factor theory for the finite thickness of the For the second test of the distributed described by Myers (Ref. 22). plate. The results bring the predicted source theory, welds were made on Myers used a set of image point heat weld pool depth into much better agree- stainless steel, titanium and aluminum at a 0.00 0.30 0.60 0.90 1.20 1.50 OPERATING PARAMETER, n OPERATING PARAMETER, n Fig 20- Comparison of the experimental weld depth on mild steel and Fig. 2 1 - Comparison of the experimental weld area on mild steel and the distributed source theory, where the dimensionless distribution the distributed source theory, where the dimensionless distribution parameter, u, represents the heat source width parameter, u, represents the heat source width 352-s I DECEMBER 1983

8 0.50 1,/ = OeQ Stainless Steel OPERATING PARAMETER, n Fig. 23 -Experimentally measured and theo- retically calculated weld width vs. operating parameter for stainless steel at constant heat distribution u = 0.9 Rosenthal solution is not changed mark- edly if temperature dependent proper- ties are considered, and Malmuth (Ref. 5 ) has shown that the latent heat has only a minor effect. Recently, Oreper and Szekely (Ref. 24) showed that convec- tion can, but does not always, play a role in determining weld pool shape. Rykalin (Ref. 25) has shown that the heat lost by 0.00 0.30 0.60 0.90 1.20 1.50 radiation is negligible. Although the distributed source theory OPERATING PARAMETER, n presented here is still very imperfect, the Fig. 22-Dimensionless weld depth as a function of operating parameter for three values of heat solution of this problem clearly shows distribution. The theoreticaltines are corrected for finite plate thickness. Compare these corrected that the point source assumption is the values with the uncorrected results of Fig. 20 greatest weakness of the Rosenthal theo- ry. While the distributed source theory cannot always predict the exact weld constant distribution parameter of and experiment is quite good pool shape, it provides a base line from IJ = 2.4 mm (0.09Jn.). Due to differences It may be questioned why the distrib- which to measure the effects of convec- in the thermal diffusivity of these metals, uted source theory, which neglects con- tion, surface depression and possibly o- is constant, but u is not constant; thus vective and radiative heat transfer as well evaporate heat loss as they influence one would expect a different curve for as temperature dependent properties weld bead shape. One of the difficulties each material. Figures 23,24 and 25 show and phase transformations, can give such in utilizing the distributed source theory is these different curves, based on the weld good agreement with experiment. The a priori prediction of the value of the width. The agreement between theory work of Grosh (Ref. 23) showed that the distribution parameter, a. This important topic is dealt with in a subsequent paper (Ref. 15). Conclusions t U = 0.06 Aluminum The travelling distributed heat source theory provides the first estimate of weld pool geometry based upon fundamentals of heat transfer. Although a number of simplifying assumptions remain, the agreement between theory and experi- ment is improved considerably over pre- vious models. The greatest value of this work does not lie in the ability to predict the abso- lute size of the weld zone. Rather, the strength of the new theory is that it gives 0.000 0,010 0.020 0.030 0.040 an accurate functional relationship OPERATING PARAMETER, n 0.0 1 .0 2.0 3.0 between both process parameters and OPERATING PARAMETER, n materials parameters. The theory pro- Fig. 24 -Experimentally measured and theo- Fig 25 -Experimentally measured and theo- vides a model that can be used to assess retically calculated weld width vs. operating retically calculated weld width vs. operating how changes in the process or in the parameter for aluminum at constant heat di'stri- parameter for titanium at constant heat distri- material will influence the weld geome- bution u = 0.06 bution u = 12 try. Such a model is essential to many WELDING RESEARCH SUPPLEMENT 1 353-s

9 automation and control strategies now Applied Physics 48(9): 3895. Dirichlet boundary condition at infinity. being considered for welding processes. 14. Glickstein, S. S., Friedman, E., and Yenis- The Green's function must satisfy the cavich, W . 1975. An investigation of alloy 600 equation: Acknowledgments welding parameters. Welding Journal 54(4):113-s to 122-S. The authors wish to express their 15, Tsai, N. S., and Eagar, T. W. Variation of appreciationto Professor Nils Christensen the heat distribution with welding parameters for several helpful discussions. They are in gas tungsten arc welding-to be pub- grateful for financial support from the lished. 16. Block-Bolten, A,, and Eagar, T. W. 1982. Department of Energy under contract Selective evaporation of metals from weld DE-AC02-78ER 94799 A.0003. pools. Trends in Welding Research in the United States. Metals Park, Ohio: American A solution of equation (AI) in spherical References Society for Metals. coordinates is: 1. Rosenthal, D. 1946. The theory of mov- 17. Glickstein, S. S., and Yeniscavich, W. ing source of heat and its application to metal 1977. Review of minor element effects on the eL(r-r') e - -v (r - r') welding arc and weld penetration. Welding treatment. Transactions ASME 43(11):849- G(r-rf)=A^1 2a 866. Research Council bulletin no. 266, p. 18. lr - r f l r - r'l 2. Christensen, N., Davies, V., and Gjer- 18.'Heiple, C. R., and Roper, j. R. 1981. rnundsen, K. 1965. The distribution of temper- Effect of selenium on GTAW fusion zone ature in arc welding. British Welding Journal geometry. Welding Journal 60(8):143-s to 12(2):54-75. 145-5. 19. Cobine, j. D., and Burger, E. E. 1955. The boundary condition that 3. Rykalin, N. N., and Nikolaev, A. V. 1971. Analysis of electrode phenomena in the high- Welding arc heat flow. Welding in the World G(r - r') = 0 at infinity requires that the 9(3/4):112-132. current arc. / Appl. Phys. 26(7):895. 20. Ghent, H. W., Roberts, D. W., Her- first term of equation (A2) vanish. Then 4. Pavelic, V. 1979 (May) Weld puddle the solution can be written as: shape and size correlation in a metal plate mance, C. E., Kerr, H. W., and Strong, A. B. welded by the GTA process. Arch Physics and 1979 (May). Arc efficiency in TIC welds. Arc Weld Pool Behavior, pp. 251-258. London: Physics and Weld Pool Behavior, International The Welding Institute Conference. Conference Proceedings. London: The Weld- 5. Malmuth, N. D., Hall, W. F., Davis, B. I., ing Institute. and Rosen, C. D. "1974. Transient thermal 21. Tsai, N. S. 1983 (February). Heat distri- phenomena and weld geometry in GTAW. bution and weld geometry in arc welding. Welding Journal 53(9):388-s to 400-5. Ph.D. thesis, Massachusetts Institute of Tech- 6. Friedman, E. 1975 (August). Thermody- nology. namic analysis of the welding process using the 22. Myers, P. S., Uyehara, 0 . A,, and Bor- A symmetric heat source located at the finite element method. ASTM Transactions; man, G. L. 1967 (July). Fundamentals of heat position (wf , y f ,-z') is needed in order journal of Pressure Vessel Technology, p. flow in welding. Welding Research Council that the second boundary condition bulletin no. 123. 206. aG(rIr')/az = 0 at z = 0 be satisfied. This 7. Peng, T. C., Sastry, 5. M. L., and O'Neal, 23. Grosh, R. J., and Trabant, E. A. 1956. Arc welding temperature. Welding journal imaginary heat source leads to the fol- J E. 1981. Exploratory study of laser processing lowing Green's function: of titanium alloys. The Metallurgical Society of 35(8):396-s to 400-s. AIME. Lasers in metallurgy, pp. 279-292. 24. Oreper, G.,Eagar, T. W and Szekely, J. 8 Linnert, G. E. 1967. Weldability of austen- 1983. Convection in arc weld pools. Welding itic stainless steels. ASTM STP 418. Journal 62(11):307-s t o 312-5. 9. Nestor, 0. H. 1967. Heat intensity and 25. Rykalin, N. N., and Beketov, A. 1. 1967. current density distribution at the anode of Calculating the thermal cycle in the heat- high current inert gas arcs. Journal of Applied affected zone from the two dimensional out- Physics 33(5):1638-1648. line of the molten pool. Weld. Prod. 14:42- 10. Shoeck, P. 1963. An investigation of 47. anode energy balance of high intensity arcs in argon. Modern development of heat transfer, ~ e Yorkw and on don: Academic Press, pp. Appendix 353-478. A. Derivation of Green's Function 11. Stakgold, I. 1979. Green's functions and boundary value problems, chapter 2. New The boundary conditions of the York: Wiley and Sons. Green's function are the same as those of 12. Carslaw, H. S., and Jaeger, J. C. 1967. Oxford University Press. Conduction of heat in equation (3) - namely: solid, pp. 255. I. aT*/az = 0 at z = 0. 13. Cline, M . E., and Anthony, T. R. 1977. 2. T* = 0 at r = oo. If one considers a heat source at the Heat treating and melting material with a These are the Neumann boundary surface z' = 0, the solution reduces to scanning laser or electron beam. Journal of conditions at the surface 2 = 0, and the equation (4). B. Superposition of Point Heat Source Solutions by Integration qdt' 1 - - 2~u~~c(- ,,,2Sm 4 it r 1~ t -a dxf Jm -a dy' exp( - [x'2($ +( - t - t',) + 354s I DECEMBER 1983

10 WELDING RESEARCH SUPPLEMENT 1 355-s

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