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. VV. 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 v(w + R) „ q 2a Introduction study is to determine what shape infor 27rkR (1) mation can be obtained from the solution It has been more than forty years since of a travelling distributed heat source. Christensen (Ref. 2) converted this to Rosenthal presented his solution of a Fortunately, several investigators have travelling point source of heat (Ref. 1) measured actual heat distributions in arcs the dimensionless form: which has been the basis for most subse on water-cooled copper anodes (Refs. 9, -
Despite these simplifications, it will be where T* = (T — T0) e , and Q* shown that the results not only agree T. W. EAGAR, Associate Professor —Materials (w,y,z,) is the heat source moving at a Engineering, and N. -S. TSAI are with the Maswith 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 | DECEMBER 1983 where <5Q is the amount of heat located Table 1—List of symbols at position (x',y',z') at time t'. The solution of an instantaneous a - thermal diffusivity Gaussian heat source is the superposition c — specific heat of a series of point heat source solutions G — Green's function k — thermal conductivity over the distributed region. By substitut n — operating parameter (n = qv/ e ing the Gaussian distributed heat source 2
WELDING RESEARCH SUPPLEMENT 1347-s 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; \fi equals produces values of a ranging from 1.6 to vy T 2a; f equals vz * 2a; r equals 4 mm (0.06 to 0.16 in.); these are in vz t + 2a; and u equals va -S- 2a reasonable agreement with the results of Rykalin (Ref. 3). Equation (10) then reduces to: Qualitatively, it can be argued that a will increase with increasing current, v-r -i/i increasing arc length (i.e., voltage) and increasing tungsten electrode tip angle. dr- 2 V^-'-Ao r-F u The effect of variations in shielding gas composition are less obvious (Ref. 14). A {2 + vl2 + 7{x + T2 f2 (11) more quantitative discussion of the varia 2r + 2u2 2r tion in arc welding conditions will be given in a subsequent publication (Ref. 0.30 0.60 1.20 1.80 2.40 3.00 Further solution of this equation 15). DIMENSIONLESS DISTANCE, il Fig. 3-Peak temperature distribution along requires a numerical procedure. For the purposes of this paper, values the dimensionless transverse distance, \f>, as a It is interesting to note that the two of a between 1.6 and 4 mm (0.06 to 0.16 primary dimensionless variables that function of the dimension/ess distribution 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 = vrj/2a value, u, represents the width of the heat of 2 mm/s (4.7 ipm), these values of a 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 0, 0.4, 0.6, O.e 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 (11), it is necessary to select per alloys are approximated better by the values of the distribution parameter, a. Rosenthal equation than are fast welds Fortunately, Nestor (Ref. 9) and Schoeck on materials such as stainless steel or l I I I I I L (Ref. 10) have measured the heat distribu titanium where u is larger. For conve 0.68 I 20 I .88 2.40 3.00 tions of arcs on water-cooled copper nience, the results presented here will DIMENSIONLESS DISTANCE, f 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, _i~, as a can be approximated as Gaussians by a Figures 3 and 4 show the dimensionless function of the dimensionless 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 UJ all solutions on a single graph. The distrib I- 10.8 tu uted heat source solution does not pre '//.OMAWCrO/ > dict an infinite centerline temperature as < '// 8AW 348-s | DECEMBER 1983 4.00 1 .83 - 3 3.20 * 0.80 — 0.8//"' 3 x r- a CL H kJ 3 2.40 — 0.60 — 0.6 / mm co " I • 60 U 40 0.40 "~ kJ 0.4 / O M 40 tn 5 0.80 fi 0.20 - ~ u = 0. • 0.00 i ^^T^ 1 1 1 1 0.00 0 . 00 0.30 0.60 0.90 1 .20 I .50 0.00 0.30 0.60 0.90 1 .20 50 OPERATING PARAMETER, n OPERATING PARAMETER, n Fig. 5 — Dimensionless weld width WW* (constant t) = I) versus oper ating parameter, n, as a function of the dimensionless distribution fig. 6 - Dimensionless weld depth WD* vs. operating parameter, 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 0=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 ln 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 representation of 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 1000 J/mm be maximum at an intermediate arc length. <* 0.400 - 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 500 J/mm 3 0.i00 - *Changes in arc length (i.e., voltage) do not Increase the net heat input as longer arcs are less efficient than short ones (Ref. 19). This is 0.000 i I i I i due to the fact that most of the heat is 8.30 0.60 8.30 I.20 0.0 2.0 4.0 6.B 8.0 transferred by the electrons falling across the OPERATING PARAMETER, n ARC LENGTH, ALCmm> anode fall space and condensing in the metal Fig. 7 - Calculated depth to width ratio versus (Ref. 20). Longer plasma 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 1349-s 5.0 1 .50 _ 1- * s z 3 a UJ 1- 4.0 — u = 0^— i- 1 .20 — S CL X O 1- - _J S^ 04____ LJ Ha UJ > 3 LU 3.0 — 0.90 — Q _l 1<- /^ 06^^^>- I O o O 1- CC < co /^^0-8. •'•""' co LU 2.0 — 0.60 LinU CO tf) UJ LU _l CC z o o M M co CO Z z 3 0.30 - LU UJ 1 .0 — > X 0_ H H O a 0.0 yi i 1 i 1 i 1 i 0.00 0.00 0.30 0.60 0.90 .20 1.50 0.00 0.30 0.60 0.90 1 .20 .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 These graphs correspond to 8 = 0.462 mize the thermal effect in the base metal always sufficient heat for melting, and a which is equivalent to T = 723°C or if one is trying to use the weld heat- narrower distribution will result in a nar (1333°F) for carbon steel. Subtracting the 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. 3.00 I .00 - j \o.8 ? ? 40 Q 0 80 X £ ^ % X X t- h- & n H 6 hi 3 1 80 — /\°' \ Q 0 60 N N < X< X CO CO cn UJ 1 20 — iii 0 40 _l / \0.4 \ \ _i z z o B H M 0.00 1 1 1 1 1 1 1 I 0.00 0.00 0.30 0.60 0.90 1.20 1.50 0.00 0.30 0.60 0.90 .20 1 .50 OPERATING PARAMETER, n OPERATING PARAMETER. Fig. 11 — Dimensionless width of the heat affected zone, HW* vs. Fig. 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 350-s | DECEMBER 1983 I .20 4.0 0 .qfi x - 0.00 0.0 0.00 0.30 0.60 0.90 .20 1 .50 0.00 0.30 0.60 0.90 1 .20 1 .50 OPERATING PARAMETER. OPERATING PARAMETER, n Fig. 13 — Dimensionless area of the weld metal, WA* vs. operating Fig- 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 The resulting weld widths, depths and Experimental Verification of the Predictions 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 _ * 2.000 - 0.8/'^ < I .s Sa0 - a. I . 20 — 0.5/7 00 0.4/ se _U = 0. 00 Al I I I I I l a.a 0.30 0.60 0.98 1.20 I.58 0.0 1.0 2.0 3.0 4.0 s.a 1.0 2.8 3.8 4.( 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. operating parameter, n, as a function Fig. 17—Dimensionless depth D* (constant function of the dimensionless distribution of the dimensionless distribution parameter, u, II = I) vs. operating parameter, n, as a function parameter, u equal to 0, 0.6, 1.2 and 1.8 of dimensionless distribution parameter, u WELDING RESEARCH SUPPLEMENT 1351-s 7.0 4.0 u = O.i z *-g 2.0 LJ a. o _! LU 4.0 > •' m— UJ u = 0.6 Q X o 2.0 cc < 5 4.0 u = 0.4 « 2.0 oCu —I LU > LU 0.0 Q X 0.0 1.0 2.0 3.0 4.0 5.0 0.00 0.30 0.60 0.90 1.20 1.50 O CC 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 LU ter, n, as a function of dimensionless distribution parameter, u CO the distributed source theory, where the dimensionless distribution LU parameter, u. represents the heat source width. The dashed lines cc represent the Rosenthal prediction. The solid lines are the distributed z source theory predictions. The experimental data points represent LU welds made at constant values of u o 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 LU prediction are due to the assumption of a file of finite thickness plates, based upon Q 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 O true plate thickness was 0.5 in. (12.7 mm). temperature enhancement factor was to arc forces, neither of which are consid 1.00 1.2 Q. u = 0.8 u = 0.8 o - ,^.—-•"" S' jf Q 0.50 — ^ „--""' • 0.6 — ^ 5 • X o 1.00 y cc « 0 • ' 1.2 < u = 0.6 u = 0.6 LU _-•- _ tf) yC^ LU ^—-- • CC — ^ — 0.6 - 5 --^ • < 5 Q. 1.00 • • 1.2 ^v"" %/ O u = 0.4 • u = 0.4 — • __ —- —' - s^ *s- * 0.50 — v^ Q .,--* ~» .-"^ 0.6 X o 5 a*"" oc ^-ar"^/^ < * |1 • ,L . 1 1 l 1 1 l^.*-"| la"-! I I I i LU 0.00 0.0 If) 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. 20 - Comparison of the experimental weld depth on mild steel and Fig. 21- 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 1.00 3.03 u = 0.8 __-.— """"" 2.50 — ,*.-*• 2 00 0.50 — y ____ Q * I.50 — ^*^^ _j^^ I .00 1.00 # * u = 0.6 0.50 — * 0.50 0.00 0. 70 I .40 2.10 2. 80 Q «*** •y^ OPERATING PARAMETER, n Fig. 23 — Experimentally measured and theo retically calculated weld width vs. operating parameter for stainless steel at constant heat 1.00 . •^ distribution u = 0.9 U = 0.4 r ^^"^* Z^-~~~~~~~^ Rosenthal solution is not changed mark * 0.50 edly if temperature dependent proper Q .— %S^* ties are considered, and Malmuth (Ref. 5) <*• has shown that the latent heat has only a — y y minor effect. Recently, Oreper and Szekely (Ref. 24) showed that convec 0.00 P AK I I I I I l 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 theoretical lines 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 CT = 2.4 mm (0.09 in.). 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 CT 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 Crosh (Ref. 23) showed that the distribution parameter, CT. This important topic is dealt with in a subsequent paper (Ref. 15). 6.0 • u = 0.06 • _ U = 1.2 Conclusions •• • Aluminum S.0 ~~— Titanium 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 * e.I 20 - •/ * 3.0 — 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 i A i. i i i i strength of the new theory is that it gives 0.000 0.010 0.020 0.030 0.040 A\ «i i i an accurate functional relationship OPERATING PARAMETER, n 0.0 I .0 2.0 3.0 between both process parameters and OPERATING PARAMETER, n materials parameters. The theory pro vides a model that can be used to assess Fig. 24 —Experimentally measured and theo Fig. 25 — Experimentally measured and theo 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 distri parameter for titanium at constant heat distri material will influence the weld geome bution u = 0.06 bution u = 1.2 try. Such a model is essential to many WELDING RESEARCH SUPPLEMENT 1353-s automation and control strategies now Applied Physics 48(9):3895. Dirichlet boundary condition at infinity. being considered for welding processes. 14. Clickstein, S. S., Friedman, E., and Yenis The Green's function must satisfy the cavich, W. 1975. An investigation of alloy 600 equation: welding parameters. Welding journal A ckno wledgments 54(4):113-s to 122-s. The authors wish to express their 15. Tsai, N. S., and Eagar, T. W. Variation of V2G(r|r') - (^)2G(r/r') = - 4*- 8 (r - t'\ appreciation to Professor Nils Christensen the heat distribution with welding parameters for several helpful discussions. They are in gas tungsten arc welding-to be pub lished. grateful for financial support from the 16. Block-Bolten, A., and Eagar, T. W. 1982. (A1) 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 (A1) in spherical References Society for Metals. coordinates is: 1. Rosenthal, D. 1946. The theory of mov 17. Glickstein, S. S., and Yeniscavich, W. 1977. Review of minor element effects on the v v (r - r') ing source of heat and its application to metal 1 welding arc and weld penetration. Welding e-(r-r') ne-- treatment. Transactions ASME 43(11):849- G(r-r') = A 866. Research Council bulletin no. 266, p. 18. 2. Christensen, N., Davies, V., and Gjer- 18. Heiple, C. R., and Roper, |. R. 1981. mundsen, K. 1965. The distribution of temper Effect of selenium on GTAW fusion zone (A2) ature in arc welding. British Welding lournal geometry. Welding journal 60(8):143-s to 12(2):54-75. 145-s. 19. Cobine, |. D., and Burger, E. E. 1955. 3. Rykalin, N. N., and Nikolaev, A. V. 1971. The boundary condition that Analysis of electrode phenomena in the high- Welding arc heat flow. Welding in the World G(r — r') = 0 at infinity requires that the current arc. /. Appl. Phys. 26(7):895. 9(3/4):112-132. first term of equation (A2) vanish. Then 20. Ghent, H. W„ Roberts, D. W., Her- 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 CTA 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 (r-O The Welding Institute Conference. Conference Proceedings. London: The Weld ing Institute. 5. Malmuth, N. D., Hall, W. F., Davis, B. I., C(r-r') = ,— and Rosen, C. D. 1974. Transient thermal 21. Tsai, N. S. 1983 (February). Heat distri phenomena and weld geometry in CTAW. bution and weld geometry in arc welding. (A3) Welding lournal 53(9):388-s to 400-s. Ph.D. thesis, Massachusetts Institute of Tech nology. 6. Friedman, E. 1975 (August). Thermody namic analysis of the welding process using the 22. Myers, P. S„ Uyehara, O. A., and Bor- A symmetric heat source located at the finite element method. ASTM Transactions; man, G. L. 1967 (July). Fundamentals of heat position (w',y',—z') is needed in order lournal of Pressure Vessel Technology, p. flow in welding. Welding Research Council that the second boundary condition bulletin no. 123. 206. dG(r|r')/cz = 0 at z = 0 be satisfied. This 23. Crosh, R. |., and Trabant, E. A. 1956. 7. Peng, T. C, Sastry, S. M. L, and O'Neal, imaginary heat source leads to the fol Arc welding temperature. Welding lournal |. E. 1981. Exploratory study of laser processing 35(8):396-s to 400-s. lowing Green's function: of titanium alloys. The Metallurgical Society of AIME. Lasers in metallurgy, pp. 279-292. 24. Oreper, G.. Eagar. T. W. and Szekely, ). 1983. Convection in arc weld pools. Welding 8. Linnert, G. E. 1967. Weldability of austen lournal 62(11):307-s to 312-s. e - ^ V(w - w')2 + (y - y')2 + (Z - z')2 itic stainless steels. ASTM STP 418. 2a 9. Nestor, O. H. 1967. Heat intensity and 25. Rykalin, N. N., and Beketov, A. |. 1967. G = 2 2 current density distribution at the anode of Calculating the thermal cycle in the heat- V(w - w') + (y - y') + (z - Af high current inert gas arcs, lournal of Applied affected zone from the two dimensional out Physics 33(5): 1638-1648. line of the molten pool. Weld. Prod. 14:42- 47. 10. Shoeck, P. 1963. An investigation of anode energy balance of high intensity arcs in e V(w - w')2 + (v - v')2 + (z + z')2 argon. Modern development of heat transfer, 2a New York and London: Academic Press, pp. Appendix 2 2 2 353-478. V(w - w') + (y - y') + (z + z') A. Derivation of Green's Function 1 I. 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 laeger, ). C. 1967. (A4) equation (3) —namely: Oxford University Press. Conduction of heat in solid, pp. 255. 1. dT*/dz = 0 at Z = 0. If one considers a heat source at the 13. Cline, M. E., and Anthony, T. R. 1977. 2. T* = 0 at r = co. 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 z = 0, and the equation (4). B. Superposition of Point Heat Source Solutions by Integration 00 x 2 2 x 2 2 2 2 2 qdt' p°° „, > r A > I l~ ' + y' , ' -2xx' + x 4- y' — 2yy + y + z 1\ dT,' = (B1) 27rff2pc(47ra[t 4a(t-t') qdt d exp + + 27rff2pc(47ra[i^r/1t *' />' ( - b'H M^VI) ,/ -2x \ ( 1 _J \ ,/ -2y XI x24-y24-z2\ n (B2) X Ua(t - t')/ + Y V2CT2 + 4a(t - t')f + V Ua(t - t')/J 4a(t - t') ) 354-s | DECEMBER 1983 qdt' ex 2iro-V(4ira[t-t']) 171 P x2-Fy2 dx, dy + 2CT24-4a(t-t') 4a(t-t o)r r '^-(i ^F?)) CL O (B3) ((X'~2CT24-4a(t-t')) +(y'~2CT2 + 4a(t-t')))) L>U a 2 2 2 2 qdt ; 7r2CT 4a(t-t') /_ x 4-y z \ T 27TCT2pc(4Tra([t -1' ))3/2 4a(t -1') + 2CT2 SXP V 2(r24-4a(t -1') 4a(t -1')/ (B4) O CC L X o cc < LU tf) LU WRC Bulletin 282 cc November, 1982 Elastic-Plastic Buckling of Axially Compressed Ring Stiffened Cylinders—Test vs. Theory 0. by D. Bushnell o Concern for the safety of nuclear plants and offshore structures has stimulated efforts to determine > buckling characteristics of stiffened cylindrical steel shells. LU In this paper, B0S0R 5 computer programs were used to predict buckling loads of forty axially o I compressed mild steel cylindrical shells previously tested at Chicago Bridge & Iron Co. o Publication of this report was sponsored by the Subcommittee on Shells of the Pressure Vessel cc Research Committee of the Welding Research Council. < LU The price of WRC Bulletin 282 is $10.75 per copy plus $3.00 for postage and handling tf) (foreign 4- $5.00). Orders should be sent with payment to the Welding Research Council, 345 E. 47th St., LU Room 1301, New York, NY 10017. ce 0. o X WRC Bulletin 287 o cr < September, 1983 LU tf) Welding of Copper and Copper-Base Alloys LU by R. J. C. Dawson cr This Interpretative Report discusses the current status of fusion welding technology for copper-base materials of major industrial importance. Current welding practices for each group of copper-base materials are discussed. Literature references and suggested further reading is also presented. O _j Publication of this report was sponsored by the Interpretative Reports Committee of the Welding LU Research Council. The price of WRC Bulletin 287 is $12.00 per copy, plus $5.00 for postage and handling. > LU Orders should be sent with payment to the Welding Research Council, Room 1301, 345 East 47th Street, a New York, NY 10017. x o cr < LU tf) UJ cr WELDING RESEARCH SUPPLEMENT | 355-s