n.t. nguyen', y.-w. mai* & a. ohta^ - wit press...analytical solution for a new hybrid...

Analytical solution for a new hybrid double-

ellipsoidal heat source in semi-infinite body

N.T. Nguyen', Y.-W. Mai* & A. Ohta

^Center for Advanced Materials & Technology (CAMT),

Department of Mechanical & Mechatronic Engineering,

The University of Sydney, Australia.Material's Strength & Life Evaluation Station

National Research Institute for Metals (NRIM), Japan.

Abstract

Weld pool simulation plays a significant role in thermal stress analysis, residualstress calculation as well as microstructure modeling of welded joints andstructures. This paper aims to describe the analytical solution for a new hybriddouble-ellipsoidal power density moving heat source in semi-infinite body & itsapplication in weld pool simulation. The solution has been obtained bysuperimposing the solutions for various parts of the proposed hybrid double-ellipsoidal heat source. The weld pool geometry subjected to various heat sourceparameters has been calculated and compared with results predicted by thedouble ellipsoidal heat source and available experimental data. Very goodagreement between the predicted weld pool geometry and the measured ones hasbeen obtained. The results also show that the hybrid heat source could give animproved prediction for weld pool depth in comparison with the earlier doubleellipsoidal heat source.

1. Introduction

Temperature history of the welded components has a significant influence on theresidual stresses, distortion and hence the fatigue behavior of the weldedstructures. Classical solutions for the transient temperature such as Rosenthal'ssolutions [1] which deal with the semi-infinite body subjected to instant point,line or surface heat source, can satisfactorily predict the temperature at adistance far enough from the heat source, but fail to do so at its vicinity. Eagar

Advances in Composite Materials and Structures VII, C.A. Brebbia, W.R. Blain & W.P. De Wilde (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-825-2

9QQ Advances in Composite Materials and Structures VII

and Tsai [2] modified Rosenthal's theory to include a two dimensional (2-D)surface Gaussian distributed heat source with an uniform distribution parameterand found an analytical solution for the temperature of a semi-infinite bodysubjected to this moving heat source. Their solution was a significant stepforward for temperature prediction at near heat source regions. Even though this2-D solution using the Gausian heat sources could predict the temperature atregions closer to the heat source, they are still limited by the shortcoming of the2-D heat source itself with no effect of penetration. This shortcoming can onlybe overcome if more general heat sources are to be developed and implemented.

Goldak et al. [3] first introduced the three dimensional (3-D) DoubleEllipsoidal moving heat source and used FEM to calculate temperature field of abead-on-plate. They showed that their 3-D heat source could overcome theshortcoming of the previous 2-D Gaussian model to predict the temperature ofthe welded joints with much deeper penetration. Nguyen et al. [5] has recentlydeveloped a closed form analytical solution for this kind of 3-D heat source andshowed that this solution can be used for weld pool geometry prediction [5] andfor the calculation of residual stresses in weld-bead-on-plate [6].

In this study, a new hybrid heat source has been introduced in order to matchanalytical solutions with measured weld pool shape more closely. Analyticalsolution for this new hybrid heat source in semi-infinite body has been obtainedby superposition of the already solved double-ellipsoidal heat sources [5].Subsequently, the hybrid heat source is used to simulate the weld pool geometryand the results are compared with those generated by double-ellipsoidal heatsource and with available experimental data.

2 Analytical solution for Goldak's heat source

Goldak et. al. [3] proposed a semi-ellipsoidal heat source in which the heat fluxis distributed in the Gaussian manner throughout the heat source's volume.Later, the two different semi-ellipsoids were combined together to give a newheat source called a double ellipsoidal heat source. The heat flux within thesemi-ellipsoids is described by two different equations. For a point (x,y,z) locatedwithin the first semi-ellipsoid, which represents the front part of the weldingarc, the heat flux equation is described as

6VTr,g f 3*2 3y2 3z=l (1)Q(x,y,z) = 7=exp - —^— - ——- —-j"OA^CA/a-Va- ^ c^r a* 6&J

and for a point (x,y,z) located within the second semi-ellipsoid representing therear section of the arc as

2(;.f,z) = . :V'_-/-exp| - — -- -1 (2)

where a bh, c /and c%, are the ellipsoidal heat source parameters as describedin Fig. 1; x, y, z are the moving coordinates of the heat source; Q(x,y,z) is the


Advances in Composite Materials and Structures III 209

heat flux Q(x,y,z) at a point (x,y,z); Q is arc heat input (Q = rjIV)\ V and / arethe welding voltage and current, respectively; r] is arc efficiency; Ay and r& areproportional coefficients representing heat apportionment in front and at back ofthe heat source, respectively and /y+ /& = 2.

It must be noted here that due to the continuity of the volumetric heat source,the values ofQ(x,y,z) given by Eqs. (1) and (2) must be equal at the x = 0 plane.From that condition, another constraint is obtained for r/ and r& as Tf/c f = n,/Cftb- Subsequently, the values for these two coefficients are determined as fy =2chf/(Chf + CM)', t*b = 2chb/ ( Chf + CM)- It is worth noting here that this doubleellipsoidal distribution heat source is described by five unknown parameters: thearc efficiency 77, and the double ellipsoidal axes: #&, b c/,/and c .

Fig. 1. Double ellipsoidal power density distributed heat source

The solution for the temperature field of a double-ellipsoidal heat source in asemi-infinite body is based on the solution for an instant point source, whichsatisfies the following differential equation of heat conduction in fixedcoordinate [7]

d TI , = -jyy- . 6 X p - • ; ^ '

where a is thermal diffusivity (a — k/cp)\ c is specific heat; k is thermalconductivity; p is mass density; f, t' are time; dTt' is transient temperature dueto the point heat source 5Q and (x',y',z') is location of the instant point heatsource SQ at time t'.

Let us consider the solution of a double-ellipsoidal heat source as a result ofsuperimposing a series of instant point heat sources over the volume of thedistributed Gaussian heat source. Substituting Eqs. (1) & (2) for the heat flux at


210 Advances in Composite Materials and Structures 111

a point source into Eq. (3) and integrating over the volume of the heat source,the analytical solution for double ellipsoidal moving heat source with a constantspeed v from time t' = 0 to time t' = t becomes

T is temperature at time t and TO is initial temperature of a point (x,y,z). Moredetails about the derivation of this solution are described elsewhere [5].

3 A new hybrid double-ellipsoidal heat source

It has been reported in earlier work by Nguyen et. al. [5] that a double-ellipsoidal heat source fails to simulate the finger-tip shape of the weld poolcross-section and underestimates the weld pool depth. In this study, an improvedheat source is introduced by superimposing several semi-ellipsoids in such a waythat they could formulate a shape similar to the crater of the weld pool as shownin Fig. 2.

Fig. 2. New hybrid double-ellipsoidal heat source


Advances in Composite Materials and Structures VII "2\\

Description of the hybrid heat sourceThe new hybrid heat source consists of two volumes of the upper & lower semi-double-ellipsoids excluding the overlapping volume. The solution for this newhybrid heat source can be obtained by superimposing the solutions that werealready obtained for the individual ellipsoids and will be described below. Let usconsider the parameters of the upper semi-double-ellipsoid (USE) are a*/, 6&y,Chfi and Chbh the lower semi-double-ellipsoid (LSE) are a , 6/,a Chp and c 2.Parameters of the third middle semi-ellipsoid (MSE) created by the overlappingzone of the first two semi-double-ellipsoids are a , bhs, c and Chbs as shown inFig. 2.

Subsequently, the heat flux equation for an internal point (x,y,z) of a frontsemi-ellipsoid is described as

Qj,(x,y,z) = — 7=— - exp -^V^a,,6^c^

and of a rear semi-ellipsoid as

TO

where / = 1 to 3 for each semi-ellipsoid shown in Fig. 2. As the origin of thesecond and third semi-ellipsoid is determined by b^ then b^ = b^i - 6&Ax asshown in Fig. 2. Furthermore, since the third semi-ellipsoid is created by theoverlap of the first two semi-ellipsoids then

= C hf 2 ~ C hf ]

C hb 3 = C hb 2 =

"V1 - (8)

(9)

As rfi and r are proportional coefficients representing heat apportionment infront and back of each semi-ellipsoid, then, using the similar assumption whichhas been used for the double-ellipsoidal heat source gives

f/i +?6i +r/2 +f%2 -r/3 -%,3 = 2 (10)

Again, the condition of continuity of the volumetric heat source must besatisfied for the new hybrid heat source and this results in several additionalboundary conditions described below:

1. The heat flux portion given by the equations corresponding to the frontparts of the hybrid heat source must be the same as that in the rear parts atthe plane x = 0. This gives


Advances in Composite Materials and Structures I '//

Qf,(0,y,z)+Qj2(0,y,z-bto)-Qj3(0,y,z-bto)=Qu(0,y,z)+Qv(0,y, z-b -Q O.y, z-bj

(11)

Equation (11) is valid for any point in plane x = 0, hence, applying this for point(0,0, W gives

(12)V /

2. The heat portion given by values of Qjt(x,y,z) and Qbt(x,y,z) in Eqs. (5) and(6), respectively, must be equal at the plane z = 6/ i.e. the followingconditions are valid for point (0, 0, b ) as

Qfi (0,0,bfa) — Qf2 (0>0>bfix) ~ 8f,3 (0>0>bhx)Qbi (0,0, W = Q/,2 (0,0,bj = Qbi (0,0,bfr) (14)

or

: 111 = [/2 (15)-exp I -—;

" " " : *" = ^3 (16)

Equation (12) can be rewritten from Eqs. (15) and (16) as

exp | ^- || —^ zi— | = 0 • (17a)

or

Z/l_= _f JL_ (17b)^ hf 1 C hblFurthermore, substituting a , cy? and 0 2 from Eqs (7), (8) and (9) into (15)and (16), then simplifying gives

f/L= I/I (18)

(19)

(20)

(21)


Advances in Composite Materials and Structures 111 213

Taking r and r from Eqs. (18) and (19); r and r from Eqs. (20) and (21)then substituting to Eq. (10) gives

(22)

where =

Subsequently, six unknown variables (/y, and r&,, for / = 1 to 3) which representthe heat portions in six quarter-ellipsoids composing the new hybrid heat sourcecan be found from 6 boundary condition equations namely Eqs. (17b) and (18)to (22) as

r / -^ ~\-\(23)

(24)

I'

,>'

"A/1 1 + J 1 -

(25)

(26)

(27)

(28)

As = b/,/,1 - the new hybrid double-ellipsoidal heat source is entirelydescribed by six geometrical parameters namely a/,/, b i, c/,/?, c ,/, b and 6 .

Solution for the new hybrid heat sourceUsing the superposition principle, the solution for the new hybrid heat source isobtained as the sum of heat contributed by the first two ellipsoids less theinfluence of the third one. Let T,, 73, Tj be parts of the temperature rise of apoint (x,y,z) subjected to upper, lower and middle semi-double-ellipsoid part ofthe hybrid heat source respectively. Then

B,


2] A Advances in Composite Materials and Structures 111

where

~

' '" 12a(f-f') + cl 12 a(f-,')+<,: 12a(f-f') + 6^

7} is temperature at time / and TO is initial temperature of a point (3cj>,z/ It isworth noting here that variable z in Eq. (29) should be replaced by (z - b for /= 2 and 3. This is due to the fact that the origin of coordinate system of both thesecond and third semi-double-ellipsoid was moved down by a distance b^compared with the first one as shown in Fig. 2. Hence the solution for the newhybrid heat source is obtained as

T-To = T, + T2-T3 (30)

where 7} are determined by Eq. (29) for each part of the semi-double-ellipsoidwhich belongs to the new hybrid heat source as described above.

4 Results and discussion

Numerical procedureIn this study, a numerical procedure is applied to calculate the solution for thetransient temperature field as described by Eqs. (29) and (30) for the hybriddouble semi-ellipsoidal distributed heat source. A Fortran?? computer programis written to facilitate the integral calculation in Eq. (29) and to allow for rapidcalculation of geometry of the weld pool based on the assumed meltingtemperature of 1520°C for steel. Since the solution was obtained for a semi-infinite body, the mirror method which combines the temperature distribution ina plate of infinite thickness and its reflected images was adopted [1]. Using thisprogram, the effects of various heat source parameters (a/,;, 6&/, c i, Chbi, 67,2 andbfa) on the predicted weld pool geometry were investigated. The followingmaterial properties for high strength steel were used for the calculation: heatcapacity, c = 600 J/kg/°C; thermal conductivity, k = 29 J/m/s/ °C; density p =7820 kg/m . The welding parameters used for calculation are voltage, U=26V;current, /= 230 A; welding speed, v = 30 cm/min and arc efficiency, 77 = 0.85.Due to the length limitation of the CADCOMP'2000 paper, fully obtainednumerical results are not reported here but will be presented at the conferenceand fully described in the upcoming Australasian Welding Journal.



Experimental workBead-on-plate specimen was fabricated by using a welding robot and Gas MetalArc Welding with the following welding parameters: voltage U = 26 V, current/ = 230 A, welding speed v = 30 cm/min. Shielding gas of 80 % Ar plus 20 %CO: was supplied at 20 l/min. The base material was Japanese high strengthsteel HT780 and filler material used was MLX-60B. Their mechanical propertiesare given in Tables 1. The geometry of the weld pool were obtained from thebead-on-plate specimen by means of the metallographs taken of the weld poolshape at the surface of the welded plate and at its transverse cross-section asshown in Fig. 8. The data for the weld pool profiles were measured directlyfrom the metallographs using Adobe Photoshop.

(a) Top view (b) Transverse cross-section

Fig. 3 Geometry of the weld pool

Table 1. Mechanical properties of the materials

Materials

HT-780MIX-60B

Yield strengthSy (MPa)821601

Ultimate tensile strengthSu(MPa)859662

Elongation(%)3128

Model verificationFigures 4(a) and (b) show comparisons between the measured and predicteddata of the top view of the weld pool shape on the welded plate and itstransverse cross-section. The predicted data were calculated using the followingparameters of the heat source which provide the best fit with the measured data:ah = 10 mm, 6& = 2mm, c/,/= 10 mm, 77 = 0.85 and CM, = 2c/,/for the doubleellipsoidal heat source; a^i = 8 mm, bhi = 2mm, c#i = 8 mm, 6&c = 1.75 mm, 6«= 6 mm and 77= 0.85, Chbi = 2^ for the hybrid heat source. These parameterswere selected based on the information of their effect on the weld pool geometryreported earlier. The heat transfer material properties used for the calculationwas selected for HT-780 steel based on its ranges [8] and they were the same asreported earlier.


216 Advances in Composite Materials and Structures VII

It can be seen from Fig. 4(a) that both the present new hybrid heat source &double ellipsoidal heat source can give very good agreement with the measureddata if suitable parameters of the heat source are carefully selected. This meansthat the predicted model can be calibrated with the experimental data byselecting its heat source parameters and can be used for various simulationpurposes. However, Fig. 4(b) shows that both models still fail to predict thecomplex crater shape of the weld pool in the transverse cross-section. Thehybrid heat source model can offer slightly better prediction in term of weld pooldepth due to its improved geometry. This means that more work needs to bedone to improve the predictability of both models to match the complicatedfinger tip shape of the weld pool transverse cross-section.

Exp. data Double-ellipsoidal heat source

Fig. 4 The weld bead geometry simulation: top view of the weld pool (top) andtransverse cross-section (bottom).

5 Conclusions

An analytical solution for the transient temperature of a semi-infinite bodysubjected to a new hybrid double-ellipsoidal moving heat source has beendeveloped and used for the simulation of the weld pool geometry. Both thenumerical and experimental results from this study have shown that the newhybrid heat source could offer reasonably good prediction for the weld poolgeometry by adjusting the various heat source parameters.



It also shows greater flexibility for the hybrid heat source to calibrate withthe various shapes of the weld pool and it could offer better prediction in termsof weld pool depth when compared with previously developed double-ellipsoidalheat source. However, more work needs to be done to improve the predictabilityof the model to match with the complicated finger tip shape of the weld pooltransverse cross-section. The new hybrid heat source developed in this study hasa great potential for use in various simulation purposes such as thermal stressanalysis, residual stress calculations and microstructure modelling.

AcknowledgementThis work is currently sponsored by the University of Sydney's U2000fellowship program. The authors would like to express their sincere thanks forthe great support provided by the Department of Mechanical and MechatronicEngineering, University of Sydney for this project. Special thanks are due toDrs. Charlie Montross and Kathy Wong for their useful comments during thepreparation of this paper.

References[1] Rosenthal D. Mathematical Theory of Heat Distribution During Welding

and Cutting. Welding Journal, 20(5) pp. 220s - 234s, 1941.[2] Eager, T.W. and Tsai, N. S. Temperature Fields Produced by Traveling

Distributed Heat Sources. Welding Journal, 62(12), pp. 346s - 355s, 1983.[3] Goldak J., Chakravarti A. and Bibby M.. A Double Ellipsoid Finite Element

Model for Welding Heat Sources, IIWDoc. No. 212-603-85, 1985.[4] Painter M.J, Davies M.H, Battersby S., Jarvis L. and Wahab M.A. A

literature review on Numerical Modelling the Gas Metal Arc WeldingProcess. Australian Welding Research, CRC. No. 15, Welding TechnologyInstitute of Australia, 1996.

[5] Nguyen, N.T., A. Ohta, K. Matsuoka, N. Suzuki and Y. Maeda. AnalyticalSolution for Transient Temperature in Semi-Infinite Body Subjected to 3DMoving Heat Sources, Weld. Res. Supp., Welding Journal, 78(8), pp. 265s-274s, 1998.

[6] Nguyen, N.T., A. Ohta, K. Matsuoka, N. Suzuki and Y. Maeda. AnalyticalSolution of Double-Ellipsoidal Moving Heat Source and Its Use forEvaluation of Residual Stresses in Bead-on-Plate. Proc. of Int. Conf. OnFracture Mechanics & Advanced Engineering Materials, Eds: Lin Ye & YiuWing Mai, Dec. 8-10, 1999, University of Sydney, Australia, pp. 143-149,1999.

[7] Carslaw, H.S and Jaeger, J.C. Conduction of heat in solids, OxfordUniversity Press, pp. 255, 1967.

[8] Radaj D. Heat Effects of Welding: Temperature Field, Residual Stress,Distortion. Springer-Verlag: Berlin, New York, London, Paris and Tokyo,pp.28, 1992.


n.t. nguyen', y.-w. mai* & a. ohta^ - wit press...analytical solution for a new hybrid...

Documents