Prediction of Carbon Distribution in Welded Joints of Steels

Acta Metallurgica Slovaca, 11, 2005, 3 (259 - 265) 259

PREDICTION OF CARBON DISTRIBUTION IN WELDED JOINTS OF STEELS Dobrovsk .1, ehkov L.1, Dobrovsk J.1, Strnsk K.2, Dobrovsk V.1 1 FMME VB TU Ostrava, Czech Republic, [email protected] 2 FME VUT Brno, Czech Republic PREDIKCE ROZLOEN UHLKU VE SVAROVCH SPOJCH OCEL Dobrovsk .1, ehkov L.1, Dobrovsk J.1, Strnsk K.2, Dobrovsk V.1 1 FMMI VB TU Ostrava, esk republika, [email protected] 2 FSI VUT Brno, esk republika Abstrakt Pochody, pi kterch difunduj intersticiln prvky asto proti vlastnmu koncentranmu spdu, se oznauj jako obrcen difuze, i difuze do kopce (up-hill diffusion). Vzhledem k tomu, e zvlt pi nzkch teplotch (500 a 800 C), kdy vtina svarovch spoj ocel v energetickch zazen dlouhodob pracuje, je difuze substitunch prvk ve srovnn s prvky intersticilnmi prakticky zanedbateln, je vhodn pouvat pi popisu difuze intersticilnch prvk ve svarech ocel termnu redistribuce (perozdlen) a o jejich difuzi v danm systmu hovoit jako o difuzi kvazistacionrn. Tmto termnem, zavedenm ji v roce 1966 [1] se rozum, e substitun prvky difunduj v tuhch roztocch eleza bez ohledu na prvky intersticiln, zatmco intersticiln prvky se mus k rozloen prvk substitunch, napklad jejich koncentraci na rozhran spoje, dokonale pizpsobit. V pspvku je pedloen inenrsk model kvazistacionrn difuze uhlku ve svarech ocel, pouiteln v praktickch aplikacch. Obecn je naznaena monost pomrn rychl predikce koncentranch charakteristik uhlku v okol svarovho rozhran ocel (1) a (2), v nm ocel (1) pedstavuje uhlkovou, poppad nzkolegovanou ocel a ocel (2) pedstavuje stedn a vysokolegovanou ocel, legovanou prvkem jako je napklad chrm, kter termodynamickou aktivitu uhlku siln sniuje. Pedloen fenomenologick een redistribuce uhlku ve svarovch spojch typu Fe-C-j, kde prvek j je substitun, umouje predikovat rozloen uhlku ve spojch tvoench ocelemi (1) a (2), kter pedstavuj tuh roztoky stejnho typu, napklad typu (1)-austenit/(2)-austenit. een je spojit pro x na intervalu x(,) a pro t 0. Abstract The processes, during which interstitial elements diffuse often against their own gradient of concentration, are labelled as reversed diffusion, or up-hill diffusion. Due to the fact that namely at low temperatures (500 to 800 C), when majority of welded joints of steels in power engineering equipment works on a long-term basis, diffusion of substitutive elements in comparison with interstitial elements is practically negligible, it is appropriate to use at description of diffusion of interstitial elements in welded joints of steel the term of redistribution and to describe their diffusion in the given system as quasi-stationary diffusion. It is understood by this term, which was introduced already in 1966 [1], that substitutive elements diffuse in solid solutions of iron regardless of interstitial elements, while interstitial elements must perfectly accommodate to a distribution of substitutive elements, e.g. by their concentration at the welded joint boundary.


The paper deals with a model of quasi-stationary diffusion of carbon in welded joints of steels. The model is presented in the form of engineering interpretation; the reason for such approach is its practical application. The possibility of relatively prompt prediction of carbon concentration characteristics in the neighbourhood of the boundary of welded joints of two steels (steels (1) and (2)) is presented in general manner. Steel (1) represents carbon or low-alloy steel and steel (2) represents medium- up to high-alloy steel, alloyed with element such as e.g. chromium which significantly minimises carbon thermodynamical activity. The presented phenomenological solution of carbon redistribution in welded joints of the type Fe-C-j, where the element j is a substitutive element, enables prediction of carbon distribution in joints formed by the steels (1) and (2), which represent solid solutions of the same type, e.g. of the type (1)-austenite/(2)-austenite. Solution is continuous for x in the interval x(,) and for t 0. Key words: Quasi-stationary diffusion, welded joints of steels, interstitial elements, carbon 1. Introduction Interstitial elements in iron and steel usually include hydrogen, boron, carbon and nitrogen. Characteristic feature of these elements consists in the fact that they diffuse under the same temperature in solid solutions of iron and steel faster by several orders than substitutive elements. This comparatively high mobility of interstitial elements leads to situations, when their concentration can accommodate by an order more rapidly to changes of concentration of substitutive elements than the other way round. This phenomenon manifests itself most distinctively at the boundaries of welded joints of steels with different chemical composition. That is to say that in dependence on thermodynamic interaction of interstitial and substitutive element at the point of contact of two solid solutions of different chemical composition the interstitial element can diffuse in one or another direction, in spite of the fact that concentration of substitutively deposited elements at the boundary of the joint practically does not change. The processes, during which interstitial elements diffuse often against their own gradient of concentration, are labelled as reversed diffusion, or up-hill diffusion. Due to the fact that namely at low temperatures (500 to 800 C), when majority of welded joints of steels in power engineering equipment works on a long-term basis, diffusion of substitutive elements in comparison with interstitial elements is practically negligible, it is appropriate to use at description of diffusion of interstitial elements in welded joints of steel the term of redistribution and to describe their diffusion in the given system as quasi-stationary diffusion. It is understood by this term, which was introduced already in 1966 [1], that substitutive elements diffuse in solid solutions of iron regardless of interstitial elements, while interstitial elements must perfectly accommodate to a distribution of substitutive elements, e.g. by their concentration at the welded joint boundary. Problem of redistribution of interstitial element between two ternary solid solutions of different chemical composition was described in the works [1] by method of thermodynamics of irreversible processes. Due to lack of necessary diffusion and thermodynamic data and to its comparative complexity, requiring numerical solution, this approach started to be employed in a more important extent only recently. In many practical tasks, which require rapid solution of redistribution of carbon in welded joints of steel, it is sufficient to use an engineer solution of the problem of quasi-stationary diffusion of carbon in steels, applications of which were described


in detail for example in the works [2] and [3]. The above mentioned solution is described phenomenologically, in analytical form and today there is available relative abundance of diffusion, thermodynamic and phase data necessary for its application. This paper describes engineer model of carbon redistribution and outlines its use in practice. 2. Phenomenology of carbon redistribution in steels The basis pre-requisite consists of the experimentally verified finding, that in solid solution Fe-C-j, where carbon is interstitial element and j is substitutive element, the substitutive element j diffuses in solid solution in such a manner that it is not influenced by interstitial carbon C, while carbon must perfectly accommodate to position and concentration of substitutive element. Distribution of substitutive element in welded joint of two semi-endless samples of steel can be expressed in dependence on time by the following equation

( )

=

tDxerfc

NN

NtxN

jjj

jj

221,

)2()1(

)2(

, (1)

where ( )txN j , is concentration of substitutive element j in dependence on the distance x and in time t, )2()1( , jj NN are initial concentrations in steels (1) and (2) of the welded joint (in time t = 0 for x (,0) is )1(jN , for x (0,) is )2(jN and in time t > 0 is )1(jN in x a )2(jN in x +), concentrations of N being expressed in atomic fractions; Dj is diffusion coefficient of substitutive element in the welded joint, while the function erfc(z) = 1 erf(z) comprises Gauss error integral erf(z) with the argument z, and boundary between the two steels has position x = 0. Equation (1) is continuous for x in the whole interval x (,) and t > 0, and it comprises the assumption that diffusion of substitutive element is driven by its concentration gradient. Solution of redistribution of interstitial carbon C is based on the presumption that driving power is gradient of its thermodynamic activity, whereas thermodynamic activity of carbon in the welded joint Fe-C-j can be expressed in dependence on the distance x and time t in the initial state t = 0 by the following equations:

( ) ),().,(, txNtxtxa CCC = (2) )1()1()1( . CCC Na = , (3) .. )2()2()2( CCC Na = (4)

In equations (2) to (4) the values ),,( txC )1(C and )2(C are coefficients of thermodynamic activity of carbon and ),,( txNC )1(CN and

)2(CN are concentrations of carbon in

atomic fractions, the significance of which, from the viewpoint of their distribution, is analogical to significance of concentrations of substitutive element j. Coefficients of thermodynamic activity of carbon, expressing influence of substitutive element, can be written in a logarithmic form with use of interaction coefficients of the first order jC defined by Wagner [4] in the form of equations

),,(.),(ln txNtx CjCC = (5) ,.ln )1()1( C

jCC N = (6)


,.ln )2()2( Cj

CC N = (7) provided that interaction coefficient of the type CC will be equal to 1=CC . By division of equations (5) to (7) by interaction coefficient jC it is possible to separate a concentration of the element j from these equations

.ln

,ln

,),(ln

),()2(

)2()1(

)1(j

C

CCj

C

CCj

C

CC NN

txtxN

=== (8)

On the left side of the equation (1) we can now substitute individual concentration members of substitutive element j with use of individual shares of logarithm of activity coefficient C and interaction coefficient jC , and interaction coefficients in the equation thus created can be reduced. As a result of this substitution and operations we get the expression

=

tDxerfc

tx

jCC

CC

221

lnln

ln),(ln)2()1(

)2(

, (9)

which expresses distribution of logarithm of coefficient of thermodynamic activity of carbon C in the given welded joint in dependence on distribution of concentration of the substitutive element j. It is possible to further modify this equation to this expression:

= tD

xerfc

C

CCC

jtx

221

)2(

)1()2(),( , (10)

which describes course of activity coefficient of carbon in the whole interval x (,) in time t 0. It equally follows from the equation (10), that activity coefficient is only function of distribution of concentration of substitutive element j. On condition of perfect redistribution of interstitial carbon it is now sufficient for distribution of substitutive element to leave propagation of thermodynamic activity of carbon in the welded joint its natural course, so that in dependence on time this thermodynamic activity of carbon smoothly equals between the two steels (1) and (2) according to diffusion and thermodynamic characteristics of carbon in the given type of the welded joint. For illustration let us now choose the concentrations and thermodynamic effects of the substitutive element j on carbon in such a manner, that the following relation is valid

)2()1(CC aa > . In such a case carbon will redistribute from the steel (1) into steel (2), where it has

lower thermodynamic activity. Let us choose the steel with activity )1(Ca to be on the left side of the welded joint, and the steel with activity )2(Ca to be on the right side. Steel (1) will be in this case decarburised and steel (2) of the same welded joint will be carburised. Carbon at the boundary of the welded joint from the left and from the right will in conformity with thermodynamic interaction accommodate to distribution of substitutive element in such an extent, that there will be created a quasi-equilibrium concentration of carbon )2()1( , qC

qC NN .


According to the second Fick law it is possible to express the re-distributed amount of carbon, passing from the left side of the welded joint (1) to its right side (2) with respect to validity of the law of conservation of mass by the equation

tDNNtDNN Cq

CCCCq

C)1()1()1()2()2()2(

=

, (11)

in which diffusion coefficients )2()1( , CC DD are related to the second Fick law and they are in general case reciprocally different. In the given case of selection of position of steels in the welded joint and thermodynamic effect of substitutive element on carbon for quasi-equilibrium concentrations the following inequalities are valid: )1()1( qCC NN > and )2()2( CqC NN > . Quasi-equilibrium concentrations of carbon fulfil in connection with the corresponding values of activity coefficients of the same element the conditions of local thermodynamic equilibrium

.)2()2()2( ),0()1()1()1(

),0( CCtxCCCtxC NaNa === (12) It follows from this equation, which is valid for t > 0 and x 0 (for discrete welded boundary of joint the quasi-equilibrium concentrations of carbon do not change in time), that activity of carbon from the left side of the welded joint is equal to the activity on the right side of the welded joint. It is possible to modify the equation of conservation of carbon mass with use of relations for its activity see equations (3) and (4) and relations (12) into this form:

)2(

)2(

)1(

)1(

)2(

)2()2(

)1(

)1()1(

)2()2()1()1(

C

C

C

C

C

CC

C

CC

qCC

qCC

DD

Da

Da

NN

+

+== . (13)

Two essential consequences follow from this equation, which was derived in different ways independently by the authors [5] and [6], for description of carbon redistribution in the joint of two solid solutions: 1) If the following relation is valid for the envisaged joint between Fick diffusion coefficients and carbon activity coefficients

)2(

)2(

)1(

)1(

C

C

C

C DD

= , (14)

then on the boundary of the welded joint formed by both steels (1) and (2) there is established a quasi-equilibrium carbon activity according to the equation

)(21)(

21 )2()2()1()1()2()1()2()2()1()1(

CCCCCCq

CCq

CC NNaaNN +=+== , (15) the validity of which was experimentally verified in welded joints of the steel of the types austenite/austenite [7].


2) If the equations (14) and (15) are valid for redistribution of interstitial carbon in the welded joint of both steels, which have the same type of solid solution, it is then possible to define in conformity with the equation (14) the thermodynamic coefficient of carbon diffusion by this relation

)2()2(

)1(

)1(

C

C

C

CaC

DDD == (16)

Thermodynamic coefficient of carbon diffusion thus defined is in both steels of the joint identical and at the same time independent on carbon activity, and it is also independent on its concentration. Coefficient aCD is, however, dependent on manner of selection of standard state for thermodynamic carbon activity. Equation (16) is in effect an expansion of the finding discovered by Birchenall and Mehl [8] at investigation of binary solid solutions Fe-C to ternary solid solutions Fe-C-j. It equally follows from this equation that for thus defined thermodynamic carbon coefficient it is possible to write for Fick coefficients the following:

2)1()1(

= CaCC DD and 2)2()2(

= CaCC DD . (17) Equations (11) to (17) were derived and also experimentally verified for discrete (very narrow, step) boundary of substitutive element j in the welded joint of both steels (1) and (2), i.e. for the case of stationary diffusion of interstitial element [9]. Definition of thermodynamic coefficient of carbon diffusion can be, nevertheless, with a certain extent of engineer tolerance expanded also to a width of zone of redistribution of substitutive element j in realistic welded joints, which are larger than in case of spot welded joints. In this case it is possible to write for Fick diffusion coefficients one definition equation [ ]2),(),( txDtxD CaCC = , (18) which can be used for the whole interval x (,) a t 0, and thus introduced coefficient of carbon diffusion is independent both on its concentration and its activity, and is it only function of distribution of substitutive element j and its impact on thermodynamic activity of carbon. In this way it is possible to use for solid solutions Fe-C-j the second Fick law modified analogically, as it was made by the authors [8] for solid solutions Fe-C already before, and to write the diffusion equation for propagation of activity in the form:

[ ]2

22),(

x

atxD

ta C

CaC

C

=

. (19) Due to the fact that activity coefficient ),( txC is only function of influence of substitutive element on carbon activity, it is possible to find solution of equation for propagation of activity for the given initial and boundary conditions of two semi-endless welded samples in the form:

=

tDtx

xerfcaa

atxa

aCCCC

CC

),(221),(

)2()1(

)2(

, (20)


where course of the function ),( txC is expressed by the equation (10). By substituting carbon activities with concentrations with use of equations (2) to (4) we obtain the final relation for distribution of carbon concentration in the welded joint

),(

),(221

),(

)2()2()1()1()2()2(

tx

tDtx

xerfcNNN

txNC

aCC

CCCCCC

C

+= , (21)

where the parameter ),( txC is determined by the above mentioned equation (10). 3. Conclusion The presented phenomenological solution of carbon redistribution in welded joints of the type Fe-C-j, where the element j is a substitutive element, enables prediction of carbon distribution in joints formed by the steels (1) and (2), which represent solid solutions of the same type, e.g. of the type (1)-austenite/(2)-austenite. Solution is continuous for x in the interval x(,) and for t 0. Examples can be found in the Ref. [3]. Acknowledgments This paper has been prepared thanks to financial support of the Ministry of Education, Youth and Sports of the Czech Republic, project No. MSM6198910013. Literature [1] Adda Y., Philibert J.: La diffusion dans les solides. Presses Universitaires de France, Paris

1966. [2] Strnsk k.: Termodynamika kvazistacionrn difze uhlku v ocelch a jej aplikace

[Thermodynamics of quasi-stationary carbon diffusion]. ACADEMIA, Praha 1977. [3] Pilous V., Strnsk k.: Structural Stability of Deposits and Welded Joints in Power

Engineering. Cambridge Internatiomal Science Publishing, London 1998, 176 pp. ISBN 1898326088.

[4] Wagner C.: Thermodynamics of Alloys. Addison-Wesley Press, Inc., Cambridge 1952, 180 pp.

[5] Stark J.P.: Metallurgical Transactions A, Vol. 16 A, 1980, p. 1797. [6] Kuera J., Strnsk K.: Metallurgical Transactions A, Vol. 13 A, 1982, No. 9, pp. 1658-

1659. [7] Kuera J., Strnsk, K.: Materials Science and Engineering, 52, 1982, No.1, pp.1-38. [8] Birchenall C.E., Mehl R.F.: Trans. AIME, 173, 1947, p. 143. [9] Kuera J., Million B., Strnsk K.: Czech. J. Phys., B35, 1985, pp. 1355-1361.

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Prediction of Carbon Distribution in Welded Joints of Steels