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WELDING RESEARCH

-S237WELDING JOURNAL

We depend in our everyday life on theperformance of vast quantities of steel,which we use without giving it a secondthought. This is possible because the ma-terial is reliable and cheap (weight forweight, some 1000 times cheaper thanpotato chips). Yet, there are remarkabletechnologies and sciences working behindthe scenes to create ever better steels thatcan be assembled into awe inspiring struc-tures.

Examples include the giant oil rigs thatpepper the North Sea, the oil and gaspipelines that traverse the frozen wastes,and the 101 Tower in Taiwan, which is nowthe tallest building in the world — Fig. 1.These are all made from steel and rely onwelding for their assembly.

Weld Design: Experiment orModel?

A weld is a heterogeneity introducedinto a carefully manufactured steel. It is a

defect that has to be managed. One wayof doing this is through a deep under-standing of metallurgy, thereby avoidingthe engineering disasters of the kind thatplagued for example, the Liberty ships. Aweld consists of distinct zones, each ofwhich is the consequence of a particularinteraction between heat flow and thephase transformation characteristics ofthe weld metal and the base metal.

Out of all these zones, the weld metalis particularly challenging to design be-cause there is little that can be done oncethe weld is completed (Refs. 1–3). Thiscontrasts with wrought steel, which can beprocessed and manicured using all the fa-cilities available in a modern steel plant.There is, therefore, a major industry de-voted to the design and manufacture ofwelding alloys. Academic activity has sup-ported this industry, both in terms of theunderlying science and in the creation ofquantitative methods for alloy design.Given all this effort, I shall use this lectureto explore whether it is any longer neces-sary to conduct experiments when devel-oping welding consumables. Are themodels sufficiently robust to be exploitedby industry without supervision?

The focus of this paper is on ferriticsteels, which form the bulk of the billionor so tons of steel consumed annually.The development of welding consum-ables involves all weld-metal tests in

which a joint is deposited such that sam-ples can be obtained without dilution withthe base plates. It is fortunate that the lit-erature is rich in data from tests of thiskind, data which have been liberally ex-ploited in the development of models.

Defining Characteristics of aWeld Metal

The essential variables needed inorder to be able to calculate the mi-crostructure and properties of steel weldmetal are illustrated in the top row of Fig.2 (Refs. 4, 5). The chemical composition,cooling conditions, and austenite grainsize capture all the essential features ofweld metal. The essence of the weldingprocess and joint geometry is expressedvia the cooling curve and there are manymodels with varying levels of sophistica-tion capable of predicting the change intemperature as a function of time and po-sition. As will be seen later, the chemicalcomposition defines the thermodynamicsof transformation and the nature of theheterogeneities that arise.

The austenite grain structure, a para-meter sometimes ignored in the interpre-tation of microstructures, has a profoundeffect.

Cooling Curve

There is a huge amount of researchthat has been devoted solely to the calcu-lation of the thermal cycle associated withwelding. The most sophisticated of mod-els account for joint geometry, gravita-tional forces, surface tension effects,buoyancy forces, electromagnetic forces,metal transfer, changes in thermophysicalproperties with temperature and turbu-lence, etc. (Ref. 6). Weld pool shapes canbe fairly accurately calculated and whenthere are deviations, empirical correc-

SUPPLEMENT TO THE WELDING JOURNAL, September 2004Sponsored by the American Welding Society and the Welding Research Council

KEY WORDS

ConsumablesCooling CurveExperimentationFerritic SteelModelingSolid-State Transformation

Reliability of Weld Microstructure andProperty Calculations

The AWS 2004 Adams Lecture explores modeling vs. experimentation fordeveloping welding consumables for new alloys

BY H. K. D. H. BHADESHIA

HARSHAD K. D. H. BHADESHIA was se-lected to present the 2004 Comfort A. Adams lec-ture at the AWS Annual Meeting, Chicago, Ill. Heis a professor of physical metallurgy in the Depart-ment of Materials Science and Metallurgy at theUniversity of Cambridge, U.K., where he receivedhis Ph.D. He has held the positions of universitydemonstrator, university lecturer, reader in physi-cal metallurgy, and Royal Society LeverhumeTrust Senior Fellow.

He specializes in the theory of solid-state trans-formations with emphasis on the prediction of mi-crostructural development in complex metallic al-loys, particularly for multicomponent steels asapplied to engineering applications.

He was elected a Fellow of the Royal Society,the Institute of Physics, the Institute of Materials,and the Royal Academy of Engineering, and thisyear was awarded Consulting Professorship byHarbin Institute of Technology, China. He has re-ceived other honors from institutions in Brazil,Japan, India, and the United States. He can becontacted at www.msm.cam.ac.uk/phase-trans.

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tions are made in a manner constrained bythe physics. From the present point ofview, it is really the cooling curve that isimportant, particularly in the regimewhere solid-state transformations occur,i.e., between about 900°C and the marten-site-start temperature; this must be thereason why the time ∆t800–500 is a popularmeasure in the welding industry. Thisgreatly simplifies the problem because thecooling rate (dT/dt) within the weld metalcan, to a high level of accuracy, be repre-sented independently of position with asimple equation (Refs. 7, 8).

(1)

where Q is the heat input per unit length,η is the transfer efficiency, and Ti the pre-heat or interpass temperature. There arenumerous weld cooling curves available inthe welding literature; for a particularprocess and weld geometry, this equationcan be fitted to derive the empirical con-stants C1 and C2. Such constants are avail-able for a large variety of weldingprocesses and this simple procedure worksrather well in practice. Rigorous calcula-tions in which the weld pool is properlymodeled make only a small difference tothe calculated microstructure. This is be-cause the cooling rate in the transforma-tion range is relatively insensitive to fluidflow phenomena.

The Austenite Grain Structure

The columnar austenite grains in theweld deposit derive, in most cases, fromcolumnar δ-ferrite grains that grow epi-taxially from the fusion surface during theearly stages of solidification. The colum-nar shape is quite different from the

equiaxed grains found in most steels andrequires a different approach in definingthe amount of austenite grain surface perunit volume (SV). It has been demon-strated that the columnar grains can be de-scribed as hexagonal prisms (length c andside a) in three dimensions. It follows thattwo stereological parameters, the meanlineal and mean areal intercepts, areneeded to quantify this anisotropic grainstructure. However, because the grainsare much longer than they are wide, agood approximation is that the mean lin-eal intercept measured on transverse sec-tions of the weld (Ltn), in a direction nor-mal to the columns, adequately describesSV (Ref. 9). There exist equations thatthen relate SV to the chemical compositionof the weld metal and the heat input.

The Alloying Elements

A weld metal may contain twenty ormore deliberate solutes, and others thatare introduced accidentally during deposi-tion: C, Mn, Si, Ni, Mo, Cr, V, Co, B, N, O….. Some of these, such as boron, may bepresent in minute quantities and yet canhave a profound effect on the microstruc-ture. Solutes act on steels by two essentialmechanisms:

1) The relative stabilities of austenite(γ) and ferrite (α) are affected via a ther-modynamic effect, which can be expressedrigorously in terms of the difference inGibbs free energies, Gα – Gγ, often calledthe “driving force” for transformation.This thermodynamic quantity feeds di-rectly into rate theory, for example in theequations governing classical nucleation.It is now routinely possible to calculatethese free energies and, of course, to ex-press them in terms of equilibrium phasediagrams. Such calculations can now beconducted routinely for multicomponent,

multiphase steels, using proprietary orfree software.

2) The second effect is more subtle be-cause it depends on the rate at whichchange occurs. The equilibrium solubilityof an alloying element is never identical inaustenite and ferrite. If circumstances per-mit, the solute will therefore tend to par-tition between the phases during thecourse of transformation. The requireddiffusion may then limit the kinetics of theprocess. This is a gross effect involvinglarge numbers of atoms and distancescomparable to the size of the transformedproduct. Another kinetic effect of equalimportance can be triggered by minuteconcentrations of solute; misfitting atomscan segregate to interfaces. In doing so,they reduce the interfacial energy per unitarea. This is the mechanism by whichboron renders austenite grain boundariesless effective as heterogeneous nucleationsites for ferrite. Traces of boron can there-fore have a huge influence on hardenabil-ity, far in excess of that expected from itsinfluence on Gα – Gγ.

Solid-State Transformations

Allotriomorphic Ferrite

Having described the three parametersthat are seminal in the development of weldmetal microstructure, we now address thecomplex array of phase changes that occuras the weld metal cools (Refs. 1–5). The es-sential features of the transformed mi-crostructure are illustrated schematically inFig. 3. It consists of allotriomorphic ferriteα, Widmanstätten ferrite αw, acicular ferriteα, and the so-called microphases, whichmight include small amounts of martensite,retained austenite, or degenerate pearlite.Bainite consisting of sheaves of parallelplatelets is also found in some weld de-

dT

dt

C T T

Q

i

C

=−( )1

2

η

Fig. 1 — The 101 Tower in Taiwan, made from steel andglass. Photograph courtesy of Thomas Sourmail.

Fig. 2 — Flowchart showing the progress of weld metal microstructure and property calcu-lations.

Bhadeshia--9/04 Corr 8/9/04 10:11 AM 38

WELDING RESEARCH


posits, particularly of the type used in thepower generation industry.

These phases all evolve during cooling,beginning with the decoration of austenitegrain boundaries with layers of allotri-omorphic ferrite. They seem to nucleatereadily and then thicken at a rate con-trolled by the diffusion of carbon in theaustenite (γ) ahead of the α/γ interface.The substitutional elements do not parti-tion; this kind of transformation is said tooccur under paraequilibrium conditions.Theory shows that the thickness (q) of theferrite varies parabolically with time (t)

(2)

where α1, the parabolic thickening rateconstant, is given by the solution of

(3)

and xγα xαγ are the paraequilibrium carbonconcentrations in austenite and ferrite re-spectively at the interface (obtained usingthe thermodynamic calculations de-scribed earlier), x is the average carbonconcentration in the alloy and D is aweighted average diffusivity of carbon inaustenite, given by

(4)

where D is the diffusivity of carbon inaustenite at a particular concentration ofcarbon. Some example calculations are il-lustrated in Fig. 4A, which shows that the

thickness is most sensitive to the carbonconcentration when the latter is close tothe solubility of carbon in ferrite; this isbecause the need to partition carbon de-creases as the average concentrationx→xαγ. We shall see later that this explainssome of the carbon equivalent equationsprevalent in industry. Figure 4B showsthat the the nucleation stage of the layersof ferrite can justifiably be neglected be-cause the volume fraction vα correlatesstrongly with α1 for a large variety of weldmetals.

Welds cool continuously, so the aboveequations need to be integrated over thetemperature range Th to Tl, the start andstop temperatures for α. Th is estimatedusing calculated time-temperature-trans-formation (TTT) diagrams (Ref. 10) (Fig.5) and Scheil’s rule. Tl is taken to be thepoint where displacive transformationsbecome kinetically favored. Once thethickness of the layers of α has been cal-culated, it is straightforward to relate it tothe volume fraction of ferrite using thegeometry of the austenite grains

(5)

so that the dependence on austenite grainsize becomes obvious.

Widmanstätten Ferrite

Widmanstätten ferrite plates grow withparaequilibrium and lengthen at a rate(G) controlled by the diffusion of carbonahead of the plate tips. Because of the dis-placements associated with the transfor-mation, it is necessary to account for strainenergy. The plates are confined within theaustenite grains in which they nucleate,and they grow so fast that crossing the

grains within a fraction of a second is typ-ical — Fig. 6A. The theory used here is rig-orous and proven, but the volume fractionof Widmanstätten ferrite (vW) hardly cor-relates with G — Fig. 6B.

The discrepancy arises because by thetime the weld has cooled to induce Wid-manstätten ferrite, acicular ferrite sproutsfrom inclusions dispersed within theaustenite grains. There is, therefore, com-petition for the austenite that remains anda strong possibility of impingement be-tween intragranularly nucleated acicularferrite and Widmanstätten ferrite. As il-lustrated in Fig. 7, when the alloy contentis small, the rapid growth of α and αW con-sumes much of the austenite, thereby re-ducing the ability to form αa. By contrast,in high-hardenability welding alloys, theacicular ferrite has an opportunity to de-velop and indeed to stifle the penetrationof Widmanstätten ferrite into the austen-ite grains.

The volume fraction of Widmanstättenferrite that can form is therefore a func-tion not just of the growth rate, but also ofthe thickness of allotriomorphic ferrite,the time required for it to grow across anaustenite grain, and the geometry of theaustenite grains.

(6)

where C4 is a constant independent ofalloy composition and t2 is a function ofthe impingement process. As will be seenlater, with this accounting, good agree-ment is obtained with experiments.

Finally, it is worth saying somethingabout acicular ferrite, which is a highly de-sirable phase (Refs. 1–5). Its microstruc-ture consists of intragranularly nucleated

v C G a q t aW = − { } ( )4 2

22

2 4 30 2( deg / tan

vq

a qaα =

{ }− { }

2 30

2 22

tan 30deg

tan deg/

DD x dx

x xx

x

= { }−∫ γα

γα

2

41

12

D x x

x x

D D

π

α α α

γα

γα αγ

−

−=

exp erfc

2

1

q t= α 11 2/

Fig. 3 — Schematic representation of the mi-crostructure of weld metal.

Fig. 4 — A — Isothermal thickening of ferrite layers as a function of the carbon concentration; B — cor-relation of parabolic rate constant against va for a variety of welds.

A B

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SEPTEMBER 2004-S240

plates, which radiate in many directionsfrom point nucleation sites. This leads toa chaotic microstructure, which is good atdeflecting cracks. It is therefore strongand tough. There is much evidence to sug-gest that acicular ferrite is intragranularlynucleated bainite, the heterogeneous nu-cleation sites being the complex non-metallic inclusions common in welds, ei-ther as impurities or as deliberateadditions. There is considerable qualita-tive understanding on the type of inclu-sions that are most favorable.

There also exist quantitative methodsof estimating the type of inclusion that willform during solidification and subsequentcooling. The quantitative details of thecalculation of acicular ferrite are dis-cussed elsewhere (Ref. 11).

Solidification-Induced Segregation

In the discussion above, it has been as-sumed that the chemical composition ofweld metal is uniform. This is not the casein practice because welds cool rapidly andthere may be uncontrolled variations inthe welding conditions. A good estimateof the magnitude of segregation comesfrom the partition coefficient ki, which isthe ratio of solute concentration i in thesolid to that in the liquid. The coefficientcan easily be calculated, and used to givethe compositions of the solute-rich andsolute-poor regions of the weld (carbon isvery mobile so it is assumed to be uni-formly distributed). This information canthen be used to estimate the effect on Thfor allotriomorphic ferrite, which then in-fluences all subsequent transformations.

To summarize, virtually every compo-nent of weld microstructure is amenableto calculation. Some examples are pre-sented in Fig. 8. Predictions like these

have been extensivelyvalidated using publishedexperimental data and bydesigning new experi-ments.

An interesting predic-tion to emerge fromthese calculations is thatthe microstructure is sen-sitive to the carbon whenits concentration is com-parable to its solubility inferrite. It seems that thewelding industry has im-plicitly recognized this byproposing two differentequations for the carbonequivalent.

(7)

(8)

For the IIW version, the dependence ofCE on substitutional solutes such as man-ganese is greater than for the CE reservedfor low-carbon steels. This is because inthe low-carbon steels, it is the carbon thathas a greater influence on the kinetics oftransformation.

Yield Strength

For an individual phase, the strengthcan be factorized into a number of intrin-sic components (Ref. 12).

(9)

where xi is the concentration of a substitu-tional solute which is represented here bya subscript i. The other terms in this equa-tion can be listed as follows:

KL strengthening due to grain size,115 MN m–1.

KD dislocation strengthening, 7.34× 10–6 MN m–1

σFe pure, annealed Fe, 219 MN m–2

at 300 KσSSi substitutional solute (i) strength-

eningσC solid solution strengthening due

to carbon

σ σ σ σ

ρ

= + + +

{ } +

∑Fe ii

ssi c c

L D D

x x

K L K 05.

Ito - Besseyo <0.18 wt -% C

CE CSi

30

Mn Cu Cr

20Ni

60

Mo

15

V

105B wt -%

= + + + + +

+ + +

IIW >0.18 wt-% C

CE CMn Si

6

Ni Cu

15Cr Mo V

5wt-%

= + + + + +

+ +

Fig. 5 — Calculated TTT diagrams for a variety of alloys. Each diagramconsists of two C-curves, the higher temperature one representing re-constructive reactions (e.g. α) and the lower curve, displacive transfor-mations (αW, αa, αb).

Fig. 6 — A — Calculated growth rate of αW; B — poor correlation be-tween volume fraction and growth rate of Widmanstätten ferrite.

A

B


WELDING RESEARCH


ρD dislocation density, typically1016 m–2

L measure of the ferrite plate size, typ-ically 0.2 µm

Given the microstructure, the yieldstrength can be estimated from that of theindividual phases, either as a rule of mix-tures, i.e., σ = ∑i σi Vi where Vi is the frac-tion of phase i, and σithe correspondingstrength. More sophisticated methods forcombining the effects of the differentphases are also available, but suffice it to saythat given the microstructure, it is possible,using well-established strengthening the-ory, to estimate the weld metal yieldstrength.

Complex Properties

The fabrication of useful devices andstructures is based on more than just thestrength. Properties such as fatigue, tough-

ness, stress-corrosion resistance, creep re-sistance, etc. are routine considerations inthe design process (Ref. 13). These proper-ties are “complex” in the sense that they canbe measured and used in design but cannotbe predicted. There is no theory that has therigor or sophistication to handle the largenumber of variables that are known to con-trol such properties.

The conventional way to approachsuch problems is to apply regression analy-sis in which experimental data are best-fit-ted to some function, which is usually lin-ear. The result is an equation in whicheach of the inputs xj is multiplied by aweight wj; the sum of all such products anda constant θ then gives an estimate of theoutput y = ∑j wj xj + θ. This is preciselyhow the carbon-equivalent equations arederived.

A neural network is a more generalmethod of regression analysis. As before,

the input data xj are multiplied by weights,but the sum of all these products forms theargument of a hyperbolic tangent (Refs.14, 15). The output y is therefore a non-linear function of xj, the function usuallychosen being the hyperbolic tangent be-cause of its flexibility. Combining many ofthese functions increases the availableflexibility. A few of the advantages of thenetwork over conventional regression canbe listed as follows:

1. There is no need to specify a functionto which the data are to be fitted. Thefunction is an outcome of the process ofcreating a network.

2. The network is able to capture al-most arbitrarily nonlinear relationships.

3. With Bayesian methods, it is possibleto estimate the uncertainty of extrapolation.

We shall now discuss these last twopoints in detail.The complexity and flexi-bility of the relationship that can be cre-

Fig. 7 — The hexagons are cross sections of columnar austenite grains and thedots represent inclusions.

Fig. 8 — An example of the agreement achieved between the measured andcalculated weld metal microstructures.

Fig. 9 — Illustration of the complexity of the function (of two inputs) that can be created using a simple neural network with just four hidden units. The two fig-ures are generated for the same mathematical function but different weights.


WELDING RESEARCH

SEPTEMBER 2004-S242

ated is illustrated here (Fig. 9) using justfour hyperbolic tangent functions (“hid-den units”) for two input variables

(10)

The shape of the surface illustratedchanges dramatically as the weights are al-tered from w = 3.5 to w = 10. Imaginewhat can be done using many more hyper-bolic tangents, variables, and weights.

The point about extrapolation is illus-trated in Fig. 10. The input values 2, 3, and4 represent experimental data. Both thelinear and nonlinear functions exactly rep-resent the experimental data, but makedifferent predictions when it comes to

input values 5 and6, i.e. when thefunctions are usedto extrapolate be-yond the experi-mental data. It isimpossible, withoutphysical under-standing, to choosebetween these two.

This could be interpreted as a cri-sis, but instead, thedifference in thepredicted valuescan be taken as anindication of theuncertainty of ex-trapolation. Thisuncertainty arisesbecause the func-tions representingthe data extrapo-late differently. It isextremely useful tohave this indicationof uncertainty when

dealing with nonlinear functions that arenot physically based. MacKay’s work hasbeen seminal in expressing neural net-works in a Bayesian framework so that themodeling uncertainties become transpar-ent (Ref. 16).

There are now examples where neuralnetworks, in combination with microstruc-tural calculations and experience, haveshort-circuited the development of weld-ing alloys. A number of these cases aredocumented in the series of books on theMathematical Modelling of Weld Phenom-ena I–VII. One desirable feature of thenetwork models is that they are readily up-dated as more experimental data becomeavailable. This is often paraphrased bysaying that the models continue to learnand extend their knowledge base.

Proposal for Alloy DevelopmentProcedures

Mathematical models of welding willnever replace experiments — the problemis too complicated to deal with. There is nodoubt, however, that so much tremendousprogress has been made that it can justifi-ably be argued that the process of weldingconsumable design should begin with cal-culations in the manner summarized inFig. 11.

Once the need for the development ofa new alloy is identified and design re-quirements have been formulated, thefirst step should be an attempt at calcula-tions. There is a plethora of freely accessi-ble software that can be used for this pur-pose. If a convincing theoretical solutionemerges, a critical experiment can be de-signed in which a consumable is manufac-tured and tested.

There are circumstances where the cal-culations will look reasonable but the un-certainties (error bars) associated with theoutcomes are large. A series of carefullydesigned experiments can be imple-mented to resolve the uncertainties. Itwould be of long-term benefit to the com-munity if the data associated with these ex-periments are published so that the mod-els can be modified for greater reliability.

If, at the decision stage (Fig. 11), cal-culations are impossible because the ap-propriate models do not exist (for exam-ple, creep-fatigue theory), it is justified toconduct empirical experiments based onexperience.

It will be fascinating to see how alloydesign progresses over the next few years,whether the practitioners are courageousenough to try using models and whetherthose who develop models have the ap-preciation necessary to recognize thecomplexity of welding. The models dis-cussed here are freely available on

y wx x w

wx z w

= −{ } + −

+

+{ } + −

+

tanh tanh

tanh tanh

1 12

2 22

2

2 1

Fig. 10 — Two functions that exactly match the experimental data (2, 4, 6) extrapolate differently.

Fig. 11 — Recommended paths in the process of alloy development.


WELDING RESEARCH


http://www.msm.cam.ac.uk/map/map-main.html.

Finally, it should be emphasized thatmuch of this work would not have beenpossible without the enormous amount ofexperimental work and contributions tounderstanding from a large number ofpeople involved in the subject. Referencesto that work are listed in Refs. 1–14. I apol-ogize that it is not possible to present themin this short article.

Acknowledgments

I am immensely grateful to the Ameri-can Welding Society for the opportunity topresent this lecture and to many col-leagues in industry and academia whohave helped my career in welding metallurgy.

References

1. Grong. Ø. 1997. Metallurgical Modelling ofWelding, 2nd edition, London: Institute of Ma-terials, pp. 1–602.

2. Easterling, K. E. 1992. Introduction to thePhysical Metallurgy of Welding, 2nd edition, But-terworths-Heineman.

3. Svensson, L.-E. 1994. Control of Mi-crostructures and Properties in Steel Arc Welds,London: CRC Press.

4. Bhadeshia, H. K. D. H., Svensson, L.-E.,and Gretoft, B. 1985. Acta Metallurgica 33:1271–1283.

5. Bhadeshia, H. K. D. H, and Svensson, L.-E. 1993. Mathematical Modelling of Weld Phe-nomena. Eds. H. Cerjak and K. E. Easterling,London: Institute of Materials, pp. 109–182.

6. Mundra, K., DebRoy, T., Babu, S. S.,David, S. A., and Paul, A. J. 1995. Modelling ofCasting, Welding and Advanced SolidificationProcesses, VII. Eds. M. Cross and J. Campbell,TMMMS, pp. 31–38.

7. Weisman, C. ed. 1981. Welding Handbook,7th edition, Miami, Fla.: American Welding Society.

8. Svensson, L.-E, Gretoft, B., andBhadeshia, H. K. D. H. 1986. ScandinavianJournal of Metallurgy 15: 97–103.

9. Bhadeshia, H. K. D. H., Svensson, L.-E.,

and Gretoft, B. 1986. Journal of Materials Sci-ence 21: 3947–3951.

10. Bhadeshia, H. K. D. H. 1982. Metal Sci-ence 16: 159–165.

11. Bhadeshia, H. K. D. H. 1992. Interna-tional Trends in Welding Research. Eds. S. A.David and J. M. Vitek, Materials Park, Ohio:ASM International, pp. 213–222.

12. Bhadeshia, H. K. D. H., and Svensson,L.-E. 1997. Mathematical Modelling of WeldPhenomena III. Eds. H. Cerjak and H. K. D. H.Bhadeshia, London: Institute of Materials, pp.229–284.

13. Bhadeshia, H. K. D. H. 1999. Trends inWelding Research. Eds. S. A. David, T. DebRoy,J. A. Johnson, H. B. Smartt, and J. M. Vitek,Materials Park, Ohio: ASM International, pp.795–804.

14. Bhadeshia, H. K. D. H. 1999. ISIJ Inter-national 39: 966–979.

15. Bhadeshia, H. K. D. H. 2001. ISIJ Inter-national 41: 626–640.

16. MacKay, D. J. C. 2003. Information The-ory, Inference and Learning Algorithms, U.K.:Cambridge University Press.

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