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FINITE ELEMENT MODELINGAND SIMULATION OFWELDING. PART 2:IMPROVED MATERIALMODELINGLars-Erik LindgrenPublished online: 29 Jul 2006.

To cite this article: Lars-Erik Lindgren (2001) FINITE ELEMENT MODELING ANDSIMULATION OF WELDING. PART 2: IMPROVED MATERIAL MODELING, Journal ofThermal Stresses, 24:3, 195-231, DOI: 10.1080/014957301300006380

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FINITE ELEMENT MODELING AND SIMULATION OFWELDING. PART 2: IMPROVED MATERIAL MODELING

Lars-Erik LindgrenLuleÔ University of Technology

and Dalarna University SE-971 87 LuleÔ, Sweden

Simulation of welding has advanced from the analysis of laboratory setups to realengineering applications during the last three decades. This development is outlinedand the directions for future research are summarized in this review, which consistsof three parts. The material modeling is maybe the most crucial and dif¢cult aspectof modeling welding processes. The material behavior may be very complex forthe large temperature range considered.

The development of welding procedures is based on performing experiments(Goldak et al. 1990) and a Welding Procedure Speci¢cation (WPS) is the ¢nal result.The evaluation of a welding procedure is based on joint integrity, absence of defects,microstructure, and mechanical testing, for example. Computational methods arerarely used in the process of developing welding procedures. It is expected thatsimulations will complement the experimental procedures for obtaining a WPS, sinceaspects like residual stresses can then be considered when comparing differentwelding procedures. Furthermore, simulations are also useful in designing the manu-facturing process as well as the manufactured component itself. Distortions areusually in focus in the ¢rst case, whereas residual stresses are of interest in the lattercase.

This review concentrates on the simulation of fusion welding processes ofmetals. These processes do have several common traits and are therefore similarto the model. Publications presenting ¢nite element simulations of the mechanicaleffects of welding appeared in the early 1970s, and simulations are currently onlyused in applications where safety aspects are very important (like aerospace andnuclear power plants) or when a large economic gain can be achieved. The scope

Journal of Thermal Stresses, 24:195^231 , 2001Copyright # 2001 Taylor & Francis0149-5739 /01 $12.00 + .00

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Received 12 August 1999; accepted 12 April 2000.This work has been ¢nanced by NUTEK (the Swedish National Board for Industrial and Technical

Development) via the Polhem Laboratory. Its completion has also been made possible by the cooperationwith ABB Atom AB and Volvo Aero Corporation in a project about multipass welding where this reviewwas one part.

Address correspondence to Professor L.-E. Lindgren, Department of Mechanical Engineering, LuleÔUniversity of Technology, SE-971 87 LuleÔ, Sweden. E-mail: lel@cad.luth.se

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of most simulations has been to obtain residual stresses and correspondingdeformations. There are also publications presenting analyses of hot crackingand other phenomena.

A common concern in the simulation of welding is to account for the interactionbetween welding process parameters, the evolution of the material microstructure,temperature, and deformation. The resultant material structure and deformationmay also be needed, in combination with in-service loads, to predict lifetime andperformance of a component. Therefore, research in this ¢eld requires the collab-orative efforts of experts in welding methods, welding metallurgy, material behavior,computational mechanics, and crack propagation, for example. It is important toobserve that uncertain material properties and net heat input make the successof simulations signi¢cantly dependent on experimental results (Lindgren 1996).

The review is split into three parts. Part 1 is ``Increased Complexity’’; this is part2, ``Improved Material Modeling.’’ Part 3 is called ``Ef¢ciency and Integration.’’They all outline the development of simulation of welding and are separated intosections where different aspects of ¢nite element modeling are the focus. Each sec-tion contains some recommendations based on the review and the experience ofthe author. It is hoped that this approach will be appropriate for those that areentering this ¢eld of research and useful as a reference for those already familiarwith this subject.

Material modeling is important for all computational models. The materialmodel and pertinent data must represent the real material behavior with suf¢cientaccuracy. What is suf¢cient depends on the focus of the performed study. This articleshows that there are different requirements depending on the scope of the analysis. Itis also shown how the effect of phase changes has been accommodated in the models.The in£uence of simpli¢cations of the material behavior has been studied in severalpapers. The simpli¢cations are necessary due to both lack of data and numericalproblems when trying to model the actual high-temperature behavior of the material.

IMPROVED MATERIAL MODELING

Material modeling is, together with the uncertain net heat input, one of the majorproblems in welding simulation (Goldak 1989, Lindgren 1996). The thermal analysisis, in general, more straightforward than the mechanical analysis. It entails fewnumerical problems, with the exception of the large latent heat during the solid-liquid transition, and it is easier to obtain the thermal rather than the mechanicalproperties of a solid. McDill et al. (1990) investigated the relative importance ofthe thermal and mechanical properties of stainless and carbon steels in weldingsimulations. Two bars of dimension 20 in 2 in 0.5 in were ``welded’’ alongthe free edge. No ¢xture was used. One bar was made of a stainless steel andthe other of a carbon steel. The resultant radius of curvature was compared withthe computed values for the material properties of carbon steel (MS) or stainlesssteel (SS). The results are shown in Table 1. The somewhat unexpected conclusionwas that the thermal properties play a more important role than the mechanicalproperties in explaining the different behavior between these steels. This is due

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to the fact that the thermal dilatation is the driving force in the deformation and thebar is free to bend. The thermal dilatation is determined by the temperature ¢eld andis, therefore, strongly in£uenced by the thermal properties.

Dependency on Temperature and Microstructure

The complete thermomechanical history of a material will in£uence its material pro-perties. However, this can be approximated to a dependency on the current tem-perature and deformation for many materials. This simpli¢cation may be toolarge for ferritic steels where solid-state phase transformations occur that will in£u-ence the thermal dilatation and the plastic behavior of the material in a way thatwill affect the residual stresses. Different approaches for modeling this dependencyare discussed. Also see the discussion about latent heats in the section ``Propertiesfor Modeling Heat Conduction’’ and about transformation-induced plasticity inthe section ``Properties for Modeling Plastic Deformation.’’ Leblond et al. (1997)discuss the consequences of phase transformations. The in£uence of the stresseson the phase transformations are usually ignored and are not discussed in this article.

The simplest and most common approach is to ignore the microstructure changeand assume that the material properties depend only on temperature. The effect ofphase changes may be ignored for austenitic steels (e.g., Ueda et al. 1986, Rybickiet al. 1977, 1978, Brown and Song 1992a, b) and materials such as copper (Lindgrenet al. 1997) and Inconel (Friedman 1975, Nickell and Hibbitt 1975, Ueda et al. 1991).But phase changes have also been ignored in the case of ferritic steels (e.g., Ueda andYamakawa 1971a, Hibbitt and Marcal 1973, Mok and Pick 1990, Shim et al. 1992,Michaleris 1996, Ravichandran et al. 1997, Lindgren et al. 1999). Then, for example,the thermal dilatation is the same during heating and cooling, as in Figure 1a. Theenvelope technique and the lumping of weld passes, which are sometimes usedin simulation of multipass welding (see the section ``Increasing Complex Models’’in Part 1), modify or even completely ignore the temperature history. In some casesthe stresses are only computed during the cooling phase (Hepworth 1980, Freeand Porter Goff 1989). Canas et al. (1996) investigated different simpli¢cations

Table 1 Curvature for combinations of thermal and mechanical properties (McDillet al. 1990)

Thermal Mechanical Radius of CurvatureTest Properties Properties [m¡1]

A MS MS 41.6B SS SS 11.9C MS SS 29.6D SS MS 16.3

Experiment MS MS 23.3Experiment SS SS 8.1

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Figure 1. Different methods for modeling dependency of thermal dilatation on phase changes in ferriticsteel.

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of welding analysis and found that an envelope technique for the welding of analuminum plate gave acceptable results for the longitudinal residual stresses.Surprisingly, they found only small differences if they used constant ortemperature-dependent mechanical properties. Mahin et al. (1988) simpli¢ed thecooling phase by arti¢cially quenching the welded plate when the maximum tem-perature reached 600 C. This was done to shorten the cooling time. They usedan explicit code and could not increase the length of their time steps during coolingbut they did not perform this in their later analysis (Mahin et al. 1991) becauseit affected the history-dependent internal variables in their material model.

However, there appeared a need to account for the phase transformations forferritic steels even if their role was not fully appreciated. The effect of the phasetransformations on the thermal dilatation was included ¢rst by Ueda andYamakawa (1971b). Andersson (1978) also accounted for this effect. The effecton yield limit and Young’s modulus was included by Ueda et al. (1976, 1977).Ueda et al. (1976) used one curve during heating and another curve for the materialproperties during cooling to account for the effect of the phase transformations onthe material properties, as shown in Figure 2. Phase transformations were assumedto occur instantaneously at speci¢c temperatures. The material became fullyaustenized at 750 C during heating, and the austenite decomposed at 600 C.The latter temperature was the average of the start temperatures for the formationof bainite and martensite. The material was assumed to lose its strength rapidly justabove 750 C during heating and gained strength just at 600 C during cooling. Asimilar approach was also used by Andersson (1978) and other researchers(Josefson 1982, Jonsson et al. 1985a, Troive et al. 1989, Karlsson and Josefson1990). It is also based on given property-temperature curves, but the curves wereless idealized than those used by Ueda et al. (1976). Different curves are chosenin the analysis depending on some characteristics of the temperature history atthe considered point in the model. Usually these characteristics are the peak tem-perature and the cooling rate between 800 C and 500 C. They are the primaryparameters that determine the obtained microstructure of steels. The cooling rateis approximately the same in the entire heat-affected zone. Therefore, differentproperty-temperature curves are chosen during the cooling phase depending onthe peak temperatures (see Figure 1b in the case of thermal dilatation). A single,common curve is used during the heating phase. Different curves are chosen duringcooling depending on the value of the peak temperature, Tpeak . It is also possible tointerpolate between these curves with respect to the peak temperature instead ofjust switching between them. The curves can be obtained from experiments wheretest specimens are subjected to some temperature histories typical of the processand tested at different temperatures (Lindgren et al. 1993) or compiled fromthe literature. Voss et al. (1998a, b) suggested that varying cooling rates in theheat-affected zone can be accounted for by constructing a maximum-temperaturecooling-time diagram. This diagram then gives the microstructure of the material.The latter can then be used to determine the properties of the material. They onlyapplied this to the hardness, which was compared with measurements. Theydid not use the diagram for any other material properties.

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Papazoglou and Masubuchi (1982) used a CCT diagram to estimate the phasechanges and the corresponding volume changes. Oddy et al. (e.g., 1989) also esti-mated the phase transformations and used a mixture rule for the austenite propertiesand the decomposition product’s properties (e.g., pearlite forms austenite, whichdecomposes back to bainite). The formation/decomposition is assumed to occurlinearly with temperature within a given temperature interval. Murty et al. (1996)used a TTT diagram to estimate the transformations that occur in a welded pipe.This information was used to predict yield stresses and volume changes due tothe phase transformations in the weld metal. This yielded an improvement inthe results when compared to the simulation of the same pipe by Lindgren andKarlsson (1988), Karlsson and Josefson (1990), and Josefson and Karlsson (1992).

Figure 2. Determination of yield stress by Ueda et al. (1976, 1977). Tp is the maximum temperature inearlier heating cycles and Tc is the maximum temperature in current heating cycle of multipass weld.

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The most £exible way to include the effect of the temperature history is to computethe evolution of the microstructure in the material. Each phase is assignedtemperature-dependent properties, and simple mixture rules are used to obtainthe macroscopic material properties. This coupling of thermal, metallurgical,and mechanical models (TMM) was used by Inoue and Wang (1983), Wang andInoue (1983, 1985), Bergheau and Leblond (1991), Devaux et al. (1991), Inoue (1996,1998), Roelens (1994, 1995a, b), and BÎrjesson and Lindgren (1998, 1999). The in£u-ence of grain size and microstructure, for example, on the mechanical properties isdiscussed by Bhadeshia (1997). An example of a mixture rule is shown in Figure1c, where the thermal expansion coef¢cient is given one constant value for austeniteand another value for the other phases. The difference in volume between theaustenite, g, and its decomposition products is accounted for by etr, which, in thiscase, is this difference extrapolated to 0 C. The phase change may be approximatedor computed according to some model as discussed previously. The microstructuremodel used by BÎrjesson and Lindgren (1998, 2000) uses the chemical compositionas input, and its prediction can be improved by supplying data from a measuredTTT diagram (Figure 3). Veri¢cation of the microstructure model is performedby comparing with results from CCT curves. The model was applied to the samemultipass weld that was studied by Lindgren et al. (1999). Lindgren et al. (1999)assumed only temperature-dependent data, whereas BÎrjesson and Lindgren (2000)computed the material properties based on the microstructure evolution. Das etal. (1993) also computed the microstructure but used this only to account fortransformation-induced plasticity (see the discussion in the section about``Properties for Modeling Plastic Deformation’’). Dufrene et al. (1996) used simplemodels for phase changes between martensite and austenite for the case ofEB-welding. This was combined with mixture rules for the material properties.However, they do not give any details of the model or the results. Ronda et al. (1995)formulates TMM models, but no simulations are given in their paper.

Myhr et al. (1997, 1998a) simulated the welding of aluminum tubes and the buttwelding of aluminum plates (Myhr et al. 1998b). They studied the importance ofaccounting for the microstructure evolution, namely, strengthening effects of hard-ening precipitates. The thermal analysis was followed by a computation of the dis-solution and recombination of precipitates and ¢nally a mechanical analysis.The importance of accounting for the in£uence of precipitates on the plastic £owwas studied. See also Grong (1994, 1997) for more details.

Recommendations. Once again, the recommendation is to aim for accuracy. Themicrostructure evolution is important to include in the material modeling of ferriticsteels since their properties can change a lot due to the phase transformations.The major in£uence is on the thermal dilatation and the yield stress, which canbe modeled as discussed previously. It is a problem to account for phase changesin the case of multipass welding. One should have the ability to compute themicrostructure and predict the effect on the material properties, or it will be necess-ary to simplify the history dependency of the material properties to one cycle ofheating and cooling (Ueda et al. 1976) or even ignore it (Lindgren et al. 1999).It should be noted that the needed number of measured curves, such as for the

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thermal dilatation of material subjected to different peak temperatures, depends onthe mesh. The intermediate curves between the heating curve and the curve corre-sponding to the highest peak temperature all are assigned to integration pointsin the heat-affected zone. Thus, it is of no use to have many curves if the modelhas just one element across this zone.

Density

Simulation of welding can usually be performed as a quasi-static analysis (see thesection ``Simulation of Welding as a Coupled Problem’’ in part 1). The density,r, is needed even for a quasi-static analysis because it is multiplied with the heatcapacity in the thermal analysis. Handbooks may give the density as a functionof temperature. This accounts for the volume change due to the thermal dilatation.However, one must know how the density is handled by the chosen ¢nite elementcode. A constant density may be suf¢cient as input if the code itself computesthe change in density when deformations are computed simultaneously with the tem-peratures (e.g., in a staggered approach). A temperature-dependent density may beused for a pure thermal analysis where no deformations are included.

Figure 3. Measured TTT diagram (solid lines) and computed TTT diagram (dashed lines) from BÎrjessonand Lindgren (1997).

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Properties for Modeling Heat Conduction

The solution of the heat conduction equation requires heat conductivity, l, andheat capacity, c. The density was discussed previously. These thermal propertiesare temperature dependent. They may also depend on the temperature historysince different phases may have different thermal properties. Furthermore, thelatent heats due to phase changes are needed. These depend on the phasetransformations and thereby on the temperature history and the currenttemperature. The discussion in the section ``Dependency on Temperature andMicrostructure’’ describes different approaches on how this latter dependencycan be handled or ignored.

The dependency on temperature is completely ignored by those usingRosenthal’s or other analytic solutions (see the section about ``Improved Modelsfor Heat Input’’ in part 1). Free and Porter Goff (1989) and Dong et al. (1997) alsoassumed constant thermal properties despite the use of the ¢nite element methodfor computing the temperatures. Their statement that large variations in thermalproperties result in very small changes of the transient temperatures is surprisingwhen considering the data in Figures 4 and 5, for example. It is also inconsistentwith the ¢ndings of McDill et al. (1990). It may be possible to obtain good residualstresses away from the heat-affected zone with this simpli¢cation (Lindgren 1986),but this is no general conclusion. Dong (1997) and Reed et al. (1997) also used con-stant thermal properties in their ¢nite element model. Ueda and Yamakawa usedconstant thermal properties in their early analyses (1971a, 1971b) but includedthe temperature dependency in later papers (e.g. Ueda et al. 1976, 1977). Goldaket al. (1985) and Moore et al. (1985) discussed and showed the effects of using con-stant or varying thermal properties.

Most numerical models include the temperature dependency of the thermalproperties but not any effects of the phase transformations other than the associ-ated latent heats. The latent heats due to the solid-solid phase changes are oftenignored and only the heat of fusion is included in the models. The effect of thelatent heats due to the solid phase transformation is shown, for example,by Dubois et al. (1984). The magnitude of these latent heats can be estimatedfrom diagrams by Pehlke et al. (1982). They give a latent heat of about75 kJ/kg for the a-to-g phase change in steels. Murty et al. (1996) used 92 kJ/kgfor the g-to-pearlite phase change and 83 kJ/kg for the g-to-martensite phasechange. The heat of fusion, for example, is 277 kJ/kg for a 1.2% C-steel withTsolidus 1387 C and Tliquidus of 1481 C. There is, in general, no problem inobtaining thermal properties, such as those in Figures 4 and 5 from Richter(1973). Pehlke et al. (1982) also give data for many materials. See also thereferences in Tables 2^4.

The latent heats give effective heat capacitiesölatent heat divided by the tem-perature interval for phase changeöwhich are large compared to the heat capacityshown in Figure 5. This makes the heat conduction equation stiff. For example,a pure metal with a given heat input will experience a temperature increase, butsuddenly, at the melting temperature, the temperature will not increase since theheat input is consumed by the phase change. This abrupt change in the input-output

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Figure 4. Heat conductivity for some steels. Data from Richter (1973).

Figure 5. Heat capacity for some steels. Data from Richter (1973).

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relation constitutes a stiff equation. This is the only numerical problem in the ¢niteelement solution of the heat conduction problem. It can be handled in different ways,including some of the following.

The enthalpy method (Andersson 1978, Jonsson et al. 1985a) can be used toreduce the aforementioned nonlinearity. Then the enthalpy, heat content, is usedas the primary unknown in the ¢nite element solution. It is de¢ned as

H…T † ˆT

0rc…t† dt …1†

This is a monotonous increasing function w.r.t. the temperature and will reduce thenonlinearity when latent heats are consumed/released (see, e.g., Karlsson (1986)).

Table 2 Reference with material properties, ferritic steels

Reference Comments

Tall (1964) No hardeningUeda and Yamakawa (1971a, b) No hardeningUeda and Nakacho (1982)Ueda et al. (1976, 1977) No hardening, power law creep during annealingUeda et al. (1988, 1991) Piecewise linear, isotropic hardeningRybicki and Stonesifer (1980)Rybicki et al. (1988)Oddy et al. (1989)Troive et al. (1989)Karlsson (1989), Karlsson and Josefson (1990)Tekriwal and Mazumder (1989, 1991a,b) Kinematic hardeningJosefson (1982) Creep druing stress reliefBae (1994)Ueda (1986, 1991)Hibbitt and Marcal (1973) Creep during stress reliefLobitz (1977)Shim et al. (1992)Brown and Song (1992a,b)Ma et al. (1995)Andersson (1978)Argyris et al. (1982, 1985) Viscoplastic, Ramberg^OsgoodLindgren and Karlsson (1988)Sheng and Chen (1992), Chen and Sheng

(1991, 1992)Viscoplastic, Bodner^Partom and Walker

Wikander et al. (1993, 1994)Yuan et al. (1995)Michaleris et al. (1995)Siva Prasad and Sankaranarayana n (1996)Murty et al. (1994, 1996)Michaleris and DeBiccari (1997) von Mises, kinematic hardeningRavichandran et al. (1997)Taljat et al. (1998)Hong et al. (1998b)Kim et al. (1998)

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Another numerical approach is the ¢ctitious heat source method (Rolph and Bathe1982) used by Murty et al. (1996). A third approach, used by Lindgren et al. (1999),is based on an effective heat capacity (Ch 5 in Lewis et al. 1996) and is easy toimplement into a ¢nite element code; it is sometimes also called an enthalpy method.The heat capacity used in the heat conduction equation is replaced by

ceff ˆHn ¡ Hn¡1

T n ¡ Tn¡1 if Tn ¡ T n¡1 6ˆ 0 …2†

where the right superscripts denote time-step counters in the ¢nite element solutionprocess.

The enthalpy method, Eq. (1), will work even for a pure metal. The othermethods work well for alloys where the melting occurs over a temperature range,but they may experience problems for pure alloys. The most straightforwardway to reduce the nonlinearity is to extend the temperature range over whichthe latent heat is released/consumed. Feng (1994) and Feng et al. (1997) used a more

Table 3 References with material properties, austenitic steels

Reference Comments

Ueda et al. (1986)Ueda et al. (1991) Piecewise linear, isotropic hardeningRybicki et al. (1977, 1978, 1979, 1986)Rybicki et al. (1988)Chidiac and Mirza (1993) ViscoplasticLejeail (1997) Kinematic hardeningTroive et al. (1998)Mahin et al. (1991), Ortega et al. (1998),

Dike et al. (1998)Rate-dependent, intern variable model with

combined hardening

Table 4 References with material properties, other materials

Material Reference Comments

Inconel Ueda et al. (1991) Piecewise linear, isotropc hardeningInconel Friedman (1975), Nickell and

Hibbitt (1975)Inconel NÌsstrÎm et al. (1989, 1992)Aluminum Canas et al. (1996) Kinematic hardeningAluminum Feng et al. (1997) Detailed model at high temperatureAluminum Dong et al. (1998a,b) ,

Yang et al. (1998, 2000)Aluminum Michaleris et al. (1997)Copper Lindgren et al. 1997

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advanced model for the release of latent heat during solidi¢cation of aluminum.Their model gives a higher release of latent heat in the upper part of the Tsolidus

to Tliquidus interval. Otherwise it is typically assumed that the latent heat is evenlydistributed over this interval.

There is a modeling consideration of the melting behavior that sometimes has tobe taken into account in the thermal analysis. The heat in the weld pool is not onlyconducted but also convected due to the £uid £ow. Many analysis imitate thisby increasing the thermal conductivity at high temperatures (e.g. Andersson 1978,Hepworth 1980, Papazoglou and Masubuchi 1982, Jonsson et al. 1985a, Leungand Pick 1986, Das et al. 1993, Michaleris and DeBiccari 1997). Ronda and Oliver(1998) and Voss et al. (1998b) used different conductivities in different directionsat varying locations in the weld pool. This was taken from Pardo and Weckman(1989).

Recommendations. There is no reason to use the analytic solutions any more sincethe numerical simulation of the thermal ¢eld is quite straightforward. It is importantand straightforward to account for the temperature dependency of the thermal pro-perties. Latent heats are also important to include. The use of effective heat capacityin handling the latent heats works well. If the chosen ¢nite element code does nothave this capability, then it might be necessary to extend the temperature range overwhich the latent heat is distributed.

Introduction to Mechanical Properties

It is assumed that the deformation can be decomposed into a number of components.The increment in total strain is computed from the incremental displacements duringa nonlinear ¢nite element analysis. The elastic part of the strain gives the stresses,and there are a number of inelastic strain components that can be accounted for.The decomposition is expressed in terms of strain rates as

_eij ˆ _eeij ‡ _ep

ij ‡ _evpij ‡ _ec

ij ‡ _ethij ‡ _etp

ij …3†

where _eij is the total strain rate, _eeij is the elastic strain rate, _ep

ij is the plastic strain ratedue to rate-independent plasticity, _evp

ij is the viscoplastic strain rate, _ecij is the creep

strain rate, _ethij is the thermal strain rate consisting of thermal expansion and volume

changes due to phase transformations, and _etpij is the transformation plasticity strain

rate. The additive decomposition of the strain rate can be derived from amultiplicative decomposition of the symmetric part of the deformation gradient(Simo 1998). The book by Simo and Hughes (1997) contains detailed discussionsabout material modeling and its mathematical framework and computationalimplications. The stress increment is computed from strain increments given bystrain rates when using a hypoelastic approach. The stress increment must comefrom an objective stress rate in the case of a large deformation analysis. Thereare several possibilities w.r.t. large deformation analyses available. The basicequations are outlined in Appendix A in Part 3 of this review.

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A welding simulation must at least account for elastic strains, thermal strains,and one more inelastic strain component in order to give residual stresses. Theplastic, viscoplastic, and creep strains are all of the same nature. They representdifferent mathematical models for the plastic deformation (See Figure 9 in the sec-tion where the varying plastic phenomena at different temperatures are discussed).The following sections deal with material properties for the thermoelastic behavior(_eij ˆ _ee

ij ‡ _ethij ), and thereafter the plastic properties are discussed that give the plastic

components (_epij ‡ _evp

ij ‡ _ecij) and then the modeling of transformation plasticity (_etp

ij ).The mechanical analysis requires much more time due to more unknowns per

node than in the thermal analysis. Furthermore, it is much more nonlinear dueto the mechanical material behavior. The mechanical properties are more dif¢cultto obtain than the thermal properties, especially at high temperatures, and they con-tribute to the numerical problems in the solution process. This is discussed sub-sequently in more detail. Nearly all papers about welding simulations includetemperature dependencies of at least some mechanical properties (an exceptionis Ca·as et al. 1994). The treatment of the dependency on the temperature historywas discussed in the section ``Dependency on Temperature and Microstructure.’’

The high-temperature mechanical behavior is modeled in an approximate waydue to several factors: experimental data is scarce, too-soft material causes numeri-cal problems (Hepworth 1980, Leung and Pick 1986), and it is found thatapproximations introduced do not signi¢cantly in£uence the resultant residualstresses. Many analyses use a cut-off temperature above which no changes inthe mechanical material properties are accounted for. It serves as an upper limitof the temperature in the mechanical analysis. The meaning of using a cut-off tem-perature may vary since some studies only apply this cut-off to some properties.Ueda and Yamakawa (1971a, b) did not heat the material higher than 600 C,and Fujita et al. (1972) limited the maximum temperature to 500 C. Ueda et al.(1985) assumed that the material did not have any stiffness above 700 C. They calledthis the mechanical rigidity recovery temperature, above which the Young’s moduluswas set to zero. It is therefore likely that this corresponds to the cut-off temperature.Hepworth (1980) used 800 C as a cut-off temperature above which he did not includeany thermal dilatation. Free and Porter Goff (1989) completely ignored the tem-perature dependency of Young’s modulus, neglected hardening, and used 900 Cas a cut-off temperature. Furthermore, they only followed the cooling phase ofthe compiled temperature envelope. Tekriwal and Mazumder (1991a) varied thecut-off temperature from 600 C up to the melting temperature. The residualtransverse stress was overestimated by 2 to 15% when the cut-off temperaturewas lowered.

Properties for Modeling Thermoelastic Deformation

The thermoelastic properties are used to compute the stresses. The stress-strainrelations have to be written in incremental form since the nonlinear analysis isperformed by a time-stepping procedure. Strain increments give stress incrementsaccording to Hooke’s law. The stress-strain relations are usually separated into volu-metric and deviatoric parts. The ¢rst part is purely thermoelastic in the case of

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incompressible plasticity (see the description of plasticity later). For convenience,only the rate-independent plastic strain is included in the subsequent equations.Assuming that the thermoelastic behavior can be described by the linear Hooke’slaw gives

skk ˆ 3Keekk ˆ 3K…ekk ¡ 3eth† …4†

sij ˆ 2Geeij ˆ 2G…eij ¡ ep

ij† …5†

where K ˆ E/3…1 ¡ 2n† is the temperature-dependent bulk modulus, G ˆ E/2…1 ‡ n†is the temperature-dependent shear modulus, E is Young’s modulus and n isPoisson’s ratio, sij is the stress tensor, sij ˆ sij ¡ skkdij/3 is the stress deviator tensor,eij is the total strain tensor, ee

ij is the elastic strain tensor, eij ˆ eij ¡ ekkdij/3 is the totalstrain deviator tensor, ee

ij is the elastic strain deviator tensor, eth is the uniaxialthermal dilatation, and ep

ij ˆ epij is the plastic strain tensor, which is equal to the

deviatoric plastic strain tensor for incompressible plasticity. These relations are usedto compute the thermoelastic stress in the absence of plasticity, ep

ij 0, but they arealso used to compute so-called trial stress in computational plasticity. The differentforms of Eqs. (4) and (5), which are needed in a nonlinear analysis, are

dskk ˆ 3K…dekk ¡ 3deth† ‡ 3dKeekk ˆ 3K…dekk ¡ 3deth† ‡ dK

Kskk …6†

dsij ˆ 2Gdeeij ‡

dGG

sij …7†

The equations should be implemented in an incremental form that gives the exactstresses if the increment is elastic, that is, when Eqs. (4) and (5) are evaluated atthe end of an increment. This requires

Dskk ˆ 3K…Tn‡1†…Dekk ¡ 3Deth† ‡ DKK…T n†sn

kk …8†

Dsij ˆ 2G…Tn‡1†Deeij ‡ DG

G…Tn† snij …9†

where the right superscripts denote time-step counters in the ¢nite element solutionprocess.

The thermoelastic models tried by Argyris et al. (1982) are not in accordancewith this. Their implementation of the thermoelastic behavior is not correct evenin their so-called mixed model since all material properties in the equations corre-sponding to Eqs. (8) and (9) are evaluated at the beginning of the increment in theirpaper.

Different assumptions regarding the high-temperature elastic properties havebeen used. Friedman (1975, 1977) assumed a constant bulk modulus. This wasobtained by decreasing the Young’s modulus with temperature and at the same timeincreasing Poisson’s ratio toward 0.5. Young’s modulus and Poisson’s ratio are

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shown in Figures 6 and 7 for lower temperatures (Richter 1973). An estimate basedon this data does not support the use of a constant bulk modulus. It seems to decreasefor higher temperatures. Increasing Poisson’s ratio and assuming incompressibilityin the liquid state have led others (e.g. Hibbitt and Marcal 1973, Andersson 1978,Josefson 1982, Jonsson et al. 1985a, Cowles et al. 1995, Troive et al. 1998) to increasePoisson’s ratio toward 0.5 near the melting temperature. However, there is no reasonto have a continuous increasing bulk modulus up to in¢nity in the liquid state sincethere is a phase change occurring during melting and the material can, therefore,have discontinuous material properties. Leung and Pick (1986) compared an analysiswith a Poisson’s ratio ranging from 0.24 at room temperature to 0.45 at meltingtemperature with an analysis for a constant value of 0.24. They found that the resultswere almost identical, but the ¢rst analysis required 50% more computer time.Tekriwal and Mazumder (1991a) also investigated the in£uence of Poisson’s ratio.They obtained nearly identical results, but the computer time was about the samefor all variations of Poisson’s ratio. The extra computer time required in the analysisby Pick and Leung (1986) was probably due to the selected ¢nite element implemen-tation and not due to the physical problem. The chosen values for Young’s modulusis an important parameter that affects the residual stresses. If the chosen value athigh temperature is too low, then the ¢nite element analysis will fail. Lindgrenet al. (1999) used 1 GPa as the lower limit for a ferritic steel.

Figure 6. Young’s modulus for some steels. Data from Richter (1973).

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The thermal dilatation, eth , is the total effect of the usual thermal expansion andthe volume changes due to phase changes. The thermal dilatation is the driving forcefor the thermal stresses and is therefore an important parameter. Typical values aregiven in Figure 8 for slow heating. The in£uence of the volume changes due to phasetransformations on the thermal dilatation was discussed earlier in the section``Dependency on Temperature and Microstructure.’’ Murty et al. (1996) assumedthat the transformation strain is 0.044 when 100% austenite is transformed into100% martensite and 0.007 if it is transformed into ferrite/pearlite. The solidi¢cationshrinkage is usually ignored. The motivation may be that it will lead to plastic strainsthat are removed anyway due to this phase change. The use of a cut-off temperaturewill also exclude some thermal dilatation in the high-temperature range. Feng (1994)and Feng et al. (1997) discuss the modeling of solidi¢cation shrinkage for aluminum.Their study is concerned with solidi¢cation cracking, so they cannot resort to toolarge simpli¢cations in the high-temperature range. Sheng and Chen (1992) andChen and Sheng (1991, 1992) also had a detailed model of the near-weld-pool regionwhere the solidi¢cation shrinkage was included.

The implementation of the computation of the thermal dilatation can be done indifferent ways. It is important to observe whether thermal dilatation is speci¢ed orthermal expansion coef¢cient. The latter can vary since some codes want the secantand other the tangent thermal expansion coef¢cient. This is due to the way theincremental thermal dilatation deth is computed. It can be computed directly byinterpolating from a given thermal dilatation curve or by the tangent thermal

Figure 7. Poisson’s ratio modulus for some steels. Data from Richter (1973).

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expansion

deth ˆ deth

dTdT ˆ atdT …10†

or the secant thermal expansion

deth ˆ an‡1s …T n‡1 ¡ Tref† ¡ an

s …Tn ¡ Tref†. …11†

ABAQUS (Wilkening and Snow 1991) also computes an initial thermal strainbecause it does not assume that the initial temperature is the zero-strain state. Thus,it is important to set this reference temperature equal to the initial temperature givento the elements added to the model when a weld is laid. Hong et al. (1998a) found thedifference between using the room temperature or the melting temperature as ref-erence temperature to be small when simulating a multipass weld using ABAQUS.It is not clear if they removed all accumulated plastic strains when the materialmelted. This will remove the previous history effects at these elements. The removalof plastic strains when a material is melting is mandatory (see the next section),otherwise the chosen reference temperature will give different plastic strains, whichmay, depending on the hardening, be visible on the residual stress ¢elds. The correctapproach is to use the zero-stress temperature of an element as the referencetemperature. This is the melting temperature for added ¢ller material.

Figure 8. Thermal dilatation for some steels. Data from Richter (1973).

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Properties for Modeling Plastic Deformation

The mechanisms for the plastic deformation vary due to the large temperature rangeinvolved. The Ashby diagram (Figure 9) illustrates the changing plastic strain ratefor different stresses and temperatures. However, the time scale is also importantwhen determining whether rate-dependent plasticity should be included. The argu-ment for using rate-independent plasticity at high temperatures is based on theinvolved time scales. This was stated clearly by Hibbitt and Marcal (1973), whoincluded creep only when considering stress relief. The same was done by Uedaet al. (1976, 1977) and Josefson (1982). The material has a high temperature duringa relatively short time of the weld thermal cycle, and therefore the accumulatedrate-dependent plasticity is neglected. Bru et al. (1997), who studied the same prob-lem as Roelens (1995a, b), used tensile data for the strain rate of 0.1 sec¡1 andthe rate-independent plasticity model. Sekhar et al. (1998) show the variation ofyield stress with temperature for two different strain rates. The difference is smallfor that particular case except around 700 to 900 C. Most studies in simulationof welding approximate the yield limit at higher temperatures and try not to makeany elaborate adjustment for expected dominant strain rates since the available dataare scarce. The plastic £ow in rate-independent plasticity is determined by the con-sistency condition, which requires that the effective stress must stay on the yield

Figure 9. Sketch of Ashby diagram (Ashby 1992) illustrating deformation mechanisms and differentstrain rates. Different deformation mechanisms in different temperature and stress regimes are shown.

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surface for a plastic process. This leads to a direct relation between change of plasticstrain and change in stress. The model that has been used most widely forrate-independent plasticity is the von Mises yield criterion together with the associ-ated £ow rule. Thus, the plastic strains are incompressible and are not dependenton the hydrostatic part of the stresses. The £ow rule states that the plastic £owis orthogonal to the yield surface. This comes from the assumption of maximumplastic dissipation or Drucker’s postulate. The £ow rule and the yield functionare given as

epij ˆ _ep

ij ˆ l0 ¶ f¶ sij

ˆ l…sij ¡ aij† …12†

where l is the plastic parameter that determines the amount of £ow, f ˆ s ¡ sy…k, T†is the yield function, s ˆ 3

2 …sij ¡ aij†…sij ¡ aij† is the effective stress, aij is the backstress or the center of the yield surface that accounts for the kinematic hardening,sy is the yield limit, and k is an internal variable for the isotropic part of thehardening. The consistency condition, f 0, is evoked to determine the plastic par-ameter l. A hardening rule is also needed to determine the amount of plastic £owand the evolution of the yield strength of the material. There are several accuratemethods for computing the stress increment for inelastic material behavior. Theeffective-stress function is a general version of the radial return method that canaccommodate different kinds of inelastic strains (Kojic and Bathe 1987, Josefsonand Lindgren 1997). See also Appendix A of Part 3.

Tall (1964) assumed the material properties to be temperature dependent but nowork hardening was accounted for. Tsuji (1967) assumed a linear decreasing yieldstress and linear, isotropic hardening. Jonsson et al. (1985b) varied the yield limit,hardening modulus, and thermal dilatation in order to study the in£uence onthe residual stresses and the change in gap width in front of the moving arc. Theyalso assumed isotropic hardening. A higher yield limit and hardening modulusat low temperatures raised the residual stresses in the weld region. It also affectedthe change in gap width as the strength of the tack welds was important in the studiedcase. The thermal dilatation had only a smaller in£uence on the residual stresses. Thelarge restraint of the plate against in-plane motion causes the yield strength to be thelimiting, and therefore important, factor for the studied case. McDill et al. (1990)investigated the in£uence of the thermal and mechanical properties on the residualcurvature of a welded bar. They only combined thermal and mechanical propertiesof a stainless steel and a low carbon steel and did not vary the individual properties.For example, a larger plastically deformed region was obtained with lower yieldstrength, but the in£uence on the curvature was counteracted by lower residualstresses. All changes affected the residual curvature of the welded bar quite a lot.It varied from 41.6 to 11.9 m¡1. The case they studied was little restrained, andtherefore it is expected that thermal dilatation together with the temperature dis-tribution is important for the deformation. The latter depends strongly on thethermal properties. This explains why they found that the thermal properties werethe most important difference between carbon and stainless steels. Bru et al. (1997)investigated the in£uence of the mechanical properties at high temperatures. It

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was a continuation of the study by Roelens (1995a,b), where microstructure evol-ution was used to compute the material properties using mixture rules. The yieldstress of the austenite was uncertain due to its dependency on grain size and its ratedependency. They used von Mises plasticity and the yield limit for a given strainrate. Furthermore, Roelens (1995a, b) used 1250 C as a cut-off temperature forstrain hardening, thermal strain, and Young’s modulus. This was raised to 1500 Cby Bru et al. (1997). All these modi¢cations did not affect the residual stresses sig-ni¢cantly; however, they affected the residual deformation. This is probably dueto the sensitivity of the welding con¢guration, butt-welded plates, to the deformationat high temperature during the ¢rst weld pass. This can be seen from Roelens(1995a, b). It seems that they did not have any restraints or applied gravity loadingthat would push the plates toward the welding table and the halves were only initiallyconnected with one weld.

The material near and in the weld is subjected to reversed plastic yielding duringthe cooling phase. Thus, using kinematic, isotropic, or combined hardening willaffect the stresses in this region. Andersson (1978) suspected that his use of isotropichardening was one reason for the deviations between experiments and simulations.However, this is not the case, as is shown in the section ``A Welding SimulationRevisited’’ in Part 3. Bammann and Ortega (1993) investigated the effect ofassuming isotropic and kinematic hardening. They found that the choice of hard-ening in£uences the residual stresses very much in the weld metal, but further awaythe different models gave identical results. They used the rate-dependent materialmodel described later in this section. Devaux et al. (1991) found only small differ-ences in residual stresses between models with isotropic or kinematic hardening.They studied weld repair with four or six beads. No hardening has been assumedin some studies (e.g. Ueda and Yamakawa 1971a, b, Hibbitt and Marcal 1973, Uedaet al. 1976, 1977, Hsu 1977, Jonsson et al. 1985a, Karlsson 1989, Karlsson andJosefson 1990). Excluding hardening is an approximation, but it can be motivatedin some cases. Karlsson (1989) argues that most plastic strains are accumulatedabove 600 C, where the material has nearly no hardening. These plastic strainsare removed during subsequent phase transformations (see discussions later, inthe studied low alloy steel). Finally, the yield limit used at room temperaturewas estimated from hardness measurements on the welded plate and thereforealready included the effect of hardening during the ¢nal cooling. Most studiesuse linear, isotropic hardening. Friedman (1975, 1977) assumed isotropic hardeningusing a power law hardening and Ueda et al. (1991) used piecewise linear, isotropichardening. Kinematic hardening (Papazoglou and Masubuchi 1982, Wang andInoue 1985, Leung and Pick 1986, Ueda et al. 1986, Mok and Pick 1990, Tekriwaland Mazumder 1989, 1991a, b, Michaleris 1996, Michaleris and DeBiccari 1997)and combined hardening (Chakravarti 1986, 1987, Murthy et al. 1994) have beenassumed in some simulations.

The rate-dependent plasticity models use a £ow potential surface that deter-mines the plastic strain rate. The effective stress can be outside this surface andthe plastic strain rate is a function of the distance to this surface. The materialbehavior is elastic if the stress state is inside the potential surface. The surface doesnot exist in the case of creep, _ec

ij , which can be considered a special case of

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viscoplasticity. A general form for the plastic £ow, where it is assumed that it isnormal to the potential surface, is

_evpij ˆ F…sij , aij, sf…k, T †† ¶ F

¶ sij…13†

where F is the £ow potential and sf is the £ow stress of the material corresponding tothe yield limit in the case of rate-independent plasticity. Some studies like those ofArgyris et al. (1982, 1985), Inoue and Wang (1983), Wang and Inoue (1985), Chidiacand Mirza (1993), and Myhr et al. (1998a, b) used a viscoplastic material model.Wang and Inoue (1985) used a model that represents elastic and viscousdeformation, pure viscous deformation, an incompressible Newtonian £uid, andinviscid behavior. The inviscid behavior is the rate-independent plasticity discussedpreviously. Goldak et al (1996, 1997a) discuss the use of different rate dependenciesfor the plastic behavior at different temperatures and stresses. They chose to usedifferent constitutive models for different temperature regions. A linear viscousmodel is used at a homologous temperature above 0.8. The homologous temperatureis de¢ned as the current temperature in Kelvin divided by the melting temperature inKelvin. Rate-dependent plasticity is used down to a homologous temperature of 0.5and von Mises plasticity for lower temperatures. Carmignani et al. (1999) discussthe formulation and numerical implementation of a uni¢ed model for plasticityand viscoplasticity. Mahin et al. (1988, 1991), Winters and Mahin (1991), Ortegaet al. (1992, 1998), and Dike et al. (1998) used a uni¢ed creep-plasticity modelby Bammann. They assumed

_evpij ˆ A…T †sinh

jsij ¡ aij j ¡ sf …k, T†B…T †

sij ¡ aij

jsij ¡ aij j…14†

where sf ˆ sf0…T † ‡ k, _k ˆ H…T†_ep¡recovery terms, _ep ˆ 23 _evp

ij _evpij is the effective

plastic strain rate, and _aij ˆ h…T †_evpij ¡recovery terms. Ronda and Oliver (1998) com-

pare three different viscoplastic models. The models give different results, but thereis no discussion about their applicability for the studied case and there is no com-parison with the often used rate-independent elastoplastic model. Sheng and Chen(1992) and Chen and Sheng (1991, 1992) used the Bodner^Partom viscoplastic modeland the Walker model. The latter accounts for kinematic hardening. Oddy andMcDill (1999) used a model where rate-independent plasticity and creep damagewere accounted for simultaneously in a simulation of burnthrough during weldingof pressurized pipelines.

They used the previously described von Mises plasticity together with

_ec ˆ f …s, T †G…T † s1 ¡ o

n

…15†

where o is the damage parameter, f is a function to be discussed, and n is a materialconstant. The damage parameter o, assumed to be 0 for undamaged and 1 at creep

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rupture, is determined by

_o ˆ C…1 ¡ o†Z Dn …16†

where D ˆ sa1s…1¡a† is a stress-dependent parameter combining the in£uence of effec-

tive stress and the largest principal stress s1, Z and n are material parameters,and a2[0, 1] is a traditional factor. The latter parameters were obtained for onetemperature and extrapolated to other temperatures by assuming that time forrupture changes with temperature in such a way that the Larson^Millar parameteris constant. This parameter is given by

L ˆ T…m ‡ log…t†† …17†

where T is in Kelvin and t is time. Then the material parameters in Eq. (16) can bedetermined at any temperature if data are given for one temperature by

uT ˆ n1T1

C…1 ‡ Z† ˆ …C1…1 ‡ Z1††T1/T

10m‰…T1/T†¡1Š

where the subscript 1 denotes test data at temperature T1. Constant data above 750were assumed due to nucleation of new grains. Oddy and McDill (1999) assumed mwas 16.4. The function f (s, T) in Eq. (15) was used to adjust the creep rate to datafor the minimum creep rate, which occurs when o ˆ 0. They used a relation whereone additional dependency on stress, except the sn term, also was included. Thefunction had the form

f …s, T† ˆ AB ‡ CT

e…¡QR/T† sinhAsT

…19†

and n ˆ 3 in Eq. (15).In general, the £ow stress decreases with increasing temperature and increases

with increasing strain rate for a constant microstructure. An Arrhenius type ofequation, e¡Q/…RT†, may describe the £ow stress decrease with absolute temperatureand it is roughly proportional to a power of strain rate. However, when plasticdeformation is accompanied by precipitation, phase changes, and recrystallization,for example, then the £ow stress changes in a complex manner. An example ofthe £ow stress for a low carbon steel is shown in Figure 10. The anomaly for higherstrain rates is due to so-called blue shortness. The £ow stress for a stainless steelis given in Figure 11. The strain rate dependency can be seen. High-temperaturedata may have to be derived from tests for similar materials due to the lack of data.The in£uence of the material composition is often less important at higher tempera-tures (Figure 12), but not when carbide formers for added high-temperature strengthare present in the material. The in£uence of the yield limit at higher temperatures onthe residual stresses is also less important. This is related to the removal ofaccumulated plastic strains during phase transformations. The exclusion of therate-dependency of the plastic £ow at higher temperatures can be combined with

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Figure 10. Flow stress for 0.15% C-steel evaluated at a total strain of 0.2 for different strain rates andtemperatures. Data from Suzuki (1968).

Figure 11. Flow stress for austenitic stainless steel evaluated at a total strain of 0.2 for different strainrates and temperatures. Data from Suzuki (1968).

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using a higher yield limit than the real yield limit. This may also be bene¢cial for thenumerical stability of the simulations, avoiding large deformations and plasticstrains.

The in£uence of the phase transformations on the thermal dilatation and theyield limit was discussed earlier in the section about ``Dependency on Temperatureand Microstructure.’’ Two other aspects of the phase transformations have to bemodeled: transformation -induced plasticity (TRIP) and change in dislocationstructure. The latter is usually accounted for in the hardening model by theaccumulated effective plastic strain. The modeling of these phenomena is discussedlater in this section.

Experimental measurements of welding residual stresses in alloys with low aus-tenite decomposition temperatures showed that longitudinal stresses can be stronglyaffected by phase transformations (Nitschke-Pagel and Wohlfahrt 1992). The com-bination of an applied stress (Figure 13) and a phase change will give rise to anadditional plastic deformation, TRIP. Fischer et al. (1996, 2000) discusses the sub-ject more thoroughly in the context of martensitic (displacive) transformations. (Seealso Bergehau and Leblond (1991).) The transformation plasticity is usually assumedto be proportional to the deviatoric stresses in the same ways as is assumed standardplasticity models. It is given by

_etpij ˆ f …_etr

a!b, sa, _Xa!b†sij …20†

where _etra!b is the volume change when transforming from phase a to b, _Xa!b is the

rate of phase change from phase a to b, and sa is the yield limit of the weaker phase

Figure 12. Flow stress for carbon steel at higher temperatures. Data from Suzuki (1968). The strain rate is10 sec¡1 and the £ow stress is evaluated at e ˆ 0.2.

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a. Argyris (1982, 1985), Josefson (1985c), and Karlsson (1989) accounted for TRIPby lowering the yield limit. Denis (1984), Dubois et al. (1984), Leblond et al. (1989),Leblond (1989), Josefson and Karlsson (1992), and Oddy et al. (1989, 1992)accounted for TRIP in a more correct way. Oddy et al. (1989) showed that thismay be the explanation for the discrepancies between simulations and measurementsby Hibbitt and Marcal (1973). The effect of different material models on the residualstresses are shown in Figure 14 for conditions resembling the case studied by Hibbittand Marcal (1973). The papers by Leblond et al. (1989) and Leblond (1989) give atheoretical framework. They did not consider the effect of the stresses on the phasechanges. The paper by Oddy et al. (1992) gives a concise description of TRIP.Bammann et al. (1995) implement this effect differently by using a multiphase statevariable constitutive model. Fischer et al. (2000) give a more detailed model byincluding both the shear associated with the martensitic transformation and its vol-ume change …_etr

a!b† to the transformation plasticity. They propose the relation

_etpij ˆ 3

2K

djdXa!b

_Xa!bsij …21†

where K ˆ 5……_etra!b†2 ‡ 3

4 g2m†/6s0

y, gm is the transformation shear because of themartensitic formation, s

0

y ˆ smy ……1 ¡ sg

y/smy †/…ln…sm

y /sgy††† is the rate of phase change

from phase a to b, smy is the yield limit of martensite, sg

y is the yield limit of austenite,

Figure 13. Effect of stress applied during martensitic transformation for uniaxial test (Cavallon et al.1997).

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and j(X) is an heuristic function. Equation (21) gives transformation plasticitybecause of an accommodation process when martensite if formed. It is calledthe ``Greenwood^Johnson’’ effect. There is also an orientation process, the ``Magee’’effect. See Fischer et al. (2000) for further details about this and its relation tokinematic hardening. Further details can be found in Berveiller and Fischer (1997).Denis (1997) discusses pearlitic, that is, a diffusive transformation, in this book.An expression of the same type as in Eq. (21) is also used for this process.

Modeling the yield strength of a material that undergoes phase changes posessome questions. Usually the yield strength is determined by a virgin yield limitand a hardening contribution due to the deformation. The hardening is due tothe increase in dislocation density. The dislocation structure is represented bythe effective plastic strain in a material model. Then there is a question abouthow the dislocation density will change during phase transformations. Severalpapers have attempted to account for these effects. Friedman (1975, 1977) andPapazoglou and Masubuchi (1982) removed the accumulated plastic strains whenthe material melted and relieved the accumulated strains by multiplying the pre-viously accumulated plastic strains with a factor due to solid-state transformations.Sarrazin et al. (1997) also included recrystallization and recovery effects, whichreduced the yield strength of aluminum. It is at least appropriate to remove all

Figure 14. Effect of transformation plasticity on welding residual stresses at top surface of weld (Oddy etal. 1989).

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accumulated strains when the material melts (Lindgren et al. 1997, Brickstad andJosefson 1998). Mahin et al. (1991) included the removal of plastic strains duringmelting because they had found that this was a source of discrepancy betweensimulations and measurements in their earlier work (Mahin et al. 1988). Ortegaet al. (1992) initialized all internal state variables to zero when the material melted.They also removed the deviatoric stresses, creating a hydrostatic stress state inthe weld pool. This was also done by Dike et al. (1998). Devaux et al. (1991) arguedthat it is reasonable that the memory of previous plastic deformation disappearsfor all solid-state phase transformations in ferritic steels with perhaps the exceptionfor the martensite formation. The latter involves very small displacements of theatoms during the transformations, which they assumed did not affect the dis-locations. Brust and Dong (Brust et al. 1997, Dong et al. 1998c) introduced rateequations applied between an anneal temperature and the melting temperaturefor anneal strain. The introduction of these strains corresponds to the removalof plastic strains as they reduce the hardening. They also applied this to the elasticstrains and thereby reduced the stress. The technique can be used with the elementbirth option for multipass welding in order to remove history effects. This is alsodiscussed by Hong et al. (1998b). However, the uniaxial case discussed by themis not completely correct. The removal of plastic strains does not imply that thethermal and elastic strains should also be set to zeroöthe latter giving zero stressat this instant in time. Models for recovery processes at higher temperatures, likeEq. (16), exist, but physical-based models for these processes during phasetransformations are lacking. Physical-based material models are models basedon observations of the actual physical processes, phase changes, and dislocationprocesses, for example, that take place during the deformation.

There are no papers where the texture created by the directional solidi¢cation inthe weld metal is taken into account. Mahin et al. (1988, 1989) discussed this aspectw.r.t. the interpretation of the neutron diffraction measurements in the weld metal.

A list of papers where some material properties are included is given in Table 2for ferritic steels, Table 3 for austenitic steels, and Table 4 for other metals. Ifnothing else is stated in the comments, then the model is von Mises rate-independentplasticity with isotropic hardening and the associated £ow rule. Most works rely onproperties compiled from the literature, and a few papers refer to experiments pre-pared especially for the welding simulations at hand. A discussion of materialmodeling with respect to obtaining residual stresses is given by Oddy and Lindgren(1997).

Recommendations. It is important to have a correct description of the materialbehavior in order to have an accurate model. The more important mechanicalproperties are Young’s modulus, thermal dilatation, and parameters for the plasticbehavior. The in£uence of these properties at higher temperatures is less pronouncedon the residual stress ¢elds. The material is soft and the thermal strains cause plasticstrains even if the structure is only restrained a little as the surrounding, initially coldmaterial acts as a restraint on the heat-affected zone. A cut-off temperature maytherefore be used. It may also be necessary to simplify the tangent matrix usedin the solution procedure if it is too ill-conditioned. Andersson (1978) and Jonsson

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et al. (1985a) used only the thermoelastic part of the stiffness matrix at hightemperatures. This reduced the convergence rate, but it may be necessary if thematrix is too ill-conditioned.

The phase transformation of ferritic steels during cooling may in£uence theresidual stresses quite a lot. Therefore, the cut-off temperature should at leastbe higher than austenization temperature, but it is better to use a value near themelting temperature. The in£uence of the material properties on stresses anddeformations is dependent on the studied con¢guration. A rigid structure willget larger changes in plastic strains, whereas a £exible structure will experience dif-ferent deformation due to different material properties.

The importance of including the effect of phase transformations on the mech-anical behavior for ferritic steels is also discussed in the recommendations in thesection ``Dependency on Temperature and Microstructure.’’ The effect of TRIPon residual stresses depends on the restraint of the structure and at what temperaturethe phase changes occur. If, for example, the shrinkage after the martensite forma-tion gives rise to plastic yielding before the weld has cooled completely, then theresidual stress is limited by the yield limit of the hardening material and the differ-ence in the residual stresses between an analysis that includes or ignores the volumechanges due to the phase transformation may be small.

The plastic strains generated at higher temperatures may be removed or reduceddue to phase transformation and thereby reduce the in£uence of high-temperatureproperties even more. It is preferable that the material is modeled as ideal plasticat this temperature; otherwise the removal of plastic strains may lower the yieldstrength, which in turn will create additional plastic strains if the stress ¢eld ison the yield surface. The plastic strains that exist during cooling will be the majorremaining effect of the properties at higher temperatures. They will in turn contrib-ute to the hardening of the material. This will also limit the in£uence of these plasticstrains. Thus, the use of approximate material data and ignoring the rate dependencyat higher temperatures has been found to affect the residual stresses very little (e.g.Free and Porter Goff 1989, Bru et al. 1997, Tekriwal and Mazumder 1991a). Manysteels have about the same properties when they become fully austenized at highertemperatures (Figure 12) if no special carbide formers have been added for specialhigh-temperature strength. It is appropriate to combine kinematic and isotropichardening in order to get accurate residual stresses in the weld metal, but theremay be a problem in obtaining the needed data.

Finally, it should be noted that the requirements of the material model are higherif the zone near the melt is of particular interest, for example, when hot cracking isstudied. This improved material model should, at the same time, be matched bya re¢ned spatial and temporal discretization.

FUTURE RESEARCH

The modeling of material behavior at higher temperatures and in the presence ofphase transformations is perhaps the most crucial ingredient in successful weldingsimulations. Progress in this ¢eld is dependent on the collaborative efforts from com-

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putational thermomechanics and material science. This is a ¢eld that must be focusedon by the research community active in the ¢eld of welding simulation. The improve-ment in material modeling and increasing availability of material parameters will, incombination with the computational development, increase the industrial use ofwelding simulations.

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