Numerical Predictions for the Thermal History , Microstructure and Hardness Distributions at the HAZ during Welding of Low Alloy Steels

Microstructure plays an essential role on the accurate determination of the residual stresses during the steels welding and it is, therefore, a subject that needs better attention due to its technological importance. Furthermore, some examples where the microstructural prediction are valuable to control the final properties (e.g., toughness of steel welds); to the knowledge of microstructural limits for process optimization (e.g., maximum welding speed) and when microstructure captures the coupling through multi-stage process, giving opportunities for new alloys and process development (e.g., effect of prior forming and heat treatment on weldability)1. Several works have been carried out and highlighted the importance of taking into account the phase transformations effects and properties evaluation developed during welding2-13. Simultaneously, the numerical methods have been a fundamental tool for supporting qualitative and quantitative analysis of these stresses, as well as on the prediction of the resultant microstructure and mechanical properties of the weldment. For example, some experimental methods to analyze the residual stresses from welding are affected by the weldment microstructure and a previous knowledge about the phase transformations that could take place during this procedure would be very helpful3,12. Thus, this study deals with the issue of kinetics transformation under non-isothermal conditions, which in turn, is a subject of great practical interest. An attempt to establish a correlation aiming at the use of data obtained from the isothermal transformations in order to calculate non-isothermal transformations was initially presented by Avrami14, through the definition of an isokinetic reaction by the condition that the nucleation and growth rates are proportional to each other, i.e., they have same temperature variation. Nevertheless, Cahn15 considered that this condition will rarely be expected to occur in practice. However, he has mentioned that in many reactions the nucleation rate can saturate earlier during the transformation, e.g.: many systems which exhibit heterogeneous nucleation or in continuous reaction and, since the growth rate in any instant will be only dependent on the temperature, the reaction will be isokinetic15. Following this matter, Reti et al.16 have proposed in the last decade a phenomenological kinetic model of Avrami-type to predict the multiphase diffusional decomposition of the Numerical Predictions for the Thermal History, Microstructure and Hardness Distributions at the HAZ during Welding of Low Alloy Steels


Introduction
Microstructure plays an essential role on the accurate determination of the residual stresses during the steels welding and it is, therefore, a subject that needs better attention due to its technological importance.Furthermore, some examples where the microstructural prediction are valuable to control the final properties (e.g., toughness of steel welds); to the knowledge of microstructural limits for process optimization (e.g., maximum welding speed) and when microstructure captures the coupling through multi-stage process, giving opportunities for new alloys and process development (e.g., effect of prior forming and heat treatment on weldability) 1 .
Several works have been carried out and highlighted the importance of taking into account the phase transformations effects and properties evaluation developed during welding [2][3][4][5][6][7][8][9][10][11][12][13] .Simultaneously, the numerical methods have been a fundamental tool for supporting qualitative and quantitative analysis of these stresses, as well as on the prediction of the resultant microstructure and mechanical properties of the weldment.For example, some experimental methods to analyze the residual stresses from welding are affected by the weldment microstructure and a previous knowledge about the phase transformations that could take place during this procedure would be very helpful 3,12 .
Thus, this study deals with the issue of kinetics transformation under non-isothermal conditions, which in turn, is a subject of great practical interest.An attempt to establish a correlation aiming at the use of data obtained from the isothermal transformations in order to calculate non-isothermal transformations was initially presented by Avrami 14 , through the definition of an isokinetic reaction by the condition that the nucleation and growth rates are proportional to each other, i.e., they have same temperature variation.Nevertheless, Cahn 15 considered that this condition will rarely be expected to occur in practice.However, he has mentioned that in many reactions the nucleation rate can saturate earlier during the transformation, e.g.: many systems which exhibit heterogeneous nucleation or in continuous reaction and, since the growth rate in any instant will be only dependent on the temperature, the reaction will be isokinetic 15 .
Following this matter, Reti et al. 16 have proposed in the last decade a phenomenological kinetic model of Avrami-type to predict the multiphase diffusional decomposition of the Numerical Predictions for the Thermal History, Microstructure and Hardness Distributions at the HAZ during Welding of Low Alloy Steels austenite in low-alloy hypoeutectoid steels during cooling.The model consists of coupled differential equations and takes into account the austenitic grain growth effects, permitting the prediction of the progress of ferrite, pearlite, upper bainite and lower bainite transformations simultaneously.Low-alloy steels are a group of steels with a very large application in engineering designs.However, these steels can be submitted to welding during a fabrication or repairing procedure and a prior knowledge on the microstructural changes that could occur during its welding would be of great practical interest, since it can directly affect their properties.Thus, the present work is an attempt towards numerically to predict the transformations occurring during the welding of low-alloy hypoeutectoid steels.For this purpose, the kinetic model of Avramy-type proposed by Reti et al. 16 was mathematically adapted and implemented using a developed finite volume method (FVM) based computational code.The model is capable of tracking the microstructural changes occurring at the HAZ of a low-alloy hypoeutectoid steel, besides of correlating it to the hardness variations observed in this weldment region.
On the other hand, this FVM based computational code differs from previous works due to its ability to handle simultaneous phase transformations, heat input effects and nonlinearities into an efficient flamework 5,6,[8][9][10][11][12] .Therefore, the approach used in this study represents a step forward on the challenging task of numerically predicts the complex phenomena and changes taking place during the steels welding and could be used to evaluate new welding procedures for the industrial practice.

Modeling
The present study deals with a model implementation, which takes into account the coupling phenomena of phase transformations, temperature evolution and hardness prediction during the welding.Autogenous welding, i.e., a fusion welding process without addition of filler material was selected to avoid the effects of bead formation and material additions into the phase transformations.The welding process is numerically simulated by solving the coupled transient equations of energy conservation, phase transformations and the model for hardness prediction.The material investigated in this work was a low-alloy hypoeutectoid steel with dimensions of 10 and 25 x 60 x 220 mm with the chemical composition presented in table 1.The material used in this study was selected due to its ability to undergo multiphase transformations during cooling, which allows the verifications of the kinetic model for phase transformations from the available data 16 .Furthermore, the computer code used in this study has been continuously developed by the authors and applied for different materials and welding conditions, which has validated the general features of the model and computer implementations [17][18][19] .In this paper, new features such as models for kinetics of phase transformations and hardness prediction for low-alloy hypoeutectoid steel are added and, thus, improving the model capability.

Model features
In this study, fundamental thermal and metallurgical phenomena occurring during the welding of a low-alloy hypoeutectoid steel were evaluated by means of numerical simulation.For this purpose, it was necessary to predict the temperature field coupled dynamically with the welding evolution and the material thermophysical properties, together with the kinetic model for phase transformations, besides the model for hardness prediction.In order to model the process, the phenomena of heat transfer by radiation, convection and conduction are taken into account coupled with mass transfer, melting and solidification and the thermophysical properties were assumed as composition and temperature dependent [20][21][22][23][24][25] .The energy equation for a general coordinate system is represented in compact form by the Equation 1 [17][18][19] .In Equation 1, ρ is the density; C p is the specific heat; k is thermal conductivity; U → is velocity field, which accounts for buoyancy driven flow in the liquid pool or moving mesh to match the geometry changes due to the metal deposition; T is the temperature field and S is the source term, which accounts for all source or sink due to phase transformations, melting and solidification.

Initial and boundary thermal conditions
The initial condition is assumed with the workpiece setup to a given temperature and composition.For each time step, the geometry is actualized after metal deposition and moving heat source according to the assumed welding speed.For the boundary conditions, the effects of convective and radiative fluxes are considered, while the heat input supplied by the torch is modeled by the power distribution given by the well-known moving Goldak's double-ellipsoid heat source model 26 (Figure1).
The model is a combination of two ellipses: one in the front quadrant of the heat source and the other in the rear quadrant.Equations 2 and 3 show the volumetric heat flux distributions inside the front and rear quadrant of the heat source, respectively.The model is defined as a function of position and time together with a number of parameters that affect the heat flux magnitude and distribution 26 .

The heat input rate
is determined by welding operational parameters current (I), voltage (V) and thermal efficiency h respectively.The factors f f and f r denote the fraction of the heat deposited in the front and rear quadrant respectively, which are setup to attain the restriction f f + f r = 2.The a,b f , c and b r , are source constant parameters that define the size and shape of the ellipses and, therefore, the heat source distribution.These parameters were estimated from the models for determination of the dimensions of a weld pool proposed by Wahab and Painter 27 and may be found in Table 2 together with the respective investigated welding parameters in this study, while the factors f f and f r were defined as 0.6 and 1.4 respectively.Thickness and preheating effects on the welding region for the investigated steel were also carried out in order to evaluate the response of the present model when applied to more general cases.Further informations on the procedures adopted in this study are also available in table 2.
The cooling boundary conditions between the workpiece and environment by means of convection and radiation are calculated by Equations 4 and 5, respectively.
( ) where T 0 (25 0 C ) is the room temperature, e (T ) is the emissivity as a function of temperature, s ( .   is the natural convective heat coefficient assumed in this study.The temperature dependency of emissivity for low carbon steels was investigated by Paloposki and Liedgust 22 , but limited for a narrow range of temperature (up to 700 o C).The available data 22 can be fitted as shown in Equation 6.However, a common approach is to assume constant emissivity during welding simulations when Equations 2 and 3 are used to dynamically impose the volumetric heat source due to the welding arc, where the heat source parameters includes the net heat transferred to the workpiece at high temperatures [5][6][7][8][9][10][11][12][13][17][18][19] . In thi model, the Equation 6, obtained by regression of the data presented by Paloposki and Liedgust 22 , is used for imposing the radiative cooling boundary conditions additionally to the double ellipsoid heat flux.Equation 6predicts constant emissivity for temperature below 300 o C and above 750 o C.This behavior is due to the range of measured data for low-alloyed steels, which is assumed to have similar behavior to the steel considered in this study.A clear shortcoming of this approach is that effects related to the surface parameters and environment temperature and composition are not considered and assumed negligible during the cooling by radiation and natural convection [5][6][7][8][9][10][11][12][13][17][18][19] .These shortcomings are usually overcame by adjusting the heat source parameters, shown in Table 2, which is dominant for the high temperature region (above 750 o C), meanwhile, the radiation cooling effects for low temperature (below 200 o C ) are negligible compared with the convection contribution. Threfore, this approach has been widely used in welding simulation [28][29][30][31] .

Phase transformations
A phenomenological kinetic model based on the austenite diffusional transformation during cooling after austenitization of low-alloy hypoeutectoid steels was used for numerically to simulate the transformations from austenite into ferrite, pearlite and bainite simultaneously.In this section will be presented some features of the model.A more detailed description concerning to its formulation can be found in elsewhere 16 .Thus, the base of the extended version of the multiphase diffusional transformation model of Avrami-type is represented by the Equation 9.
[ ] The temperature-dependent parameters B i and m i are estimated from the TTT diagram for the investigated steel from the Equations 10 and 11.

( )
where t s and t f are the times correspondent to 1 and 99% of transformation, respectively.Due to its importance during the austenite diffusional decomposition process, the austenitic grain growth was predicted by the temperature-dependent kinetic equation (Equation 12).

( ) ( )
In order to take into account the effect of austenitic grain size in the model, the B i parameter in Equation 9 is defined as ( 14) In Equations 12 to 14, k A was assumed as 6,087x10 7 , n A was assumed as 2,44, E A is the activation energy for the growth process (317 kJ mol -1 ) 16,21 , R is the universal gas constant (8,314 kJ K -1 mol -1 ), D 0 is the initial grain diameter at 3 c A temperature (0.00794 mm) 16 , B i (T) parameter is obtained from the isothermal diagram of the investigated steel by means of Equation 11, D ref corresponds to the reference grain diameter (0.0159 mm) 16 , e i are positive constants varying between 0.6 and 1.3 and depends upon the transformation type, i.e., ferrite, pearlite or bainite 16 .
After some modifications in order to take into account the coupling effects among the individual phase transformation process, the multiphase model has its final form represented by the coupled system of the differential Equations 15 to 19.
[ ] ( ) ( ) [ ] ( ) ( ) [ ] ( ) where y 1 , y 2 , y 3 and y 4 correspond to the products from austenite transformation, namely, ferrite, pearlite, upper and lower bainite, respectively; H (x) is the Heaviside function (in order to take into account the irreversibility of process); m 1 , m 2 , m 3 and m 4 are parameters temperature-dependent obtained from the isothermal diagram of the investigated steel using Equation 10; Y Fe , Y Pe and Y Ba correspond to maximum volume fractions of ferrite, pearlite and bainite (Figure 4); K 1 , K 2 , K 3 and K 4 are functions defined by Equation 20.

( ) ( )
Meanwhile, the volume fraction of martensite was calculated using a novel model proposed by Lee and Van Tyne 32 (Equation 21), which has been based on the well-known Koistinen-Marburger model.Koistinen-Marburger model has been optimized by means of the introduction of two Figure 4 -Estimated maximum volume fractions of ferrite, pearlite and bainite for the investigated steel 16 .
Numerical Predictions for the Thermal History, Microstructure and Hardness Distributions at the HAZ during Welding of Low Alloy Steels parameters, K LV and n LV , which can be adjusted to take into account the several effects of the steel composition on the kinetic.
( ) where V m is the volume fraction of martensite; T is the absolute temperature; M s is the martensite start temperature and, -

Hardness
The hardness distribution at the HAZ of investigated steel was calculated by using the rule of mixtures (Equation 24). ( ) where Hv is the hardness (Vickers); X M , X B , X F and X P are the volume fractions of martensite, bainite, ferrite and pearlite, respectively; Hv M , Hv B and Hv F+P are the hardness of martensite, bainite and the mixture of ferrite and pearlite, respectively.
For the calculating of Hv M , Hv B and Hv F+P were used the formulae developed by Maynier et al. 33 where Vr is the cooling rate at 700 o C in °C.h - .At this point, is convenient to summarizes and point out the main features of the FVM computational code considered in this work and their shortcomings.Firstly, it is important to emphasize that the FVM computational code is able to handle complex geometries and non-linear temperature dependence on both boundary conditions and thermophysical properties, which are very important for practical applications.The main shortcomings, however, are related to the accuracy of the available data for thermophysical properties for the complete range of phase compositions developed by the steels during welding.However, the FVM computational code has been successfully applied and verified for different materials and welding conditions and has been continuously updated for new applications and studies [17][18][19] .Another feature is that the FVM computational code is an open source code, which can be updated using newly available data for properties and kinetic equations.

Thermal features
The numerical model for thermal analysis during welding, based in the mentioned FVM computational code, has been previously validated for the temperature field predictions and previously published [17][18][19] and new features incorporated in this study.In order to demonstrate the accuracy of the prediction model, the Figures 5 (a) and (b) present a comparison between the calculated and measured values for temperature and welding zones in the plate.The temperature acquisition using thermocouples shows good agreement with the calculated results for a steel whose thermophysical and metallurgical features are quite representative of the steel investigated in this study, as shown in Figure 5 (a).As can be observed in Figure 5 (a), close agreement for the measured and calculated thermal history was obtained, allowing the applicability of the model for temperature predictions and welding zone (see Figure 5 (b)) and, accordingly, in providing of reliable data for the calculations of phase volume fractions.
Figure 6 shows the results for three dimensional transient temperature distributions for the plates with different heat input and thickness.The results are shown for times that correspond to the temperature distribution on the plates when the welding heat source had already travelled an identical distance along of the different workpieces in each welding condition considered in this study (see Table 2).Thus, the corresponding welding times are 11.7 and 70 s for the heat inputs evaluated in this study, i.e., 0.5 and 3.0 kJ mm -1 , respectively.
Vertical planes in the transversal direction on the plates in Figure 7 shows the effects of heat input, workpiece thickness and preheating temperature on the thermal profile within the plates.These planes correspond to the time when the heat source is located exactly in half the length of the plate, i.e., after welding times of 9.2 and 55 s, considering the heat inputs of 0.5 and 3.0 kJ mm -1 , respectively (see Table 2).Thermal profile will set the extensions and outline the boundaries among several regions of the weld, e.g.: HAZ and fusion zone (FZ), where important metallurgical reactions may occur and directly affect the weldment properties.As can be seen in Figures 7(a) and (b), the workpiece with preheating leads to only a slight changes on the thermal profile and, accordingly, on the extensions of the HAZ and FZ, but it has imposed a significant influence on the cooling rates (Figures 8 and 9).On the other hand, the workpiece thickness has a larger influence on the thermal profile and, accordingly, on the extension of the HAZ and FZ (Figures 7(c) and (d)), as well as on the cooling rate (Figures 8 and 9).In the former, both thickness and heat input were kept unchanged and only the preheating effect was imposed, whereas in the latter, the heat input was kept unchanged and no preheating was attributed, but the workpiece thickness has changed.When the thickness increases, the effect of the temperature distribution is more concentrated around the HAZ and FZ and higher thermal gradients occur.These phenomena will have a direct influence on the resulting microstructure and, accordingly, on the hardness distribution at the HAZ.   Figure 8 shows the profile and the intensity of the cooling and heating rates acting along of the workpieces in a plane corresponding to its centerline at the welding direction.The welding times corresponding to the location of the heat source are 10.8 and 55.4 s for the heat inputs evaluated in this study, i.e., 0.5 and 3.0 kJ mm -1 respectively (see table 2).
The preheating effect has resulted in lower cooling rates as can be seen in Figures 8(a) and (b) while keeping unchanged both the workpiece thickness and the heat input.As previously mentioned, higher thermal gradients are observed when thickness increases and greater cooling rates are reached accordingly.This behavior can be confirmed by comparing the Figures 8(c) and (d).In this case, only the workpiece thickness has changed.Figures 8(a) and (d) also allow a direct comparison from the influence of heat input on the cooling rates when the workpiece thickness is kept unchanged and no preheating has been used, i.e., it can be observed that very lower cooling rate were reached when higher heat input was used.Figure 9 could complement the interpretation of these results.
Figure 9 shows the thermal cycles located at the HAZ where the peak temperature has reached about 1100 o C in each welding condition investigated (see Table 2).where are also plotted some calculated cooling curves considering each investigated welding condition in this study (see Table 2).The cooling curves have started from a local at the HAZ where the peak temperature has reached about 1100 o C, with the volume fraction of the constituents and the hardness indicated in Figure 10 calculated using the mathematical formulation presented in this study.

Phase transformations and grain growth
Figures 11 and 12 present more comprehensive results depicting the HAZ and the resultant microstructure.
The results are shown for a specific plane on the transversal section of the plates after cooling of weldment to room temperature.Furthermore, the boundaries between FZ and HAZ have been outlined in order to distinguish both weld regions, since the HAZ is the focus of this study.The only constituents predicted to occur at the HAZ using the numerical simulation were martensite and bainite.Indeed, a more comprehensive analysis based on that presented in Figure 10 can be carried out in order to correlate the different cooling rates locally reached within the HAZ with the microstructures predicted by the CCT diagram, which support both qualitatively and quantitatively the results shown in Figures 11 and 12.
Figure 13 shows the final grain size at the HAZ as consequence of thermal evolution during the welding taking into account the variables adopted in this study, whereas Figure 14 shows the effects of heating rate on the grain growth when comparing two different heat input and identical workpiece thickness.Heating rates in Figure 14 were calculated along transversal section of the workpiece referring to temperature of 1500 o C, i.e., immediately below of the FZ.Due to the comparatively shorter time exposed to elevated temperatures, the final grain size was significantly lower when higher heating rates were reached, i.e., with decreasing heat input.Effects from the cooling rates caused by different welding conditions on the hardness profile along the transversal plane to the workpiece and crossing the HAZ can be seen in Figure 16.Accordingly, greater hardness levels were comparatively obtained with increasing cooling rates since it favors the formation of harder constituents as the martensite.

Conclusions
In this study, a phenomenological model to predict the multiphase diffusional decomposition of the austenite during continuous cooling was implemented in a developed FVM computational code and applied for numerically to simulate the microstructure in the HAZ of a low-alloy hypoeutectoid steel.Thus, the results are summarized below.
a) The used methodology was able of satisfactorily to predict the phase transformations and the hardness distribution at the HAZ of the steel investigated.
For this purpose, it was necessary to predict the temperature field coupled dynamically with the welding evolution and the material thermophysical properties, together with the kinetics model for phase transformations and the model for hardness prediction.
b) Different welding conditions were simulated in order to evaluate the effectiveness of the methodology employed.Heat input, workpiece thickness and preheating temperature were some of the investigated welding variables, which showed to play a fundamental effect on the thermal history from welding and, accordingly, on the resultant microstructure and the hardness distribution at the HAZ.
c) Grain growth at the HAZ showed be dependent on the heating rate, i.e., larger grain size was noticed to occur when lower heating rates were reached, i.e., with increasing heat input.
d) Elevated cooling rates have contributed for attaining higher hardness values in the HAZ, since it has favored the formation of harder constituents as the martensite.

Figure 2 -
Figure 2 -Temperature-dependent of specific heat for the investigated steel with individual phases during welding considered in this mode l4,24,25 .

Figure 3 -
Figure 3 -Temperature-dependent thermal conductivity for the investigated steel and individual phases considered in this mode l4,24,25

Figure 5 -
Figure 5 -(a) Comparison for temperature evolution measured by thermocouples located at the bottom and centerline of the weldment and model predictions: Thermocouple (T1) and Thermocouple (T2) at 60 and 120 mm from origin of welding respectively and (b) Numerical and experimental comparison for the fusion zone (FZ) and heat affected zone (HAZ) in a transversal section located at 60 mm in the welding direction.Steel: 0.38%C, 0.96%Mn, 0.22%Si, 0.016%Cu, 0.12%Ni, 0.028%Cr, 0.03%Mo.Plate dimensions 10 x 60 x 220 mm.Heat input 1.5 kJ mm -1 .

Figure 10
Figure 10 corresponds to the CCT diagram for the steel evaluated16 where are also plotted some calculated cooling curves considering each investigated welding condition in this study (see Table2).The cooling curves have started from a local at the HAZ where the peak temperature has reached about 1100 o C, with the volume fraction of the constituents and the hardness indicated in Figure10calculated using the mathematical formulation presented in this study.Figures11 and 12present more comprehensive results depicting the HAZ and the resultant microstructure.

Figure 15
Figure 15 shows the final hardness distribution at the HAZ.Boundaries between FZ and HAZ have been again outlined by the same reasons previously mentioned.Calculated hardness values are compatible with those expected for the resultant microstructure and with those predicted by the CCT diagram of the steel investigated (see Figures 10 to 12).Effects from the cooling rates caused by different welding conditions on the hardness profile along the transversal plane to the workpiece and crossing the HAZ can be seen in Figure16.Accordingly, greater hardness levels were comparatively obtained with increasing cooling rates since it favors the formation of harder constituents as the martensite.
Figure 15 shows the final hardness distribution at the HAZ.Boundaries between FZ and HAZ have been again outlined by the same reasons previously mentioned.Calculated hardness values are compatible with those expected for the resultant microstructure and with those predicted by the CCT diagram of the steel investigated (see Figures 10 to 12).Effects from the cooling rates caused by different welding conditions on the hardness profile along the transversal plane to the workpiece and crossing the HAZ can be seen in Figure16.Accordingly, greater hardness levels were comparatively obtained with increasing cooling rates since it favors the formation of harder constituents as the martensite.
(Equations 25 to 27), which take into account the steel composition and the cooling rate.