1
Combined finite element and phase field method for simulation of austenite grain growth
in the heat-affected zone of a martensitic steel weld
1L. Shi, S. A. Alexandratos, N. P. O'Dowd*
School of Engineering, Bernal Institute, University of Limerick, Limerick V94 T9PX, Ireland
*
Corresponding author, Tel.: +353 61 20 2545; E-mail: [email protected]
Abstract Engineering components operating at high temperature often fail due to the initiation and
growth of cracks in the heat-affected zone (HAZ) adjacent to a weld. Understanding the effects of microstructural evolution in the HAZ is important in order to predict and control the final properties of welded joints. This study presents a combined finite-element (FE) and phase-field (PF) method for simulation of austenite grain growth in the HAZ of a tempered martensite (P91) steel weld. The FE method is used to determine the thermal history of the HAZ during gas tungsten arc welding (GTAW) of a P91 steel plate. Then, the calculated thermal history is included in a PF model to simulate grain growth at various positions in the HAZ. The predicted mean grain size and grain distribution match well with experimental data for simulated welds from the literature. The work lays the foundation for optimising the process parameters in welding of P91 and other ferritic/martensitic steels in order to control the final HAZ microstructure.Keywords
:
Phase field method (PFM); finite-element method (FEM); heat-affected zone (HAZ); austenite grain growth; P91 tempered martensitic steel; welding thermal process1.
Introduction
Environmental concerns are a strong driving force for the development of new power plant materials which can withstand higher operating temperatures, with resultant increase in thermal efficiency and reduction in emissions [1, 2]. P91 steel, a modified 9Cr-1Mo-V-Nb steel, has been widely used for pressure vessels and super heaters in fossil fuel power plants due to its high corrosion resistance and creep resistance [3, 4]. However, the manufacturing history (e.g. welding, heat treatment) of this material may lead to microstructural changes and negatively influence the performance under creep or creep-fatigue conditions [5, 6]. Therefore, understanding the effects of welding processes on microstructure degradation and the subsequent impact on in-service performance are crucial to improve the sustainability of engineering structures through microstructurally-based design [7]. However, a full understanding of the detailed effects of welding processes on microstructural degradation is lacking and the effects on mechanical response of P91 are to a large extent unquantified [1].
During welding, thermal gradients in the material adjacent to the weld metal (WM) result in heterogeneous microstructures near the weld fusion line, a region known as the heat affected zone (HAZ) [6]. These regions typically experience a peak temperature above Ac1 during welding (when the
base martensite starts to transform to austenite as shown in Fig. 1, adapted from [8]). The HAZ of a P91 weld is divided into a number of sub-zones: coarse-grain (CGHAZ, 1100 oC < Tp < Ts), fine-grain
1 Research presented at the IUTAM Symposium Multi-scale Fatigue, Fracture and Damage of Materials in Harsh Environments in Galway, Ireland, August 28-September 1, 2017
2 (FGHAZ, Ac3 <Tp < 1100
o
C) and intercritical (ICHAZ, Ac1 < Tp <Ac3) based on the austenite grain size
resulting from the peak temperature Tp reached during welding in these regions [7]. A simplified
schematic of the microstructural evolution of the different sub-zones of the HAZ during welding of P91 is shown in Fig. 2. Here, we ignore the relatively small + region of the phase diagram and assume no phase change occurs in the CGHAZ during welding. The initial microstructure of P91 consists of prior austenite grains with a packet/block microstructure (Fig. 2a). During heating, these initial microstructures recover and begin to transform to austenite above Ac1 (Fig. 2b), resulting in a
partially transformed microstructure of over-tempered martensite (’) and newly formed austenite (). If the temperature increases, full transformation to austenite occurs above Ac3 (see Fig. 1),
resulting in a fine austenite grain structure (Fig. 2c). As the temperature increases above 1100 oC, the precipitates (particularly M23C6 and MX) that pin grain boundaries and restrict grain growth dissolve in the matrix [9], resulting in a coarse grain austenite microstructure (Fig. 2d). Following rapid cooling during welding, the austenite transforms back to martensite below Ms (martensite start temperature) and precipitation of M23C6 and MX occurs resulting in the final microstructures as shown in Figs. 2e to 2g. A small amount of -ferrite may be present in the CGHAZ near the weld metal depending on the thermal conditions; these -ferrite precipitates typically distribute along the PAG boundaries with low volume fraction [8] and are not considered here. Table 1 summarises the microstructure in the three HAZ sub-zones.
It is reported that crack initiation has been observed mainly in the FGHAZ or ICHAZ of P91 steels [7, 9]. The austenite grain size and morphology determine the final martensite microstructure (packet/block/lath size and morphology) which influences the final properties of the welded joint [10] and it is reported that an optimum prior austenite grain (PAG) size appears to exist for a creep strength enhanced Cr-Mo-V rotor steel [11]. Thus, understanding the austenite grain evolution during welding is crucial for optimising and controlling the performance of welded steels.
Several methods have been implemented to simulate grain growth in welds [12, 13, 14], such as the Monte-Carlo (MC) method, the cellular automata (CA) method and the phase field (PF) method. In [12] the MC method has been used to simulate grain growth in the HAZ of Ti-6Al-4V during GTAW; it was found that the predicted mean grain diameter near the WM was about four to twelve times larger than the base metal, depending on the heat input. Recently, the MC method has been combined with a thermo-fluid model to predict grain growth in the WM and HAZ of copper [15]. Grain growth in the HAZ of 12 wt% Cr ferritic stainless steel during laser-pulsed gas metal arc hybrid welding, has also been examined using the MC method [13]. A 2D model combining the FE and CA method has been used in [16] to predict dendritic grain growth in WMduring a laser engineering net shaping (LENS) process. Similarly, a 3D coupled CA-FE method was used to predict WM microstructures during multiple passes GTAW and gas metal arc welding (GMAW) process [14]. These mesoscale MC and CA methods are implemented by considering a time-dependent probability of grain growth due to thermal history. However, it is difficult to scale the numerical time step of these two methods to the physical time step. The microscale PF method therefore seems more suitable for simulation of grain growth and prediction of the grain morphology in the HAZ since the thermal history can be directly included in the analysis. Although the PF method has been widely used for to predict grain growth to investigate the grain topology and morphology [17, 18, 19], there
3
has been limited studies on the application of PF to analyse grain growth in the presence of thermal gradient. In [20] the PF method was used to predict the austenite grain size of AISI 316L stainless steel following GTAW. The mobility of grain boundary is treated as temperature dependent to imply the loss of pinning effects due to precipitate coarsening during welding. However, the effect of precipitates dissolution at high temperature, which is relevant to the P91 material being studied here, has not been considered in the model [20]. An extended approach was adopted in [21] to simulate austenite grain growth in the HAZ of X80 linepipe steel, with precipitate dissolution at higher temperature implicitly included in the model. In [21] the welding thermal history was obtained using a simplified Rosenthal solution for a moving point heat source, which is generally not accurate enough to predict the temperature in the HAZ of the welds [22]. In this study, a combined FE and PF approach to predict austenite grain growth in the HAZ during GTAW of P91 steel has been developed. The thermal history during welding is determined using a commercial FE package (Abaqus [23]). This thermal history is then implemented within an open-source PFM code (OpenPhase [17]) to predict austenite grain growth. The effects of precipitates on grain boundary mobility are implicitly considered in the PF model and the predicted grain structure and mean grain diameter at different peak temperatures are calibrated and validated by using independent experimental data.
2.
Modelling framework
2.1 Finite element modelling of temperature distribution in welding
The welding process involves complex thermo-physical phenomena such as heat conduction, convection, radiation, solidification and microstructural evolution. Completely describing these phenomena with a transport mechanism based model is costly and time-consuming and therefore a thermal conduction model with convection/radiation boundary conditions is implemented in this work. This study focuses on the characteristics of the thermal history and its influence on austenite grain growth in the HAZ of P91 during welding. The temperature profiles during welding are simulated by a thermal conduction model of a single pass bead-on-plate [24, 25]. Thus, the energy conservation equation is represented as follows:
q
z
T
y
T
x
T
k
t
T
c
p
2 2 2 2 2 2
(1)where is the mass density, cp is the specific heat capacity, T is the temperature, k is the thermal
conductivity, and q is the heat source from the welding torch, which is expressed as a Gaussian ellipsoidal heat flux distribution [24]. The initial temperature of the plate and the weld bead is set to be the ambient temperature and symmetry boundary conditions are applied to the longitudinal mid-plane [24]. The boundary conditions for welding heat transfer model include the heat loss due to convection and radiation on all external surfaces of the plate. Thus, the boundary condition at the surfaces of the plate can be defined as:
4
0 4 0T
T
T
T
h
n
T
k
, (2)4
where h is the convection coefficient, is the emissivity and the Stefan-Boltzmann constant (5.67 × 10-8 Wm2K4 ). The element birth technique has been utilised to simulate the weld filler deposition in the welding simulation [24, 26]. The unknown temperature field during welding is determined from Eqs. 1-2 using Abaqus [23].
2.2 Phase field modelling of austenite grain growth in HAZ
The phase field methodology to simulate grain growth in polycrystalline materials is described in detail in the literature, e.g. [19]. Only the relevant aspects of the method are reviewed here. The solution of the PF problem involves the tracking of a moving boundary or interface implicitly. Here a diffuse interface between grains is introduced, where each grain i is represented by its phase field variable i. For austenite grain growth the interfacial energy is the dominant contribution to the total free energy [17]. The total free energy depends on the total length (in 2D) or area (in 3D) of grain boundaries in the domain . The reduction in grain boundary length/area minimises total free energy and is the driving force for grain growth. The total free energy of the system, F, is:
d
f
F
int (3)where
f
int is the interfacial energy density. For a system consisting of N grains, the interfacialenergy density can be written as [17],
i j i j N j i j i sf
2 , 1 , int4
(4) where s is the grain boundary energy and is the grain boundary width. To obtain a solution the domain is divided into elements or sub-domains. The value of i represents whether an elementis within grain i or not: if the element is inside grain i,i = 1, otherwise i = 0. The region where0 <i < 1 represents the boundary of grain i as shown in Fig. 3. Within the grain boundary region of
width , the value of i changes continuous from 0 to 1. Each element within the domain is constrained such that
Ni1
i1. For example, for a binary grain boundary between i and j,i+j = 1. Equation 4 implies that the interfacial energy density is zero within the grain, while it is non-zero at the grain boundary. The evolution of the phase field variable i can be obtained using the relationship,
j i N j i F F N t
1 2 4 (5)where
F
i is the variational derivative of the total free energy functional F with respect to thephase field variable i and is the grain boundary mobility. From Eq. 5, we obtain the thermodynamically consistent phase field evolution equations for grain growth,
N j j i j i s iN
t
1 2 2 2
(6) In the initial P91 microstructure, precipitates of MX type are distributed throughout the microstructure and precipitates of M23C6 type are dispersed along grain boundaries [27]. These5
precipitates hinder the mobility of grain boundaries. However, the high temperatures experienced during welding lead to coarsening or even dissolution of the precipitates [28, 29, 30].Thus, to account for the loss of grain boundary pinning effects due to dissolution or coarsening of precipitates, it is necessary to modify the grain boundary mobility, . An Arrhenius relationship of temperature-dependent grain boundary mobility is employed in this model to account for the effects of precipitates evolution on grain growth which is expressed as follows [21],
RT
Q
exp
0
(7)where Q is the activation energy, R is the gas constant (8.314 J/mol/K) and 0 is a pre-exponential factor. In the analysis, two regimes are considered: T < 1100 oC, where the effect of precipitate pinning is important and the values of Q and 0 are 170 kJ/mol and 2.42 × 10
-4
m4/J/s, respectively, and T ≥ 1100 oC, where precipitates have dissolved and no longer pin the grain boundaries, and Q and 0 are 225 kJ/mol and 2.10 × 10-2 m4/J/s, respectively. These values have been calibrated from experimental data, as discussed in the subsequent sections.
2.3 Analytical solution of austenite grain growth in HAZ
Austenite grain growth has been modelling using analytical models, e.g. [31]. A widely used model comes from the classical relationship for isothermal grain growth, which can be used to simulate grain growth in the HAZ over discrete thermal cycles. In that case, each time step ti is considered as an isothermal grain growth process and the overall growth is obtained from the integration over time. The classical relationship of isothermal grain growth is expressed as [31]:
RT
Q
t
k
d
d
n n sexp
0 0 (8)where d0 and d are the initial and final mean grain diameter, respectively, k0 is a kinetic constant, t is time at temperature T, n is the time exponent and Qs is the activation energy for grain boundary movement. From Eq. 8, we can derive an explicit relationship for grain growth during welding as follows [31]: n t s n
RT
Q
t
k
d
d
1 0 0 0exp
(9)The value of Qs for austenite grain growth has been taken to be 53.9 kJ/mol, from [31]. The calibrated values of k0 and n (dimensionless) are 1.5 × 10-10m
n
/s and 2.25, respectively. Note that Q in Eq. (7) is not equal to Qs in the analytical grain growth model.
3.
Results and Discussion
The solution to Eq. 6 is obtained using OpenPhase [17]. The initial average austenite grain after full austenization at Ac3 is set to be 5 m, following [32], and the initial RVE has N = 400grains. The
grains are randomly introduced in the calculation domain by Voronoi tessellation, Voro++ [33]. Periodic boundary conditions are employed in the calculation domain. The computation parameters are chosen following [21]: the element (sub-domain) size x= y= z= 0.2 m, the grain boundary
6
thickness, = 0.6 m the time step t= 1.0 × 10-4 s. The grain boundary mobility is coupled with the thermal history during the welding process using Eq. 7 as discussed in previous Section 2.2.
3.1 Model calibration and validation
Data from simulated welding experiments (Gleeble tests) [32, 34] were used to calibrate and validate the PF model for austenite grain growth in P91. First, the P91 simulated weld data from [34] were used to calibrate the parameters of Q and 0 in the grain boundary mobility, (T). Next, to provide independent validation, the calibrated model parameters were used to predict the mean grain diameter in a P91 weld using the thermal history from [32]. The fitted data for grain boundary mobility in the HAZ of P91 steel are summarised in Table 2 and illustrated in Fig. 4. A two term and one term model are considered, as discussed in Section 2.2, the former taking into account the full dissolution of precipitates above 1100 oC, resulting in higher mobility relative to the one term model. Figure 5 shows an example of grain evolution at different times with a small grain (Labelled as `1’ in Fig. 5) eventually vanishing as the other grains become larger. As discussed in Section 2.2, the total free energy of the system depends on the total length/area of grain boundaries in the calculation domain. Grain growth is driven by the thermal activation of the grain boundary motion during welding so as to reduce the total free energy of the system [17]. This result in the initially larger grains expanding while the smaller grains contract and may eventually vanish as shown in Fig. 5, leading to a decrease in the number of grains with the RVE and an overall increase in grain size.
Figure 6 illustrates the final predicted grain morphology for the one term and two term grain mobility models. For comparison, two grains initially of the same size are labelled by dotted lines in Fig. 6(a) and (b). As expected, due to the enhanced grain boundary mobility above 1100 oC in the two term model, these two grains (Fig. 6b) are predicted to be larger than the corresponding grains in the one term model (Fig. 6a). Figure 7 shows the corresponding evolution of grain number and mean grain diameter for the two models. The time scale is made non-dimensional by dividing by total time (4 seconds). Figure 7 shows that the grains grow with a decrease in the total grain number in the calculation domain from an initial 400 grains to 130 for the one term model, and 89 for the two term model. The final mean grain diameters in the one term and two term model are approximately 9 m and 11 m, respectively. In what follows, the two term model is used throughout, as it provides a better representation of the grain growth process and closer agreement with measured grain sizes.
The predicted evolution of the austenite grains morphology in the CGHAZ using a thermal cycle from [34] with a heating rate of 120 oC/s up to 1,200o C and a cooling rate of 19.2 oC/s, is shown in Fig. 8. The calculation domain is 130 m × 175 m, with an initial 1,000 grains randomly distributed within the RVE using the VT method. During heating from Ac3 to its peak value, grain growth occurs
with a decrease of grain number from 1,000 to 376 as shown in Fig. 8(b) and corresponding increase in mean grain size to 13 m. Thereafter, grain growth continues during cooldown from peak temperature to 1,100 oC with a decrease of the grain number to 89 (Fig. 8c, mean grain size = 21 m). Below 1,100 oC grain growth occurs at a slower rate (compare Fig. 8c with 8d, mean grain size = 23 m, grain number = 69). Figure 9 shows a comparison of the predicted austenite grain morphology in the CGHAZ (Fig. 9a) with the experimentally measured one (Fig. 9b) [34]. The
7
predicted mean grain diameter is 23 m and the measured value is 25 m. Figure 10 shows the comparison of the predicted grain distribution frequency and cumulative frequency using the PF model with the data from [34]. Both predicted and experimental grain sizes were measured using the ASTM E1382 linear intercept length method [35, 36]. Figure 10(a) shows that both grain distributions are well described by the lognormal distributions [37] and the lognormal fit of the predicted distribution agrees very closely with the experimental data. The predicted and experimentally measured cumulative frequencies of grain diameters are presented in Fig. 10(b). The experimental measured maximum grain diameter is 66 m, while the predicted maximum grain diameter is 60 m.
The calibrated grain boundary mobility values were then used to predict the mean grain diameter for different peak temperatures to compare with the measured data from simulated welds in [32]. For these analyses, the heating rate is 53.8 oC/s, cooling rate is 7.5 oC/s with a hold time of 0.5 s at peak temperature.Figure 11 shows the comparison of predicted mean grain diameter with experimental data at various peak temperatures [32]. Excellent agreement is seen between the experimental measurements and PF simulations over the temperature range of the FGHAZ and CGHAZ (900 ˚C ≤ T < 1400 ˚C). The simulation somewhat overestimates the grain size at the highest temperature and this may be due to the phase change which occurs above 1300 ˚C (see Fig. 1) which is not accounted for in the model.
3.2 Results from the welding thermal model of a bead on plate simulation
A three-dimensional FE analysis was carried out using the thermal model introduced above for a single pass bead-on-plate geometry [24]. The specific heat capacity and thermal conductivity were temperature dependent [26, 38]. The welding efficiency was 0.75, welding speed was 2.27 mm/s and welding heat input was 844 J/mm, representative of bead-on-plate welds on 316 stainless steel described in [25].
Figure 12 shows a typical temperature contour 26 s after welding has started. Note that only half of the weld is modelled since the plate, weld bead and arc torch are symmetrical with respect to the welding path. It is seen that a significant region of the plate experiences temperatures in excess of Ac1 (approximately 830 oC). Temperature contours at the top surface of the plate, along the longitudinal section and the transverse section are shown in Fig. 13. The general features of the calculated temperature field are consistent with the results reported in the literature [24].
According to the phase diagram (Fig. 1), the simulated HAZ region is determined by the temperature range from Ac1 (830
o
C) to solidus (1500 oC). Figure 14 shows two typical thermal histories corresponding to points in the plate in the CGHAZ and FGHAZ. The `jump’ in the thermal history at t ≈ 12 s corresponds to the time when the weld bead is deposited and the weld torch is closest to the material point in question. These two representative thermal cycles drive the PF analysis in the following section.
3.3 Predicted austenite grain growth in HAZ
Figure 15 shows the initial randomly generated grain structure and the corresponding final grain structure due to the thermal histories shown in Fig. 14. Figure 16 shows the predicted evolution of
8
mean grain diameter and total grain number with non-dimensional time. The dotted line is the predicted grain size using the analytical model described in Section 2.3 (Eq. 8). The grain number from the analytical model in Fig. 6(b) is obtained by using the total RVE area divided by the average grain size. Figure 16(a) shows that the mean grain diameter in the CGHAZ starts to increase and reaches a steady state with a final mean grain diameter of approximately 10 m. The corresponding total grain number decreases from 400 to 107, as shown in Fig. 16(b). The PF model predicts a small amount of grain growth in the FGHAZ is limited and the final mean grain diameter is approximately 6 m with the grain number = 340. It may be seen that the analytical model provides a reasonable agreement with the PF simulation, particularly for grain growth in the CGHAZ. This is due to the choice of parameters used in the analytical model which were taken from [31]. An appropriate set of parameters could be chosen to fit the FGHAZ data, but a single set of parameters cannot predict both sets of results. A modified grain growth model, along the lines of the two term grain mobility, Eq. (7) would be expected to give an improved result, and is left for future work.
Figure 17 shows the predicted grain distribution frequency in the FGHAZ and CGHAZ after experiencing the thermal history shown in Fig. 14. All grain distributions are well described by lognormal distributions. Figure 18 shows the predicted HAZ region (Fig. 18a) from the FE analysis (900 ˚C ≤ T < 1400 ˚C) and the corresponding predicted mean grain size (Fig. 18b) at different regions in the HAZ from the PF analysis. As expected, the grain size decreases from the WM region (a) towards the unaffected base metal (b). The region near the weld metal experienced higher peak temperature and relatively longer time in austenization zone, which leads to a larger mean grain diameter. Away from the weld metal, the peak temperature decreases along with shorter exposure time resulting in finer grains. Thus, the combined FE method and PF method can capture the austenite grain growth in different sub-zones of the HAZ of P91 steel during welding process.
4.
Conclusion
A methodology combining a finite element thermal analysis with the phase field method has been used to predict austenite grain growth in the heat affected zone (HAZ) of P91 steel. The model can capture grain growth in different sub-zones of the HAZ. The prediction of mean grain size and grain size distribution by this methodology are in good agreement with the experimentally measured data from the literature. This type of multiscale methodology combining the finite element and phase field method, as illustrated here, should be generally applicable in process-microstructure-property prediction of polycrystalline materials, specifically the impact of microstructural evolution during manufacturing processes and their final influence on mechanical response. Further work on a martensite transformation model is required to obtain more quantitative information of the final martensitic microstructure in the HAZ and its influence on the final mechanical response of P91 joints.
Acknowledgements
The authors are grateful to Dr Oleg Shchyglo of the Interdisciplinary Centre for Advanced Materials Simulation, Ruhr University Bochum, Germany for useful discussions on implementing the OpenPhase code. Helpful discussions with Mr Edward Meade and Dr Fengwei Sun of the Bernal
9
Institute at University of Limerick, Ireland; Dr Richard Barrett, Prof Sean Leen, Mr. Ming Li, and Prof Padraic O’Donoghue of National University of Ireland, Galway; and Dr David Allen, Mr Bartosz Polomski and Mr Rod Vanstone of the Impel project team are gratefully acknowledged.
Funding
The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This publication has emanated in part from research conducted with the financial support of Science Foundation Ireland (Grant No. SFI/14/IA/2604). The modelling work was supported by the Irish Centre for High-End Computing (ICHEC).
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[38] Yaghi AH, Hyde TH, Becker AA, Williams JA, Sun W, Residual stress simulation in welded sections of P91 pipes. J Mater Process Technol 2005; 167: 480-487.
Table 1 Summary of the microstructure evolution in sub-zones of HAZ
Sub-zone Temperature PAG size Precipitatesat peak
temperature Phase
ICHAZ Ac1 < Tp < Ac3 Fine Few coarse Over-tempered ’with fine FGHAZ Ac3 <Tp < 1100
o
C Fine Few coarse Fine
CGHAZ 1100 oC < Tp < Ts Coarse None Coarse
Table 2 Calibrated data of grain boundary mobility in the simulation Temperature region Q(kJ mol-1) 0 (m4J-1s-1)
T < 1100 oC 170 2.42 x 10-4 T ≥ 1100 o
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Figure 1. Schematic of the sub-zone of the HAZ corresponding to the calculated equilibrium phase diagram of P91 steel, adapted from [6,8]
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Figure 2. Schematic of microstructure for different sub-zones of the HAZ during welding of P91 steel. The heating rate is approximately twice as large as the cooling rate and a short dwell at peak temperature is indicated. The dotted line in the temperature-time graph corresponds to the IC-HAZ, the dashed line the FGHAZ and the dot-dash line the CGHAZ. (The small amount of delta-ferrite distribute along austenite grain boundary in CGHAZ near the weld metal has not been considered in this schematic. The length scale indicated in the schematic is approximate.)
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Figure 3. Schematic profiles of two field variables for two grain
Figure 4. The grain boundary mobility, , at different temperature for the one term and two term model
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Figure 5. An example of grain evolution with a small grain (labelled 1 in the figure) eventually vanishing; The simulation times are (a) 0.6 s, (b) 0.7 s, (c) 0.8 s, (d) 0.9 s.
Figure 6. Final austenite grain morphology in (a) one term model and (b) two term model. Initial average grain size is approximately 5 m.
(a) (b)
Figure 7. Comparison of predicted (a) grain number and (b) mean grain diameter in one term model and two term model. The time scale is made non-dimensional by dividing by total time (4 seconds).
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Figure 8. Predicted grain evolution in CGHAZ at different temperature (a) initial grain structure at T = 920 oC during heating, (b) at T = 1200 oC (peak value), (c) at T = 1100 oC during cooling, (d) at T = 1200 oC during cooling.
Figure 9. Comparison of (a) simulation (23 m) and (b) experimentally measured (25 m) grain structure from Ref. [34] in CGHAZ.
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(a) (b)
Figure 10. Comparison of the model predicted (a) grain distribution frequency, and (b) cumulative frequency with experimental data from Ref. [34]
Figure 11. Comparison of the model predicted mean grain diameter and the experimental data [32] for various peak temperature.
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Figure 13. Temperature distribution at t = 26 s (a) top surface of plate, (b) longitudinal cross-section, (c) transverse cross-section at z = 30 mm.
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Figure 15. Predicted grain structure (a) initial grain morphology, (b) final grain morphology in FGHAZ, (c) final grain morphology in CGHAZ. Total evolution time = 5 s.
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(a) (b)
Figure 16. Predicted grain evolution in FGHAZ and CGHAZ (a) mean grain diameter, (b) total grain number. The analytical solution is that obtained from Eq. 9.
(a)
(b) (c)
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Figure 18. Predicted grain size for different regions in HAZ (a) temperature profile near the weld, (b) predicted grain size in the HAZ (along line ab in figure a).
