选区激光熔化镍基合金IN625数值仿真热源模型比较研究

李斌训; 孙玉晶; 杜劲; 夏岩; 苏国胜; 张庆; LI Binxun; SUN Yujing; DU Jin; XIA Yan; SU Guosheng; ZHANG Qing

Transactions of Nanjing University of Aeronautics & Astronautics

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Comparison of Heat Source Model for Numerical Modeling of Selective Laser Melting of IN625 Superalloy PDF

- ORCID：
LI Binxun ^1,2
✉
- ORCID：
SUN Yujing ^1,2
- ORCID：
DU Jin ^1,2
- ORCID：
XIA Yan ^1,2
- ORCID：
SU Guosheng ^1,2
- ORCID：
ZHANG Qing ³

1. School of Mechanical Engineering， Qilu University of Technology （Shandong Academy of Sciences）， Jinan 250353， P.R. China； 2. Shandong Institute of Mechanical Design and Research， Jinan 250031， P.R. China； 3. Engineering Training Center， Shandong University， Jinan 250002， P.R. China

CLC： TG456.7

Updated：2024-05-16

DOI：10.16356/j.1005-1120.2024.02.004

Abstract

The selection of heat source model is very important to accurately predict the distribution of temperature field and melting pool geometry in the numerical modeling of additive manufacturing process. The surface model， volumetric model and double-ellipsoid model are selected for comparison and analysis. These three heat source models are progarmmed as user-defined subroutines with Abaqus/Standard simulation software to predict the peak temperature and melting pool geometry during selective laser melting （SLM） of IN625. The comparison between simulation and experimental results shows that double-ellipsoid model can predict the melting pool geometry well， while the volumetric model provides comparative peak temperature predictions. In contrast， the surface model exhibits significant deviations in both melting pool geometry and peak temperature. The findings in this research highlight the need for model calibration or modification to enhance efficiency and accuracy before further research can be conducted.

Keywords

heat source model; numerical modeling; IN625; temperature; melting pool geometry

0 Introduction

Metal additive manufacturing （AM）， as one of the emerging advanced manufacturing technologies， has demonstrated the great capability of manufacturing components with intricate geometry and free-form surfaces in comparison to conventional manufacturing processes， which has found successful application in various industries such as aerospace， automotive and medical devices^［

1-3］. However， despite its benefits， additively manufactured components face significant challenges in terms of quality and mechanical properties， primarily due to internal defects， distortion， and high thermal residual stress^{［Reference 4

Baidu Scholar}4］.

In terms of additive manufacturing， selective laser melting （SLM） is one of the commonly utilized techniques for metal additive manufacturing including widely used nickel-based superalloys in the aerospace industry， which involves high-density energy， melting， liquidus material flowing， vaporization， and solidification^［

5］. Therefore， it is unrealistic to observe the complicated interaction experimentally between laser and metal powders. Under this condition， the finite element modeling is proposed as an efficient method to fulfill the purpose， and as far as AM modeling is concerned， various numerical modeling methods have been developed and used by many researchers. These methods can be generally classified as mesh-based numerical method^{［Reference 6-9}6-9］ and mesh-free modeling strategy^{［Reference 10-12}10-12］， to be capable of feedbacking high fidelity solutions. Since the laser heat source typically features with local concentration， transient and fast-moving， it develops the tendency to generate a large gradient of non-uniform temperature distribution as well as the thermal-induced stress. However， despite the numerical modeling methods， there is no doubt that the proper heat source model selection plays an important role in reliably describing the thermodynamic behavior of the melting pool.

So far， several heat source models have been developed by many scholars to define heat transfer in the melting pool and powder bed， including Gaussian surface heat source， double-ellipsoidal heat source， and many other geometrically modified volumetric heat sources^［

13-15］. Considering the quite limited layer thickness of the powder， the surface heat source model has been widely adopted to simulate the SLM process^{［Reference 16-19}16-19］. Chaurasia et al.^{［Reference 20

Baidu Scholar}20］ used a 2-D surface heat source model with a Gaussian shape to model laser surface melting of IN625 with and without considering fluid dynamics and obtain comparatively reasonable results. In addition， Fu and Guo^{［Reference 21

Baidu Scholar}21］ simulated the melting pool dimensions in terms of pool width， length， and depth in SLM of Ti-6Al-4V using surface heat flux with the highlighted extremely higher surface temperature. In practice， the laser heat flux will penetrate through the shallow powder thickness due to heat conductivity and particle surface reflection and therefore results in the remelting of the previous solidification layer^{［Reference 22

Baidu Scholar}22］. Based on the surface heat model， the volumetric heat surface model with Gaussian distribution was developed^{［Reference 23-24}23-24］. When modeling laser butt welding of stainless steel， the presented volumetric model in this study predicted the peak temperature and residual stress profile against the measured values^{［Reference 25

Baidu Scholar}25］. In addition， another volumetric heat source model commonly employed by researchers is the double-ellipsoid type^{［Reference 26-29}26-29］. Dunbar et al.^{［Reference 30

Baidu Scholar}30］ utilized a double-ellipsoid model combining mechanical analysis to simulate the built part distortion and the model results matched well with the experimental measurements. Moreover， the double-ellipsoid heat model proposed first by Goldak et al.^{［Reference 31

Baidu Scholar}31］ has also been validated against many experimental investigations in metal AM^{［Reference 32-33}32-33］. It can be seen that， although various heat source models have been adopted by many scholars in the numerical modeling of AM processes， even for the same metal materials and AM technique， the efficiency and accuracy of different types of heat flux models deserve further investigation and comparison.

In this study， a three-dimensional finite element model for the fulfillment of SLM IN625 is built in Abaqus/Standard. The surface heat source model， volumetric model， and double-ellipsoid model are respectively presented and programmed as a user-defined subroutine implemented into Abaqus for modeling heat transfer in the IN625 powder bed. The predicted melting pool geometry and peak temperature are extracted validating against the experimental results cited from open literatures.

1 Heat Source Model for SLM

To model the laser heat distribution and transfer on the powder bed， the three most commonly used heat source models including the surface heat source model， volumetric heat source model， and double-ellipsoid heat source model have been selected for comparison in terms of efficiency and accuracy.

The frequently utilized surface heat flux equation with Gaussian distribution is listed as

q (x, y) = \frac{2 A P}{π ω^{2}} e x p (- \frac{2 ({(x - x_{0})}^{2} + (y - y_{0})^{2})}{ω^{2}})

(1)

where q is the laser intensity， A the coefficient of laser absorption， P the nominal laser power， ω the laser spot radius， and （x₀， y₀） the coordinate of the laser spot center.

Based on the surface heat source model， the volumetric heat source model with Gaussian form takes into account the penetration of the laser beam into the powder bed， as shown in Fig.1（a）， which is defined as

\begin{array}{l} q (x, y, z) = \frac{2 A P}{π ω^{2} η} e x p (- 2 \frac{(x - x_{0})^{2} + (y - y_{0})^{2}}{ω^{2}}) \cdot \\ e x p (- \frac{|z|}{η}) \end{array}

(2)

Fig.1 Schematic of heat source models

where η is the depth of laser beam penetration.

The double-ellipsoid heat source model was presented by Goldak et al.^［

31］ to accurately capture the melting pool geometry， as shown in Fig.1（b）， and the equation of this model includes two parts， named the front part and rear part， respectively.

The front part of the double-ellipsoid can be written as

q (x, y, z) = \frac{6 \sqrt[]{3} f_{f} A P}{a_{f} b c π \sqrt[]{π}} e x p (- \frac{3 {(x - x_{0})}^{2}}{a_{f}^{2}} + \frac{3 {(y - y_{0})}^{2}}{b^{2}} + \frac{3 {(z - z_{0})}^{2}}{c^{2}}) \begin{matrix} x \geq 0 \end{matrix}

(3)

While the rear part can be expressed as

q (x, y, z) = \frac{6 \sqrt[]{3} f_{r} A P}{a_{r} b c π \sqrt[]{π}} e x p (- \frac{3 {(x - x_{0})}^{2}}{a_{r}^{2}} + \frac{3 {(y - y_{0})}^{2}}{b^{2}} + \frac{3 {(z - z_{0})}^{2}}{c^{2}}) \begin{matrix} x < 0 \end{matrix}

(4)

where a_f and a_r denote the semi-axes of the front and rear ellipsoidal， respectively； $f_{f}$ and $f_{r}$ the ratios of controlling heat flux flows into the front and rear parts of the heat source， respectively； b and c the lengths of semi-axes along the y and z directions， respectively. The model constants associated with the above-mentioned equations are cited and listed in Table 1.

Table 1 Constants for three heat source models

Heat source model	Variable	Value	Source
Surface model	A	0.3	Ref.[21]
Volumetric model	A	0.3	Ref.[21]
Volumetric model	η	0.4	Ref.[34]
Double⁃ellipsoid model	a_f/μm	276	Refs.[35⁃36]
	a_r/μm	1 520
	b/μm	160
	c/μm	160
	f_f	1.4
	f_r	0.6

2 Finite Element Modeling

In this research， the surface type， volumetric type， and double-ellipsoid type are selected to compare their efficiency and accuracy during the modeling of the SLM of IN625 nickel-based alloy. As shown in Fig.2， a 3D FE model with a geometrical size of 10 mm × 2 mm × 1 mm is created using Abaqus/Standard. Taking into account both the computation time and grid independence， the region located in the laser impact domain is meshed with 100 μm in length and 25 μm in width and a bias along the depth direction， with a total number of 193 456 elements. The element type of DC3D8 is assigned to the FE model. The “birth and death element control” function built-in Abaqus is activated to simulate the conversion from powder to solid. The above-mentioned three heat source models are respectively programmed as user-defined subroutine DFLUX and subsequently integrated into Abaqus. To simplify the modeling procedures， the assumptions of the numerical model include （i） The liquid in the melting pool is considered as viscous incompressible Newtonian fluid，（ii） composition changes and elements loss due to evaporation and spattering during melting are ignored，（iii） the coefficient of surface tension in the melting pool has not been considered in the model. The required thermo-mechanical properties of IN625 are listed in Table 2.

Fig.2 3D FE model in SLM of IN625

Table 2 Thermo⁃mechanical properties of IN625 powder

Material property	Value
Density ρ/(kg·m^-3)	8 440
Liquidus temperature T_L/℃	1 450
Solidus temperature T_S/℃	1 290
Specific heat C_p/(J·kg^-1·℃^-1)	338.98 + 0.243 7T (T ≤ T_S) 735 (T ≥ T_L)
Thermal conductivity k/(W·m^-1·℃^-1)	5.331 + 0.001 5 T (T ≤ T_S) 30.05 (T ≥ T_L)
Latent heat of fusion/(J·kg^-1)	227 × 10³
Thermal expansion coefficient β_T/℃^-1	1.28 × 10^-5

The SLM processing parameters include the scanning speed of 100 mm/s and laser powder of 300， 400， and 500 W， respectively， which correspond to those used in Ref.［

20］. The laser spot diameter is 500 μm. The simulated molten pool geometry and temperature using various heat source models are extracted to compare with the experimental data for accuracy validation.

3 Results and Discussion

During metal AM， the melting pool in which the solid particles change to liquid provides substantial information for a profound understanding of the AM process^［

37-38］. In general， the melting pool geometry （width and depth） and peak temperature are the typical indexes used for model validation. A typical simulated temperature contour of SLMed IN625 is shown in Fig.3， indicating the melting pool shape and geometrical characteristics.

Fig.3 Characterization of melting pool geometry

Fig.4 shows the top view （x-y plane） of the transient temperature distribution using various heat source models. As the laser heat source moves from left to right， a “comet tail” profile can be noticed. In addition， the steep temperature gradient is highly prominent ahead of the laser beam， and the good thermal conductivity as well as the Marangoni’s flow induced by strong surface tension results in the expanded melting pool size^［

39-40］.

Fig.4 Top view of molten pool geometry with different heat source models (P=500 W)

Fig.5 shows the simulated temperature distribution contour of the cross-section using different heat source models under various SLM processing parameters. The white dot lines within the contour image are the liquidus line corresponding to the liquidus temperature 1 350 ℃ of IN625 powder. According to the legend， the predicted highest surface temperature is corresponding to the surface model regardless of SLM processing parameters. The findings are expected to be the same as the reports in Ref.［

20］. The temperature induced by laser heat flux increases with the increasing of laser power. Since the surface model neglects the heat flux distribution beneath the top surface， all the heat energy focuses on the surface eventually， resulting in the observed extremely high temperature and quite shallow melting pool shape. In contrast， the volumetric and double-ellipsoid models consider the heat penetration in depth due to particle surface reflection， the transient temperature distribution along the Z-direction is much deeper， and thus the expected melting pool depth increases. It should be highlighted that the highest temperature of 1 339 ℃ using volumetric model under laser power 300 W is lower than the liquidus temperature of 1 350 ℃， which suggests the inadequate melting of the powder particles in SLM processes and is supposed to form a mushy zone.

Fig.5 Cross-section view of molten pool geometry with different heat source models at different laser powers

As far as the melting pool geometry and peak temperature are concerned， some detailed information is listed in Table 3. Based on the data in Table 3， the predicted melting pool width and depth associated with the double-ellipsoid model show a relatively good agreement with the experimental results in Ref.［

20］， while a large discrepancy appears with surface and volumetric models. Specifically， the predicted melting pool widths using the volumetric model are 324， 372 and 500 μm corresponding to laser power 300， 400 and 500 W， respectively. And the experimental data is 307， 385 and 455 μm with an absolute error of 5.5%， 3.4% and 9.89%， respectively. For the melting pool depth， the predicted values are 225， 246 and 287 μm and correspondingly the experimental data are 179， 274 and 321 μm with an absolute error of 25.7%， 10.2% and 10.59%， which proves the accuracy of the double-ellipsoid model in predicting the melting pool dimensions. After all， Eqs.（2—4） themselves describing the volumetric models consider the heat penetration depth. In terms of the peak temperature， it is the volumetric model that is in agreement with experimental data in Ref.［20］， while the simulated temperatures with double-ellipsoid and surface models show great deviation. To be more specific， the absolute errors between simulation with volumetric model and experiment under different laser powers are 30.1%， 26.5%， and 21.48%， respectively. Although the estimated absolute errors are all greater than the commonly accepted value of 15%， the volumetric model is still proved to be superior to the other two models. It can be derived that， none of the single heat source models presented in this study can satisfy the melting pool geometry and peak temperature predictions against experiments simultaneously. In contrast， the Gaussian heat source model feedbacks the worst predictions. In Ref.［23］， the correlation between melt pool geometry or the peak temperature with the energy deposition has been proved. Similarly， Chukkan et al.^{［Reference 25

Baidu Scholar}25］ conducted a combination of 3D conical and cylindrical shell heat source models and produced more accurate results， which confirmed the essentiality of authentic heat source shape description.

Table 3 Geometrical dimensions of the melting pool and peak temperature with different heat source models at different laser powers

Heat source model	Width /μm			Depth /μm			Peak temperature/℃
Heat source model	300 W	400 W	500 W	300 W	400 W	500 W	300 W	400 W	500 W
Surface model	482	500	550	72	82	100	6 383	8 486	10 730
Volumetric model	—	176	230	12	87	166	1 339	1 596	1 875
Double⁃ellipsoid model	324	372	500	225	246	287	4 837	6 270	7 760
Ref.[20]	307	385	455	179	274	321	1 916	2 172	2 388

Fig.6 plots the temperature distribution profile starting from the top surface deep into the powder bed. The heat source model has a great impact on the temperature gradient. The observed dramatic temperature gradient is found to be related to the surface model， while the insignificant temperature gradient change corresponds to the volumetric model. With the laser power increasing， the melting pool depth increases simultaneously. Meanwhile， it should be pointed out that every single powder layer thickness is merely limited to several ten microns. As a consequence， the input laser flux will cause the remelting of the previously solidified layer. As the melting pool depth continues to increase， more solidified material will be remelted. Material remelting has been proven to have a significant impact on microstructure evolution and residual stress^［

41-42］.

Fig.6 Simulated temperature distribution along Z-direction at different laser powers

Fig.7 plots the temperature variation trend along the Y-direction crossing the melting pool center. Determined by the shape of the laser heat source， it is not hard to imagine that the temperature distribution is symmetrical. The difference in temperature distribution across the melting pool along the Y-direction with various heat source models can be vividly exhibited and a rough comparison of melting pool width can be obtained. In addition， the maximum temperature increases with the rise of laser power from 300 W to 500 W.

Fig.7 Simulated temperature distribution across the molten pool at different laser powers

Fig.8 provides the simulated melting pool transverse-section profile comparison against the experimentally obtained image. It directly reveals that the melting pool shape using the double-ellipsoid model is comparable regarding the pool width and depth.

Fig.8 Comparison of the transverse section of the melting pool at laser power of 500 W

4 Conclusions

Three commonly employed heat source models including surface， volumetric， and double-ellipsoid models are selected for accuracy and efficiency comparison in the SLM of IN625 nickel-based superalloys. A 3D FE model with implemented user-defined subroutines DFLUX is created in Abaqus/Standard for numerical modeling and analysis. The main conclusions are as follows：

（1） Concerning the melting pool geometry， the double-ellipsoid model provides acceptable results in comparison to the experimental result cited in the literature， while both the surface and volumetric models show large discrepancies.

（2） The simulated peak temperature with the volumetric model is relatively close to the reported value in the literature in comparison to surface and double-ellipsoid models， even though the absolute errors are all beyond 20% under various SLM processing parameters.

（3） Melting pool dimensions and peak temperature cannot be accurately captured simultaneously with either a double-ellipsoid or volumetric model. To guarantee the numerical model results during the SLM of IN625， the heat source model selection， calibration or modification is a basic prerequisite.

Contributions Statement

Dr. LI Binxun designed the study， conducted the analysis， interpreted the results and wrote the manuscript. Dr. SUN Yujing contributed to the discussion and background of the study. Prof. DU Jin contributed to the data analysis and funding acquisition. Dr. XIA Yan contributed to the finite element modeling. Prof. SU Guosheng contributed to the draft and language modifica⁃tion. Mr. ZHANG Qing contributed to the subroutine written. All authors commented on the manuscript draft and approved the submission.

Acknowledgements

This work was supported by the Natural Science Foundation of Shandong Province （No.ZR2021QE230）， the Talent Research Project of Qilu University of Technology （Shandong Academy of Sciences）（No.2023RCKY118）， and the National Natural Science Foundation of China （Nos.52275438， 52205480）.

Conflict of Interest

The authors declare no competing interests.

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