Remelting process has been utilized to mitigate the residual stress level in the selective laser melting (SLM) process in recent years. However, the complex consolidation mechanism of powder and the different material behavior after the first laser melting hinder the direct implementation of the remelting process. In this work, the effects of remelting on the temperature and residual stress evolution in the SLM process are investigated using a thermomechanically coupled finite element model. The degree of consolidation is incorporated in the energy balance equation based on the thermodynamicallyconsistent phasefield approach. The drastic change of material properties due to the variation of temperature and material state is also considered. Using the proposed simulation framework, the singletrack scanning is simulated first to predict the melt pool dimension and validate the proposed model with the existing experimental data. The obtained thermal histories reveal that the highest cooling rate is observed at the end of the local solidification time which acts as an important indicator for the alleviation of temperature gradient. Then, the scanning of a whole single layer that consists of multiple tracks is simulated to observe the stress evolution with several remelting processes. After the full melting of powder material in the first scanning process, the increase of residual stress level is observed with one remelting cycle. Moreover, the predicted stress level with the remelting process shows the variation trend attributable to the accumulated heat in the tracks. The numerical issues and the detailed implementation process are also introduced in this paper.
The additive manufacturing (AM) process is an advanced manufacturing technique to build a threedimensional part using computeraideddesign (CAD) with successive melting/fusing/curing processes of distributed raw material on a solid substrate. Among the AM processes defined by ASTM [
Despite the advantages of the SLM process, the inability to fully observe and model the complex nature of the process hinders the wide implementation of SLM. During the SLM process, highly concentrated heat energy induced by a laser beam is absorbed by powder particles leading to localized melting which forms a molten region called melt pool. Then, the rapid consolidation of the molten material occurs as the center of the heat source moves to the scanning direction. The heating and the consolidation mechanism in the SLM process include various physical phenomena (melting, vaporization, and melt pool dynamics such as Marangoni convection) [
To mitigate these defects and improve the mechanical properties of the SLM printed parts, laser remelting procedure has been employed as a solution. Yasa et al. [
In recent years, numerous numerical models have been proposed and used to investigate the temperature and stress evolutions during the SLM process as well as the effect of the process parameters on the quality of the printed part. Simulation methods with various scales have been proposed from the mesoscale approach that considers individual powder particles to macroscale analysis with an entire part geometry. Chen et al. [
Although there have existed considerable efforts to study and optimize the SLM process, the thermal history and the consolidation behavior with the remelting process have not been studied in detail. To investigate the heat transfer mechanisms and the generation of residual stress with several remelting cycles, a numerical approach could be an effective solution to provide an insight into the process. However, most of the previous studies seldom consider (or partially apply) the drastic change of material properties and the degree of consolidation which can lead to insufficient reflection of the actual process. Furthermore, the computational issues and implementation methods in finite element analysis also have not been introduced in detail.
In this study, a simulation framework for the thermomechanical analysis of the SLM process considering the temperaturedependent material properties, and the degree of consolidation is proposed using commercial finite element analysis software ABAQUS. A heat transfer model based on the phasefield approach considering the consolidation behavior and latent heat of vaporization is implemented with the provided usersubroutines. The proposed model is then validated with the simulation results of the singletrack scanning process using the experimentally measured melt pool dimensions from existing results. Then, a singlelayer scanning process with several remelting cycles is simulated and the stress field is obtained by conducting sequentially coupled thermomechanical analysis. The model considers the strain hardening and the annealing effect with the element deactivation techniques applied to the powder state material. The different heat absorption mechanisms for the powder and dense state material are also considered using both of the surface and volumetric heat flux models. Finally, the variation of the stress level and the cooling rate after the local solidification with the several remelting processes are discussed to evaluate the process.
To build a numerical model of the SLM process, complex physical phenomena including the moving heat flux, the phase change, and the consolidation of the powder material with severe temperature change must be considered. The consolidated material can undergo several heating cycles and be remelted as the laser beam passes the neighboring region during the process. Furthermore, the thermal and mechanical properties of each material state (powder, liquid, and solid) are significantly different from one another. In this section, the numerical models for the thermomechanical analysis of SLM are described including the governing equations for continuumbased finite element formulations, the reference heat source models, thermal and mechanical material properties with effective values for powder/liquid states.
In this study, the powder bed is treated as a continuum material with effective material properties. The fluid dynamic is not considered thus the melt flow and the recoil pressure due to the vaporization are neglected for the following thermomechanical analysis. The governing equation of heat transfer which is the energy balance equation for an arbitrary material point in a volume
where
Wang et al. [
where
The phase parameter
where
where
Lastly, the vaporization term is added in the energy density
where
The material properties required for SLM process simulation are highly dependent on the temperature and the material state due to the severe temperature fluctuation. The recommended thermophysical properties are often used with some uncertainties near or above the melting temperature as reported by many previous studies [
Descriptions  Values 

Conductivity (powder) 

Conductivity (dense) 

Volumetric heat capacity (dense) 

Volumetric heat capacity (liquid) 

Latent heat of fusion 

Solidus temperature 

Liquidus temperature 

Latent heat of vaporization 

Liquidus temperature (liquid–vapor) 

Vaporized temperature 
The main heat transfer mechanism in the SLM process consists of heat radiation to the powder layer from the laser beam, heat conduction, and heat convection at the free surface. The typical boundary and initial conditions for the SLM process are shown in
where
For the simulation of the SLM process and other laser beam applications, numerical heat source models are widely used to model the moving laser beam. In the numerical model, the laser beam is modeled as a moving heat flux determined by the distance from the center of the virtual beam and other required beam parameters including the laser spot diameter and the scanning velocity.
The representative numerical heat source models are generally divided into 2 types (2D or 3D) as shown in
Model equation  Description  Reference 

2D Gaussian distribution  [ 

2D Gaussian distribution with absorptivity and characteristic beam radius  [ 

2D Gaussian distribution with radiation reduced by reflection and characteristic beam radius  [ 

A different form of 2D Gaussian distribution in a single metal powder layer  [ 

Equivalent form (numerical average) of 2D Gaussian distribution and characteristic beam radius  [ 
Model equation  Description  Reference 

3D uniform energy distribution with the optical penetration depth (OPD)  [ 

3D conical moving heat flux with Gaussian distribution and decaying penetration function  [ 

3D moving heat flux with Gaussian distribution and decaying penetration function using BeerLambert attenuation law  [ 

3D moving heat flux: Goldak’s semiellipsoid model  [ 

3D moving heat flux: Goldak’s doubleellipsoid model  [ 
The penetration depth and the other geometric parameters for volumetric heat source models are often determined empirically and need calibration process which requires some experimental results beforehand. Moreover, the obtained temperature fields based on these models are highly affected by the geometric shape of the heat source rather than the applied process parameters and the absorption mechanisms.
In order to obtain a more analytical solution of the heat flux in the SLM process, Gusarov et al. [
where
In addition, the 2D Gaussian distributed heat flux is applied at the top surface with the absorptivity
The governing equation for static structural analysis which is the conservation of linear momentum equation is expressed as follows
where
where
where
where
If
If
The mechanical properties of solid Ti6Al4V used in this study are listed in
Temperature 
Young’s modulus 
Yield stress 
Poisson ratio 
Plastic tangent modulus 
Coefficient of thermal expansion (CTE) 

297  125  1000  0.34  0.7  8.78 
367  110  630  0.35  2.2  9.83 
478  100  630  0.35  2.2  10.01 
590  100  525  0.36  2.2  10.71 
701  80  500  0.36  1.9  11.10 
812  74  446  0.37  1.9  11.22 
923  55  300  0.38  1.9  11.68 
1034  27  45  0.40  2  12.21 
1145  20  25  0.42  2  12.29 
1367  5  5  0.44  2  12.37 
1923  0.1  0.1  0.43  0.1  12.5 
To obtain the predictions for temperature and stress fields in the SLM process, including subsequent results such as melt pool dimensions and residual stress field, a sequentially coupled finite element analysis framework is developed using the commercial software ABAQUS/Standard. ABAQUS provides the interface for programming userdefined material behavior and boundary conditions. The specific features and numerical models of SLM introduced in Section 2 are implemented using the provided user subroutines [
The heat source can be modeled using DFLUX subroutine in ABAQUS. The position of the flux integration point is passed to the subroutine and the distance from the center of a virtual laser beam needs to be calculated to use the heat source models introduced in Section 2.3. The unit of the heat flux is
The variational statement of the energy balance equation (a 1D problem for simplicity) and the Jacobian (or tangent stiffness) contributions of each term after time integration using the backward difference in ABAQUS/Standard is as follows [
and the linearized equation using Newton’s method with the convection boundary condition is
where
Because the state variable
It is worth noting that different forms of energy balance equation based on thermodynamic consistency can be implemented in the same way. To validate the model implemented in ABAQUS, the variation of energy density at a material point (for both initially porous and dense cases) with temperature is extracted as shown in
The time increment also needs to be carefully determined to obtain proper results depending on the choice of the fixed domain methods to solve the phase change problems. The time required for melt interface to reach the steadystate is on the order of
For predictions of stress field and deformation in the SLM process, the temperature field obtained in the previous thermal analysis is passed as a function of time to the mechanical analysis. The time increment can be controlled differently because the convergence criteria are now different from the previous thermal analysis. Besides, it is possible to use a fixed time increment for both thermal and mechanical analysis, but not preferred when the problems are highly nonlinear. For the case of the singletrack scanning process considered in this study, the time required for solidification in SLM is generally in the units of
As the state variables are updated and saved using USDFLD subroutine in the previous thermal analysis, the state variable is used as an additional field variable to apply statedependent elastic and plastic properties with the annealing temperature as introduced in Section 2.4. To define the thermal expansion induced by the temperature variation, UEXPAN subroutine is used to apply isotropic thermal strain increment with temperaturedependent CTE.
To validate the analysis methodology introduced in this paper, an SLM process with a single powder layer is simulated. The finite element model with the given initial material states for analysis is as shown in
Based on the fact that the selective laser melting process is a sum of singlelayer formations, and the thermal cycles of the previous layers have little influence on the temperature profile of subsequent layers [
The proposed thermal model is validated by comparing the predicted melt pool dimensions with the experimentally results obtained in the literature [
Descriptions  Values 

Laser power 

Laser scanning speed 

Laser spot radius 

Layer thickness 
The local temperature evolution with the high cooling rate is the key feature of the PBF process which is closely related to the mechanical properties of the SLM printed part and the process stability. The typical local temperature history in the SLM process is as shown in
The transient temperature field, the cooling rate, and the state variables versus time at the middle point of the singlescan track on the solid substrate (which is initially porous) with the varying laser power are depicted in
The material state can be tracked using the adopted state variables as shown in
The liquid lifetime can be observed using the state graph as shown in
After conducting the simulation of the singletrack scanning process, a simulation of a single layer scanning process with multiple scanning tracks is conducted to obtain the stress fields involving several remelting processes. The laser tracks are defined as shown in
The singlelayer scanning process which involves 0–3 remelting processes are defined as SLM + LR0, SLM + LR1. SLM + LR2, and SLM + LR3 in this study. For the scanning process of one layer and each remelting process, the scanning time of 0.01 s is applied.
As well as the thermal gradients, the cooling rate is an important factor in the SLM process which determines the degree of the residual stress. The maximum cooling rates observed after the local solidification for the different process conditions are shown in
Using the obtained temperature field from the thermal analysis, the stress field is obtained to investigate the effect of the remelting process on the residual stress formation.
To investigate the effect of the remelting process on the degree of the residual stress, the average values of Mises’ stress and S11 in the scanned layer are computed as shown in
In this study, a thermomechanical analysis framework for SLM process simulation with the heat transfer model based on the phasefield approach, and the statedependent thermal and mechanical material properties is proposed with the detailed implementation process. Using the introduced methodology, other thermodynamically consistent heat transfer models can be implemented in the commercial finite element tool ABAQUS. The heat transfer model is validated using the existing experimental data with varying laser power (
The heat transfer model based on thermodynamically consistent phasefield approach is implemented successfully in ABAQUS/Standard using the usersubroutine and used in a thermomechanical simulation of the SLM process
The predicted melt pool dimension has a similar trend and good agreement with the experimental results, but the discrepancy becomes larger as the higher laser power is applied if the consolidation of the material and the vaporization is not considered.
From the simulation results of the singletrack scanning process with varying laser powers (20–80 W), the minimum cooling rate is observed with the laser power of 80 W, and the maximum cooling rate is observed at the time after the local solidification with the laser power of
The variation of the stress level with remelting processes is investigated. The result shows that the stress level increases with one remelting process, and decreases with the subsequent remelting processes due to the different heat transfer mechanisms between the porous and dense material which is also in a good agreement with the experimental result.
The obtained maximum cooling rates for cases with a different number of remelting processes show a similar trend with the variation of the stress level. However, the stress level decreases lower than that of the process without remelting after two times of remelting cycles whereas the cooling rate is higher for the case with the remelting process.
Based on the proposed thermomechanical simulation framework, the SLM process can be evaluated considering various analysis results (melt pool dimensions, phase field, cooling rates, and residual stress field). For future work, simulation in a wider range of process windows will be investigated and the sensitivity analysis for different process parameters and the remelting cycles will be conducted with extensive computations.