Applied Physics A

, 124:313 | Cite as

Thermal dynamic behavior during selective laser melting of K418 superalloy: numerical simulation and experimental verification

  • Zhen Chen
  • Yu Xiang
  • Zhengying Wei
  • Pei Wei
  • Bingheng Lu
  • Lijuan Zhang
  • Jun Du


During selective laser melting (SLM) of K418 powder, the influence of the process parameters, such as laser power P and scanning speed v, on the dynamic thermal behavior and morphology of the melted tracks was investigated numerically. A 3D finite difference method was established to predict the dynamic thermal behavior and flow mechanism of K418 powder irradiated by a Gaussian laser beam. A three-dimensional randomly packed powder bed composed of spherical particles was established by discrete element method. The powder particle information including particle size distribution and packing density were taken into account. The volume shrinkage and temperature-dependent thermophysical parameters such as thermal conductivity, specific heat, and other physical properties were also considered. The volume of fluid method was applied to reconstruct the free surface of the molten pool during SLM. The geometrical features, continuity boundaries, and irregularities of the molten pool were proved to be largely determined by the laser energy density. The numerical results are in good agreement with the experiments, which prove to be reasonable and effective. The results provide us some in-depth insight into the complex physical behavior during SLM and guide the optimization of process parameters.

1 Introduction

K418 is a precipitation-strengthened Ni-based superalloy developed in China in the 1970s. It has been extensively used in high-temperature environments, such as hot end turbocharger turbine wheels in aero engines, automotive and shipping industry due to its excellent high-temperature mechanical properties and oxidative stability at elevated temperature, as well as good stability and reliability over a wide temperature range [1]. However, it is difficult to manufacture K418 material using conventional machining methods at atmospheric temperature due to high cutting temperature, large plastic deformation, easy work hardening, excessive tool wear, and low material removal rates. Moreover, the application of the K418 components is very complicated with the shape of the labyrinth cavity, thin-walled complex surface or overhang structure, which is difficult to be manufactured through a single conventional method [2]. With the development of aerospace technology, the demand for sophisticated-geometry and high-performance components is increasing. However, advances in high-performance aerospace components would not be possible without the synchronous improvement of processing technology [3].

Recently, the newly developed selective laser melting (SLM) has been regarded as one of the most promising metal additive manufacturing (AM) technologies for fabricating the geometrically complex components [4, 5]. SLM makes it possible to manufacture arbitrary complex components by melting a randomly packed powder bed with a high-energy density laser beam. However, this may be inconceivable for traditional manufacturing methods [6].

Owing to the uniquely non-equilibrium mechanism in the SLM process, the powder material undergoes a complete melting and solidification corresponding to a rapid solidification rate of 105–108 K/s [7]. In addition, the SLM process involves a series of complex physical metallurgy processes, such as energy absorption, melting and solidification, re-melting of the previously solidified layer, heat conduction and heat radiation, diffusion and heat conduction in the molten pool, curing shrinkage, wetting effect, recoil pressure, etc. [8]. The molten pool is highly dynamic due to the rapid scanning of the Gauss laser beam, resulting in typical process defects such as porosity, balling, spattering, and denudation of single track [9, 10]. These defects may subsequently be inherited into the SLM fabricated parts through a track-by-track scanning process, ultimately reducing the mechanical properties of the final parts.

Generally, a successful SLM process is characterized by a well-bonded adjacent scanning track and adequate melting and solidification between adjacent layers [11]. It is well-known that the SLM parts are built from multi-single tracks and layers, therefore it is essential to fully understand the dynamic physical mechanisms during single track formation before fabricating a multi-layer component [12]. Actually, the processing parameters window for the SLM produced parts is quite narrow. In order to obtain the desired high quality and high-performance parts, it is critical to predict the thermal behavior and flow characteristics of the molten pool. However, relying solely on experiments is far from sufficient to reveal the mechanism of the interaction between the laser beam and the K418 powders, the dynamic thermal behavior, as well as the variation in the morphology of melted tracks [13].

In recent years, there are various approaches to modeling of SLM process including macroscopic, mesoscopic and microscopic models. Mesoscopic numerical models are commonly used to predict the thermal dynamic behavior of the molten pool during SLM. While the mesoscopic models considered so far have been discretized by means of either the FEM, FVM, or the LBM [14, 15, 16, 17]. Patil and Yadava [18] established a 2D finite element model (FEM) to calculate the temperature distribution during SLS. Kolossov et al. and Hussein et al. [19, 20] developed a 3D finite element model (FEM) to calculate the variation of temperature and thermal stresses by ANSYS, taking into account the non-linear behavior of specific heat and thermal conductivity. Unfortunately, most of the FEM models are based on similar assumptions—treating the powder bed as a homogeneous continuum body with the temperature independent thermal–physical properties, and the shrinkage effect of a real random powder bed is ignored due to the collapse of the powder during solidification and re-melting. Other numerical methods used include the finite volume method (FVM) and the Lattice Boltzmann method (LBM), which take into account the fluid dynamics of the molten pool [21]. Gu et al. [22, 23] developed a 3D finite volume method (FVM) to investigate the thermodynamic mechanisms and migration behavior during SLM using a commercial computational fluid dynamics (CFD) software. Khairallah et al. [24, 25] presented a 3D FVM mesoscopic model to reveal the defect mechanism during SLM by applying the ALE3D multi-physics code. In the simulations, the solidification behavior of the discrete powder particles and the discontinuity due to Plateau–Rayleigh instability were also studied. Körner et al. and Klassen et al. [12, 26, 27] coupling the sequential addition packing algorithm into Lattice Boltzmann model (LBM) to investigate melting and solidification of the powder bed randomly distributed on a solid substrate.

Although the thermal behavior of the molten pool during SLM process has been numerically simulated, these studies are far from enough. However, the physical mechanism of the SLM process is not yet fully understood, and the thermophysical parameters of individual materials vary widely, resulting in different thermal behavior and ultimate melt morphology. At present, most of the studies focused on Ti6Al4V [12], AlSi10Mg [28], 316L stainless steel [29], Inconel 718 [30]; however, the researches on SLM process of K418 have not been reported yet.

In this work, a 3D transient mesoscopic model was developed to predict the thermal behavior and flow mechanism of the molten pool, as well as the morphology of the melted tracks during SLM process of the K418 material. In the simulation, a 3D randomly packed powder bed consisting of spherical particles was created using the discrete element method (DEM), taking into account the effects of individual powder particles. Volume shrinkage and the temperature-dependent thermos-physical parameters were also taken into account. The free surface of solid–liquid was tracked by VOF. Furthermore, experiments were carried out to investigate the morphology of the single melted tracks under various processing parameters to verify the feasibility and reliability of the established model.

2 Numerical model

2.1 Mathematical–physical considerations

In order to describe the complex physical mechanisms during SLM, the real physical phenomena (depicted in Fig. 1) that occur during the process should be simplified so that the dominant mechanisms are considered. The assumptions used in this study are as follows [28]: (1) assuming that the melt in the molten pool is an incompressible homogeneous Newtonian fluid and is a laminar flow, the fluid flow and heat transfer are axisymmetric [31]; (2) except the specific heat, thermal conductivity, viscosity and surface tension, some other thermophysical constants are considered temperature independent; (3) the mushy zone is considered as isotropic in the solid–liquid phase transition; (4) the particle size of the spherical powder has a Gauss distribution; (5) the thermal conductivity of the powder and the shielding argon is considered constant. It should be noted that some other modelling aspects were not considered in the present model. For example, the unmelted powder particles are assumed to be spatially fixed and the 3D randomly packed powder bed is only solved for the pure thermal problem based on an irradiation of Gaussian laser beam during SLM. Different thermophysical properties and laser absorption of the substrate and powder are not considered, and the wetting behavior of the liquid metal on the solid substrate has not been modeled. Also, the surrounding argon phase and the fluid dynamics induced in this phase have not been modeled.

Fig. 1

Physical phenomena during selective laser melting

2.2 Governing equations

A three-dimensional finite difference method (FDM) was developed to investigate the dynamic thermal behavior of the randomly packed K418 powder bed under an irradiation of Gaussian laser beam during SLM. The free surface of the solid–liquid in the molten pool was tracked by the volume of fluid (VOF) method. Generally, the dynamic flow field within the molten pool was governed by Navier–Stokes and energy equations, which is expressed as follows according to [32, 33]:

Mass continuity equation:

For incompressible fluid with constant volume, the density is constant, the mass continuity equation is
$$\frac{{\partial u}}{{\partial t}}+\frac{{\partial \left( {\rho u} \right)}}{{\partial X}}+\frac{{\partial \left( {\rho v} \right)}}{{\partial Y}}+\frac{{\partial \left( {\rho w} \right)}}{{\partial Z}}=0,$$
where ρ is the liquid density, u, v, and w represent the components of the velocity vector along X, Y, and Z axes, respectively.
Momentum conservation equation:
$$\frac{{\partial \vec {V}}}{{\partial t}}+\left( {\vec {V} \cdot \nabla } \right)\vec {V}= - \frac{1}{\rho }\nabla P+\mu {\nabla ^2}\vec {V}+\vec {g}\left[ {1 - \beta \left( {T - {T_{\text{m}}}} \right)} \right],$$
where \(\mu\) is the fluid viscosity, P is the hydrodynamic pressure, g is the gravitational acceleration, \({T_{\text{m}}}\) is the melting temperature of the K418 material, T is the fluid temperature, \(\beta\) is the volumetric thermal expansion coefficient of the K418 material.

Energy conservation equation:

The conservation of energy satisfies the following equation
$$\frac{{\partial H}}{{\partial t}}+\left( {\vec {V} \cdot \nabla } \right)H=\frac{1}{\rho }\left( {\nabla \cdot k\nabla T} \right)+{S_{\text{U}}},$$
where \(k\) is the thermal conductivity, \({S_{\text{U}}}\) is the energy source term, \(H\) is the enthalpy which is considered to be a linear function of temperature T, depending on the solid fraction of the fluid.
$$H=\int {{C_{\text{p}}}T{\text{d}}T} +{L_{{\text{sl}}}}{f_{\text{s}}},$$
where Cp is the specific heat of the liquid metal, \({L_{{\text{sl}}}}\) is the solid–liquid phase change latent heat, fs is the phase change fraction. In this work, the latent heat L associated with metal melting is controlled by fs between the liquidus temperature Tl and the solidus temperature Ts. Fs is expressed as
$${f_{\text{s}}}=\left\{ \begin{gathered} 0\quad T \leqslant {T_{\text{s}}} \hfill \\ \frac{{T - {T_{\text{s}}}}}{\begin{gathered} {T_{\text{l}}} - {T_{\text{s}}} \hfill \\ 1\quad T \geqslant {T_{\text{l}}} \hfill \\ \end{gathered} }\quad {T_{\text{s}}} \leqslant T \leqslant {T_{\text{l}}} \hfill \\ \end{gathered} \right.$$
To simplify the calculation, the enthalpy is expressed as a function of temperature, which can be calculated according to the following formula
$$T - {T_{\text{m}}}=\left\{ \begin{gathered} \frac{{H - {H_{\text{s}}}}}{{{C_{\text{s}}}}} - \varepsilon \quad H \leqslant {H_{\text{s}}} \hfill \\ \frac{{H - {C_{\text{p}}}{T_{\text{m}}} - {L_{{\text{sl}}}}/2}}{{C{}_{{\text{p}}}+{L_{{\text{sl}}}}/2\varepsilon }}\quad {H_{\text{s}}} \leqslant H \leqslant {H_{\text{l}}} \hfill \\ \frac{{H - {H_{\text{l}}}}}{{{C_{\text{l}}}}}+\varepsilon \quad H \geqslant {H_{\text{l}}}, \hfill \\ \end{gathered} \right.$$
where Hs and Hl are the saturation enthalpy of the solid phase and liquid phase; Cs and Cl are the specific heat capacity of the solid and liquid phase; \({T_{\text{s}}}\) and \({T_{\text{l}}}\) are the solidus and liquidus temperature of K418, respectively, ε is the phase change radius \(\left( {\varepsilon =\left( {{T_{\text{l}}} - {T_{\text{s}}}} \right)/2} \right)\).

The VOF method defines a fluid volume function F, which represents the fraction of the volume fraction of the liquid metal filled in the cell, as shown in Fig. 2. F = 1 indicates that the grid is full of liquid, and F = 0 is completely empty. 0 < F < 1, suggesting that the grid is on a free surface.

Fig. 2

Schematic diagram of VOF

VOF method was applied to reconstruct the evolution of the free surface when particles are penetrated into the molten pool. The VOF function was originally derived by Hirt and Nichols [34]. The following equations are satisfied:
$$\frac{{DF}}{{Dt}}=\frac{{\partial F}}{{\partial t}}+\frac{{\partial \left( {uF} \right)}}{{\partial x}}+\frac{{\partial \left( {vF} \right)}}{{\partial y}}+\frac{{\partial \left( {wF} \right)}}{{\partial z}}=0.$$

2.3 Boundary conditions

2.3.1 Thermal boundary condition of free interface

There is a complex energy exchange between the free interface and argon atmosphere, which includes the input of laser energy, the absorption of heat by the powder bed, the heat dissipated by heat convection and heat radiation, and the heat taken away by melt evaporation, as shown in Figs. 3.

Fig. 3

Thermophysical phenomena at the interface of molten pool in the SLM process

The thermal boundary condition of the free interface is given by Dai et al. [35]:
$$\kappa \frac{{\partial T}}{{\partial z}}=Aq\left( {x,y,z,t} \right) - {h_{\text{c}}}\left( {T - {T_0}} \right) - {\varepsilon _{\text{r}}}{\sigma _{\text{s}}}\left( {{T^4} - T_{0}^{4}} \right) - {q_{{\text{ev}}}},$$
where \(A\) is the laser absorptivity of K418 powder, q(x,y,z,t) is the energy input of the Gauss laser beam, hc is the natural convection heat transfer coefficient, \({\varepsilon _{\text{r}}}\) is the thermal radiation coefficient and \({\sigma _{\text{s}}}\) is the Stefan–Boltzmann constant, \({q_{{\text{ev}}}}\) is the heat taken away by melt evaporation, \({T_0}\) is the ambient temperature.
In this work, a rotating volumetric Gaussian distributed laser source with a wavelength of 1064 nm for ytterbium fiber laser (IPG, YLR-500-WC, Germany) was used in the SLM process, Fig. 4 shows the Gaussian distributed laser heat source modele adopted in this work. The moving beam described by a Gaussian surface heat equation is given by Yin et al. and Gu et al. [30, 36, 37]:
$$q\left( {x,y,z} \right)=\frac{{2AP}}{{\pi {\varpi ^2}}}\exp \left( { - 2\frac{{{r^2}}}{{{\varpi ^2}}}} \right),$$
where P is the laser rated power, \(\omega\) is the radius of the Gaussian laser beam, q(r) means the heat flux at a distance r from the center of the circle
$$r=\sqrt {{{\left( {\left| x \right| - \left| {v \cdot t} \right|} \right)}^2}+{{\left| y \right|}^2}} ,$$
where \(\left| x \right|\) and \(\left| {\text{y}} \right|\) denote the distance along the X and Y axes, respectively, and v is the laser scanning speed.
Fig. 4

The intensity distribution of the Gaussian laser beam with p = 320 W and a spot diameter of 70 µm

The absorptivity of the material to the laser beam is relative to the thermal physical parameters of the material, material porosity, the laser wavelength, and laser incident angle. Due to the smaller laser scanning area of 100 mm × 100 mm and the larger focal length (H = 330 mm) of the F − θ lens as seen in Fig. 5, the laser beam can be approximately considered as perpendicularly irradiate on a substrate pass through an F − θ lens, i.e., the incident angle is 90°in simulation. In this work, the absorptivity \(A\) of the fiber laser beam by nickel-based alloy powder bed was settled as 0.6 according to Gusarov et al. [38].

Fig. 5

The schematic diagram of the focusing of F − θ lens

The full heat-flux boundary condition at the free surface is given by
$${q_{{\text{in}}}}=Aq\left( {x,y,z,t} \right) - {h_{\text{c}}}\left( {T - {T_0}} \right) - {\varepsilon _{\text{r}}}{\sigma _{\text{s}}}\left( {{T^4} - T_{0}^{4}} \right),$$
where the Stefan–Boltzmann constant \({\sigma _{\text{s}}}\) is 5.67 × 10− 8 W/(m2 K4).
During the SLM process, at higher energy inputs, some liquid metal evaporates and accumulates above the molten pool, forming a so-called Knudsen layer [39, 40]. The vaporized metal vapor takes away part of the heat and exerts a recoil pressure on the molten pool, affecting the dynamic properties of the molten pool. The heat evaporated by the metal is given by the expression [40]
$${q_{{\text{ev}}}}=0.82\frac{{\Delta {H^*}}}{{\sqrt {2\pi MRT} }}{P_0}\exp \left( {\Delta {H^*} \cdot \frac{{T - {T_{{\text{lv}}}}}}{{RT{T_{{\text{lv}}}}}}} \right),$$
where R is the ideal gas constant, P0 is the ambient pressure, M is the molar mass, ΔH* is the enthalpy of escaping metal vapor, Tlv is the boiling point of the metal melt.

2.3.2 Pressure boundary condition of free interface

The computational region with the temperature between liquid and solidus temperature can be considered as a mushy zone. The volume change associated with phase transitions is ignored, and the drag coefficient is approximated as a function of the local solids fraction and is used to solve the solid–liquid phase change. When the material is in the solid phase, the drag should be effectively infinite. In the mushy zone, the resistance should be an intermediate value. The drag coefficient \({F_{\text{d}}}\) is expressed as Carman–Kozeny equation, which is derived from Darcy model as given by Voller and Prakash [41]:
$${F_{\text{d}}}={F_0}\frac{{f_{{\text{s}}}^{2}}}{{{{\left( {1 - {f_{\text{s}}}} \right)}^3}}},$$
where \({F_0}\) is a constant drag coefficient, \({f_{\text{s}}}\) is the solid fraction in a given cell. The solid free surface of the grid is considered as the wall boundary condition
$$k\frac{{\partial T}}{{\partial n}}={h_{\text{c}}}\left( {T - {T_0}} \right).$$
In order to investigate the Marangoni flow caused by the temperature gradient on the free surface in the molten pool, the shear stress should be balanced with the free surface boundary conditions, as given by Cho et al. [42]:
$$\begin{gathered} - \mu \frac{{\partial u}}{{\partial z}}=\frac{{\partial \gamma }}{{\partial T}}\frac{{\partial T}}{{\partial x}} \hfill \\ - \mu \frac{{\partial v}}{{\partial z}}=\frac{{\partial \gamma }}{{\partial T}}\frac{{\partial T}}{{\partial y}}, \hfill \\ \end{gathered}$$
where \(\mu\) is the dynamic viscosity, \({\raise0.7ex\hbox{${\partial \gamma }$} \!\mathord{\left/ {\vphantom {{\partial \gamma } {\partial T}}}\right.\kern-0pt}\!\lower0.7ex\hbox{${\partial T}$}}\) is the surface tension gradient. The surface pressure boundary condition is expressed by Masmoudi and Coddet [33]
$$- P+2\mu \frac{{\partial {{\vec {v}}_{\text{n}}}}}{{\partial n}}= - {P_{{\text{Laser}}}}+\sigma \left( {{\rho _x}+{\rho _x}} \right),$$
where \({P_{{\text{Laser}}}}\) is the laser recoil pressure, \({\vec {v}_{\text{n}}}\) is the normal velocity vector, \(\sigma\) is the surface tension, and \({\rho _{\text{x}}}\) and \({\rho _{\text{y}}}\) represent the curvature radius along the X and Y directions, respectively.
$${P_{{\text{laser}}}}=0.54{P_0}\exp \left( {{L_{{\text{lv}}}} \cdot \frac{{T - {T_{{\text{lv}}}}}}{{RT{T_{{\text{lv}}}}}}} \right).$$

2.4 Materials and modeling of powder bed

In this work, the gas atomized spherical K418 alloy powder was supplied by Beijing AMC Powder Metallurgy Technology Co., Ltd. The chemical composition is shown in Table 1. The scanning electron microscope (SEM) morphology of the K418 powder is shown in Fig. 6a, indicating that the powder particles have a nearly spherical morphology. The particle size distribution of the powder is obtained by a laser diffraction particle size analyzer (Sympatec GmnH, HELOS H3185, Germany). As shown in Fig. 6b, the powder has a particle size range from 15 to 53 µm and an average size of 28.59 µm.

Table 1

The chemical composition of K418, according to GB/T 14992-2005 (wt%)











Std. (wt%)








≤ 1.0


Actual (wt%)










Fig. 6

a SEM micrograph, and b powder size distribution of K418 powder

In the simulation, a discrete element method (DEM) was firstly used to generate the powder particles, which contains the particle stack information (e.g., the size, shape, and coordinates of individual particles). The tap density of the K418 powder was set to 55%. The transient simulation was performed in a 3D computation domain with 200 × 500 × 120 µm, as shown in Fig. 7. The height of the substrate and the thickness of the powder layer were both 30 µm. The entire domain was gridded by 180 × 80 × 60 cells. These information were then entered into input the Flow-3D software to solve thermal behavior and the fluid flow of the K418 materials during SLM by a developed 3D finite difference method (FDM).

Fig. 7

Model of randomly packed K418 powder bed with Gaussian distributed particles

2.5 Material thermal properties and the processing parameters

During SLM, thermal properties of the K418 material, such as thermal conductivity, specific heat, viscosity, and density were obtained by JmatPro, as shown in Fig. 8. A commercialized numerical software Flow-3D was used in the simulation and the applied parameters are given in Table 2.

Fig. 8

Thermos-physical parameters of K418: a thermal conductivity, b specific heat, c viscosity, and d surface tension

Table 2

Material parameters and processing conditions used in this work [23]



Ambient temperature

293.15 K

Laser power, P

240 W, 280 W, 320 W, 360 W

Scanning speed, v

800–3600 mm/s (the interval is 200 mm/s)

Laser beam spot size, d

70 µm

Stefan–Boltzmann constant

5.67 × 10−8 W/(m2 K4)

Heat transfer coefficient, hc

80 W/(m2 K)

Radiation emissivity, ε


Power size distribution

0–53 µm, Gaussian

K418 solidus point, Ts

1573 K

K418 liquids point, Tl

1623 K

Shielding gas


Oxygen content, ppm

< 200

Gas pressure, Pa


In the simulation, the explicit hydrodynamic formulation was used to solve the motion of the powder and the molten pool. we firstly solve the temperature field, when the temperature of the powder bed reached the melting temperature of the K418 material, the flow of the molten pool occurred. The temperature field was then provided as the input for the fluid mechanics solver in a staggered manner to obtain the flow field and dynamic behavior of the molten pool. In order to ensure the stability of the simulation, the pressure solver, heat transfer solver, and advection were all set to be implicit. Moreover, in order to ensure sufficient calculation accuracy and less computational time, the minimum time step and the maximum time step were e−11 and 3e−8, respectively. As the calculation proceeds, the scale of the time step was generally stable at e−9–2e−8 to simulate the mesoscopic physical time scale between µs and ms. Calculation was carried out with Flow3D version 10.1.4 using 4 cores on an Intel® Xeon® processors E5-2696 and 4 GB RAM and a calculated timespan of 13 h.

3 Results and discussion

3.1 Dynamic thermal behavior of liquid metal

In the process of SLM, the variation of the process parameters leaded to different local solidification conditions, i.e., the temperature gradient G at the solid–liquid interface, solidification rate V in the molten pool, which evidently affected the microstructure and properties of the final components [43]. In addition, the temperature gradient G in the molten pool caused the surface tension gradients dγ/dT and the associated Marangoni flow of the fluid [25]. Under the Marangoni convection, the liquid at free surface in the molten pool was pulled to the sideways at a high velocity, while the cold liquid in the vicinity of the molten pool flew to the center from its edge, creating dynamic fluctuations (clockwise and anticlockwise vortices) in the pool. Therefore, the dynamic thermal behavior within the molten pool in the SLM process largely determined the final forming accuracy and quality of the SLM-processed components, which can be visually reflected in the metallurgical defects such as porosity, balling, spattering, denudation, etc., and the morphological features of the single melted track.

To investigate the morphological evolution of the molten pool quantitatively and intuitively, and to reveal the underlying physical mechanisms of the interaction between K418 powder and the laser beam during SLM, the SLM process of the K418 powder was divided into three stages. Figure 9 depicts the dynamic development of the molten pool during single track scanning at P = 320 W and v = 2400 mm/s. In the initial stage, the randomly packed K418 powder bed was stacked on a 316 L stainless steel substrate with a thickness of 30 µm, as shown in Fig. 9a. In the second stage, the powder layer absorbed the energy from the Gaussian laser source, was heated, and eventually melted. Under the irradiation of the Gauss heat source, the temperature of the molten pool increased sharply up to the melting temperature of the K418 material. Simultaneously, a complex and strongly fluctuating pool was created with the noticeable recoil effect, capillary, gravity and wetting forces, as shown in Fig. 9b. In the third stage, with the laser beam moving, the molten pool presented a highly dynamic fluctuation due to the high surface tension and the low viscosity of liquid metals until solidification, resulting in an irregular appearance of the molten pool, as indicated in Fig. 9c. Marangoni convection intensity can be written as [26]:

Fig. 9

The dynamic evolution of the molten pool during SLM process of K418 at P = 320 W and v = 2400 mm/s: a absorption laser, b melting and flow, and c solidified

$${\text{Ma}}=\frac{{{\text{d}}\gamma }}{{{\text{d}}T}}\frac{{{l_0}{G_{\text{T}}}}}{{\mu \alpha }},$$
where l0 is the characteristic length of the molten pool, GT is the temperature gradient, α is the thermal diffusion coefficient. It is shown that the intensity of Marangoni convective essentially depends on the surface tension gradient and the temperature gradient in the molten pool. At a high laser energy density, the powder bed absorbed sufficient heat and the temperature of the pool rose rapidly, resulting in an increase in the temperature gradient GT from the center of the molten pool to its side and the surface tension gradient dγ/dT, thereby aggravating the Marangoni convection. However, excessive laser energy density leads to a deteriorating Marangoni convection, resulting in dramatic fluctuations in the molten pool and a broken melted track, which was detrimental to the SLM processing; whereas, too low laser energy input leads to a rapid decrease in surface tension gradient dγ/dT and temperature gradient GT, corresponding weakening the Marangoni flow, and may also adversely affect the final morphology of the single melted track.

In fact, due to the randomly packed particles, the dynamic thermal evolution of the molten pool was non-uniform, which significantly affected the characteristics the molten pool. The transient temperature and velocity evolution at different moments can be observed at P = 320 W and v = 2400 mm/s during SLM in Fig. 10. According to the different melting/solidification states of the molten pool, the temperature field can be divided into three regions: the red region was distributed at the center of the instantaneous laser beam, the average temperature reached 2560 K, which exceeded the liquidus temperature of the K418 material (1623 K), and appeared a nearly circular shape. The transition region was a solid–liquid mushy zone (1508 K < T < 1638 K). The temperature of the powder bed increased rapidly under an applied Gaussian heat source until it exceeded the melting temperature, the molten pool and the heat affected zone (HAZ) were generated around the loose powders [6]. With the solidification of the tail of the molten pool, the molten pool was well-mixed and the macro-particles were entrapped in the solidifying liquid metal, and the molten pool was elongated with the trailing edge increasing in length compared to the leading edge. This shape was consistent with a moving Gaussian laser beam [44].

Fig. 10

Temperature and velocity fields in the molten pool at P = 320 W and v = 2400 mm/s, a t = 1 µs, b t = 4 µs, c t = 12 µs

As depicted in Fig. 10, at the beginning of the laser scanning the powder bed (t = 1 µs), the peak temperature in the molten pool rose rapidly up to the melting temperature of the K418 material, resulting in a high-temperature gradient and Marangoni convection. In addition, due to the short duration of the laser beam on the k418 powder, less heat was accumulated within the molten pool. The hot melt tended to flow from the center of the molten pool to its side carrying the cold particles. Therefore, the velocity towards the center of the pool, as shown in Fig. 10a. With the laser beam moving (at t = 4 µs), the center temperature of the molten pool further increased. Due to a large amount of heat accumulation at the tail of the trailing melt pool, more liquid phase flew to the forefront of the pool, corresponding the irregular macro-particles at the leading edge of the pool trapped in the solidifying metal. These particles melted rapidly and flew toward the center of the molten pool, resulting in a reverse velocity field (see in Fig. 10b). The flow of the molten pool reached a relatively stable state at about t = 12 µs, at which the peak temperature of the molten pool was maintained approximately 2850 K and the average velocity was approximately 2800 mm/s. The width of the molten pool stabilized at about 100 µm, as depicted in Fig. 10c.

3.2 Variation of thermal behavior with laser processing conditions

Since the powder layer thickness was fixed and no hatch space during single track scanning, the laser power P and scanning speed v were considered to be the dominant process parameters during SLM scanning of an individual melted track. To quantitatively describe the influence of the laser power P and scanning speed v on the dynamic thermal behavior and flow behavior, a comprehensive parameter of the linear energy density (LED) was given to quantify the laser energy input, according to Gu et al. [6]:
where P is the laser power (W), v is the laser scanning speed (mm/s).

Figure 11 shows the influence of the different linear energy densities LED with a fixed power of 320 W on dimensions and the dynamic thermal behavior of the molten pool. It was obvious that with the energy density increased, more heat accumulation generated due to the more heat was absorbed by the powder, creating a larger heat affected zone, thus widening the molten pool (~ 65.8 µm for 89 J/m, and ~ 110.5 µm for 267 J/m). At a low LED of 89 J/m, suggesting a low temperature of powder bed and the powder in the vicinity of the molten pool can not be completely melt due to the insufficient heat, resulting in a lower Marangoni convection and a higher viscosity in the molten pool, leading to a larger corrugation at the free surface of liquid metal. The balling risk of the melt pool increased due to the Plateau–Rayleigh instability, resulting in a discontinuous melted track and poor surface finishes [11, 45] (Fig. 11a). With the LED increased up to 267 J/m, the surface temperature in the local area of the molten pool rapidly approached the evaporation temperature of the K418 material (3200 K). The recoil pressure \({P_{\text{r}}}\) at the free surface of the molten pool increased exponentially, accelerating the liquid metal escaped away from the molten pool at a relatively high velocity (Fig. 11c). The recoil pressure Pr caused by the vaporization, which can be calculated by kinetic theory according to Bäuerle [11]:

Fig. 11

Temperature and velocity evolution in the molten pool at different LEDs (P = 320 W, v = 1200, 2400, 3600 mm/s): a LED = 89 J/m, b LED = 133 J/m, c LED = 267 J/m

$${P_{\text{r}}}=0.56{P_{{\text{Ar}}}} \cdot \exp \left[ {\lambda \left( {\frac{1}{{{T_{\text{b}}}}} - \frac{1}{{{T_{\text{s}}}}}} \right)} \right],$$
where \({P_{{\text{Ar}}}}\) is the argon atmosphere pressure, \(\lambda\) is the evaporation energy of an individual atom, and Tb is evaporation temperature, and Ts is the surface temperature of the pool.

Figure 12 shows the variation of cooling rate and temperature on the free surface of the molten pool under different laser energies. Obviously, the temperature curve showed very sharp peaks at all different energy densities, i.e., the temperature rapidly rose up to a peak and immediately dropt afterward and then essentially stabled near the melting temperature, suggesting that SLM process was an extremely non-equilibrium process with a rather large cooling rate. The positive cooling rate means cooling and solidification process, while the negative cooling rate means heating and melting process. With the LED increases from 89 to 267 J/m, the peak cooling rate decreased from 2.2 × 108 to 4 × 107 K/s, while the maximum temperature increases from ~ 3050 to ~ 3900 K.

Fig. 12

The curve of temperature and cooling rate on the free surface of the molten pool at different LEDs (P = 320 W, v = 1200, 2400, 3600 mm/s): a LED = 89 J/m, b LED = 133 J/m, c LED = 267 J/m

In addition, the duration of the molten pool increased from ~ 200 to ~ 300 µs. This indicated that a low LED with a high scanning speed shortened the lifetime of the molten pool. In addition, the heat conduction in the molten pool was relieved so that the powders can’t be melted completely, even the previous solidified layer can’t be remelted, leading to a discontinuous melting track and a shallow molten pool. However, due to a longer interaction time between the K418 powders and the laser beam at the high LED, the evaporation effect due to the excessive energy input leaded to an irregular and denudated melted track.

Fig. 13

Surface tension on the free surface of the molten pool at different LEDs (P = 320 W, v = 1200, 2400, 3600 mm/s): a LED = 89 J/m, b LED = 133 J/m, c LED = 267 J/m

The dynamic behavior of the liquid within the molten pool was actuated by the low viscosity that corresponded with the high surface tension and of liquid metals, thus affecting the surface quality [26]. Figure 13 shows the effect of LED on the surface tension at the free surface of the molten pool. As the energy density LED increased from 89 to 267 J/m, the temperature gradient increased while the surface tension gradient decreases, which promoted Marangoni convection and accelerated the heat and mass transfer in the molten pool. According to [30], the instability of the molten caused by the surface tension gradient would result in an uneven melted track. As a result, causing a fluctuation (Plateau–Rayleigh instability) of fluid in the molten pool and an increased molten pool in width. At a low LED with higher scanning speed, those effects were more serious and might even lead to balling effect and discontinuity. On the other hand, the surface shear force \(\left( {{F_{{\text{surface}}}}=\frac{{{\text{d}}\sigma }}{{{\text{d}}T}}\frac{{{\text{d}}T}}{{{\text{d}}x}}} \right)\) which related to the surface tension gradient would decrease when the LED decreased from 267 to 89 J/m, as a result, reduced the acceleration of molten fluid [28]. The surface tension was overcome by recoil pressure, resulting in spatter and material dissipation. However, at a higher LED of 267 J/m, the peak temperature in combination with the maximum of the surface tension curve resulted in inward Marangoni convection. This inward flow caused a deeper penetration molten pool. As a result, a smaller surface tension might lead to a better surface quality. Similar results have also been found by Khairallah et al. [24].

3.3 Experiment verification

To validate the accuracy and validity of the numerical model, a single track laser scanning experiment was carried out using a self-developed SLM equipment (Xi’an Jiaotong University, SLM-100, China), as shown in Fig. 14b. During the SLM process, the build chamber was filled with argon to protect the K418 material from oxidation at elevated temperature. The residual oxygen in the chamber was controlled within 500 ppm and the pressure was controlled in the range of 0–1500 Pa. The SLM machine was equipped with a ytterbium fiber laser (IPG, YLR-500-WC, Germany) with a wavelength of 1064 nm, a scanning galvanometer (ScanLAB, hurrySCAN 30, Germany), an f − θ lens (Sill Optics GmbH, f 330, Germany). During the process, the laser beam was always perpendicular to the substrate with a diameter of 70 µm, as shown in Fig. 14a. Before the experiment, the accuracy of the equipment was calibrated to meet the requirements of the experiment.

Fig. 14

a The schematic diagram of SLM process, and b the self-developed SLM-100

During SLM, the K418 powders were located on a 304 stainless steel substrate, which had a good wettability of the applied powder. A laser power ranging from 240 to 360 W was applied corresponding to a scan speed ranging from 800 to 3600 mm/s. After SLM, the top view of the as-SLMed single scan tracks was firstly evaluated by an optical microscope (OM, MA-200, Nikon) without any treatment to observe the melted morphology. Then the cross section of single melted tracks was prepared into standards metallographic specimen through the transparent resin inlaid to measure the depth and height. The specimen was firstly ground with metallographic sandpaper and then polished by polishing fabric with a diamond polishing agent. And then the cross section of the melted tracks was etched in aqua regia with the composition of 30 ml HCl and 10 ml HNO3 for 30 s. The microstructures of the specimen were obtained by a scanning electron microscope (SEM, S3000, Tescan).

The typical morphology of the SLM-processed K418 single melted track under various linear energy densities LED is shown in Fig. 15. Similar to the numerical result, the width of the melting track increased with the LED increasing from 89 to 267 J/m. Discontinuities and irregularities on the top view of the melted tracks were observed at a low LED of 89 J/m. In addition, lack fusion between melted tracks and substrate or adjacent tracks due to the insufficient laser energy input, resulting in a considerable reduction in the melting depth (Fig. 15a). However, the energy density increased up to 267 J/m, excess energy input caused the evaporation effect on the surface of the molten pool, resulting in the void defects of SLM-processed parts, taking into consideration the denudation and melt track formation due to the over-burning phenomenon. On the other hand, the excessive energy caused a dramatic fluctuation of the liquid metal in the molten pool, which tended to splatter under the recoil pressure, resulting in a poor surface finish (Fig. 15c). While at an optimum LED of 89 J/m, the fluid flow within the molten pool became stable, indicating a good wettability between the molten pool and substrate, and sufficient depth of penetration beyond the molten pool and into the substrate, as a result, a good and relatively smooth scan track was obtained (Fig. 15b).

Fig. 15

Typical morphologies of melted tracks at different LEDs (P = 320 W, v = 1200, 2400, 3600 mm/s): a LED = 89 J/m, b LED = 133 J/m, c LED = 267 J/m

Figure 16 shows the pool morphology on the cross section caused by Gaussian laser irradiation in the simulation and experiment at a laser power of 320 W and scanning speed of 2400 mm/s. The depth of the molten pool was measured vertically from the lowest to the highest point of the molten pool, while the half width as the distance from the estimated centerline to the outermost of the molten pool boundary, as shown in Fig. 16a. The numerical depths and widths of the molten pool between experimental results under the different process parameters are shown in Fig. 16b. Obviously, the numerical results were proved to be in good agreement with the experiments. However, complete error avoidance was almost impossible due to the inherent experimental and numerical errors. As shown in Fig. 16b, the maximum error was less than 15%, indicating that the developed 3D finite difference method (FDM) can well predict the three-dimensional morphology of the molten pool.

Fig. 16

Experimental and numerical results of a morphology of the molten pool at P = 320 W and v = 2400 mm/s and, b width and depth of the molten pool at different LEDs (P = 320 W, v = 1200, 2400, 3600 mm/s): a LED = 89 J/m, b LED = 133 J/m, c LED = 267 J/m

To visually depict the influence of the process parameters, i.e., laser power P and scanning speed v, on the morphology of the molten pool, a processing window map is shown in Fig. 17. The straight lines in Fig. 17 represent energy densities of 200 and 100 J/m, respectively. The morphology of the melted track was almost invariable along the line with a constant energy density. According to the three-dimensional morphologies of the melted tracks, the processing parameters map was divided into three areas: Zone 1, overheating; Zone 2, good processing; Zone 3, disconnection and balling. Taking into account all experimental and numerical results, Zone 2 represented the stable processing parameters, corresponding to a laser linear energy density of 100–200 J/m. The characteristics of the molten pool were obviously distinct in three regions, which mainly dominated by the laser linear energy density. Therefore, although the dynamic behavior of the molten pool was quite strong, the linear energy density determined the final morphology of the molten pool [46].

Fig. 17

Processing window for K418 single melted track formation

4 Conclusions

In this work, considering the temperature-dependent thermophysical parameters and the shrinkage effect of the real 3D randomly packed powder bed, a 3D finite difference method (FDM) was developed to investigate the dynamic thermal behavior during SLM process of the K418 material. Numerical results and experimental measurements were compared to verify the feasibility and reliability of the established model. The main findings are summarized as follows:

  1. 1.

    During SLM process of K418, the laser linear energy density LED was introduced to quantitatively describe the influence of process parameters on the dynamic thermal behavior and the morphology of the melted tracks. The results show that the LED plays a crucial role in determining the dynamic thermal behavior and the final forming quality, which can be visually reflected in the metallurgical defects such as porosity, balling, spattering, denudation, etc., and the morphological features of the single melted track.

  2. 2.

    The peak temperature of the molten pool increased with the applied LED. At a low laser density (LED < 89 J/m), the powder cannot be completely melted due to the insufficient heat, resulting in a discontinuous melting track and a shallow molten pool. However, an increase in laser density (LED > 267 J/m) actually resulted in high temperature and Marangoni convection, causing a dramatic fluctuation of the melt in the molten pool, and then worsen the surface finish due to the defects such as sparking, spattering and denudation.

  3. 3.

    The numerical results were in good agreement with the experiments, which proved to be reasonable and effective. And a processing parameters window was defined that shows dependence on the laser energy density: Zone 1, overheating; Zone 2, good processing; Zone 3, disconnection and balling. Finally, the optimum linear energy density LED threshold was determined as 100–200 J/m, in which a good bonding and a relatively smooth melting track were obtained.




The research is financially supported by Science Challenge Project of China, Dongguan University of Technology high-level talents (innovation team) research project (project number: KCYCXPT2016003), National Natural Science Foundation of China under Grant No. 51775420, and Science and Technology Planning Project of Guangdong Province Grant No. 2017B09091101.


  1. 1.
    M.G. Yan, China Aviation Materials Handbook, Volume II: Superalloy. (China Standard Press, Beijing, 2002)Google Scholar
  2. 2.
    Z.X. Shi, J.X. Dong, M.C. Zhang et al., Solidification characteristics and segregation behavior of Ni-based superalloy K418 for auto turbocharger turbine. J. Alloys Compd. 571(18), 168–177 (2013)CrossRefGoogle Scholar
  3. 3.
    H. Attia, S. Tavakoli, R. Vargas et al., Laser-assisted high-speed finish turning of superalloy Inconel 718 under dry conditions. CIRP Ann. Manuf. Technol. 59(1), 83–88 (2010)CrossRefGoogle Scholar
  4. 4.
    J.P. Kruth, L. Froyen, J.V. Vaerenbergh et al., Selective laser melting of iron-based powder. J. Mater. Process. Technol. 149(1), 616–622 (2004)CrossRefGoogle Scholar
  5. 5.
    M. Cloots, P.J. Uggowitzer, K. Wegener, Investigations on the microstructure and crack formation of IN738LC samples processed by selective laser melting using Gaussian and doughnut profiles. Mater. Des. 89(5), 770–784 (2016)CrossRefGoogle Scholar
  6. 6.
    D.D. Gu, W. Meiners, K. Wissenbach et al., Laser additive manufacturing of metallic components: materials, processes and mechanisms. Int. Mater. Rev. 57(3), 133–164 (2012)CrossRefGoogle Scholar
  7. 7.
    F. Calignano, D. Manfredi, E.P. Ambrosio et al., Direct fabrication of joints based on direct metal laser sintering in aluminum and titanium alloys. Proc. CIRP 21, 129–132 (2014)CrossRefGoogle Scholar
  8. 8.
    P. Yuan, D.D. Gu, D.H. Dai, Particulate migration behavior and its mechanism during selective laser melting of TiC reinforced Al matrix nanocomposites. Mater. Des. 82(5), 46–55 (2015)CrossRefGoogle Scholar
  9. 9.
    A. Simchi, Direct laser sintering of metal powders: mechanism, kinetics and microstructural features. Mater. Sci. Eng. A 428(1–2), 148–158 (2006)CrossRefGoogle Scholar
  10. 10.
    Y.J. Shi, H. Shen, Z.Q. Yao et al., An analytical model based on the similarity in temperature distributions in laser forming. Opt. Lasers Eng. 45(1), 83–87 (2007)CrossRefGoogle Scholar
  11. 11.
    Y. Li, D.D. Gu, Parametric analysis of thermal behavior during selective laser melting additive manufacturing of aluminum alloy powder. Mater. Des. 63(2), 856–867 (2014)CrossRefGoogle Scholar
  12. 12.
    C. Körner, E. Attar, P. Heinl, Mesoscopic simulation of selective beam melting processes. J. Mater. Process. Technol. 211(6), 978–987 (2011)CrossRefGoogle Scholar
  13. 13.
    Z.Y. P.Wei, Z. Wei, Chen et al., The AlSi10Mg samples produced by selective laser melting: single track, densification, microstructure and mechanical behavior. Appl. Surf. Sci. 408(30), 38–50 (2017)ADSCrossRefGoogle Scholar
  14. 14.
    C. Meier, R.W. Penny, Y. Zou et al., Thermophysical Phenomena in Metal Additive Manufacturing by Selective Laser Melting: Fundamentals, modeling, Simulation and Experimentation (2017).
  15. 15.
    W.J. Sames, F.A. List, S. Pannala et al., The metallurgy and processing science of metal additive manufacturing. Int. Mater. Rev. 6608, 1–46 (2016)Google Scholar
  16. 16.
    W.E. King, A.T. Anderson, R.M. Ferencz et al., Laser powder bed fusion additive manufacturing of metals; physics, computational, and materials challenges. Appl. Phys. Rev. 2, 041304 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    C. Korner, E. Attar, P. Heinl, Mesoscopic simulation of selective beam melting processes. J. Mater. Process. Technol. 211(6), 978–987 (2011)CrossRefGoogle Scholar
  18. 18.
    R.B. Patil, V. Yadava, Finite element analysis of temperature distribution in single metallic powder layer during metal laser sintering. Int. J. Mach. Tools Manuf. 47(7–8), 1069–1080 (2007)CrossRefGoogle Scholar
  19. 19.
    A. Foroozmehr, M. Badrossamay, E. Foroozmehr et al., Finite element simulation of selective laser melting process considering optical penetration depth of laser in powder bed. Mater. Des. 89(5), 255–263 (2016)CrossRefGoogle Scholar
  20. 20.
    A. Hussein, L. Hao, C. Yan, R. Everson, Finite element simulation of the temperature and stress fields in single layers built without-support in selective laser melting. Mater. Des. 52(24), 638–647 (2013)CrossRefGoogle Scholar
  21. 21.
    B. Schoinochoritis, D. Chantzis, K. Salonitis, Simulation of metallic powder bed additive manufacturing processes with the finite element method: a critical review. Proc. Inst. Mech. Eng Part B J. Eng. Manuf. 231(1), 96–117 (2015)CrossRefGoogle Scholar
  22. 22.
    D.H. Dai, D.D. Gu, Thermal behavior and densification mechanism during selective laser melting of copper matrix composites: Simulation and experiments. Mater. Des. 55(6), 482–491 (2014)CrossRefGoogle Scholar
  23. 23.
    M.J. Xia, D.D. Gu, G.Q. Yu et al., Selective laser melting 3D printing of Ni-based superalloy: understanding thermodynamic mechanisms. Chin. Sci. Bull. 61(13), 1013–1022 (2016)Google Scholar
  24. 24.
    S.A. Khairallah, A.T. Anderson, Mesoscopic simulation model of selective laser melting of stainless steel powder. J. Mater. Process. Technol. 214(11), 2627–2636 (2014)CrossRefGoogle Scholar
  25. 25.
    S.A. Khairallah, A.T. Anderson, A. Rubenchik et al., Laser powder-bed fusion additive manufacturing: physics of complex melt flow and formation mechanisms of pores, spatter, and denudation zones. Acta Mater. 108(15), 36–45 (2016)CrossRefGoogle Scholar
  26. 26.
    C. Körner, A. Bauereiß, E. Attar, Fundamental consolidation mechanisms during selective beam melting of powders. Model Simul. Mater. Sci. Eng. 21(8), 5011 (2014)Google Scholar
  27. 27.
    A. Klassen, T. Scharowsky, C. Körne, Evaporation model for beam based additive manufacturing using free surface lattice Boltzmann methods. J. Phys. D Appl. Phys. 47(27), 275303 (2014)ADSCrossRefGoogle Scholar
  28. 28.
    X. Ding, L. Wang, Heat transfer and fluid flow of molten pool during selective laser melting of AlSi10Mg powder: Simulation and experiment. J. Manuf. Process. 26, 280–289 (2017)CrossRefGoogle Scholar
  29. 29.
    G. Miranda, S. Faria, F. Bartolomeu et al., Predictive models for physical and mechanical properties of 316L stainless steel produced by selective laser melting. Mater. Sci. Eng. A 657, 43–56 (2016)CrossRefGoogle Scholar
  30. 30.
    M.J. Xia, D.D. Gu, D.H. Dai et al., Porosity evolution and its thermodynamic mechanism of randomly packed powder-bed during selective laser melting of Inconel 718 alloy. Int. J. Mach. Tools Manuf. 116, 96–106 (2017)CrossRefGoogle Scholar
  31. 31.
    J.H. Ferziger, M. Peric, Computational methods for fluid dynamics. (World Book Inc, Beijing, 2012)zbMATHGoogle Scholar
  32. 32.
    Q. Zhang, Y. Huo, Z. Rao, Numerical study on solid-liquid phase change in paraffin as phase change material for battery thermal management. Chin. Sci. Bull. 61(5), 1–10 (2016)Google Scholar
  33. 33.
    A. Masmoudi, R. Bolot, C. Coddet, Investigation of the laser–powder–atmosphere interaction zone during the selective laser melting process. J. Mater. Process. Technol. 225, 122–132 (2015)CrossRefGoogle Scholar
  34. 34.
    C.W. Hirt, B.D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39(1), 201–225 (1981)ADSCrossRefzbMATHGoogle Scholar
  35. 35.
    K. Dai, L. Shaw, Finite element analysis of the effect of volume shrinkage during laser densification. Acta Mater. 53(18), 4743–4754 (2005)CrossRefGoogle Scholar
  36. 36.
    J. Yin, H.H. Zhu, L.D. Ke et al., Simulation of temperature distribution in single metallic powder layer for laser micro-sintering. Comput. Mater. Sci. 53(1), 333–339 (2012)CrossRefGoogle Scholar
  37. 37.
    D.H. Dai, D.D. Gu, Influence of thermodynamics within molten pool on migration and distribution state of reinforcement during selective laser melting of AlN/AlSi10Mg composites. Int. J. Mach. Tools Manuf. 100, 14–24 (2016)CrossRefGoogle Scholar
  38. 38.
    A.V. Gusarov, J.P. Kruth, Modelling of radiation transfer in metallic powders at laser treatment. Int. J. Heat Mass Transf. 48(16), 3423–3434 (2005)CrossRefzbMATHGoogle Scholar
  39. 39.
    J.H. Cho, S.J. Na, Theoretical analysis of keyhole dynamics in polarized laser drilling. J. Phys. D Appl. Phys. 40(24), 7638 (2007)ADSCrossRefGoogle Scholar
  40. 40.
    V. Semak, A. Matsunawa, The role of recoil pressure in energy balance during laser materials processing. J. Phys. D Appl. Phys. 30(18), 2541 (1999)ADSCrossRefGoogle Scholar
  41. 41.
    V.R. Voller, C. Prakash, A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems. Int. J. Heat Mass Transf. 30(8), 1709–1719 (1987)CrossRefGoogle Scholar
  42. 42.
    H.C. Min, C.L. Yong, D. Farson, Simulation of weld pool dynamics in the stationary pulsed gas metal arc welding process and final weld shape. Weld. J. 85(12), 271–283 (2006)Google Scholar
  43. 43.
    X.B. Liu, G. Yu, J.G. Guo et al., Research on laser welding of cast Ni-based superalloy K418 turbo disk and alloy steel 42CrMo shaft. J. Alloys Compd. 453(1), 371–378 (2008)ADSCrossRefGoogle Scholar
  44. 44.
    P. Wei, Z. Wei, Z. Chen et al., Thermal behavior in single track during selective laser melting of AlSi10Mg powder. Appl. Phys. A. 123(9), 604 (2017)ADSCrossRefGoogle Scholar
  45. 45.
    W.E. King, H.D. Barth, V.M. Castillo et al., Rubenchik, observation of keyhole-mode laser melting in laser powder-bed fusion additive manufacturing. J. Mater. Process. Technol. 214(12), 2915–2925 (2014)CrossRefGoogle Scholar
  46. 46.
    E. Attar, C. Körner, Lattice Boltzmann model for thermal free surface flows with liquid–solid phase transition. Int. J. Heat Fluid Flow. 32(1), 156–163 (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Zhen Chen
    • 1
  • Yu Xiang
    • 1
  • Zhengying Wei
    • 1
  • Pei Wei
    • 1
  • Bingheng Lu
    • 1
    • 2
  • Lijuan Zhang
    • 3
  • Jun Du
    • 1
  1. 1.State Key Laboratory of Manufacturing System EngineeringXi’an Jiaotong UniversityXi’anChina
  2. 2.School of Mechanical EngineeringDongguan University of TechnologyDongguanChina
  3. 3.Xi’an National Institute of Additive Manufacturing Co., Ltd.Xi’anChina

Personalised recommendations