Multi-physical simulation of selective laser melting

Metal Powder Report  Volume 00, Number 00  May 2016

Multi-physical simulation of selective laser melting K.-H. Leitz, P. Singer, A. Plankensteiner, B. Tabernig, H. Kestler and L.S. Sigl Plansee SE, Metallwerk-Plansee-Straße 71, 6600 Reutte, Austria

Multi-physical simulations are a powerful tool to gain a deeper understanding of selective laser melting as they allow analyzing the influence of process and material parameters on process dynamics and processing result. Challenges in selective laser melting of refractory metals In selective laser melting (SLM) a laser beam is applied to build up a workpiece by locally melting up powder layer by layer. The technology principle and the process during operation are shown in Fig. 1. SLM offers unique design possibilities at small lot sizes and is highly material efficient [1]. Especially, SLM of metals offers great potential for future manufacturing technology as it allows the fabrication of fully functional machine and tool parts. Whereas for materials like steel SLM is already well established, commercial systems are available and the technology is on the transition from pure prototype production to industrial fabrication, SLM of refractory metals still poses a challenge, due to their high melting temperature and high thermal conductivity. Nevertheless, SLM offers great potential for the consolidation of high melting point materials as it allows the fabrication of geometrically complex objects that can hardly be fabricated in a classical way. Fig. 2 shows examples of thin-walled grid structures and back tapered machine parts that have been successfully built up by SLM of molybdenum.

Multi-physical modeling of laser beam-matter interaction In order to overcome the challenges arising during SLM of refractory metals, numerical multi-physical simulations are a powerful tool. They allow to analyze the process dynamics at high spatial and temporal resolution and to study the influence of process parameters. However, modeling laser beam-matter interaction is a challenging task, as it requires a coupling between optics, thermo- and fluid dynamics. Approaches for such a coupled E-mail address: [email protected].

thermo-fluid dynamical modeling of laser material processing have been firstly developed by Mazumder et al. [2] and have been extended to a variety of laser processes [3–6] including SLM [7]. In the following multi-physical modeling is applied in order to analyze the influence of process parameters and material on process dynamics and melt track width in SLM of steel and molybdenum. The present article is based on results presented at the Euro PM2015 conference in Reims [8].

Simulation model Model description The thermo-fluid dynamical multi-phase model for SLM is based on the computational fluid dynamics and the heat transfer module of the finite element (FE) simulation software package Comsol Multiphysics. It includes absorption of laser radiation on the metal surface, conductive and convective heat transfer in the metal and the ambient atmosphere as well as melting, solidification, evaporation and condensation processes. The fluid dynamical description in melt and vapor is incompressible and it is distinguished between four phases: solid, liquid and vapor metal as well as ambient air atmosphere. The major simplifications of the model are that with respect to absorption angle dependency, shadowing effects and multiple reflections are neglected, an idealized spherical, simple cubic packed, monomodal powder is assumed and the discretization was chosen relatively coarse to keep computational times in a manageable scale. The model geometry consists of a powder layer on a massive base plate. The laser radiation is absorbed on the top surface of the metal powder. The material heats up and a melt pool is formed. Depending on the energy input evaporation can occur. The laser moves along the powder bed and when the material solidifies

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Metal Powder Report  Volume 00, Number 00  May 2016

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Selective laser melting: (a) technology principle; (b) process during operation.

again, a molten track is formed. Fig. 3 shows a typical simulation result at a certain time instant of a SLM process. It shows the interfaces between the different phases and the color reflects the temperature. The model utilizes symmetry along the direction of laser movement. The FE-mesh consists of 91,270 tetrahedral elements with a characteristic size of 5 mm in the particle layer and 20 mm in the boundary regions, respectively. Typical computation times for a molten track of s = 360 mm were found to be in the range of 12–48 h on 4 cores of a dual CPU Intel Xeon E5-2630 v2 Dell Precision T5610 workstation.

metal and the absorption coefficient A [] describing the energy input by the laser beam. Furthermore, thermal radiation losses Qrad [W/m2] according to Stefan–Boltzmann law are considered on the metal surface. The melting enthalpy Hm [J/kg] and evaporation

Fundamental equations The thermo-fluid dynamical multi-phase description of the model is based on a coupled system of four differential equations: the heat conduction equation, the continuity equation, the Navier– Stokes equation and the Cahn–Hillard equation. The heat conduction equation ~p rC

@T ~p urT ¼ rðkrTÞ þ G top AQ þ rC Laser G Q rad @t

(1)

with the density r [kg/m3], the thermal conductivity k [W/(m  K)] and the interface function G [1/m] is the base for the calculation of the temperature field T [K]. It includes a volume heat source with the intensity QLaser [W/m2] on the top surface Gtop [1/m] of the

FIGURE 3

Simulation model for the SLM process.

FIGURE 2

Molybdenum demonstrator parts fabricated by SLM: (a) grid structures; (b) machine parts. 2 Please cite this article in press as: K.H. Leitz, et al., Met. Powder Rep. (2016), http://dx.doi.org/10.1016/j.mprp.2016.04.004

MPRP-591; No of Pages 8 Metal Powder Report  Volume 00, Number 00  May 2016

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r

@u þ rðurÞu ¼ rp þ mr2 u þ rg þ F s @t

(2) 2

with the velocity u [m/s], the pressure p [N/m ], the viscosity m [Pa  s], the gravity constant g [m/s2] and the surface tension force Fs [N/m3]. Based on the temperature in the metal phase it is distinguished between solid, liquid and vapor phase. In the solid phase the viscosity m[Pa  s] is strongly increased and the surface tension force Fs [N/m3] is restricted to the surface of the liquid metal. The continuity equation ru ¼ Q v

(3)

assures the conservation of mass and includes a source term ! 1 1 _   (4) Q v ¼ mG ½1=s rvap rmet with the evaporation rate m_ ½kg=ðm2 sÞ, the vapour density rvap ½Kg=m3  and the metal density rmet [kg/m3] accounting for the density change during evaporation. The vapor density rvap ½Kg=m3  is calculated on the base of the ideal gas law. The multi-phase description of the model follows the phase field approach and is based on the Cahn–Hillard equation for the phase function F[] @F gl þ urF ¼ r 2 c þ Q 0v @t e

(5)

c ¼ re2 rF þ ðF2 1ÞF

(6)

with the mobility g [(m  s)/kg], the mixing energy density l½N, the surface tension coefficient s[N/m] and the capillary width pffiffiffi e ¼ 2 2l=ð3sÞ ½m. The source term ! 1f F _  þ (7) Q 0v ¼ mG ½1=s rvap rmet accounts for the evaporation process.

The described implementation of evaporation by source terms in continuity and Cahn–Hillard equation follows an approach described by Sun and Beckermann in [9]. The evaporation rate m_ ½kg=ðm2 sÞ is calculated on the base of the Hertz–Knudsen formula [10]: rffiffiffiffiffiffiffiffiffiffiffiffiffi M m_ ¼ ’vap ðpvap p0 Þ (8) 2pRT    HvM 1 1   pvap ¼ p0 exp R Tv T

(9)

with the vapor phase fraction ’vap ½, the vapour pressure pvap ½N=m2 , the molar mass M [kg/mol] and the ideal gas constant R [J/(mol  K)].

Results Simulation results The multi-physical simulation model described in the previous paragraph was used to analyze the influence of laser power P [W] and energy input rE [J/m] on process dynamics and melt track width in SLM of stainless steel (1.4404) and pure molybdenum. In the present study a simple cubic monomodal layer of spheroidal powder (powder size d = 40 mm) irradiated with a Gaussian laser beam (focus size dfocus = 130 mm) was regarded. The scan length was s = 360 mm. Temperature dependent material properties were applied and have been taken from Refs. [11–18], see Table 1. Figs. 4 and 5 show the simulation results at different process stages for the two investigated materials at standard parameters for volume buildup (standard power P0 [W], standard energy input rE0 [J/m]). A comparison of the results makes obvious that SLM of steel and molybdenum operate in different process regimes. Whereas for steel a long melt pool is observed and evaporation plays a significant role, for molybdenum the melt pool size is restricted to the focal spot area and temperatures are far below evaporation temperature. This has its origin in the different phase transition temperatures and thermal conductivities of the two materials (compare Table 1). In SLM of molybdenum due to the high thermal conductivity heat losses are significantly higher, restricting the size of the melt pool. Besides that, the high heat losses in combination with the high evaporation temperature prevent the occurrence of evaporation.

TABLE 1

Temperature dependent material data (blue: low temperature regime, red: high temperature regime) [11–18].

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enthalpy H v ½J=kg are considered by an adopted heat ~p ¼ Cp þ dðTT m ÞH m þ dðTT v ÞH v ½J=ðkgKÞ. The funccapacity C tion d [1/K] is a Dirac pulse accounting for the latent heats at the melting temperature Tm [K] and the evaporation temperature T v ½K. The fluid mechanical description of the model is based on the Navier–Stokes equation

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Metal Powder Report  Volume 00, Number 00  May 2016

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SLM of steel at standard parameters for volume buildup (standard power P0, process stages.

steel

[W], standard energy input rE0, steel [J/m]): simulation results at different

FIGURE 5

SLM of molybdenum at standard parameters for volume buildup (standard power P0, different process stages.

In order to analyze the influence of process parameters for both materials power P [W] and energy input rE [J/m] were varied with respect to standard parameters for volume buildup. The simulation results for steel and molybdenum are shown in Figs. 6 and 7. A look at the mere process appearance at different parameter sets makes obvious that for steel there is a dependence on both power P [W] and energy input rE [J/m], with the energy input dependency being more pronounced. For molybdenum a completely different picture can be observed. There is only a minor dependency on the energy input rE [J/m], however a strong power dependency. This behavior becomes even more obvious when comparing the calculated widths of the molten tracks for the both materials. These are plotted vs. power P [W] respectively energy input rE [J/m] in Figs. 8 and 9. Whereas for steel the calculated widths of the molten tracks show a strong dependence on the energy input and only a weak power dependency, for molybdenum the opposite situation is found and the power P [W] is the dominant process parameter. This behavior can be qualitatively understood when considering the Taylor evolution of the temperature distribution of a moving point heat source of power P [W] and

Mo

[W], standard energy input rE0,Mo [J/m]): simulation results at

velocity vx ½m=s on a semi-infinite substrate [19]: v  2P 2P 2Pv x x  exp  TðyÞ ¼ 4pyrCp k 4pyrCp k 8pyrCp k2 2k ¼

2P 2P2  4pyrCp k 8pyrCp k2 rE

(10)

Whereas for steel with a thermal conductivity k [W/(m  K)] approximately one order of magnitude smaller than that of molybdenum, the second term of Eq. (10) is dominant and the width of the temperature field respectively the molten track depends both on power P [W] and energy input rE [J/m], for molybdenum with a high thermal conductivity, the pure power dependency of the first term is more dominant.

Experimental verification In order to demonstrate that the presented multi-physical simulation is able to describe material specific process characteristics of SLM, the calculated widths of the molten tracks were compared to experimental data. In Figs. 6 and 7 for each parameter set a scanning electron microscope (SEM) image of a molten track

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FIGURE 6

Influence of laser power P [W] and energy input rE [J/m] on SLM of steel (P0,steel [W] and rE0,steel [J/m] are standard power and energy input for volume buildup from steel): results of simulation and verification experiments (magnification scale bar for electron micrographs 100 mm).

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Metal Powder Report  Volume 00, Number 00  May 2016

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Influence of laser power P [W] and energy input rE [J/m] on SLM of molybdenum (P0,Mo [W] and rE0,Mo [J/m] are standard power and energy input for volume buildup from molybdenum): results of simulation and verification experiments (magnification scale bar for electron micrographs 100 mm).

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Metal Powder Report  Volume 00, Number 00  May 2016

FIGURE 8

Width of the molten track in SLM of steel – simulation vs. experiment (P0,steel [W] and rE0,steel [J/m] are standard power and energy input for volume buildup from steel): (a) energy input dependency; (b) power dependency.

FIGURE 9

Width of the molten track in SLM of molybdenum – simulation vs. experiment (P0,Mo [W] and rE0,Mo [J/m] are standard power and energy input for volume buildup from molybdenum): (a) energy input dependency; (b) power dependency.

produced from a single powder layer on a solid base plate is shown. The widths of the molten tracks were determined from the SEM images of n = 3 identical experiments. For steel an excellent agreement between experimental data and simulation results can be observed. For molybdenum, the calculated widths of the molten tracks are generally smaller than observed experimentally. It is assumed that this has its origin in the idealized simple cubic particle arrangement and discretization errors in the simulation model, both enhancing heat conduction losses. Nevertheless, the order of magnitude and the trend for power and energy input dependency show good agreement.

Whereas for steel a long melt pool is predicted, a significant amount of evaporation occurs and the energy input is the dominant process parameter, for molybdenum, due to the high thermal conductivity, the melt pool size is predicted to be much smaller, no evaporation occurs and the power dependency is severely pronounced. The results demonstrate that SLM is a promising approach for processing of molybdenum. The small melt pool offers the potential for productivity enhancement by an increase of laser power and energy input. Due to the generally smaller melt pool it can be expected that melt pool distortion effects restricting the feed rate in SLM of steel [20] are less dominant for molybdenum.

Conclusions Multi-physical simulations are found to be a powerful tool to gain a deeper process understanding of laser based manufacturing processes like SLM. The presented thermo-fluid dynamical multi-phase simulation model is able to describe process dynamics and material specific differences in SLM of steel and molybdenum.

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