Analytical Modeling of Three-Dimensional Temperature Distribution of Selective Laser Melting of Ti-6 Al-4 V

Selective laser melting (SLM) is one of the widely used techniques in metallic additive manufacturing, in which high-density laser powder is utilized to selectively melting layers of powders to create geometrically complex parts. Temperature distribution and molten pool geometry directly determine the balling effect, and concentrated balling phenomenon significantly deteriorates surface integrity and mechanical properties of the part. Finite element models have been developed to predict temperature distribution and molten pool geometry, but they were computationally expensive. In this paper, the three-dimensional temperature distributions are predicted by analytical models using point moving heat source and semi-ellipsoidal moving source respectively. The molten pool dimensions under various process conditions are obtained from the three-dimensional temperature predictions and experimentally validated. Ti-6Al-4V alloy is chosen for the investigation. Good agreements between the predictions and the measurements are observed. The presented models are also suitable for other metallic materials in the SLM process.


Introduction
Additive manufacturing (AM), alternatively named 3D printing, has been extensively studied in the past two decades.AM process is a cost-effective process for a single part or small batches, and it is capable of producing geometrical complex parts [1].
Selective laser melting (SLM) is widely used in metallic additive manufacturing, in which a highdensity laser powder is utilized to fully melting layers of powders to create geometrically complex parts.However, consistency of the part quality for batch production is difficult to control due to intense heat input and repeatedly rapid melting and solidification.
Problems such as balling effect, deteriorated surface finish, undesired residual stress and part distortion have been frequently observed in the SLM process.It has been reported that the process parameters such as laser power and scanning speed have considerable influence on the molten pool geometry, and the molten pool geometry directly determines the balling effect and thus affects surface integrity [2].A large molten pool length to width ratio (L/W > π) indicates that balling effect is at the concentrated condition.
The balls form in various shapes due to the inadequate energy input or the molten pool splashing under high scan speed [3].Balling effect at concentrated conditions considerably deteriorates the surface roughness.A post-processing procedure such as polishing becomes necessary and thus decreases the dimensional accuracy.In addition, a large number of pores tends to be formed due to the discontinuity of the formed balls, resulting in poor mechanical properties [4].

Materials and Method
In this paper, the three-dimensional temperature distributions in SLM of Ti-6Al-4V under varying process conditions are investigated by analytical models.The temperature profile near the moving heat source in a single-track scanning was predicted by two analytical models separately.One analytical model uses a point moving heat source assumption; the other analytical model uses a three-dimensional semi-ellipsoidal moving heat source assumption.In addition, the medium is assumed to be isotropic, homogeneous, and semi-infinite in the analytical models.The materials properties are temperature-dependent and adopted as they are at room temperature as given in Table 1.
Table 1.Physical and thermal properties of Ti-6Al-4V [14,15] Density (Kg/m3) 4428 Specific heat (J/Kg-K) 580 Thermal conductivity (W/m-K)  Moreover, varying laser power magnitudes are used in the predictions to investigate the influence of laser power on the molten pool size.The laser is continuous at the wavelength ( = 1.06 μ) that is typical in the SLM process.The energy absorption coefficient of Ti-6Al-4V powder is assumed to be the same as pure titanium powder, given as 0.77 [16].The process parameters in SLM are given in Table 2.The general convection-diffusion equation can be expressed as where u is internal energy, h is enthalpy, ρ is density, k is conductivity, and ̇ is a volumetric heat source, t is time, V is heat source moving speed, and T is temperature.
When V = 0, the Equation (1) becomes the heat conduction equation.The heat conduction can be expressed as the following with  =  , where c is heat capacity.
The steady state equation with constant velocity can be simplified using the continuity equation.The continuity equation is expressed as The heat conduction equation then becomes The heat conduction equation can be solved with assumptions of a point moving heat source and a semi-ellipsoidal moving heat source for isotopic and semi-infinite medium [18].

Point Moving Heat Source
A point moving heat source solution for the temperature field in a three-dimensional semi-infinite body is presented by Carslaw and Jaeger [19].The solution is expressed as where  is the dimensionless temperature,  is thermal diffusivity, R is the distance from the heat source location.They are defined as the following: Numerical analyses based on finite element analysis (FEA) and analytical models have been developed to investigate temperature distribution in the SLM process.Different processing techniques have been applied in FEA simulation to predicted temperature distribution.Peyre P. et al. developed a FEA model using steady-state calculations to predict temperature distribution and molten pool geometry in printing a thin wall structure using a laser-based direct metal deposition process [5].Paul S. et al. developed another FEA model to predict the temperature distribution and molten pool geometry using element birth technique with uniform moving heat source and Gaussian powder distribution.Other multiple-physics numerical models have also been developed to predict the temperature distribution and molten pool geometry [6-10].Although the developed FEA models have made considerable progress in predicting temperature distribution and molten pool geometry in AM processes, the expensively computational cost and time are still the main drawbacks.Analytical models have also been developed to predict temperature distribution using closed-form solutions.Li JF et al. developed an analytical model with the Green function method to predict the temperature distribution with a volumetric heat source and surface heat source respectively [11].The hydrodynamics of the molten pool is neglected by implementing proper volume heat flux.Batut B. et al. developed another analytical model to predict twodimensional temperature distribution using modified point moving heat source solution with arbitrarily shaped laser source [12].Fergani O. et al. also presented an analytical model to predict twodimensional temperature distribution using point moving heat source [13].Isotropic and homogeneous material and semi-infinite medium were assumed in the analytical model.Li's model has demonstrated the capability of three-dimensional temperature prediction in AM process, but the integrated Green function dramatically increased the mathematical complexity and resulting in a decreased computational efficiency.Batut's models and Fergani's model only predicted two-dimensional temperature distributions in AM process.In this work, three-dimensional temperature distributions under varying process conditions in SLM are predicted by two analytical models.A point moving heat source and a semi-ellipsoidal heat source are used in the two analytical models respectively.Molten pool dimensions are obtained from the predicted three-dimensional temperature distribution under each process condition.The predicted molten pool dimensions including melting length, melting depth, and melting width are validated with experimental measurements adopted from literature.

Figure 1 .
Figure 1.Schematic drawing of a molten pool in SLM process.Laser travels along x-direction [11].
+  +  Semi-ellipsoidal Moving Heat Source A moving heat source solution in a threedimensional semi-infinite body with a threedimensional semi-ellipsoidal heat source was developed by Nguyen [20].This solution is expressed as 2√6;  =  2√6;  =  2√6;  =  4  ( − ) where  ,  ,  are heat source parameters. and  are assumed to be the same as laser spot radius. is obtained based on the given heat flux at given source location using the following equation, given as 7temperature profiles in SLM of Ti-6Al-4V were predicted by analytical models using point moving heat source and semiellipsoidal moving heat source respectively.The temperature profiles on the top surface and cross sections at heat source location were plotted to illustrate the molten pool geometry and heat affected zone due to a single-track scanning.The molten pool geometry was determined as space where predicted temperatures were higher than melting temperature.Molten pool length, width, and depth were obtained from the three-dimensional temperature profile.The molten pool width and depth were validated by experimental measurements.The experimental measurements were adopted from literature, in which the molten pool size was measured by optical microscopy based on the solidified structure [17].The three-dimensional temperature profiles predicted with the analytical model using point moving heat source are shown in Figure 1.Isotropic and homogeneous material and semi-infinite body are assumed in the prediction.The laser power and scanning speed are 20 W and 200 mm/s respectively.Laser travels along the x-direction.Isothermal lines are plotted at temperature levels of 30, 100, 500,1000, and 1655 in Celsius.

Figure 2 .
Figure 2. Temperature profile using point moving heat source with P = 20 w, V = 200mm/s.The temperature profile in top view (a), front view of the cross-section (b), right view of the cross-section.Note cross sections are at heat source location.Heat source moves along the xdirection.

Figure 3 .
Figure 3. Temperature profile using semi-ellipsoidal moving heat source with P = 20 w, V = 200mm/s.The temperature profile in top view (a), front view of the cross-section (b), right view of the cross-section.Note

Figure 4 .
Figure 4. Predicted molten pool dimensions and experimental measurements under varying process conditions.(a) molten pool depth, (b) molten pool width, (c) molten pool length, (d) molten pool volume.