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Finite element analysis of temperature distribution in single metallic powder layer during metal laser sintering
ARTICLE IN PRESS
International Journal of Machine Tools & Manufacture 47 (2007) 1069–1080 www.elsevier.com/locate/ijmactool
Finite element analysis of temperature distribution in single metallic powder layer during metal laser sintering Rahul B. Patil, Vinod Yadava Mechanical Engineering Department, Motilal Nehru National Institute of Technology, Allahabad, UP—211 004, India Received 23 November 2005; received in revised form 15 May 2006; accepted 19 September 2006 Available online 9 November 2006
Abstract The metal laser sintering (MLS) is used to make strong or hard metallic models for tools and dies directly from metallic powders. Thermal distortion is the serious problem after cooling of the solidified part rapidly. Uncontrolled temperature distribution in the metallic powder layer leads to thermal distortion of the solidified part. The study of temperature distribution within the metallic layer during MLS is important from the quality of the layer point of view. The high temperature generated in the powder layer leads to thermal distortion of the part and causes thermal as well as residual stresses in the part. In this paper the powder layer is assumed to be subjected to plane stress type of temperature variation and a transient finite element method-based thermal model has been developed to calculate the temperature distribution within a single metallic layer during MLS. A finite element code has been developed and validated with the known results from the literature. The obtained results of temperature distribution show the temperature and temperature gradient variation along X- and Y-axis. The effect of process parameters such as laser power, beam diameter, laser on-time, laser off-time and hatch spacing on temperature distribution within a model made of titanium during MLS is studied. The results computed by the present model agree with experimental results. Temperature increases with increase in laser power and laser on-time but temperature decreases with increase in laser off-time and hatch spacing. r 2006 Elsevier Ltd. All rights reserved. Keywords: Rapid prototyping; Metal laser sintering; FEM; Temperature distribution
1. Introduction Rapid prototype manufacturing is an emerging technology that provides an integrated way of manufacturing three-dimensional components from computer aided design (CAD) files to finished parts [1], using only additive processing. Additive processing implies that the structures are made by cumulative deposition of material, without using any hard tooling. In metal laser sintering (MLS) process parts are usually built by sintering a thin layer of metal powder using either CO2 or Nd: YAG laser beam. The interaction of the laser beam with the powder raises the temperature to the point of melting, resulting in particle bonding fusing the particles to themselves and the previous Corresponding author. Tel.: +91 0532 2271812; fax: +91 0532 2445101, +91 0532 2445077. E-mail address: [email protected] (V. Yadava).
0890-6955/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2006.09.025
layer to form the solid. The power of the laser beam is adjusted to bring the selected powder areas to a temperature just sufficient for the particles in the powder to get attached (i.e. get sintered). Sintering may occur through partial melting or softening of powder particles themselves, or of coating on the particles. An initial cross-section of the object under fabrication is traced on the layer of powder by means of XY controlled pulsed laser beam. Sintering occurs when the grain viscosity drops with temperature creating an interfacial kitting of the grains without full melting. The next layer is then built up directly on top of the sintered layer after an additional layer of powder is deposited via a roller mechanism on top of the previously formed layer. In the unsintered areas, powder remains loose and serves as a natural support for the next layer of powder and the object under fabrication [2]. Fischer et al. [3] have measured temperature on the top surface of titanium powder layer during MLS. Nd:YAG
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Nomenclature
r
[A] {B} [B] [C] Cp(T) Cs Cl {f} [GC] [GK] {GF} I(r) [K] ks(T)
s T Tm Ti {T} fT g t x, y
keff kg Lm {N} P Q q0 q(r) R
matrix of A vector of B matrix of derivative of shape functions capacitance matrix temperature-dependent specific heat (J/kg K) specific heat of solid (J/kg K) specific heat of powder in liquid state (J/kg K) right-hand side force vector global capacitance matrix global stiffness matrix global right side force vector laser intensity stiffness matrix temperature dependent thermal conductivity of solid (W/m K) effective thermal conductivity of powder layer (W/m K) thermal conductivity of the surrounding gaseous environment (W/m K) latent heat of melting (kJ/kg) column vector of 2-D bilinear shape functions laser power (W) internal heat generated (W/m3) heat flux at the center of the laser beam (W/m2) input heat flux (W/m2) radius of laser beam (m)
laser is focused at a specified location on the powder layer. Temperature at the top surface of the powder layer has been measured with the help of an infrared camera with two different laser parameter sets, pulsed and continuous laser modes. The measured temperature field in each case has been compared with each other. They observed that the peak skin temperature in pulsating mode is much higher than that in continuous mode. Deckard and Williams [4] have carried out experiment on bisphenol—a polycarbonate. The effect of machine parameters namely laser power, beam velocity, hatch spacing, beam spot size and scan line length on part density and strength of the part have been studied. Their findings provide understanding about interaction of laser irradiation with the powder layer. Kolossov et al. [5] have developed a finite element method (FEM)-based numerical model for determination of temperature distribution on the top surface of titanium powder layer during MLS. The FEM model was developed considering sintering potential and taking specific heat as a function of temperature. Temperature distribution on the top surface of the powder layer has been determined when laser beam is arrived at a particular location while moving. They also carried out physical experiment and measured temperature on the top surface of the powder layer with an infrared camera at the same location. Shiomi and Osakada
distance of a point from the center of the laser beam (m) thickness of the powder layer (m) temperature (K) melting temperature of titanium room temperature (K) nodal temperature vector time derivative of temperature time (s) cartesian coordinate x and y
Greek letters r rbed f a
Density (kg/m3) Density of the powder bed (kg/m3) Empirical coefficient Constant
Subscripts l s i
Liquid Solid Initial at time ¼ 0
Superscripts e
Element
[6] have developed a FEM-based heat conduction model to find weight of the solidified part made of metallic powder. Their experimental and FEM-based results show that weight of solidified part is affected by peak power rather than duration of laser irradiation. Shiomi and Osakada [7] have developed a thermal model to calculate temperature and stress distribution for various track lengths. They have shown that the cracking of layer can be avoided by using smaller track length. In the present paper a FEM-based thermal model is developed to determine the non-uniform transient temperature distribution in the single powder layer of titanium due to laser irradiation for different laser power, beam diameter, laser on-time, laser off-time and hatch spacing. 2. Mathematical modeling Localized heating of a small volume of powder during MLS is shown schematically in Fig. 1 [4]. The duration of laser beam at any powder particle is short, typically between 0.5 and 25 ms. Therefore, the thermally induced binding reactions must be kinetically rapid. In principle, both single- and two-component powders can be subjected to MLS process [8]. In case of single-component powders the liquid phase arises due to the surface melting of particles and the powder is sintered by joining the solid
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Laser moving direction Heat lost by radiation
Laser beam
Heat lost by convenction
Unsintered Region
Sintered Region
Heat conducted to the powder bed
Previous Layer (a)
(b) Fig. 1. (a) Interaction of the laser irradiation and metal powder [4]. (b) Schemes of inter-particle contacts formation during metal laser sintering of single component powders [8].
non-melted cores of particles. Fig. 1(b) shows the possible ways of contact formation during MLS of single-component powders.
gaseous environment (i.e. air) and initial relative density of the powder bed. 2.2. Governing equation
2.1. Assumptions Owing to random and complex nature of MLS process, following assumptions are made to make the problem mathematically tractable: 1. The input heat flux due to laser irradiation is treated as an internal heat generation in the powder layer. The heat flux from laser beam is taken as Gaussian distributed heat flux and given directly to the top of the powder layer. 2. The powder layer is assumed to be subjected to plane stress type of temperature variation because of the thickness of the powder layer is very small. 3. The heat flux due to laser irradiation is given to the fully covered finite element under the laser beam at a time. 4. Laser beam spot is assumed to be circular in shape. 5. To simplify the calculation the whole powder layer is considered to be homogeneous and continuous. 6. Only the part of powder material having temperature equal to or greater than melting temperature is assumed to be sintered. 7. All the solid particles in the powder are assumed to be spherical in shape and equal in size. 8. The effective thermal conductivity of the powder layer is used in the calculation, which incorporates solid thermal conductivity, thermal conductivity of the surrounding
In MLS, the powder layer is spread on the table surface with a pre-defined very small layer thickness. Powder layer is considered as plane stress type of temperature variation governed by following thermal heat diffusion equation with internal heat generation in a Cartesian domain [9]: rbed C p ðTÞ
qT ¼ rðkeff rTÞ þ Qg qt
within the domain ABCD;
(1) where rbed is the density of the powder bed, Cp(T) is specific heat, T is temperature, t is time, keff effective thermal conductivity of the powder bed, Q internal heat generated due to laser irradiation and X and Y are coordinates axes as shown in Fig. 2. 2.3. Initial and boundary conditions The initial temperature Ti is taken as equal to room temperature within domain ABCD. Thus, T ¼ Ti
in ABCD at
t ¼ 0.
(2)
The temperature gradient is taken as zero across the boundaries of the domain (ABCD) because these boundaries AB, BC, CD and AD are at such a large distance from the laser scanning area (area EFGH in Fig. 2), so that there is no heat transfer across it. Hence, the boundary
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Typical area element
e Y
Typical boundary element Domain
B
C
b
Scanning area of laser heating F
G
3 1
4 2
Common node
H
E
2.5. Calculation of bed density (rbed) Elements affected by Gaussian heat flux X
A
where Q is internal heat generated and s is the thickness of the powder layer. Gaussian distributed heat flux, which is distributed over four area elements at a time, is also shown in Fig. 2. The Gaussian distributed peak of the heat flux is considered at the junction node, which is common to four area elements. The dark dot at the center of the laser beam is the common node to the area elements 1, 2, 3 and 4. To satisfy the necessary condition to have at least four area elements under the laser beam. The elemental length is selected in such a way that the laser beam diameter becomes a whole number multiple of the elemental length because of computational simplicity.
D
Fig. 2. Finite element mesh of thermal model of metal laser sintering.
conditions can be written as qT keff ¼ 0, qn
(3)
on boundaries AB, BC, CD and AD. Here n is the outward normal. 2.4. Calculation of internal heat generation due to laser irradiation (Q) Most researchers [6,7] working in the area of MLS have considered uniformly distributed heat source of laser intensity. This assumption is far from reality as it is known from the literature that laser beam intensity profile I(r) is Gaussian [10]. In the present model, Gaussian heat flux distribution is taken same as laser intensity. If the laser input power P and laser beam radius R are known, then the laser intensity I(r) at a distance r from the center of the laser beam is given by [11] 4:55P 4:5ðr=RÞ2 e . (4) pR2 This is the input heat flux from the laser beam given to the powder layer. Hence, it can be written as [11]
The relative bed density is calculated [12] using Eq. (7) with the assumption that all the solid particles in the powder layer are spherical in shape and equal in size and density arranged in a cubic array: pr rbed ¼ rR rs ¼ s , (7) 6 where rR is initial relative bed density, rbed is the bed density and rs is density of the solid. 2.6. Calculation of the effective thermal conductivity (keff) of powder layer Heat transfer through powders mainly occurs through conduction and radiation because inter-particle distances are too small to permit convection heat transfer. The powdered bed mainly consists of solid particles separated by a gas. Since gases have much smaller thermal conductivities (0.0065 W/m K at room temperature to 0.049 W/m K at 325 1C for air) than those of good metallic conductors (400 W/m K). The thermal conductivity of a powder bed is mainly dictated by the thermal conductivity of the gas embedded within the voids [10]. The effective thermal conductivity of the powder bed is defined as follows [13]:
2
IðrÞ ¼ I 0 e4:5ðr=RÞ ¼
4:55P 4:5ðr=RÞ2 e . (5) pR2 Here, q(r) is the input heat flux at a distance r from laser beam center, q0 is the laser heat flux at the center of the laser beam and R is the radius of the laser beam. This incoming heat flux is treated as internal heat generation in the powder layer. Hence, quantity of heat generation (W/m3) can be written as [11] qðrÞ , (6) Q¼ s 2
qðrÞ ¼ q0 e4:5ðr=RÞ ¼
keff ¼
rR ks ðTÞ , 1 þ f ks ðTÞ kg
(8)
where rR is the initial relative density of the powder layer, ks is temperature-dependent solid thermal conductivity, kg is thermal conductivity of the surrounding gaseous environment, i.e. air and f is empirical coefficient, f ¼ 0:02 102ð0:7rR Þ . 2.7. Temperature-dependent material properties In general thermal conductivity and specific heat both change with respect to temperature. Therefore, in the present model, thermal conductivity and specific heat both are taken as temperature dependent. The graphs of
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temperature-dependent thermal properties are taken from Kolossov et al. [5]. 2.8. Phase change consideration When the temperature of the material exceeds the melting point, latent heat due to melting is needed. The latent heat required for phase change (for solid to liquid) is accounted by modifying the expression for the specific heat. Considering enthalpy before and after phase change, the latent heat of melting is incorporated to get modified specific heat [14]: Cl ¼ Cs þ
Lm 2DT
for T m DTpTpT m þ DT,
(9)
where Tm is the melting temperature of titanium (1933 K) and Lm is the latent heat of melting. 3. Finite element formulations The Galerkin’s finite element formulation [15] has been applied to obtain temperature distribution within the square domain due to laser irradiation as shown in Fig. 2. The following expressions are obtained for elemental capacitance matrix [C]e, conductivity matrix [K]e and boundary flux vector {f}e, when Galerkin’s method is applied to Eqs. (1)–(3) 9 R ½K e ¼ De keff ½BeT ½Be dx dy; > > = R e e eT ½C ¼ De rbed C p fN g fN g dx dy; . (10) > e R e > e eT ; f ¼ De fN g fN g qg dx dy: Here, {N}e is the nodal shape function vector, [B]e is the matrix relating temperature derivative with its nodal values and {qg}e is vector of nodal heat generation. The integrals in the expression (10) are computed numerically using Gaussian quadrature with three points in each direction. When the elemental quantities of Eq. (10) are assembled, the following differential equations are obtained: no ½GCnnmnnm T þ ½GKnnmnnm fT gnnm1 ¼ fGFgnnm1 ,
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4. Results and discussion Formulation of Section 3 is used for the development of a FEM-based code in MATLAB 7.0. The connectivity matrix and nodal coordinates of the elements are generated using ANSYS 9.0. To check the accuracy of present FEMbased model for the determination of temperature distribution within the single layer of powder during MLS, the temperature distribution on the top surface of powder layer is determined, and then compared with experimental results of Fischer et al. [3] as well as with numerical results of Kolossov et al. [5]. Here the present model is used with same conditions as taken in Refs. [3,5]. 4.1. Comparison 4.1.1. Comparison with experimental results Fischer et al. [3] have measured temperature on the top surface of titanium powder layer during MLS. Gaussian heat flux, effective thermal conductivity is used for the calculation of the temperature distribution. The laser related parameters and powder material properties are given in Tables 1 and 2, respectively. Fig. 3 shows the top surface temperature variation along X-axis at Y ¼ 0.55 mm as well as along Y-axis at X ¼ 0.35 mm considering different state conditions of density and thermal conductivity of titanium powder material. Figs. 3(a) and (b) show a comparison between the experimental and computed results along X- and Y-axis, respectively. The highest top surface temperature measured during experiment was 1500 K where as top surface temperature in similar conditions using present FEM model is determined as 2134.8 K. The computed top surface temperature using present code is determined as 2978.8 K if keff ¼ 0.2154 W/m K and rbed ¼ 2931.5 kg/m3. The discrepancy between the computed and experimental results could be assigned to the difference in the effective thermal conductivity of powder used in the present model in place of thermal conductivity of solid used by Fischer et al. [3] (0.2154 W/m K in place of Table 1 Laser-related parameters [3]
nnm1
(11) where [GC] is the global capacitance matrix, [GK] is the global stiffness matrix, {GF} is the global right side force vector, {T} is global temperature vector and fT g is time derivative of {T}. Eq. (11) represents a set of ordinary differential equations in the variable {T} as a function of time t. These equations are converted to set of algebraic equations by the application of implicit finite difference method [15]. Here, the solution marches in time, in steps of Dt until the desired final time is reached. In the present model, Dt is divided into two time steps Dt1 and Dt2. Here, Dt1 and Dt2 pulse on time and off-time of the laser beam, respectively.
Variable
Notion
Value
Laser power (W) Spot radius (m) Laser spot speed (mm/s)
P R V
2 50 106 1
Table 2 Titanium material properties [3] Variable
Notion
Value
Thermal conductivity (W/m K) Density (kg/m3) Specific heat (J/kg K)
k r C
1.45 2931.5 335
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Present FEM based using ρs and ks Present FEM based using ρs and keff Experimental peak temperature Present FEM based peak temperature
peaks are observed in the area, which comes under the laser beam. Fig. 4(b) shows the contour plot of isotherms at the top surface of the area in the powder layer from X ¼ 0.25 to 0.45 mm and from Y ¼ 0.45 to 0.675 mm. The thick line shows the isotherm of the melting temperature of titanium powder. The area inside the thick isotherm line is having temperature greater than the melting temperature of titanium. It is assumed in the present model that the powder particles having temperature greater than the melting temperature are sintered together by liquid-phase sintering. Therefore the powder particles inside the area of thick isotherm undergo liquid-phase sintering at the surface and remaining particles are remained unsintered.
3000
Temperature (K)
2500
2000
1500
1000 0.15
0.20
0.25
(a)
0.30 0.35 0.40 X at Y = 0.55 (mm)
0.45
0.50
0.55
Present FEM based using ρs and ks Present FEM based using ρs and keff Experimental peak temperature Present FEM based peak temperature 3000
Temperature (K)
2500
2000
1500
1000 0.40 (b)
0.45
0.50
0.55
0.60
0.65
0.70
0.75
Y at X = 0.35 (mm)
Fig. 3. Variation of top surface temperature (a) along X-axis at Y ¼ 0.55 mm (b) along Y-axis at X ¼ 0.35 mm at time t ¼ 1 s.
1.45 W/m K), unknown number of pulses for which experimental results are reported. Another cause may be the measurement of temperature using camera, which is not able to capture the temporarily higher skin temperature. Here, theoretically calculated temperature value is higher as compared to the experimental value if the effect of powder bed density and effective thermal conductivity of the powder layer (rbed and keff) are considered together. After time t ¼ 1 s, the temperature on the top surface of the powder layer calculated by present FEM model marches toward the experimental temperature. Fig. 4(a) shows the surface plot of top surface temperature distribution of the area in the powder layer from X ¼ 0.20 to 0.50 mm and from Y ¼ 0.40 to 0.70 mm. This figure shows that the maximum surface temperature (2978.8 K) is observed at the center of the laser beam. Some
4.1.2. Comparison with numerical results Kolossov et al. [5] have developed a three-dimensional FEM-based numerical model for predication of temperature on the top surface of titanium powder layer during MLS. Present model is used with Gaussian heat flux, effective thermal conductivity, temperature-dependent thermal conductivity and specific heat and phase change consideration for the calculation of the temperature distribution. The laser related parameters are given in Table 3. The density of the material is 2900 kg/m3. Temperature distribution is predicted when the laser beam is focused at the position X ¼ 0.225 mm, Y ¼ 0.300 mm after time t ¼ 1.0 s. Fig. 5 shows the top surface temperature distribution along X-axis at different Y distances (Y ¼ 0.250, 0.2625, 0.275, 0.300 mm) as well as along Y-axis at different X distances (X ¼ 0.150, 0.1625, 0.175, 0.225 mm) in the powder layer, due to irradiation of laser beam, respectively. Fig. 5(a) shows that the peak top surface temperature by numerical model [5] under the laser beam was 2806 K, where as using the present FEM model; the peak top surface temperature under the laser beam is determined as 3029.7 K. Also, it can be observed that the area, which is nearer to the laser heat source, has high temperature. The temperature gradually decreases on both sides of laser center. This pattern is followed in the subsequent Y distances. Fig. 5(b) shows temperature profile along Y-axis. Here temperature profile follows the pattern as that of X-axis temperature variation but it is not symmetric. Temperature variation along Y-axis is not very steep as that of X-axis. This is so because laser beam is moving along Y-axis that means the nodes lying on laser path and in the close vicinity of laser beam are getting heat frequently than those lying outside the laser path. The peak temperature location is slightly shifted in present FEM model but still within the laser beam area. This discrepancy could be assigned to very small domain size and the scheme of laser movement after interaction time. Fig. 6(a) shows the surface plot of top surface temperature distribution of the powder layer after time t ¼ 1.0 s. It is seen from the figure that the peak temperature is observed not at the center of the laser beam but slightly above the center of the laser beam. This is so
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Center of the laser beam
(a)
0.67 0.65 0.62
Y - axis (mm)
0.60 0.57 0.55 0.52 0.50 0.47 0.45 0.25 0.27 0.29 0.31 0.33 0.35 0.37 0.39 0.41 0.43 0.45 X-axis (mm)
(b)
Fig. 4. Top surface temperature distribution between X ¼ 0.2 and 0.5 mm and Y ¼ 0.4 and 0.7 mm considering rbed and keff into account at time t ¼ 1 s: (a) surface plot and (b) contour plot. Table 3 Laser-related parameters [5] Variable
Notion
Value
Laser power (W) Spot radius (m)
P R
2 25 106
because the laser is moving along Y-axis and insulated boundary is on decreasing X-axis side. As across this boundary thermal gradient is zero, the temperature is accumulating along this boundary increasing the temperature of the nodes nearer to this boundary. Another possible
reason for this is very small domain size. Fig. 6(b) shows the contour plot of isotherms of the entire powder layer area after time t ¼ 1.0 s. The thick line (1933 K) shows the isotherm of the melting temperature of titanium. The area inside the thick isotherm line is having temperature greater than the melting temperature of titanium. It can be seen from the contour plot that temperature of the left hand side area exceeds melting temperature and is in the liquid phase. Also the temperature of the entire domain is very high because the domain size is very small and laser is operating in continuous wave mode. As the heat is given to the powder layer continuously without sufficient time to solidify, maximum powder layer area is in liquid phase.
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Y = 0.2500 Y = 0.2750 Y = 0.3000 Y = 0.2625 Present FEM based peak temperature Peak temperature by Kolossov et. al. [5]
3100 3000
Temperature (K)
2900 2800 2700 2600 2500 2400 2300 2200 2100 0.075
0.150
0.225
0.300
0.375
Along X - axis (mm)
(a)
X = 0.1500 X = 0.1750 X = 0.2250 X = 0.1625 Present FEM based peak temperature Peak temperature by Kolossov et. al. [5] 3100 3000 Temperature (K)
2900 2800 2700 2600 2500 2400 2300 2200 0.15 (b)
0.20
0.25 0.30 0.35 Along Y - axis (mm)
0.40
0.45
Fig. 5. Variation of top surface temperature along (a) X-axis at Y ¼ 0.2, 0.2625, 0.275, 0.3 mm and (b) Y-axis at X ¼ 0.15, 0.1625, 0.175, 0.225 mm at time t ¼ 1.0 s.
4.2. Temperature distribution in the powder layer MLS process is affected by several process parameters. These independent process parameters are laser power, laser beam diameter, laser on-time, laser off-time and hatch spacing. In the present work the effect of above-mentioned parameters on the temperature distribution are examined. Fig. 7(a) shows the effect of laser power on temperature distribution along X-axis at different Y ¼ 0.3 mm distance in the powder layer due to irradiation of laser beam. The material properties and other data are given in Table 4. Laser power is changed between 2 and 4 W with 1 W increment. The nature of temperature distribution graph shows that there is an increase in temperature with increase
in laser power. Temperature gradient along X-axis increases from 6.9 to 9.5 K/m. As the temperature increases thermal gradient also increases. Also the heat flux given to the powder layer is more with increasing laser power, Eq. (5). The temperature gradually decreases on both sides of laser center. Fig. 7(b) shows the effect of laser power on temperature distribution along Y-axis at X ¼ 0.20 mm distance in the powder layer due to irradiation of laser beam. The nature of temperature distribution graph shows that there is an increase in temperature with increase in laser power. Temperature gradient along Y-axis increases from 7.4 to 10.1 K/m. But the increase in temperature gradient is less along Y-axis as compared to the temperature gradient along X-axis. This is because the laser beam is moving along Y-axis and some nodes on the laser path are repeatedly heated. In all cases peak temperature is not observed at the center of the laser beam X ¼ 0.225 mm, Y ¼ 0.300 mm but slightly toward decreasing X-axis and increasing Y-axis side still within the laser beam area. Fig. 8(a) shows the effect of laser beam diameter on temperature distribution along X-axis at Y ¼ 0.3 mm distance in the powder layer due to irradiation of laser beam. Laser beam diameter is changed between 0.05 and 0.15 mm. The temperature increases with increase in beam diameter up to the value D ¼ 0.10 mm but after that the temperature decreases as we go on increasing the beam diameter. Temperature decreases after D ¼ 0.15 mm because more and more number of elements are coming under the laser beam area causing decrease in the thermal gradient. Temperature gradient at D ¼ 0.15 mm is 6.55 K/m where as temperature gradient at D ¼ 0.05 mm is 6.96 K/m. This implies that temperature gradient decreases with increase in beam diameter. Also at D ¼ 0.15 mm, the peak temperature is observed not at the center of the laser beam but slightly shifted to the side of increasing X-axis. This happens because of the fact that the elements lying on the increasing X-axis side getting heated frequently. Fig. 8(b) shows the effect of laser beam diameter on temperature distribution along Y-axis at different X distances in the powder layer due to irradiation of laser beam. The temperature increases with increase in beam diameter up to the value D ¼ 0.10 mm but after that the temperature decreases as we go on increasing the beam diameter. The increase or decrease in the temperature gradient along Y-axis is not as steep as the temperature gradient along X-axis. Fig. 8 shows that as we increase the beam diameter, the peak temperature (3696 K) also increases up to the beam diameter value of 0.10 mm after that peak temperature (3492 K) decreases. This peak temperature is less than the peak temperature (5273 K) observed due to laser power effect. Fig. 9(a) shows the effect of different laser on-time on temperature distribution along X-axis at Y ¼ 0.3 mm distance in the powder layer due to irradiation of laser beam. Laser on time is changed between 50 and 70 ms with
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(a) 0.63
0.55
0.48
Y - axis (mm)
0.40
0.33
0.25
0.18
0.10
0.03 0.03
(b)
0.10
0.18
0.25
0.33
0.40
0.48
0.55
0.63
X - axis (mm)
Fig. 6. Top surface temperature distribution at time t ¼ 1.0 s: (a) Surface plot and (b) contour plot.
10 ms increment. The nature of temperature distribution graph shows that temperature increases with increase in laser on time. Laser on time is the time within which the heat in laser beam is given to the powder layer. If we increase laser on time, the interaction time of the heat flux from the laser beam with the powder layer increases. Thus, increase in temperature causes increase thermal gradient. The temperature gradient increases from 6.32 to 6.96 K/m. The temperature gradually decreases on both sides of laser center. Fig. 9(b) shows the effect of different laser on-time on temperature distribution along Y-axis at X ¼ 0.20 mm
distances in the powder layer due to irradiation of laser beam. The nature of temperature distribution graph shows that temperature increases with increase in laser on time. Temperature increases because heat is supplied for more period of time. The temperature gradient increases from 5.45 to 6.1 K/m. But this temperature gradient along Y-axis is not as steep as the temperature variation along X-axis. So due to rapid change in temperature gradient along X-axis may induce more thermal stresses in the powder layer. In all cases peak temperature is not observed at the center of the laser beam X ¼ 0.225 mm, Y ¼ 0.300 mm but slightly toward decreasing X-axis and increasing Y-axis
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P = 2W P = 3W P = 4W
D = 0.050 mm D = 0.100 mm D = 0.150 mm
4000
5200
3800
4800
3600
4400
3400
Temperature (K)
Temperature (K)
5600
4000 3600 3200 2800
3200 3000 2800 2600 2400
2400
2200 0.10
0.15
0.20
0.25
0.30
0.35
0.40
Distance along X - axis (mm)
(a)
5500
0.05
0.10
P = 2W P = 3W P = 4W
0.15
0.20
0.25
0.30
0.35
0.40
Distance along X - axis (mm)
(a)
4000
D = 0.050 mm D = 0.100 mm D = 0.150 mm
Temperature (K)
Temperature (K)
5000 4500 4000 3500 3000
2800
0.15 (b)
0.15
3200
2400
2500
(b)
3600
0.20
0.25
0.30
0.35
0.40
0.20
0.25
0.30
0.35
0.40
0.45
Distance along Y - axis (mm)
0.45
Distance along Y - axis (mm)
Fig. 7. Variation of temperature with different laser power in the titanium powder during metal laser sintering along (a) X-axis at Y ¼ 0.30 mm and (b) Y-axis at X ¼ 0.20 mm. Table 4 Material properties and process parameters used in the present model Powder material
Titanium
Laser power (W) Pulse duration (ms) Laser beam diameter (mm) Track length (mm) Hatch spacing (mm) Thickness of powder bed (mm) Thermal conductivity of solid (W/m/K) Effective thermal conductivity of powder layer (W/m/K) Specific heat (J/kg/K) Latent heat (kJ/kg) Density (kg/m3) Initial relative density Initial temperature of powder (K)
2 50 0.050 0.625 0.025 1.0 22.5 0.2933 523 296 2931.5 0.523 293
side still within the laser beam area. This is because the laser beam is moving along Y-axis and some nodes on the laser path are repeatedly heated.
Fig. 8. Variation of temperature with different beam diameter in the titanium powder during metal laser sintering along (a) X-axis at Y ¼ 0.30 mm and (b) Y-axis at X ¼ 0.20 mm.
Fig. 10(a) shows the effect of laser off-time on temperature distribution along X-axis at Y ¼ 0.3 mm distance in the powder layer due to irradiation of laser beam in MLS process. Laser off-time is changed between 1 and 5 ms with 2 ms variation. The nature of temperature distribution graph shows that temperature decreases with increase in laser off-time. When the laser is off, no heat is supplied to the powder layer. If we increase laser off-time, as no heat is supplied during the off-time, heat supplied during on-time gets more time to diffuse in the powder layer through conduction decreasing the temperature gradient and overall temperature in the powder layer. Temperature gradient decreases from 2.92 to 2.84 K/m with decrease in temperature. Fig. 10(b) shows the effect of laser off-time on temperature distribution along Y-axis at different X ¼ 0.3 mm distance in the powder layer due to irradiation of laser beam. The nature of temperature distribution graph shows that temperature decreases with increase in laser off-time. This decreases in temperature causes
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3600
toff = 1 ms toff = 3 ms toff = 5 ms
2200 2100
3200
Temperature (K)
Temperature (K)
3400
ton = 50 ms ton = 60 ms ton = 70 ms
3000 2800 2600
1079
2000 1900 1800 1700
2400 1600 2200 0.05
0.10
0.15
0.20
0.25
0.30
0.35
Distance along X - axis (mm)
(a)
3600
1500 0.05
0.40
ton = 50 ms ton = 60 ms ton = 70 ms
0.25
0.30
0.35
0.40
2100 Temperature (K)
Temperature (K)
0.20
toff = 1 ms toff = 3 ms toff = 5 ms
2150
3200 3000 2800 2600
2050 2000 1950 1900 1850 1800 1750
2400
0.15 (b)
0.15
Distance along X - axis (mm) 2200
3400
0.15
0.10
(a)
0.20
0.25
0.30
0.35
0.40
0.45
(b)
0.20
0.25
0.30
0.35
0.40
0.45
Distance along Y - axis (mm)
Distance along Y - axis (mm)
Fig. 9. Variation of temperature with different laser on-time in the titanium powder during metal laser sintering along (a) X-axis at Y ¼ 0.30 mm and (b) Y-axis at X ¼ 0.20 mm.
decrease in the temperature gradient. Temperature gradient along Y-axis decreases from 5.45 to 4.59 K/m. This decrease in temperature gradient along Y-axis is steeper when compared with temperature gradient along X-axis. Fig. 11(a) shows the effect of hatch spacing on temperature distribution along X-axis at Y ¼ 0.3 mm distance in the powder layer. Hatch spacing is the distance moved by laser beam after each time step. Hatch spacing of 0.025 and 0.05 mm is taken to study the effect on temperature. Further values of hatch spacing cannot be taken into consideration because maximum hatch spacing distance is equal to laser diameter (0.050 mm in present case). The nature of temperature distribution graph shows that temperature decreases with increase in hatch spacing. When hatch spacing is 0.025 mm, every element in the laser path get heat from laser four times while scanning the powder layer. This is because of the scheme followed (Fig. 2) in the present model that only four elements are under the laser beam. This means the elements get heated
Fig. 10. Variation of temperature with different laser off-time in the titanium powder during metal laser sintering along (a) X-axis at Y ¼ 0.30 mm and (b) Y-axis at X ¼ 0.20 mm.
more frequently thus increasing their temperature. When hatch spacing is 0.050 mm, temperature decreases drastically as every element gets heated only twice. This decrease in temperature causes decrease in thermal gradient from 6.96 to 6.19 K/m. Fig. 11(b) shows the effect of hatch spacing on temperature distribution along Y-axis at X ¼ 0.20 mm distance in the powder layer. Hatch spacing of 0.025 and 0.05 mm is taken to study the effect on temperature. The nature of temperature distribution graph shows that temperature decreases with increase in hatch spacing. When hatch spacing is 0.05 mm, temperature decreases drastically as every element gets heated only twice. Also, peak temperature in case of hatch spacing of 0.025 mm is 3010 K changes to 2300 K for hatch spacing of 0.05 mm. Temperature gradient decreases from 5.4 to 4.6 K/m. So, due to rapid change in temperature gradient along X-axis may induce more thermal stress on the powder layer.
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Hatch Spacing = 0.025 mm Hatch Spacing = 0.050 mm
3200
2.
3000
Temperature (K)
2800 2600
3.
2400 2200 2000
4.
1800
5.
1600
the temperature distribution in the single powder layer accurately when compared with experimental results. The peak temperature is observed not exactly at the center of the laser beam but slightly toward decreasing X-axis and increasing Y-axis but still within the laser beam area. The temperature initially increases with increase in beam diameter but after 0.15 mm of beam diameter the temperature decreases. The temperature increases with increase in the laser ontime but decreases with increase in the laser off-time. The temperature decreases by increasing hatch spacing.
1400 0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Distance along X - axis (mm)
(a)
Hatch Spacing = 0.025 mm Hatch Spacing = 0.050 mm
3200
Temperature (K)
3000 2800 2600 2400 2200 2000 1800 0.15 (b)
0.20
0.25 0.30 0.35 0.40 Distance along Y - axis (mm)
0.45
Fig. 11. Variation of temperature with different hatch spacing in the titanium powder during metal laser sintering along (a) X-axis at Y ¼ 0.30 mm and (b) Y-axis at X ¼ 0.20 mm.
5. Conclusions In the present paper, a finite element-based model of MLS is developed for the determination of temperature distribution in the single powder layer. Temperature distribution in the powder layer due to laser irradiation has been studied by varying input parameters such as laser power, beam diameter, laser on-time, laser off-time and hatch spacing. On the basis of these computational investigations, following conclusions can be drawn: 1. The present thermal model is able to predict temperature distribution in the metallic layer including the effect of Gaussian heat flux distribution of laser beam, effective thermal conductivity of the powder layer, bed density and temperature-dependent thermal properties of powder material. Also the developed model predicts
References [1] C.C. Kai, L.K. Fai, Rapid Prototyping Principles and Applications, Wiley, New York, 1997. [2] D. Kochan, Solid Freeform Manufacturing, Elsevier Science Publicaters, London, 1993. [3] P. Fischer, M. Locher, S. Kolossov, R. Glardon, V. Romano, H.P. Weber, Temperature measurement during selective laser sintering of titanium powder, International Journal of Machine Tools and Manufacture 44 (2004) 1293–1296. [4] J.D. Williams, C.R. Deckard, Advances in modeling the effects of selected parameters on the SLS process, Rapid Prototyping Journal 4 (1998) 90–100. [5] S. Kolossov, E. Boillat, R. Glardon, P. Fischer, M. Locher, 3D FE simulation for temperature evaluation in the selective laser sintering process, International Journal of Machine Tools and Manufacture 44 (2004) 117–123. [6] M. Matsumoto, M. Shiomi, F. Abe, K. Osakada, Finite element analysis of single layer forming on metallic powder bed in rapid prototyping by selective laser processing, International Journal of Machine Tools and Manufacture 42 (2002) 61–67. [7] M. Shiomi, A. Yoshidome, K. Osakada, F. Abe, Finite element analysis of melting and solidifying processes in laser prototyping of metallic powders, International Journal of Machine Tools and Manufacture 39 (1999) 237–252. [8] N. Tolochko, S. Mozzharov, T. Laoui, L. Froyen, Selective laser sintering of single and two component metal powders, Rapid Prototyping Journal 9 (2) (2003) 68–78. [9] R.C. Sachdeva, Fundamental of Engineering Heat and Mass Transfer, New Age International (P) Limited, New Delhi, 1988. [10] J. Beaman, J. Barlow, D. Bourell, R. Crawford, H. Marcus, K. McAlea, Solid Freeform Fabrication: A New Direction in Manufacturing, Kluwer Academic Publicaters, New York, 1997. [11] R.B. Patil, Finite element analysis of temperature in selective laser sintering, M.Tech. Thesis,Motilal Nehru National Institute of Technology, Allahabad, India, 2005. [12] K. Patri, Venuvinod, Ma. Weiyin, Rapid Prototyping Laser-Based and other Technologies, Kluwer Academic Publishers, United Kingdom, 2004. [13] G.M.C. Bugeda, G. Lombera, Numerical prediction of temperature and density distribution in metal laser sintering process, Rapid Prototyping Journal 5 (1999) 21–26. [14] J.N. Reddy, D.K. Gartling, The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press, London, 2000. [15] J.N. Reddy, An Introduction to the Finite Element Method, McGraw-Hill Company, Singapore, 2003.
International Journal of Machine Tools & Manufacture 47 (2007) 1069–1080 www.elsevier.com/locate/ijmactool
Finite element analysis of temperature distribution in single metallic powder layer during metal laser sintering Rahul B. Patil, Vinod Yadava Mechanical Engineering Department, Motilal Nehru National Institute of Technology, Allahabad, UP—211 004, India Received 23 November 2005; received in revised form 15 May 2006; accepted 19 September 2006 Available online 9 November 2006
Abstract The metal laser sintering (MLS) is used to make strong or hard metallic models for tools and dies directly from metallic powders. Thermal distortion is the serious problem after cooling of the solidified part rapidly. Uncontrolled temperature distribution in the metallic powder layer leads to thermal distortion of the solidified part. The study of temperature distribution within the metallic layer during MLS is important from the quality of the layer point of view. The high temperature generated in the powder layer leads to thermal distortion of the part and causes thermal as well as residual stresses in the part. In this paper the powder layer is assumed to be subjected to plane stress type of temperature variation and a transient finite element method-based thermal model has been developed to calculate the temperature distribution within a single metallic layer during MLS. A finite element code has been developed and validated with the known results from the literature. The obtained results of temperature distribution show the temperature and temperature gradient variation along X- and Y-axis. The effect of process parameters such as laser power, beam diameter, laser on-time, laser off-time and hatch spacing on temperature distribution within a model made of titanium during MLS is studied. The results computed by the present model agree with experimental results. Temperature increases with increase in laser power and laser on-time but temperature decreases with increase in laser off-time and hatch spacing. r 2006 Elsevier Ltd. All rights reserved. Keywords: Rapid prototyping; Metal laser sintering; FEM; Temperature distribution
1. Introduction Rapid prototype manufacturing is an emerging technology that provides an integrated way of manufacturing three-dimensional components from computer aided design (CAD) files to finished parts [1], using only additive processing. Additive processing implies that the structures are made by cumulative deposition of material, without using any hard tooling. In metal laser sintering (MLS) process parts are usually built by sintering a thin layer of metal powder using either CO2 or Nd: YAG laser beam. The interaction of the laser beam with the powder raises the temperature to the point of melting, resulting in particle bonding fusing the particles to themselves and the previous Corresponding author. Tel.: +91 0532 2271812; fax: +91 0532 2445101, +91 0532 2445077. E-mail address: [email protected] (V. Yadava).
0890-6955/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2006.09.025
layer to form the solid. The power of the laser beam is adjusted to bring the selected powder areas to a temperature just sufficient for the particles in the powder to get attached (i.e. get sintered). Sintering may occur through partial melting or softening of powder particles themselves, or of coating on the particles. An initial cross-section of the object under fabrication is traced on the layer of powder by means of XY controlled pulsed laser beam. Sintering occurs when the grain viscosity drops with temperature creating an interfacial kitting of the grains without full melting. The next layer is then built up directly on top of the sintered layer after an additional layer of powder is deposited via a roller mechanism on top of the previously formed layer. In the unsintered areas, powder remains loose and serves as a natural support for the next layer of powder and the object under fabrication [2]. Fischer et al. [3] have measured temperature on the top surface of titanium powder layer during MLS. Nd:YAG
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Nomenclature
r
[A] {B} [B] [C] Cp(T) Cs Cl {f} [GC] [GK] {GF} I(r) [K] ks(T)
s T Tm Ti {T} fT g t x, y
keff kg Lm {N} P Q q0 q(r) R
matrix of A vector of B matrix of derivative of shape functions capacitance matrix temperature-dependent specific heat (J/kg K) specific heat of solid (J/kg K) specific heat of powder in liquid state (J/kg K) right-hand side force vector global capacitance matrix global stiffness matrix global right side force vector laser intensity stiffness matrix temperature dependent thermal conductivity of solid (W/m K) effective thermal conductivity of powder layer (W/m K) thermal conductivity of the surrounding gaseous environment (W/m K) latent heat of melting (kJ/kg) column vector of 2-D bilinear shape functions laser power (W) internal heat generated (W/m3) heat flux at the center of the laser beam (W/m2) input heat flux (W/m2) radius of laser beam (m)
laser is focused at a specified location on the powder layer. Temperature at the top surface of the powder layer has been measured with the help of an infrared camera with two different laser parameter sets, pulsed and continuous laser modes. The measured temperature field in each case has been compared with each other. They observed that the peak skin temperature in pulsating mode is much higher than that in continuous mode. Deckard and Williams [4] have carried out experiment on bisphenol—a polycarbonate. The effect of machine parameters namely laser power, beam velocity, hatch spacing, beam spot size and scan line length on part density and strength of the part have been studied. Their findings provide understanding about interaction of laser irradiation with the powder layer. Kolossov et al. [5] have developed a finite element method (FEM)-based numerical model for determination of temperature distribution on the top surface of titanium powder layer during MLS. The FEM model was developed considering sintering potential and taking specific heat as a function of temperature. Temperature distribution on the top surface of the powder layer has been determined when laser beam is arrived at a particular location while moving. They also carried out physical experiment and measured temperature on the top surface of the powder layer with an infrared camera at the same location. Shiomi and Osakada
distance of a point from the center of the laser beam (m) thickness of the powder layer (m) temperature (K) melting temperature of titanium room temperature (K) nodal temperature vector time derivative of temperature time (s) cartesian coordinate x and y
Greek letters r rbed f a
Density (kg/m3) Density of the powder bed (kg/m3) Empirical coefficient Constant
Subscripts l s i
Liquid Solid Initial at time ¼ 0
Superscripts e
Element
[6] have developed a FEM-based heat conduction model to find weight of the solidified part made of metallic powder. Their experimental and FEM-based results show that weight of solidified part is affected by peak power rather than duration of laser irradiation. Shiomi and Osakada [7] have developed a thermal model to calculate temperature and stress distribution for various track lengths. They have shown that the cracking of layer can be avoided by using smaller track length. In the present paper a FEM-based thermal model is developed to determine the non-uniform transient temperature distribution in the single powder layer of titanium due to laser irradiation for different laser power, beam diameter, laser on-time, laser off-time and hatch spacing. 2. Mathematical modeling Localized heating of a small volume of powder during MLS is shown schematically in Fig. 1 [4]. The duration of laser beam at any powder particle is short, typically between 0.5 and 25 ms. Therefore, the thermally induced binding reactions must be kinetically rapid. In principle, both single- and two-component powders can be subjected to MLS process [8]. In case of single-component powders the liquid phase arises due to the surface melting of particles and the powder is sintered by joining the solid
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Laser moving direction Heat lost by radiation
Laser beam
Heat lost by convenction
Unsintered Region
Sintered Region
Heat conducted to the powder bed
Previous Layer (a)
(b) Fig. 1. (a) Interaction of the laser irradiation and metal powder [4]. (b) Schemes of inter-particle contacts formation during metal laser sintering of single component powders [8].
non-melted cores of particles. Fig. 1(b) shows the possible ways of contact formation during MLS of single-component powders.
gaseous environment (i.e. air) and initial relative density of the powder bed. 2.2. Governing equation
2.1. Assumptions Owing to random and complex nature of MLS process, following assumptions are made to make the problem mathematically tractable: 1. The input heat flux due to laser irradiation is treated as an internal heat generation in the powder layer. The heat flux from laser beam is taken as Gaussian distributed heat flux and given directly to the top of the powder layer. 2. The powder layer is assumed to be subjected to plane stress type of temperature variation because of the thickness of the powder layer is very small. 3. The heat flux due to laser irradiation is given to the fully covered finite element under the laser beam at a time. 4. Laser beam spot is assumed to be circular in shape. 5. To simplify the calculation the whole powder layer is considered to be homogeneous and continuous. 6. Only the part of powder material having temperature equal to or greater than melting temperature is assumed to be sintered. 7. All the solid particles in the powder are assumed to be spherical in shape and equal in size. 8. The effective thermal conductivity of the powder layer is used in the calculation, which incorporates solid thermal conductivity, thermal conductivity of the surrounding
In MLS, the powder layer is spread on the table surface with a pre-defined very small layer thickness. Powder layer is considered as plane stress type of temperature variation governed by following thermal heat diffusion equation with internal heat generation in a Cartesian domain [9]: rbed C p ðTÞ
qT ¼ rðkeff rTÞ þ Qg qt
within the domain ABCD;
(1) where rbed is the density of the powder bed, Cp(T) is specific heat, T is temperature, t is time, keff effective thermal conductivity of the powder bed, Q internal heat generated due to laser irradiation and X and Y are coordinates axes as shown in Fig. 2. 2.3. Initial and boundary conditions The initial temperature Ti is taken as equal to room temperature within domain ABCD. Thus, T ¼ Ti
in ABCD at
t ¼ 0.
(2)
The temperature gradient is taken as zero across the boundaries of the domain (ABCD) because these boundaries AB, BC, CD and AD are at such a large distance from the laser scanning area (area EFGH in Fig. 2), so that there is no heat transfer across it. Hence, the boundary
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Typical area element
e Y
Typical boundary element Domain
B
C
b
Scanning area of laser heating F
G
3 1
4 2
Common node
H
E
2.5. Calculation of bed density (rbed) Elements affected by Gaussian heat flux X
A
where Q is internal heat generated and s is the thickness of the powder layer. Gaussian distributed heat flux, which is distributed over four area elements at a time, is also shown in Fig. 2. The Gaussian distributed peak of the heat flux is considered at the junction node, which is common to four area elements. The dark dot at the center of the laser beam is the common node to the area elements 1, 2, 3 and 4. To satisfy the necessary condition to have at least four area elements under the laser beam. The elemental length is selected in such a way that the laser beam diameter becomes a whole number multiple of the elemental length because of computational simplicity.
D
Fig. 2. Finite element mesh of thermal model of metal laser sintering.
conditions can be written as qT keff ¼ 0, qn
(3)
on boundaries AB, BC, CD and AD. Here n is the outward normal. 2.4. Calculation of internal heat generation due to laser irradiation (Q) Most researchers [6,7] working in the area of MLS have considered uniformly distributed heat source of laser intensity. This assumption is far from reality as it is known from the literature that laser beam intensity profile I(r) is Gaussian [10]. In the present model, Gaussian heat flux distribution is taken same as laser intensity. If the laser input power P and laser beam radius R are known, then the laser intensity I(r) at a distance r from the center of the laser beam is given by [11] 4:55P 4:5ðr=RÞ2 e . (4) pR2 This is the input heat flux from the laser beam given to the powder layer. Hence, it can be written as [11]
The relative bed density is calculated [12] using Eq. (7) with the assumption that all the solid particles in the powder layer are spherical in shape and equal in size and density arranged in a cubic array: pr rbed ¼ rR rs ¼ s , (7) 6 where rR is initial relative bed density, rbed is the bed density and rs is density of the solid. 2.6. Calculation of the effective thermal conductivity (keff) of powder layer Heat transfer through powders mainly occurs through conduction and radiation because inter-particle distances are too small to permit convection heat transfer. The powdered bed mainly consists of solid particles separated by a gas. Since gases have much smaller thermal conductivities (0.0065 W/m K at room temperature to 0.049 W/m K at 325 1C for air) than those of good metallic conductors (400 W/m K). The thermal conductivity of a powder bed is mainly dictated by the thermal conductivity of the gas embedded within the voids [10]. The effective thermal conductivity of the powder bed is defined as follows [13]:
2
IðrÞ ¼ I 0 e4:5ðr=RÞ ¼
4:55P 4:5ðr=RÞ2 e . (5) pR2 Here, q(r) is the input heat flux at a distance r from laser beam center, q0 is the laser heat flux at the center of the laser beam and R is the radius of the laser beam. This incoming heat flux is treated as internal heat generation in the powder layer. Hence, quantity of heat generation (W/m3) can be written as [11] qðrÞ , (6) Q¼ s 2
qðrÞ ¼ q0 e4:5ðr=RÞ ¼
keff ¼
rR ks ðTÞ , 1 þ f ks ðTÞ kg
(8)
where rR is the initial relative density of the powder layer, ks is temperature-dependent solid thermal conductivity, kg is thermal conductivity of the surrounding gaseous environment, i.e. air and f is empirical coefficient, f ¼ 0:02 102ð0:7rR Þ . 2.7. Temperature-dependent material properties In general thermal conductivity and specific heat both change with respect to temperature. Therefore, in the present model, thermal conductivity and specific heat both are taken as temperature dependent. The graphs of
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temperature-dependent thermal properties are taken from Kolossov et al. [5]. 2.8. Phase change consideration When the temperature of the material exceeds the melting point, latent heat due to melting is needed. The latent heat required for phase change (for solid to liquid) is accounted by modifying the expression for the specific heat. Considering enthalpy before and after phase change, the latent heat of melting is incorporated to get modified specific heat [14]: Cl ¼ Cs þ
Lm 2DT
for T m DTpTpT m þ DT,
(9)
where Tm is the melting temperature of titanium (1933 K) and Lm is the latent heat of melting. 3. Finite element formulations The Galerkin’s finite element formulation [15] has been applied to obtain temperature distribution within the square domain due to laser irradiation as shown in Fig. 2. The following expressions are obtained for elemental capacitance matrix [C]e, conductivity matrix [K]e and boundary flux vector {f}e, when Galerkin’s method is applied to Eqs. (1)–(3) 9 R ½K e ¼ De keff ½BeT ½Be dx dy; > > = R e e eT ½C ¼ De rbed C p fN g fN g dx dy; . (10) > e R e > e eT ; f ¼ De fN g fN g qg dx dy: Here, {N}e is the nodal shape function vector, [B]e is the matrix relating temperature derivative with its nodal values and {qg}e is vector of nodal heat generation. The integrals in the expression (10) are computed numerically using Gaussian quadrature with three points in each direction. When the elemental quantities of Eq. (10) are assembled, the following differential equations are obtained: no ½GCnnmnnm T þ ½GKnnmnnm fT gnnm1 ¼ fGFgnnm1 ,
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4. Results and discussion Formulation of Section 3 is used for the development of a FEM-based code in MATLAB 7.0. The connectivity matrix and nodal coordinates of the elements are generated using ANSYS 9.0. To check the accuracy of present FEMbased model for the determination of temperature distribution within the single layer of powder during MLS, the temperature distribution on the top surface of powder layer is determined, and then compared with experimental results of Fischer et al. [3] as well as with numerical results of Kolossov et al. [5]. Here the present model is used with same conditions as taken in Refs. [3,5]. 4.1. Comparison 4.1.1. Comparison with experimental results Fischer et al. [3] have measured temperature on the top surface of titanium powder layer during MLS. Gaussian heat flux, effective thermal conductivity is used for the calculation of the temperature distribution. The laser related parameters and powder material properties are given in Tables 1 and 2, respectively. Fig. 3 shows the top surface temperature variation along X-axis at Y ¼ 0.55 mm as well as along Y-axis at X ¼ 0.35 mm considering different state conditions of density and thermal conductivity of titanium powder material. Figs. 3(a) and (b) show a comparison between the experimental and computed results along X- and Y-axis, respectively. The highest top surface temperature measured during experiment was 1500 K where as top surface temperature in similar conditions using present FEM model is determined as 2134.8 K. The computed top surface temperature using present code is determined as 2978.8 K if keff ¼ 0.2154 W/m K and rbed ¼ 2931.5 kg/m3. The discrepancy between the computed and experimental results could be assigned to the difference in the effective thermal conductivity of powder used in the present model in place of thermal conductivity of solid used by Fischer et al. [3] (0.2154 W/m K in place of Table 1 Laser-related parameters [3]
nnm1
(11) where [GC] is the global capacitance matrix, [GK] is the global stiffness matrix, {GF} is the global right side force vector, {T} is global temperature vector and fT g is time derivative of {T}. Eq. (11) represents a set of ordinary differential equations in the variable {T} as a function of time t. These equations are converted to set of algebraic equations by the application of implicit finite difference method [15]. Here, the solution marches in time, in steps of Dt until the desired final time is reached. In the present model, Dt is divided into two time steps Dt1 and Dt2. Here, Dt1 and Dt2 pulse on time and off-time of the laser beam, respectively.
Variable
Notion
Value
Laser power (W) Spot radius (m) Laser spot speed (mm/s)
P R V
2 50 106 1
Table 2 Titanium material properties [3] Variable
Notion
Value
Thermal conductivity (W/m K) Density (kg/m3) Specific heat (J/kg K)
k r C
1.45 2931.5 335
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Present FEM based using ρs and ks Present FEM based using ρs and keff Experimental peak temperature Present FEM based peak temperature
peaks are observed in the area, which comes under the laser beam. Fig. 4(b) shows the contour plot of isotherms at the top surface of the area in the powder layer from X ¼ 0.25 to 0.45 mm and from Y ¼ 0.45 to 0.675 mm. The thick line shows the isotherm of the melting temperature of titanium powder. The area inside the thick isotherm line is having temperature greater than the melting temperature of titanium. It is assumed in the present model that the powder particles having temperature greater than the melting temperature are sintered together by liquid-phase sintering. Therefore the powder particles inside the area of thick isotherm undergo liquid-phase sintering at the surface and remaining particles are remained unsintered.
3000
Temperature (K)
2500
2000
1500
1000 0.15
0.20
0.25
(a)
0.30 0.35 0.40 X at Y = 0.55 (mm)
0.45
0.50
0.55
Present FEM based using ρs and ks Present FEM based using ρs and keff Experimental peak temperature Present FEM based peak temperature 3000
Temperature (K)
2500
2000
1500
1000 0.40 (b)
0.45
0.50
0.55
0.60
0.65
0.70
0.75
Y at X = 0.35 (mm)
Fig. 3. Variation of top surface temperature (a) along X-axis at Y ¼ 0.55 mm (b) along Y-axis at X ¼ 0.35 mm at time t ¼ 1 s.
1.45 W/m K), unknown number of pulses for which experimental results are reported. Another cause may be the measurement of temperature using camera, which is not able to capture the temporarily higher skin temperature. Here, theoretically calculated temperature value is higher as compared to the experimental value if the effect of powder bed density and effective thermal conductivity of the powder layer (rbed and keff) are considered together. After time t ¼ 1 s, the temperature on the top surface of the powder layer calculated by present FEM model marches toward the experimental temperature. Fig. 4(a) shows the surface plot of top surface temperature distribution of the area in the powder layer from X ¼ 0.20 to 0.50 mm and from Y ¼ 0.40 to 0.70 mm. This figure shows that the maximum surface temperature (2978.8 K) is observed at the center of the laser beam. Some
4.1.2. Comparison with numerical results Kolossov et al. [5] have developed a three-dimensional FEM-based numerical model for predication of temperature on the top surface of titanium powder layer during MLS. Present model is used with Gaussian heat flux, effective thermal conductivity, temperature-dependent thermal conductivity and specific heat and phase change consideration for the calculation of the temperature distribution. The laser related parameters are given in Table 3. The density of the material is 2900 kg/m3. Temperature distribution is predicted when the laser beam is focused at the position X ¼ 0.225 mm, Y ¼ 0.300 mm after time t ¼ 1.0 s. Fig. 5 shows the top surface temperature distribution along X-axis at different Y distances (Y ¼ 0.250, 0.2625, 0.275, 0.300 mm) as well as along Y-axis at different X distances (X ¼ 0.150, 0.1625, 0.175, 0.225 mm) in the powder layer, due to irradiation of laser beam, respectively. Fig. 5(a) shows that the peak top surface temperature by numerical model [5] under the laser beam was 2806 K, where as using the present FEM model; the peak top surface temperature under the laser beam is determined as 3029.7 K. Also, it can be observed that the area, which is nearer to the laser heat source, has high temperature. The temperature gradually decreases on both sides of laser center. This pattern is followed in the subsequent Y distances. Fig. 5(b) shows temperature profile along Y-axis. Here temperature profile follows the pattern as that of X-axis temperature variation but it is not symmetric. Temperature variation along Y-axis is not very steep as that of X-axis. This is so because laser beam is moving along Y-axis that means the nodes lying on laser path and in the close vicinity of laser beam are getting heat frequently than those lying outside the laser path. The peak temperature location is slightly shifted in present FEM model but still within the laser beam area. This discrepancy could be assigned to very small domain size and the scheme of laser movement after interaction time. Fig. 6(a) shows the surface plot of top surface temperature distribution of the powder layer after time t ¼ 1.0 s. It is seen from the figure that the peak temperature is observed not at the center of the laser beam but slightly above the center of the laser beam. This is so
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Center of the laser beam
(a)
0.67 0.65 0.62
Y - axis (mm)
0.60 0.57 0.55 0.52 0.50 0.47 0.45 0.25 0.27 0.29 0.31 0.33 0.35 0.37 0.39 0.41 0.43 0.45 X-axis (mm)
(b)
Fig. 4. Top surface temperature distribution between X ¼ 0.2 and 0.5 mm and Y ¼ 0.4 and 0.7 mm considering rbed and keff into account at time t ¼ 1 s: (a) surface plot and (b) contour plot. Table 3 Laser-related parameters [5] Variable
Notion
Value
Laser power (W) Spot radius (m)
P R
2 25 106
because the laser is moving along Y-axis and insulated boundary is on decreasing X-axis side. As across this boundary thermal gradient is zero, the temperature is accumulating along this boundary increasing the temperature of the nodes nearer to this boundary. Another possible
reason for this is very small domain size. Fig. 6(b) shows the contour plot of isotherms of the entire powder layer area after time t ¼ 1.0 s. The thick line (1933 K) shows the isotherm of the melting temperature of titanium. The area inside the thick isotherm line is having temperature greater than the melting temperature of titanium. It can be seen from the contour plot that temperature of the left hand side area exceeds melting temperature and is in the liquid phase. Also the temperature of the entire domain is very high because the domain size is very small and laser is operating in continuous wave mode. As the heat is given to the powder layer continuously without sufficient time to solidify, maximum powder layer area is in liquid phase.
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Y = 0.2500 Y = 0.2750 Y = 0.3000 Y = 0.2625 Present FEM based peak temperature Peak temperature by Kolossov et. al. [5]
3100 3000
Temperature (K)
2900 2800 2700 2600 2500 2400 2300 2200 2100 0.075
0.150
0.225
0.300
0.375
Along X - axis (mm)
(a)
X = 0.1500 X = 0.1750 X = 0.2250 X = 0.1625 Present FEM based peak temperature Peak temperature by Kolossov et. al. [5] 3100 3000 Temperature (K)
2900 2800 2700 2600 2500 2400 2300 2200 0.15 (b)
0.20
0.25 0.30 0.35 Along Y - axis (mm)
0.40
0.45
Fig. 5. Variation of top surface temperature along (a) X-axis at Y ¼ 0.2, 0.2625, 0.275, 0.3 mm and (b) Y-axis at X ¼ 0.15, 0.1625, 0.175, 0.225 mm at time t ¼ 1.0 s.
4.2. Temperature distribution in the powder layer MLS process is affected by several process parameters. These independent process parameters are laser power, laser beam diameter, laser on-time, laser off-time and hatch spacing. In the present work the effect of above-mentioned parameters on the temperature distribution are examined. Fig. 7(a) shows the effect of laser power on temperature distribution along X-axis at different Y ¼ 0.3 mm distance in the powder layer due to irradiation of laser beam. The material properties and other data are given in Table 4. Laser power is changed between 2 and 4 W with 1 W increment. The nature of temperature distribution graph shows that there is an increase in temperature with increase
in laser power. Temperature gradient along X-axis increases from 6.9 to 9.5 K/m. As the temperature increases thermal gradient also increases. Also the heat flux given to the powder layer is more with increasing laser power, Eq. (5). The temperature gradually decreases on both sides of laser center. Fig. 7(b) shows the effect of laser power on temperature distribution along Y-axis at X ¼ 0.20 mm distance in the powder layer due to irradiation of laser beam. The nature of temperature distribution graph shows that there is an increase in temperature with increase in laser power. Temperature gradient along Y-axis increases from 7.4 to 10.1 K/m. But the increase in temperature gradient is less along Y-axis as compared to the temperature gradient along X-axis. This is because the laser beam is moving along Y-axis and some nodes on the laser path are repeatedly heated. In all cases peak temperature is not observed at the center of the laser beam X ¼ 0.225 mm, Y ¼ 0.300 mm but slightly toward decreasing X-axis and increasing Y-axis side still within the laser beam area. Fig. 8(a) shows the effect of laser beam diameter on temperature distribution along X-axis at Y ¼ 0.3 mm distance in the powder layer due to irradiation of laser beam. Laser beam diameter is changed between 0.05 and 0.15 mm. The temperature increases with increase in beam diameter up to the value D ¼ 0.10 mm but after that the temperature decreases as we go on increasing the beam diameter. Temperature decreases after D ¼ 0.15 mm because more and more number of elements are coming under the laser beam area causing decrease in the thermal gradient. Temperature gradient at D ¼ 0.15 mm is 6.55 K/m where as temperature gradient at D ¼ 0.05 mm is 6.96 K/m. This implies that temperature gradient decreases with increase in beam diameter. Also at D ¼ 0.15 mm, the peak temperature is observed not at the center of the laser beam but slightly shifted to the side of increasing X-axis. This happens because of the fact that the elements lying on the increasing X-axis side getting heated frequently. Fig. 8(b) shows the effect of laser beam diameter on temperature distribution along Y-axis at different X distances in the powder layer due to irradiation of laser beam. The temperature increases with increase in beam diameter up to the value D ¼ 0.10 mm but after that the temperature decreases as we go on increasing the beam diameter. The increase or decrease in the temperature gradient along Y-axis is not as steep as the temperature gradient along X-axis. Fig. 8 shows that as we increase the beam diameter, the peak temperature (3696 K) also increases up to the beam diameter value of 0.10 mm after that peak temperature (3492 K) decreases. This peak temperature is less than the peak temperature (5273 K) observed due to laser power effect. Fig. 9(a) shows the effect of different laser on-time on temperature distribution along X-axis at Y ¼ 0.3 mm distance in the powder layer due to irradiation of laser beam. Laser on time is changed between 50 and 70 ms with
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(a) 0.63
0.55
0.48
Y - axis (mm)
0.40
0.33
0.25
0.18
0.10
0.03 0.03
(b)
0.10
0.18
0.25
0.33
0.40
0.48
0.55
0.63
X - axis (mm)
Fig. 6. Top surface temperature distribution at time t ¼ 1.0 s: (a) Surface plot and (b) contour plot.
10 ms increment. The nature of temperature distribution graph shows that temperature increases with increase in laser on time. Laser on time is the time within which the heat in laser beam is given to the powder layer. If we increase laser on time, the interaction time of the heat flux from the laser beam with the powder layer increases. Thus, increase in temperature causes increase thermal gradient. The temperature gradient increases from 6.32 to 6.96 K/m. The temperature gradually decreases on both sides of laser center. Fig. 9(b) shows the effect of different laser on-time on temperature distribution along Y-axis at X ¼ 0.20 mm
distances in the powder layer due to irradiation of laser beam. The nature of temperature distribution graph shows that temperature increases with increase in laser on time. Temperature increases because heat is supplied for more period of time. The temperature gradient increases from 5.45 to 6.1 K/m. But this temperature gradient along Y-axis is not as steep as the temperature variation along X-axis. So due to rapid change in temperature gradient along X-axis may induce more thermal stresses in the powder layer. In all cases peak temperature is not observed at the center of the laser beam X ¼ 0.225 mm, Y ¼ 0.300 mm but slightly toward decreasing X-axis and increasing Y-axis
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P = 2W P = 3W P = 4W
D = 0.050 mm D = 0.100 mm D = 0.150 mm
4000
5200
3800
4800
3600
4400
3400
Temperature (K)
Temperature (K)
5600
4000 3600 3200 2800
3200 3000 2800 2600 2400
2400
2200 0.10
0.15
0.20
0.25
0.30
0.35
0.40
Distance along X - axis (mm)
(a)
5500
0.05
0.10
P = 2W P = 3W P = 4W
0.15
0.20
0.25
0.30
0.35
0.40
Distance along X - axis (mm)
(a)
4000
D = 0.050 mm D = 0.100 mm D = 0.150 mm
Temperature (K)
Temperature (K)
5000 4500 4000 3500 3000
2800
0.15 (b)
0.15
3200
2400
2500
(b)
3600
0.20
0.25
0.30
0.35
0.40
0.20
0.25
0.30
0.35
0.40
0.45
Distance along Y - axis (mm)
0.45
Distance along Y - axis (mm)
Fig. 7. Variation of temperature with different laser power in the titanium powder during metal laser sintering along (a) X-axis at Y ¼ 0.30 mm and (b) Y-axis at X ¼ 0.20 mm. Table 4 Material properties and process parameters used in the present model Powder material
Titanium
Laser power (W) Pulse duration (ms) Laser beam diameter (mm) Track length (mm) Hatch spacing (mm) Thickness of powder bed (mm) Thermal conductivity of solid (W/m/K) Effective thermal conductivity of powder layer (W/m/K) Specific heat (J/kg/K) Latent heat (kJ/kg) Density (kg/m3) Initial relative density Initial temperature of powder (K)
2 50 0.050 0.625 0.025 1.0 22.5 0.2933 523 296 2931.5 0.523 293
side still within the laser beam area. This is because the laser beam is moving along Y-axis and some nodes on the laser path are repeatedly heated.
Fig. 8. Variation of temperature with different beam diameter in the titanium powder during metal laser sintering along (a) X-axis at Y ¼ 0.30 mm and (b) Y-axis at X ¼ 0.20 mm.
Fig. 10(a) shows the effect of laser off-time on temperature distribution along X-axis at Y ¼ 0.3 mm distance in the powder layer due to irradiation of laser beam in MLS process. Laser off-time is changed between 1 and 5 ms with 2 ms variation. The nature of temperature distribution graph shows that temperature decreases with increase in laser off-time. When the laser is off, no heat is supplied to the powder layer. If we increase laser off-time, as no heat is supplied during the off-time, heat supplied during on-time gets more time to diffuse in the powder layer through conduction decreasing the temperature gradient and overall temperature in the powder layer. Temperature gradient decreases from 2.92 to 2.84 K/m with decrease in temperature. Fig. 10(b) shows the effect of laser off-time on temperature distribution along Y-axis at different X ¼ 0.3 mm distance in the powder layer due to irradiation of laser beam. The nature of temperature distribution graph shows that temperature decreases with increase in laser off-time. This decreases in temperature causes
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toff = 1 ms toff = 3 ms toff = 5 ms
2200 2100
3200
Temperature (K)
Temperature (K)
3400
ton = 50 ms ton = 60 ms ton = 70 ms
3000 2800 2600
1079
2000 1900 1800 1700
2400 1600 2200 0.05
0.10
0.15
0.20
0.25
0.30
0.35
Distance along X - axis (mm)
(a)
3600
1500 0.05
0.40
ton = 50 ms ton = 60 ms ton = 70 ms
0.25
0.30
0.35
0.40
2100 Temperature (K)
Temperature (K)
0.20
toff = 1 ms toff = 3 ms toff = 5 ms
2150
3200 3000 2800 2600
2050 2000 1950 1900 1850 1800 1750
2400
0.15 (b)
0.15
Distance along X - axis (mm) 2200
3400
0.15
0.10
(a)
0.20
0.25
0.30
0.35
0.40
0.45
(b)
0.20
0.25
0.30
0.35
0.40
0.45
Distance along Y - axis (mm)
Distance along Y - axis (mm)
Fig. 9. Variation of temperature with different laser on-time in the titanium powder during metal laser sintering along (a) X-axis at Y ¼ 0.30 mm and (b) Y-axis at X ¼ 0.20 mm.
decrease in the temperature gradient. Temperature gradient along Y-axis decreases from 5.45 to 4.59 K/m. This decrease in temperature gradient along Y-axis is steeper when compared with temperature gradient along X-axis. Fig. 11(a) shows the effect of hatch spacing on temperature distribution along X-axis at Y ¼ 0.3 mm distance in the powder layer. Hatch spacing is the distance moved by laser beam after each time step. Hatch spacing of 0.025 and 0.05 mm is taken to study the effect on temperature. Further values of hatch spacing cannot be taken into consideration because maximum hatch spacing distance is equal to laser diameter (0.050 mm in present case). The nature of temperature distribution graph shows that temperature decreases with increase in hatch spacing. When hatch spacing is 0.025 mm, every element in the laser path get heat from laser four times while scanning the powder layer. This is because of the scheme followed (Fig. 2) in the present model that only four elements are under the laser beam. This means the elements get heated
Fig. 10. Variation of temperature with different laser off-time in the titanium powder during metal laser sintering along (a) X-axis at Y ¼ 0.30 mm and (b) Y-axis at X ¼ 0.20 mm.
more frequently thus increasing their temperature. When hatch spacing is 0.050 mm, temperature decreases drastically as every element gets heated only twice. This decrease in temperature causes decrease in thermal gradient from 6.96 to 6.19 K/m. Fig. 11(b) shows the effect of hatch spacing on temperature distribution along Y-axis at X ¼ 0.20 mm distance in the powder layer. Hatch spacing of 0.025 and 0.05 mm is taken to study the effect on temperature. The nature of temperature distribution graph shows that temperature decreases with increase in hatch spacing. When hatch spacing is 0.05 mm, temperature decreases drastically as every element gets heated only twice. Also, peak temperature in case of hatch spacing of 0.025 mm is 3010 K changes to 2300 K for hatch spacing of 0.05 mm. Temperature gradient decreases from 5.4 to 4.6 K/m. So, due to rapid change in temperature gradient along X-axis may induce more thermal stress on the powder layer.
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Hatch Spacing = 0.025 mm Hatch Spacing = 0.050 mm
3200
2.
3000
Temperature (K)
2800 2600
3.
2400 2200 2000
4.
1800
5.
1600
the temperature distribution in the single powder layer accurately when compared with experimental results. The peak temperature is observed not exactly at the center of the laser beam but slightly toward decreasing X-axis and increasing Y-axis but still within the laser beam area. The temperature initially increases with increase in beam diameter but after 0.15 mm of beam diameter the temperature decreases. The temperature increases with increase in the laser ontime but decreases with increase in the laser off-time. The temperature decreases by increasing hatch spacing.
1400 0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Distance along X - axis (mm)
(a)
Hatch Spacing = 0.025 mm Hatch Spacing = 0.050 mm
3200
Temperature (K)
3000 2800 2600 2400 2200 2000 1800 0.15 (b)
0.20
0.25 0.30 0.35 0.40 Distance along Y - axis (mm)
0.45
Fig. 11. Variation of temperature with different hatch spacing in the titanium powder during metal laser sintering along (a) X-axis at Y ¼ 0.30 mm and (b) Y-axis at X ¼ 0.20 mm.
5. Conclusions In the present paper, a finite element-based model of MLS is developed for the determination of temperature distribution in the single powder layer. Temperature distribution in the powder layer due to laser irradiation has been studied by varying input parameters such as laser power, beam diameter, laser on-time, laser off-time and hatch spacing. On the basis of these computational investigations, following conclusions can be drawn: 1. The present thermal model is able to predict temperature distribution in the metallic layer including the effect of Gaussian heat flux distribution of laser beam, effective thermal conductivity of the powder layer, bed density and temperature-dependent thermal properties of powder material. Also the developed model predicts
References [1] C.C. Kai, L.K. Fai, Rapid Prototyping Principles and Applications, Wiley, New York, 1997. [2] D. Kochan, Solid Freeform Manufacturing, Elsevier Science Publicaters, London, 1993. [3] P. Fischer, M. Locher, S. Kolossov, R. Glardon, V. Romano, H.P. Weber, Temperature measurement during selective laser sintering of titanium powder, International Journal of Machine Tools and Manufacture 44 (2004) 1293–1296. [4] J.D. Williams, C.R. Deckard, Advances in modeling the effects of selected parameters on the SLS process, Rapid Prototyping Journal 4 (1998) 90–100. [5] S. Kolossov, E. Boillat, R. Glardon, P. Fischer, M. Locher, 3D FE simulation for temperature evaluation in the selective laser sintering process, International Journal of Machine Tools and Manufacture 44 (2004) 117–123. [6] M. Matsumoto, M. Shiomi, F. Abe, K. Osakada, Finite element analysis of single layer forming on metallic powder bed in rapid prototyping by selective laser processing, International Journal of Machine Tools and Manufacture 42 (2002) 61–67. [7] M. Shiomi, A. Yoshidome, K. Osakada, F. Abe, Finite element analysis of melting and solidifying processes in laser prototyping of metallic powders, International Journal of Machine Tools and Manufacture 39 (1999) 237–252. [8] N. Tolochko, S. Mozzharov, T. Laoui, L. Froyen, Selective laser sintering of single and two component metal powders, Rapid Prototyping Journal 9 (2) (2003) 68–78. [9] R.C. Sachdeva, Fundamental of Engineering Heat and Mass Transfer, New Age International (P) Limited, New Delhi, 1988. [10] J. Beaman, J. Barlow, D. Bourell, R. Crawford, H. Marcus, K. McAlea, Solid Freeform Fabrication: A New Direction in Manufacturing, Kluwer Academic Publicaters, New York, 1997. [11] R.B. Patil, Finite element analysis of temperature in selective laser sintering, M.Tech. Thesis,Motilal Nehru National Institute of Technology, Allahabad, India, 2005. [12] K. Patri, Venuvinod, Ma. Weiyin, Rapid Prototyping Laser-Based and other Technologies, Kluwer Academic Publishers, United Kingdom, 2004. [13] G.M.C. Bugeda, G. Lombera, Numerical prediction of temperature and density distribution in metal laser sintering process, Rapid Prototyping Journal 5 (1999) 21–26. [14] J.N. Reddy, D.K. Gartling, The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press, London, 2000. [15] J.N. Reddy, An Introduction to the Finite Element Method, McGraw-Hill Company, Singapore, 2003.