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1 Copyright © 2005 by ASME

Proceedings of HT2005 2005 ASME Summer Heat Transfer Conference

July 17-22, 2005, San Francisco, California, USA

HT2005-72161

PARTIAL MELTING AND RESOLIDIFICATION OF SINGLE-COMPONENT METAL POWDER WITH A MOVING LASER BEAM

Bin Xiao and Yuwen Zhang

Department of Mechanical and Aerospace Engineering University of Missouri-Columbia

Columbia, MO 65211

ABSTRACT Partial melting and resolidification of single-component

metal powders with a moving laser beam is investigated numerically. Since laser processing of metal powder is a very rapid process, the liquid layer and solid core of a partially molten powder particle may not at thermal equilibrium and have different temperatures: the temperature of the liquid part is higher than the melting point, and the temperature of the solid core is below the melting point. Therefore, the local temperature of regions with partial molten particles is within a range of temperature adjacent to the melting point, instead of at the melting point. The partial melting of the metal powder is also accompanied by shrinkage that drives out the gas in the powder bed and the powder structure is supported by the solid core of the partially melted powder particles. Melting with shrinkage and resolidification are described using a temperature transforming model. The convection driven by capillary and gravity forces in the melting liquid pool is formulated by using Darcy’s law. The effects of laser beam intensity and scanning velocity on the shape and size of the heat affected zone and molten pool are investigated.

INTRODUCTION

Emerging in the late 1980s, Rapid Prototyping (RPT) is termed as a class of technologies such as Stereolithography [1], Laminated Object Manufacturing [2] and Fused Deposition Modeling [3] by which the complex, convoluted and near-impossible shapes can be automatically constructed from Computer-Aided Design (CAD) data in a short time. Selective Laser Sintering [4-6] is one of the rapid prototyping technologies allowing fabrication of three-dimensional freeform objects directly from polycarbonate, nylon and metal powder with a scanning laser beam. Various modeling has been explored by researches in order to understand the physical mechanism during the complex SLS process. In the early literature the sintering process was modeled as a pure

conduction problem [7-11], which is applicable to the sintering of amorphous polymeric material since it has little crystallinity and a near zero latent heat of fusion during melting.

Heat conduction in a semi-infinite region or finite slab with moving heat source has been investigated intensively in the past. Cline and Anthony [7] examined the heat conduction problem using Green’s function for a Gaussian beam moving at a constant velocity. Moody and Hendel [8] numerically solved the heat conduction in an infinite medium during laser heating. Kant [9] studied the heating of a multilayered cylindrical medium resting on a half-space with a stationary laser beam. Modest and Abakians [10] examined the heat conduction in a moving semi-infinite solid under a pulsed, penetrating, Gaussian laser source. Kar and Mazumder [11] presented a three-dimensional transient thermal analysis on finite slabs moving at a constant velocity by considering temperature-dependent thermophysical properties and laser irradiation with a Gaussian beam.

In contrast with SLS of amorphous polymeric materials, the latent heat of fusion in metal SLS is usually very large and therefore melting and resolidification have significant effects on the temperature distribution during sintering. In addition, convective flow in the molten pool strongly influences the energy transport, the propagation of the melting front during metal SLS process. Recently quantitative mathematical studies of metal SLS have been carried out by researchers with the consideration of convection effect. Pak and Plumb [12] presented a 1-D model of melting in a two-component powder bed with the consideration of the liquid motion driven by gravity and capillary forces. It showed that the constant porosity model yielded results compared well with the experimental data. Iwamonto et al. [13] examined a computational model describing the mechanism for the formation of topographic features during pulsed laser texturing of metal substrates by focused laser beams. Capillary forces acting on the surface of the molten material and driven by


temperature gradients were shown to produce surface deformation profiles. Han and Liou [14] simulated the laser melting process for different laser beam modes. Flow pattern, which is driven mainly by the thermo-capillary force and recoil pressure, together with temperature distribution are investigated and compared.

During metal SLS process, the powder bed experiences a significant density change because the pore space initially occupied by gases is taken up by the liquid. The shrinkage phenomena in metal SLS caused by density change have been considered by some researchers. Zhang and Faghri [15] analytically solved a one-dimensional melting problem in a powder bed containing a powder mixture under a boundary condition of the second kind. The results show that the shrinkage effect on the melting of the powder bed is not negligible. Zhang and Faghri [16] numerically examined the melting and resolidification of a subcooled mixed powder bed with moving Gaussian heat source. The results with and without shrinkage effect are compared. Furthermore, Zhang et al. [17] numerically and experimentally investigated the sintering process of two-component metal powders under irradiation of a stationary or a moving Gaussian laser beam. The liquid velocities caused by capillary and gravity forces and solid velocities caused by the shrinkage are all considered in the study.

In principle, both single- and two-component metal powders can be used in metal SLS [18]. In case of two-component metal sintering, a liquid phase is formed by melting the powders possessing the lower melting point, which infiltrates into the voids between the solid powders with higher melting point and binds them. The disadvantage of two-component metal approach is that the final product exhibits the mechanical properties and characteristics of their weakest component. Thus the interest in the production of metallic objects using single component powders has been increased in the recent years [19-21].

To process the single-component metal sintering, a partial melting approach can be employed. During SLS process of single-component powders, the surfaces of particles are molten and the powders are sintered by binding the solid non-melted cores of particles. Control of solid-liquid ratio and the melt viscosity are critical in successful sintering. Excessive molten material or too low melt viscosity causes “balling” phenomenon [22] which reduces the effective load carrying capacity of a material and acts as stress concentrations and effective crack initiation sites. So the parameters of laser processing must be carefully selected in order to ensure the partial melting. Since SLS processing is very rapid, the liquid layer and solid core of a partially molten powder particle have different temperatures. In order to model SLS of single-component metal powder using equilibrium model, melting of single-component metal powder bed can be assumed to be occurred in a range of temperature and the degree of melting is function of temperature in the range. To understand the fundamental partial melting mechanisms of single-component

metal system, this paper will present a 3-D model to investigate the effect of laser beam intensity and scanning velocity on temperature distribution, solid fraction change and the shapes of heat affected zone and molten pool. The shrinkage induced during the melting process will also be considered in the governing equation and the phase change is described using a temperature transforming model. The liquid velocities induced by capillary and gravity forces and the solid velocities caused by shrinkage of density change during the melting process will also be taken into consideration.

NOMENCLATURE b source term B dimensionless source term Bi Biot number Bo Bond number C dimensionless heat capacity C heat capacity ( KmJ ⋅3/ )

pC specific heat ( KkgJ ⋅/ )

pd diameter of the solid core of the powder particle ( m )

slh latent heat of melting or solidification ( kgJ / ) k thermal conductivity ( KmW ⋅/ ) K dimensionless thermal conductivity or permeability ( 2m )

rK relative permeability Ma Marangoni number

iN dimensionless moving laser beam intensity

RN radiation number

tN temperature ratio pc capillary pressure ( 2/ mN ) P laser power (W ) Pc dimensionless capillary pressure R radius of the laser beam ( m ) s location of interface between solid zone and mushy zone ( m )

0s location of surface ( m ) S dimensionless location of interface between solid and mushy zones

0S dimensionless location of heating surface T temperature ( K ) t time ( s ) u liquid velocity in X direction ( sm / )

bu scanning velocity of laser beam ( sm / ) U dimensionless liquid velocity in the X-direction

bU dimensionless scanning velocity of laser beam v liquid velocity in the Y-direction ( sm / ) V volume ( 3m ) or dimensionless liquid velocity in Y direction w liquid velocity in the z-direction ( sm / )


sw velocity induced by shrinkage ( sm / ) W dimensionless liquid velocity in the Z-direction

sW dimensionless velocity induced by shrinkage zyx ,, coordinates ( m ) ZYX ,, dimensionless coordinates

Greek symbol α thermal diffusivity ( 2 /m s )

aα absorptivity θ dimensionless temperature

T∆ one-half of phase change temperature range ( K ) θ∆ one-half of dimensionless phase change temperature

range ε porosity (volume fraction of void ), )/()( slglg VVVVV +++

eε emissivity µ dynamic viscosity ( smkg ⋅/ ) ρ density ( 3/ mkg ) σ Stefan-Boltzman constant ( 428 /1067.5 KmW ⋅× − ) τ dimensionless time

gϕ volume fraction of gas, /( )g g sV V V V+ +

ϕ volume fraction of liquid, /( )g sV V V V+ +

sϕ volume fraction of solid, /( )s g sV V V V+ +

ψ saturation Subscripts eff effective g gas f fusion p powder material s solid

MATHEMATICAL MODEL Problem description

HAZ

Liquid Pool

Unsintered Zone

xy

z z

x

y

Laser Beam

zD

yD

xD

Velocity, Ub

Fig. 1 Physical model

Consider a powder bed with initial temperature, iT , below the melting point, fT , in a cavity with a size of DDD zyx ××2

(length width height) as illustrated in Fig. 1. A Gaussian laser beam scans the surface of the powder bed in the positive x-direction with a constant velocity, bu . The liquid pool forms when the laser beam with high energy irradiates the powders and scans the surface of the powder bed. Then the liquid pool cools and resolidifies to form a densified heat affect zone (HAZ). Since the diameter of laser beam scanning the surface of powder bed is very small compared to the size of the powder bed, the sintering process appears to be quasi-steady state from the standpoint of the laser beam. Consequently, the SLS of metal powder becomes a three-dimensional steady-state problem in a coordinate system that is attached to and move with the laser beam. Since laser processing of metal powder is a very rapid process, the liquid layer and solid core of a partially molten powder particle may not at thermal equilibrium and have different temperatures during melting. That is, the liquid surface of a partially molten powder particle has a higher temperature than melting point while the solid core has a temperature below the melting point [23]. Therefore, if the temperature of the partially molten particles is represented by a single temperature, it is within a range of temperature adjacent to the melting point, depending on the degree of melting. Thus the melting of the single component metal powder bed can be modeled as melting occurring in a range of temperature, instead of a fixed temperature point. The liquid infiltrates into the void in the unsintered region of the powder bed which affects the melting process and causes the density change. The non-molten cores of the particles cannot sustain the powder bed and will move downward to cause shrinkage of the whole powder bed. A constant porosity model [12] is employed to predict the melting problem. During the melting process, the following assumptions are made:

1. The density, specific heat and thermal conductivities of both liquid and solid phases of the powder material are the same and are independent of the temperature. 2. The contribution of the gas to the density and heat capacity to the powder bed is negligible. 3. The velocity of the solid induced by the shrinkage of the powder bed is only in the direction and the liquid flows in all directions.

Velocities and fractions in the mushy zone Once partial melting in the powder bed is initiated, the

volume initially occupied by the gas will be partially taken up by the liquid because only solid can sustain the powder structure. The porosity, defined as the total volume of void, including the volumes of gas and liquid, relative to the total volume occupied by the solid matrix and void volume [ ( )/( )g g s gV V V V Vε ϕ ϕ= + + + = + ; 12, 24], remains constant while

volume of the bed continuously changes because the gas is driven out by the liquid generated during melting. In other words, the volume of the gas driven out from the powder bed is


equal to the volume of liquid produced during partial melting in order to maintain a constant porosity. The shrinkage accompanying the partial melting will induce a downward velocity in the z-direction.

The degree of melting can be measured by the solid fraction, f : a value of 1 indicates full solid, 0 indicates full liquid, and the value between 1 and 0 indicates mixture of solid and liquid. The solid fraction is related to the local temperature by

T

TTTf f

∆

−∆+=

2 (1)

It should be pointed out that the constant porosity model is valid only if the solid fraction, f , is less than or equal to 1 ε− . Since the porosity, ε, is a constant in the constant porosity model, the volume fraction of unmelted solid,

εϕ −=1s , remains unchanged during partial melting. According to the definitions of the solid mass fraction,

/( )p s p s pf V V Vρ ρ ρ= + , and the liquid volume fraction,

/( )s gV V V Vϕ = + + , the following equation can be obtained:

(1 ) (1 )(1 )sf ff fϕ εϕ

− − −= = (2)

The liquid velocities must satisfy the following continuity equation

( ) ( )( ) 0s s swt z

ϕ ϕ ϕϕ

∂ + ∂+∇ ⋅ + =

∂ ∂v (3)

Substituting εϕ −=1s into eq. (3), one obtains

( ) (1 ) 0swt zϕ ϕ ε

∂∂+∇ ⋅ + − =

∂ ∂v (4)

The Darcy’s law is utilized to obtain the liquid velocities in three directions,

( )rs c

KKw p gρϕ µ

− = ∇ +v k k (5)

The permeability, K , is related to the diameter of the solid core, pd , and porosity, ε , by Carman-Kozeny equation [17]:

2 3

2180(1 )pd

Kεε

=−

(6)

The relative permeability, rlK , is equal to 3e

ψ [17], where is

the normalized saturation:

⎪⎩

⎪⎨⎧

≤

>−−

=

ir

irir

ir

eψψ

ψψψψψ

ψ0

1 (7)

The capillary pressure is 2[1.417(1 ) 2.12(1 )c e ep ψ ψ= − − − 3 01.263(1 ) ] /e Kψ γ ε+ − [24],

where the surface tension of liquid metal, 0γ , is expressed as a

linear function of temperature, 0 0[1 ( )]m fT Tγ γ γ ′= − − [17].

Energy Equation The temperature transforming model which combines the

advantages of the enthalpy model and equivalent heat capacity

models is used to describe the heat transfer process during melting and resolidification [25]. The energy equation for the physical problem is [ ] [ ]( ) ( )s s sC T C T w C T

t zϕ ϕ ϕ ϕ∂ ∂

+ + ∇ ⋅ +∂ ∂

v

[ ] [ ]( ) ( ) ( )s s sk T b b w bt z

ϕ ϕ ϕ ϕ∂ ∂⎧ ⎫= ∇ ⋅ ∇ − + +∇ ⋅ +⎨ ⎬∂ ∂⎩ ⎭

v (8)

The effective heat capacity can be expressed as

2

p p f

p sp p f f

p p f

c T T Th

C c T T T T TT

c T T T

ρρ

ρ

ρ

≤ − ∆⎧⎪⎪= + − ∆ < < + ∆⎨ ∆⎪

≥ + ∆⎪⎩

(9)

and b in eq.(8) is defined as

0

12

f

p s f f

p s f

T T T

b h T T T T T

h T T T

ρ

ρ

≤ − ∆⎧⎪⎪= − ∆ < < + ∆⎨⎪

≥ + ∆⎪⎩

(10)

The thermal conductivity of the powder bed is calculated by

( )2

eff f

effeff f f f

f

k T T Tk k

k k T T T T T T T TT

k T T T

≤ − ∆⎧⎪ −⎪= + − + ∆ − ∆ < < + ∆⎨

∆⎪≥ + ∆⎪⎩

(11) where

effk is the effective thermal conductivity of the unsintered

powder bed. It can be calculated using the empirical correction proposed by Hadley [26]. When melting begins, the liquid infiltrated between the unmolten cores of the particles and the contact area is significantly increased. The thermal conductivity of the liquid pool and resolidified part is therefore calculated using

( )s pk kϕ ϕ= + (12)

The boundary and initial conditions of the energy equation are

( )2 2

4 42 2exp ( )a

ePT x yk T T h T T

z R Rα

ε σπ ∞ ∞

⎡ ⎤∂ +− = − − − − −⎢ ⎥∂ ⎣ ⎦

(13a) ,0=

∂∂

zT Dzz = (13b)

,0=∂∂

xT Dxx ,0= (13c)

,0=∂∂

yT Dyy ,0= (13d)

,iTT = 0=t (13e) Dimensionless governing equations

By defining the following dimensionless variables

2Rtpα

τ = , R

zyxZYX ),,(,, = , if

f

TTTT−−

=θ , if TT

T−∆

=∆θ ,

pk

kK = , p

effeff k

kK = , ( )s

p

CCc

ϕ ϕρ+

= , ( )( )

s

p f i

bB

c T Tϕ ϕ+

=−

,


( )p s

u RU ϕα ϕ ϕ

=+

, ( )p s

v RV ϕα ϕ ϕ

=+

, ( )( )

s s

p s

w w RW ϕ ϕα ϕ ϕ

+=

+,

/m

pPkγ ε

= , ( )p f i

s

c T TSc

hρ−

= , )( ifp

ai TTRk

PN

−=

α ,

3( )

( )e f i

p f i

R

T T RN

Rk T T

ε σ

π

−=

−,

if

ft TT

TN

−= ,

pkhRBi = ,

p

bb

RuU

α= (14)

The energy equation (8) is nondimensionalized as )()()()()(

XK

XZWC

YVC

XUC

XCUb ∂

∂∂∂

=∂

∂+

∂∂

+∂

∂+

∂∂

−θθθθθ

)()(Z

KZY

KY ∂

∂∂∂

+∂∂

∂∂

+θθ ( ) ( ) ( )

bB UB VB WBUX X Y Z∂ ∂ ∂ ∂⎧ ⎫− − + + +⎨ ⎬∂ ∂ ∂ ∂⎩ ⎭

(15) Where

( )1( )(1 )

2( )

s

s

s

C TSc

ϕ ϕ θ θ

ϕ ϕ θ θθ

ϕ ϕ θ θ

+ ≤ −∆⎧⎪⎪= + + −∆ < < ∆⎨ ∆⎪

+ ≥ ∆⎪⎩

(16)

0

2s

s

B TSc

Sc

θ θϕ ϕ

θ θ

ϕ ϕθ θ

⎧⎪ ≤ −∆⎪

+⎪= −∆ < < ∆⎨⎪

+⎪≥ ∆⎪⎩

(17)

( )( )

2( )

eff

s effeff

s

KK

K K T

θ θϕ ϕ

θ θ θ θθ

ϕ ϕ θ θ

≤ −∆⎧⎪ + −⎪= + + ∆ −∆ < < ∆⎨ ∆⎪

+ ≥ ∆⎪⎩

(18)

where the dimensionless velocities of the mushy zone, which are calculated by the dimensionless form of Eq. (5)

3 2 3

2180(1 )180(1 )e o e

s cMa MaBW Pε ψ ε ψ

ε ψε ψ− = ∇ +

−−v k k (19)

Equation (19) should satisfy the nondimensional continuity equation of the mushy zone, which can be obtained by nondimensionalizing eq. (4):

( ) (1 ) 0sWZ

ϕ ϕ ετ

∂∂+∇⋅ + − =

∂ ∂v (20)

The boundary condition at the top surface of the powder bed, eq. (13a) is nondimensionalized as

[ ] [ ]4422 )()(exp ttRi NNNYX

NK +−+−−−=∂∂

− ∞θθπτ

θ )( ∞−− θθiB

),(0 YXSZ = (21) The dimensionless form of other equations can also be obtained using the dimensionless variables defined in eq. (14). NUMERICAL SOLUTION

The melting and resolidfication problem governed by eqs. (15-21) is a steady-state three-dimensional, nonlinear problem. Since the locations of the solid-liquid interface and the surface of mushy zone are unknown a priori, a false transient method is employed to locate various interfaces. Steady-state solution is declared when the temperature distribution and locations

various interfaces do not vary with the false time. Equation (16) with the false transient term can be solved by the finite volume method [27]. The computational domain is the whole powder bed and a block-off technique [27] is employed to simulate the existence of the empty space created by shrinkage of the powder bed after melting. The density and thermal conductivity in the empty space are set to be zero. The power-law scheme is applied to discrete the convection-diffusion terms.

Due to axisymmetry with respect to the center plan (Y=0), the velocity and temperature fields were calculated on only one side of the center plan (Y>0). The analysis of the sintering process involves solutions of: (1) the velocities and the volume fractions of the solid particles; (2) the velocities and the volume fractions of the liquid; (3) the temperature distribution and the location of the surface of mushy zone, the solid-liquid interface and the sintering interface. Since the eq. (16), (20) and eq. (21) are conjugated, an iterative solution procedure with underrelaxation is employed and the underrelaxation factor is 0.2. The grid number used in computation is 122×52×122 (in the X, Y, Z directions respectively) and the false time step is 0.1. A non-uniform grid was used, i.e., fine grids near the heat source and coarse grid away from the center of the laser beam.

RESULTS AND DISCUSSIONS

The porosity of the unsintered powder bed plays an important role since the thermal properties of the powder bed depend on the value of the porosity. For uniform-sized spherical particles, the porosity is independent of the particle size but relates to only the structural arrangement of the particles in the powder bed. As to simple-cubic (6 contact points), face-centered cubic (8 contact points) and body-centered cubic (12 contact points) close-pack arrangements, the porosities are 0.476, 0.32, and 0.26, respectively. For randomly packed powder bed, the porosity is around 0.4 [24]. Therefore, the initial porosity of the unsintered powder bed is set as 0.4.

In order to demonstrate the validity of the simulation code, the calculation of temperature distribution for pure conduction in the powder bed with a moving laser beam is made to compare with the analytical solution for the pure conduction in a semi-infinite body with a moving point heat source [28].

⎥⎦⎤

⎢⎣⎡ +−

=−απ

α2

)(exp2

xrurk

PTT

eff

ais

(22)

Using the dimensionless variables defined in eq. (14), the dimensionless form of eq. (22) can be rewritten as the following

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +−−=+

eff

b

eff

i

KXRU

RKN

2)()1(

exp2

1ε

πθ (23)

where 222 ZYXR ++= . To simulate the conduction problem in the powder bed

with a moving point heat source on the surface, The Gussian distribution laser beam is replaced by a top-hat (uniform distribution) laser beam because the former one diffuses the


energy in a wide circle. Thus the boundary condition illustrated in eq. (21) is changed to the following equation:

2 2

2 2

0 1, 0

0 0 1, 0i

eff

N Z X YK

Z Z X Y

τθ

τ

⎧ = + ≤ >∂ ⎪− = ⎨∂ = + > >⎪⎩

(24)

As can be seen from Fig. 2, the agreement between the numerical and analytical solutions is very good except area around 0=X . This deviation is caused by the difference between the two heat sources used in the numerical and analytical models. The one used in analytical model is infinitesimal while the one used in numerical model is of finite size. Considering this point, the overall agreement between the numerical and analytical solution is good.

X

-20 -10 0 10 20

θ W

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Numerical SolutionAnalytical Solution

Ni=0.1Ub=0.18

ε=0.4

Fig. 2 Comparison between analytical and numerical solutions for pure conduction

Table 1 Thermophysical properties of AISI 304

Nomenclature symbol unit value

Density pρ kg/m3 7200

Thermal conductivity pk W/m·K 14.9 Specific heat

pc J/kg·K 462.6 Melting point

fT K 1670 Latent heat of fusion sh kJ/kg 247 Viscosity µ kg/s·m 0.05 Surface tension at melting point γ N/m 1.943 Temperature range T∆2 K 160

Numerical calculation is performed for sintering of AISI

304 stainless steel powder. Table 1 lists the material physical properties for AISI 304 [29-30]. In order to minimize the balling effect and geometric distortion (curling or delaminating), a pre-heated powder bed with initial temperature of 380°C is used in the simulation and the corresponding subcooling parameter is Sc=2.0. Since partial melting of the powder particle is desirable, selection of laser processing parameters must be strict in order to ensure not complete but

only surface melting of particles. Therefore, the heat source intensity and the scan velocity at which sintering occurs vary over a narrow range.

Figure 3 illustrates the surface temperature distribution of the powder bed. The dimensionless intensity of laser beam iN is 0.19 and the dimensionless scanning velocity bU is 0.12. It can be seen that the peak temperature at the powder bed surface is near the trailing edge of the laser beam rather than at the center of the laser beam due to motion of the laser beam. Because the thermal conductivity in the molten pool is much larger than that in the unsintered zone, the temperature changes smoothly in the molten pool while sharply in the unsintered zone near the molten pool. Figure 4 shows the three-dimensional shape of the powder bed surface, molten pool and HAZ subjected to irradiation of a moving laser beam in the moving coordinates system at the same condition of Fig. 3. The shrinkage at the track of the center of the laser beam is most significant and a groove is formed on the surface of the powder bed. And the depths of surface and HAZ decrease with the increasing X in the laser scanning direction Y.

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

-4

-2

0

2

4

0.51.0

1.52.0

2.5

T

X

Y Fig. 3 Three-dimensional surface temperature distribution

on the powder bed surface

0.0

0.1

0.2

0.3

0.4

0.5

-2.0-1.5

-1.0-0.5

0.00.5

1.01.5

2.0

0.0

0.5

1.0

1.5

S

X

Y

Fig. 4 Three-dimensional shape of the HAZ

( 19.0=iN , 12.0=bU )


Figure 5 (a) and 5 (b) illustrate the contour of solid fraction at the surface of the powder bed and the X-Z plane at the same condition as Fig. 3. The solid fraction in the liquid pool represents the mass fraction of the unmelted solid in the total mass of the powder materials. In the resolidified region that contains resolidified solid and the solid core that did not undergo melting and resolidification, the solid fraction represents the fraction of solid did not undergo melting and resolidification. It can be seen from Fig. 5(a) that the solid fraction contour is in a horse shoe shape. The solid fraction near the center of the laser beam is the lowest where the degree of melting is highest. As required by constant porosity model, the shrinkage at the center of the laser beam is also the most significant. As the result of the laser scanning, what left behind the surface of the powder bed is a groove (see Fig. 4) and the density of the HAZ at the bottom of the groove is highest. The solid fraction contour in the X-Z plane is shown in Fig. 5(b). It can be seen that the solid fraction near the surface of the heat

0.960.920.880.840.800.760.72

X

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Y

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Solid Fraction Contour

Sc=2.0

Ni=0.19

Ub=0.12

(a) Powder bed surface

0.960.92

0.880.84

0.80

0.76

0.72

X

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0.0

0.1

0.2

0.3

0.4

0.5


Sc=2.0

Ni=0.19

Ub=0.12

Z

(b) X-Z plane

Fig. 5 Contour of solid fraction ( 19.0=iN , 12.0=bU )

affected zone is the lowest because the degree of melting at the surface is highest.

Figure 6 (a) and (b) present the cross-sectional shapes of surface, molten pool and HAZ at X-Z and X-Y planes at

19.0=iN . It can be seen that, the depths of the surface and HAZ decrease and the molten pool becomes narrower with the increasing scanning velocity because the interaction time between the moving heat source and the powder surface is decreased when the heat source moves faster. In the meantime, the whole molten pool moves towards the opposite direction of the laser scanning as the scanning velocity increases. Figure 7 (a) and 7 (b) illustrate the contour of solid fraction at the surface of the powder bed and X-Z plane at 19.0=iN and

24.0=bU . Compared with Fig. 5, the solid fraction contour shifts towards the opposite direction of the laser scanning because the enhanced advection flow caused by the increasing scanning velocity. As the interaction time of the laser beam on laser beam and powder bed surface shortens with the increasing scanning velocity, the degree of melting decreases so the curve with equal solid fraction near the center of the laser beam, such as 72.0=sf , is pushed inward.

X

-2 -1 0 1 2

S0,S

0.0

0.1

0.2

0.3

0.4

0.5

Ub=0.12Ub=0.18Ub=0.24

Sc=2.0

Ni=0.19

S0

S

(a) X-Z plane

Y

-2 -1 0 1 2

S0,S

0.0

0.1

0.2

0.3

0.4

0.5

Ub=0.12Ub=0.18Ub=0.24

Sc=2.0Ni=0.19

S0

S

(b) X-Y Plane

Fig. 6 Cross-sectional shape of surface, molten pool and HAZ ( 19.0=iN )


0.960.920.880.840.800.760.72

X

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Y

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5


Sc=2.0Ni=0.19

Ub=0.24


0.960.920.88

0.84

0.80

0.76

0.72

X

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Z

0.0

0.1

0.2

0.3

0.4

0.5


Sc=2.0N i=0.19

U b=0.24

(b) X-Z Plane


Figure 8 (a) and (b) show the cross-sectional shapes of

surface, molten pool and HAZ at X-Z and X-Y planes at 196.0=iN . Similar to the trend demonstrated in Fig. 6, the

depths of surface, molten pool and HAZ decrease and the molten pool becomes narrower with the increasing scanning velocity. The whole molten pool shifts towards the opposite direction of the laser scanning as the scanning velocity increases. Fig.6 shows that the sintering depth increases with the increasing laser intensity. Since shrinkage is assumed to occur in the Z-direction only, increase of the depth of molten pool is much more significant than the increase of the molten pool width in the Y- direction. Figure 9 (a) and 9 (b) show the contour of solid fraction at the powder bed surface and X-Z plane at 196.0=iN and 24.0=bU . Compared with Fig. 7, as

the laser intensity increases, more solid is melted, thus the curve with equal solid fraction near the center of the laser beam, such as 72.0=sf , is pushed outward.

X

-2 -1 0 1 2

S0,S

0.0

0.1

0.2

0.3

0.4

0.5

Ub=0.18Ub=0.24Ub=0.30

Sc=2.0Ni=0.196

S0

S

(a) X-Z plane

Y

-2 -1 0 1 2

S0

0.0

0.1

0.2

0.3

0.4

0.5

Ub=0.18Ub=0.24Ub=0.30

S

S0,S

Sc=2.0Ni=0.196

(b) X-Y plane


0.960.920.880.840.800.760.72

X

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Z

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5


Sc=2.0Ni=0.196Ub=0.24



0.960.920.880.84

0.80

0.76

0.72

X

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Z

0.0

0.1

0.2

0.3

0.4

0.5


Sc=2.0Ni=0.196Ub=0.24

(b) X-Z Plane


X

-2 -1 0 1 2

S0,S

0.0

0.1

0.2

0.3

0.4

0.5

Ub=0.24Ub=0.30Ub=0.36

S0

S Sc=2.0Ni=0.2

(a) X-Z plane

Y

-2 -1 0 1 2

S0,S

0.0

0.1

0.2

0.3

0.4

0.5

Ub=0.24Ub=0.30Ub=0.36

S0

S Sc=2.0Ni=0.2

(b) X-Y plane


Figure 10(a) and (b) show the cross sectional shapes of

surface, molten pool and HAZ in the X-Z and X-Y planes at 2.0=iN . The trends similar to Fig. 6 and Fig. 8 can be

observed in Fig. 10. Figure 11 (a) and (b) illustrate the contour of solid fraction in surface of the powder bed and X-Z plane at

2.0=iN and 24.0=bU . Compared with Fig. 9, as the laser intensity increases, more solid is melted, thus the curve with equal solid fraction near the center of the laser beam such as

72.0=sf is pushed outward.

0.960.920.880.840.800.760.720.68

X

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Y

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5


Sc=2.0

Ni=0.20

Ub=0.24


0.920.880.84

0.80

0.76

0.720.68

X

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Z

0.0

0.1

0.2

0.3

0.4

0.5


Sc=2.0

Ni=0.20

Ub=0.24

(b) X-Z Plane



Figure 12 shows the sintering depths (at Y=0) for different scanning velocities and laser intensity. It can be seen that the sintering depth increases with increasing laser intensity or decreasing scanning velocity. At fixed laser intensity, reduced scanning velocity leads to increased laser-powder interaction time and ultimately results in greater sintering depth. On the other hand, when the scanning velocity is fixed, i.e., the laser-powder interaction time is fixed, increasing laser intensity results in greater sintering depth amount of laser energy delivered to the powder bed is increased.

Ub

0.10 0.15 0.20 0.25 0.30 0.35 0.40

S

0.28

0.30

0.32

0.34

0.36

0.38

0.40

0.42

Ni=0.19Ni=0.196Ni=0.20

Fig. 12 Sintering depths at different laser intensities and scanning velocities

CONCLUSION Three-dimensional sintering of single component metal

powder by partial melting with moving laser beam is investigated numerically. Since the liquid and solid of the partially molten particles during the sintering process is not at the same temperature, melting during sintering is assumed to occur over a range of temperature instead of a fixed point. Numerical simulation of sintering of AISI 304 powder is carried out and a parametric study is performed. The results showed that the laser intensity and scanning velocity play the important roles in the temperature distribution, the shapes of the sintering depth and the solid fraction distribution. Decreasing laser intensity resulted in decrease of the sintering depth and the volume of the molten pool. Increasing the scanning velocity decreased the sintering depth and the location and shape of the molten pool is affected significantly.

ACKNOWLEDGMENTS Support for this work by the Office of Naval Research

(ONR) under grant number N00014-04-1-0303 is gratefully acknowledged.

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