Multi-scale Multi-physics Modeling of Electron Beam

     

      

  

 

   

          Wing Kam Liu 

)

Multi-scale Multi-physics Modeling of Electron Beam Selective Melting Process

Thesis Submitted to Tsinghua University in partial fulfillment of the requirement for the professional degree of Doctor of Philosophy by Wentao Yan ( Material Science and Engineering )

Thesis Supervisor : Professor Feng Lin Cooperate Supervisor : Professor Wing Kam Liu

March, 2017

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Abstract

Abstract Metallic Additive Manufacturing technologies have proven to be very promising in recent years, which is believed to lead a new technological revolution. However, the manufacturing process, which consists of multiple complex physical phenomena over a broad range of time and length scales, poses a significant challenge for accurate experimental observations and measurement. Moreover, the process involves tens of parameters that could affect the manufacturing quality. Therefore, the quality inconsistency is the bottleneck for the wide industrialization of Additive Manufacturing technologies. In this dissertation, a multi-scale multi-physics modeling framework is proposed to simulate the Electron Beam Selective Melting (EBSM) process, which is a typical metallic powder-based Additive Manufacturing process. To the best of my knowledge, this is the first report of multi-scale model for the EBSM process. This framework mainly consists of three models ranging from micro-scale to meso-scale to macro-scale. • A micro-scale electron-atom interaction model using the Monte Carlo method, is aimed to deriving a new heat source model for the electron beam, by tracking the collision and energy transition between electrons and atoms. The heat source model, named as "double-Gaussian" heat source model, is material-dependent and experimental set-up specific, which is able to guide the process design and provide insight into uncertainty in experiments. • A meso-scale powder evolution modelwhich incorporates multiple physical phenomena and takes into a variety of influencing factors, is capable of reproducing the complex melt-flow-solidify process of individual powder particles. The formation mechanisms of the single track defects, including balling effect and single track distortion, are systematically investigated, which reveals the dominance of surface tension. The melting processes along various scan paths in multiple powder layers are reproduced in 3D simulations to reveal the influence of successive tracks and layers, which is the first report. Thanks to the multi-layer multi-track simulation, the energy efficiency is quantitatively discussed, and the formation mechanism of surface roughness is studied. Additionally, a simplified powder-scale heat transfer model is developed to provide rapid prediction if powder particles can fully melt under given process parameters. II

Abstract

• A macro-scale heat transfer model, in which the loosely packed powder bed is simplified as an effective continuum material, is a powerful tool to rapidly reproduce the experimental fabrication process. To ensure a fair prediction accuracy, the simplifications are made based on the meso-scale simulations. Thus, the macroscale model is more physically-enriched and predictive for more quality indices, which is validated by experiments. Besides shedding light on the fundamental physical mechanisms, the multi-scale model is feasible in design and optimization of the manufacturing process. A simulationdriven parameter selection scheme is proposed, and then demonstrated for an example case of the Functionally Graded Materials, which agrees well with experimental results. Key words: Additive Manufacturing; Electron Beam;Multi-scale Modeling; Heat Transfer; Molten Pool Flow

III





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1.1    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.2  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.1

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1.3.2

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2

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2.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.3 

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2.2.4  Marangoni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.5 $ # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.6

 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.2   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.3    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 "  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4.2

 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 IV





3.4.4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5 "  % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.5.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.5.2 "   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5.3 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.6  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4

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4.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2.1 "   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2.2



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4.2.3 B B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2.4

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4.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3.2   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3.4   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4.3   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4.4

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.6   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.7  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.2  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 V





5.4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6

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6.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.2     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.3    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.3.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3.2 8   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7

 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.2  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 

 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110



 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

VI

2

 AM

 (Additive Manufacturing)

EBSM

Electron Beam Selective Melting

SLM

Selective Laser Melting

LENS

 Laser Engineered Net Shaping

FEM

Finite Element Method

FVM

 Finite Volume Method

VOF

  Volume of Fluid

CFD

Computational Fluid Dynamics

FGM

Functional Graded Material

Q



q

 

T



t



Rb

 

P



w1



w2



wh

Hatching distance

VII

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'   rain drop model %     



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# [61] %2016%



%

[60]

Cellular AutomatonCA ) [62] *    Lawrence Livermore National Lab LLNL Wayne King [38]  LLNL$BALE3D %  8     [37,44,63]    %   $       $

 #  

Marangoni effect %   denudation* LLNL

    "% 

1.6   [64] a  b  +

ESIB 3 BCFD-ACE+* %SLM [39]    '

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% LENS [64]   1.6  

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SLMEBSM6   SLS ' [71] *     &6-? ?

 

 

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#  . %   -?&   +  - &  16

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/  /"        #   * =    * =  & /  **     (  

  /  / $   



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/    ? ?    #  # [77]  !  *    [78]   

 ##      Monte Carlo  +   17

2  

 3 2.2.2 

   

1



 ∂ −v I) = q +  · (kT) (ρI) +  · (ρ→ ∂t ρTTk  

(2-1) −v 

→

  q I

'  I = cT + (1 − fs )L

(2-2)

c L  fs        B   * ?   B kT = αb σb (T 4 − Ta4 )

(2-3)

αb $

σb = 5.67×10−8 W/( m2 · K4 )Stefan-Boltzmann

Ta   *            -?&

  '   (  

B   '

 "& '





  " %        &     

18

2  

2.2.3 

 -& %    

?B

1.  2-4  −v ) = 0  · (ρ→ 2.

(2-4)

Navier-Stokes 2-5  ∂ → −v ⊗ → −v ) =  · (μ→ −v ) − P + ρ→ −g (ρ−v ) +  · (ρ→ ∂t

(2-5)

3.   2-1 B 



       %       −g      Marangoni    #    2-5 P   →  μ

Marangoni #

B 8 2.2.4  Marangoni

%     2.2a 1 P  = 2 × σs sin( dθ)/r dθ = σs /r 2

(2-6)

r  !  σs 

 %

   2-7 σs = σs0 − σsT (T − T0 )

(2-7)

σs0 T0 

σsT '

 '1K'  # 2.2b $ Marangoni 19



2  

2.2

a

bMarangoni T1 > T2

 * *  *-? $* -?&   ( ' [79] " Marangoni

   *

B$  $ -  Marangoni"  #



 [76] & Marangoni9 2.2.5

 

  # $





 [76] 

# Klassen4 [61]



%$ %.   "   :  ($    

$



$#

%     $      

%

-? !

 

$    $    •    + 20

2  

•  NV =

N  % V

   02-8  

2-9 v x = v y = vz = 0

(2-8)

1 vx2 = vy2 = vz2 = v 2 3

(2-9)

- (1 *  ΔA $  2mvix  Δt, 12 NV i ΔAvix Δt  $   0

 1 (NV i ΔAvix Δt) × (2mvix )  1 2 2 = mNV i vix P= = NV mv 2 ΔtΔA 3 i i

(2-10)

mvix x NV i  vi    

N R nRT = T = NV kT V V NA

P=

(2-11)

n molR = 8.31J/(mol · K)

,N A = 6.02 × 1023  

k =

R NA

= 1.38 × 10−23 

-

%    

 v2

1 

=

++

3kT = m



3RT = mN A



3RT M



21

(2-12)





2  

2.3

 0

-

  T

F(v ) =

d3 N  N dvx dvy dvz

 f (v) = g(vx ) =

dN xyz N dv dN g(vy ) = NdN g(vz ) = NdN  -  N dvx dvy dvz

+- F(v )dvx dvy dvz =

d3 N = g(vx )dvx · g(vy )dvy · g(vz )dvz N

F(v ) = g(vx )g(vy )g(vz )

(2-13)

(2-14)

 % ?F+ *  F(vx2 + vy2 + vz2 ) = g(vx )g(vy )g(vz )

(2-15)

- * g(vx ) = a1 exp(a2 vx2 ) F(v ) = a1 exp(a2 vx2 ) · a1 exp(a2 vy2 ) · a1 exp(a2 vz2 ) = a13 exp(a2 v 2 )

(2-16)

* *   f (v) = F(v ) · 4πv 2 2-162-17

a1 a2 ?B 22

(2-17)

2  

∫∞ • B 0 f (v)dv = 1 ∫   ∞ •   v 2 = 0 f (v)v 2 dv = 3kT/m 2-12 a1 = m/2kT πa2 = −m/2kT(% 2-18  ⎧ m 3/2 ⎪ F(v ) = ( 2πkT ) exp(−mv 2 /2kT) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m 3/2 2 ⎪ ⎪ ) v exp(−mv 2 /2kT) f (v) = 4π( 2πkT ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ m 1/2 g(vx ) = ( 2πkT ) exp(−mvx2 /2kT) ⎪ ⎪ ⎪ ⎪ ⎪ m 1/2 ⎪ ⎪ g(vy ) = ( 2πkT ) exp(−mvy2 /2kT) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ g(vz ) = ( m )1/2 exp(−mv 2 /2kT) z ⎩ 2πkT

(2-18)

    Ps $ $

   *: 

 $ Γe f f = Γvap − Γdep = 0   $  +x vx < 0  2-10 ∫ Γdep = mNV

0

−∞





$  

g(vx )vx dvx = Ps

m 2πkT

(2-19)

T-($ "$   

10−3 Pa "  

$  



& $ 9T $ Ps  $ "& * √2πmkT ∫0  2-19 −∞ g(vx )vx dvx   

Nv   2-10-   $   Pa   Pa  $ * √2πmkT

   Pa *Ps − Pa  +(Pa  $  Pa   √2πmkT

23

2  

$   Γe f f = (Ps − Pa )

m = αPs 2πkT



m 2πkT

(0  α  1)

(2-20)

 Langmuir α$

α = 0 α = 1   

$ $+ 

$

=$$ 6  & $  Knudsen Kn = λ/Lλ    L     Kn  [80]  • 0.01 < Kn < 0.1Kn = 0.01  

 [81]   9

8

 8

     ; • 0.1 < Kn < 10      • Kn > 10    -   

    

 2.4A Δt   v = ∫∞ √ f (v)vdv*;   2v $ 0 √ 2   2vΔtπd NV -   λ=√

vΔt

1 kT =√ =√ 2vΔtπd 2 NV 2πd 2 NV 2πd 2 P

(2-21)

d Ti  4×10−10 m Al  3.6×10−10 m ?ARCAM 3   0.2Pa0-10−3 Pa [82]  -3000K(Ti-6Al-4V1928K)   0.16m-0.24m 0.5mKn = 0.32 − 0.48-32m-48m* Kn = 64 − 96 [61] $  *Kn     $  " 24

2  

2.4

  

  -Gibbs   *      'Gibbs '0 dGl dG v = dT dT

(2-22)

dG dP = −S + V dT dT

(2-23)

Gibbs 

-     Clapeyron*  ' dPs Sv − Sl Hv − Hl = = dT Vv − Vl T(Vv − Vl )

(2-24)

Hv Hl Vv Vl  * Vv  Vl Vv = RT/Ps  dPs Ps (Hv − Hl ) = dT RT 2

(2-25)

 Clausius-ClapeyronClapeyron - dPs (Hv − Hl ) = dT Ps RT 2

25

(2-26)

2  

Hv Hl    

   Lvap = (Hv − Hl )/M

-

Lvap M 1 1 ( − ) Ps = Ps0 exp R T0 T

(2-27)

Ps0 T0 2-27 2-20-   Lvap M 1 m 1 ( − ) − Pa = Ps0 exp R T0 T 2πkT 

Γe f f



(2-28)

 %?'%$ * 

2.5

 #

 2.5  :0 *   

  -?

$ 2-29 2-12 -# 2-30 Precoil = Γe f f

Precoil

1 = Γe f f · 3



 1 2 · v 3

3kT 1 = Γe f f m 3

(2-29)



3RT M

  * 

(2-30)

  

# *     +$ "  . [83]  [84] #.   26

2  





-! *-$

 ;       + $ 2.2.6 

 Ti-6Al-4V  2.1 [57]-  ?%    ?  2.1

Ti-6Al-4V

properties

value

Density at liquidus state (ρl )

4000 kg/m3

Solidus temperature (Ts ) Liquidus temperature (Tl ) Latent heat of melting (Lm ) Latent heat of evaporation (Lv ) Saturated vapor pressure (Ps0 ) at T0 =3315K

1878K 1928K 2.86×105 J/kg 9.7×106 J/kg 1.013×105 Pa

Specific heat (c) Thermal conductivity at solidus state (k) Thermal conductivity at liquidus state (k) Surface radiation coefficient (αb ) Surface tension coefficient (σ)

872J/(K·kg) 16W/(m·K) 32W/(m·K) 0.4 1.68N/m

Temperature sensitivity of surface tension coefficient (σsT ) Viscosity (μ)

0.00026 0.005Pa·s

2.3     • Lagrangian     *  ? '   • Eulerian  

  *   " 

 % 27

2  

2.3.1 

Finite Element Method, FEM  [85] %

  * 

* :  

    -?

   

strong form  weak form ?   ⎧ ⎪ ρc ∂T − ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T =T ⎪

∂ (k x ∂T ) ∂x ∂x

−

∂ (k ∂T ) ∂y y ∂y

−

∂ (k ∂T ) ∂z z ∂z

−Q = 0

in Ω

at ΓT

⎪ ⎪ ⎪ (k x ∂T )nx + (k y ∂T )ny + (k z ∂T )nz = q ⎪ ∂x ∂y ∂z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T(x, y, z, t) = T0 (x, y, z) at t = t0 ⎩

(2-31) at Γq

ΩΓT Γq B B  * * %'

∂T ∂T ∂T ∂ ∂ ∂ ∂T − (k x ) − (k y ) − (k z ) − Q dΩ δT ρc ∂t ∂x ∂x ∂y ∂y ∂z ∂z Ω

∫ ∫   ∂T ∂T ∂T + δT T − T dΓT + δT (k x )nx + (k y )ny + (k z )nz − q dΓq = 0 ∂x ∂y ∂z ΓT Γq ∫



  B   ∫

∂T δT ρc dΩ+ ∂t Ω

∫ Ω

∫ ∫ ∂δT ∂T ∂δT ∂T ∂δT ∂T kx + ky + kz dΩ− δTQdΩ− δT qdΓq = 0 ∂x ∂x ∂y ∂y ∂z ∂z Ω Γq (2-32)

  [86]  1.   2. * %* N(x, y, z)" *  i Ti (t)(x, y, z) T(t) =

n  i=1

28

Ni (x, y, z)Ti (t)

(2-33)

2  

n  ' n ⎧ ⎪ δT = ⎪ i=1 Ni (x, y, z)δTi ⎪ ⎪ ⎪ ⎪ n ∂Ni (x,y,z) ⎪ ∂T ⎪ ⎪ ⎨ ∂x = i=1 ∂x Ti ⎪ n ∂Ni (x,y,z) ⎪ ∂T ⎪ ⎪ = i=1 Ti ⎪ ∂y ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂T = n ∂Ni (x,y,z) Ti i=1 ⎩ ∂z ∂z

(2-34)

 2-34  2-32- 

n  ∂Tj  − Ni (x, y, z)δTi Q − Ni (x, y, z)δTi q ∂t Γq i=1   ∂Ni (x, y, z)   ∂Ni (x, y, z) ∂N j (x, y, z) ∂N j (x, y, z) + δTi k x Tj + δTi k y Tj ∂x ∂x ∂y ∂y   ∂Ni (x, y, z) ∂N j (x, y, z) δTi k z Tj = 0 + ∂z ∂z

Ni δTi N j ρc

  K C / 2-35    ∂Ni ∂N j ⎧ ⎪ K = k + ij ⎪ ∂x x ∂x ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎨ Ci j = ρcNi N j ⎪ n ⎪ ⎪ ⎪ PQ = i=1 Ni (x, y, z)Q ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Pq =  Ni (x, y, z)q Γq ⎩

∂Ni ∂y

ky

∂N j ∂y

+

∂Ni ∂z

kz

∂N j ∂z



(2-35)

   CT + KT = PQ + Pq

(2-36)

3.   4. B 5.   6. ;'  &* -   * '

 '?

  *  29

2  

 B'       BABAQUS  $ABAQUS   DFLUX   USDFLD   UMAT UEL )-)$

   2.3.2 

 Finite Volume Method, FVM  (Control Volume Method)    

    

  

 2.6         :  [87] 

2.6 

  6

  ;    

[88]

?  ∂u ∂u dt + A =0 ∂t ∂x

(2-37)

 f = Au j - ∫

x j+1/2

x j−1/2

∂u + f j+1/2 − f j−1/2 = 0 ∂t

30

(2-38)

2  

j − 1/2 j + 1/2=  j f j−1/2  f j+1/2 =  - ∫

x j+1/2

(u

(n+1)Δt

∫ −u

nΔt

)dx +

x j−1/2

(n+1)Δt

∫ f j+1/2 dt −

nΔt

(n+1)Δt

f j−1/2 dt = 0

(2-39)

nΔt

%      ⎧ nΔt ⎪ ⎪ ⎨ uj = ⎪

∫

x j+1/2 x j−1/2

u nΔt dx

Δx

⎪ ⎪ ⎪ f nΔt = ⎩ j+1/2

(2-40)

∫ (n+1)Δt nΔt

fj+1/2 dt Δt

% " 2-41 u(n+1)Δt − unΔt j j Δt

nΔt

+

nΔt

f j+1/2 − f j−1/2 Δx

=0

(2-41)

  ? nΔt 1.  nΔt unΔt (x) j  ( u

   

 [88]  2. nΔt      unΔt (x)       [nΔt, (n ∫ (n+1)Δt nΔt Au j+1/2 (t)dt  1)Δt]u j+1/2 (t) f j+1/2 = Δt1 nΔt

+

- 3.   2-41((n + 1)Δtu(n+1)Δt j

 &*%  Computational Fluid Dynamics       

 

BFLOW-3D  $  

# 2.3.3



  Volume of Fluid, VOF [89] +!     FVM% %%



     1.   31

2  

2. ' Lagrangian !   

 '*

'

 ? Eulerian"D -Arbitrary Lagrangian-Eulerian, ALE    



!     VOF      * F 0 ≤ F ≤ 1 F = 0 F = 1 0 < F < 1 !    F'$   +   -(      -%(     (Piecewise Linear Interface Calculation, PLIC)   )   [90] 

2.7     F  (nx , ny , nz )%  [90]

  *   2-42% '      [91] 8 

∂F −v ) = 0 +  · (F→ ∂t 32

(2-42)

2  

  VOF**  ;! Marker and Cell, MAC * Level set&   

2.4

 %

  

 +  

 

 

%



Marangoni #8%   *     

  -  '  * 

  

 *!    

33

3

  +

3   3.1   *    $ [92]    [93] &  "   106 K/s?

-106 K/m [50] 

 -

 ' 

    -   

''.    "

 % "-   -?!Rosenthal [94] 1946(/     * 1969Pavelic4 [95] (%    3.1 (a)  794 %  [96] (%  

7   9  [97] *  Goldak [98] 1984(" %( 



 -?(   * 

 -? ;     +%*  &  $



# " *   

  

*

 +  % 

 *"

%    

 66#(    "   (:  ($ ! +    

34

3

  +

3.1    a  [43] b"  [98] c   [97]

3.2

 ! 

B  !

    0!    8!     ! 3.2 $ $  !  1   ( $  ! -   2 &    -      

keV

MeV!/  ! 

 0  3  0 *   *



 '  40 *  * B "B  '

 

'  % 

  BB      1= $ 35

3

  +

   2  

 3



    "& ?  ?   [99] -?  



. !0   !B- $' *   -& "- %    ARCAM 3 %Automatic Beam Calibration [100]   * &     $ ?

  *" [100] *

 (powder smoke) [101]   

-(

 +

3.2

a!   b$B 36

3

  +

3.3  Monte Carlo [102]      3.3 (a)    

  

   *  ! 

  





 3.3 (b) $  $' 

  

% " 

   $        

  

 Gauvin4 [103] ( 

  3-1  ⎧ ⎪ ⎪ ⎨ E j+1 = E j + ⎪ ⎪ ⎪ ⎪ ⎩ E

dE dS

=

dE L dS

−7.8×10−3 ρ Ej

×

n

Ej Ci Zi i=1 Fi ln(1.116( J j

(3-1) + Ki ))

 j j L 

Ci Zi  Ji Ki i   & 

    1  (  3.3 (b)

2 2



  50eV?  3.3 (b) 1 ? • * )  20-100μm   400μm 107  01mA1ns   %107 ? - • Ti-6Al-4VTiAlV&

         6  37

3

3.3

  +

 a!  b +

Ti-6Al-4VTi-47Al-2Cr-2Nb +%    3.1   3.1

Ti-6Al-4VTi-47Al-2Cr-2Nb

Ti-6Al-4V Ti-47Al-2Cr-2Nb

Ti

Al

V

Cr

Nb

W

87.9 56.8

7.4 31.3

4.7 0

0 2.6

0 4.7

0 4.6

3.4   * 60kV

Ti-6Al-4V Monte Carlo  3.4  3.4a  

 





-?(-

20μm*   50um-? (





 "   ? 

*

 

    "   # 





+      38

3

  +

3.4  60kV

Ti-6Al-4V a   bXZ c r d Ze 

XZ  3.4b=    



4-5μm4

      -( ?      *.   39

3

  +

 ( *   "     3.4c d  %  Rb  Zmax Δr = 380.5nmΔz = 134.7nm % r ∼ (r + Δr)z ∼ (z + Δz)% ?*  2 ⎧ ⎪ ⎪ Intensit y(r) = 2.008 × exp(− (r−1450) 2 ) ⎪ 22870 ⎨ ⎪

(3-2)

2 ⎪ Intensit y(z) = 1.542 × exp(− (z−4397) ⎪ 2 ) ⎪ 5712 ⎪ ⎩

 Intensit y% zrnm  %   ∫ r+Δr ⎧ ⎪ ⎪ (Intensit y(r))% = fr (ξ)dξ ⎪ r ⎨ ⎪ ∫ z+Δz ⎪ fz (ξ)dξ (Intensit y(z))% = ⎪ ⎪ z ⎪ ⎩

(3-3)

 fr  fz * nm−1 ΔrΔz * - fr  fz -  60kV

Ti-6Al-4V*  ⎧ ⎪ ⎪ f = ⎪ ⎨ r ⎪ ⎪ f = ⎪ ⎪ ⎪ z ⎩

(I ntensity(r))% Δr

r = 0.000053 × exp(− (r−1450) ) ≈ 0.000053 × exp(− 22870 2) 228702

(I ntensity(z))% Δz

= 0.00114 × exp(− (z−4397) ) 57122

2

2

2

(3-4)

 •  3-4B%  ∫ Rb ∫ 50000nm ⎧ r2 ⎪ ⎪ f dr = 0.000053 × exp(− 22870 r 2 )dr ≈ 1 ⎪ 0 ⎨ 0 ⎪ ∫ Zma x ∫ 20000nm 2 ⎪ f dz = 0.00114 × exp(− (z−4397) )dz ≈ 1 ⎪ z ⎪ 57122 0 ⎪ 0 ⎩ • 'ΔrΔz% '' 40

(3-5)



3

  +

%'Intensit y(r)Intensit y(z)*  $ ' fr  fz * $D' • * Rb    fr  fz * $' -?

   3-6 

1

 q MT −3 L −1 SIW/m3  2η 3 Q M L 2T −3 SIW 4 * fz  L −1 SIm−1  * fr  L −2 SIm−2  3-2-(   

"  :  q = Q × η × fz × fr

(3-6)

3.4.1 

 3-4-?( fr *      * '

 



   )       C C · r2 exp − 2 fr = πRb2 Rb

(3-7)

  C C · [(x − xb )2 + (y − yb )2 ] exp − fr = πRb2 Rb2

(3-8)

 '

xb yb 



 C

 (1 − e

−C

) Rb 

"=r → ∞ fr *09*0"  ?  [104] .  95%C = 3. % 99% 41

3

  +

C = 4.6 % ∫

Rb

∫ fr · 2πrdr =

0

0

Rb

C C · r2 exp(− ) · 2πrdr = 1 − e−C πRb2 Rb2

(3-9)

(  !$" !$B-&   '* &  .-  %#-?      3.4.2  

 3-4-?(z = 4397nm 3.4b* - 

(z − zT (x, y) − z0 )2 fz = ∫ +∞ exp − δ2 δ − z0 exp(−t 2 )dt 1

 (3-10)

δ

 zT (x, y)z z0   δ (  L  1      '' δ ∫ +∞z exp(−t 2 )dt − δ0 ∫

"  * z Fz dz = 1 *Ti-6Al-4V

 zT (x, y) = 0 z0 = 4397nm δ = 5712nm δ ∫ +∞z 0.00114nm−1 .   3-10 3-4

− δ0

1 exp(−t 2 )dt

=

  z0   δ * 100kV 3.5(d)(e) 

    

 



' 

   ! &    % 60kV!*   42



3

  +

 3.2

   

material

acceleration voltage

incidence angle

z0

δ

Ti-6Al-4V Ti-6Al-4V Ti-47Al-2Cr-2Nb

60kV 100kV 60kV

0o 0o 0o

4400nm 10900nm 5000nm

5700nm 13900nm 6700nm

3.5    a

b c  100kV

Ti-6Al-4Vd    e 60kV30

Ti-6Al-4Vfg

43

3

  +

3.4.3 

*)X-?& -?  #  -?( ηB  

  η = 1 − ηB   3.4e -  *  3.4a" # (!  %     -?(  3-11 FB % [EB , EB + ΔEB ]  ⎧ ⎪ ⎪ E = 0E0 EB · FB ⎪ ⎨ B ⎪ ⎪ η =1− ⎪ ⎪ ⎪ ⎩

(3-11)

η B ·E B E0

      

  

ηB   EB  η3.3 -?(  

* 

    

#"   3.3

    

material

acceleration voltage

incidence angle

ηB

EB

η

Ti-6Al-4V

60kV

0o

0.1677

37.66keV

0.895

Ti-6Al-4V Ti-6Al-4V Ti-47Al-2Cr-2Nb

60kV 100kV 60kV

30o 0o 0o

0.2000 0.1547 0.1727

37.65keV 61.60keV 37.70keV

0.874 0.905 0.891

3.4.4  

Q* UI Q = UI

44



(3-12)

3

  +

   &!+    %$$  $    ! !  [104]  - *$:$ -  I  -?  !-?  

3.5

   3-8 3-10 3-11 3-12  3-6%" 

  %' -?    (%!      1 ηB · EB (z − zT (x, y) − z0 )2 ∗ ∫ +∞ q(x, y, z) = UI ∗ 1 − exp − E0 δ2 δ − z0 exp(−t 2 )dt δ   C C · [(x − xb )2 + (y − yb )2 ] ∗ exp − πRb2 Rb2 

3.5.1 

      3.5 1

 2 3   2   

x yz  XY Z+  '  3-13  ⎧ ⎪ ⎨ x = X cos θ − Z sin θ ⎪ ⎪ ⎪ z = X sin θ + Z cos θ ⎩

(3-13)

 3.5 (c)   60kV30o Ti-6Al-4V

 3-13' - X0 = 4.4μm × sin 30o = 2.0μmZ0 = 4.4μm × cos 30o = 3.9μm    3.5 (f)(g)  45

3

  +

 %  

   



& 

  *   4.2 "  η* " 3-10zT (x, y)*  zT (x, y)

*( (xc ,yc ,zc ) Rc ) zT (x, y) = zc +



Rc2 − (x − xc )2 − (y − yc )2

(3-14)

3.5.2 

:   "*8  % 

  "   "*   48

3.6

  a  b

* 3.6 (a) 

      3.6 (b)  t