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-
z=-L
z=0
z=+L
T=T(z)
y
z
x 2L
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T=T(z)
z=0
z
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0
T=T(z)
undeformed state
T (z)rem
T (z)asym
T_
0
0 = 0
0 = 0 =
=
=
+
+
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0
E(t-0)
t
E(t- )
E(t)
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t=t
t=0
T(z,t)z
T(z)
t0
z=L
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_
_
+
_
_
z
E(t)
t0 t
t'0
T(z) res( z,t )1 res(z,t )2
t2t1
E(t)
0
0E(t-t )=E(t')
+
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0 =( )
t
T t1
E(t, )=E( )Tref
E(t,T)E(t,T) T >Tref
Tref
( )=1Tref
log T( )
log (t)
E(t,T) ; T >Tref
t1 t2
2
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t
T(z,t)
z=L
T=T
t
z=L
T=Tg
T(z,t)
z
g
z
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Mathcad implementation procedure:F_left( )x cot( )..2 x L F_right( )x
x
.2 H
H
.2 x
roots_( )N guess.1
2
.2 L
for
guess guess
.2 L
n
root ,F_left guess F_right guess guess
guess n
n ..0 N 1
. m1
=roots_( )3
277.565
557.186
840.345
1
m
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0
50
100
150
200
temperature(oC)
cooling time (s)
0 60 120 180 240
z=5.00 mm (top surface)
z=3.75 mm
z=2.50 mm
z=1.25 mm
z=0.00 mm (middle surface)
-5.0
-2.5
0.0
2.5
5.0
thicknesscoordinatez(mm)
temperature ( oC)
0 50 100 150 200
201
510
2030
6090
4050
120
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0
50
100
150
200
temperature(oC)
cooling time (s)
0 120 240 360 480 600
z=10.00 mm (top surface)
z=7.50 mm
z=5.00 mm
z=2.50 mm
z=0.00 mm (bottom surface)
0.0
2.5
5.0
7.5
10.0
thicknesscoordinatez(mm)
temperature ( oC)
0 50 100 150 200
20
15
3060
120180
240300
360
1020
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Tg
0
50
100
150
200
temperature(oC)
cooling time (s)
0 60 120 180 240
z=5.00 mm (top surface)
z=3.75 mm
z=2.50 mm
z=1.25 mm
z=0.00 mm (middle surface)
42.5 s
113.9 s
-5.0
-2.5
0.0
2.5
5.0
thickne
sscoordinatez(mm)
temperature ( oC)
-50 0 50 100 150 200
60
0
113.9
10014080
42.5
20
( ) T zresTg
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0
50
100
150
200
temperature(oC)
cooling time (s)
0 60 120 180 240 300
full temperature solution (n=100)
aproximate solution (n=1)
middle surface
top/bottom surface
Tg 2
Tg1
Tg 3
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- -
- - -
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sol
t=const
'frozen-in'gradientfield
(t+dt)=const
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symmetryplane
T=Tg
T=Tg
gradT
symmetry
plane
gradT
gradT
orthogonal lines
-
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A
B B
A
B
A A
a) Solidification fronts b) Orthogonal lines
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a) Solidification fronts b) Orthogonal lines
weld line
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-
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P
f
S
y
z
x
r
N
CV
f
-
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P Nf
nfdPN
(a) orthogonal mesh
P Nf
nfdPN
(b) non-orthogonal mesh
nf
o
k
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-
-
-
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F
x
y
y
u
x
uyy
>>
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