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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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-

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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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-

-

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-

-

-

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-

-

-

-

- - -

-

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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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-

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-

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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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-

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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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-

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F

x

y

y

u

x

uyy

>>

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