Mori-Tanaka Model

CONFIDENTIAL

2

Mori-Tanaka model

3

Mori-Tanaka model

> The dilute strain-concentration tensor

> To get composite stiffness:

> Where:– C: Stiffness

– v: Volume fraction

– I: Identify matrix

– S: Eshelby’s tensor

– Superscript, Subscript

• m: matrix

• f: fiber

11

mfmEshelby

Eshelbyf

CCCSIA

A

1 Eshelby

fmEshelbyf

fm

m vvvv AIACCC

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Eshelby’s Tensor:Fiber inclusion (a1=a2<a3)

5

Assumption

> Domain of an ellipsoidal inclusion:

> Assume the fiber is transversely isotropic:

123

23

22

22

21

21

a

x

a

x

a

x

c

a

acaa

aaa

,,

,

31

321

6

Eshelby’s Tensor (Tensor form): Fiber inclusion (a1=a2<a3)

> P-Basis form (Tensor form)

106

105

104

103

102

101

654321

6

1

4242222

2

1

2220

22

2

2

2

2

421

412

2211

21

12

4312

1

,,,,,

116

3

14

1

116

1

18

3

18

331

2

118

4

1

4

1

2

11

12

1

2

1

2,

1

1ln

1

11

11ln

1

1,1

ffs

ffs

ffs

ffs

ffs

ffs

ssssssPsS

ghf

ghf

gghg

t

tijkltijkl

lkjiijkl

ikljilkjjklijlkiijkl

kljiijkl

lkijijkl

klijjkiljlikijkl

klijijkl

ijijij

jiij

i

nnnnP

nnnnnnnnP

nnP

nnP

P

P

nn

n

6

5

4

3

2

1

22

4

1

2

1

2

1

1

n

material.matrix theof ratio spoison' theis

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Eshelby’s Tensor (Matrix form): Fiber inclusion (a1=a2<a3)

> P-Basis form (Matrix form)

6

1

131313231312133313221311

231323232312233323222311

121312231212123312221211

331333233312333333223311

221322232212223322222211

111311231112113311221111

*13

*23

*12

*33

*22

*11

131313231312133313221311

231323232312233323222311

121312231212123312221211

331333233312333333223311

221322232212223322222211

111311231112113311221111

13

23

12

33

22

11

*

222222

222222

222222

222222

222222

222222

t

tttttt

tttttt

tttttt

tttttt

tttttt

tttttt

t

C

C

C

C

C

C

C

PPPPPP

PPPPPP

PPPPPP

PPPPPP

PPPPPP

PPPPPP

s

SSSSSS

SSSSSS

SSSSSS

SSSSSS

SSSSSS

SSSSSS

S

S

8

[Example] P-Basis: n=(0,0,1)

> n=(0,0,1)

1

1

1

4

111

1

1

1

11

11

2

1

11

11

654

321

PPP

PPP