Zhi Wang1

Xianyu Jin*1

Nanguo Jin1

Abid A Shah2

Bei Li1

DHD

2(1 2 ) tr( ) 1tr( )

6 1- 2

D D

H

v v

E D E

σHσ Hσ

E v

Dσ H

HD

1tr( )

3

D σ σ σ

1

2( )

H I D1

tr3

HD D

1

2( )

I D ( )diagI D1( ) ( )diag

I D P I D P

1/ 2 1/ 2 1( ) ( )diag

I D P I D PijI P

( )I D

2 2

1 2

tr( ) tr( )(1 2 ) 1[ ] tr( )

6 1- 1- 2

D D

H H

v v

E D D E

σ σHσ Hσ

max(0, )x x

min(0, )x x

+ -

1 2

tr( ) tr( )(1 2 ) 1= [ ] ( )

3 1- 1-

D D

H H

v v

E D D E

σ σε I Hσ H

σ

I

1

2

3

1

1

1

1 1

1

2 2

1

3 3

11 2

13 1

1

1 121 13

1 1 1

31 1

1 13

1 1

e

e

He

D

v

E D

D D

E D D

D D

1(1 )HK D K

D

1 2 3, ,x x x

1

2

3

D

D

D

D

1 2 3

1( , , )D D D

E

ε B σ

B

1X B

12 1311 12 21 13 311 1 1

2322 23 32 332 2 3

, , ,

, , ,

E E EB B B v B B v

E E E

E E EB B B v B

E E E

iE i

1X 2X 1E 2E 12

1 1 1121

2

1 (1 )(2 )E E

DE E

2 2 1122

1

1 (1 )[2 (1 ) ]E E

DE E

1

12

1 21

1 2H

ED

E

:σ C ε 1 1: tr( )

v v

E E

ε C σ σ σ I C

σ σ

+ -

1 2

tr( ) tr( )1[ ] [( )]

3 1- 1-

D D D

H HD D

σ σσ I Hσ Hσ

1/ 2 1/ 2 2( ) : 1( ) ( ) ( ) [(1 tr ) tr (1 tr ) tr ]

3 tr 3 3

I - D σσ I - D σ I - D I - D D σ D σ I

D

eq eq

+

2

+ += =eq

ε ε ε：

= (tr ) 0eqf D

0tr(tr ) tan[ arctan( )]a

aA a

DD

0f 0f 0f

(tr ) D

(tr ) D0 (0)

(tr ) D

a A

E v0 (0)

( ; ) :F F

Y ε Y ε

tr( ) tr( )F

D ε D ε

Y

0f 0f

1tr ( )eq D

1( )tr( )

eqeq

eqd

D

1[ ( )]tr( )

tr( ) tr( )

eqeq

eq

d

d

D

ε ε

1( ( ))tr( )

tr( ) tr( )

eqeq

eq

d

d

D

D ε εε ε

2

2tr( ) ( )eq

ε eq

12

2

( ) eqeq

eq eq

Dd

ε

1X

1 0 0

0 0 0

0 0 0

ε

1

2

2

0 0

0 0 0

0 0 0

ε

1 0 0

0 0 0

0 0 0

D

D

1X

2

2

0 0 0

0 0

0 0

ε2

2

2 2

2

0 0 0

0 0

0 0

ε

2

2

0 0 0

0 0

0 0

D

D

D

E

1 2

tr trtr

3 (1 tr ) 3 (1 tr )3 3

K K

σ σε

D D

K

tr 1D

tr 0σ 3K

3 (1 tr )K D

1 2

tr trtr

3 3K K

σ σ

ε

1 tr3

cD

D

1 21 2(1 tr ), (1 tr )

3 3K K K K

D D

1

2

3tr

3

cc

DD

D

21 2(1 ), (1 )

3c cK K D K K D

2

3tr cD

D

1 2(1 ), (1 )c cK K D K K D

cD

2

D ε

( )D D D D cD n

n n

n n cD D n

2 (1 ) [(1 ) tr (1 ) tr ]D

c c cG D K D D

σ ε ε ε I

/ 2(1 )G E

( 1) 1 1:eq n n n ε ε

( 1) (tr )trial eq n nf D

0f 1n n D D 0f

2

2

1 1

1

2

( 1)

tr tr

n n n

n n

eq n

D D D ε

D D

D2

1 1n n n D D ε

1 1:n n σ C ε

11/ 2 1/ 2 111 1 1 1

1

21 11 1

( ) :( ) ( ) ( )

3 tr( )

1[(1 tr ) tr (1 tr ) tr ]

3 3

nnnn n n n

n

n nn n

I D σσ I D I D I D

D

D σ D σ I

2385m / kg

19.98μm

21 2 C

24GPa, 0.2,E v 1=3 2 0.5 2.88605e 4,a

7.7e3,A 0 1.35e 4,k E v

1 2 0, ,a A k

24MPa, 0.2E v

3.21MPa

3.21MPa

D

C

:σ C ε

tr 0,ε

1/ 2 1/ 2 ( ) ( )2 [( ) ( ) ] (1 tr )

3 trG K

I D I DC I D I D D I I

D

tr 0,ε

1/ 2 1/ 2 ( ) ( )2 [( ) ( ) ]

3 trG K

I D I DC I D I D I I

D

/[2(1 )]G E /[3(1 2 )]G K

A B ( )ijkl ik jlA B A B

B

11

1 2 3 1

(1 ) 4 1 1 1 2( )

9 1 1 1 3(1 )H

BD D D D

12 21

1 2 3 1

(1 ) 2 2 1 1 2( )

9 1 1 1 3(1 )H

B BD D D D

13 31

1 2 3 1

(1 ) 2 1 2 1 2( )

9 1 1 1 3(1 )H

B BD D D D

22

1 2 3 1

(1 ) 1 4 1 1 2( )

9 1 1 1 3(1 )H

BD D D D

23 32

1 2 3 1

(1 ) 1 2 2 1 2( )

9 1 1 1 3(1 )H

B BD D D D

33

1 2 3 1

(1 ) 1 1 4 1 2( )

9 1 1 1 3(1 )H

BD D D D

B

11

1 2 3 2

(1 ) 4 1 1 1 2( )

9 1 1 1 3(1 )H

BD D D D

12 21

1 2 3 2

(1 ) 2 2 1 1 2( )

9 1 1 1 3(1 )H

B BD D D D

13 31

1 2 3 2

(1 ) 2 1 2 1 2( )

9 1 1 1 3(1 )H

B BD D D D

22

1 2 3 2

(1 ) 1 4 1 1 2( )

9 1 1 1 3(1 )H

BD D D D

23 32

1 2 3 2

(1 ) 1 2 2 1 2( )

9 1 1 1 3(1 )H

B BD D D D

33

1 2 3 2

(1 ) 1 1 4 1 2( )

9 1 1 1 3(1 )H

BD D D D