PSI - Issue 2_B

Maria Kashtalyan et al. / Procedia Structural Integrity 2 (2016) 3377–3384 Author name / Structural Integrity Procedia 00 (2016) 000– 00

3380 4

11        ( ) 11 c ( 

       

( ) 2 c L s l L x L s l L x   2 ( ) 2 c ( ) 1 c 2 ( ) 1 c ( (

cosh

( ) ~ 1 1 c

( ) 22 c

),









22

22

( ) 1 c

L

cosh

)

(7a)

    

cosh

( ) ~ 1 1 c

( ) 12 c









12

12

( ) 2 c

L

cosh

)

( ) 1 c h L K 

( ) 22 m

( ) c S a S  22 (

( ) m

( ) c

( ) 2 c h L K 

( ) m

( ) c

[

)],

(

),

S

S

S

S

 









2

1

1 12

12

66

66

m

m

h K h K

h K

2    (1 ( ) 11 c

( ) m S a S  12

( ) m

2    (1 ( ) 22 c

( ) 22 m S a S 

( ) m

)(

),

)(

),

(7b)





1 11

1 12

m

m

( ) m

( ) c

S S

 

 

1    ( ) 12 c

1 ( ) 66 m S a S   ,

(1 ) 

12

12

( ) m

( ) c

S

11

11

m

The in-plane layer stresses in the ceramic layer containing multiple transverse cracks can be used to evaluate the reduction of stiffness properties of the single domain sample due to multiple cracking. Instead of a cracked sample we consider a layered solid, in which the damaged layer is replaced with an equivalent homogeneous layer with degraded stiffness properties. The constitutive equations of the equivalent homogeneous layer are { } [ ]{ } ( ) ( ) ( ) c c c Q    (8) The in-plane macrostresses ( ) c ij  and microstrains ( ) c ij  in the equivalent homogeneous layer are determined as         s s c ij ij m ij c ij s s c ij c ij dx s dx s 2 ( ) ( ) ( ) 2 ( ) ( ) ~ 2 1 , ~ 2 1       (9) The reduced in-plane stiffness matrix ] [ ( ) c Q of the equivalent homogeneous layer is related to the in-plane stiffness matrix [ ˆ ] ( ) c Q of the undamaged ceramic layer via the In-situ Damage Effective Functions (IDEFs) ( ) 66 ( ) 22 , c c   , as

       

       

( ) 2 12

( ) ˆ ( ˆ ) c c Q Q 22

ˆ

  ( ) ( ) 22 c c c ( ) 22 12

0

( ) Q Q Q ˆ ˆ c c

  ( ) ( ) 22 c c ( ) 22 22

( ) c Q Q [

  ( ) c

] [ ˆ ]

0

(10)

12

ˆ Q

(2)

( ) 66 c

0

0



66

( ) c c   can be expressed in terms of macrostresses ( ) c

( ) c ij  as

The IDEFs

ij  and macrostrains

( ) 66

22 ,

  

  

c ( ) 1

(1

)

D

D





1

tanh



w

c ( ) 1

D



c ( )



(11a)

   c ( ) 22

1 ˆ

  1

22

ˆ

  

  

c ( ) 1

c ( ) 1

1

(1

)

 Q Q c c ( ) ( ) 

( )

( )

D

D

c c

D





 1





c ( )

tanh

1 



w

w

12

11

22

22

c ( ) 1

D

D





w

  

  

c ( ) 2

(1

)

D

D





1

tanh



w

c ( ) 2

D



c ( )



(11b)

   c ( ) 66

1 ˆ

  1

12

  

  

c ( ) 2

c ( ) 2

1

(1

)

( )

( )

D

D

c c

D





 1



Q



c ( ) 2

tanh





w

w

66

12

c ( ) 2

D

D





w