PSI - Issue 2_B

K.B. Ustinov / Procedia Structural Integrity 2 (2016) 3439–3446 K.B. Ustinov / Structural Integrity Procedia 00 (2016) 000 – 000

3445

7



 







  36 3 2 1 log      3

2

3 h I 

3 I d h ds   1 A

(2) E J

3 h M I N E a  (2) ,

6 1 / 

2 1 / 

 

 

 



A

1

33

3









(2) E a

3 h I

(2) E a E a  (2)

3 h I

2

I

12 1 / 

,

6 1 / 

,

 

 

 

A

(24)

A

A

A

22

1

23

32

1

1

ds s



0

  

  

3 1 log tan d s   3

3

3

s

s h ds

s

sinh cosh s

s s 

sinh cosh hs

hs h 

I

h

,









A

0 

A

3

3

2

2

2 2 sinh ( ) hs hs 

ds

s

s s 

1

sinh







For shear separation we have    sin cos K p 







/ p p p d

/ h p h p h p d 

sin cos

 



1

(25)

(2)

 

0 



  ,0 u x u x e dx F p      ,0   (2) (1) , px

  ,0

  F p E  

px x e dx 





0 





xy



x

2





The energy release rate (which may be found elementary for this case) and the only available for this case coefficient of matrix of elastic compliance for which are

  

hs hs ds   

tanh sinh cosh s s 

h

s s 

sinh cosh hs

1 /  

 (2) 2 1 / J T h E    2





a

log







0 

,

 

 

 

11

(2)

2

2

2 sinh ( ) hs hs 

2

2

E

s s 

s

1

sinh







4. Application of the obtained solution to the problem of coatings delamination

10 h  the

The most important coefficients of clamping compliance are presented in figure 3. It is seen that for ) h  , which, in particular, proves that the formulae (Goldstein et al. 2011, Ustinov 2015) obtained for the critical value of compressive stresses cr  corresponding to buckling of the delaminated part of coating remain valid through not very thick substrates   2 2 (2) 2 2 0 0 3 0 0 2 1 2 3 ... , 12 cr h E h d d b b            (26) solution becomes very close to the case of semi-infinite substrate (

Fig. 3. Coefficients of matrix of elastic compliance with respect to ratio of Young moduli of the coating and substrate. Solid lines correspond to the exact solution of matrix problem for h =1; dotted lines correspond to scalar solution for h =1; dashed (short) lines correspond to scalar solution for h =2; dashed (long) lines correspond to scalar solution for h =10; dot-dashed lines correspond to matrix solution for h  .