PSI - Issue 28

Mohamed Khodjet Kesba et al. / Procedia Structural Integrity 28 (2020) 864–872 KHODJET KESBA Mohamed/ Structural Integrity Procedia 00 (2019) 000–000

867

4

assumption will be in the form:

(8)



G G 

xz



G G

1

 

0 xz xz

b) In the case when the variation of the longitudinal displacement is supposed progressive in 0 0 -layer: We use the function  z t in the (7), the shear modulus for progressive shear

   

   

 ( ) sinh

1 cosh

f z

   t



 

t



t

assumption will be in the form:

(9)



G

G



xz

1 ) 1  

(tanh



G G

 



1 3 

t

t

xz

2 

0

t

xz

2.2 Variational approach The variational approach consist that the normal ply stresses in load direction are constant over ply thickness. We thus construct admissible stress fields which satisfy equilibrium and all boundary and interface conditions (Vingradov and Hashin (2010)). Expressions for x-axis stress components are following: (1 ( )) 1 x xx        (10) ( )) (1 2 0 0 x xx      (11) Where   and 0  are the stress in β° and 0° layers before cracking respectively. are unknown functions. Next, we denote (12) And express in term of due to equilibrium condition in x direction: ( ) 0 ( ) 2 0 0 1   t x t x       (13) Final expression for complementary energy will be in form of:                    a c d d C d d t C C C d u           2 11 2 2 2 22 02 2 00 2 2 ' 2 1 (14) 1 2 ,   ( ) ( ) 1 x x    ( ) 2 x  ( ) x 

   

   

a

Where,

1 1 E E 

(15)

C

  

00

0



3 3 2    

0 

    

(16)



C



02

E

E

0



1

   1 3 12 8 2    

(17)



C







22

60

 E

   

   

3 1

1



(18)

C





11

G G

0



t t 0 

(19)





The function which minimizes the complementary energy

is the fourth order differential equation of Euler

' c u



Lagrange:

4

2 2

d d



 

0

d p d

   q



(20)

4

