PSI - Issue 5

Khodjet-Kesba Mohamed et al. / Procedia Structural Integrity 5 (2017) 271–278 Khodjet-Kesba Mohamed et al / Structural Integrity Procedia 00 (2017) 000 – 000

273

3

Stress strain relations in the 0° layers and 90° layer are taken in the forms:

du

0

0

with

(3)

xx 0  

0

E  





xx

xx

0

dx du

90 xx

90 xx

with

(4)

xx 90  

90

E









90

dx

Where 0 E and 90 E are the Young’s moduli of the 0° and 90° layers, respectively. Taking into account the previous relations, the classical differential equation will be:

E

2 90

d



(5)

xx

2 90    xx  

2  



0 90

C

2

dx

E

x

0

0 0 90 t t E E G t t E x  2 90 90 0 ) 3 (

(6)

2  

0 x E is the Young’s mo dulus of the undamaged laminates.

Where

The general solution of differential equation (5) is:

(7)

xx 90

A

x B 

E x E 

cosh

sinh











0 90

C

x

The constants A and B are determined to satisfy the symmetry conditions. Finally, the stress and displacement distribution are expressed in the form:

l a x

cosh



E E

1

(8)

0

0

2

f z

( , ) x z

( ) x

( )













0 0 0 90

xx

xx

C

' E G t f t x xz 90

a

cosh



( ) 90

l a x

cosh



   t

  

2 90 E G E x

z

3 1

(9)

90 xx

2

90 xx

( , ) x z

( ) x















C

0 90

2 90

a

cosh

2



xz

l a x

cosh



E

(10)

x ( )









0 90

C

a

E

cosh



x

2.2. Variational approach There is another method, relatively simple, the variational approach (Hashin 1985), which satisfies equilibrium and all boundary and interface conditions to find an optimal approximation to the principle of minimum complementary energy. By assumption, the normal stress in load direction σ xx are constants depending on the thickness (z) and the width (y) in the 90° and 0° layers respectively. They may be expressed in the form: (1 ( )) 1 90 90 x xx      (11) ( )) (1 2 0 0 x xx      (12) Where 90  and 0  are the stress in 90° and 0° layers before cracking, 1 2 ,   are unknown functions.

Final expression for complementary energy will be in form of: