PSI - Issue 83

Victor Rizov et al. / Procedia Structural Integrity 83 (2026) 105–114

109

The equilibrium of the linear dashpot and spring leads to the following differential equation:

1

E

Ei   





i 

i

,

(15)

Ei

i 

i 

i  is given by Eq. (10). The solution of Eq. (15) is

where

E

   .

     i i t E

i i 

 

i

t



i 

e





i

(16)

Ei

2

E

E

i

i

The constitutive law written in the next equation is applied here for treating the behavior of the non-linear spring in the model in Fig. 2 (Varbanov et al. (1992)).     i nli i nli B       1 1 . (17)

nli  - stress; nli  - strain; i B - parameter; i  - parameter.

The notations in the above equations are:

Since

i nli    ,

(18)

from Eq. (17) we derive

1

t

   

  

i 

i 

1 1

 



.

(19)

nli

B

i

From Eqs. (11), (16) and (19) we obtain

1

E

  

     i i t E

t

   

  

i i 

 



i

t



i 

i 

1 1 

e

i 



i B

i

.

(20)

2

E

E

i

i

i

The combination of Eqs. (10) and (20) gives

1

E

     i i t E t

  

   

  

i i  E t 2

 

i 

i

t



i 

i 

1 1 

e



i 

i

,

(21)

E

B

i

i

i

where n 1 1,2,...,  (here n is the number of layers in the stepwise functionally graded beam). The above equation represents the stress-strain-time relationship applied to treat the non-linear viscoelastic behaviour of the moving beam under inertia load. The beam in Fig. 1 is delaminated in portion, 2 3 R R . The delamination is analyzed by the J -integral (Broek (1986)). The contour, L , is used for the integration. Therefore, the solution of the J -integral is