PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 402–415 Author name / Structural Integrity Procedia 00 (2019) 000–000

407

6

v t    .

(13)

 v , along the thickness of the right-hand delamination crack arm is expressed as

The distribution of the speed,

2

v

DRH 

v

z

 

,

(14)

1

h

where

1 h z h   

.

(15)

2

2

In formula (14), DRH v  is the speed at the lower surface of the same crack arm. For the viscoelastic model shown in Fig. 4, the variation of the stress,

i  , with the time in the i -th layer is

written as (Popov (1998))

i n t

   

   







1

E v t n E E  

e v

 i





,

(16)

1

pri

i

i

pri





pri E and i n , which are involved in (16) are expressed as (Popov

 v is found by (14). The quantities,

where

(1998))

i i pri E E E E E 2 1 1 2   i

,

(17)

i



n

i 1  i i E E 

.

(18)

2

i

In the present paper, a time-dependent solution to the strain energy release rate, G , that accounts for the viscoelastic behaviour of the material is obtained by differentiating of the strain energy, U , cumulated in the mulilayered functionally graded beam with respect to the crack area, A , i.e.

dA G dU  ,

(19)

where

hda dA  .

(20)

In formula (19), da is an elementary increase of the delamination crack length. By combining of (19) and (20), one obtains