PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 103–114 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

105 3

where 0  t . The constitutive law of the viscoelastic model in Fig. 2 is obtained in the following manner. 0  is the stress at

Fig. 1. Viscoelastic beam configuration with a longitudinal crack.

2   , in the dashpot with coefficient of viscosity, 2  , is found as  .

First, the stress,

   2

(3)

2   is expressed as

Besides,

    ,



(4)

2 

2 

2

where 2    is the first derivative of the strain in the dashpot with respect to time. By combining of (3) and (4), one derives t       0 2 2   . (5)

Fig. 2. Viscoelastic model.

The solution of (5) is found as

t 

0 



C







.

(6)

1

2 

ln  



2

1 C , in (6) is determined by using the following initial condition:

The integration constant,

  0 0 2 

 

.

(7)

From (6) and (7), one derives

1



1   C

0

.

(8)

ln

  

2

By substituting of (8) in (6), one obtains