PSI - Issue 41

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

106 4





 1

     t 





ln 2 0

.

(9)

2 

The strain, 1   , in the dashpot with coefficient of viscosity, 1  , is found from the following differential equation: t         2 1 1 1     , (10) where

1   E 

,

(11)

1

0     .

1 (12) It should be noted that equation (10) is composed by considering the equilibrium of the spring and the dashpot with coefficient of viscosity, 1  . The solution of (10) is derived as 2

1   t C e









,

(13)

2

1 

1 

1   is a particular solution of (14).

1   is written in the form

where

t 2    , 



1  (14) where 1  and 2  are parameters. These parameters are determined by substituting of (14) in (10). The result is 1

2 

1 



,

(15)

ln



  

1

   2 .

(16)

2 C , is written as

The initial condition for determination of the integration constant,

  0 0 1 

 

.

(17)

After substituting of (13) in (17), one determines 1 2    C . By combining of (13), (14) and (18), one obtains   t t e e 1 2 1 1         .

(18)

(19)

The strain,  , is found as

 

   

.

(20)

2 

1

By substituting of (9) and (19) in (20), one derives   t t e e 1 2 1          1 ln 2 0   t       . (21) Formula (21) is the constitutive law of the viscoelastic model in Fig. 2. The time-dependent modulus of elasticity, * E , of the model is found as

  

* E

.

(22)

By substituting of (2) and (21) in (22), one obtains

          ln 1 1   t t t

E



.

(23)

*

 e e 2  t





ln

  



1

1

2