PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1212–1225 Author name / Structural Integrity Procedia 00 (2019) 000–000

1216

5

The linear viscoelastic model shown in Fig. 2a consists of a linear spring with shear modulus, 0 G , connected consequently to a linear dashpot of coefficient of viscosity, 1  . The variation of the shear stress,  , with the time, t , is expressed as (Zubchaninov (1990))

G t

1 0



 

0 G e





,

(7)

where 1  is the coefficient of viscosity. The distribution of the coefficient of viscosity in radial direction of the beam is written as

4 R b R C e    , 1

(8)

1

where

4 0 R R   .

(9)

In (8), C 1  is the value of 1  in the centre of the beam cross-section, b is a parameter that controls the distribution of the coefficient of viscosity in radial direction. The distribution of C 1  along the beam length is expressed as

l f x

D e 1    ,

(10)

1

C

where

x l   0 .

(11)

In (10), D 1  is the value of C 1  at the free end of the beam, f is a parameter that controls the distribution of the coefficient of viscosity along the beam length. The time dependent shear modulus is defined as

   ( )

0 G t

.

(12)

By substituting of (7) in (12), one obtains

1 0  G t

0 0 ( ) G t G e  

.

(13)

The linear viscoelastic model under a constant applied shear strain shown in Fig. 2b consists of a linear spring that is connected consecutively to a parallel combination of a second linear spring of shear modulus, 02 G , and a linear dashpot of coefficient of viscosity, 2  . The evolution of the shear stress,  , with the time, t , is written as (Zubchaninov (1990))