PSI - Issue 25

Victor Rizov / Procedia Structural Integrity 25 (2020) 88–100 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

92

5

rheological models for a constant applied stress are used. The models are shown schematically in Fig. 1. First, a rheological model consisting of a linear spring and a linear dashpot which are connected consecutively is used in the analysis of the strain energy release rate (Fig. 1a). The strain – time relationship for this rheological model is written as (Arkulis and Dorogobid (1987))

ij    , t

ij

  

(11)

ij

E

j

j

where j  is the coefficient of viscosity in the j -th layer of the beam, t is time. The strain energy release rate in the multilayered beam configurations is obtained also by applying a linear rheological model in which a linear spring is mounted parallel to a linear dashpot (Fig. 1b). For this rheological model, the strain – time relationship is expressed as (Arkulis and Dorogobid (1987))

pj pj E t 

      pj ij E  1

   



e



,

(12)

ij

where pj E and pj  are the modulus of elasticity and the coefficient of viscosity in the j -th layer. A linear rheological model consisting of a linear spring that is connected consecutively to a parallel combination of a second linear spring and a linear dashpot is also applied in the analysis of the strain energy release rate (Fig. 1c). The strain-time relationship is written as (Arkulis and Dorogobid (1987))

pj pj E t 

       pj ij j ij E E   1

   



e



.

(13)

ij

Finally, a linear rheological model constructed through consequently connection of a linear dashpot, a linear spring and a parallel combination of a second linear spring and a second linear dashpot is used (Fig. 1d). For this model, the strain – time relationship is obtained as (Arkulis and Dorogobid (1987))

pj pj E t 

        pj ij j ij j ij E t E     1

   



e



.

(14)

ij

The time dependent modulus of elasticity that is used in the delamination analysis is written as



ij

E t

( )



,

(15)

j

( ) t



ij

where ( ) t ij  is obtained by (11), (12), (13) or (14) depending on the viscoelastic model used.

3. Influence of the viscoelestic behaviour on the strain energy release rate The approach for analyzing the strain energy release rate developed in the previous section of the paper is applied