PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 134–144 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

138

5

2 1 2 3     Q 1 2 6    H 1 2 6     R 3 4

,

(16)

,

(17)

.

(18)

0.5

1.0 2   , and curve 3 – at

1  (curve 1 – at

2  

Fig. 4. Variation of the strain energy release rate with increase of

, curve 2 – at

2.0

2  

).

The stress,  , in the viscoelastic model is determined as 2 E ng nf        .

(19)

By using (2), (3), (4), (7), (9) and (19), one derives

s

3

3

 

       2 E s

         2 3 D t E P m



t

 3    2 6 1 E

3 E E

6

E

1   

2    2 t



t

1 (20) Equation (20) represents the constitutive law of the model in Fig. 2. This law relates stress, strain and time. Apparently, the law (20) is non-linear. The constitutive law (20) is applied for modelling the non-linear viscoelastic behaviour of the beam in Fig. 1. The beam exhibits continuous material inhomogeneity along its thickness. The changes of material properties along the beam thickness are described as 3 1 2 1 1 4 1 1      t e t    . 2 1 2 