PSI - Issue 28

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

1221

10





a  0

l

a 

1

2

pr

pr

dx

dx







.

(42)

R

R

3

4

By substituting of (26), (29), (30), (36) and (42) in (41), one derives

R

  u RdR R 02 0 4





  

   

1

  

  

3

0 

1

2

pr

pr

G

T

u RdxdR 01





,

(43)

2

R

R R



3

3

4

where 1 pr  , 2 pr  , 01 u , 02 u and 4 R are determined at x a  . The integration in (43) is carried-out by using the MatLab computer program. It should be mentioned that the strain energy release rate obtained by (43) matches exactly that found by using (39). This fact is a verification of the fracture analysis.

0.5  f , curve 2 – at

0.7  f

Fig. 4. The strain energy release rate in non-dimensional form presented in a function of b (curve 1 – at

0.9  f ).

and curve 3 – at

It should be noted that the solutions derived in the present paper can be applied to analyze the strain energy release rate at any value of time. 3. Numerical results The results shown here are obtained by applying the solutions to the strain energy release rate derived in the previous section of the paper. The strain energy release rate is presented in non-dimensional form by using the formula   1 / G G G R D N  . It is assumed that 0.200  l m and 0.005 1  R m. The time-dependent longitudinal fracture behaviour of the inhomogeneous beam due to the stress relaxation phenomenon is analyzed first by using the linear viscoelastic model shown in Fig. 2a. For this purpose, the strain energy release rate in non-dimensional form is presented in a function of the non-dimensional time in Fig. 3 at three 2 1 / R R ratios. It should be mentioned that the time is expressed in non-dimensional form by using the formula