PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 402–415 Author name / Structural Integrity Procedia 00 (2019) 000–000

413

12

t v DRH  and

t v DUN  in (37), one obtains

DRH  and DUN  with

By replacing of

)] ( 2 [ v ta v t l a h DUN DRH       .

(38)

By substituting of (35), (36) and (38) in (34), one derives

v

 t v

t

    RH i n i 1 ( )





G

i z dA 1





.

(39)

DRH

DUN

2

h

A

i

The MatLab computer program is used to carry-out the integration in (39). The strain energy release rate obtained by (39) is exact match of that found by (33). This fact is a verification of the time-dependent solution to the strain energy release rate derived in the present paper with considering of the viscoelastic behaviour of the multilayered cantilever beam configuration. 3. Numerical results The numerical results obtained in the present section of the paper illustrate the variation of the strain energy release rate with the time due to the viscoelastic behaviour of the material. The influences of the material gradient and the location of the delamination crack along the beam width on the strain energy release rate are also studied. For this purpose, calculations of the strain energy release rate are performed by applying the time-dependent solution to the strain energy release rate derived in previous section of the paper. The strain energy release rate is expressed in non-dimensional form by using the formula ) /( 1 1 G G E b L N  . Two three-layered functionally graded cantilever beam configurations are analyzed in order to evaluate the influence of the delamination crack location along the width of the beam on the strain energy release rate (Fig. 4). A delaminatio crack of length, a , is located between layers 2 and 3 in the beam configuration depicted in Fig. 4a. A beam with a delamination crack between layers 1 and 2 is also considered (Fig. 4b). The width of each layer in both beam configurations is s , the beam thickness is h (Fig. 4). It is assumed that 0.003  s m and 0.015  h m. The variation of the strain energy release rate with the time is analyzed for both three-layered beam configurations shown in Fig. 4. For this purpose, calculations of the strain energy release rate are carried-out at various values of the time. The results obtained are shown in Fig. 5 where the strain energy release rate in non dimensional form is plotted against the non-dimensional time for both beam configurations. It should be mentioned that the time is expressed in non-dimensional form by using the formula 1 1 1 / L L N t t E   . It is evident from Fig. 5 that the strain energy release rate increases with the time (this finding is attributed to the viscoealstic behaviour of the material). It can also be observed in Fig. 5 that the strain energy release rate in the beam configuration in which the delamination crack is located between layers 2 and 3 is higher than that in the beam in which the delamination crack is between layers 1 and 2. The influence of the material gradient on the strain energy release rate is also evaluated. The three-layered functionally graded cantilever beam configuration with a delamination crack located between layers 2 and 3 is considered (Fig. 4a). The material gradient along the width of layer 1 is characterized by 11 q , 21 q and 31 q . The influences of 11 q and 21 q on the strain energy release rate are illustrated in Fig. 6 where the strain energy release rate in non-dimensional form is potted against 11 q at three values of 21 q . The curves in Fig. 6 indicate that the strain energy release rate decreases with increasing of 11 q . One can observe also in Fig. 6 that the increase of 21 q leads to decrease of the strain energy release rate. The influence of 31 q on the strain energy release rate is evaluated too. For this purpose, the strain energy release rate is plotted against 31 q in Fig. 7 at three values of  v . The three-layered functionally graded beam with a