PSI - Issue 51

S. Zhao et al. / Procedia Structural Integrity 51 (2023) 69–75

74

6

S. Zhao et al./ Structural Integrity Procedia 00 (2022) 000–000

The lower material: elastic constants (GPa): 11 234.33 c  , 12 57.41 c  , 13 66.63 c  , 33 232.22 c  , 44 70.19 c  ,

1 122 K  ,

24

2 K 

,

1 3 0.8846 R R R    ; piezoelectric constants ( dielectric constants ( 2

4.4

, 33 18.6 e  , 15 11.6 e  , 15 1.16 e   , 33 1.86 e   ;

31 e 

-2 Cm ):

11 5   , 33 10   .

-9 2 -1 -2 10 C N m ):

(23)

The variation of the ERR at the right crack tip for 100 mm a  and different values of the remote shear stress   are presented in Fig. 2. Line I corresponds to the bi-material mentioned above, lines II and III are drawn for the particular cases of the homogeneous material with the characteristics equal to the lower and upper materials, respectively. It can be seen that the ERR for the composite material are approximately equal to the mean value of the energy release rates for the homogeneous materials which characteristics are equal to those of each components of the composite. The similar variation of the ERR for 1 MPa    , 0    , 100 mm a  and different values of H  are drawn in Fig. 3. Line I corresponds to the bi-materials (22) and (23), lines II and III are obtained for the same cases of the homogeneous material as in Fig. 2. It is interesting to mention that usually the values of ERR increase with increasing of H  . However, this growing is rather weak in the whole considered interval for the homogeneous material (22) (line II ), it is almost constant in the interval   0,0.5 for the bi-material case (line I ) and the ERR even decreases at the initial section of the considered region for the homogeneous material (23) (line III ). 1 MPa    , 0 H   ,

16

0 1 2 3 4 5 6 7

ERR

ERR

12

III

I

III

I

8

4

II

II

0

5 10 H 

6 10  

0

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

Fig. 2. The variation of the ERR N/m with respect to the remote phonon shear stress   ( Pa ).

Fig. 3. The variation of the ERR ( N/m ) with respect to the phason stress H  ( Pa ).

To confirm the correctness of the obtained results we found the ERR for the case of homogeneous QC with characteristics (23), in which we assumed 13 94 c  GPa , 33 234.33 c  GPa , 1 2 3 0 R R R    and all piezoelectric constants approximately equal to 0. In this case we got the uncoupled isotropic material with Young’s modulus 180 E  GPa and Poison’s ratio 0.286   . The calculation of the ERR for such material with 1MPa    , 0    , 0 H   and 100mm a  gives 1.598 G  N/m. This value completely coincides with the known analytical result obtained on the formula     2 2 1 / a E      . By the way the value of the ERR for this loading and homogeneous QC with characteristics (23) is 1.527 N/m (see line II of Fig. 2).

5. Conclusions

The problem of electrically permeable crack between dissimilar 1D hexagonal quasicrystals with piezoelectric effect is considered. Mixed mode phonon and phason loading and electric displacement can be prescribed remotely from the crack. The main attention is devoted to the analytical determination of the energy release rate at the crack