PSI - Issue 6

Anna Kudimova et al. / Procedia Structural Integrity 6 (2017) 301–308 Author name / Structural Integrity Procedia 00 (2017) 000–000

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element to join the granule. There are up to six possible directions in which we can add new element. Random dis tribution for each candidate allows us to construct granules of natural shape. 3D examples of di ff erent representative volumes for granular composite with 3–0 connectivity are shown in Fig. 2.

5. Numerical Examples and Conclusion

By using the model described in the previous section and the models of other types [Nasedkin and Shevtsova (2011, 2013)], we can obtain e ff ective material properties of porous piezoceramic and ceramic polycrystalline piezocompos ites. Thus, after solving static piezoelectric problems (1)–(5) with di ff erent boundary conditions (6), (12), (13) or (14) for these representative volumes numerically, we will find the full set of the e ff ective moduli for two phase composite piezoceramic. Such approach allows us to determine the properties of the intrinsic structure of pores or inclusions and their distribution.

Table 1. E ff ective moduli of porous piezoceramic PZT-4, p = 20%. Material moduli [Nasedkin (2015)], 3–0,

3–0,

3–0,

3–3,

3–3,

3–3,

n = 20

n = 8

n = 16

n = 32

n = 8

n = 16

n = 32

10 (N / m 2 ) 10 (N / m 2 ) 10 (N / m 2 ) 10 (N / m 2 ) 10 (N / m 2 )

˜ c 11 ∗ 10 ˜ c 12 ∗ 10 ˜ c 13 ∗ 10 ˜ c 33 ∗ 10 ˜ c 44 ∗ 10 ˜ e 33 (C / m − ˜ e 31 (C / m ˜ e 15 (C / m

9.23 4.66 4.28 7.29 1.84 3.14 8.89 588 488 0.99 0.90 1.34 1.22 11.45

9.27 4.82 4.52 7.53 1.92 3.08 9.27 571 496 1.01 0.94 1.34 1.25 11.60

8.40 4.27 3.95 7.03 1.84 2.51 8.70 566 498 0.99 0.89 1.35 1.21 11.60

7.99 3.98 3.67 6.56 1.77 2.01 8.31 571 497 0.99 0.85 1.39 1.19 11.28

9.28 4.72 4.48 7.66 1.93 3.23 9.31 576 496 0.99 0.93 1.31 1.23 11.80

8.82 4.30 4.01 7.00 1.82 2.90 8.78 584 488 0.99 0.89 1.35 1.22 11.39

8.35 4.05 3.71 6.65 1.75 2.54 8.35 581 487 0.99 0.86 1.38 1.20 11.24

2 )

2 )

2 )

˜ κ 11 /ε 0 ˜ κ 33 /ε 0 r ( d 33 ) r ( d 31 ) r ( g 33 ) r ( g 31 )

Table 2. E ff ective moduli of porous piezoceramic PZT-4, p = 80%. Material moduli [Nasedkin (2015)],

3–3,

3–3,

3–3,

n = 20

n = 8

n = 16

n = 32

10 (N / m 2 ) 10 (N / m 2 ) 10 (N / m 2 ) 10 (N / m 2 ) 10 (N / m 2 )

˜ c 11 ∗ 10 ˜ c 12 ∗ 10 ˜ c 13 ∗ 10 ˜ c 33 ∗ 10 ˜ c 44 ∗ 10 ˜ e 33 (C / m − ˜ e 31 (C / m ˜ e 15 (C / m

0.68 0.13 0.10 0.47 1.77 1.24 0.10 0.44 121

0.75 0.23 0.19 0.63 1.93 1.57 0.10 0.82 116 0.99 0.52 9.04 4.80 87

0.44 0.12 0.11 0.41 1.82 1.09 0.02 0.50

0.31 0.07 0.06 0.22 1.75 0.61 0.01 0.29

2 )

2 )

2 )

86 76

82 66

˜ κ 11 /ε 0 ˜ κ 33 /ε 0 r ( d 33 ) r ( d 31 ) r ( g 33 ) r ( g 31 )

75

0.98 0.39

1.03 0.49

1.02 0.36

10.78

11.57

15.09

4.30

5.57

5.39

In Tables 1, 2 we give the calculation results for the e ff ective moduli and the relative values of the main piezomoduli r ( d 3 j ) = ˜ d 3 j / d 3 j and r ( g 3 j ) = ˜ g 3 j / g 3 j ( j = 1 , 3) for porous piezoceramic PZT-4 in comparison with the results [Nased-