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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kin (2015)], obtained from calculations in ANSYS using similar approaches for a representative volume supporting the connectivity of only one phase. Here d 3 j and g 3 j are the corresponding values for the dense piezoceramic, d = e · c − 1 , g = ( ε T ) − 1 · d , ε T = κ + e · d ∗ . For dense piezoceramic PZT-4 we take the following mechanical, dielectric and piezoelectric constants : c E 11 = 13 . 9 · 10 10 ; c E 12 = 7 . 78 · 10 10 ; c E 13 = 7 . 43 · 10 10 ; c E 33 = 11 . 5 · 10 10 ; c E 44 = 2 . 56 · 10 10 (N / m 2 ); e 31 = − 5 . 2; e 33 = 15 . 1; e 15 = 12 . 7 (C / m 2 ); ϵ S 11 = 730 ε 0 ; ϵ S 33 = 635 ε 0 ; ε 0 = 8 . 85 · 10 − 12 (F / m). As we can conclude from Tables 1, 2, for the representative volumes with 3–0 and 3–3 connectivity for various pore sizes or for various cross-section areas of percolation paths we obtain slightly di ff erent values of the e ff ective moduli. The connectivity model 3–3 is slightly sti ff er than the 3–1 model, and a decrease of the relative pore sizes while increasing their number leads to a reduction of the overall rigidity of the representative volume. We can also note, that the charge coe ffi cient ˜ d 33 practically does not decrease with the porosity increment, but at the same time the piezomodule ˜ d 31 decreases su ffi ciently fast. Such properties of di ff erent porous piezoceramics were also obtained from experimental data [Getman and Lopatin (1996); Rybyanets (2011)] and from another computational results [Getman and Lopatin (1996); Nasedkin (2015); Nasedkin and Shevtsova (2011, 2013)]. On the contrary, the voltage coe ffi cients ˜ g 33 and ˜ g 33 increase with the porosity increment. These dependencies of the piezomoduli suggest important practical applications of porous piezoceramics in many areas [Rybyanets (2011)]. As a result, from the finite element solutions we can conclude that high-porous piezoceramic is a high-performance material for quasistatic and low-frequency applications in piezoelectric energy generators, because its piezo-sensitivity rapidly increases with the porosity growth. We can also note that porous ceramic is preferable for di ff erent hydro acoustic ultrasonic applications [Kara et al. (2003); Ringgaard et al. (2015); Rybyanets (2011)]. In Table 3 we give the results of calculations of e ff ective moduli for piezoceramic PZT-4 with inclusions of sapphire ( α –corundum) crystallites Al 2 O 3 . The e ff ective moduli for polycrystalline inclusions are calculated as the average moduli of monophase polycrystallite of trigonal system¡ and we assume α –corundum as an isotropic phase with c E 11 = 46 . 88 · 10 10 ; c E 12 = 14 . 22 · 10 10 (N / m 2 ); ϵ S 11 = 730 ε 0 ; ϵ S 33 = 10 ε 0 .

Table 3. E ff ective moduli of piezoceramic PZT-4 with α –corundum as inclusions, p = 20%. Material moduli 3–0, 3–0,

3–0,

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

19.22

18.10

17.82

8.71 8.28

8.64 8.22

8.67 8.28

16.94

16.84

16.55

4.78

4.53

4.40

2 )

11.72

11.59

11.45

2 )

4.20

4.44 9.94 545 482 0.42 0.44 0.77 0.80

4.50 9.97 540 473 0.43 0.45 0.79 0.84

2 )

10.13

561 488 0.41 0.40 0.75 0.75

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

As expected, with the addition of a more rigid material, the sti ff ness of the piezoceramic increases. At the same time, with the same proportion of inclusions for smaller dimensions, but with a larger number of them, the sti ff ness of the composite decreases. Thus, ACELAN-COMPOS package allows us to determine the properties of two-phase piezoceramic materials with 3–0 and 3–3 connectivity. Other possibilities of ACELAN package, which is now being actively developed, will be associated with the di ff erent types of the representative volumes and with the modeling of surface e ff ects and surface finite elements for piezoelectric composite media on the micro- and nanoscale [Eremeyev and Nasedkin (2017)].