Verification of the present homogenisation procedures

An annular PFC actuator for shear mode piezoelectric actuation of plane structures of

5.4 Results and discussions

5.4.1 Verification of the present homogenisation procedures

Since a similar shear mode PFC actuator (in cylindrical coordinates) is not available in the literature, the present FE procedure for the numerical homogenization of the PFC actuator is first verified by taking the RVE (Fig.

5.5(b)) of the active (2-2 PFC) layer in the Cartesian coordinate system. If the radial and circumferential spans of the RVE are of small values, then a high value of the inner radius yields its (RVE) geometry similar to that in the Cartesian coordinate system. The material and geometrical properties of this RVE are taken from (Benjeddou and Al-Ajmi, 2011; Trindade and Benjeddou 2011) and the magnitudes of the effective coefficients are evaluated using the present FE procedure. These results are illustrated in Table 5.2 together with the similar analytical and FE results available in the literature (Benjeddou and Al-Ajmi, 2011; Trindade and Benjeddou, 2011). It may be observed from Table 5.2 that the present FE results are in good agreement with the available analytical/FE results (Benjeddou and Al-Ajmi, 2011; Trindade and Benjeddou 2011). This comparison verifies the present FE procedure for estimation of the effective coefficients of the 2-2 PFC layer. Next, for the construction of the RVE (Fig. 5.5(b)) in the cylindrical coordinate system, the magnitudes of the effective coefficients are computed using the present FE procedure. The magnitudes of the same effective coefficients are also computed using the present analytical expressions (Eq. (5.5c)). These results are illustrated in Table 5.3 (Shear mode 2- 2 PFC in cylindrical coordinate system) for an FVF (v_f ) and other geometrical properties of the RVE (Fig. 5.5(b)) as, v_f 0.8, r_i 0.2 m, (r_or_i) 500 µm, hf  500 µm, h_c h_f , w_f h_f / 5 , θf ^sin1



wf ^{/ 2}ri



,



_c



_f / 0.8. It may be observed from Table 5.3 (Shear mode 2-2 PFC in cylindrical coordinate system) that the results obtained from the FE procedure are very close to the results

obtained from the analytical expressions. This comparison verifies the numerical accuracy in the present computation of effective coefficients of the shear mode 2-2 PFC (Fig. 5.5(b)). Next, The magnitudes of the effective coefficients of a typical sub-volume (Fig. 5.4(a)) are computed using the present FE procedure.

These results are illustrated in Table 5.3 (Sub-volume of the shear mode PFC actuator). The FVF (v_f ) and other geometrical properties of the sub-volume are taken as, v_f 0.76, r_i  0.2 m, (r_or_i) 500 µm, h_f  500 µm, h_c h_f / 0.95,

f f / 5

w h , θf ^sin1



wf ^{/ 2}ri



,



_c



_f / 0.8 . Table 5.3 also contains the magnitudes of the same effective coefficients obtained from the present analytical expressions (Eq. (5.6b)).It may be observed from these results that the

FE results are in excellent agreement with the analytical results, and thus the numerical accuracy in the present computation of effective coefficients of a typical sub-volume is verified. The FE results (Table 5.3) for the effective properties of the sub-volume are obtained by applying zero or non-zero homogeneous electric field across the thickness of the active (2-2 PFC) layer.

These FE results are verified with the analytical results (Table 5.3), and this verification infers that the numerical homogenization of the sub-volume may be carried out by assuming the electric field in the active layer as the effective electric field within the domain of the sub-volume. For further verification of the present numerical homogenization procedure due to this consideration of the effective electric field, an electric field (E_z^a  V_a /h_f , V_a (

 

_a _b),

 

_b  _t 0) is Table 5.2 Verification of the present FE procedure for homogenization of the

2-2 PFC layer.

Parameters FE results (ANSYS) (Trindade and Benjeddou, 2011)

UFM (Benjeddou

and Al-Ajmi, 2011)

Present FE

E1 or E₂ (GPa) 50.09 52.76 52.74

E3 (GPa) 20.52 22.49 22.72

G23or G₁₃(GPa) 6.49 5.89 6.49

12 0.43 0.59 0.51

13 or ₂₃ 0.2 0.19 0.1

e15 (C/m²) 3.13 2.84 3.13

11

T (nF/m) 14.85 14.85 14.85

Chapter 5: Design of shear mode annular PFC actuator

applied across the thickness of the 2-2 PFC layer within the heterogeneous sub- volume. The corresponding volume-average free shear strain (_rz ) over the volume of the heterogeneous sub-volume is computed using the FE model of the RVE. In parallel, an electric field with the same magnitude of the previous appli- Table 5.3 Magnitudes of the effective coefficients of the shear mode 2-2 PFC or a typical sub-volume of the shear mode PFC actuator (C_ij in GPa, e_ij in C/m²,

ij in 1e-9 F/m).

Shear mode 2-2 PFC in

cylindrical coordinate system Sub-volume of the shear mode PFC actuator

Coefficients UFM FE UFM FE

C

11 56.15 56.05 47.49 47.42

C

22 17.38 17.31 16.15 16.27

C

33 67.99 67.89 37.31 36.55

C

44 4.68 4.67 4.026 4.085

C

55 18.61 18.59 10.425 10.320

C

66 _4.67 5.02 4.824 4.899

C

12 10.74 10.69 8.543 8.629

C

13 31.55 31.49 17.24 16.96

C

23 10.26 10.21 6.132 6.216

e35 13.62 13.61 7.25 7.17

33 12.08 12.07 15.92 15.91

-ed electric field ( E^a_z  V_a/h_f ) is applied across the thickness (h_c ) of the homogenised sub-volume by assuming a potential difference (V_c) as, V_c E h^a_{z c}. The material properties of the homogenised sub-volume are taken from the FE results as presented in Table 5.3 and the volume-average free shear strain (_rz) over the volume of the asymptotically homogeneous sub-volume is computed using the FE model. These results are illustrated in Table 5.4 for three different values of the applied electric field (E_z E_z^a). It may be observed from Table 5.4 that the magnitude of the volume-average free strain (_rz) of the heterogeneous sub-volume is very close to that of the homogenised sub-volume for any value of the applied electric field. This comparison study shows that the applied electric field across the thickness of the active (2-2 PFC) layer may be assumed as the effective electric field within the domain of the sub-volume, and the

homogenization of the sub-volume may be carried out on the basis of this electric field.

Table 5.4 Verification of the volume-average free shear strain (_rz) (PFC-Hom:

Homogenised sub-volume, PFC-Het: Heterogeneous sub-volume).

Ez

(Volt/m)

Free strain (_rz) of PFC-Het

Free strain (_rz) of PFC-Hom

100e3 0.694e-4 0.691e-4

200e3 1.389e-4 1.382e-4

300e3 2.084e-4 2.073e-4