Two-step simulation of piezoelectric properties of porous PZT according to porosity

Figures & data

Figure 1. Finite element model for calculating d_ij of KICET-PZT8 ceramics: (a) 2D conceptual model and (b) 3D model.

Figure 2. Flowchart for the property optimization of KICET-PZT8 using the finite element method.

Table 1. Effect of frequency of KICET-PZT8 in radial and thickness modes on various parameters.

Increased Parameter	Radial mode				Thickness mode
	fr1(rad)	fa1(rad)	fr2(rad)	fa2(rad)	fr(TE)	fa(TE)
${\mathbf{s}}_{11}^E$	←	←	←	←	←	←
$\left|{\mathbf{s}}_{12}^E\right|$	→	→	→	→	→	→
$\left|{\mathbf{s}}_{13}^E\right|$	-	-	-	-	→	→
${\mathbf{s}}_{33}^E$	-	-	-	-	←	←
${\mathbf{s}}_{66}^E$	-	-	-	-	-	-
${\mathbf{s}}_{11}^E$+${\mathbf{s}}_{12}^E$	←	←	←	←	←	←
$\left|d_{31}\right|$	-	→	-	-	-	←
$d_{33}$	-	-	-	-	→	→
${\epsilon }_{11}^T$/${\epsilon }_0$	-	-	-	-	-	←
${\epsilon }_{33}^T$/${\epsilon }_0$	-	←	-	-	-	←
Poisson’s Ratio	-	←	←	←	-	-

Figure 3. Micrographs of sintered KICET-PZT8 ceramics with different porosities: (a) 7% (b) 13% (c) 22% (d) 30% (e) 39%.

Table 2. Pore characteristics of the fabricated porous KICET-PZT8 ceramics according to PMMA vol%.

PMMA Vol (%)	10	20	30	40	50	60
Porosity (%)	7	13	22	30	39	48
average pore size (μm)	42.08	40.81	33.84	34.16	37.77	33.93

Figure 4. Calculated change in d_ij of KICET-PZT8 according to the number of pores at 30% porosity: (a) d₃₃ (b) d_31.

Figure 5. Simulation results of piezoelectric constant at various porosities: (a) d₃₃ and d₃₁ (b) d_h (c) g_h and (d) d_h*g_h.

Table 3. Normalized mechanical and piezoelectric properties of KICET-PZT8 calculated using h-parametric estimation.

Porosity Property		0%	7%	13%	22%	30%	39%
Mechanical properties	$s_{11}^E$ (${10}^{-10}{m}^2/)$	1.00	1.10	1.31	1.65	2.09	2.94
	-$s_{12}^E$ (${10}^{-10}{m}^2/N)$	1.00	1.09	1.48	1.79	2.35	3.30
	-$s_{13}^E$ (${10}^{-10}{m}^2/N)$	1.00	1.02	1.09	1.25	1.29	1.72
	$s_{33}^E$ (${10}^{-10}{m}^2/N)$	1.00	1.03	1.25	1.75	2.21	3.61
	$s_{66}^E$ (${10}^{-10}{m}^2/N)$	1.00	1.10	1.36	1.69	2.16	3.03
Piezoelectric constants	$d_{33}$ (pC/N)	1.00	0.97	0.95	0.93	0.91	0.91
	-$d_{31}$ (pC/N)	1.00	0.94	0.80	0.51	0.45	0.31
	$d_h$ (pC/N)	1.00	1.07	1.49	2.43	2.55	3.04
	$g_h$ (${10}^{-3}Vm/N$)	1.00	1.13	1.81	3.98	4.95	7.59
	$d_h\ast g_h$ (${10}^{-15}{m}^2/N)$	1.00	1.21	2.69	9.67	12.62	23.06
Dielectric constants	${\epsilon }_{33}^T/{\epsilon }_0$	1.00	0.95	0.87	0.61	0.51	0.40
	${\epsilon }_{33}^S/{\epsilon }_0$	1.00	0.96	0.92	0.81	0.67	0.56
Frequency Constants	$N_D\left(Hz\ast m\right)$	1.00	0.98	0.96	0.88	0.83	0.76
	$N_t\left(Hz\ast m\right)$	1.00	0.95	0.84	0.68	0.61	0.49
Density ($kg/m^3)$		1.00	0.93	0.87	0.78	0.70	0.61

* Each result was normalized with respect to the value at 0% porosity.

Figure 6. Fabricated KICET-PZT8 samples with various vibration modes to derive piezoelectric properties using a resonance method based on IEEE standards.

Figure 7. Comparison of the mechanical properties of KICET-PZT8 optimized through parametric estimation and experimental results with various porosities: (a) $s_{11}^E$ (b) $s_{12}^E$ (c) $s_{13}^E$ (d) $s_{33}^E$ and (e) $s_{66}^E$.

Figure 8. Comparison of piezoelectric and dielectric constants of KICET-PZT8 optimized through parametric optimization and experimental results with various porosities: (a) $d_{33}$ (b) $d_{31}$ (c) $d_h$ (d) $g_h$ (e) $d_h\ast g_h$ (f) ${\epsilon }_{33}^S/{\epsilon }_0$ and (g) ${\epsilon }_{33}^T/{\epsilon }_0$.

Figure 9. Comparison of coupling factors and frequency constants of KICET-PZT8 optimized through parametric optimization and experimental results with various porosities: (a) $k_t$ (b) $k_{33}$ (c) $N_t$, and (d) $N_D$.