Three-dimensional static analysis of a viscoelastic rectangular functionally graded material plate embedded between piezoelectric sensor and actuator layers

Figures & data

Figure 1. Sandwich plate with FGM layer sandwiched between two piezoelectric layers.

Table 1. Material properties of piezoelectric sensor and actuator (Alibeigloo 2010).

Elasticity moduli [10^9N m^−2]	$Q_{11}$	$Q_{12}$	$Q_{13}$	$Q_{22}$	$Q_{23}$	$Q_{33}$	$Q_{44}$	$Q_{55}$	$Q_{66}$
(PZT-4)	139	78	74	139	74	115	25.6	25.6	30.5
$(B{a}_{2}NaN{b}_5{O}_{15})$	239	104	50	274	52	135	65	66	76
Piezoelectric coefficients [cm^−2]	$e_{31}$	$e_{32}$	$e_{33}$	$e_{24}$	$e_{15}$
(PZT-4)	−5.2	−5.2	15.1	12.7	12.7
$(B{a}_{2}NaN{b}_5{O}_{15})$	−0.4	−0.3	4.3	3.4	2.8
Permittivities [10^−9 As V^−1m^−1]	${\epsilon }_{11}$	${\epsilon }_{22}$	${\epsilon }_{33}$
(PZT-4)	6.5	6.5	5.6
$(B{a}_{2}NaN{b}_5{O}_{15})$	1.96	2.01	0.28

Figure 2. Distribution of transverse normal stress, shear stress, and deflection along the thickness direction for different half wave numbers (m, n), using 15 fictitious layers for the FGM core layer.

Table 2. Comparison of stresses and displacements of an elastic square FGM plate under a mechanical load.

	$\frac{a}{h}=4$		$\frac{a}{h}=10$
	(Brischetto et al. 2008)	Analytical	(Brischetto et al. 2008)	Analytical
${\overline{w}}_{(h)}$	−13.46	−13.38	−168.9	−168.8
${\overline{w}}_{(\frac{h}{2})}$	−13.70	−13.77	−170.7	−170.39
${\overline{w}}_{(0)}$	−12.73	−12.80	−168.5	−168.3
${\overline{u}}_{(h)}$	4.02	4.09	26.17	26.165
${\overline{u}}_{(\frac{h}{2})}$	−0.08998	−0.090	0.7108	0.7121
${\overline{u}}_{(0)}$	−4.069	−4.08	−24.72	−24.78

Figure 3. Through-the-thickness distribution of the non-dimensional transverse normal stress, inplane normal stress, and transverse shear stress for different numbers of mathematical layers for the FGM core.

Figure 4. Effect of length to thickness ratio, a/h, on through-thickness distribution of nondimensional in-plane and transverse shear stresses and longitudinal displacement.

Figure 5. Effect of piezoelectric thick on distribution of non-dimensional in-plane and transverse shear stresses and longitudinal displacement.

Figure 8. Time history of longitudinal displacement, transverse normal and shear stresses at the height of $\frac{(h_p+h_f+h_p)}{2}$ for different time constants.

Table 3. Maximum stresses and displacements of the sandwich FGM plate in elastic ($\mathrm{\tau }=0$) and viscoelastic ($\mathrm{\tau }\mathrm{>}0$) cases for different $\frac{a}{h}$ ratio.

$\frac{a}{h}$	$\tau$	${\overline{\sigma }}_x$	$\overline{u}$	$\overline{w}$	${\overline{\tau }}_{xz}$	${\overline{\tau }}_{xy}$
10	0 (s)	33.53	0.3	1.7	5.0	−22.55
	5 (s)	36.62	0.2	0.8	5.1	−24.68
	10 (s)	36.76	0.2	0.8	5.1	−24.77
20	0 (s)	133.78	2.7	25.7	10	−90.39
	5 (s)	143.13	1.3	12.5	10.3	−98.78
	10 (s)	146.70	1.2	11.9	10.3	−99.17
40	0 (s)	534.91	21.3	407.1	20.0	−362.05
	5 (s)	584.31	10.4	198	20.6	−395.54
	10 (s)	586.57	9.9	188	20.6	−397.10

Alibeigloo, A. 2010. Thermo-elasticity solution of functionally graded plates integrated with piezoelectric sensor and actuator layers. Journal of Thermal Stresses 33 (8):754–74. doi:10.1080/01495739.2010.482350.

 Web of Science ®Google Scholar

Brischetto, S., R. Leetsch, E. Carrera, T. Wallmersperger, and B. Kröplin. 2008. Thermo-mechanical bending of functionally graded plates. Journal of Thermal Stresses 31 (3):286–308. doi:10.1080/01495730701876775.

 Web of Science ®Google Scholar