Hygro-thermo-mechanical bending of FG piezoelectric plates using quasi-3D shear and normal deformations theory

Zenkour, Ashraf M.; Hafed, Zahra S.

doi:10.1590/1679-78255396

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Latin American Journal of Solids and Structures

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ORIGINAL ARTICLE • Lat. Am. j. solids struct. 16 (07) • 2019 • https://doi.org/10.1590/1679-78255396 copy

Hygro-thermo-mechanical bending of FG piezoelectric plates using quasi-3D shear and normal deformations theory

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

A simple quasi-3D sinusoidal shear and normal deformations theory for the hygro-thermo-mechanical bending of functionally graded piezoelectric (FGP) plate is developed under simply-supported edge conditions. The governing equations are deduced based on the principle of virtual work. The exact solutions for FGP plate are obtained. The current study investigates the effect of some parameters, like piezoelectricity, hygrothermal parameter, gradient index and electric loading on the mechanical and electric displacements, electric potential and stresses. They are explored analytically and numerically presented and discussed in detail. The numerical results clearly show the effect of piezoelectric and hygrothermal parameter on the FGP plate.

Keywords:
Functionally graded; hygrothermal piezoelectric plates; bending; quasi-3D; analytical solutions

Methods	$a / h = 10$
Methods	$\bar{w}$	${\bar{σ}}_{11}$	${\bar{σ}}_{22}$	${\bar{σ}}_{12}$	${\bar{σ}}_{33}$
Benyoucef et al. 2010Benyoucef, S., Mechab, I., Tounsi, A., Fekrar, A., Atmane, H.A. and Adda Bedia, E.A., (2010). Bending of thick functionally graded plates resting on Winkler-Pasternak elastic foundations. Mechanics of Composite Materials 46: 425-434. Thai and Vo 2013Thai, H.T., Vo, T.P., (2013). A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates. Applied Mathematical Modelling 37: 3269-81. Al-Khateeb and Zenkour 2014Al Khateeb, S.A., Zenkour, A.M., (2014). A refined four-unknown plate theory for advanced plates resting on elastic foundations in hygrothermal environment. Composite Structures 111: 240-248. Present	0.2960 0.2960 0.2936 0.2933	1.9955 1.9955 2.0211 2.1247	1.3121 1.3121 1.3240 1.2905	0.7065 0.7065 0.6932 0.6985	- - 0.1489 0.3623

$a / b$	$k$	$a / h = 2$				$a / h = 10$
$a / b$	$k$	FG	$V_{0} = - 0.5$	$V_{0} = 0$	$V_{0} = 0.5$	FG	$V_{0} = - 0.5$	$V_{0} = 0$	$V_{0} = 0.5$
1/3	0.5	1.1722	0.8264	0.5748	0.3232	0.7232	0.4290	0.4179	0.4068
	1.0	0.9057	0.8301	0.4425	0.0550	0.5605	0.3432	0.3261	0.3090
	1.5	0.6961	0.7824	0.3382	-0.1059	0.4333	0.2741	0.2545	0.2349
1/2	0.5	0.9701	0.6896	0.4680	0.2464	0.5733	0.3408	0.3309	0.3211
	1.0	0.7493	0.7012	0.3599	0.0186	0.4443	0.2734	0.2582	0.2430
	1.5	0.5758	0.6657	0.2745	-0.1167	0.3434	0.2189	0.2015	0.1841
1	0.5	0.4678	0.3417	0.2088	0.0758	0.2280	0.1369	0.1308	0.1247
	1.0	0.3611	0.3644	0.1596	-0.0450	0.1766	0.1114	0.1020	0.9258
	1.5	0.2771	0.3553	0.1207	-0.1138	0.1365	0.0904	0.7955	0.0687

$a / b$	$k$	$a / h = 2$				$a / h = 10$
$a / b$	$k$	FG	$V_{0} = - 0.5$	$V_{0} = 0$	$V_{0} = 0.5$	FG	$V_{0} = - 0.5$	$V_{0} = 0$	$V_{0} = 0.5$
1/3	0.5	1.4383	1.4460	1.2670	1.0881	6.3022	5.7601	5.7241	5.6880
	1.0	1.7085	1.7859	1.4639	1.1419	7.4271	6.6412	6.5766	6.5121
	1.5	2.0216	2.2925	1.6861	1.0797	8.7298	7.6770	7.5562	7.4353
1/2	0.5	1.2189	1.4306	1.1001	0.7695	5.2003	4.9286	4.8619	4.7951
	1.0	1.4490	1.7440	1.2723	0.8006	6.1262	5.6803	5.5848	5.4893
	1.5	1.7156	2.2146	1.4666	0.7185	7.2006	6.5681	6.4170	6.2658
1	0.5	0.6841	1.4517	0.6695	-0.1127	2.5027	2.7786	2.6197	2.4608
	1.0	0.8169	1.6922	0.7789	-0.1344	2.9439	3.1954	3.0072	2.8190
	1.5	0.9699	2.0600	0.9010	-0.2579	3.4602	3.6981	3.4564	3.2147

$a / b$	$k$	$a / h = 2$				$a / h = 10$
$a / b$	$k$	FG	$V_{0} = - 0.5$	$V_{0} = 0$	$V_{0} = 0.5$	FG	$V_{0} = - 0.5$	$V_{0} = 0$	$V_{0} = 0.5$
1/3	0.5	0.6986	2.2745	0.8808	-0.5129	2.6542	3.9477	3.6658	3.3839
	1.0	0.8285	2.5440	1.0207	-0.5025	3.1040	4.5114	4.1991	3.8868
	1.5	0.9803	2.9258	1.1809	-0.5639	3.6446	5.1901	4.8270	4.4639
1/2	0.5	0.7239	2.0841	0.8435	-0.3970	2.7683	3.7415	3.4904	3.2392
	1.0	0.8599	2.3483	0.9782	-0.3919	3.2441	4.2819	4.0006	3.7193
	1.5	1.0181	2.7285	1.1314	-0.4656	3.8103	4.9313	4.5986	4.2658
1	0.5	0.6841	1.4517	0.6695	-0.1127	2.5027	2.7786	2.6197	2.4608
	1.0	0.8169	1.6922	0.7789	-0.1344	2.9439	3.1954	3.0072	2.8190
	1.5	0.9699	2.0600	0.9010	-0.2579	3.4602	3.6981	3.4564	3.2147

$a / b$	$k$	$a / h = 2$				$a / h = 10$
$a / b$	$k$	FG	$V_{0} = - 0.5$	$V_{0} = 0$	$V_{0} = 0.5$	FG	$V_{0} = - 0.5$	$V_{0} = 0$	$V_{0} = 0.5$
1/3	0.5	-0.2758	0.3112	-0.1438	-0.5989	0.7061	0.5113	0.4250	0.3387
	1.0	-0.3266	0.2861	-0.1639	-0.6140	0.6502	0.4883	0.4051	0.3219
	1.5	-0.3821	0.2426	-0.1839	-0.6104	0.5923	0.4593	0.3798	0.3002
1/2	0.5	-0.3272	0.4368	-0.1692	-0.7753	0.8362	0.6181	0.5030	0.3880
	1.0	-0.3868	0.4061	-0.1921	-0.7905	0.7699	0.5904	0.4794	0.3685
	1.5	-0.4502	0.3518	-0.2136	-0.7790	0.7013	0.5555	0.4494	0.3434
1	0.5	-0.2607	0.6234	-0.1304	-0.8844	0.6504	0.5339	0.3901	0.2462
	1.0	-0.3068	0.5924	-0.1462	-0.8850	0.5990	0.5107	0.3720	0.2332
	1.5	-0.3524	0.5317	-0.1575	-0.8469	0.5454	0.4814	0.3487	0.2160

$k$	$a / h = 5$			$a / h = 10$			$a / h = 20$
$k$	$\bar{w}$	${\bar{σ}}_{11}$	${\bar{σ}}_{33}$	$\bar{w}$	${\bar{σ}}_{11}$	${\bar{σ}}_{33}$	$\bar{w}$	${\bar{σ}}_{11}$	${\bar{σ}}_{33}$
ceramic	0.18344	0.8776	-3.6692	0.16789	2.1380	-1.9652	0.16387	4.4688	-1.2452
1	0.20830	0.9301	-3.7520	0.21143	2.4227	-2.2562	0.21234	5.1285	-1.8898
2	0.21498	1.0374	-3.8338	0.22347	2.5620	-2.3813	0.22571	5.3700	-2.1220
3	0.22318	1.1308	-3.9006	0.23025	2.6654	-2.4664	0.23212	5.5335	-2.2664
4	0.23103	1.2091	-3.9570	0.23547	2.7509	-2.5347	0.23658	5.6680	-2.3800
5	0.23808	1.2748	-4.0044	0.23989	2.8245	-2.5931	0.24034	5.7880	-2.4788
6	0.24428	1.3303	-4.0440	0.24375	2.8895	-2.6420	0.24357	5.8935	-2.5648
7	0.24967	1.3782	-4.0758	0.24713	2.9461	-2.6857	0.24639	5.9865	-2.6384
8	0.25369	1.4187	-4.0944	0.24999	2.9942	-2.7173	0.24890	6.0685	-2.7009
9	0.22865	1.4183	-3.6682	0.24542	3.0230	-2.5193	0.24942	6.1320	-2.6432
metal	0.34951	0.8495	-3.6552	0.30302	2.1237	-1.9589	0.29119	4.4614	-1.2422

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