Static and dynamic analysis of functionally graded fluid-infiltrated porous skew and elliptical nanoplates using an isogeometric approach

doi:10.1007/s11709-023-0918-5

Front. Struct. Civ. Eng.

2023, Vol. 17

Issue (3) : 477-502 https://doi.org/10.1007/s11709-023-0918-5

RESEARCH ARTICLE

Static and dynamic analysis of functionally graded fluid-infiltrated porous skew and elliptical nanoplates using an isogeometric approach

Tran Thi Thu THUY(

)

Faculty of Mechanical Engineering, Hanoi University of Industry, Hanoi 100000, Vietnam

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Abstract

The analysis of static bending and free and forced vibration responses of functionally graded fluid-infiltrated porous (FGFP) skew and elliptical nanoplates placed on Pasternak’s two-parameter elastic foundation is performed for the first time using isogeometric analysis (IGA) based on the non-uniform rational B-splines (NURBSs) basis function. Three types of porosity distributions affect the mechanical characteristics of materials: symmetric distribution, upper asymmetric distribution, and lower asymmetric distribution. The stress–strain relationship for Biot porous materials was determined using the elastic theory. The general equations of motion of the nanoplates were established using the four-unknown shear deformation plate theory in conjunction with the nonlocal elastic theory and Hamilton’s principle. A computer program that uses IGA to determine the static bending and free and forced vibration of a nanoplate was developed on MATLAB software platform. The accuracy of the computational program was validated via numerical comparison with confidence assertions. This set of programs presents the influence of the following parameters on the static bending and free and forced vibrations of nanoplates: porosity distribution law, porosity coefficient and geometrical parameters, elastic foundation, deviation angle, nonlocal coefficient, different boundary conditions, and Skempton coefficients. The numerical findings demonstrated the uniqueness of the FGFP plate’s behavior when the porosities are saturated with liquid compared with the case without liquid. The findings of this study have significant implications for engineers involved in the design and fabrication of the aforementioned type of structures. Furthermore, this can form the basis for future research on the mechanical responses of the structures.

Keywords static bending free and forced vibrations nonlocal theory isogeometric analysis fluid-infiltrated porous nanoplates

Corresponding Author(s): Tran Thi Thu THUY

Just Accepted Date: 06 February 2023 Online First Date: 06 May 2023 Issue Date: 24 May 2023

Cite this article:

Tran Thi Thu THUY. Static and dynamic analysis of functionally graded fluid-infiltrated porous skew and elliptical nanoplates using an isogeometric approach[J]. Front. Struct. Civ. Eng., 2023, 17(3): 477-502.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-023-0918-5
https://academic.hep.com.cn/fsce/EN/Y2023/V17/I3/477

Fig.1 FGFP skew-nanoplates model supported by an elastic foundation.

Fig.2 Shape of function

Γ z

and elastic modulus

E (G P a)

change based on the thickness of the plate. (a) Function

Γ z

; (b)

n z = 0.5

; (c)

n z = 1

; (d)

n z = 5

Tab.1 Convergence of central dimensionless deflection and natural frequency

Ω 1

of SSSS homogeneous nanoplate with various nonlocal coefficients subjected to sinusoidally distributed load (h/a = 0.1)

n_z	$μ = 0 n m$				$μ = 2 n m$
	$w 1$		$Ω 1$		$w 1$		$Ω 1$
	[42]	present	[42]	present	[42]	present	[42]	present
0	2.9603	2.9603	1.9318	1.9317	5.2977	5.2977	1.4441	1.4441
0.5	5.4971	5.4843	1.4969	1.4982	9.8374	9.8146	1.1189	1.1200
2.5	8.8382	8.8055	1.2572	1.2594	15.8166	15.7581	0.9397	0.9416
5.5	10.0219	9.9793	1.2087	1.2113	17.9350	17.8586	0.9035	0.9056
10.5	11.1361	11.0835	1.1609	1.1638	19.9288	19.8346	0.8678	0.8701

Tab.2 Central dimensionless deflection

w 1

and non-dimensional natural frequency

Ω 1

of SSSS

A l A l 2 O 3

square nanoplates subjected to a sinusoidal distributed load (

a / h = 10

)

$b / a$	$a / h$	BC	method	Ω₁
$b / a$	$a / h$	BC	method	$φ$ = 0°	$φ$ = 15°	$φ$ = 30°	$φ$ = 45°	$φ$ = 60°
1	1000	SSSS	[49]	2.000	2.1147	2.5294	3.5800	6.7179
			this study	2.000	2.1147	2.5294	3.5803	6.7220
			error (%)^*	0	0	0	0.0084	0.061
		CCCC	[49]	3.6460	3.8691	4.6698	6.6519	12.3399
			this study	3.6434	3.8662	4.6658	6.6425	12.2883
			error (%)	0.072	0.075	0.0857	0.1415	0.4199
	10	SSSS	[49]	1.9317	2.0379	2.4195	3.3548	5.9281
			this study	1.9317	2.0387	2.4218	3.3698	6.0253
			error (%)	0	0.0392	0.0950	0.4451	1.6132
		CCCC	[49]	3.2921	3.4746	4.1145	5.6040	9.2949
			this study	3.3367	3.5229	4.1766	5.7042	9.5268
			error (%)	1.3548	1.3901	1.5093	1.7880	2.4949
2	1000	SSSS	[49]	1.2500	1.3279	1.6113	2.3306	4.4575
			this study	1.2500	1.3279	1.6107	2.3282	4.4527
			error (%)	0	0	0.0372	0.1030	0.1077
		CCCC	[49]	2.4902	2.6568	3.2617	4.7952	9.3628
			this study	2.4887	2.6552	3.2597	4.7914	9.3547
			error (%)	0.0602	0.0602	0.0613	0.0792	0.0865
	10	SSSS	[49]	1.2224	1.2969	1.5654	2.2337	4.1201
			this study	1.2227	1.2972	1.5659	2.2363	4.1352
			error (%)	0.0245	0.0231	0.0319	0.1164	0.3665
		CCCC	[49]	2.3074	2.4502	2.9583	4.1785	7.3628
			this study	2.3288	2.4739	2.9904	4.2346	7.5050
			error (%)	0.9275	0.9673	1.0851	1.3426	1.9313

Tab.3 Non-dimensional natural frequency

Ω 1

of isotropic skew plates

$a / h$	boundary conditions	method	Ω₁
$a / h$	boundary conditions	method	β = 0.1	β = 0.3	β = 0.5	β = 0.7
5	SSSS	this study	0.2109	0.2137	0.2164	0.2190
		[48]	0.2127	0.2156	0.2186	0.2216
		error (%)^*	0.8535	0.8891	1.0166	1.1872
	CCCC	this study	0.3310	0.3343	0.3375	0.3405
		[48]	0.3304	0.3373	0.3370	0.3401
		error (%)	0.1813	0.8974	0.1481	0.1175
20	SSSS	this study	0.0147	0.0150	0.0152	0.0154
		[48]	0.0147	0.0150	0.0153	0.0155
		error (%)	0.0000	0.0000	0.6579	0.6494
	CCCC	this study	0.0265	0.0269	0.0273	0.0277
		[48]	0.0266	0.0270	0.0275	0.0278
		error (%)	0.3774	0.3717	0.7326	0.3610

Tab.4 Non-dimensional natural frequency

Ω 1

of FG porous square plates (

λ = 0

)

$φ$	porosity	μ (nm)	$w 1$
$φ$	porosity	μ (nm)	β = 0	β = 0.2	β = 0.4	β = 0.6	β = 0.8
0°	FGFP I	0	6.0598	5.9297	5.8087	5.6960	5.5907
		1	6.8370	6.6961	6.5648	6.4423	6.3276
		2	8.8382	8.6754	8.5232	8.3804	8.2462
	FGFP II	0	6.3703	6.2348	6.1087	5.9912	5.8813
		1	7.1738	7.0277	6.8915	6.7643	6.6452
		2	9.2268	9.0600	8.9037	8.7570	8.6192
	FGFP III	0	6.2358	6.1028	5.9790	5.8637	5.7559
		1	7.0283	6.8845	6.7506	6.6256	6.5085
		2	9.0595	8.8945	8.7401	8.5953	8.4592
15°	FGFP I	0	5.5157	5.3959	5.2846	5.1809	5.0841
		1	6.2750	6.1441	6.0223	5.9087	5.8024
		2	8.2315	8.0780	7.9345	7.8000	7.6737
	FGFP II	0	5.8017	5.6767	5.5605	5.4522	5.3511
		1	6.5880	6.4521	6.3255	6.2073	6.0967
		2	8.5984	8.4408	8.2932	8.1549	8.0249
	FGFP III	0	5.6780	5.5554	5.4415	5.3353	5.2362
		1	6.4531	6.3195	6.1951	6.0790	5.9704
		2	8.4407	8.2850	8.1394	8.0028	7.8745
30°	FGFP I	0	4.0537	3.9630	3.8788	3.8006	3.7276
		1	4.7448	4.6427	4.5478	4.4594	4.3769
		2	6.5289	6.4030	6.2854	6.1754	6.0722
	FGFP II	0	4.2702	4.1751	4.0868	4.0047	3.9281
		1	4.9892	4.8826	4.7836	4.6913	4.6050
		2	6.8303	6.7004	6.5790	6.4653	6.3587
	FGFP III	0	4.1774	4.0843	3.9979	3.9176	3.8427
		1	4.8847	4.7802	4.6831	4.5926	4.5080
		2	6.7018	6.5737	6.4541	6.3422	6.2372
45°	FGFP I	0	2.1864	2.1358	2.0890	2.0456	2.0053
		1	2.7253	2.6645	2.6081	2.5558	2.5070
		2	4.1204	4.0376	3.9605	3.8886	3.8213
	FGFP II	0	2.3060	2.2526	2.2031	2.1573	2.1146
		1	2.8699	2.8060	2.7468	2.6917	2.6404
		2	4.3175	4.2315	4.1514	4.0765	4.0065
	FGFP III	0	2.2559	2.2038	2.1556	2.1109	2.0693
		1	2.8096	2.7471	2.6892	2.6354	2.5853
		2	4.2354	4.1508	4.0721	3.9987	3.9299

Tab.5 Dimensionless deflection

w 1

of SSSS FGFP skew-nanoplates resting on elastic foundations with respect to porosity, nonlocal coefficient, and Skempton coefficient index

$φ$	porosity	μ (nm)	Ω₁
$φ$	porosity	μ (nm)	β = 0	β = 0.2	β = 0.4	β = 0.6	β = 0.8
0°	FGFP I	0	1.8377	1.8578	1.8771	1.8956	1.9134
		1	1.7141	1.7321	1.7494	1.7660	1.7820
		2	1.4838	1.4978	1.5112	1.5240	1.5364
	FGFP II	0	1.8000	1.8196	1.8383	1.8563	1.8736
		1	1.6807	1.6982	1.7149	1.7311	1.7466
		2	1.4587	1.4722	1.4852	1.4976	1.5096
	FGFP III	0	1.8089	1.8286	1.8474	1.8656	1.8830
		1	1.6882	1.7058	1.7227	1.7390	1.7546
		2	1.4636	1.4772	1.4903	1.5029	1.5150
15°	FGFP I	0	1.9260	1.9474	1.9678	1.9874	2.0063
		1	1.7881	1.8071	1.8253	1.8428	1.8597
		2	1.5357	1.5503	1.5644	1.5778	1.5908
	FGFP II	0	1.8861	1.9068	1.9267	1.9458	1.9641
		1	1.7528	1.7712	1.7889	1.8060	1.8223
		2	1.5094	1.5235	1.5371	1.5502	1.5628
	FGFP III	0	1.8956	1.9164	1.9364	1.9557	1.9741
		1	1.7608	1.7794	1.7972	1.8144	1.8309
		2	1.5146	1.5289	1.5426	1.5557	1.5684
30°	FGFP I	0	2.2453	2.2709	2.2954	2.3189	2.3416
		1	2.0503	2.0728	2.0943	2.1150	2.1348
		2	1.7158	1.7326	1.7487	1.7642	1.7791
	FGFP II	0	2.1972	2.2222	2.2461	2.2691	2.2911
		1	2.0085	2.0304	2.0514	2.0715	2.0908
		2	1.6853	1.7017	1.7173	1.7324	1.7469
	FGFP III	0	2.2088	2.2339	2.2579	2.2810	2.3031
		1	2.0182	2.0402	2.0613	2.0815	2.1010
		2	1.6915	1.7080	1.7237	1.7389	1.7535
45°	FGFP I	0	3.0469	3.0828	3.1171	3.1500	3.1816
		1	2.6780	2.7083	2.7373	2.7651	2.7918
		2	2.1260	2.1475	2.1682	2.1880	2.2070
	FGFP II	0	2.9803	3.0154	3.0491	3.0813	3.1123
		1	2.6222	2.6519	2.6803	2.7075	2.7337
		2	2.0874	2.1084	2.1286	2.1479	2.1665
	FGFP III	0	2.9960	3.0313	3.0650	3.0973	3.1282
		1	2.6350	2.6647	2.6932	2.7205	2.7467
		2	2.0952	2.1163	2.1366	2.1560	2.1746

Tab.6 Dimensionless natural frequency

Ω 1

of SSSS FGFP skew-nanoplates resting on elastic foundations with respect to porosity, nonlocal coefficient, and Skempton coefficient index

$φ$	porosity	μ (nm)	$w 1$
$φ$	porosity	μ (nm)	λ = 0	λ = 0.2	λ = 0.4	λ = 0.6	λ = 0.8
0°	FGFP I	0	2.9875	3.2239	3.5044	3.8524	4.3412
		1	3.1099	3.3456	3.6230	3.9630	4.4319
		2	3.4350	3.6671	3.9349	4.2545	4.6780
	FGFP II	0	2.9875	3.3934	3.9906	4.9810	7.0464
		1	3.1099	3.5153	4.1047	5.0641	6.9945
		2	3.4350	3.8346	4.3981	5.2736	6.8936
	FGFP III	0	2.9875	3.3229	3.8192	4.6541	6.4639
		1	3.1099	3.4446	3.9350	4.7473	6.4552
		2	3.4350	3.7648	4.2359	4.9864	6.4535
15°	FGFP I	0	2.6914	2.9073	3.1642	3.4842	3.9363
		1	2.8129	3.0289	3.2839	3.5976	4.0324
		2	3.1356	3.3503	3.5985	3.8958	4.2915
	FGFP II	0	2.6914	3.0601	3.6035	4.5081	6.4095
		1	2.8129	3.1826	3.7211	4.6006	6.3833
		2	3.1356	3.5034	4.0229	4.8326	6.3398
	FGFP III	0	2.6914	2.9970	3.4498	4.2137	5.8787
		1	2.8129	3.1190	3.5681	4.3140	5.8901
		2	3.1356	3.4399	3.8753	4.5703	5.9342
30°	FGFP I	0	1.9279	2.0888	2.2823	2.5265	2.8774
		1	2.0424	2.2056	2.3999	2.6419	2.9825
		2	2.3473	2.5143	2.7089	2.9444	3.2622
	FGFP II	0	1.9279	2.1975	2.5970	3.2683	4.7102
		1	2.0424	2.3166	2.7178	3.3791	4.7464
		2	2.3473	2.6285	3.0276	3.6543	4.8401
	FGFP III	0	1.9279	2.1538	2.4902	3.0614	4.3239
		1	2.0424	2.2717	2.6097	3.1743	4.3825
		2	2.3473	2.5822	2.9195	3.4606	4.5325
45°	FGFP I	0	1.0210	1.1124	1.2244	1.3693	1.5844
		1	1.1145	1.2098	1.3251	1.4722	1.6852
		2	1.3637	1.4670	1.5893	1.7403	1.9492
	FGFP II	0	1.0210	1.1672	1.3849	1.7544	2.5702
		1	1.1145	1.2677	1.4929	1.8676	2.6621
		2	1.3637	1.5309	1.7690	2.1460	2.8740
	FGFP III	0	1.0210	1.1468	1.3351	1.6569	2.3777
		1	1.1145	1.2459	1.4403	1.7670	2.4739
		2	1.3637	1.5065	1.7123	2.0436	2.7044

Tab.7 Dimensionless deflection

w 1

of CSCS FGFP skew-nanoplates resting on elastic foundations with respect to porosity, nonlocal coefficient, and porosity coefficient

$φ$	porosity	μ (nm)	Ω₁
$φ$	porosity	μ (nm)	λ = 0	λ = 0.2	λ = 0.4	λ = 0.6	λ = 0.8
0°	FGFP I	0	2.5443	2.5332	2.5310	2.5432	2.5807
		1	2.3461	2.3393	2.3412	2.3572	2.3980
		2	1.9858	1.9873	1.9973	2.0211	2.0692
	FGFP II	0	2.5443	2.4803	2.4045	2.3114	2.1929
		1	2.3461	2.2925	2.2301	2.1558	2.0694
		2	1.9858	1.9524	1.9159	1.8781	1.8526
	FGFP III	0	2.5443	2.4918	2.4258	2.3392	2.2187
		1	2.3461	2.3025	2.2483	2.1790	2.0886
		2	1.9858	1.9591	1.9276	1.8911	1.8576
15°	FGFP I	0	2.6787	2.6657	2.6616	2.6720	2.7077
		1	2.4573	2.4490	2.4494	2.4641	2.5034
		2	2.0628	2.0635	2.0727	2.0958	2.1431
	FGFP II	0	2.6787	2.6101	2.5286	2.4280	2.2980
		1	2.4573	2.4000	2.3332	2.2531	2.1581
		2	2.0628	2.0272	1.9882	1.9473	1.9175
	FGFP III	0	2.6787	2.6220	2.5506	2.4569	2.3253
		1	2.4573	2.4104	2.3521	2.2772	2.1784
		2	2.0628	2.0342	2.0002	1.9607	1.9231
30°	FGFP I	0	3.1565	3.1363	3.1252	3.1287	3.1566
		1	2.8439	2.8301	2.8252	2.8345	2.8673
		2	2.3235	2.3212	2.3276	2.3479	2.3915
	FGFP II	0	3.1565	3.0719	2.9707	2.8442	2.6743
		1	2.8439	2.7743	2.6924	2.5928	2.4686
		2	2.3235	2.2809	2.2337	2.1825	2.1392
	FGFP III	0	3.1565	3.0848	2.9945	2.8754	2.7054
		1	2.8439	2.7855	2.7127	2.6188	2.4920
		2	2.3235	2.2883	2.2464	2.1968	2.1460
45°	FGFP I	0	4.3078	4.2681	4.2368	4.2188	4.2203
		1	3.7233	3.6953	3.6757	3.6689	3.6805
		2	2.8821	2.8724	2.8713	2.8832	2.9149
	FGFP II	0	4.3078	4.1861	4.0402	3.8553	3.5958
		1	3.7233	3.6269	3.5129	3.3720	3.1868
		2	2.8821	2.8256	2.7624	2.6920	2.6239
	FGFP III	0	4.3078	4.1990	4.0625	3.8834	3.6262
		1	3.7233	3.6383	3.5327	3.3968	3.2108
		2	2.8821	2.8329	2.7743	2.7048	2.6298

Tab.8 Dimensionless natural frequency

Ω 1

of CSCS FGFP skew-nanoplates resting on elastic foundations with respect to porosity, nonlocal coefficient, and porosity coefficient

Fig.3 Influence of plate geometry and different boundary conditions on dimensionless deflection of FGFP skew-nanoplates resting on elastic foundations with the porosity type I. (a)a/h–w₁; (b) μ–w₁; (c) n_z–w₁; (d)

φ

–w₁.

Fig.4 Influence of plate geometry and different boundary conditions on dimensionless natural frequency

Q 1

of FGFP skew-nanoplates resting on elastic foundations with the porosity type I. (a) a/h–Ω₁; (b) μ–Ω₁; (c) n_z–Ω₁; (d)

φ

–Ω₁.

Fig.5 Influence of material factors, elastic foundation, and different boundary conditions on dimensionless deflection

w 1

of FGFP skew nanoplates resting on elastic foundations with the porosity type I. (a)λ–w₁; (b) β–w₁; (c) K_w–w₁; (d) K_r–w₁.

Fig.6 Influence of material factors, elastic foundation, and different boundary conditions on dimensionless natural frequency

Q 1

of FGFP skew-nanoplates resting on elastic foundations the porosity type I. (a) λ–Ω₁; (b) β–Ω₁; (c) K_w–Ω₁; (d) K_r–Ω₁.

Fig.7 Shape of first four vibration modes of FGFP skew-nanoplates resting on elastic foundations with FGFP I porosity and SSSS boundary conditions. (a) Mode 1,

φ = 0 ∘

; (b) Mode 1,

φ = 30 ∘

; (c) Mode 2,

φ = 0 ∘

; (d) Mode 2,

φ = 30 ∘

; (e) Mode 3,

φ = 0 ∘

; (f) Mode 3,

φ = 30 ∘

; (g) Mode 4,

φ = 0 ∘

; (h) Mode 4,

φ = 30 ∘

Fig.8 Comparison between deflection time-history of the center of the FG square plate under a uniform load.

Fig.9 Types of load: sinusoidal load, step load, triangular load, and explosive blast load with

ω 1 = 5.24 THz

Fig.10 Effect of the nonlocal coefficient

μ (n m)

on the dynamic response of dimensionless deflection

W 1

of SSSS FGFP I skew-nanoplates resting on elastic foundation. (a) Sinusoidal load; (b) step load; (c) triagular load; (d) explosive blast load.

Fig.11 Effect of the Skempton coefficient index

β

on the dynamic response of dimensionless deflection

W 1

of SSSS FGFP I skew-nanoplates resting on elastic foundation. (a) Sinusoidal load; (b) step load; (c) triagular load; (d) explosive blast load.

Fig.12 Effect of the porosity coefficient

λ

on the dynamic response of dimensionless deflection

W 1

of SFSS FGFP I skew-nanoplates resting on elastic foundation. (a) Sinusoidal load; (b) step load; (c) triagular load; (d) explosive blast load.

Fig.13 Effect of the deviation angle

φ

on the dynamic response of dimensionless deflection

W 1

of CCCC FGFP I skew-nanoplates resting on elastic foundation. (a) Sinusoidal load; (b) step load; (c) triagular load; (d) explosive blast load.

Fig.14 Effect of stiffness foundation

K w

and

K s

on the dynamic response of dimensionless deflection

W 1

of SFSS FGFP I skew-nanoplates resting on elastic foundation. (a) Sinusoidal load; (b) step load; (c) triagular load; (d) explosive blast load.

$R y / R x$	porosity	$μ (n m)$	$w 1$
$R y / R x$	porosity	$μ (n m)$	β = 0	β = 0.2	β = 0.4	β = 0.6	β = 0.8
$2$	FGFP I	0	11.5210	11.2903	11.0751	10.8740	10.6855
		1	12.2219	11.9882	11.7700	11.5655	11.3737
		2	13.9308	13.6975	13.4785	13.2725	13.0783
	FGFP II	0	12.1118	11.8751	11.6539	11.4470	11.2529
		1	12.8166	12.5782	12.3550	12.1459	11.9493
		2	14.5161	14.2811	14.0603	13.8523	13.6560
	FGFP III	0	11.7521	11.5192	11.3019	11.0986	10.9081
		1	12.4560	12.2206	12.0006	11.7945	11.6010
		2	14.1625	13.9288	13.7093	13.5027	13.3079
$1$	FGFP I	0	5.2028	5.0702	4.9480	4.8352	4.7307
		1	5.8620	5.7184	5.5859	5.4633	5.3495
		2	7.4170	7.2575	7.1095	6.9717	6.8431
	FGFP II	0	5.5184	5.3791	5.2507	5.1321	5.0221
		1	6.1992	6.0494	5.9111	5.7830	5.6640
		2	7.7821	7.6186	7.4666	7.3250	7.1927
	FGFP III	0	5.3286	5.1935	5.0691	4.9542	4.8477
		1	5.9972	5.8514	5.7168	5.5923	5.4766
		2	7.5632	7.4024	7.2531	7.1141	6.9844
$0.5$	FGFP I	0	1.0864	1.0590	1.0337	1.0105	0.9890
		1	1.5092	1.4723	1.4383	1.4070	1.3779
		2	2.5339	2.4794	2.4290	2.3821	2.3385
	FGFP II	0	1.1560	1.1266	1.0996	1.0747	1.0517
		1	1.6028	1.5636	1.5274	1.4940	1.4631
		2	2.6704	2.6134	2.5606	2.5116	2.4659
	FGFP III	0	1.1167	1.0885	1.0627	1.0388	1.0168
		1	1.5503	1.5125	1.4778	1.4457	1.4160
		2	2.5934	2.5381	2.4869	2.4394	2.3951

Tab.9 Dimensionless deflection

w 1

of SSSS FGFP elliptical nanoplates resting on elastic foundations with respect to porosity, nonlocal coefficient, and Skempton coefficient index

$R y / R x$	porosity	$μ (n m)$	Ω₁
$R y / R x$	porosity	$μ (n m)$	β = 0	β = 0.2	β = 0.4	β = 0.6	β = β = 0.8
$2$	FGFP I	0	2.1555	2.1795	2.2026	2.2246	2.2458
		1	2.0401	2.0619	2.0828	2.1028	2.1219
		2	1.8122	1.8293	1.8457	1.8614	1.8765
	FGFP II	0	2.1080	2.1316	2.1541	2.1757	2.1964
		1	1.9977	2.0190	2.0394	2.0589	2.0777
		2	1.7805	1.7971	1.8130	1.8283	1.8430
	FGFP III	0	2.1233	2.1469	2.1695	2.1912	2.2119
		1	2.0112	2.0325	2.0530	2.0726	2.0914
		2	1.7894	1.8061	1.8222	1.8375	1.8523
$1$	FGFP I	0	3.0687	3.1037	3.1370	3.1690	3.1995
		1	2.7863	2.8167	2.8458	2.8736	2.9002
		2	2.3162	2.3385	2.3598	2.3802	2.3998
	FGFP II	0	3.0007	3.0352	3.0681	3.0996	3.1297
		1	2.7278	2.7577	2.7863	2.8137	2.8399
		2	2.2751	2.2969	2.3178	2.3378	2.3570
	FGFP III	0	3.0213	3.0556	3.0883	3.1197	3.1496
		1	2.7456	2.7755	2.8040	2.8313	2.8574
		2	2.2864	2.3083	2.3292	2.3492	2.3684
$0.5$	FGFP I	0	7.0703	7.1412	7.2084	7.2722	7.3328
		1	5.5953	5.6501	5.7022	5.7517	5.7989
		2	4.0222	4.0571	4.0902	4.1217	4.1518
	FGFP II	0	6.9565	7.0283	7.0965	7.1612	7.2228
		1	5.5051	5.5605	5.6130	5.6630	5.7106
		2	3.9670	4.0019	4.0351	4.0668	4.0970
	FGFP III	0	6.9541	7.0235	7.0893	7.1518	7.2112
		1	5.5136	5.5674	5.6185	5.6671	5.7133
		2	3.9729	4.0070	4.0395	4.0704	4.0999

Tab.10 Dimensionless natural frequency

Ω 1

of CCCC FGFP elliptical nanoplates resting on elastic foundations with respect to porosity, nonlocal coefficient, and Skempton coefficient index

Fig.15 Geometry and element mesh of an elliptical nanoplate. (a) Geometric configuration; (b) control point net and 11 × 11 cubic elements.

Fig.16 Effect of some parameters on the dynamic response of dimensionless deflection

∇ 1

of SSSS FGFP I elliptical nanoplates resting on elastic foundation under triangular load. (a) Nonlocal coefficient

μ (n m)

; (b) Skempton coefficient index

β

; (c)

R y / R x

ratio; (d) structural drag coefficient

ξ

Fig.17 First four vibration modes shape of FGFP elliptical nanoplate resting on elastic foundations with FGFP I porosity and SSSS boundary conditions. (a) Mode 1,

R y / R x = 1

; (b) Mode 1,

R y / R x = 2

; (c) Mode 2,

R y / R x = 1

; (d) Mode 2,

R y / R x = 2

; (e) Mode 3,

R y / R x = 1

; (f) Mode 3,

R y / R x = 2

; (g) Mode 4,

R y / R x = 1

; (h) Mode 4,

R y / R x = 2

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