Free vibration analysis of functionally graded porous curved nanobeams on elastic foundation in hygro–thermo–magnetic environment

doi:10.1007/s11709-023-0916-7

Front. Struct. Civ. Eng.

2023, Vol. 17

Issue (4) : 584-605 https://doi.org/10.1007/s11709-023-0916-7

RESEARCH ARTICLE

Free vibration analysis of functionally graded porous curved nanobeams on elastic foundation in hygro–thermo–magnetic environment

Quoc-Hoa PHAM¹, Parviz MALEKZADEH², Van Ke TRAN³, Trung NGUYEN-THOI^4,^5,⁶(

)

¹. Faculty of Engineering and Technology, Nguyen Tat Thanh University, Ho Chi Minh City 700000, Vietnam
². Department of Mechanical Engineering, Persian Gulf University, Bushehr 7516913817, Iran
³. Faculty of Mechanical Engineering, Le Quy Don Technical University, Hanoi 100000, Vietnam
⁴. Laboratory for Applied and Industrial Mathematics, Institute for Computational Science and Artificial Intelligence, Van Lang University, Ho Chi Minh City 700000, Vietnam
⁵. Faculty of Mechanical-Electrical and Computer Engineering, School of Technology, Van Lang University, Ho Chi Minh City 700000, Vietnam
⁶. Bualuang ASEAN Chair Professor, Thammasat School of Engineering, Thammasat University, Pathumtani 12120, Thailand

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Abstract

Herein, a two-node beam element enriched based on the Lagrange and Hermite interpolation function is proposed to solve the governing equation of a functionally graded porous (FGP) curved nanobeam on an elastic foundation in a hygro–thermo–magnetic environment. The material properties of curved nanobeams change continuously along the thickness via a power-law distribution, and the porosity distributions are described by an uneven porosity distribution. The effects of magnetic fields, temperature, and moisture on the curved nanobeam are assumed to result in axial loads and not affect the mechanical properties of the material. The equilibrium equations of the curved nanobeam are derived using Hamilton’s principle based on various beam theories, including the classical theory, first-order shear deformation theory, and higher-order shear deformation theory, and the nonlocal elasticity theory. The accuracy of the proposed method is verified by comparing the results obtained with those of previous reliable studies. Additionally, the effects of different parameters on the free vibration behavior of the FGP curved nanobeams are investigated comprehensively.

Keywords functionally graded porous material curved nanobeam hygro–thermo–magnetic enriched finite element method

Corresponding Author(s): Trung NGUYEN-THOI

About author:

^* These authors contributed equally to this work.

Just Accepted Date: 22 February 2023 Online First Date: 26 May 2023 Issue Date: 25 June 2023

Cite this article:

Quoc-Hoa PHAM,Parviz MALEKZADEH,Van Ke TRAN, et al. Free vibration analysis of functionally graded porous curved nanobeams on elastic foundation in hygro–thermo–magnetic environment[J]. Front. Struct. Civ. Eng., 2023, 17(4): 584-605.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-023-0916-7
https://academic.hep.com.cn/fsce/EN/Y2023/V17/I4/584

Fig.1 Modeling of FGP curved nanobeams embedded in an elastic medium.

material	properties
material	E (GPa)	ρ (kg/m³)	v	αK^?1	β (%H₂O)^?1
aluminum $(Al)$ : metal	70	2707	0.3	23 × 10^?6	0.44
alumina $(A l 2 O 3)$ : ceramic	380	3800	0.3	7 × 10^?6	0.001

Tab.1 Properties of the FG material [40]

$a / h$	$μ$	$N e$	first mode			second mode
$a / h$	$μ$	$N e$	CLT	FSDT	HSDT	CLT	FSDT	HSDT
5	0	1	4.4844	4.3109	4.2788	31.3667	27.7451	27.1676
		2	4.4844	4.3109	4.2788	31.3543	27.5203	27.1520
		3	4.4845	4.3109	4.2788	31.2444	27.3764	26.9898
		Navier-present	4.4845	4.3109	4.2788	31.2444	27.3764	26.9898
		FEM [39]	4.4844	4.3109	4.2791	31.1728	27.5439	26.9737
	1	1	4.2783	4.1127	4.0821	26.5551	23.4881	22.9989
		2	4.2783	4.1127	4.0821	26.4854	23.3672	22.7918
		3	4.2783	4.1127	4.0821	26.4132	23.2811	22.7175
		Navier-present	4.2783	4.1127	4.0821	26.4132	23.2811	22.7175
		FEM [39]	4.2783	4.1127	4.0824	26.3950	23.3223	22.8395
	2	1	3.7971	3.6502	3.6230	19.5247	17.2686	16.9088
		2	3.7971	3.6502	3.6230	19.4130	17.1505	16.8419
		3	3.7971	3.6502	3.6230	19.4105	17.1193	16.7931
		Navier-present	3.7971	3.6500	3.6230	19.4105	17.1193	16.7931
		FEM [39]	3.7971	3.6502	3.6232	19.4105	17.1509	16.7958
50	0	1	4.5630	4.5610	4.5606	36.8989	36.8136	36.7966
		2	4.5614	4.5595	4.5591	33.2697	33.2089	33.4682
		3	4.5614	4.5595	4.5591	33.2726	33.1971	33.1985
		4	4.5614	4.5595	4.5591	33.2726	33.1971	33.1985
		Navier-present	4.5614	4.5595	4.5591	33.2726	33.1971	33.1985
	1	1	4.3532	4.3514	4.3510	31.1425	31.0698	31.0554
		2	4.3517	4.3499	4.3495	28.1876	28.1214	28.1542
		3	4.3517	4.3498	4.3495	28.1703	28.1149	28.1083
		4	4.3517	4.3498	4.3495	28.1703	28.1149	28.1083
		Navier-present	4.3517	4.3498	4.3495	28.1703	28.1149	28.1083
	2	1	3.8636	3.8620	3.8616	22.8142	22.7604	22.7497
		2	3.8623	3.8607	3.8603	20.7188	20.6813	20.6899
		3	3.8623	3.8606	3.8603	20.7152	20.6847	20.6774
		4	3.8623	3.8606	3.8603	20.7152	20.6847	20.6774
		Navier-present	3.8623	3.8606	3.8603	20.7152	20.6847	20.6774

Tab.2 Convergence and comparison of first and second nondimensional frequencies of SS curved nanobeam (

K w = 0,

K s = 0,

Δ T = 0,

Δ C = 0,

H x * = 0,

φ = 120 o

)

$a / h$	$μ$	$N e$	first mode			second mode
$a / h$	$μ$	$N e$	CLT	FSDT	HSDT	CLT	FSDT	HSDT
5	0	1	3.8016	3.7365	3.7240	14.8505	13.2765	13.0321
		2	3.8013	3.7358	3.7233	14.8480	13.2174	12.9765
		4	3.8013	3.7355	3.7231	14.8480	13.1886	12.9595
		5	3.8013	3.7355	3.7231	14.8480	13.1829	12.9581
		6	3.8013	3.7355	3.7231	14.8480	13.1791	12.9575
		FEM [39]	3.8014	3.7357	3.7234	14.8483	13.2013	12.9772
	1	1	3.7112	3.6492	3.6373	13.4017	12.0400	11.8249
		2	3.7109	3.6486	3.6366	13.4003	11.9963	11.7837
		4	3.7109	3.6484	3.6365	13.4003	11.9746	11.7709
		5	3.7109	3.6483	3.6365	13.4003	11.9703	11.7698
		6	3.7109	3.6483	3.6365	13.4003	11.9675	11.7694
		FEM [39]	3.7110	3.6485	3.6368	13.4006	11.9842	11.7847
	2	1	3.4731	3.4186	3.4081	10.6816	9.6765	9.5133
		2	3.4728	3.4181	3.4075	10.6810	9.6533	9.4914
		4	3.4728	3.4180	3.4075	10.6810	9.6416	9.4845
		5	3.4728	3.4180	3.4075	10.6810	9.6393	9.4839
		6	3.4728	3.4180	3.4075	10.6810	9.6377	9.4837
		FEM [39]	3.4728	3.4181	3.4077	10.6812	9.6469	9.4927
50	0	1	3.8496	3.8488	3.8487	16.1445	16.1196	16.1146
		2	3.8430	3.8423	3.8421	16.0594	16.0350	16.0301
		4	3.8430	3.8422	3.8421	16.0576	16.0327	16.0278
		5	3.8430	3.8422	3.8421	16.0576	16.0326	16.0276
		6	3.8430	3.8422	3.8421	16.0576	16.0326	16.0276
		FEM [39]	3.8500	3.8493	3.8491	16.0845	16.0598	16.0549
	1	1	3.7582	3.7574	3.7573	14.3471	14.3268	14.3227
		2	3.7515	3.7508	3.7506	14.2954	14.2755	14.2715
		4	3.7514	3.7508	3.7506	14.2939	14.2736	14.2696
		5	3.7514	3.7508	3.7506	14.2939	14.2736	14.2695
		6	3.7514	3.7508	3.7506	14.2939	14.2735	14.2695
		FEM [39]	3.7583	3.7576	3.7575	14.3177	14.3177	14.2936
	2	1	3.5175	3.5168	3.5167	11.2194	11.2054	11.2026
		2	3.5106	3.5100	3.5099	11.1944	11.1807	11.1779
		4	3.5106	3.5100	3.5099	11.1933	11.1794	11.1767
		5	3.5106	3.5100	3.5099	11.1933	11.1794	11.1766
		6	3.5106	3.5100	3.5099	11.1933	11.1794	11.1766
		FEM [39]	3.5169	3.5163	3.5162	11.2120	11.1982	11.1955

Tab.3 Convergence and comparison of first and second nondimensional frequencies of CF curved nanobeams with respect to nonlocal coefficient

(φ = 120 o)

$a / h$	$μ$	$N e$	first mode			second mode
$a / h$	$μ$	$N e$	CLT	FSDT	HSDT	CLT	FSDT	HSDT
5	0	1	34.3848	33.1324	32.9307	43.4625	35.9557	34.9067
		2	34.3725	33.0894	32.8827	43.0508	35.3280	34.3044
		4	34.3723	33.0793	32.8756	43.0508	35.1833	34.2203
		5	34.3723	33.0774	32.8750	43.0508	35.1542	34.2130
		6	34.3723	33.0761	32.8748	43.0508	35.1348	34.2101
		FEM [39]	34.3728	33.0841	32.8848	43.0525	34.0420	33.1013
	1	1	31.9526	30.6165	30.3815	38.4106	31.3819	30.4234
		2	31.9332	30.5105	29.8599	37.8902	30.7985	30.2652
		4	31.9329	30.4826	29.8140	37.8900	30.7164	30.2472
		5	31.9329	30.4772	29.8101	37.8900	30.6998	30.2457
		6	31.9329	30.4736	29.8087	37.8900	30.6888	30.2452
		FEM [39]	31.9336	29.7348	28.8511	37.8919	30.4949	30.2667
	2	1	26.7310	23.6354	22.9030	28.9802	24.8990	24.5287
		2	26.6851	23.1268	22.4050	28.4327	24.6205	24.2346
		4	26.6847	23.1030	22.3923	28.4323	24.5578	24.1995
		5	26.6847	23.0982	22.3913	28.4323	24.5458	24.1970
		6	26.6847	23.0950	22.3909	28.4323	24.5378	24.1960
		FEM [39]	26.6855	22.4400	21.7350	28.4340	24.5848	24.2400
50	0	1	69.8713	69.3862	69.2911	133.8935	133.2674	133.1444
		2	51.8940	51.6881	51.6473	102.5468	101.7561	101.6006
		4	51.8887	51.6792	51.6376	102.1855	101.3698	101.2097
		5	51.8886	51.6783	51.6366	102.1846	101.3626	101.2014
		6	51.8886	51.6778	51.6360	102.1846	101.3583	101.1964
		FEM [39]	51.9687	51.7241	51.6761	102.3418	101.5397	101.3836
	1	1	59.9935	59.5779	59.4964	99.8114	99.3304	99.2360
		2	43.8841	43.7191	43.6864	74.9147	74.3757	74.2696
		4	43.8716	43.7053	43.6723	74.7069	74.1595	74.0517
		5	43.8715	43.7048	43.6718	74.7059	74.1562	74.0480
		6	43.8715	43.7046	43.6715	74.7058	74.1545	74.0462
		FEM [39]	43.9396	43.7467	43.7089	74.8226	74.2803	74.1745
	2	1	44.8539	44.5442	44.4834	65.1654	64.8473	64.7849
		2	32.1459	32.0310	32.0082	48.1910	47.8594	47.7940
		4	32.1240	32.0090	31.9862	48.0829	47.7489	47.6831
		5	32.1239	32.0088	31.9860	48.0820	47.7473	47.6813
		6	32.1239	32.0088	31.9859	48.0819	47.7467	47.6807
		FEM [39]	32.1742	32.0428	32.0170	48.1584	47.8262	47.7613

Tab.4 Convergence and comparison of first and second nondimensional frequencies of CC curved nanobeam with respect to nonlocal coefficient

(φ = 120 o)

Fig.2 Effect of magnetic ?eld

H x *

on nondimensional natural frequency

Ω 1

of FGP curved nanobeam for different boundary conditions. (a) SS; (b) CC; (c) CS; (d) CF.

Fig.3 Effect of temperature change

Δ T (K)

on nondimensional natural frequency

Ω 1

of FGP curved nanobeam for different boundary conditions. (a) SS; (b) CC; (c) CS; (d) CF.

Fig.4 Effect of moisture change

Δ C (% H 2 O)

on nondimensional natural frequency

Ω 1

of FGP curved nanobeam for different boundary conditions. (a) SS; (b) CC; (c) CS; (d) CF.

$a / h$	$μ$	$H x ∗$	$φ = 30 °$						$φ = 120 °$
			first mode			second mode			first mode			second mode
			CLT	FSDT	HSDT	CLT	FSDT	HSDT	CLT	FSDT	HSDT	CLT	FSDT	HSDT
5	0	0	21.5141	19.7337	19.7590	42.8951	38.1425	38.0645	29.8907	29.2992	29.2422	35.8631	31.1756	31.1084
		1	21.8903	20.1165	20.1463	43.1143	38.7018	38.6226	30.1690	29.6070	29.5488	36.1882	31.6447	31.5831
		5	23.3263	21.5653	21.6121	43.7915	40.6603	40.5735	31.2464	30.7681	30.7080	37.3728	33.3495	33.3073
	1	0	20.9515	19.5806	19.5489	38.4708	34.5793	34.3341	28.2295	27.7091	27.6428	32.4778	29.2289	28.9938
		1	21.4293	20.0964	20.0639	39.0596	35.4936	35.2418	28.5871	28.1114	28.0457	33.0371	30.0031	29.7669
		5	23.2371	22.0356	22.0005	40.9055	38.6566	38.3976	29.9551	29.5933	29.5300	35.0221	32.7139	32.4820
	2	0	19.7966	18.9733	18.9288	31.0209	29.1894	28.9669	24.8824	24.5489	24.4808	27.1880	25.7044	25.4856
		1	20.4808	19.7099	19.6646	32.0718	30.4988	30.2722	25.4116	25.1375	25.0726	28.1184	26.8423	26.6255
		5	23.0091	22.4083	22.3606	35.5774	34.8672	34.6542	27.3689	27.2215	27.1646	31.2795	30.6026	30.4088
10	0	0	23.8283	23.3256	23.3202	48.7068	45.5215	45.4756	40.6980	38.0988	38.0394	49.5461	48.4382	48.4425
		1	24.1872	23.6839	23.6791	49.3106	46.1569	46.1124	41.2274	38.6558	38.5979	49.7953	48.7466	48.7505
		5	25.5636	25.0530	25.0506	51.6449	48.6019	48.5625	43.2555	40.7790	40.7267	50.7521	49.9087	49.9106
	1	0	22.9384	22.5098	22.5014	41.5120	39.2161	39.1534	35.6881	33.7668	33.6929	45.0260	43.4628	43.4199
		1	23.4035	22.9805	22.9721	42.3994	40.1686	40.1052	36.4974	34.6373	34.5628	45.4629	44.0689	44.0265
		5	25.1710	24.7665	24.7583	45.7687	43.7614	43.6966	39.5378	37.8853	37.8096	47.0307	46.1256	46.0902
	2	0	21.0459	20.7346	20.7259	32.2472	30.9029	30.8534	28.6534	27.5235	27.4638	36.0096	34.4498	34.3731
		1	21.7456	21.4451	21.4366	33.6024	32.3402	32.2905	29.9290	28.8766	28.8168	37.0019	35.7050	35.6355
		5	24.3377	24.0726	24.0644	38.5399	37.5307	37.4801	34.5166	33.6990	33.6386	40.1012	39.4696	39.4250
20	0	0	30.0208	29.9189	29.9180	45.1104	44.0779	44.0620	37.9211	37.0420	37.0236	70.2272	67.7246	67.6879
		1	30.3330	30.2335	30.2327	45.8052	44.7837	44.7681	38.5688	37.6996	37.6815	70.8253	68.3605	68.3246
		5	31.5364	31.4438	31.4432	48.4775	47.4931	47.4786	41.0442	40.2071	40.1900	73.1215	70.8001	70.7677
	1	0	27.7161	27.5959	27.5959	35.3201	34.4525	34.4420	30.1160	29.3533	29.3407	50.6477	48.3777	48.3623
		1	28.1492	28.0381	28.0373	36.4240	35.5849	35.5736	31.1801	30.4453	30.4319	51.8867	49.7259	49.6472
		5	29.7816	29.6906	29.6902	40.5353	39.7819	39.7703	35.1014	34.4479	34.4339	56.4902	54.6206	54.5826
	2	0	20.4515	19.7982	19.7737	21.8997	19.7120	19.7054	17.1689	16.4826	16.4732	26.1301	24.4830	24.4749
		1	22.6595	21.9978	21.9863	22.8930	21.9924	22.5073	19.4235	18.8179	18.8085	28.9282	26.6264	26.8116
		5	25.9017	25.7997	25.7999	29.8998	29.4047	29.3988	26.5756	26.1419	26.1340	37.9705	36.7990	36.7734

Tab.5 First two nondimensional fundamental frequencies

Ω

of CC FGP curved nanobeam with porosity on an elastic foundation in hygro–thermo–magnetic environment

Fig.5 Effect of porosity coefficient

χ

on nondimensional natural frequency

Ω 1

of FGP curved nanobeam for different boundary conditions. (a) SS; (b) CC; (c) CS; (d) CF.

$a / h$	$μ$	$H x ∗$	$φ = 30 °$						$φ = 120 °$
			first mode			second mode			first mode			second mode
			CLT	FSDT	HSDT	CLT	FSDT	HSDT	CLT	FSDT	HSDT	CLT	FSDT	HSDT
5	0	0	13.1874	13.0853	13.0829	32.7304	30.6188	30.5769	9.9088	9.8835	9.8830	28.2733	26.5989	26.5827
		1	13.6630	13.5670	13.5646	33.4384	31.4163	31.3736	10.3505	10.3276	10.3270	28.9858	27.3959	27.3782
		5	15.4160	15.3399	15.3374	36.1284	34.4154	34.3697	11.9442	11.9287	11.9281	31.6650	30.3602	30.3376
	1	0	13.0055	12.9135	12.9112	29.0372	27.4124	27.3754	9.8475	9.8255	9.8250	25.2679	24.0034	23.9869
		1	13.4871	13.4010	13.3987	29.8320	28.2985	28.2606	10.2905	10.2708	10.2702	26.0595	24.8791	24.8612
		5	15.2587	15.1916	15.1892	32.8128	31.5847	31.5440	11.8869	11.8741	11.8735	28.9936	28.0870	28.0646
	2	0	12.6038	12.5346	12.5325	24.0310	23.1185	23.0887	9.7092	9.6944	9.6939	21.2248	20.5498	20.5335
		1	13.0991	13.0351	13.0330	24.9810	24.1552	24.1246	10.1545	10.1416	10.1411	22.1494	21.5500	21.5325
		5	14.9117	14.8643	14.8621	28.4502	27.8944	27.8622	11.7531	11.7456	11.7451	25.4761	25.1028	25.0828
10	0	0	12.5216	12.4847	12.4842	33.2845	32.4151	32.4040	9.2266	9.2158	9.2157	28.7495	28.0418	28.0361
		1	13.0434	13.0082	13.0077	34.0706	33.2261	33.2150	9.7241	9.7141	9.7140	29.5549	28.8720	28.8662
		5	14.9487	14.9189	14.9184	37.0480	36.2886	36.2778	11.4981	11.4902	11.4900	32.5756	31.9750	31.9690
	1	0	12.3242	12.2902	12.2898	29.1917	28.4871	28.4778	9.1636	9.1539	9.1538	25.3754	24.8075	24.8026
		1	12.8539	12.8216	12.8211	30.0848	29.4065	29.3973	9.6641	9.6551	9.6549	26.2839	25.7416	25.7366
		5	14.7835	14.7563	14.7558	33.4187	32.8267	32.8176	11.4461	11.4391	11.4389	29.6367	29.1760	29.1708
	2	0	11.8883	11.8610	11.8606	23.5523	23.0919	23.0852	9.0258	9.0183	9.0182	20.7702	20.4080	20.4042
		1	12.4364	12.4106	12.4102	24.6502	24.2165	24.2099	9.5327	9.5258	9.5257	21.8688	21.5316	21.5277
		5	14.4212	14.3998	14.3994	28.6220	28.2692	28.2627	11.3316	11.3264	11.3262	25.7905	25.5267	25.5226
20	0	0	7.8224	7.8062	7.8060	27.3746	27.0661	27.0626	4.1345	4.1273	4.1273	22.5161	22.2468	22.2444
		1	8.6430	8.6283	8.6281	28.3572	28.0599	28.0565	5.1660	5.1603	5.1602	23.5815	23.3251	23.3228
		5	11.3464	11.3353	11.3352	31.9869	31.7251	31.7221	8.0649	8.0614	8.0614	27.4317	27.2136	27.2115
	1	0	7.4990	7.4836	7.4834	22.0294	21.7538	21.7507	3.9929	3.9862	3.9861	17.8007	17.5554	17.5533
		1	8.3514	8.3376	8.3374	23.2391	22.9785	22.9756	5.0533	5.0481	5.0480	19.1306	18.9033	18.9014
		5	11.1259	11.1156	11.1155	27.5516	27.3339	27.3313	7.9931	7.9899	7.9898	23.7148	23.5342	23.5325
	2	0	6.7511	6.7376	6.7375	13.3462	13.0981	13.0954	3.6698	3.6641	3.6640	9.7832	9.5382	9.5362
		1	7.6869	7.6751	7.6750	15.2605	15.0448	15.0424	4.8021	4.7978	4.7977	12.0350	11.8379	11.8363
		5	10.6360	10.6276	10.6275	21.2584	21.1063	21.1045	7.8362	7.8337	7.8337	18.4732	18.3485	18.3473

Tab.6 First two nondimensional fundamental frequencies

Ω

of SS FGP curved nanobeam with porosity on an elastic foundation in hygro–thermo–magnetic environment

Fig.6 Effect of power-law index

p z

on nondimensional natural frequency

Ω 1

of FGP curved nanobeam for different boundary conditions. (a) SS; (b) CC; (c) CS; (d) CF.

$a / h$	$μ$	$H x ∗$	$φ = 30 °$						$φ = 120 °$
			first mode			second mode			first mode			second mode
			CLT	FSDT	HSDT	CLT	FSDT	HSDT	CLT	FSDT	HSDT	CLT	FSDT	HSDT
5	0	0	15.3992	14.8528	14.8490	24.2246	24.1131	24.1143	10.1183	9.7755	9.7771	28.2503	26.0575	26.0497
		1	15.7846	15.2458	15.2438	24.2988	24.1942	24.1960	10.4124	10.0642	10.0679	28.6238	26.5708	26.5664
		5	17.1610	16.6492	16.6534	24.6339	24.5349	24.5394	11.4596	11.0877	11.0990	29.9704	28.4009	28.4054
	1	0	15.2015	14.7665	14.7469	23.7715	23.6603	23.6562	10.1904	9.9227	9.9104	26.2344	24.5310	24.4183
		1	15.6324	15.2161	15.1962	23.8521	23.7539	23.7518	10.5194	10.2579	10.2454	26.7473	25.2101	25.1003
		5	17.1608	16.8143	16.7932	24.2116	24.1270	24.1296	11.6853	11.4463	11.4336	28.5144	27.5101	27.4173
	2	0	14.7587	14.4759	14.4562	22.6286	22.5089	22.4954	10.3435	10.1721	10.1591	22.7542	21.8134	21.7001
		1	15.2851	15.0243	15.0044	22.7358	22.6513	22.6438	10.7471	10.5871	10.5741	23.5177	22.7183	22.6102
		5	17.1159	16.9308	16.9107	23.1583	23.1038	23.1039	12.1521	12.0284	12.0157	25.9770	25.5781	25.4977
10	0	0	15.2642	15.0721	15.0702	39.5424	37.8179	37.7920	9.9192	9.8054	9.8042	32.8405	31.3264	31.2995
		1	15.7315	15.5398	15.5382	40.1463	38.4742	38.4495	10.2783	10.1618	10.1609	33.4810	31.9934	31.9679
		5	17.4570	17.2624	17.2621	42.3162	40.8832	40.8631	11.5722	11.4427	11.4430	35.9018	34.5040	34.4837
	1	0	14.9580	14.7916	14.7885	34.5821	33.2031	33.1654	9.9313	9.8315	9.8292	29.1054	27.9587	27.9181
		1	15.4793	15.3162	15.3130	35.4256	34.0973	34.0594	10.3356	10.2360	10.2336	29.9276	28.8243	28.7833
		5	17.3940	17.2413	17.2381	38.5388	37.3989	37.3611	11.7872	11.6880	11.6855	32.9777	32.0178	31.9769
	2	0	14.2893	14.1672	14.1645	27.5661	26.7082	26.6788	9.9564	9.8819	9.8797	23.8266	23.1319	23.0986
		1	14.9273	14.8107	14.8080	28.7655	27.9601	27.9309	10.4620	10.3904	10.3882	24.9655	24.3194	24.2862
		5	17.2349	17.1353	17.1327	33.0922	32.4507	32.4216	12.2527	12.1906	12.1884	29.0247	28.5235	28.4906
20	0	0	11.3305	11.2687	11.2700	35.5662	34.9403	34.9324	6.6224	6.5871	6.5883	28.8009	28.2651	28.2564
		1	11.9919	11.9326	11.9338	36.3779	35.7628	35.7552	7.2120	7.1788	7.1800	29.6232	29.0983	29.0900
		5	14.3053	14.2507	14.2519	39.4553	38.8757	38.8691	9.1412	9.1097	9.1107	32.7019	32.2115	32.2041
	1	0	10.4501	10.3903	10.3913	28.2722	27.7384	27.3230	6.1024	6.0689	6.0684	22.9485	22.4821	22.3953
		1	11.2234	11.1655	11.1633	29.3951	26.1136	15.7215	6.8109	6.7798	6.7791	24.0935	23.4771	23.4696
		5	13.8628	13.8131	13.8145	33.5132	33.0569	33.0518	9.0422	9.0136	9.0138	28.2113	27.8223	27.8163
	2	0	8.3326	8.2748	8.2774	16.7827	16.3140	16.3087	4.6952	4.6552	4.6580	13.0545	12.6044	12.6079
		1	9.4313	9.3923	9.3823	18.8005	18.3805	18.3741	5.7920	5.7638	5.7628	15.2049	14.8183	14.8153
		5	12.8886	12.8486	12.8506	25.3168	25.0108	25.0090	8.8102	8.7872	8.7869	21.7864	21.5258	21.5231

Tab.7 First two nondimensional fundamental frequencies

Ω

of CS FGP curved nanobeam with porosity on an elastic foundation in hygro–thermo–magnetic environment

Fig.7 Effect of stiffness foundation (

K w, K s

) on nondimensional natural frequency

Ω 1

of FGP curved nanobeam for different boundary conditions. (a) CC; (b) CS; (c) CC; (d) CS.

$a / h$	$μ$	$H x ∗$	$φ = 30 °$						$φ = 120 °$
			first mode			second mode			first mode			second mode
			CLT	FSDT	HSDT	CLT	FSDT	HSDT	CLT	FSDT	HSDT	CLT	FSDT	HSDT
5	0	0	9.9385	9.8629	9.8663	19.7107	18.8497	18.8854	8.0310	7.9047	7.9124	16.4703	15.5572	15.5923
		1	10.1599	10.0719	10.0762	20.1758	19.3791	19.4181	8.1737	8.0391	8.0481	17.0531	16.1367	16.1758
		5	10.9549	10.8238	10.8322	21.3954	20.8538	20.8984	8.7079	8.5406	8.5546	18.9967	18.0852	18.1375
	1	0	9.9285	9.8717	9.8717	18.7436	18.1008	18.0918	8.0294	7.9351	7.9361	15.7307	15.1062	15.0960
		1	10.1591	10.0953	10.0952	19.3126	18.7256	18.7187	8.1874	8.0907	8.0918	16.3432	15.7385	15.7284
		5	10.9981	10.9152	10.9148	20.8785	20.5118	20.5162	8.7785	8.6734	8.6749	18.4125	17.8833	17.8754
	2	0	9.8875	9.8547	9.8544	16.7352	16.3711	16.3557	7.9917	7.9274	7.9291	14.3029	13.9712	13.9616
		1	10.1351	10.1000	10.0996	17.4132	17.0835	17.0694	8.1787	8.1137	8.1157	14.9367	14.6264	14.6179
		5	11.0538	11.0141	11.0135	19.5200	19.3115	19.3071	8.8699	8.8023	8.8052	17.1044	16.8586	16.8551
10	0	0	9.5354	9.5201	9.5210	20.8339	20.4042	20.4046	7.7913	7.7610	7.7619	16.5605	16.2092	16.2100
		1	9.7847	9.7655	9.7664	21.7586	21.3211	21.3222	7.9465	7.9136	7.9146	17.3620	17.0007	17.0022
		5	10.6523	10.6190	10.6203	24.9824	24.5086	24.5125	8.5280	8.4840	8.4856	20.1015	19.7001	19.7045
	1	0	9.5144	9.5013	9.5020	19.0542	18.7497	18.7457	7.7623	7.7362	7.7367	15.5373	15.2781	15.2746
		1	9.7704	9.7546	9.7552	19.9804	19.6794	19.6752	7.9342	7.9064	7.9070	16.3527	16.0941	16.0904
		5	10.6777	10.6533	10.6538	23.2555	22.9627	22.9582	8.5765	8.5430	8.5436	19.1813	18.9244	18.9203
	2	0	9.4533	9.4443	9.4449	16.2400	16.0709	16.0679	7.6706	7.6495	7.6504	13.7550	13.6067	13.6046
		1	9.7218	9.7115	9.7121	17.1330	16.9726	16.9697	7.8756	7.8533	7.8542	14.5620	14.4195	14.4176
		5	10.7020	10.6886	10.6892	20.3084	20.1736	20.1708	8.6297	8.6032	8.6044	17.3872	17.2623	17.2611
20	0	0	5.1602	5.1052	5.1108	12.1899	12.0697	12.0722	6.4757	6.4728	6.4738	7.1245	6.9839	6.9889
		1	6.6891	6.6729	6.6761	13.6609	13.5368	13.5395	6.6817	6.6769	6.6778	9.3709	9.2609	9.2647
		5	8.8974	8.8944	8.8958	18.9712	18.8508	18.8532	7.4119	7.4049	7.4058	14.8796	14.7826	14.7847
	1	0	7.1245	6.9839	6.9889	10.6945	10.6021	10.6015	6.0106	5.9034	5.9015	6.4422	6.4124	6.4132
		1	9.3709	9.2609	9.2647	12.0087	11.9019	11.8774	6.5127	6.5042	6.5048	8.3828	8.2714	8.2888
		5	14.8796	14.7826	14.7847	17.1069	17.0153	17.0179	7.3445	7.3378	7.3388	13.7729	13.6958	13.6981
	2	0	5.0693	5.0002	5.0067	8.4487	8.4026	8.4045	3.9856	3.8478	3.8730	6.0286	5.9601	5.9901
		1	6.7221	6.7138	6.7038	9.4541	9.4069	9.3930	5.7889	5.7549	5.7564	6.9522	7.0178	6.8797
		5	8.8033	8.8007	8.8022	14.2587	14.2017	14.2044	7.1682	7.1622	7.1633	11.9341	11.8847	11.8872

Tab.8 First two nondimensional fundamental frequencies

Ω

of CF FGP curved nanobeam with porosity on an elastic foundation in hygro–thermo–magnetic environment

Fig.8 First two vibration mode shapes of CC FGP curved nanobeam on an elastic medium. (a) Mode 1; (b) Mode 2.

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