The C1 Isogeometric analysis procedure for analysis of free vibration of GPLR-FG-TPMS nanoplates embedded in Pasternak foundation

Do, Ngoc-Tu; Hoang, Nhan Thinh; Tuan, Vo Thanh

doi:10.1590/1679-7825/e8600

Properties	Factor	TPMS types
Properties	Factor	P	G	IWP
𝐸	$C_{1}^{E}$	0.317	0.596	0.597
	$n_{1}^{E}$	2.006	2.351	1.782
	$n_{2}^{E}$	1.715	1.982	2.188
	$k_{m}^{E}$	0.25	0.45	0.35
𝐺	$C_{1}^{G}$	0.705	0.777	0.529
	$n_{1}^{G}$	1.189	1.544	1.287
	$n_{2}^{G}$	1.715	1.982	2.188
	$k_{m}^{G}$	0.25	0.45	0.35
$v$	$a_{1}$	0.314	0.192	2.597
	$b_{1}$	-1.004	-1.349	-0.157
	$a_{2}$	0.152	0.402	0.201
	$k_{ν}$	0.55	0.50	0.13

Material	Young’s Modulus (GPa)	Poisson’s Ratio	Density (kg/m³)
Si3N4	348.43	0.3	2370
SUS304(2)	201.04	0.24	8166
GPL	1010	0.186	1062.5
Copper	130	0.34	8960
SUS304	200	0.3	8000

TPMS types	Mesh grid	$ρ_{0} = 0.75$		$ρ_{0} = 0.85$		$ρ_{0} = 0.95$
TPMS types	Mesh grid	PD1	PD2	PD1	PD2	PD1	PD2
P	$5 \times 5$	3.7053	4.1912	3.4545	4.3888	3.1585	4.4558
	$7 \times 7$	3.7052	4.1911	3.4544	4.3887	3.1584	4.4558
	$9 \times 9$	3.7052	4.1911	3.4544	4.3887	3.1584	4.4558
	$11 \times 11$	3.7052	4.1911	3.4544	4.3887	3.1584	4.4558
	$13 \times 13$	3.7052	4.1911	3.4544	4.3887	3.1584	4.4558
	Tran et al. (2024)	3.7051	4.1936	3.4543	4.3894	3.1584	4.4619
G	$5 \times 5$	3.6611	4.0849	3.3617	4.2469	3.0434	4.3005
	$7 \times 7$	3.6610	4.0848	3.3616	4.2468	3.0433	4.3005
	$9 \times 9$	3.6610	4.0848	3.3616	4.2468	3.0433	4.3005
	$11 \times 11$	3.6610	4.0848	3.3616	4.2468	3.0433	4.3005
	$13 \times 13$	3.6610	4.0848	3.3616	4.2468	3.0433	4.3005
	Tran et al. (2024)	3.6608	4.0872	3.3614	4.2496	3.0432	4.3135
IWP	$5 \times 5$	3.7604	4.1446	3.5141	4.3021	3.2079	4.3472
	$7 \times 7$	3.7603	4.1445	3.5140	4.3020	3.2078	4.3472
	$9 \times 9$	3.7603	4.1445	3.5140	4.3020	3.2078	4.3472
	$11 \times 11$	3.7603	4.1445	3.5140	4.3020	3.2078	4.3472
	$13 \times 13$	3.7603	4.1445	3.5140	4.3020	3.2078	4.3472
	Tran et al. (2024)	3.7602	4.1469	3.5139	4.3042	3.2076	4.3584

$h / a$	$μ$	$n = 1$
$h / a$	$μ$	Sobhy and Radwan (2017)	Ahmed A Daikh and Zenkour (2020)	Present	Sobhy and Radwan (2017)	Ahmed A Daikh and Zenkour (2020)	Present
0.1	0	0.82250	0.82296	0.81658	0.66485	0.66510	0.65828
	0.52	0.80292	0.80338	0.79715	0.64903	0.64927	0.64262
	1	0.75165	0.75208	0.74624	0.60758	0.60781	0.60158
	1.52	0.68443	0.68483	0.67951	0.55325	0.55346	0.54778
	22	0.61484	0.61520	0.61041	0.49699	0.49719	0.49208
0.05	0	0.21083	0.21098	0.20921	0.17077	0.17086	0.16888
	0.52	0.20581	0.20596	0.20423	0.16671	0.16679	0.16486
	1	0.19267	0.19280	0.19119	0.15606	0.15614	0.15434
	1.52	0.17544	0.17556	0.17409	0.14210	0.14218	0.14053
	22	0.15760	0.15771	0.15639	0.12765	0.12772	0.12624

Pattern types	Frequencies	PD1			PD2			PD3
Pattern types	Frequencies	GPL1	GPL2	GPL3	GPL1	GPL2	GPL3	GPL1	GPL2	GPL3
P	$ω_{1}^{*}$	6.5048	6.4759	6.4961	6.8709	6.8175	6.8287	6.5525	6.5179	6.5179
	$ω_{2}^{*}$	8.4141	8.3461	8.3943	9.0049	8.9628	8.941	8.5408	8.4574	8.4574
	$ω_{3}^{*}$	8.4141	8.3461	8.3943	9.0049	8.9628	8.941	8.5408	8.4574	8.4574
	$ω_{4}^{*}$	8.9944	8.9922	8.9285	9.0238	8.9816	8.9598	8.5524	8.5524	8.5502
	$ω_{5}^{*}$	8.9944	8.9922	8.9285	9.0238	8.9816	8.9598	8.5524	8.5524	8.5502
	$ω_{6}^{*}$	9.0132	9.011	8.9472	9.3102	9.1916	9.217	8.5703	8.5703	8.5681
G	$ω_{1}^{*}$	6.4853	6.4602	6.4783	6.8365	6.7858	6.7964	6.5343	6.5021	6.5021
	$ω_{2}^{*}$	8.4059	8.3399	8.3871	8.6277	8.5588	8.5255	7.9607	7.9607	7.9586
	$ω_{3}^{*}$	8.4059	8.3399	8.3871	8.6277	8.5588	8.5255	7.9607	7.9607	7.9586
	$ω_{4}^{*}$	8.6092	8.6093	8.5073	8.6458	8.5768	8.5433	7.9774	7.9774	7.9753
	$ω_{5}^{*}$	8.6092	8.6093	8.5073	8.6458	8.5768	8.5433	7.9774	7.9774	7.9753
	$ω_{6}^{*}$	8.6272	8.6273	8.5251	9.2445	9.1328	9.1572	8.5248	8.4426	8.4426
IWP	$ω_{1}^{*}$	6.5232	6.4914	6.5131	6.8649	6.8133	6.8242	6.5519	6.5175	6.5175
	$ω_{2}^{*}$	8.5244	8.4398	8.4348	8.5529	8.4876	8.4557	7.7358	7.7358	7.7338
	$ω_{3}^{*}$	8.5244	8.4398	8.4348	8.5529	8.4876	8.4557	7.7358	7.7358	7.7338
	$ω_{4}^{*}$	8.5408	8.5318	8.4525	8.5708	8.5054	8.4734	7.752	7.752	7.750
	$ω_{5}^{*}$	8.5408	8.5318	8.4525	8.5708	8.5054	8.4734	7.752	7.752	7.750
	$ω_{6}^{*}$	8.5587	8.5497	8.497	9.331	9.2185	9.2437	8.5995	8.5087	8.5088

Nonlocal factor ( $μ$ )	Pattern types	PD1			PD2			PD3
Nonlocal factor ( $μ$ )	Pattern types	GPL1	GPL2	GPL3	GPL1	GPL2	GPL3	GPL1	GPL2	GPL3
0	P	7.8441	7.7807	7.8062	7.9227	7.8431	7.8566	7.8015	7.752	7.7584
	G	7.829	7.7701	7.7932	7.8966	7.8243	7.8367	7.7849	7.7394	7.7455
	IWP	7.8505	7.7881	7.8121	7.917	7.8424	7.8552	7.7994	7.7513	7.7577
1	P	7.8031	7.7487	7.7709	7.8706	7.8023	7.8142	7.7675	7.7252	7.7310
	G	7.7903	7.7398	7.7600	7.8485	7.7864	7.7973	7.7535	7.7146	7.7200
	IWP	7.8086	7.7551	7.7761	7.8659	7.8017	7.8131	7.7658	7.7247	7.7303
2	P	7.7732	7.7254	7.7453	7.8326	7.7725	7.7833	7.7428	7.7058	7.7110
	G	7.7621	7.7177	7.7359	7.8133	7.7587	7.7687	7.7307	7.6966	7.7016
	IWP	7.7781	7.7311	7.7498	7.8285	7.7721	7.7824	7.7413	7.7053	7.7105
4	P	7.7325	7.6937	7.7105	7.7807	7.732	7.7413	7.7093	7.6795	7.684
	G	7.7239	7.6878	7.7032	7.7655	7.7212	7.7298	7.6998	7.6723	7.6766
	IWP	7.7367	7.6985	7.7143	7.7776	7.7319	7.7407	7.7082	7.6792	7.6837

Length scale ( $l$ )	Pattern types	PD1			PD2			PD3
Length scale ( $l$ )	Pattern types	GPL1	GPL2	GPL3	GPL1	GPL2	GPL3	GPL1	GPL2	GPL3
0	P	6.6777	6.6241	6.6605	6.9497	6.857	6.8871	6.6994	6.6449	6.6510
	G	6.6645	6.6148	6.6486	6.9127	6.8279	6.8556	6.6813	6.6309	6.6367
	IWP	6.6989	6.6410	6.6790	6.9373	6.8515	6.8798	6.6980	6.6445	6.6507
0.5	P	6.6881	6.6322	6.6701	6.9714	6.8751	6.9063	6.7106	6.6538	6.6602
	G	6.6743	6.6224	6.6576	6.9329	6.8448	6.8736	6.6918	6.6392	6.6452
	IWP	6.7102	6.6498	6.6895	6.9585	6.8693	6.8988	6.7093	6.6534	6.6598
1	P	6.7189	6.656	6.6986	7.0351	6.9283	6.9628	6.7440	6.6802	6.6873
	G	6.7034	6.6451	6.6846	6.9923	6.8945	6.9264	6.7228	6.6638	6.6705
	IWP	6.7437	6.6759	6.7204	7.0207	6.9218	6.9544	6.7425	6.6798	6.6869
2	P	6.8370	6.7484	6.8081	7.2720	7.1280	7.1742	6.8717	6.7821	6.7918
	G	6.8154	6.7329	6.7885	7.2145	7.0820	7.1249	6.8421	6.7589	6.7680
	IWP	6.8721	6.7767	6.8391	7.2530	7.1194	7.1632	6.8699	6.7817	6.7914

\frac{E (z)}{E_{s}} = \{\begin{matrix} C_{1}^{E} \times ρ^{n_{1}^{E}} ρ \leq k_{m}^{E} \\ C_{2}^{E} \times ρ^{n_{2}^{E}} + C_{3}^{E} ρ > k_{m}^{E} \end{matrix}

\frac{G (z)}{G_{s}} = \{\begin{matrix} C_{1}^{G} \times ρ^{n_{1}^{G}} ρ \leq k_{m}^{G} \\ C_{2}^{G} \times ρ^{n_{2}^{G}} + C_{3}^{G} ρ > k_{m}^{G} \end{matrix}

v (z) = \{\begin{matrix} a_{1} \times e^{b_{1} \times ρ} + d_{1} ρ \leq k_{v} \\ a_{2} \times ρ^{2} + b_{2} \times ρ + d_{2} ρ > k_{v} \end{matrix}

\{\begin{matrix} ρ (z) = ρ_{s} \times ρ_{m a x} [1 - ρ_{0} + ρ_{0} \times {(\frac{z}{h} + \frac{1}{2})}^{n_{1}}] & {P D}_{1} (a s y m m e t r i c) \\ ρ (z) = ρ_{s} \times ρ_{m a x} [1 - ρ_{0} + ρ_{0} \times {[1 - \cos (\frac{π z}{h})]}^{n_{2}}] & {P D}_{2} (s y m m e t r i c) \\ ρ (z) = ρ_{s} \times [1 - g_{m} ψ_{g}] & {P D}_{3} (u n i f o r m) \end{matrix}

\{\begin{matrix} V_{r}^{1} (z) = S_{1} [1 - \cos (\frac{π z}{h})] & {G P L}_{1} (s y m m e t r i c) \\ V_{r}^{2} (z) = S_{2} [1 - \cos (\frac{π z}{2 h} + \frac{π}{4})] & {G P L}_{2} (a s y m m e t r i c) \\ V_{r}^{3} (z) = S_{3} & {G P L}_{3} (u n i f o r m) \end{matrix}

S_{i} = \frac{V_{r}^{t o t a l} \int_{- h / 2}^{h / 2} {ρ (z) / ρ}_{s} d z}{\int_{- h / 2}^{h / 2} {V_{r}^{i} {(z) / S}_{i} \times ρ (z) / ρ}_{s} d z}

E_{s} = [\frac{3}{8} \frac{1 + ξ_{L} η_{L} V_{r}}{1 - η_{L} V_{r}} + \frac{5}{8} \frac{1 + ξ_{w} η_{w} V_{r}}{1 - η_{w} V_{r}}] E_{m}

\int_{t_{0}}^{t_{1}} [\int_{V} (σ_{i j} - λ \nabla^{2} σ_{i j}) δ ε_{i j} d V + \int_{V} (1 - μ \nabla^{2}) ρ {\ddot{u}}_{i} δ u_{i} d V - \int_{V} (1 - μ \nabla^{2}) f_{i} δ u_{i} d V] d t = 0

u = \{\begin{matrix} u \\ v \\ w \end{matrix}\}, u_{0} = \{\begin{matrix} u_{0} \\ v_{0} \\ w_{0} \end{matrix}\}, u_{1} = \{\begin{matrix} {- β}_{x} \\ {- β}_{y} \\ 0 \end{matrix}\}, u_{2} = \{\begin{matrix} θ_{x} \\ θ_{y} \\ 0 \end{matrix}\}

ε_{0} = \{\begin{matrix} u_{0, x} \\ v_{0, y} \\ u_{0, y} + v_{0, x} \end{matrix}\}; ε_{1} = - \{\begin{matrix} β_{x, x} \\ β_{y, y} \\ β_{x, y} + β_{y, x} \end{matrix}\}; ε_{2} = \{\begin{matrix} θ_{x, x} \\ θ_{y, y} \\ θ_{y, x} + θ_{x, y} \end{matrix}\};

\{\begin{matrix} σ_{x x} \\ σ_{y y} \\ \begin{matrix} τ_{x y} \\ \begin{matrix} τ_{x z} \\ τ_{y z} \end{matrix} \end{matrix} \end{matrix}\} = [\begin{matrix} C_{11} & C_{12} & 0 \\ C_{21} & C_{22} & 0 \\ \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} & \begin{matrix} C_{66} \\ 0 \\ 0 \end{matrix} \end{matrix} \begin{matrix} 0 & 0 \\ 0 & 0 \\ \begin{matrix} 0 \\ C_{55} \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \\ C_{44} \end{matrix} \end{matrix}] \{\begin{matrix} ε_{x x} \\ ε_{y y} \\ \begin{matrix} γ_{x y} \\ γ_{x z} \\ γ_{y z} \end{matrix} \end{matrix}\}

δ \int_{t_{0}}^{t_{1}} \int_{Ω} [\frac{1}{2} {\hat{ε}}^{T} D_{b} \hat{ε} + \frac{1}{2} γ^{T} D_{s} γ - \frac{λ}{2} {(\nabla^{2} \hat{ε})}^{T} D_{b} \hat{ε} + \frac{λ}{2} {(\nabla^{2} γ)}^{T} D_{s} γ + \frac{1}{2} (1 - μ \nabla^{2}) {\dot{\hat{u}}}^{T} m \dot{\hat{u}} - (1 - μ \nabla^{2}) R w] d Ω d t = 0 .

\hat{ε} = \{\begin{matrix} ε_{0} \\ ε_{1} \\ ε_{2} \end{matrix}\}, γ = \{\begin{matrix} ε_{s 0} \\ ε_{s 1} \end{matrix}\}, D_{b} = [\begin{matrix} A & B & H \\ B & D & F \\ H & F & L \end{matrix}], D_{s} = [\begin{matrix} A^{s} & B^{s} \\ B^{s} & C^{s} \end{matrix}]

\hat{ε} = {{\begin{matrix} ε_{0} & ε_{1} & ε_{2} \end{matrix}}}^{T} = \sum_{I = 1}^{n \times m} [{\begin{matrix} B_{I}^{0} & B_{I}^{1} & B_{I}^{2}] \end{matrix}}^{T} q_{I} = \sum_{I = 1}^{n \times m} {\bar{B}}_{I}^{b} q_{I}

γ = {\begin{matrix} ε_{s 0} & ε_{s 1} \end{matrix}}^{T} = \sum_{I = 1}^{n \times m} {[\begin{matrix} B_{I}^{s 0} & B_{I}^{s 1}] \end{matrix}}^{T} q_{I} = \sum_{I = 1}^{n \times m} {\bar{B}}_{I}^{s} q_{I}

B_{I}^{1} = - [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \begin{matrix} 0 & 0 & N_{I, x} \\ 0 & 0 & 0 \\ 0 & 0 & N_{I, y} \end{matrix} \begin{matrix} 0 \\ N_{I, y} \\ N_{I, x} \end{matrix}];

B_{I}^{2} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \begin{matrix} N_{I, x} & 0 & 0 \\ 0 & N_{I, y} & 0 \\ N_{I, y} & N_{I, x} & 0 \end{matrix} \begin{matrix} 0 \\ 0 \\ 0 \end{matrix}];

\hat{u} {= {\begin{matrix} u_{0} & u_{1} & u_{2} \end{matrix}}}^{T} = \sum_{I = 1}^{n \times m} {[\begin{matrix} N_{I}^{0} & N_{I}^{1} & N_{I}^{2}] \end{matrix}}^{T} q_{I} = \sum_{I = 1}^{n \times m} {\bar{N}}_{I} q_{I}

K = \int_{Ω}^{} {({\bar{B}}^{b})}^{T} {\hat{D}}^{b} {\bar{B}}^{b} d Ω + \int_{Ω}^{} {({\bar{B}}^{s})}^{T} {\hat{D}}^{s} {\bar{B}}^{s} d Ω - λ \int_{Ω}^{} {(\nabla^{2} {\bar{B}}^{b})}^{T} {\hat{D}}^{b} {\bar{B}}^{b} d Ω - λ \int_{Ω}^{} {(\nabla^{2} {\bar{B}}^{s})}^{T} D^{s} {\bar{B}}^{s} d Ω

K_{}^{f} = k_{w} \int_{Ω}^{} [{(N_{I}^{w})}^{T} (N_{I}^{w}) + μ [{(N_{I, x}^{w})}^{T} (N_{I, x}^{w}) + {(N_{I, y}^{w})}^{T} (N_{I, y}^{w})]] d Ω + k_{s} \int_{Ω}^{} [{(N_{I, x}^{w})}^{T} (N_{I, x}^{w}) + {(N_{I, y}^{w})}^{T} (N_{I, y}^{w}) + μ [{(N_{I, x x}^{w})}^{T} (N_{I, x x}^{w}) + {(N_{I, y y}^{w})}^{T} (N_{I, y y}^{w})]] d Ω