A six-variable quasi-3D isogeometric approach for free vibration of functionally graded graphene origami-enabled auxeticmetamaterial plates submerged in a fluid medium

doi:10.1007/s10483-025-3207-6

Abstract

Abstract:

This paper presents, for the first time, an effective numerical approach based on the isogeometric analysis (IGA) and the six-variable quasi-three dimensional (3D) higher-order shear deformation theory (HSDT) to study the free vibration characteristics of functionally-graded (FG) graphene origami (GOri)-enabled auxetic metamaterial (GOEAM) plates submerged in a fluid medium. The plate theory incorporates the thickness stretching and the effects of transverse shear deformation without using any shear correction factors. The velocity potential function and Bernoulli's equation are used to derive the hydrodynamic pressure acting on the plate surface. Both horizontally and vertically immersed plate configurations are considered here in the form of inertia effects. The plates are composed of multilayer GOEAMs, with the GOri content varying through the plate's thickness in a layer-wise manner. This design results in graded auxetic growth. The material properties are evaluated by mixing rules and a genetic programming (GP)-assisted micromechanical model. The governing equations of motion for the FG-GOEAM plates immersed in a fluid medium are derived by Hamilton's principle. After validating the convergence and accuracy of the present model, a comprehensive parametric study is carried out to examine the effects of the GOri content, GOri distribution pattern, GOri folding degree, fluid level, immersed depth, and geometric parameter on the natural frequencies of the FG-GOEAM plates. The results show that the natural frequencies for the four GOri distribution patterns increase with the increase in the layer number when the lay number is fewer than 10, and then stabilize after the layer number reaches 10. Besides, in general, the natural frequency of the FG-GOEAM plate in a vacuum or fluid increases when the GOri content increases, while decreases when the GOri folding degree increases. Some additional findings related to the numerical results are presented in the conclusions. It is believed that the present results are useful for the precise design and optimization of FG-GOEAM plates immersed in a fluid medium.

Key words: functionally-graded (FG) graphene origami (GOri)-enabled auxetic metamaterial (GOEAM) plate, hydrodynamic pressure, quasi-three dimensional (3D) higher-order shear deformation theory (HSDT), isogeometric analysis (IGA), free vibration

2010 MSC Number:

O321

Wei CHEN, Zhihong TANG, Yufen LIAO, Linxin PENG. A six-variable quasi-3D isogeometric approach for free vibration of functionally graded graphene origami-enabled auxeticmetamaterial plates submerged in a fluid medium. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 157-176.

TrendMD

Figures/Tables 13

Fig. 1

Fig. 2

Table 1

Convergence study on dimensionless fundamental frequencies of SSSS isotropic plates horizontally submerged in the fluid, where a=1, a/b=1, and a/h=5"

Method	Mesh	Control point	In a vacuum	Fluid level $(h 1 / a)$
Method	Mesh	Control point	In a vacuum	0.0	0.3	0.5	1.0	2.0
Quadratic-IGA	$5 × 5$	49	1.779 5	1.589 1	1.464 8	1.451 4	1.448 6	1.448 6
	$9 × 9$	121	1.771 4	1.581 9	1.458 1	1.444 8	1.442 0	1.441 9
	$13 × 13$	225	1.769 3	1.580 0	1.456 4	1.443 1	1.440 3	1.440 2
	$17 × 17$	361	1.768 5	1.579 3	1.455 7	1.442 4	1.439 6	1.439 6
Cubic-IGA	$5 × 5$	64	1.767 3	1.578 2	1.454 7	1.441 4	1.438 6	1.438 6
	$9 × 9$	144	1.767 2	1.578 1	1.454 6	1.441 3	1.438 5	1.438 5
	$13 × 13$	256	1.767 2	1.578 1	1.454 6	1.441 3	1.438 5	1.438 5
	$17 × 17$	400	1.767 2	1.578 1	1.454 6	1.441 3	1.438 5	1.438 5
Quartic-IGA	$5 × 5$	81	1.767 2	1.578 1	1.454 6	1.441 3	1.438 5	1.438 5
	$9 × 9$	169	1.767 2	1.578 1	1.454 6	1.441 3	1.438 5	1.438 5
	$13 × 13$	289	1.767 2	1.578 1	1.454 6	1.441 3	1.438 5	1.438 5
	$17 × 17$	441	1.767 2	1.578 1	1.454 6	1.441 3	1.438 5	1.438 5
Pham et al.^[25]	–	–	1.778 5	1.578 1	1.449 5	1.435 8	1.432 9	1.432 9

Table 1

Table 2

Validation analysis on the dimensionless fundamental frequencies of SSSS isotropic plates vertically submerged in the fluid with various length-to-thickness ratios, where a=1, a/b=1, and d1=d2"

$a / h$	Method	In a vacuum	Immersed depth ( $z 2 / b)$
$a / h$	Method	In a vacuum	0.1	0.3	0.5	0.7	1
5	Pham et al.^[25]	1.778 5	1.714 6	1.610 5	1.535 4	1.482 9	1.432 9
5	Present	1.767 2	1.707 3	1.609 0	1.537 2	1.486 8	1.438 5
10	Pham et al.^[25]	1.937 0	1.796 9	1.598 8	1.473 5	1.393 0	1.320 7
10	Present	1.933 7	1.795 9	1.600 2	1.476 0	1.396 0	1.324 1
100	Pham et al.^[25]	2.002 1	1.226 1	0.830 4	0.687 1	0.614 8	0.558 7
100	Present	1.999 7	1.224 6	0.829 5	0.686 3	0.614 1	0.558 0

Table 2

Table 3

Validation analysis on the first six dimensionless frequencies of CCCC FG-GPLRC plates with various GPL distribution patterns, where a=1, a/b=1, and a/h=10"

GPL pattern	Theory	Mode
GPL pattern	Theory	1	2	3	4	5	6
UD	HSDT^[42]	0.207 8	0.397 2	0.397 2	0.557 7	0.657 9	0.663 6
	Quasi-3D^[43]	0.231 4	0.433 9	0.433 9	0.602 2	0.708 3	0.711 5
	$C 0$ -HSDT^[44]	0.206 7	0.394 2	0.397 4	0.555 9	0.654 1	0.663 3
	Present	0.213 5	0.404 0	0.406 2	0.564 6	0.660 1	0.673 9
FG-O	HSDT^[42]	0.178 8	0.347 7	0.347 7	0.493 5	0.586 7	0.591 2
	Quasi-3D^[43]	0.196 7	0.376 5	0.376 5	0.531 4	0.630 6	0.632 6
	$C 0$ -HSDT^[44]	0.178 0	0.345 7	0.348 5	0.493 0	0.584 6	0.592 6
	Present	0.184 0	0.355 8	0.356 9	0.503 4	0.595 9	0.604 3
FG-X	HSDT^[42]	0.226 5	0.423 0	0.423 0	0.586 3	0.685 3	0.691 7
	Quasi-3D^[43]	0.259 7	0.478 6	0.478 6	0.652 9	0.760 3	0.764 8
	$C 0$ -HSDT^[44]	0.224 7	0.418 4	0.421 7	0.582 4	0.678 7	0.688 4
	Present	0.231 9	0.427 0	0.430 6	0.588 6	0.678 2	0.697 8
FG-A	HSDT^[42]	0.192 8	0.370 5	0.370 5	0.521 9	0.617 2	0.622 5
	Quasi-3D^[43]	0.216 0	0.407 9	0.407 9	0.569 0	0.670 8	0.674 2
	$C 0$ -HSDT^[44]	0.191 9	0.368 2	0.371 3	0.521 1	0.614 8	0.623 4
	Present	0.201 0	0.382 7	0.384 5	0.536 6	0.629 4	0.641 4

Table 3

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

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