On the buckling and vibration behavior of carbon nanotube-reinforced bioinspired composite plates: a combined microstructural and hygrothermal investigation via isogeometric analysis

doi:10.1007/s10483-025-3321-8

Abstract

Abstract:

Inspired by the structural adaptations of natural biological organisms, helicoidal composite architectures have shown significant potential for enhancing toughness, strength, and weight efficiency in engineering applications. However, temperature and moisture’s adverse effects pose challenges during service, potentially compromising their overall performance. This study meticulously analyzes the buckling and vibration behavior of carbon nanotube (CNT)-reinforced bioinspired helicoidal composite plates under different hygrothermal conditions. A novel aspect of this study lies in the proposition of a multiscale analysis combining the analytical and numerical techniques to assess the effects of temperature, moisture, weight fraction of CNTs, layup configurations of bioinspired designs, aspect ratios, loading and boundary conditions, and geometric shapes of bioinspired helicoidal composite structures on their vibration and buckling characteristics. In this context, the stiffness properties are computed with the Halpin-Tsai model, incorporating the size-dependent features of CNTs along with their waviness and agglomeration. In addition, the Chamis micro-mechanical equations are used to determine the elastic properties of individual layers constituting bioinspired composites, considering the effects of temperature and moisture. The kinematics of the laminated bioinspired structures are captured with the third-order shear deformation theory (TSDT) within the isogeometric framework employing the non-uniform rational B-splines (NURBSs) as the basis functions. The isogeometric framework ensures higher-order inter-element continuity and provides an exact geometric representation, offering various advantages over the traditional finite element method. The developed framework is validated against the existing literature, and thereafter several numerical examples are presented, providing valuable insights for the design and optimization of bioinspired composite structures, with potential benefits for various engineering fields, including marine and aerospace sectors.

Key words: bioinspired, hygrothermal, carbon nanotube (CNT)-reinforced, higher-order shear deformation theory (HSDT), isogeometric analysis (IGA), buckling, vibration

2010 MSC Number:

S. SAURABH, S. K. SINGH, V. S. CHAUHAN, R. KIRAN. On the buckling and vibration behavior of carbon nanotube-reinforced bioinspired composite plates: a combined microstructural and hygrothermal investigation via isogeometric analysis. Applied Mathematics and Mechanics (English Edition), 2025, 46(12): 2317-2340.

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Figures/Tables 20

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Table 1

Various helicoidal layup schemes (all angles are in degrees)[29]"

Layup scheme	Number of layers $N L = 12$
HR $(o 1 / o 2 / o 3 ⋯ / o n = o n − 1 + α (n − 1)),$ $α = 1$	[0/1/3/6/10/15/15/10/6/3/1/0]
HE $(o 1 / o 2 / o 3 ⋯ / o n = β n),$ $β = 2$	[2/4/8/2/4/8/8/4/2/8/4/2]
HS $(o 1 / o 2 / o 3 ⋯ / o n = θ max 2 − (γ (n − 1) − θ max) 2),$ and $θ max = 45$	[0/27/36/41.3/44.1/45/45/44.1/41.3/36/27/0], $γ = 9$
LH (with a twisting (or pitch) angle of $4 ∘$ )	[0/4/8/12/16/20/20/16/12/8/4/0]
FH $(F n = F n − 1 + F n − 2, F 0 = 0, F 1 = 10)$	[0/10/10/20/30/50/50/30/20/10/10/0]
QI	[0/90/45/ $−$ 45/0/90/90/0/ $−$ 45/45/90/0]

Table 1

Fig. 5

Table 2

The results of λ¯cr for square SSSS CNT-reinforced bioinspired composite plates at T=300 K for uniaxial stress conditions"

$M m$ /%	Layup scheme	$λ ¯ cr$
		$W cnt$ /% ( $a / h = 10$ )				$W cnt$ /% ( $a / h = 100$ )
		0.0	2.5	5.0	7.5	0.0	2.5	5.0	7.5
2	HR	14.685 3	23.195 6	25.080 5	25.813 0	21.209 9	26.858 8	28.494 4	29.158 5
	HE	14.648 1	23.170 8	25.056 6	25.789 0	21.149 3	26.844 7	28.483 6	29.148 2
	HS	17.082 6	25.865 6	27.678 5	28.375 1	24.516 4	30.365 3	31.893 9	32.520 0
	LH	14.866 9	23.399 1	25.280 5	26.011 1	21.450 0	27.120 4	28.750 0	29.411 2
	FH	15.287 8	23.858 1	25.728 7	26.454 0	21.981 2	27.698 6	29.313 8	29.968 4
	QI	18.841 6	26.865 6	28.622 3	29.311 7	25.393 2	30.524 7	32.026 8	32.644 6
4	HR	14.638 6	23.148 9	25.034 3	25.767 2	21.188 1	26.819 7	28.453 1	29.116 5
	HE	14.601 4	23.124 1	25.010 5	25.743 2	21.127 2	26.805 5	28.442 2	29.106 2
	HS	17.032 1	25.820 1	27.634 6	28.331 6	24.490 6	30.328 4	31.855 6	32.471 0
	LH	14.820 0	25.352 5	25.544 5	25.965 4	21.427 9	27.081 4	28.708 9	29.369 4
	FH	15.240 3	23.811 7	25.683 0	26.408 6	21.958 4	27.659 9	29.273 1	29.927 0
	QI	18.795 6	26.822 2	28.579 1	29.268 4	25.373 3	30.489 1	31.988 6	32.605 4

Table 2

Table 3

The results of λ¯cr for square SSSS CNT-reinforced bioinspired composite plates at T=340 K for uniaxial stress conditions"

$M m$ /%	Layup scheme	$λ ¯ cr$
		$W cnt$ /% ( $a / h = 10$ )				$W cnt$ /% ( $a / h = 100$ )
		0.0	2.5	5.0	7.5	0.0	2.5	5.0	7.5
2	HR	5.887 6	13.479 5	15.373 1	16.119 4	18.364 7	20.678 4	21.542 5	21.928 7
	HE	5.871 8	13.442 2	15.336 4	16.083 4	18.241 7	20.608 8	21.487 0	21.878 6
	HS	7.292 0	15.774 8	17.824 0	18.624 1	20.751 0	23.874 3	24.906 0	25.348 8
	LH	6.000 9	13.654 3	15.558 4	16.308 4	18.535 8	20.909 9	21.787 2	22.177 9
	FH	6.271 0	14.060 5	15.986 9	16.744 5	18.908 4	21.421 4	22.328 8	22.729 9
	QI	8.448 6	17.638 8	19.515 4	20.238 0	22.790 3	24.906 7	25.697 1	26.049 4
4	HR	4.534 7	11.463 5	13.327 7	14.068 2	17.987 2	19.921 8	20.615 9	20.929 9
	HE	4.527 0	11.428 2	13.290 5	14.030 9	17.855 9	19.837 2	20.545 2	20.864 7
	HS	5.698 0	13.568 5	15.609 5	16.414 5	20.220 6	22.913 5	23.797 1	24.181 2
	LH	4.629 7	11.625 6	13.501 6	14.246 5	18.147 4	20.137 8	20.846 3	21.165 7
	FH	4.861 0	12.003 9	13.905 8	14.660 0	18.495 8	20.613 4	21.355 2	21.687 1
	QI	6.378 3	15.549 0	17.485 1	18.229 8	22.452 4	24.212 7	24.849 5	25.137 1

Table 3

Table 4

The results of λ¯cr for square CCCC CNT-reinforced bioinspired composite plates at T=300 K for uniaxial stress conditions"

$M m$ /%	Layup scheme	$λ ¯ cr$
		$W cnt$ /% ( $a / h = 10$ )				$W cnt$ /% ( $a / h = 100$ )
		0.0	2.5	5.0	7.5	0.0	2.5	5.0	7.5
2	HR	30.073 9	59.033 2	65.009 0	67.246 01	87.976 77	108.132 9	113.503 4	115.692 7
	HE	29.908 9	58.792 9	64.761 4	66.995 26	87.298 02	107.632 8	113.029 1	115.225 7
	HS	30.205 3	56.875 4	62.460 8	64.562 72	77.307 26	99.320 4	104.957 4	107.214 5
	LH	30.081 1	58.891 1	64.842 8	67.071 12	87.178 43	107.512 8	112.910 5	115.107 5
	FH	30.121 7	58.592 6	64.486 5	66.695 03	85.395 48	106.099 4	111.553 1	113.765 5
	QI	33.771 3	62.528 9	68.072 7	70.164 61	87.524 31	106.300 8	111.490 6	113.617 1
4	HR	29.925 0	58.881 0	64.866 8	67.107 30	87.884 40	108.004 1	113.367 5	115.553 7
	HE	29.760 6	58.640 9	64.619 1	66.856 70	87.204 60	107.503 3	112.892 7	115.086 3
	HS	30.067 5	56.733 7	63.327 1	64.432 20	77.206 30	99.183 6	104.816 5	107.072 0
	LH	29.932 9	58.739 6	64.700 9	66.932 90	87.085 20	107.382 2	112.774 1	114.968 1
	FH	29.975 2	58.442 6	64.345 9	66.558 00	85.300 50	105.968 2	111.415 5	113.652 3
	QI	33.609 8	62.388 2	67.939 8	70.034 50	87.442 40	106.176 9	111.358 8	113.481 9

Table 4

Table 5

The results of λ¯cr for square CCCC CNT-reinforced bioinspired composite plates at T=340 K for uniaxial stress conditions"

$M m$ /%	Layup scheme	$λ ¯ cr$
		$W cnt$ /% ( $a / h = 10$ )				$W cnt$ /% ( $a / h = 100$ )
		0.0	2.5	5.0	7.5	0.0	2.5	5.0	7.5
2	HR	8.755 71	26.336 3	32.302 9	34.785 5	71.289 0	85.646 6	89.358 2	90.904 2
	HE	8.743 19	26.189 3	32.127 7	34.601 3	70.489 3	84.943 1	88.694 3	90.256 8
	HS	9.600 48	26.733 3	32.263 5	34.549 9	60.665 8	74.776 3	78.821 0	80.522 6
	LH	8.793 48	26.361 5	32.297 8	34.767 8	70.454 3	84.827 7	88.573 0	90.134 2
	FH	8.956 57	26.442 7	32.312 9	34.753 5	68.624 6	83.003 0	86.816 9	88.409 3
	QI	10.510 40	29.720 4	36.186 4	38.879 8	73.513 6	85.475 0	88.754 4	90.144 0
4	HR	6.601 90	20.648 7	25.882 3	28.132 4	67.626 3	81.953 5	85.360 7	86.770 6
	HE	6.613 87	20.535 3	25.737 8	27.976 4	66.826 5	81.213 4	84.654 2	86.079 0
	HS	7.283 98	21.379 3	26.309 5	28.405 3	57.609 7	70.856 7	74.468 3	75.993 0
	LH	6.612 90	20.697 4	25.909 9	28.149 3	66.829 1	81.108 6	84.539 4	85.961 3
	FH	6.751 80	20.831 6	25.995 7	28.211 3	65.091 8	79.233 2	82.710 1	84.156 1
	QI	8.119 40	23.546 1	29.228 5	31.667 3	70.621 9	82.294 2	85.225 8	86.459 4

Table 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Table 6

The results of ω¯ for square CNT-reinforced bioinspired composite plates with Mm=2% at T=300 K"

Boundary condition	Layup scheme	$ω ¯$
		$W cnt$ /% ( $a / h = 10$ )				$W cnt$ /% ( $a / h = 100$ )
		0.0	2.5	5.0	7.5	0.0	2.5	5.0	7.5
SSSS	HR	11.974 2	15.031 6	15.628 1	15.854 0	14.471 1	16.282 7	16.770 9	16.965 1
	HE	11.968 6	15.031 1	15.627 9	15.853 7	14.464 3	16.287 4	16.776 0	16.970 2
	HS	13.059 9	16.001 2	16.541 2	16.744 4	15.787 0	17.467 4	17.887 6	18.055 1
	LH	12.056 8	15.105 9	15.698 5	15.922 8	14.570 4	16.373 3	16.856 6	17.048 9
	FH	12.249 2	15.273 7	15.856 8	16.077 2	14.789 2	16.572 8	17.045 1	17.233 0
	QI	13.577 9	16.183 3	16.700 6	16.899 4	15.835 6	17.359 8	17.781 5	17.952 1
CCCC	HR	20.454 1	28.399 9	29.783 5	30.285 8	35.718 1	40.231 7	41.321 6	41.753 1
	HE	20.433 0	28.372 3	29.755 7	30.258 0	35.649 7	40.183 6	41.276 7	41.709 2
	HS	20.894 6	28.374 6	29.700 3	30.185 0	34.641 5	39.413 5	40.547 8	40.994 4
	LH	20.477 3	28.396 9	29.776 9	30.278 1	35.645 2	40.176 6	41.269 8	41.702 3
	FH	20.557 1	28.404 4	29.774 7	30.272 9	35.481 2	40.051 6	41.151 8	41.586 6
	QI	23.133 8	30.014 9	31.229 7	31.679 5	36.282 9	40.537 0	41.599 3	42.022 7

Table 6

Table 7

The results of ω¯ for square CNT-reinforced bioinspired composite plates with Mm=2% at T=340 K"

Boundary condition	Layup scheme	$ω ¯$
		$W cnt$ /% ( $a / h = 10$ )				$W cnt$ /% ( $a / h = 100$ )
		0.0	2.5	5.0	7.5	0.0	2.5	5.0	7.5
SSSS	HR	7.591 9	11.474 5	12.249 9	12.542 2	13.467 1	14.288 8	14.584 0	14.713 9
	HE	7.589 9	11.468 5	12.244 8	12.537 5	13.441 3	14.279 5	14.578 6	14.710 1
	HS	8.539 4	12.557 0	13.335 7	13.626 3	14.618 8	15.594 3	15.903 1	16.034 0
	LH	7.667 5	11.557 0	12.332 4	12.624 5	13.553 7	14.387 2	14.683 6	14.813 8
	FH	7.849 9	11.750 3	12.524 1	12.815 1	13.743 5	14.603 8	14.903 2	15.033 8
	QI	9.393 6	13.142 5	13.815 6	14.066 0	15.003 6	15.683 4	15.929 9	16.038 5
CCCC	HR	11.332 2	19.188 6	21.171 4	21.942 7	31.574 2	35.152 1	36.049 6	36.416 8
	HE	11.336 4	19.169 4	21.149 3	21.919 6	31.491 7	35.080 8	35.982 9	36.352 0
	HS	12.024 1	19.698 8	21.569 8	22.294 4	30.478 6	34.048 4	34.990 6	35.378 6
	LH	11.354 0	19.215 0	21.192 5	21.961 3	31.502 1	35.077 4	35.977 9	36.346 5
	FH	11.477 7	19.305 1	21.266 1	22.027 9	31.342 8	34.909 7	35.816 4	36.188 0
	QI	13.783 5	21.972 3	23.778 4	24.461 8	32.652 7	35.770 7	36.585 3	36.922 3

Table 7

Table 8

The results of ω¯ for square CNT-reinforced bioinspired composite plates with Mm=4% at T=300 K"

Boundary condition	Layup scheme	$ω ¯$
		$W cnt$ /% ( $a / h = 10$ )				$W cnt$ /% ( $a / h = 100$ )
		0.0	2.5	5.0	7.5	0.0	2.5	5.0	7.5
SSSS	HR	11.955 2	15.016 5	15.613 8	15.839 9	14.463 6	16.270 9	16.758 7	16.952 9
	HE	11.949 7	15.016 0	15.613 5	15.839 6	14.456 7	16.275 5	16.763 8	16.958 0
	HS	13.040 8	15.987 4	16.528 3	16.731 8	15.779 2	17.457 1	17.877 2	18.044 5
	LH	12.037 8	15.090 9	15.684 3	15.908 9	14.562 9	16.361 6	16.844 6	17.036 8
	FH	12.230 2	15.259 0	15.842 8	16.063 5	14.718 7	16.561 3	17.033 3	17.221 2
	QI	13.561 5	16.170 3	16.688 0	16.887 0	15.829 4	17.349 7	17.770 9	17.941 3
CCCC	HR	20.405 2	28.363 8	29.751 2	30.254 9	35.695 8	40.205 0	41.294 6	41.726 0
	HE	20.384 1	28.336 2	29.723 4	30.227 0	35.627 3	40.156 9	41.249 6	41.681 9
	HS	20.848 5	28.340 2	29.669 3	30.155 1	34.618 0	39.385 5	40.520 7	40.966 4
	LH	20.428 5	28.360 9	29.744 7	30.247 3	35.622 8	40.149 9	41.242 7	41.675 1
	FH	20.508 7	28.368 7	29.742 7	30.242 2	35.458 6	40.024 6	41.124 6	41.559 2
	QI	23.089 5	29.983 6	31.201 0	30.651 7	36.262 7	40.511 1	41.572 9	41.996 0

Table 8

Table 9

The results of ω¯ for square CNT-reinforced bioinspired composite plates with Mm=4% at T=340 K"

Boundary condition	Layup scheme	$ω ¯$
		$W cnt$ /% ( $a / h = 10$ )				$W cnt$ /% ( $a / h = 100$ )
		0.0	2.5	5.0	7.5	0.0	2.5	5.0	7.5
SSSS	HR	6.663 9	10.585 4	11.410 0	11.721 2	13.328 2	14.025 4	14.267 3	14.375 4
	HE	6.664 4	10.579 1	11.403 9	11.715 4	13.299 7	14.011 6	14.257 6	14.367 3
	HS	7.541 5	11.654 3	12.491 8	12.805 7	14.441 5	15.302 1	15.571 0	15.686 6
	LH	6.735 2	10.667 4	11.492 5	11.803 8	13.412 2	14.121 4	14.365 5	14.474 3
	FH	6.909 2	10.861 0	11.685 9	11.996 7	13.596 4	14.332 5	14.581 8	14.692 0
	QI	8.371 0	12.348 1	13.085 8	13.358 3	14.892 2	15.463 8	15.665 4	15.755 6
CCCC	HR	9.902 8	17.074 7	19.029 0	19.807 2	30.668 3	34.240 0	35.082 1	35.426 1
	HE	9.917 0	17.059 6	19.010 1	19.787 1	30.586 2	34.164 5	35.010 5	35.356 3
	HS	10.537 6	17.684 3	19.547 6	20.284 2	29.647 9	33.104 2	33.975 4	34.335 0
	LH	9.912 2	17.105 1	19.055 8	19.832 1	30.602 4	34.163 3	35.007 2	35.352 3
	FH	10.032 6	17.209 9	19.147 2	19.917 4	30.457 8	33.991 6	34.839 1	35.186 2
	QI	12.072 2	19.949 4	21.823 4	22.544 2	31.917 6	34.955 5	35.707 7	36.018 1

Table 9

Fig. 11

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