Two-phase nonlocal integral model with bi-Helmholtz kernel for free vibration analysis of multi-walled carbon nanotubes considering size-dependent van der Waals forces

doi:10.1007/s10483-025-3313-8

Abstract

Abstract:

Current studies on carbon nanotube (CNT) size effects predominantly employ Eringen’s differential nonlocal model, which is widely recognized as ill-suited for bounded domains. This paper investigates the free vibration of multi-walled CNTs (MWCNTs) with mathematically well-posed two-phase strain-driven and stress-driven nonlocal integral models incorporating the bi-Helmholtz kernel. The van der Waals (vdW) forces coupling MWCNT layers are similarly modeled as size-dependent via the bi-Helmholtz two-phase nonlocal integral framework. Critically, conventional pure strain-driven or stress-driven formulations become over-constrained when nonlocal vdW interactions are considered. The two-phase strategy resolves this limitation by enabling consistent coupling. Each bi-Helmholtz integral constitutive equation is equivalently transformed into a differential form requiring four additional constitutive boundary conditions (CBCs). The numerical solutions are obtained with the generalized differential quadrature method (GDQM) for these coupled higher-order equations. The parametric studies on double-walled CNTs (DWCNTs) and triple-walled CNTs (TWCNTs) elucidate the nonlocal effects predicted by both formulations. Additionally, the influence of nonlocal parameters within vdW forces is systematically evaluated to comprehensively characterize the size effects in MWCNTs.

Key words: multi-walled carbon nanotube (MWCNT), two-phase nonlocal integral elasticity, bi-Helmholtz kernel, free vibration, generalized differential quadrature method (GDQM), nonlocal van der Waals (vdW) force

2010 MSC Number:

O343.1

Chang LI, Rongjun CHEN, Cheng LI, Hai QING. Two-phase nonlocal integral model with bi-Helmholtz kernel for free vibration analysis of multi-walled carbon nanotubes considering size-dependent van der Waals forces. Applied Mathematics and Mechanics (English Edition), 2025, 46(11): 2095-2114.

TrendMD

Figures/Tables 13

Fig.?1

Fig.?2

Fig.?3

Table?1

Table?2

Comparison of the first three lowest frequency parameters Ω of DWCNTs with different EBCs predicted by ED-TPNIM with ξ1=ξ2→1, κ2=κ4=0 and the pure strain-driven nonlocal model"

Mode	Model	$κ 1 = κ 3 → 0$				$κ 1 = κ 3 = 0.1$
Mode	Model	C-C	C-S	S-S	C-F	C-C	C-S	S-S	C-F
1st	Present	4.711 3	3.913 6	3.313 3	1.872 9	4.592 2	3.822 6	3.078 5	1.836 7
1st	Ref. [67]	4.726 0	3.925 0	3.141 0	1.875 0	4.590 0	3.819 0	3.068 0	1.879 0
2nd	Present	7.714 1	6.964 6	6.205 8	4.652 6	7.521 1	6.793 2	6.055 2	4.546 0
2nd	Ref. [67]	7.796 0	7.035 0	6.265 0	4.690 0	7.105 0	6.444 0	5.770 0	4.544 0
3rd	Present	10.466 4	9.798 6	9.104 6	7.661 4	9.625 1	8.967 8	8.390 1	7.476 1
3rd	Ref. [67]	10.654 0	9.981 0	9.276 0	7.797 0	9.123 0	8.557 0	7.976 0	7.111 0

Table?2

Table?3

Fig.?4

Fig.?5

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Fig.?7

Fig.?8

Fig.?9

Fig.?10

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Model	C-C	S-S	C-S	C-F
ED-TPNIM	1.051 3	0.465 0	0.725 4	0.166 1
SD-TPNIM	1.051 3	0.465 0	0.725 4	0.166 1
Ref. [57]	1.051 5	0.468 3	0.732 7	0.167 6

Material	C-C	S-S	C-S	C-F	C-G	S-G
DWCNT	35.916	15.885	24.783	5.675	9.035	3.987
TWCNT	46.573	20.907	32.320	7.908	12.326	5.459