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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
Received date: 2025-06-11
Revised date: 2025-09-16
Online published: 2025-10-29
Supported by
Project supported by the National Natural Science Foundation of China (Nos. 12172169 and 12272064), the Natural Science Foundation of Jiangsu Province of China (No. BK20241773), and the Priority Academic Program Development of Jiangsu Higher Education Institutions of China
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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.
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[J]. Applied Mathematics and Mechanics, 2025 , 46(11) : 2095 -2114 . DOI: 10.1007/s10483-025-3313-8
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