Applicability and limitations of Donnell shell theory for vibration modelling of double-walled carbon nanotubes

Strozzi, Matteo; Elishakoff, Isaac; Manevitch, Leonid I.; Gendelman, Oleg

doi:10.1016/j.tws.2022.109532

Cited by 8 publications

(10 citation statements)

References 42 publications

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“…In the case of anisotropic elastic material, the dimensionless stresses ( σ x,i , σ θ,i , τ xθ,i ) of the i-th cylindrical shell are related to the corresponding dimensionless strains by means of the following constitutive Equation [46]:…”

Section: Stress-strain Relationshipsmentioning

confidence: 99%

“…where the dimensionless material parameters Y jk,i , which substitute Young's modulus and Poisson's ratio of the corresponding isotropic model, represent anisotropic surface elastic constants, in the form [46]:…”

Section: Stress-strain Relationshipsmentioning

confidence: 99%

“…which allow the chirality effects to be considered in the anisotropic elastic constitutive Equation ( 5), where G lj,i , G lk,i , H lj,i , H lk,i are dimensionless constants and µ is a dimensionless constant parameter [46]:…”

Section: Stress-strain Relationshipsmentioning

confidence: 99%

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Applicability and Limitations of Ru’s Formulation for Vibration Modelling of Double-Walled Carbon Nanotubes

Strozzi

2022

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In this paper, a comparison is conducted between two different formulations of the van der Waals interaction coefficient between layers, as applied to the vibrations of double-walled carbon nanotubes (DWCNTs); specifically, the evaluation of the natural frequencies is achieved through Ru’s and He’s formulations. The actual discrete DWCNT is modelled by means of a couple of concentric equivalent continuous thin cylindrical shells, where Donnell shell theory is adopted to obtain strain-displacement relationships. In order to take into account the chirality effect of DWCNT, an anisotropic elastic shell model is considered. Simply supported boundary conditions are imposed and the Rayleigh–Ritz method is used to obtain approximate natural frequencies and mode shapes. A parametric analysis considering different values of diameters and numbers of waves along longitudinal and circumferential directions is performed by adopting Ru’s and He’s formulations. From the comparisons, it is evident that Ru’s formulation provides unsatisfactory results for relatively low values of diameters and relatively high numbers of circumferential waves with respect to the more accurate He’s formulation. This behaviour is observed for every number of longitudinal half-waves. Therefore, Ru’s formulation cannot be used for the vibration modelling of DWCNTs in a large range of diameters and wavenumbers.

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Section: Stress-strain Relationshipsmentioning

confidence: 99%

Section: Stress-strain Relationshipsmentioning

confidence: 99%

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Applicability and Limitations of Ru’s Formulation for Vibration Modelling of Double-Walled Carbon Nanotubes

Strozzi

2022

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“…Adopting this anisotropic elastic model, linear vibrations of single-walled and multiwalled carbon nanotubes for different geometries and wave numbers were investigated on the basis of Donnell [35], Sanders [36] and Flügge [37][38] thin shell theories, separately.…”

Section: Introductionmentioning

confidence: 99%

A Comparison of Shell Theories for Vibration Analysis of Single-Walled Carbon Nanotubes Based on an Anisotropic Elastic Shell Model

Strozzi¹,

Elishakoff²,

Bochicchio³

et al. 2023

Preprint

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In the present paper, the comparison is conducted between three classical shell theories as applied to the linear vibrations of single-walled carbon nanotubes (SWCNTs); specifically, the evaluation of the natural frequencies is conducted via Donnell, Sanders and Flügge shell theories. The actual discrete SWCNT is modelled by means of a continuous homogeneous cylindrical shell considering equivalent thickness and surface density. In order to take into account the intrinsic chirality of carbon nanotubes (CNTs), a molecular based anisotropic elastic shell model is considered. Simply supported boundary conditions are imposed and complex method is applied to solve the equations of motion and to obtain the natural frequencies. Comparisons with the results of molecular dynamics simulations available in literature are performed to check the accuracy of the three different shell theories, where Flügge shell theory is found to be the most accurate. Then, a parametric analysis evaluating the effect of diameter, aspect ratio and number of waves along the longitudinal and circumferential directions on the natural frequencies of SWCNTs is performed in the framework of the three different shell theories. Assuming the results of Flügge shell theory as references, it is obtained that Donnell shell theory is not accurate for relatively low longitudinal and circumferential wavenumbers, for relatively low diameters and for relatively high aspect ratios. On the other hand, it is obtained that Sanders shell theory is very accurate for all the considered geometries and wavenumbers, and therefore it can be correctly adopted instead of the more complex Flügge shell theory for the vibration modelling of SWCNTs.

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“…Adopting this anisotropic elastic model, the linear vibrations of single-walled and multi-walled carbon nanotubes for different geometries and wave numbers were investigated separately via the Donnell [37], Sanders [38], and Flügge [39,40] thin shell theories.…”

Section: Introductionmentioning

confidence: 99%

A Comparison of Shell Theories for Vibration Analysis of Single-Walled Carbon Nanotubes Based on an Anisotropic Elastic Shell Model

et al. 2023

Self Cite

View full text Add to dashboard Cite

In the present paper, a comparison is conducted between three classical shell theories as applied to the linear vibrations of single-walled carbon nanotubes (SWCNTs); specifically, the evaluation of the natural frequencies is conducted via Donnell, Sanders, and Flügge shell theories. The actual discrete SWCNT is modelled by means of a continuous homogeneous cylindrical shell considering equivalent thickness and surface density. In order to take into account the intrinsic chirality of carbon nanotubes (CNTs), a molecular based anisotropic elastic shell model is considered. Simply supported boundary conditions are imposed and a complex method is applied to solve the equations of motion and to obtain the natural frequencies. Comparisons with the results of molecular dynamics simulations available in literature are performed to check the accuracy of the three different shell theories, where the Flügge shell theory is found to be the most accurate. Then, a parametric analysis evaluating the effect of diameter, aspect ratio, and number of waves along the longitudinal and circumferential directions on the natural frequencies of SWCNTs is performed in the framework of the three different shell theories. Assuming the results of the Flügge shell theory as reference, it is obtained that the Donnell shell theory is not accurate for relatively low longitudinal and circumferential wavenumbers, for relatively low diameters, and for relatively high aspect ratios. On the other hand, it is found that the Sanders shell theory is very accurate for all the considered geometries and wavenumbers, and therefore, it can be correctly adopted instead of the more complex Flügge shell theory for the vibration modelling of SWCNTs.

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Applicability and limitations of Donnell shell theory for vibration modelling of double-walled carbon nanotubes

Cited by 8 publications

References 42 publications

Applicability and Limitations of Ru’s Formulation for Vibration Modelling of Double-Walled Carbon Nanotubes

Applicability and Limitations of Ru’s Formulation for Vibration Modelling of Double-Walled Carbon Nanotubes

A Comparison of Shell Theories for Vibration Analysis of Single-Walled Carbon Nanotubes Based on an Anisotropic Elastic Shell Model

A Comparison of Shell Theories for Vibration Analysis of Single-Walled Carbon Nanotubes Based on an Anisotropic Elastic Shell Model

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