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Finite element predictions on vibrations of laminated composite plates incorporating the random orientation, agglomeration, and waviness of carbon nanotubes

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Abstract

In the present article, the vibration analysis of laminated composite plates strengthened with multi-walled carbon nanotubes (MWCNTs) is performed, employing the finite element method. Rule of mixtures and the Halpin–Tsai model are utilized to ascertain the elastic properties of the nanocomposite matrix. The effect of random orientation, the curved state, and the agglomeration of carbon nanotubes in the epoxy resin on mechanical properties is investigated. Classical lamination theory is also implemented to theoretically estimate the natural frequencies for validation purposes. Then, the natural frequencies of the laminated composite plates are determined by the numerical method for various parameters involved in the design process. The effect of MWCNTs’ inclusion on the natural frequencies of the carbon nanotube-based laminated plate, taking into account the reformed micromechanical Halpin–Tsai model, is finally investigated. The outcomes of the research are in compliance with experimental values and theoretical evidence available in the bibliography.

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Abbreviations

A :

Amplitude of a curved CNT

a, b :

Plate dimensions in x, y directions

D ij :

Bending stiffness coefficients

d :

Nanotube exterior diameter

E :

Elastic modulus

f R :

Random orientation factor of CNTs

f W :

Waviness factor of CNTs

f A :

Agglomeration factor of CNTs

f mn :

Natural frequency (Hz)

G :

Shear modulus

h :

Total plate thickness

[K]:

Global stiffness matrix

L :

Nanotube length

[M]:

Global mass matrix

m, n :

Mode shape integers of vibration

P :

Associated mechanical property

\({Q}_{ij}\) :

Reduced stiffness coefficients

\({\overline{Q} }_{ij}\) :

Transformed reduced stiffness coefficients

t :

Time

{u i}:

Mode shape vector

V :

Volume fraction

W :

Half-wavelength of a curved CNT

w :

Transverse displacement component

x, y :

Coordinates parallel to ply midplane

z :

Coordinate normal to ply midplane

α, β :

Variables of CNT agglomeration factor

γ xy :

Shear strain component

ε x , ε y :

Normal strain components

\(\theta\) :

Fiber orientation angle in ply (deg)

ξ :

Strengthening factor

ρ :

Mass density

σ x , σ y :

Normal stress components

τ xy :

Shear stress component

υ :

Poisson’s ratio

ω :

Natural circular frequency (rad s−1)

\(\overline{\omega }\) :

Dimensionless natural circular frequency

1, 2, 3:

Property or variable referred to material axes

c:

Composite property

cnt:

Carbon nanotube property

f:

Fiber property

m:

Matrix property

m, n :

Mode shape integers of vibration

m-cnt:

Modified/hybrid matrix property

k:

Number of layer

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Authors and Affiliations

  1. Laboratory for Advanced Materials, Structures and Digitalization, Department of Aerospace Science and Technology, National and Kapodistrian University of Athens, Psachna, Evvoia, Greece

    Stelios K. Georgantzinos, Panagiotis Antoniou, Stylianos Markolefas & Georgios Giannopoulos

  2. General Department, National and Kapodistrian University of Athens, Psachna, Evvoia, Greece

    Panagiotis Antoniou

  3. Marine Engineering Department, Military Technological College, Muscat, Sultanate of Oman

    Stylianos Markolefas

  4. Department of Mechanical Engineering, School of Engineering, University of Peloponnese, Patras, Greece

    Georgios Giannopoulos

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  1. Stelios K. Georgantzinos
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  2. Panagiotis Antoniou
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  3. Stylianos Markolefas
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  4. Georgios Giannopoulos
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Correspondence to Stelios K. Georgantzinos.

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Georgantzinos, S.K., Antoniou, P., Markolefas, S. et al. Finite element predictions on vibrations of laminated composite plates incorporating the random orientation, agglomeration, and waviness of carbon nanotubes. Acta Mech 233, 2031–2059 (2022). https://doi.org/10.1007/s00707-022-03179-6

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