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Frequency simulation of viscoelastic multi-phase reinforced fully symmetric systems

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Abstract

Honeycomb structures have the geometry of the lattice network to allow the minimization of the amount of used material to reach minimal material cost and minimal weight. In this regard, this article deals with the frequency analysis of imperfect honeycomb core sandwich disk with multiscale hybrid nanocomposite (MHC) face sheets rested on an elastic foundation. The honeycomb core is made of aluminum due to its low weight and high stiffness. The rule of the mixture and modified Halpin–Tsai model are engaged to provide the effective material constant of the composite layers. By employing Hamilton’s principle, the governing equations of the structure are derived and solved with the aid of the generalized differential quadrature method (GDQM). Afterward, a parametric study is done to present the effects of the orientation of fibers (\(\theta_{{\text{f}}} /\pi\)) in the epoxy matrix, Winkler–Pasternak constants (\(K_{{\text{w}}}\) and \(K_{{\text{p}}}\)), thickness to length ratio of the honeycomb network (\(t_{{\text{h}}} /l_{{\text{h}}}\)), the weight fraction of CNTs, value fraction of carbon fibers, angle of honeycomb networks, and inner to outer radius ratio on the frequency of the sandwich disk. The results show that it is true that the roles of \(K_{{\text{w}}}\) and \(K_{{\text{p}}}\) are the same as an enhancement, but the impact of \(K_{{\text{w}}}\) could be much more considerable than the effect of \(K_{{\text{p}}}\) on the stability of the structure. Additionally, when the angle of the fibers is close to the horizon, the frequency of the system improves.

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Abbreviations

h, R i, and R o :

Thickness, the inner and outer radius of the disk, respectively

CNTs:

Carbon nanotubes

F and NCM:

Fiber and nanocomposite matrix, respectively

\(\rho ,E,\nu \;{\text{and}}\;G\) :

The density, Young’s modulus, Poisson’s ratio, and shear parameter, respectively

V NCM, V F :

Volume fractions of the nanocomposite matrix and fiber, respectively

l CNT, t CNT, d CNT, E CNT and V CNT :

The length, thickness, diameter, Young’s modulus, and volume fraction of carbon nanotubes, respectively

\(V_{\rm CNT}^{*}\), W CNT :

Effective volume fraction and weight fraction of the CNTs, respectively

Nt, V CNT :

Layer number and volume fraction of CNTs

\(E_{1}^{*}\) and \(E_{2}^{*}\) :

Young’s modulus in R and \(\theta\) directions, respectively

\(\nu_{12}^{*}\) and \(\nu_{21}^{*}\) :

Poisson’s ratio in R and \(\theta\) directions, respectively

\(G_{12}^{*}\) :

In-plane shear modulus

\(E_{S}\) and \(\rho_{S}\) :

Young’s modulus and mass density of the base material, which is aluminum for the honeycomb core, respectively

t m, h H, l m , and \(\theta_{{\text{h}}}\) :

The cell wall thickness, the sides of the hexagonal cell, and the angle of honeycomb core, respectively

U, V, W :

Displacement fields of a disk

u, v, w, u1, and v 1 :

The displacements of the mid-surface of the disk

\(\varepsilon_{RR}\) and \(\varepsilon_{\theta \theta }\) :

The corresponding normal strains in \(R\) and θ directions, respectively

\(\gamma_{RZ} ,\,\,\,\gamma_{R\theta } \,{\text{and}}\,\,\,\gamma_{\theta Z}\) :

The shear strain in the RZ, R\(\theta\) and \(\theta\)Z plane

U *, T *, and W * :

Corresponding strain energy of the system, kinetic energy of the system, and the work done, respectively

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

Stiffness elements, stiffness elements related to orientation angle, and the orientation angle, respectively

\(\theta_{{\text{f}}}\) :

The lamination angle concerning the disk R axis

K W and K P :

Winkler and Pasternak foundation coefficient

N r and N θ :

The number of grid points along the radial and circumferential directions, respectively

d, b, and \(\delta\) :

d As a subscript stands for the domain grid points, b as a subscript stands for boundary grid points and the displacement vector, respectively

M ij and K ij :

Components of mass and stiffness matrices, respectively

M ij * and K ij * :

Components of mass and stiffness matrices in the GDQ method, respectively

\(\omega_{n} \;{\text{and}}\,\,\overline{\omega }_{n}\) :

Dimensional and non-dimensional value of natural frequency

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Funding

National Natural Science Foundation of China (51675148). The Outstanding Young Teachers Fund of Hangzhou Dianzi University (GK160203201002/003). National Natural Science Foundation of China (51805475). This research was supported by the 2020 scientific promotion funded by Jeju National University.

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Correspondence to Mostafa Habibi, Dong won Jung or Abdelouahed Tounsi.

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Al-Furjan, M.S.H., Habibi, M., Ni, J. et al. Frequency simulation of viscoelastic multi-phase reinforced fully symmetric systems. Engineering with Computers 38 (Suppl 5), 3725–3741 (2022). https://doi.org/10.1007/s00366-020-01200-x

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