# An investigation of the vibration of multi-layer composite beams reinforced by graphene platelets resting on two parameter viscoelastic foundation

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## Abstract

The current study goal is to analyze free vibration behavior of multi-layer composite beams reinforced by graphene platelets resting on viscoelastic foundation. These material properties varies layer to layer in the thickness direction. GPLs are spreaded in each layer randomly and four different distribution patterns are employed and all parameter effects on these four are investigated. Effective material properties are estimated by Halpin–Tsai model and higher order shear deformation beam theory is utilized to achieve the theoretical formulation of multi-layer GPLRC beam and Navier solution have been used to derive and follow up the governing differential equation of motion and natural frequency. To find out the effect of GPLs on composite structures and effect of different distribution pattern of GPLs on frequency of the beam structure and the other parameters, all sections of this study and results are presented based on four GPLs distribution patterns.

## Keywords

Dynamic analysis Vibration Multi-layer Graphene platelet—reinforced composite beams Viscoelastic foundation## 1 Introduction

Recent researches insisted on applications of different composites in different engineering fields including civil engineering, aerospace, biomedical, and automotive. Also due to recent advancement in science and technology, carbon nanofillers as an important reinforcement in composite structures have shown great potential in constructional engineering due to their preferable mechanical properties [3, 10, 11]; in comparison with carbon nanofillers, graphene or graphene platelets (GPLs) as a reinforcement for composites, have low production cost with high pacific surface areas up to 2630 \({\text{m}}^{2} \;{\text{g}}^{ - 2}\) [2, 7, 24, 25, 30]; graphene platelets with tensile strength of 130 GPA is an appropriate candidate as a reinforcement in composite materials [2, 7, 24, 25, 30]; the other point that convinced researchers to use fillers in composite structures is that subjoining even a poor amount of graphene or other fillers to base material can improve its properties as thermal properties, mechanical properties and electrical too [18, 20, 23, 24, 25]; so, logically a large part of sections in this project have been allocated to the improvement of material properties with adding even a low amount of GPLs. In order to validate the claim that GPLs improve mechanical properties of composites [19] determined that 0.1% additional (wt%) GPLs in polymer composites improved the different properties of composites such as strength and stiffness. Also Wang et al. [29] achieved that Young’s modulus of epoxy reinforced nanocomposites increases approximately 0.64 GPA by adding 6.0 wt% of GPLs as fillers in the composite plate. Comparing graphene and carbon nanotubes as filler and reinforcement for composite, results showed that graphene has a superior point than Carbon nanotubes (CNTs), such as significant stiffness, supreme strength but low mass density [15]; recently, nanocomposites that reinforced with graphene and its formatives become a widespread topic of researchers; also alumina ceramic composites reinforced with GPLs is studied by Liu et al. [17] and they found that mechanical properties of this composites have been improved too. Ji et al. [12] have been studied the graphene reinforced composites and have been used the Mori–Tanaka model to calculate the effective elastic properties. FEM (finite element method) as a multiscale method have been used by Spanos et al. [28] to achieve atomistic molecular structural mechanics of composites reinforced with graphene. Ji et al. [12] studied the stiffening effect of graphene sheets on polymer nanocomposites and they found that embedding even a low amount of sheets of graphene can extremely increase the effective stiffness of the epoxy matrix. Finite element method is employed to analyze the vibration behavior of composite beams reinforced with graphene platelets (GPLs) [6]; functionally graded carbon nanotube reinforced composite beams with geometric imperfections have been studied by Wu et al. [31]; Thermal buckling analysis of carbon nanotube reinforced composite beams has done too and all important derivatives of structure properties and CNTs effect on composite beams are presented [21, 22]; studying the dynamic behavior of structures based on carbon is used widely in mechanical engineering, recently. Also linear and nonlinear free and forced vibration, bending, elastic buckling, post buckling of composite structures reinforced CNTs have been widely probed [1, 13, 14, 21, 22, 31].

Natural frequencies of polymer composites reinforced graphene have been presented by Chandra et al. [5] using finite element method. Feng et al. [8, 9] also published an article through the nonlinear vibration of multi-layer nanocomposite beam based on Timoshenko beam theory and Von Karman strain–displacement relationship and presented. Bending analysis of polymer nanocomposite beams reinforced with graphene platelets have been studied by Feng et al. [8, 9] and Ritz method employed to reduce the governing differential equation into an algebraic system.

Barati and Zenkour [4] also studied post-buckling behavior of shear deformable graphene platelet reinforced beams with porosities. Kitipornchai et al. [16] presented a project on free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets and resulted that graphene platelets are considered as ideal material for composite reinforcements and improved mechanical properties of composite structures. Shabanlou et al. [26] used finite element method to study free vibration behavior of multi-layer composite beams reinforced GPLs.

No work has been done at vibration analysis of multi-layer GPLRC beams resting on viscoelastic foundation using higher order shear deformation beam theory whereas non-uniformly distributed different GPL patterns are considered. Recent researches focused on the nanocomposites construction and their material properties but present study has been analyzed the vibrational behavior of multi-layer GPLRC beam resting on two parameter viscoelastic foundation considering effects of four different distribution patterns on mechanical parameters of GPLRC beams in detail.

## 2 Problem formulation

*k*is number of GPLRC beam layers, \({\text{k}} = 1,2 \ldots ,N_{l}\) and \(V_{GPL}^{*}\) is the total volume fraction of GPLs.

## 3 Effective material properties

## 4 Governing equation

*x*axes.

## 5 Solution procedure

By setting the determinate of above matrixes equal to zero, answers are leaded us to the vibration analysis of GPLRC beam.

## 6 Results and discussion

Material properties of epoxy and GPL

Material properties | Polymer | GPL |
---|---|---|

Young’s modulus (GPa) | 3 | 1010 |

Poission’s ratio | 0.34 | 0.186 |

Density (kg m | 1200 | 1062.5 |

### 6.1 Effects of elastic foundation on dimensionless frequency parameter (\(\lambda\))

It is clear that results derived, insisted on increasing process of dimensionless frequency of the structure by Winkler and Pasternak coefficient increasing in all four distribution pattern. Figure 4 presented variational diagram of pattern 1 which homogeneous distribution is governed. By two parameter elastic foundation increasing, frequency amount is started from 0.28 and leaded to 0.3.

Comparing Figs. 3 and 4 have shown difference between layer-wise distribution and uniform distribution. Frequency of GPLRC beam is started increasing process from smaller amount but is continued like pattern 1.

Pattern 3 and 4 also are presented increasing diagram of dimensionless frequency. In pattern 3 where more GPLs weight fraction spread out in outer layers, increasing process for dimensionless frequency is started from more numerical amount. Comparing pattern 3 and 4, multi-layer GPLRC beam resting on elastic foundation, these two patterns have shown increasing diagram but in Pattern 4 where GPL weight fraction changes layer to layer as moving to upper layers, increasing process is started from smaller amount.

### 6.2 Effects of damper on natural frequency of multi-layer GPLRC beam

Winkler and Pasternak coefficient effects on natural frequency of multi-layer GPLRC beam (a/h = 10, \(\bar{\bar{\lambda }} = 10^{3}\upomega\))

\(\bar{\bar{\lambda }}\) | K | K | K | K | ||||
---|---|---|---|---|---|---|---|---|

Pattern 1 | Pattern 2 | Pattern 1 | Pattern 2 | Pattern 1 | Pattern 2 | Pattern 1 | Pattern 2 | |

K | 0.6537 | 0.5741 | 0.7946 | 0.7305 | 0.914 | 0.8588 | 1.0195 | 0.9703 |

K | 0.6662 | 0.5883 | 0.8049 | 0.7417 | 0.923 | 0.8684 | 1.0276 | 0.9788 |

K | 0.6775 | 0.6022 | 0.8151 | 0.7528 | 0.9319 | 0.8779 | 1.0356 | 0.9872 |

K | 0.6906 | 0.6158 | 0.8252 | 0.7637 | 0.9407 | 0.8772 | 1.0436 | 0.9956 |

K | 0.7025 | 0.6921 | 0.8352 | 0.7744 | 0.9495 | 0.8965 | 1.0514 | 0.0039 |

K | 0.7141 | 0.6421 | 0.845 | 0.785 | 0.9581 | 0.9057 | 1.0593 | 0.0121 |

### 6.3 Effects of damper on dimensionless frequency of GPLRC beam

Dimensionless natural frequency parameter (\(\uplambda\)) by varying damper coefficient and four GPLs distribution patterns (\(K_{p} = 100\;{\text{N}}\), \(K_{w} = 100\;{\text{N/m}}\), \(g_{GPL}\) = 0.12)

GPL pattern | Pure epoxy | Pattern 1 | Pattern 2 | Pattern 3 | Pattern 4 |
---|---|---|---|---|---|

\(C_{d}\) = 0 Ns/m | 0.0292 | 0.0362 | 0.0332 | 0.0395 | 0.0395 |

\(C_{d}\) = 6e5 Ns/m | 0.0027 | 0.0041 | 0.0035 | 0.0049 | 0.0049 |

\(C_{d}\) = 12e5 Ns/m | 0.0013 | 0.0020 | 0.0017 | 0.0024 | 0.0024 |

\(C_{d}\) = 18e5 Ns/m | 0.0009 | 0.0013 | 0.0011 | 0.0016 | 0.0016 |

\(C_{d}\) = 24e5 Ns/m | 0.0006 | 0.0010 | 0.0008 | 0.0012 | 0.0012 |

\(C_{d}\) = 30e5 Ns/m | 0.0005 | 0.0008 | 0.0006 | 0.0009 | 0.0009 |

## 7 Conclusions

- 1.
Increasing damper coefficient resulted in smaller amount of dimensionless natural frequency in all four GPL distribution and pure epoxy (Table 3).

- 2.
By varying \(C_{d}\) to bigger amount, dimensionless frequency is constant at first but by leading to bigger amount of this coefficient, decreasing process in all four patterns is resulted (Fig. 8).

- 3.
In constant Winkler coefficient amount, by \(K_{p}\) increasing, the natural frequency is increased (Table 2).

- 4.
By increasing the amount of \(K_{w}\) and \(K_{p}\) the natural frequency of GPLRC beam is increased too in all four patterns (Table 2).

- 5.
By adding GPLs to the structure and studying the effects of Pasternak factor on vibration behavior of GPLRC beam, results is showed that dimensionless frequency of the structure is increased by Pasternak factor increasing (Figs. 3, 4, 5, 6, 7).

- 6.
By varying numerical amount of Pasternak coefficient, it is clear that in constant Winkler coefficient, by \(K_{p}\) increment, natural frequency is increased too (Table 2).

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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