Nonlinear vibrations of graphene piezoelectric microsheet under coupled excitations

doi:10.1016/j.ijnonlinmec.2020.103498

International Journal of Non-Linear Mechanics

Volume 124, September 2020, 103498

https://doi.org/10.1016/j.ijnonlinmec.2020.103498 Get rights and content

Abstract

Investigated herein are the nonlinear vibration characteristics of a rectangular microsheet comprising micro fiber composite with graphene (GP) skin under coupled excitations. Owing to the good thermal conductivity of GP, the heat sink performance of the GP skin is also considered. In this study, the constitutive relation for the microsheet is established by applying the Eringen theory and the governing equations for nonlinear dynamics of the microsheet are obtained from the principle of conservation of energy. The Green and Naghdi’s generalized thermo-elasticity theory and the Galerkin weighted residual method are employed to describe the thermo-elasticity coupling effect of the MFC-GP microsheet. The positive piezoelectric effects of graphene and MFC are also taken into consideration. Next, the coupled ordinary differential equations of thermo-elasticity-electricity are obtained by the Galerkin discrete method. Finally, the nonlinear frequency response equations of the microsheet are derived through the multi-scale method. By using the global residual harmonic balance method, nonlinear vibration characteristic of the MFC-GP microsheet are discussed with respect to nonlocal parameters, volume fraction of graphene, and aspect ratio, which have a significant impact on the dynamic behaviors of the microsheet. The findings will provide a theoretical guidance for engineering design of such microsheets.

Introduction

With the rapid development of microelectronics integration, high-power density devices, heat dissipation performance and power dissipation density become a key factor for stability and operation life of electronic components. Graphene (GP) film is a new type of heat conduction and heat dissipation material. Owing to its high in-plane thermal conductivity, low density, low thermal expansion coefficient, good mechanical properties and other excellent characteristics [1], GP film has burgeoned as one of the new heat dissipation materials. The pursuit of high performance dramatically increases the power consumption in integrated circuits and this leads to a challenging problem in how to dissipate heat in electronics systems. Moreover, the application of graphene for solving the problem of heat dissipation is mainly concentrated in the field of microelectronic devices. In these devices, graphene composites not only have good conductive abilities, but they also have to withstand high temperatures and disturbances. Therefore, the dynamical stability of such micro structures in coupled physical fields are issues worthy of further investigations.

Recently, many scholars discussed the thermal conductivity of GP film. Balandin et al. [2] verified that the thermal conductivity of a single-layer graphene could reach up to 5400 W/(m K) from experiments. Guo et al. [3] pointed out that tensile strain could reduce the thermal conductivity of interlayer-bonded graphene further by up to 50%. Li et al. [4] compared three kinds of graphene heat dissipation films, including NCSG (non-continuous single layer single-crystal graphene), CSG (continuous single layer graphene) and CDG (continuous double layer graphene), and they found that NCSG film made the temperature of a disk center dropped by 1 °C, while CSG made the temperature dropped by 6 °C when the supplied voltage was 10 V.

Meanwhile, many scholars have investigated the introduction of appropriate graphene for improving the thermal properties of composite structures [5], [6], [7]. Ren et al. [8] found that the heat transfer of composite materials may be improved when small graphene sheets were placed in the interspaces of large graphene sheets. Altay et al. [9] investigated the influence of graphene on the thermal conductivity, thermal expansion coefficient and thermal stability of PP/graphite/graphene composite and their results showed that the highest thermal conductivity of the structure was 4.6 W/(m K) at 50 wt% synthetic graphite and 3 wt% graphene. Azizi et al. [10] explored composites of low-density polyethylene (LDPE) and a graphene-like material for their thermal and electrical properties to prove that there was an alignment among the electrical, thermal and rheological properties. Liu et al. [11] introduced a functionalized hybrid Al2O3-GO into the TPI (trans-1, 4-polyisoprene) matrix to enhance the interfacial interaction of the nanocomposites, which led to an improvement of the thermodynamic properties of the TPI.

Although the thermoelastic coupling has been a concern for decades, it is still a challenging problem for nanoscale structures. Zenkour et al. [12] and Abouelregal et al. [13] employed the nonlocal theory to analyze the vibration characteristics of nanobeams. Lin et al. [14] proposed a method to solve the thermostatically problem of FG-CNT reinforced composites by considering the aerodynamic heating and transient heat conduction. Yu et al. [15] adopted both the size effect of heat conduction and elasticity in discussing the effects of size-dependent characteristic lengths and material constants of each layer on the transient responses of a bi-layered structure. Zhao et al. [16], [17] investigated the coupled thermoelastic vibration characteristics of cracked Euler–Bernoulli beams and Timoshenko beams by using Green’s function method. In addition, there are many articles written on thermoelastic coupling characteristics of different structures that include resonators, thin beams and plates, such as Cao et al. [18], Zhang et al. [19], Abouelregal et al. [20] and Guo et al. [21].

Moreover, Heydarpour et al. [22] applied the Lord–Shulman thermo-elasticity theory to discuss the transient mechanical properties of the graphene reinforced composite plates. Hosseini et al. [23] carried out coupled thermo-elastic analysis of functionally graded multilayer graphene reinforced nanocomposite plates through the Green–Naghdi theory with energy dissipation. Deng et al. [24] discussed the thermo-elastic damping of nanomaterials by considering the size effects and heat conduction effects of nano structures. Rashahmadi et al. [25] employed the nonlocal theory to model the size dependent thermo-elastic energy dissipation of graphene nano resonators.

It is clear that there have been many papers published on the thermoelastic coupled vibration of the structures, such as graphene nanocomposites, beams, and resonators. However, the mechanical, electrical and thermal properties of graphene are rarely considered and analyzed in a comprehensive way. In this paper, against the backdrop of micro-electronic radiator knowledge, we consider the thermo-elastic coupling and piezoelectric effect of the structure, and simulate the vibration characteristics of the structure more realistically. This provides a reliable support for electronic device designers to select parameters according to the dynamical stability of the structure.

In this paper, Section 2 presents the constitutive relation for the MFC-GP microsheet. In Section 3, the ordinary differential coupling equations of thermo-elasticity-electricity of the microsheet are established by applying the Hamilton principle, the Green and Naghdi’s generalized thermo-elasticity theory without energy dissipation and the Galerkin discrete method. Moreover, the global residual harmonic balance method is applied to investigate the influence of different parameters (such as the nonlocal parameter and graphene volume fraction) on the nonlinear frequency of the microsheet in Section 4. The dynamic stability of the MFC-GP microsheet is discussed in Section 5. Finally, Section 6 presents the conclusion of this research.

Section snippets

Constitutive equation for MFC-GP microsheet

Based on the manufacturing method of 1–3 micro piezoelectric fiber plate (MFC) by Han et al. [26], a new three-phase composite material microsheet model is established as shown in Fig. 1, where MFC films are composed of 1–3 type layers deposited with multilayer graphene on both upper and lower surfaces.

Based on Ref. [27], the physical parameters of the MFC-GP microsheet are obtained by the mixing rule $E_{c} = E_{g} a_{g} + (1 - a_{g}) (E_{p} a_{p} + (1 - a_{p}) E_{m})$ where $E_{i} (i = c, g, p, m)$ is the Young’s modulus, and the subscripts $c, g, p$ and m

Governing equations and boundary conditions

The mechanics model of the rectangular MFC-GP microsheet is shown in Fig. 2 with length a, width b and uniform thickness h. The edges of the microsheet are simply-supported. The sheet is subjected to the transverse harmonic excitation $F cos Ω t$ , where $F$ is the force amplitude, $Ω$ is the excitation frequency and t is the time. The microsheet is placed in a varying temperature field $T (x, y, z, t)$ . The displacements $u_{0}$ , $v_{0}$ and $w_{0}$ of the neutral surface of the microsheet are along the x, y and z

Nonlinear frequency analysis of the MFC-GP microsheet

Here, nonlinear frequencies of free vibration of the microsheet are analyzed. Then, Eq (25a) are rewritten as ${\ddot{w}}_{1} + r w_{1} {\dot{w}}_{1}^{2} + ξ w_{1} + ψ w_{1}^{3} = 0$ where $ξ$ and $ψ$ are given by $ξ = η_{1} + η_{3} T_{0 l} + η_{4} E_{3} + η_{5} T_{0 l} E_{3}, ψ = η_{2}$

It can be seen from Ref. [33] that the global residual harmonic balance method is an effective method for determining the nonlinear frequency of structures. Here, a new independent variable $t = ω τ$ is introduced and the initial transverse displacement and velocity of the microsheet are set to zero, i.e. $w_{1} (0) = A, {\dot{w}}_{1} (0) =$

Stability analysis and numerical simulations

The main resonance occurs in the MFC-GP microsheet when the external excitation frequency is equal to the natural frequency of the microsheet. In this section, the method of multi-scale based on the Ref. [47] is applied to obtain the frequency response equation of the microsheet.

Assume the solution of Eq. (25a) has the following form $w_{1} = w_{10} (T_{0}, T_{1}) + ε w_{11} (T_{0}, T_{1}) + \dots$ where $T_{0} = t$ and $T_{1} = ε t$ .

The main resonance relationship is expressed as $Ω = ω + ε σ$ where $σ$ is the detuning parameter.

The differential operators are

Conclusion

In this study, the nonlinear vibration characteristics of the MFC-GP microsheet are investigated with a focus on graphene as the heat dissipation film for microelectronic devices. The structural dynamic equations considering nonlocal elasticity parameters are obtained by Hamilton’s principle. Next, the heat conduction equations are calculated based on Green and Naghdi’s generalized thermo-elasticity theory. The Galerkin weighted residual method is then applied to establish the thermo-elasticity

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through grant Nos. 11772010 and 11832002, the Funding Project for High level teachers’ team construction in Beijing municipal colleges and Universities .

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Nonlinear vibrations of graphene piezoelectric microsheet under coupled excitations

Abstract

Introduction

Section snippets

Constitutive equation for MFC-GP microsheet

Governing equations and boundary conditions

Nonlinear frequency analysis of the MFC-GP microsheet

Stability analysis and numerical simulations

Conclusion

Declaration of Competing Interest

Acknowledgments

Carbon

Carbon

Composites A

Composite Struct.

Eur. J. Mech. A Solids

Int. J. Mech. Sci.

Int. J. Mech. Sci.

Int. J. Heat Mass Transfer

Composite B

Eng. Anal. Bound. Elem.

Compos. Struct.

Internat. J. Engrg. Sci.

Chaos Solitons Fractals

J. Sound Vib.

Compos. Struct.

Compos. Struct.

Compos. Struct.

Appl. Math. Model.

Appl. Math. Model.

Appl. Math. Model.

Superior thermal conductivity of single-layer graphene

Nano Lett.

Tuning the thermal conductivity of multi-layer graphene with interlayer bonding and tensile strain

Appl. Phys. A

Graphene heat dissipation film for thermal management of hot spot in electronic device

J. Mater. Sci., Mater. Electron.