# Vibration of a Thermoelastic Microbeam due to the Thermoelectrical Effect of a Strip of Graphene

Authors: Ahmed N. M. Alahmadi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this work, an analysis of thermoelastic, homogeneous, and isotropic microbeams was conducted in the context of the non-Fourier heat conduction law. The first end of the microbeam was based on a graphene strip, which was connected to an electrical voltage source. Under simple boundary conditions maintained by fixed side ratios, the Lord–Shulman model of generalized thermoelasticity was applied. The microbeam was thermally loaded with a heat source due to the thermal effect of the electrical current that ran through the strip of graphene. Laplace transformation of the time variable was utilized to solve controlled differential equations. All the solutions were found in the Laplace transform domain. Tzou's approximation approach, which relies on an iteration formula, was used to numerically calculate the Laplace transform inversions. For different values of the electrical voltage and the resistance of the graphene strip, different graphs were used to show the numerical results. The electrical voltage and the electrical resistance were reported to have significant influences on all the studied functions of the microbeam. Thus, controlling a microbeam’s vibration and energy could be accomplished by tuning the electrical resistance and applied voltage.


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## Body

## 1. Introduction

Fourier's law of heat conduction is the basis for linked thermoelasticity, which is a type of heat conduction based on two partial differential equations, one for motion and another for conserving energy [1–3]. In the case of an isotropic body, Lord and Shulman amended Fourier's rule of heat conduction by incorporating the relaxation time into the equation [4]. Specifically, Cattaneo's heat conduction, which is also known as non-Fourier heat conduction law, has been modified to incorporate both the heat flux and its time derivative, and it has been considered as a substitute for the standard Fourier's law of heat conduction. Accordingly, there are no infinite propagation problems, as the heat equation is hyperbolic [5–9].The most fundamental type of microbeam resonator is the vibration microbeam. There have been numerous studies on the vibration and heat transfer processes of microbeams [10–19]. The Green functions and their properties were utilized by Kidawa-Kulka to investigate the transversal vibrations caused by a moving heat source and the effects of internal and external dampening [15]. Manolis and Beskos used a computational analysis approach to evaluate the elastic dynamic response of a beam structure to heat loading [16].The investigation of visco-thermoelastic materials, which are known to often have relaxation properties, has become increasingly important in mechanics. Biot has written about the theory of visco-thermoelasticity, as well as vibrational principles in thermodynamics, among other things [20, 21]. Drozdov developed a constitutive model of the visco-thermoelasticity behaviour of polymers when they are subjected to finite strains [22]. Using a new visco-thermoelasticity model for isotropic media, Ezzat and El-Karamany investigated the consequences of relaxation on the volume features of viscoelastic materials to study the transversal vibrations caused by a moving heat source and the effects of internal and external dampening [23]. A numerical approach for wave simulations in elastic media was developed by Carcione et al., and it was based on the Kelvin–Voigt mechanical model [24]. The resonance of a microscale visco-thermoelastic beam and its transverse vibrations were studied by Grover [25–27]. In addition, an analysis applying closed-form equations for thermoelastic thin beam transverse vibrations with micro or nanoscale vacancies was carried out by Sharma and Grover [10]. Using a dual phase-lagging model, Grover and Seth investigated visco-thermoelastic microscale beam resonators [28].Since the ground-breaking observations of Novoselov et al., graphene has continued to pique the interest of scientists around the world [29]. The electronic properties of graphene as a single-layer material are intriguing and include ballistic transport, a quantum Hall effect at ambient temperature, and a size-dependent band gap, all of which have the potential to enable new types of microscale electronic devices and sensors [30].In this work, for the first time, an analysis for thermoelastic, homogeneous, and isotropic microbeams was conducted in the context of the non-Fourier heat conduction law when the first end of the microbeam was based on a graphene strip connected to an electrical voltage. Because of the thermal effect of the electrical current running on the graphene strip, the microbeam was thermally loaded with a heat source. The microbeam was electrically isolated using an electrical isolator with high thermal conductivity, as shown in Figure1.Figure 1
Rectangular thermoelastic microbeam based on a strip of graphene.


## 2. Problem Formulation

For a narrow thermoelastic microbeam with length ofℓ0≤x≤ℓ, a width of b−b/2≤y≤b/2, and a thickness of h−h/2≤z≤h/2, we considered the flexural deflections to be very small.As shown in Figure1, the x, y, and z axes of the beam are specified as the longitudinal, width, and thickness directions of the beam, respectively.We considered there was no damping and no stress or strain, and that the reference temperatureT0 was constantly in an equilibrium condition for the microbeam [5]. At first, while any plane’s cross section is bent perpendicular to a beam's axis (the “neutral surface”), it will remain perpendicular to the beam's axis and plane, as determined by the Euler–Bernoulli equation [25]. Consequently, displacement elements are formed [25, 27, 28, 31]:(1)ux,y,z,t=−z∂wx,t∂x,vx,y,z,t=0,wx,y,z,t=wx,t.The flexural moment of the cross section and the equation of motion are given by [25, 27, 28, 31](2)Mx,t=λ+2μI∂2wx,t∂x2+βMTx,t.The above equation contains the thermal momentMT about the beam axis, the elastic term due to the vibration of the beam ∂2wx,t/∂x2, and the total flexural moment of the cross section of the beam, where the equation of motion is in the following form [25, 27, 28, 31]:(3)∂2Mx,t∂x2+ρA∂2wx,t∂t2=0.The thermal moment of the microbeam'sx-axis, MT, is calculated using [25–28, 31, 32](4)MTx,t=b∫−h/2h/2Tx,z,tzdz,where I=bh3/12 denotes the cross-sectional moment of inertia about the x-axis, λandμ are the elastic constants, ρ is the density, A is the cross-sectional area, and β=3λ+2μαT.Therefore, the equation of motion that results in lateral vibrations of the microbeam due to thermally induced vibrations has the following form [25]:(5)λ+2μI∂4wx,t∂x4+ρA∂2wx,t∂t2+β∂2MTx,t∂x2=0.The cross-sectional area is given byA=hb.The heat conduction equation of Lord–Shulman has the following form [25]:(6)∂2Tx,z,t∂x2+∂2Tx,z,t∂z2=∂∂t+τ∂2∂t2ρCυKTx,z,t+βT0Kex,z,t−1+τ∂∂tQx,z,tK,where τ is the thermal relaxation time parameter, Cυ is the specific heat at a constant strain, Q is the heat source, and K is the thermal conductivity.The cubical dilatation (volumetric strain) has the following form:(7)ex,z,t=∂ux,z,t∂x+∂vx,z,t∂y+∂wx,z,t∂z.Thus, from (1), we have(8)ex,z,t=−z∂2wx,t∂x2.We then obtain(9)σxxx,z,t=λ+2μex,z,t−βθx,z,t.Since there is no heat transfer at the upper and bottom sides of the beam,∂Tx,z,t/∂z|z=±h/2=0. As a result, we may assume that the temperature relies on the function of sinpz in the direction of the beam’s thickness, where p=π/h, which yields [33](10)θx,z,t=Tx,z,t−T0=ϑx,tsinpz,(11)Qx,z,t=qx,tsinpz.The temperature increment is maintained atθx,z,t.As a result of incorporating (4), (5), and (10), the following result was obtained:(12)∂4wx,t∂x4+12ρh2λ+2μ∂2wx,t∂t2+12βh3λ+2μ∂2ϑx,t∂x2∫−h2h2sinpzzdz=0.By using (6) and (10), we obtained(13)∂2ϑx,t∂x2−p2ϑx,tsinpz=∂∂t+τ∂2∂t2εϑx,tsinpz+T0βKz∂2wx,t∂x2−1+τ∂∂tqx,tKsinpz,where ε=ρCυ/K.(12) has the following form when the integrations have been concluded:(14)∂4wx,t∂x4+12ρh2λ+2μw¨x,t+24βhπ2λ+2μ∂2ϑx,t∂x2=0.After multiplying both sides byz and integrating z from −h/2toh/2 in (13), we obtained(15)∂2ϑx,t∂x2−p2ϑx,t=∂∂t+τ∂2∂t2εϑx,t−T0hπ2β24K∂2wx,t∂x2−1+τ∂∂tqx,tK.(9) takes the following form:(16)σxx=λ+2μe−βϑsinpz.The non-dimensional variables specified below were utilized [21]:(17)x′,w′,h′=εc0x,w,h,t′,τ′=εc02t,τ,σ′=σλ+2μ,ϑ′=ϑT0,q′=qT0Kε2c02,c02=λ+2μρ.Following that, we obtained(18)∂4wx,t∂x4+ε1w¨x,t+ε2∂2ϑx,t∂x2=0,(19)∂2ϑx,t∂x2−ε3ϑx,t=∂∂t+τ∂2∂t2ϑx,t−ε4∂2wx,t∂x2−1+τ∂∂tqx,t,(20)σxxx,z,t=ex,z,t−ε5ϑx,tsinpz,where ε1=12/h2,ε2=24βT0/hπ2λ+2μ,ε3=p2,ε4=π2hβ/24Kε, and ε5=βT0/λ+2μ (the primes have been dropped for convenience).Laplace transform with the following definition was applied:(21)f¯x,s=∫0∞fx,te−stdt.The Laplace transform’s inverse is expressed as [34](22)L−1f¯s=ft≈eκtt12f¯κ+Re∑n=1N−1nf¯κ+inπt.Many investigations have verified that the value of the parameterκ would fit the relationship κt≈4.7 [34], to ensure a faster convergence, where “Re” denotes the real elements and “i” denotes the imaginary number unit.Equations (18)–(20) then take the following forms:(23)d4w¯dx4+ε1s2w¯+ε2d2ϑ¯dx2=0,d2ϑ¯dx2−ε3ϑ¯=s+τs2ϑ¯−ε4d2w¯dx2−1+τsq¯,σ¯xx=e¯−ε5ϑ¯sinpz,(24)e¯=−zd2w¯dx2.Consider an electrical current connected to the strip of graphene, which has an electrical resistanceReΩ, where the graphene strip is attached at the starting end of the microbeam, as shown in Figure 1. The beam is subjected to specific heating resulting from the thermal effect caused by the connection of the graphene strip to an electric voltage source (Joule’s equation of electrical heating) V(V).The heat flux is then calculated using Joule’s equation of electrical heating as follows [29]:(25)qx,t=V2Ret,where V is the voltage and Re is the electrical resistance.Then, after using the Laplace transform, we have(26)q¯=V2Res2,which gives(27)d2ϑ¯dx2−ε3ϑ¯=s+τs2ϑ¯−ε4d2w¯dx2−1+τsV2s2Re.We can then rewrite equations (23) and (24) in the following forms:(28)D4+ε1s2w¯+ε2D2ϑ¯=0,(29)ε4s+τs2D2w¯+D2−ε3+s+τs2ϑ¯=−1+τsV2s2Re,where Dr=dr/dxr.By elimination between (28) and (29), we obtain(30)D6−LD4+MD2−Nw¯=0,(31)D6−LD4+MD2−Nϑ¯=−ψ,where L=ε2ε4s+τs2+ε3+s+τs2,M=ε1s2,N=ε1s2ε3+s+τs2, and ψ=ε11+τsV2/Re.The general solution of (30) is as follows:(32)w¯=∑j=13Ajsinhkjℓ−x.The general solution of equation (33) is as follows:(33)ϑ¯=ψN+∑j=13Bjsinhkjℓ−x.To obtain the relationship between the parametersAj and Bj, we use the relationship in (28), which gives(34)Kj4+ε1s2Aj+ε2Kj2Bj=0,j=1,2,3.We then have(35)ϑ¯=ψN−1ε2∑j=13Kj4+ε1s2Kj2Ajsinhkjℓ−x.The boundary conditions were(36)∑j=13Ajsinhkjℓ=0,(37)∑j=13kj2Ajsinhkjℓ=0,(38)∑j=13Kj4+ε1s2Kj2Ajsinhkjℓ=ε2ψN.By solving equations (38)–(40), we obtained the parameters A1,A2,A3 as follows.A1=ε2ψ/ε1s2k12−k22k12−k32sinhk1ℓ, A2=ε2ψ/ε1s2k22−k12k22−k32sinhk2ℓ, and A3=ε2ψ/ε1s2k32−k12k32−k22sinhk3ℓ.This completes the solution of the Laplace transform domain.The lateral deflection function is as follows:(39)w¯x,s=ε2ψε1s2sinhk1ℓ−xk12−k22k12−k32sinhk1ℓ+sinhk2ℓ−xk22−k12k22−k32sinhk2ℓ+sinhk3ℓ−xk32−k12k32−k22sinhk3ℓ.The temperature increment function is as follows:(40)θ¯x,s=ψsinpzε1s2ε3+s+τs2+ε2ψsinpzε1s2K14+ε1s2sinhk1ℓ−xK12k12−k22k12−k32sinhk1ℓ+K24+ε1s2sinhk2ℓ−xK22k22−k12k22−k32sinhk2ℓ+K34+ε1s2sinhk3ℓ−xK32k32−k12k32−k22sinhk3ℓ.From (24) and (39), we obtained the deformation in the following form:(41)ex,z,s=−zε2ψε1s2k12sinhk1ℓ−xk12−k22k12−k32sinhk1ℓ+k22sinhk2ℓ−xk22−k12k22−k32sinhk2ℓ+k32sinhk3ℓ−xk32−k12k32−k22sinhk3ℓ.The strain-energy density function through the microbeam is given by [35–37](42)ϖx,z,t=∑i,j312σijx,z,teijx,z,t=12σx,z,tex,z,t.


## 3. Numerical Results and Discussion

As copper is a thermoelastic material, the following physical constants were utilized to describe the material [27, 38–40]: ρ=8954kgm−3, αT=1.7810−5K−1, k=386Wm−1K−1, λ=77.6×109Nm−2, τ0=4.32×10−13s, Cυ=383.1Jkg−1K−1, T0=300K, and μ=38.6×109Nm−2.The resistance of the graphene at the microscale has the valueRe=500Ω [30].The microbeam's aspect ratios were set atℓ/h=8 and b=h/2. However, a microbeam length range of ℓ1−100×10−6m was used, while the original time t and the relaxation time τ0 were on the scale of 10−12 and 10−14 s, respectively.The dimensionless variables for microbeam lengthℓ=1.0, θ0=1.0, z=h/4, and t=1.0 were used to generate the figures.The numerical results of the problem have been illustrated in two groups of figures. The first group shows the distributions of the vibration (lateral deflection), cubical deformation, temperature increment, stress, and strain-energy density when the electrical resistance of strip of the graphene was constant and had the valueRe=500Ω with three different values of the electrical voltage V=10,11,12V. The second group represents the distributions of the same functions when the value of the electrical voltage was constant at a value of V=10V but with three different values of electrical resistance, Re=500,600,700Ω.Figure2 shows that the value of the electrical voltage had a considerable impact on the distributions of vibration (lateral deflection), temperature increment, cubical deformation, stress, and strain-energy density. An increase in the value of the electrical voltage led to an increase in the temperature increment and the absolute values of vibration (lateral deflection), cubical deformation, stress, and strain-energy density.Figure 2
(a) The distribution of the temperature increment with different values of voltage. (b) The distribution of the vibration with different values of voltage. (c) The distribution of the cubical deformation with different values of voltage. (d) The distribution of the stress with different values of voltage. (e) The distribution of strain-energy density with different values of voltage.
(a)(b)(c)(d)(e)In Figure2(a), the three curves show that the value of the temperature increment increased sharply for a distance of not more than 0.1 due to the thermoelectrical effect at this end; then, the curves entered a stage of stability until the end but with different values, depending on the voltage value. In Figures 2(b), 2(c), and 2(e), the value of the peak point in each curve increased as the value of the voltage increased. Along the x-axis, the values of the peak points decreased as the distance x increased. In Figure 2(d), each curve of the stress curves had only one peak point, and their absolute values increased when the voltage value increased.The distributions of temperature increment, vibration (lateral deflection), cubical deformation, stress, and strain-energy density were all significantly affected by the electrical resistance value, as shown in Figure3. The temperature increment and the absolute values of vibration (lateral deflection), cubical deformation, stress, and strain-energy density decreased with an increase in the electrical resistance.Figure 3
(a) The distribution of the temperature increment with different values of resistance. (b) The distribution of the vibration with different values of resistance. (c) The distribution of the cubical deformation with different values of resistance. (d) The distribution of the stress with different values of resistance. (e) The distribution of the strain-energy density with different values of resistance.
(a)(b)(c)(d)(e)Figure3(a) shows that the value of temperature increment increased sharply for a distance of not more than 0.1 due to the thermoelectrical effect at this end; then, the curves progressed to the stage of stability until the end, but with different values depending on the value of electrical resistance. Figures 3(b), 3(c), and 3(e) show that the value of the peak point in each of the three curves decreased as the value of electrical resistance increased. The values of the peak points decreased with an increase in the distance x along the x-axis. Figure 3(d) shows that the stress curves had one peak point, and their absolute values reduced as the electrical resistance increased.Thus, we can say that the electrical resistance could be used to tune the vibration as well as the energy which has been generated along the microbeam.


## 4. Conclusions

Non-Fourier heat conduction was used to analyse thermoelastic microbeams in this study. An electrical voltage source was connected to the graphene strip at the initial end of the microbeam. The Lord–Shulman model of generalized thermoelasticity was applied. As a result of the electrical current running in a strip of graphene, the microbeam was thermally loaded.In all the functions studied, electrical voltage and electrical resistance were found to have a significant impact. The temperature increment and the absolute values of the vibration (lateral deflection), cubical deformation, stress, and strain-energy density increased as the electrical voltage increased. With increasing electrical resistance, the temperature and the absolute values of vibration (lateral deflection), cubical deformation, stress, and strain-energy density increased. Thus, the electrical voltage and resistance of the graphene strip might be utilized as a tuner to control the vibration and energy of a microbeam.In general, the results agree with the results in other papers such as references [10, 13, 16, 25, 28, 35]. In the future work, electrical voltage as a heat source will be applied on different types of beams and in the context of different types of heat conduction laws.



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