# Influence of the Static Pre-Stress in Micro-Viscothermoelastic Resonators Based on Dual-Phase-Lagging Heat Conduction

Authors: Najat A. Alghamdi; Hamdy M. Youssef
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Thermal and mechanical relaxation times play vital roles in the values of the quality factor of micro/nanoresonators. They can control the energy dissipation across the coupling of mechanical and thermal behavior. In this paper, we introduce an analytical model that considers a pre-stress in a micro-viscothermoelastic resonator to modify the thermal and mechanical relaxation times and thus higher the quality factor. The impacts of length scale and static pre-stress on the quality factor have been discussed. The model expects that significant improvement in terms of quality factors is possible by tuning the pre-stress and the thermal and mechanical relaxation times parameters, and the isothermal value of frequency have significant effects on the thermal quality factor of the resonators.


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## Body

## 1. Introduction

Many applications based on microelectromechanical resonators are essential in different fields, such as mechanical signal processing, scanning probe microscopes, and ultrasensitive mass detection. The most critical parameter of micro-viscothermoelastic resonators is the quality factorQ. Higher Q factor indicates less energy. The study of the energy dissipation mechanism is significant for the improvement of the design of micro/nanoelectromechanical resonators [1–4].Zener [5–7] is the first who introduced the Q-factor in thermoelastic dissipation, and he gave an approximate analytical form of it. He has studied the thermoelastic damping in beams by treating the viscoelastic material. Many research studies have been carried out on the thermoelastic damping in which the classical theory of thermoelasticity based on the Fourier heat law of heat conduction has been used [8]. Lifshitz and Roukes [9] introduced an exact expression for thermoelastic damping. Sun et al. [10] study thermoelastic damping of a beam resonator based on a non-Fourier heat equation. Sharma and Sharma [11] studied damping in micro-scale circular plate resonators using the Lord–Shulman theory of generalized thermoelasticity theory (L-S).Tzou [12, 13] proposed a mathematical model to study heat conduction known as dual-phase-lag (DPL). This model established the temperature gradient and heat flux. Many scientists used this model in heat transfer problems [8, 14], physical systems [15–19], and thermoelastic damping vibration [20, 21]. Guo et al. [22, 23] studied the thermoelastic damping theory of micro- and nanomechanical resonators by using the DPL model.The study of viscoelastic materials has become essential in mechanics. Biot [24, 25] discussed the theory of viscothermoelasticity in thermodynamics. A thermoviscoelasticity model of polymers at finite strains was derived by Drozdov [26]. A new model of thermoviscoelasticity for isotropic media was established by Ezzat and El-Karmany [27].As the size of a flexural resonator is reduced, its natural frequency increases and thermoelastic damping also increases in the process. The natural frequency of beams can also be changed by the application of an axial force [3, 28]. A compressive force decreases in the natural frequency, whereas a tensile force increases in the natural frequency [8]. Experimental results have been utilizing the frequency change to tune resonators. Furthermore, these experiments also suggest an increase in the Q-factor with the application of tensile stress [3, 29].In the present study, we present an analytical model that considers a pre-stress in a micro-viscothermoelastic resonator to quantify the quality factorQ. We also analyze the results with respect to the resonator size and pre-stress and then compare the analytical results with the experimental results available in the literature.


## 2. Basic Equations and Model Formulation

According to Zener’s model of a linear anelastic solid, the stress equation takes the following form [6]:(1)σ=σ0+σ1eiωt,where σ0 is the static pre-stress, and the second part represents the dynamic-stress due to the ensuing vibration and ω is the natural frequency.The resultant strain for a thermoelastic solid is given as [6, 8](2)ε=σ0ER−σ0E2e−t/τσ+ε1eiωt.Making derivatives of equations (1) and (2) with respect to time and inserting them into the equation for a thermoelastic solid are given as follows:(3)σ+τεσ˙=ERε+τσε˙,where ER is relaxed elastic modulus and τε and τσ are constants strain and constant stress relaxation time constants; we obtain(4)σ0+σ1eiωt+τεiωσ1eiωt=ERσ0ER−σ0E2e−t/τσ+ε1eiωt+τσ1τσσ0E2e−t/τσ+iωε1eiωt.All the terms due to static stressσ0 get canceled from both sides. That results in static stress independence of Zener’s thermoelastic solid model. This model assumes no contribution of a stress field to the thermal relaxation time in solids. Thus, we decided to follow Lifshitz–Roukes’ approach to the exact solution for a pre-stressed beam resonator. The constitutive relationship for Euler–Bernoulli’s beam with dimensions length L, width b, and depth h along the x, y, and z-axes, respectively, the moment of inertia I, and applied axial force F and flexural displacement wx,y,z,t can be written as follows:(5)σxx=σ0−Ey∂2w∂x2−EαTθ,where σ0=F/bh is the stress due to the applied axial force, E is Young’s Modulus, and αT is the coefficient of thermal expansion. The equilibrium temperature of the beam is T0, and the ensuing vibration will result in a temperature field T=T0+θ.By deriving the equation of motion for flexural vibration, we obtain(6)EI∂4w∂x4−F∂2w∂x2+EαT∂2IT∂x2+ρA∂2w∂t2=0,where(7)I=∫−b/2b/2∫−h/2h/2y2dydz=bh312,(8)IT=∫−b/2b/2∫−h/2h/2yθdydz.The dual-phase-lag heat equation takes the form(9)K1+τT∂∂t∇2θ=∂∂t+τq∂2∂t2ρCνθ+EαTT01−2νε,where K is the thermal conductivity, υ is Poisson’s ratio, Cυ is the specific heat at constant strain, and τqandτT are the thermal relaxation of the temperature and its gradient, respectively.The volumetric deformations are as follows:(10)ε=εxx+εyy+εzz,where(11)Eεxx=σ0−yE∂2w∂x2,Eεyy=Eεzz=−υσ0+υyE∂2w∂x2+1+υαTEθ.Then, we have(12)Eε=1−2υσ0−1−2υyE∂2w∂x2+21+υαTEθ.Substituting from equation (11) into equation (9), we obtain(13)1+τT∂∂t∇2θ=η∂∂t+τq∂2∂t2θ+αTT0K∂∂t+τq∂2∂t221+υαT1−2υEθ−yE∂2w∂x2,which gives(14)1+τT∂∂t∇2θ+αTT0EKy∂∂t+τq∂2∂t2∂2w∂x2=∂∂t+τq∂2∂t2η+2ET0αT21+υK1−2νθ,where η=ρCυ/K.For viscothermoelastic materials, we consider Young’s modulus in the form [24, 27](15)E=E01+E1∂∂t,where E0 is Young’s modulus value for the usual case, while E1 is the mechanical relaxation time (relaxation time means the return of a perturbed system into equilibrium or the time between repeating mechanical waves).Hence, equation (14) takes the form(16)1+τT∂∂t∇2θ+ΔEαT1+E1∂∂t∂∂t+τq∂2∂t2y∂2w∂x2=∂∂t+τq∂2∂t2η+2ΔE1+υ1−2υ1+E1∂∂tθ,where ΔE=T0αT2E0/K.Because no gradient exists in thez-direction, then ∇2θ=∂2θ/∂y2; therefore, we have(17)1+τT∂∂t∂2θ∂y2+ΔEαTy1+E1∂∂t∂∂t+τq∂2∂t2∂2w∂x2=∂∂t+τq∂2∂t2η+2ΔE1+υ1−2υ1+E1∂∂tθ.We consider the following functions:(18)wx,t=Wxeiωt,θx,y,t=Θx,yeiωt.Hence, equation (17) will be in the form(19)1+iτTω∂2Θ∂y2+ΔEαTy1+iE1ωiω−τqω2d2Wdx2=iω−τqω2η+2ΔE1+υ1−2υ1+iE1ωΘ,which gives(20)∂2∂y2−λ2Θx,y=−αyd2Wxdx2,where(21)λ=λω=iω−τqω21+iτTωη+2ΔE1+v1−2v1+iE1ω,α=αω=ΔEαT1+iE1ωiω−τqω21+iτTω.The general solution of the differential equation (20) takes the form(22)Θx,y=Acoshλy+Bsinhλy+αλ2yd2Wxdx2.The boundary conditions are as follows:(23)∂Θx,y∂yy=0=∂Θx,y∂yy=h=0.Hence, we obtain(24)Θ=y−sinhλyλcoshλh/2αλ2d2Wdx2.From equations (6), (8), and (18), we obtain(25)E01+iωE1IρA∂4Wx∂x4+E01+iωE1αTρA∂2∂x2∫−b/2b/2∫−h/2h/2yΘx,ydydz−FρA∂2Wx∂x2=ω2Wx.Substituting from equation (24) in equation (25), we obtain(26)E01+iωE1IρA∂4Wx∂x4+E01+iωE1αTαρAλ2∂4Wx∂x4∫−b/2b/2∫−h/2h/2yy−sinλyλcosλh/2dydz−FρA∂2Wx∂x2=ω2Wx.Finally, we obtain(27)IEωρA∂4Wx∂x4−FIEω∂2Wx∂x2=ω2Wx,where Eω=E01+iωE1fω and fω=1+αTbα/λ2Ih3/12+h/λ2−2/λ3tanλh/2.For a simply supported beam, the exact analytical solution for the natural frequency is given as [28](28)ω=πLρAπ2IE0L21+iω0E1fω0+F,where ω0 is the isothermal value of fundamental frequency given by [10, 30](29)ω0=q2bL2E012ρ.For beams with both ends clamped,q=4.73, while q=π for beams with both ends simply support.Hence, we obtain(30)ω=π2L2IE0ρA1+iE1ω01+ΔEbh31+iE1ω012Iε11+12ε2h2ε1−24h3tanhλh2ε11+iτTω0−3/2+L2π2I2E0F,where ε1=η+2ΔE1+υ1+iE1ω0/1−2υandε2=1+iτTω0/iω0−τqω02The quality factorQ−1 is defined as follows:(31)Q−1=2ImωReω.


## 3. Numerical Results and Discussion

Now, the relationships between the variation of the thermoelastic damping with the beam heighth, and the beam length in different values of the applied axial force F on the microbeam resonator, which is made of silicon and clamped at two ends, will be explored.Material properties of silicon (Si) have been taken as follows [10, 30]:T0=300K,ρ=2330kgm−3,K=141.04Wm−1k−1,Cυ=1.64×106Jm−3K−1,E=165GPa,andα=2.6×10−6K−1. The aspect ratios of the beam are fixed as L/h=50 and b=5h. For the micro-scale beam, we will take the beam’s thickness h10−11m≤h≤10−9m.Figure1 represents the quality factor of the resonator with a different value of the force F=0.0,1.0,5.0×10−20Kgms−2 of the static pre-stress. It is noted that the force due to the static pre-stress has a significant effect on the quality factor Q−1. The value of the quality factor Q−1 decreases when the value of static pre-stress increases until the position h=6.0×10−10m; then, the three cases are very closed. In Figure 1, for the case F=0.0Kgms−1, the curve has the same attitude of the curves of Figure 3 in [30], Figure 2 in [31], and Figure 8 in [32].Figure 1
The quality factor of the resonator with a different value of static pre-stress.Figure 2
The quality factor of the resonator with a different value of mechanical relaxation time parameter.Figure 3
The quality factor of the resonator with a different value of the isothermal value of frequencyω0(qn).Figure2 represents the inverse of the quality factor Q−1 of the resonator with a different value of the mechanical relaxation time E1=0.0,10.0,20.0×10−22s of the static pre-stress. It is noted that the value of mechanical relaxation time has a significant effect on the quality factor. In essence, the value of the inverse of the quality factor Q−1 decreases when the value of the mechanical relaxation time increases. In Figure 2, for the case E1=0.0s, the curve agrees with the curves of Figure 3 in [30], Figure 2 in [31], and Figure 8 in [32].Figure3 represents the quality factor Q−1 of the resonator with a different value of the isothermal value of frequency ω0q, where q=π,4.73,7.85. It is noted that the value of the isothermal value of frequency has a significant effect on the quality factor Q−1 of the thermal damping; the value of the quality factor Q−1 increases when the isothermal value of frequency decreases.Figure4 represents the inverse of the quality factor Q−1 of the resonator with a different value of the ratio of the thermal relaxation times τq/τT=20,25,30, respectively. It is noted that the values of the ratio of the thermal relaxation times have a significant effect on the quality factor Q−1 of the thermal damping. The value of the quality factor Q−1 decreases when the value of the ratio τq/τT increases. In Figure 4, the curves agree with the curves of Figure 3 in [30], Figure 2 in [31], and Figure 8 in [32].Figure 4
The quality factor of the resonator with a different ratio of thermal relaxation times.Figure5 represents the quality factor Q−1 of the resonator with a different value of the size ratio L/h=30,40,50, respectively. It is noted that the values of the size ratio have a significant effect on the quality factor Q−1; the value of the quality factor Q−1 increases when the value of the size ratio L/h increases.Figure 5
The quality factor of the resonator with a different value of size ratioL/h.


## 4. Conclusion

An analytical model for viscothermoelastic relaxation times in ﬂexural resonators under axial pre-stress has been developed in the micro-scale. The model confirms that a significant reduction in the quality factor as a tensile axial pre-stress is applied. This reduction in damping can be correlated with a decrease in the relaxation rate as compared to the natural frequency of the resonator. The force due to the static pre-stress, isothermal frequency, thermal relaxation times, mechanical relaxation time, and size ratio have a significant effect on the quality factor of the nanobeams.



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*Source: 1316281-2020-07-27.xml*