# Thermoelastic Damping Limited Quality Factor Enhancement and Energy Dissipation Analysis of Rectangular Plate Resonators Using Nonclassical Elasticity Theory

Authors: Resmi R.; Suresh Babu V.; Baiju M. R.
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

Among the different energy dissipation mechanisms, thermoelastic damping plays a vital role and needs to be alleviated in vibrating resonators to mitigate parameters by improving the thermoelastic damping limited quality factor,QTED. The maximum energy dissipation is also interrelated with the critical dimension hc of the plates, and by optimizing the dimensions, the peaking of energy dissipation can be diminished. As the size of the devices is scaled down, classical continuum theories become incompetent to explain the size-effect related mechanical nature at the micron and submicron levels, and, as a result, nonclassical continuum theories have been pioneered with the inception of internal length scale parameters. In this work, an analysis of isotropic rectangular microplates based on the Kirchhoff model and a higher order theory like Modified Couple Stress Theory is utilized to study size-dependent thermoelastic damping and its impact on the quality factor and critical dimensions. The Hamilton principle is adapted to derive the governing equations of motion, and the coupled heat conduction equation is employed to formulate the thermoelastic damping limited quality factor of the plates. Five different structural materials (PolySi, diamond, Si, GaAs, and SiC) are used for optimizing QTED and hc, which depends on two material performance index parameters: the thermoelastic damping index (TDI) and the material thermal diffusion length, lT. According to this work, the maximum QTED is attained for PolySi with the lowest TDI, and hcmax is obtained for Si with the maximum lT. The impacts of the dimensionless length-scale parameters (l/h), vibration modes, and boundary conditions (clamped-clamped and simply supported) on QTED and hc are also investigated. From the current analysis, QTED can be further enhanced by selecting higher vibration modes and clamped-clamped boundary conditions. QTED can be maximized by fixing the internal length scale parameter (l) and making the thickness of the beam equal to l. The analytical study is numerically simulated by using MATLAB 2015 software. Prior knowledge of QTED and hc will help designers to produce high-performance and low-loss resonators for the futuristic technological applications.


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## Body

## 1. Introduction

### 1.1. Implications of MEMS/NEMS-Based Rectangular Plate Resonators

MEMS/NEMS resonators have been developed as pivotal components for sensing [1–3] and communication [4–8] applications, where increasing the sensitivity and resolution of devices is a decisive issue. Micro/nanoplates [9–13] are one of the paramount structures used extensively as micro/nanoresonators due to the advantages of their reduced dimensions, lower weight, surplus sensitivity, enhanced reliability, and elevated S/V ratios [14–16]. Small-scaled elements and structures like micro/nanoplates have attracted extensive attention due to their unique features and broad area of applications such as sensors [17–21], actuators [22, 23], switches [24, 25], filters [26–29], and pumps [30, 31]. MEMS and NEMS plates can be made of various materials, such as metallic, polymeric, semiconductor-based, and even functionally graded materials.


### 1.2. Energy Dissipation Mechanisms and Quality Factor of MEMS/NEMS Resonators

One of the salient features of resonators used mainly as sensors and filters is their quality factor (QF), which portrays the depleted energy in the system due to the heterogeneous dissipation mechanisms. As the dissipated energy increases, the quality factor declines, which deteriorates the sensitivity, resolution, and reliability of micro/nanodevices. To model high quality factor resonators, all dissipation mechanisms that cause the quality factor to diminish must be identified. The different energy dissipation mechanisms are classified into extrinsic (e.g., anchor damping [32–34] and squeeze film damping [35, 36]) and intrinsic losses (e.g., thermoelastic damping [37–39]). Extrinsic losses are easily controlled by changing the operating conditions, but intrinsic losses are not able to be managed as fully as extrinsic ones. Thermoelastic damping is a crucial energy dissipation mechanism at the micro and nanoscales that limit the maximum achievable quality factor in the resonator, which is denoted by QTEDMAX. To manage the proliferate demands in achieving microstructures with high quality factors, several optimizations are made based on the relevant geometry and materials. The QTED is surpassed mainly by properly selecting the structural materials, dimensions, and operating temperature of the resonator. Selection of the vibrating modes and boundary conditions has only negligible influence in enhancing QTED.


### 1.3. Evolvement of Thermoelastic Damping and Impacts of Size Effects

Zener identified the existence of Thermoelastic Damping (TED) as a prominent energy loss mechanism in homogeneous and isotropic Euler–Bernoulli microbeams. He presented an analytical solution for energy dissipation based on resonator properties [40–42]. The effect of thermoelastic internal friction on the quality factor of silicon resonators is studied by Roszhart [43]. An exact solution for thermoelastic energy dissipation based on coupled thermoelasticity is derived by Lifshitz et al. [44]. A major breakthrough in thermoelastic damping analysis occurred when Evoy and Duwel demonstrated experimentally that TED is a predominant source of damping in micro/nanostructures [45, 46]. Thermoelastic energy dissipation in MEMS resonators with two-dimensional heat conduction is presented by Prabhakar and Vengallatore [47].The analysis of thermoelastic damping in microplates is an interesting issue in micro/nanosystems. Nayfeh and Younis [48] expressed QTED analytically in terms of their structural mode shapes by utilizing the perturbation method under electrostatic loading and residual stresses for microplates with general shapes and boundary conditions. The Laplace transform method is employed by Sun and Saka to perform a vibration analysis of the pulsed laser heating of a microbeam under various boundary conditions [49]. Sun et al. also analysed the equations of the coupled thermoelastic condition to study TED in circular plate resonators for axisymmetrical out-of-plane vibrations [50, 51]. Li et al. presented a set of analytical approximate models to obtain TED in rectangular and circular microplates by evaluating the expended energy per cycle of vibration, with diverse boundary conditions based on Rayleigh’s method [52]. Zuo et al. studied TED in multilayered (bilayered) rectangular and circular microplate resonators and derived analytical expressions for the same [53]. Fang et al. also presented analytical solutions for TED in rectangular microplates using Rayleigh’s method and analysed three common boundary conditions [54]. Liu et al. also provided an expression for analysing TED in bilayered circular microplates [55].The size dependence effects of materials have been experimentally observed by scaling down the microstructure size. Indeed, the size effect is essential in mechanical analyses of various structures. Voigt’s works provided complicated differential equations set about the kinematics, laws of balancing, and constitutive relations. The size effects in engineering materials are first explored by Cauchy [56] and Voigt [57]. Sutherland et al. published a detailed investigation on the size and scale effects involved in forecasting the strength of fiber-reinforced plastic composite materials and structures [58]. Tabiei and Sun conducted a study on laminated composites and presented a theoretical analysis of the size effect correlated with the length and thickness [59].Classical theories are insufficient to predict the size effects due to the absence of the material length scale parameter; hence, higher order continuum theories must be used to predict the size dependencies. Accordingly, to include size effects in the vibration analysis, nonclassical continuum theories known as high order elasticity theories are used. The primary nonclassical continuum theory is couple stresses theory (CST) which includes higher order theories developed by Mindlin and Tiersten [60], Koiter [61],and others including Toupin [62].Among the different higher order theories, the most common is the Modified Couple Stress Theory (MCST) developed by Yang et al. [63]. MCST incorporates a single material length scale parameter to capture small sizes and scaling effects. The strain gradient theory in microstructures is evolved by Lam et al. [64]. Eringen and Edelen [65] developed nonlocal elasticity and proposed that the microstructure body is a collection of deformable particles, each possessing finite size and inner structure. Compared to other nonclassical continuum theories, MCST is the most accurate and simple theory compared to other nonclassical continuum theories like modified strain gradient theory (MSGT) and nonlocal theory. For MSGT, three length scale parameters are included which makes the analysis laborious. As far as nonlocal theory is concerned, the concept is based on an axisymmetric condition at local zones which is not true and reduces its accuracy.The dynamic behaviours of the micro Kirchhoff plate based on the modified couple stress theory are analysed by Yin et al. [66]. Ke and Wang investigated the size effects in strain gradient thin plates adopting Kirchhoff’s theory with the aid of bending analysis [67]. The authors modified Von Karman’s nonlinear equations and presented a strain gradient geometrically nonlinear plate model to investigate the size effects. Rahaeifard et al. studied the size effects on deflection utilizing MCST in microcantilevers with silicon as the structural material [68]. Ke et al. analysed the size dependency and free vibrations of Mindlin microplates [69]. The authors also analysed the static pull-in voltage and removed the gap between the experimental observations and the results of classical theories. The length scale parameter associated with the silicon material is also identified. Guo et al. investigated TED in circular microplates using a nonclassical heat conduction model of dual phase lagging and derived an explicit expression for the thermoelastic damping limited quality factor [70]. Zhong et al. investigated thermoelastic damping in the size-dependent microplate resonators based on modified couple stress theory and developed a closed-form expression for TED in fully simply supported rectangular microplates [71]. Tsiatas and Yiotis analysed isotropic microplates based on the Kirchhoff plate model and suggested applying modified couple stress theory for static analyses that need only one material length scale parameter, which is enough to capture the size effect [72]. Razavilar et al. analysed the TED of a rectangular microplate based on the MCST and studied the significance of the length scale parameter [73]. Resmi et al. explored thermoelastic damping limited quality factor, thermoelastic frequency, and figure of merit of rectangular microplates using MCST [74]. Borjalilou and Asghari analysed thermoelastic damping limited quality factor of rectangular microplates using MCST and MSGT [75]. Devi and Kumar investigated TED and frequency shift in Kirchhoff plates based on MCST [10]. Thermoelastic damping study in rectangular micro/nanoplate resonators based on nonlocal strain gradient theory is done by Xiao et al. [13]. Yang et al. [76] explored small scale effects on the rectangular microplates via nonlocal elasticity theory.From the literature review, thermoelastic damping analysis in micro/nanorectangular plates using five different structural materials (PolySi, diamond, Si, GaAs, and SiC) under various conditions for reduction of energy dissipation and optimization of quality factor is not available. The major purpose of the investigation is an attempt to fill the gap according to previous research works by exploring thermoelastic damping and related quality factor of isotropic Kirchhoff model based rectangular microplates with diverse structural materials and various conditions. Nonclassical elasticity theory like MCST is used to enquire size dependency on thermoelastic damping because of its simplicity and accuracy as discussed in the previous section.The governing equations of motion are derived by conforming to the Hamilton principle and the coupled heat conduction equation is employed to formulate the thermoelastic damping limited quality factor of the rectangular plates. Using complex frequency approach, from the real and imaginary parts of the frequency, an expression for quality factor is derived incorporating size effects. Hence an analytical expression comprising small-scale characteristic parameters is derived to quantify TED, a comparison is made between the results extracted based on present model and those provided via classical formulation. The significance of capturing size effect and the influences of the different parameters, such as material length scale parameter, material properties, boundary conditions, and the mode switching, are investigated in the current analysis. The critical thickness (hc), which corresponds to the peak energy dissipation under the abovementioned conditions, is also explored. The impacts of material properties, size scaling, boundary conditions, and mode switching on critical thickness using both classical theory and nonclassical theory (MCST) is also investigated.In this paper, Section2 gives the basic formulation of MCST according to Yang et al. [63]. In Section 3, the expressions for the thermoelastic damping limited quality factor of the microplates applying MCST are derived in terms of its structural material properties. The analytical expressions for thermoelastic energy dissipation and associated factor factor based on classical theory like Lifshitz and Roukes formulations are also presented by neglecting the size effects. Section 4 presents the results and discussions, in which the various techniques for enhancing QTED are analysed quantitatively by applying MCST. The impacts of size scaling, mode switching, boundary conditions, material properties, and the internal length scale parameter are discussed in Section 4. The conclusions of the work are given in Section 5.


## 1.1. Implications of MEMS/NEMS-Based Rectangular Plate Resonators

MEMS/NEMS resonators have been developed as pivotal components for sensing [1–3] and communication [4–8] applications, where increasing the sensitivity and resolution of devices is a decisive issue. Micro/nanoplates [9–13] are one of the paramount structures used extensively as micro/nanoresonators due to the advantages of their reduced dimensions, lower weight, surplus sensitivity, enhanced reliability, and elevated S/V ratios [14–16]. Small-scaled elements and structures like micro/nanoplates have attracted extensive attention due to their unique features and broad area of applications such as sensors [17–21], actuators [22, 23], switches [24, 25], filters [26–29], and pumps [30, 31]. MEMS and NEMS plates can be made of various materials, such as metallic, polymeric, semiconductor-based, and even functionally graded materials.


## 1.2. Energy Dissipation Mechanisms and Quality Factor of MEMS/NEMS Resonators

One of the salient features of resonators used mainly as sensors and filters is their quality factor (QF), which portrays the depleted energy in the system due to the heterogeneous dissipation mechanisms. As the dissipated energy increases, the quality factor declines, which deteriorates the sensitivity, resolution, and reliability of micro/nanodevices. To model high quality factor resonators, all dissipation mechanisms that cause the quality factor to diminish must be identified. The different energy dissipation mechanisms are classified into extrinsic (e.g., anchor damping [32–34] and squeeze film damping [35, 36]) and intrinsic losses (e.g., thermoelastic damping [37–39]). Extrinsic losses are easily controlled by changing the operating conditions, but intrinsic losses are not able to be managed as fully as extrinsic ones. Thermoelastic damping is a crucial energy dissipation mechanism at the micro and nanoscales that limit the maximum achievable quality factor in the resonator, which is denoted by QTEDMAX. To manage the proliferate demands in achieving microstructures with high quality factors, several optimizations are made based on the relevant geometry and materials. The QTED is surpassed mainly by properly selecting the structural materials, dimensions, and operating temperature of the resonator. Selection of the vibrating modes and boundary conditions has only negligible influence in enhancing QTED.


## 1.3. Evolvement of Thermoelastic Damping and Impacts of Size Effects

Zener identified the existence of Thermoelastic Damping (TED) as a prominent energy loss mechanism in homogeneous and isotropic Euler–Bernoulli microbeams. He presented an analytical solution for energy dissipation based on resonator properties [40–42]. The effect of thermoelastic internal friction on the quality factor of silicon resonators is studied by Roszhart [43]. An exact solution for thermoelastic energy dissipation based on coupled thermoelasticity is derived by Lifshitz et al. [44]. A major breakthrough in thermoelastic damping analysis occurred when Evoy and Duwel demonstrated experimentally that TED is a predominant source of damping in micro/nanostructures [45, 46]. Thermoelastic energy dissipation in MEMS resonators with two-dimensional heat conduction is presented by Prabhakar and Vengallatore [47].The analysis of thermoelastic damping in microplates is an interesting issue in micro/nanosystems. Nayfeh and Younis [48] expressed QTED analytically in terms of their structural mode shapes by utilizing the perturbation method under electrostatic loading and residual stresses for microplates with general shapes and boundary conditions. The Laplace transform method is employed by Sun and Saka to perform a vibration analysis of the pulsed laser heating of a microbeam under various boundary conditions [49]. Sun et al. also analysed the equations of the coupled thermoelastic condition to study TED in circular plate resonators for axisymmetrical out-of-plane vibrations [50, 51]. Li et al. presented a set of analytical approximate models to obtain TED in rectangular and circular microplates by evaluating the expended energy per cycle of vibration, with diverse boundary conditions based on Rayleigh’s method [52]. Zuo et al. studied TED in multilayered (bilayered) rectangular and circular microplate resonators and derived analytical expressions for the same [53]. Fang et al. also presented analytical solutions for TED in rectangular microplates using Rayleigh’s method and analysed three common boundary conditions [54]. Liu et al. also provided an expression for analysing TED in bilayered circular microplates [55].The size dependence effects of materials have been experimentally observed by scaling down the microstructure size. Indeed, the size effect is essential in mechanical analyses of various structures. Voigt’s works provided complicated differential equations set about the kinematics, laws of balancing, and constitutive relations. The size effects in engineering materials are first explored by Cauchy [56] and Voigt [57]. Sutherland et al. published a detailed investigation on the size and scale effects involved in forecasting the strength of fiber-reinforced plastic composite materials and structures [58]. Tabiei and Sun conducted a study on laminated composites and presented a theoretical analysis of the size effect correlated with the length and thickness [59].Classical theories are insufficient to predict the size effects due to the absence of the material length scale parameter; hence, higher order continuum theories must be used to predict the size dependencies. Accordingly, to include size effects in the vibration analysis, nonclassical continuum theories known as high order elasticity theories are used. The primary nonclassical continuum theory is couple stresses theory (CST) which includes higher order theories developed by Mindlin and Tiersten [60], Koiter [61],and others including Toupin [62].Among the different higher order theories, the most common is the Modified Couple Stress Theory (MCST) developed by Yang et al. [63]. MCST incorporates a single material length scale parameter to capture small sizes and scaling effects. The strain gradient theory in microstructures is evolved by Lam et al. [64]. Eringen and Edelen [65] developed nonlocal elasticity and proposed that the microstructure body is a collection of deformable particles, each possessing finite size and inner structure. Compared to other nonclassical continuum theories, MCST is the most accurate and simple theory compared to other nonclassical continuum theories like modified strain gradient theory (MSGT) and nonlocal theory. For MSGT, three length scale parameters are included which makes the analysis laborious. As far as nonlocal theory is concerned, the concept is based on an axisymmetric condition at local zones which is not true and reduces its accuracy.The dynamic behaviours of the micro Kirchhoff plate based on the modified couple stress theory are analysed by Yin et al. [66]. Ke and Wang investigated the size effects in strain gradient thin plates adopting Kirchhoff’s theory with the aid of bending analysis [67]. The authors modified Von Karman’s nonlinear equations and presented a strain gradient geometrically nonlinear plate model to investigate the size effects. Rahaeifard et al. studied the size effects on deflection utilizing MCST in microcantilevers with silicon as the structural material [68]. Ke et al. analysed the size dependency and free vibrations of Mindlin microplates [69]. The authors also analysed the static pull-in voltage and removed the gap between the experimental observations and the results of classical theories. The length scale parameter associated with the silicon material is also identified. Guo et al. investigated TED in circular microplates using a nonclassical heat conduction model of dual phase lagging and derived an explicit expression for the thermoelastic damping limited quality factor [70]. Zhong et al. investigated thermoelastic damping in the size-dependent microplate resonators based on modified couple stress theory and developed a closed-form expression for TED in fully simply supported rectangular microplates [71]. Tsiatas and Yiotis analysed isotropic microplates based on the Kirchhoff plate model and suggested applying modified couple stress theory for static analyses that need only one material length scale parameter, which is enough to capture the size effect [72]. Razavilar et al. analysed the TED of a rectangular microplate based on the MCST and studied the significance of the length scale parameter [73]. Resmi et al. explored thermoelastic damping limited quality factor, thermoelastic frequency, and figure of merit of rectangular microplates using MCST [74]. Borjalilou and Asghari analysed thermoelastic damping limited quality factor of rectangular microplates using MCST and MSGT [75]. Devi and Kumar investigated TED and frequency shift in Kirchhoff plates based on MCST [10]. Thermoelastic damping study in rectangular micro/nanoplate resonators based on nonlocal strain gradient theory is done by Xiao et al. [13]. Yang et al. [76] explored small scale effects on the rectangular microplates via nonlocal elasticity theory.From the literature review, thermoelastic damping analysis in micro/nanorectangular plates using five different structural materials (PolySi, diamond, Si, GaAs, and SiC) under various conditions for reduction of energy dissipation and optimization of quality factor is not available. The major purpose of the investigation is an attempt to fill the gap according to previous research works by exploring thermoelastic damping and related quality factor of isotropic Kirchhoff model based rectangular microplates with diverse structural materials and various conditions. Nonclassical elasticity theory like MCST is used to enquire size dependency on thermoelastic damping because of its simplicity and accuracy as discussed in the previous section.The governing equations of motion are derived by conforming to the Hamilton principle and the coupled heat conduction equation is employed to formulate the thermoelastic damping limited quality factor of the rectangular plates. Using complex frequency approach, from the real and imaginary parts of the frequency, an expression for quality factor is derived incorporating size effects. Hence an analytical expression comprising small-scale characteristic parameters is derived to quantify TED, a comparison is made between the results extracted based on present model and those provided via classical formulation. The significance of capturing size effect and the influences of the different parameters, such as material length scale parameter, material properties, boundary conditions, and the mode switching, are investigated in the current analysis. The critical thickness (hc), which corresponds to the peak energy dissipation under the abovementioned conditions, is also explored. The impacts of material properties, size scaling, boundary conditions, and mode switching on critical thickness using both classical theory and nonclassical theory (MCST) is also investigated.In this paper, Section2 gives the basic formulation of MCST according to Yang et al. [63]. In Section 3, the expressions for the thermoelastic damping limited quality factor of the microplates applying MCST are derived in terms of its structural material properties. The analytical expressions for thermoelastic energy dissipation and associated factor factor based on classical theory like Lifshitz and Roukes formulations are also presented by neglecting the size effects. Section 4 presents the results and discussions, in which the various techniques for enhancing QTED are analysed quantitatively by applying MCST. The impacts of size scaling, mode switching, boundary conditions, material properties, and the internal length scale parameter are discussed in Section 4. The conclusions of the work are given in Section 5.


## 2. The Modified Couple Stress Theory and Coupled Thermoelasticity Equations

### 2.1. Basic Formulation of MCST

In classical elasticity theory, rotation gradients are not considered, and only the strain tensor is included in the stress and strain energy, which is not the case with nonclassical continuum theories. Using MCST, Yang et al. [63] applied a higher order equilibrium condition to predict the size dependencies at the micro/nanoscales. The total strain energy is considered as a quadratic function of both the symmetrical strain tensor and the symmetrical curvature tensor, where the second one represents the added measure of deformation. The symmetrical strain tensor and symmetric curvature tensor are conjugated with the stress tensor and coupled stress tensor, respectively. To interpret the size effect related to the material microstructures, a single material length scale is included by applying MCST for a Kirchhoff model based rectangular plate.The strain energyU in a deformed isotropic linear elastic material occupying region Ӑ is given by(1)U=∫Ăσijεij+mijηijdĂi,j=1,2,3.From (1), it is clear that the deformation energy density depends only on the symmetrical part of the displacement gradient (conventional strain tensor) and the rotation gradient (symmetrical curvature tensor). The antisymmetrical part of displacement gradient and the rotation gradient has no contribution to the deformation energy. The strain tensor and symmetrical curvature tensor are the deformation measures conjugated to the symmetrical stress tensor σ and the deviatoric couple stress tensor m, respectively. The different tensors are defined by coupled stress, which can be estimated from torsion tests of slim cylinders or bending tests of thin beams at micro/nanoscales.The symmetrical part of Cauchy’s stress tensor is(2)σij=λtrεijI+2μεij.The strain tensor is(3)εij=12∇u+∇uT.The deviatoric part of the symmetrical couple stress tensor is(4)mij=2l2μχij,and the deviatoric part of the symmetrical curvature tensor is(5)χij=12∇ψ+∇ψT,where u is the displacement vector, λ and μ Lame’s constants, and λ=Eν/np, μ=E/2n, where E, ν are the Young’s modulus and Poisson’s ratio of the plate material, respectively; l is the material length scale parameter regarded as the material property characterizing the size effect. In addition to the conventional parameters, two new material dependent parameters, p and n, are introduced, which are derived from the Poisson’s ratio, n=1+ν and p==1−2ν.Infinitesimal rotation vector:(6)ψi=12curlui.Based on (3) and (5), the strain tensor εij and curvature tensor χij are symmetrical, and, consequently, based on (2) and (4), the stress tensor σij and coupled stress tensor mij are also symmetrical.A schematic diagram of a microplate of length a, widthb, and thickness h defined in the rectangular coordinate system is shown in Figure 1. When a thin plate is considered for analysis, it is assumed to be under the plane stress condition, and the thickness h and width b of the plate are taken to be small compared to the length a; i.e., b<5h.Figure 1
Schematic diagram of a rectangular microplate: (a) clamped-clamped (CC) and (b) simply supported (SS) boundary conditions.
(a)(b)


### 2.2. Governing Equations of Motion and the Coupled Thermoelasticity of Microplates

Letψx and ψy denote the transverse normal rotations about the y and x axes:(7)ψxx,t=−∂Wx,y,t∂x,ψyx,t=−∂Wx,y,t∂y,ux=−zWx,y,t.According to Kirchhoff’s plate theory, the displacement fieldux,uy,uz of an arbitrary point (x, y, z) is given by(8)ux=zψxx,t=−z∂Wx,y,t∂x,uy=zψyx,t=−z∂Wx,y,t∂y,uz=Wx,y,t,where Wx,y,t denotes the transverse deflection of the microplate.From (1) and (2), the rotational vector ψ can be rewritten as(9)ψx=12∂uz∂y−∂uy∂z=12W,y,ψy=12∂ux∂z−∂uz∂y=−12W,x,ψz=12∂uy∂x−∂ux∂y=−z2W,xy.The symmetrical rotation gradient tensorχij can be expanded using (5) as(10)χij=12∇ψ+∇ψT.The nonzero components of the symmetrical parts of the rotation gradient tensors are(11)χxxs=W,xy,χyys=−W,xy,χxys=12W,yy−W,xx,χzz=χyz=χzx=0.The nonzero components of the strain are(12)εxx=−zW,xx,εyy=−zW,yy,εxy=−zW,xy.The stress-strain relationships for the thermoelastic materials are(13)εxx=1Eσxx−vσyy+αtT−T0=1Eσxx+m−n2σyy+αtT−T0,εyy=1Eσyy−vσxx+αtT−T0=1Eσyy+m−n2σxx+αtT−T0,εzz=vEσxx+σyy+αtT−T0=n−m2Eσxx+σyy+αtT−T0,where m=0.774,0.931,0.72,0.69,0.808n=0.548,0.862,0.44,0.38,0.616 for PolySi, diamond, Si, GaAs, and SiC, respectively, which are derived from Poisson’s ratio.For a thin beam, assume a plane stress condition, whereσzz=0,(14)σxx=Emnεxx+vεyy−1mEαθ=Emnεxx+n−m2εyy−1mEαθ,σyy=Emnεyy+vεxx−1mEαθ=Emnεyy+n−m2εxx−1mEαθ,εzz=−vmεxx+εyy+nmαθ=m−n2mεxx+εyy+nmαθ.Using (11), (13) is rewritten as(15)σxx=−EzmnW,xx+vW,yy−1mEαθ=−EzmnW,xx+n−m2W,yy−1mEαθ,σyy=−EzmnW,yy+vW,xx−1mEαθ=−EzmnW,yy+n−m2W,xx−1mEαθ,εzz=−vzmW,xx+W,yy+nmαθ=m−nz2mW,xx+W,yy+nmαθ.In rectangular plates, bending momentsmxx,myy,mxy and torsion moments Yxx,Yyy,Yxy exist and are expressed as(16)mxx=El2nW,xy,myy=−El2nW,xy,mxy=El2nW,yy−W,xx,mzz=myz=mzx=0,Yxx=D×−zuz,xx−αθ,Yyy=D×−zuz,yy−αθ,Yxy=−EnZuz,xy,where D=Em/np.Including size effects, the strain energy is(17)Е=12∫0a∫0b∫−h2h2Dzh2F1+EnF2dxdydz,where(18)F1=W,xx2+W,yy2+αθW,xx+W,yy,F2=W,xy2zh2+lh2+lh2W,yy−W,xx2,and the kinetic energy of the microplate is(19)K=12∫ba∫0bρhW˙2+ρh312ψ˙x2+ψ˙y2dxdy,where ρ is the density of the plate material.According to the variational principle,(20)δ∫t1t2K−Udt=0.For clamped-clamped and simply supported rectangular microplates, the integral of the Gaussian curvature is zero,(21)∫0a∫0bW,xxW,yy−W,xy2dxdy=0.The moment of inertia is(22)IT=12αh3∫−h/2h/2zθdz.The dynamic governing equation of the microplate in terms ofWx,t is given by(23)ℬ+ℬ′∇4W+ℬ2∇2IT−ρhW¨=0,where(24)∇2=∂2∂x2+∂2∂y2,∇4=∂4∂x4+2∂4∂x∂y+∂4∂y4,ℬ=Dh3,ℬ'=Dl2hp2m.The temperature distribution in the microplate due to thermoelastic coupling is [74](25)θ˙=χ∇2θ−αEbTpCV∂∂t∑iεjj,where CV is the heat capacity coefficient at a constant volume, and χ denotes the thermal diffusivity of the solid.The relaxation strength of the thermoelastic solid can be expressed as(26)ŕ=Ead−EE=Eα2T0ρCp,where E is the isothermal or relaxed value, and Ead is the adiabatic or unrelaxed value of Young’s modulus.The linearized form of the heat equation can be expressed as(27)1+2ŕnpθ˙=χθ,zz+zŕαW˙,xx+W˙,yy.


## 2.1. Basic Formulation of MCST

In classical elasticity theory, rotation gradients are not considered, and only the strain tensor is included in the stress and strain energy, which is not the case with nonclassical continuum theories. Using MCST, Yang et al. [63] applied a higher order equilibrium condition to predict the size dependencies at the micro/nanoscales. The total strain energy is considered as a quadratic function of both the symmetrical strain tensor and the symmetrical curvature tensor, where the second one represents the added measure of deformation. The symmetrical strain tensor and symmetric curvature tensor are conjugated with the stress tensor and coupled stress tensor, respectively. To interpret the size effect related to the material microstructures, a single material length scale is included by applying MCST for a Kirchhoff model based rectangular plate.The strain energyU in a deformed isotropic linear elastic material occupying region Ӑ is given by(1)U=∫Ăσijεij+mijηijdĂi,j=1,2,3.From (1), it is clear that the deformation energy density depends only on the symmetrical part of the displacement gradient (conventional strain tensor) and the rotation gradient (symmetrical curvature tensor). The antisymmetrical part of displacement gradient and the rotation gradient has no contribution to the deformation energy. The strain tensor and symmetrical curvature tensor are the deformation measures conjugated to the symmetrical stress tensor σ and the deviatoric couple stress tensor m, respectively. The different tensors are defined by coupled stress, which can be estimated from torsion tests of slim cylinders or bending tests of thin beams at micro/nanoscales.The symmetrical part of Cauchy’s stress tensor is(2)σij=λtrεijI+2μεij.The strain tensor is(3)εij=12∇u+∇uT.The deviatoric part of the symmetrical couple stress tensor is(4)mij=2l2μχij,and the deviatoric part of the symmetrical curvature tensor is(5)χij=12∇ψ+∇ψT,where u is the displacement vector, λ and μ Lame’s constants, and λ=Eν/np, μ=E/2n, where E, ν are the Young’s modulus and Poisson’s ratio of the plate material, respectively; l is the material length scale parameter regarded as the material property characterizing the size effect. In addition to the conventional parameters, two new material dependent parameters, p and n, are introduced, which are derived from the Poisson’s ratio, n=1+ν and p==1−2ν.Infinitesimal rotation vector:(6)ψi=12curlui.Based on (3) and (5), the strain tensor εij and curvature tensor χij are symmetrical, and, consequently, based on (2) and (4), the stress tensor σij and coupled stress tensor mij are also symmetrical.A schematic diagram of a microplate of length a, widthb, and thickness h defined in the rectangular coordinate system is shown in Figure 1. When a thin plate is considered for analysis, it is assumed to be under the plane stress condition, and the thickness h and width b of the plate are taken to be small compared to the length a; i.e., b<5h.Figure 1
Schematic diagram of a rectangular microplate: (a) clamped-clamped (CC) and (b) simply supported (SS) boundary conditions.
(a)(b)


## 2.2. Governing Equations of Motion and the Coupled Thermoelasticity of Microplates

Letψx and ψy denote the transverse normal rotations about the y and x axes:(7)ψxx,t=−∂Wx,y,t∂x,ψyx,t=−∂Wx,y,t∂y,ux=−zWx,y,t.According to Kirchhoff’s plate theory, the displacement fieldux,uy,uz of an arbitrary point (x, y, z) is given by(8)ux=zψxx,t=−z∂Wx,y,t∂x,uy=zψyx,t=−z∂Wx,y,t∂y,uz=Wx,y,t,where Wx,y,t denotes the transverse deflection of the microplate.From (1) and (2), the rotational vector ψ can be rewritten as(9)ψx=12∂uz∂y−∂uy∂z=12W,y,ψy=12∂ux∂z−∂uz∂y=−12W,x,ψz=12∂uy∂x−∂ux∂y=−z2W,xy.The symmetrical rotation gradient tensorχij can be expanded using (5) as(10)χij=12∇ψ+∇ψT.The nonzero components of the symmetrical parts of the rotation gradient tensors are(11)χxxs=W,xy,χyys=−W,xy,χxys=12W,yy−W,xx,χzz=χyz=χzx=0.The nonzero components of the strain are(12)εxx=−zW,xx,εyy=−zW,yy,εxy=−zW,xy.The stress-strain relationships for the thermoelastic materials are(13)εxx=1Eσxx−vσyy+αtT−T0=1Eσxx+m−n2σyy+αtT−T0,εyy=1Eσyy−vσxx+αtT−T0=1Eσyy+m−n2σxx+αtT−T0,εzz=vEσxx+σyy+αtT−T0=n−m2Eσxx+σyy+αtT−T0,where m=0.774,0.931,0.72,0.69,0.808n=0.548,0.862,0.44,0.38,0.616 for PolySi, diamond, Si, GaAs, and SiC, respectively, which are derived from Poisson’s ratio.For a thin beam, assume a plane stress condition, whereσzz=0,(14)σxx=Emnεxx+vεyy−1mEαθ=Emnεxx+n−m2εyy−1mEαθ,σyy=Emnεyy+vεxx−1mEαθ=Emnεyy+n−m2εxx−1mEαθ,εzz=−vmεxx+εyy+nmαθ=m−n2mεxx+εyy+nmαθ.Using (11), (13) is rewritten as(15)σxx=−EzmnW,xx+vW,yy−1mEαθ=−EzmnW,xx+n−m2W,yy−1mEαθ,σyy=−EzmnW,yy+vW,xx−1mEαθ=−EzmnW,yy+n−m2W,xx−1mEαθ,εzz=−vzmW,xx+W,yy+nmαθ=m−nz2mW,xx+W,yy+nmαθ.In rectangular plates, bending momentsmxx,myy,mxy and torsion moments Yxx,Yyy,Yxy exist and are expressed as(16)mxx=El2nW,xy,myy=−El2nW,xy,mxy=El2nW,yy−W,xx,mzz=myz=mzx=0,Yxx=D×−zuz,xx−αθ,Yyy=D×−zuz,yy−αθ,Yxy=−EnZuz,xy,where D=Em/np.Including size effects, the strain energy is(17)Е=12∫0a∫0b∫−h2h2Dzh2F1+EnF2dxdydz,where(18)F1=W,xx2+W,yy2+αθW,xx+W,yy,F2=W,xy2zh2+lh2+lh2W,yy−W,xx2,and the kinetic energy of the microplate is(19)K=12∫ba∫0bρhW˙2+ρh312ψ˙x2+ψ˙y2dxdy,where ρ is the density of the plate material.According to the variational principle,(20)δ∫t1t2K−Udt=0.For clamped-clamped and simply supported rectangular microplates, the integral of the Gaussian curvature is zero,(21)∫0a∫0bW,xxW,yy−W,xy2dxdy=0.The moment of inertia is(22)IT=12αh3∫−h/2h/2zθdz.The dynamic governing equation of the microplate in terms ofWx,t is given by(23)ℬ+ℬ′∇4W+ℬ2∇2IT−ρhW¨=0,where(24)∇2=∂2∂x2+∂2∂y2,∇4=∂4∂x4+2∂4∂x∂y+∂4∂y4,ℬ=Dh3,ℬ'=Dl2hp2m.The temperature distribution in the microplate due to thermoelastic coupling is [74](25)θ˙=χ∇2θ−αEbTpCV∂∂t∑iεjj,where CV is the heat capacity coefficient at a constant volume, and χ denotes the thermal diffusivity of the solid.The relaxation strength of the thermoelastic solid can be expressed as(26)ŕ=Ead−EE=Eα2T0ρCp,where E is the isothermal or relaxed value, and Ead is the adiabatic or unrelaxed value of Young’s modulus.The linearized form of the heat equation can be expressed as(27)1+2ŕnpθ˙=χθ,zz+zŕαW˙,xx+W˙,yy.


## 3. Solution of the Coupled Thermoelastic Equations Based on MCST and Quality Factor

The solutions for the equation of motion and the coupled heat equation are [75](28)Wx,y,t=w0x,yeiwt,θx,y,z,t=θ0x,y,zeiωt.The temperature profile in the microplate is derived as(29)θ0x,y,z=ŕαw0,xx+w0,yyz−singzgcosgh/2.The thermal moment of inertia is(30)IT=12ŕh3h312+hg2−2g3tanhg2∇2ω0.Using (26) and (27) in (22), the equation of motion will be(31)ℬ+ℬ'+ℬ2ŕ1+fω∇4W−ρhW¨=0,where(32)fω=fgω=24h3g3hg2−tanhg2.The isothermal value of the eigenfrequency of the microplate is(33)ω0=π2ℬ/ρhm12a2+m22b2forSS,π2ℬ/ρhm1+1/22a2+m2+1/22b2;forCC,(34)ωω02=1+ℬ'ℬ+ŕ21+fω.From the above equation, the frequency is shown to be complex and is expressed as(35)Reω=ω0B4+ŕB1−6ξ3sinhξ+sinξcoshξ+cosξ,Imω=ω04ŕB6ξ3sinhξ+sinξcoshξ+cosξ−6ξ2,where(36)ξ=bω02χ.(37)B=161+ℬ′ℬ.TED, in terms of the inverse of the quality factor, is expressed as [75](38)Q−1=2ImωReω.In the analysis,(39)QMCST−1=2ŕ6/ξ3sinhξ+sinξ/coshξ+cosξ−6/ξ2B2B/16+ŕ/B1−6/ξ3sinhξ+sinξ/coshξ+cosξ.According to Lifshitz and Roukes [44], (37) is reduced to the well-known Lifshitz and Roukes expression when the size effect is not included:(40)QLR−1=ŕ6ξ3sinhξ+sinξcoshξ+cosξ−6ξ2.QMCST−1 in terms of QLR−1 is expressed as(41)QMCST−1=2QLR−1ŕ6/ξ3sinhξ−sinξ/coshξ+cosξ−1−B/4.


## 4. Results and Discussions

The numerical results are used to analyse the influence of size scaling, mode switching, boundary conditions, and material properties on thermoelastic energy dissipation and resulting quality factor.The obtained results for thermoelastic energy dissipation based on the given formulation are compared with those obtained for the classical continuum theories. The most accurate and simple theory among the higher order theories is MCST and hence opted for incorporating size effects. The achieved results of this work can be a reference for future researches in micro/nanoplate resonators to evolve great accomplishment in developing high performance MEMS/NEMS resonators for the leading edge technology.To validate the analytical study, plate resonators with lengtha=200μm, width b=200μm, and thickness h = 10 μm using five different structural materials are examined at a temperature of T0=298K. The experimentally reported mechanical and thermodynamic properties of all the materials are taken from [74]. As a common practice, the majority of the microstructures are fabricated with silicon or polysilicon, which can withstand large bending stresses, making failure due to excessive stress unlikely to occur. Besides the frequently used Si and PolySi, three different MEMS materials, diamond, GaAs, and SiC, are used in the case study and analysis according to their reported advantages and applications. The different boundary conditions selected for analysis are simply supported (SS) and clamped-clamped (CC), with the first two modes designated as (1, 1) and (1, 2). The numerical results are analysed to ascertain the influence of various parameters like material length scale parameter, boundary conditions, structural material properties, and mode switching on thermoelastic energy dissipation along with small-scale effects. The attained results for TED based on the given formulation applying MCST are compared with those based on the classical continuum theories (Lifshitz and Roukes Theory).


### 4.1. Effects of Size Scaling

The size dependence effect, which has been experimentally observed, is an inherent property of certain materials when the structures are reduced to the order of microns and submicrons. This effect plays an essential role in mechanical analyses of such structures and cannot be ignored in the optimal design of MEMS devices. The size effect in micro/nanostructures can be included by introducing the material length scale parameterl and made dimensionless by taking l/h, which is known as the dimensionless length scale parameter.The size effect study is characterized by changing the dimensionless length parameterl/h from 0 to 1, where l = 0 denotes the classical thermoelastic plate model, and varying l corresponds to the nonclassical continuum model of the Modified Couple Stress Theory (MCST). Size-dependent behaviour is apparent as l/h increases and becomes maximized when the characteristic size, such as the thickness, is comparable to the material length scale parameter, and the two theories converge as l/h decreases.The quality factors predicted by considering modified couple stress theory diverge from the classical theory, and the quality factor is maximum when the thickness and length scale parameter are equal, i.e.,l/h = 1. By comparing the results of l/h = 0 (CT) and l/h = 1 (MCST), it is verified that the values of the quality factor and the critical thickness have a great disparity between the two theories. The presence of the internal material length scale parameter is the main cause of the enhanced QTED and diminished hc in micro/nanoresonators, as shown in Tables 1 and 2.Table 1
Thermoelastic damping limited quality factor,QTED, of a microplate with different structural materials, dimensionless length scale parameters, vibrating modes, and boundary types.
MaterialBoundary conditionPolySiDiamondSiGaAsSiCl/h = 0Mode-ICC282225257603835032700SS281405225588334992688Mode-IICC283215343607535592710SS282445286603335132702l/h = 0.5Mode-ICC5866110108986054943597SS5780310000983954933581Mode-IICC58480103211000055153879SS5814010285990154984815l/h = 1Mode-ICC14836821630191831047110000SS1470592000018868104169866Mode-IICC14877422222192681048210256SS1481002176719194103579936Table 2
Critical thicknesshc of a microplate for different structural materials, dimensionless length scale parameter, vibrating modes, and boundary conditions.
l/h00.51Boundary conditionSSCCSSCCSSCCMode-IPolySi3.642.783.242.462.762.12Diamond43.063.462.642.922.24Si4.883.724.383.343.782.88GaAs4.023.083.642.783.162.4SiC3.242.462.842.182.421.86Mode-IIPolySi2.682.262.3822.041.7Diamond2.942.462.542.142.161.8Si3.63.023.222.72.782.32GaAs2.962.482.682.242.321.94SiC2.3822.11.761.81.5


#### 4.1.1. Quality Factor Analysis

The thermoelastic damping limited quality factor of a microplate resonator showing the size effect is clearly illustrated in Table1. The thermoelastic damping limited quality factor can be augmented by incrementing the dimensionless parameter l/h and can be maximized by choosing an internal length scale parameter (l) equal to the thickness of the resonating plate structure. The value of l is unique for each material, which can be experimentally proven, and much research work has been conducted for determining the material length scale parameter. The significance of l on QTED is analysed by examining the variation in energy dissipation with thickness for different values of l/h using different structural materials. By incorporating the internal material length scale parameter, even at elevated temperatures, energy dissipation is suppressed. The numerical values obtained for the simulation are summarized in Table 1, which will be very helpful for designing a low loss resonator with appropriate materials.The impact of the scaling effect onQTED is remarkable, and quality factor is significantly enhanced. The reason behind this noteworthy effect is explained as follows. The size effect parameter (l/h) increases the bending rigidity of the microplate, which enhances the size-dependent frequencies and quality factors of the resonators. This tendency shows a large amount of trimming in energy dissipation due to thermoelastic damping. The percentage increase in QTED with size effect (i.e., by varying l/h) is shown in Table 3.Table 3
Percentage increase inQTED with size effect (l/h).
l/h0–0.50.5–10–1Boundary conditionSSCCSSCCSSCCMode-IPolySi105.4104107.8548154.4118152.9228422.5882425.7122Diamond91.3792.2556100113.9943282.74311.4161Si67.2569963.2850291.7735894.57126220.7547217.7057GaAs56.9757256.8038289.6458390.60733197.6979198.8796SiC33.2294833.23741175.4933178267.0383270.4Mode-IIPolySi105.8488106.4912154.7318154.4037424.3624425.3214Diamond94.5878793.17783111.6277115.3063311.8018315.9241Si64.1386164.693.8579792.67823218.1958217.1484GaAs56.4932954.9385188.3654290.07338194.7792194.4969SiC41.1672843.1527160.4336164.3761267.6471278.4615Figure2 represents the variation in energy dissipation, Q−1 with the thickness of the microplate for two boundary conditions (simply supported-SS and clamped-clamped -CC) and first two vibrating modes {Mode I-(1, 1) and Mode II-(1, 2)}. In Figure 2, Q−1 and hc both are apparently decreased when the l/h value is varied from 0 to 1.Figure 2
Energy dissipation,Q−1 vs. thickness, h of a rectangular plate with different modes and boundary conditions: (a) mode (I) simply supported (SS); (b) mode II, simply supported (SS); (c) mode (I) clamped-clamped (CC), and (d) mode II, clamped-clamped (CC).
(a)(b)(c)(d)


#### 4.1.2.hc Analysis

As the size effect is incorporated, the critical thicknesshc is found to be reduced, which indicates that as the nondimensional parameter l/h is increased, the critical dimensions are reduced. From Figure 2, it is obvious that Q−1 and hc are both decreased when the l/h value varies from 0 to 1.Table2 shows the variation of critical thickness for different materials with size effects. Thus, the material order for critical thickness change is, in descending order, Si > GaAs > diamond > polySi > SiC. From the current analysis hc is the highest in Si with l/h = 0 and the lowest in SiC with l/h = 1.Table4 gives the percentage increase in critical thickness with the size effect variations. The impact of size effect is analysed by the transitions of l/h values. The percentage change in hc is the highest when l/h is varied from 0 to 1 and the lowest for 0 to 0.5 transition which verifies the size effect impact. Among the different structural materials, the percentage increase is the highest for diamond (27%) and the lowest for GaAs (21.393%). The maximum percentage increase in critical thickness is different for various structural materials and the sequential order in which the percentage change diminishes is diamond > SiC > PolySi > Si > GaAs.Table 4
Percentage increase (%) of critical thickness of a microplate for different structural materials with size effect.
l/h0–0.50.5–10–1Boundary conditionSSCCSSCCSSCCMode-IPolySi12.3456813.0081317.391316.0377424.1758223.74101Diamond15.6069415.9090918.4931517.857142726.79739Si11.4155311.3772515.8730215.9722222.5409822.58065GaAs10.4395610.7913715.1898715.8333321.3930322.07792SiC14.0845112.8440417.3553717.204325.3086424.39024Mode-IIPolySi12.605041316.6666717.6470623.880624.77876Diamond15.7480314.9532717.5925918.8888926.5306126.82927Si11.8012411.8518515.8273416.3793122.7777823.17881GaAs10.4477610.7142915.5172415.4639221.6216221.77419SiC13.3333313.6363616.6666717.3333324.3697525


### 4.2. Effects of Boundary Conditions

The quality factor clearly increases with the clamped-clamped (CC) boundary condition compared to the simply supported (SS) case. The impact of boundary conditions on the quality factor with simply supported boundary conditions is shown in Figures2(a) and 2(b), and that with clamped-clamped is shown in Figures 2(c) and 2(d).


#### 4.2.1. Quality Factor Analysis

The boundary conditions slightly change theQTED. Table 1 depicts that the quality factor is enhanced when the clamped-clamped (CC) boundary condition is selected for the microplate compared to simply supported (SS) boundary condition. In the analysis, the maximum attained quality factor QTED=148368 is for PolySi-based microplates with clamped-clamped boundary conditions and the minimum (2687.955) is obtained for simply supported microplates with SiC structural material. Table 5 shows the percentage increase in QTED with size effects when the boundary condition is changed.Table 5
Variation ofQTED (%) of a microplate for different boundary conditions.
Boundary conditionSS-CCl/h00.51Mode-IPolySi0.2906891.4841320.890208Diamond0.6098521.0754438.14786Si2.6446080.2070391.668905GaAs0.1261250.0164810.52356SiC0.44006500.4460431.36Mode-IIPolySi0.2718780.5847950.455249Diamond1.0792330.3467852.091065Si0.71688910.385356GaAs1.3098420.3033141.212788SiC0.2710031.681243.220513The maximum percentage (8.14786%) inQTED is obtained for diamond at l/h = 1 and the minimum (0.01648%) is obtained for GaAs at l/h = 0.5. Figure 3(a) shows the variation of Q−1 with the thickness of the plate for two boundary conditions at l/h = 0 for the first mode (1, 1). There is only slight variation in the quality factor when boundary conditions are different.Figure 3
Comparison of energy dissipation,Q−1 vs. thickness, h of a rectangular plate for simply supported (SS) and clamped-clamped (CC) boundary conditions vibrating in Mode-I with (a) l/h = 0, (b) l/h = 0.5, and (c) l/h = 1.
(a)(b)(c)Figures4(a)–4(c) show the variation of Q−1 with the thickness of the beam for different boundary conditions with l/h = 0, l/h = 0.5, and l/h = 1. As l/h increases, the value of QTED is enhanced, as shown in Figures 4(a) and 4(c).Figure 4
Comparison of energy dissipation,Q−1 vs. thickness, h of a rectangular plate for simply supported (SS) and clamped-clamped (CC) boundary conditions vibrating in second mode with varying dimensionless length scale parameters (l/h): (a) l/h = 0, (b) l/h = 0.5, and (c) l/h = 1.
(a)(b)(c)Figures4(a)–4(c) show the size effect variation of plate resonators for Mode-II (1, 2). The variation of the quality factor with mode switching is negligible here.


#### 4.2.2.hc Analysis

The maximum critical thickness is attained for Si-based microplates with simply supported boundary condition (4.88μm), and the minimum hc (1.5 μm) is obtained for clamped-clamped, SiC-based microplates as shown in Table 2. The maximum percentage change (31.70732%) in critical thickness is obtained for PolySi-based microplate at l/h = 0.5 vibrating in first mode. The minimum percentage increase (15.67641%) in hc is found for PolySi at l/h = 0 with the second vibration mode.


### 4.3. Effects of Mode Switching

In the case of microplate resonators, the quality factor increases with the material length scale parameter, which is applicable for all modes. The size-dependent quality factor is greater than the classical ones and can be further increased by increasing the mode number, i.e., the thermoelastic damping limited quality factor is enhanced slightly when the microplate resonators are operated at higher modes.


#### 4.3.1. Quality Factor Analysis

As the vibration modes of the resonators are increased, energy dissipation decreases due to the effect of thermal relaxation time. From Table1, the maximum QTED is achieved for PolySi (148774.1) vibrating in the second mode, and the minimum quality factor (2687.955) is obtained for SiC-based microplates with Mode I.While analysing various modes, the maximum percentage increase (8.8328%) in QTED is obtained with l/h = 0.5 for diamond-based microplates with clamped-clamped boundary condition. The minimum percentage increase (0.09%) obtained in hc is for a simply supported GaAs-based microplate with l/h = 0.5 and is found to be very small. The decreasing order for the maximum percentage variation for different structural materials is diamond > SiC > Si > GaAs > PolySi.


#### 4.3.2.hc Analysis

The effect of critical thickness changes for each mode is shown in Figure5 for two boundary conditions. The maximum (4.88 μm) and minimum (1.5 μm) values of hc are obtained for the first and second modes, respectively.Figure 5
Comparison of energy dissipation,Q−1 vs. thickness, h of a rectangular plate for two modes (mode I and mode II) with (a) simply supported boundary condition and varying l/h values: (A) l/h = 0, (B) l/h = 0.5, and (C) l/h = 1 and (b) clamped-clamped boundary condition and varying l/h values (D) l/h = 0, (E) l/h = 0.5, and (F) l/h = 1.
(a)(b)Maximum percentage increase (26.5896%) in hc is obtained for simply supported diamond-based microplates withl/h = 0.5 and the minimum percentage increase (18.69919%) is obtained for SiC-based microplate with clamped-clamped boundary condition and l/h = 0.A difference in boundary conditions changes critical thickness significantly, as shown in Table2. The change in quality factor is not significant, as verified by comparing Figures 5A–5C, which illustrates the energy dissipation curves corresponding to the simply supported case, with Figures 5D–5E, which illustrate the clamped-clamped case.


### 4.4. Micro/Nanoplate Resonator Modelling Applying FEM

COMSOL finite element analysis is also employed to investigate the thermoelastic damping behaviour of the plates and related displacement and temperature coupling are explored. The mean value of the displacements and temperature amplitudes at different points along the plate resonator are shown in Figures6(a) and 6(b), respectively. The results obtained from the FEM analysis are also in accordance with those attained through numerical simulations.Figure 6
(a) Temperature distribution of a rectangular microplate with thermoelastic damping and (b) the displacement of a rectangular microplate with thermoelastic damping.
(a)(b)


## 4.1. Effects of Size Scaling

The size dependence effect, which has been experimentally observed, is an inherent property of certain materials when the structures are reduced to the order of microns and submicrons. This effect plays an essential role in mechanical analyses of such structures and cannot be ignored in the optimal design of MEMS devices. The size effect in micro/nanostructures can be included by introducing the material length scale parameterl and made dimensionless by taking l/h, which is known as the dimensionless length scale parameter.The size effect study is characterized by changing the dimensionless length parameterl/h from 0 to 1, where l = 0 denotes the classical thermoelastic plate model, and varying l corresponds to the nonclassical continuum model of the Modified Couple Stress Theory (MCST). Size-dependent behaviour is apparent as l/h increases and becomes maximized when the characteristic size, such as the thickness, is comparable to the material length scale parameter, and the two theories converge as l/h decreases.The quality factors predicted by considering modified couple stress theory diverge from the classical theory, and the quality factor is maximum when the thickness and length scale parameter are equal, i.e.,l/h = 1. By comparing the results of l/h = 0 (CT) and l/h = 1 (MCST), it is verified that the values of the quality factor and the critical thickness have a great disparity between the two theories. The presence of the internal material length scale parameter is the main cause of the enhanced QTED and diminished hc in micro/nanoresonators, as shown in Tables 1 and 2.Table 1
Thermoelastic damping limited quality factor,QTED, of a microplate with different structural materials, dimensionless length scale parameters, vibrating modes, and boundary types.
MaterialBoundary conditionPolySiDiamondSiGaAsSiCl/h = 0Mode-ICC282225257603835032700SS281405225588334992688Mode-IICC283215343607535592710SS282445286603335132702l/h = 0.5Mode-ICC5866110108986054943597SS5780310000983954933581Mode-IICC58480103211000055153879SS5814010285990154984815l/h = 1Mode-ICC14836821630191831047110000SS1470592000018868104169866Mode-IICC14877422222192681048210256SS1481002176719194103579936Table 2
Critical thicknesshc of a microplate for different structural materials, dimensionless length scale parameter, vibrating modes, and boundary conditions.
l/h00.51Boundary conditionSSCCSSCCSSCCMode-IPolySi3.642.783.242.462.762.12Diamond43.063.462.642.922.24Si4.883.724.383.343.782.88GaAs4.023.083.642.783.162.4SiC3.242.462.842.182.421.86Mode-IIPolySi2.682.262.3822.041.7Diamond2.942.462.542.142.161.8Si3.63.023.222.72.782.32GaAs2.962.482.682.242.321.94SiC2.3822.11.761.81.5


### 4.1.1. Quality Factor Analysis

The thermoelastic damping limited quality factor of a microplate resonator showing the size effect is clearly illustrated in Table1. The thermoelastic damping limited quality factor can be augmented by incrementing the dimensionless parameter l/h and can be maximized by choosing an internal length scale parameter (l) equal to the thickness of the resonating plate structure. The value of l is unique for each material, which can be experimentally proven, and much research work has been conducted for determining the material length scale parameter. The significance of l on QTED is analysed by examining the variation in energy dissipation with thickness for different values of l/h using different structural materials. By incorporating the internal material length scale parameter, even at elevated temperatures, energy dissipation is suppressed. The numerical values obtained for the simulation are summarized in Table 1, which will be very helpful for designing a low loss resonator with appropriate materials.The impact of the scaling effect onQTED is remarkable, and quality factor is significantly enhanced. The reason behind this noteworthy effect is explained as follows. The size effect parameter (l/h) increases the bending rigidity of the microplate, which enhances the size-dependent frequencies and quality factors of the resonators. This tendency shows a large amount of trimming in energy dissipation due to thermoelastic damping. The percentage increase in QTED with size effect (i.e., by varying l/h) is shown in Table 3.Table 3
Percentage increase inQTED with size effect (l/h).
l/h0–0.50.5–10–1Boundary conditionSSCCSSCCSSCCMode-IPolySi105.4104107.8548154.4118152.9228422.5882425.7122Diamond91.3792.2556100113.9943282.74311.4161Si67.2569963.2850291.7735894.57126220.7547217.7057GaAs56.9757256.8038289.6458390.60733197.6979198.8796SiC33.2294833.23741175.4933178267.0383270.4Mode-IIPolySi105.8488106.4912154.7318154.4037424.3624425.3214Diamond94.5878793.17783111.6277115.3063311.8018315.9241Si64.1386164.693.8579792.67823218.1958217.1484GaAs56.4932954.9385188.3654290.07338194.7792194.4969SiC41.1672843.1527160.4336164.3761267.6471278.4615Figure2 represents the variation in energy dissipation, Q−1 with the thickness of the microplate for two boundary conditions (simply supported-SS and clamped-clamped -CC) and first two vibrating modes {Mode I-(1, 1) and Mode II-(1, 2)}. In Figure 2, Q−1 and hc both are apparently decreased when the l/h value is varied from 0 to 1.Figure 2
Energy dissipation,Q−1 vs. thickness, h of a rectangular plate with different modes and boundary conditions: (a) mode (I) simply supported (SS); (b) mode II, simply supported (SS); (c) mode (I) clamped-clamped (CC), and (d) mode II, clamped-clamped (CC).
(a)(b)(c)(d)


### 4.1.2.hc Analysis

As the size effect is incorporated, the critical thicknesshc is found to be reduced, which indicates that as the nondimensional parameter l/h is increased, the critical dimensions are reduced. From Figure 2, it is obvious that Q−1 and hc are both decreased when the l/h value varies from 0 to 1.Table2 shows the variation of critical thickness for different materials with size effects. Thus, the material order for critical thickness change is, in descending order, Si > GaAs > diamond > polySi > SiC. From the current analysis hc is the highest in Si with l/h = 0 and the lowest in SiC with l/h = 1.Table4 gives the percentage increase in critical thickness with the size effect variations. The impact of size effect is analysed by the transitions of l/h values. The percentage change in hc is the highest when l/h is varied from 0 to 1 and the lowest for 0 to 0.5 transition which verifies the size effect impact. Among the different structural materials, the percentage increase is the highest for diamond (27%) and the lowest for GaAs (21.393%). The maximum percentage increase in critical thickness is different for various structural materials and the sequential order in which the percentage change diminishes is diamond > SiC > PolySi > Si > GaAs.Table 4
Percentage increase (%) of critical thickness of a microplate for different structural materials with size effect.
l/h0–0.50.5–10–1Boundary conditionSSCCSSCCSSCCMode-IPolySi12.3456813.0081317.391316.0377424.1758223.74101Diamond15.6069415.9090918.4931517.857142726.79739Si11.4155311.3772515.8730215.9722222.5409822.58065GaAs10.4395610.7913715.1898715.8333321.3930322.07792SiC14.0845112.8440417.3553717.204325.3086424.39024Mode-IIPolySi12.605041316.6666717.6470623.880624.77876Diamond15.7480314.9532717.5925918.8888926.5306126.82927Si11.8012411.8518515.8273416.3793122.7777823.17881GaAs10.4477610.7142915.5172415.4639221.6216221.77419SiC13.3333313.6363616.6666717.3333324.3697525


## 4.1.1. Quality Factor Analysis

The thermoelastic damping limited quality factor of a microplate resonator showing the size effect is clearly illustrated in Table1. The thermoelastic damping limited quality factor can be augmented by incrementing the dimensionless parameter l/h and can be maximized by choosing an internal length scale parameter (l) equal to the thickness of the resonating plate structure. The value of l is unique for each material, which can be experimentally proven, and much research work has been conducted for determining the material length scale parameter. The significance of l on QTED is analysed by examining the variation in energy dissipation with thickness for different values of l/h using different structural materials. By incorporating the internal material length scale parameter, even at elevated temperatures, energy dissipation is suppressed. The numerical values obtained for the simulation are summarized in Table 1, which will be very helpful for designing a low loss resonator with appropriate materials.The impact of the scaling effect onQTED is remarkable, and quality factor is significantly enhanced. The reason behind this noteworthy effect is explained as follows. The size effect parameter (l/h) increases the bending rigidity of the microplate, which enhances the size-dependent frequencies and quality factors of the resonators. This tendency shows a large amount of trimming in energy dissipation due to thermoelastic damping. The percentage increase in QTED with size effect (i.e., by varying l/h) is shown in Table 3.Table 3
Percentage increase inQTED with size effect (l/h).
l/h0–0.50.5–10–1Boundary conditionSSCCSSCCSSCCMode-IPolySi105.4104107.8548154.4118152.9228422.5882425.7122Diamond91.3792.2556100113.9943282.74311.4161Si67.2569963.2850291.7735894.57126220.7547217.7057GaAs56.9757256.8038289.6458390.60733197.6979198.8796SiC33.2294833.23741175.4933178267.0383270.4Mode-IIPolySi105.8488106.4912154.7318154.4037424.3624425.3214Diamond94.5878793.17783111.6277115.3063311.8018315.9241Si64.1386164.693.8579792.67823218.1958217.1484GaAs56.4932954.9385188.3654290.07338194.7792194.4969SiC41.1672843.1527160.4336164.3761267.6471278.4615Figure2 represents the variation in energy dissipation, Q−1 with the thickness of the microplate for two boundary conditions (simply supported-SS and clamped-clamped -CC) and first two vibrating modes {Mode I-(1, 1) and Mode II-(1, 2)}. In Figure 2, Q−1 and hc both are apparently decreased when the l/h value is varied from 0 to 1.Figure 2
Energy dissipation,Q−1 vs. thickness, h of a rectangular plate with different modes and boundary conditions: (a) mode (I) simply supported (SS); (b) mode II, simply supported (SS); (c) mode (I) clamped-clamped (CC), and (d) mode II, clamped-clamped (CC).
(a)(b)(c)(d)


## 4.1.2.hc Analysis

As the size effect is incorporated, the critical thicknesshc is found to be reduced, which indicates that as the nondimensional parameter l/h is increased, the critical dimensions are reduced. From Figure 2, it is obvious that Q−1 and hc are both decreased when the l/h value varies from 0 to 1.Table2 shows the variation of critical thickness for different materials with size effects. Thus, the material order for critical thickness change is, in descending order, Si > GaAs > diamond > polySi > SiC. From the current analysis hc is the highest in Si with l/h = 0 and the lowest in SiC with l/h = 1.Table4 gives the percentage increase in critical thickness with the size effect variations. The impact of size effect is analysed by the transitions of l/h values. The percentage change in hc is the highest when l/h is varied from 0 to 1 and the lowest for 0 to 0.5 transition which verifies the size effect impact. Among the different structural materials, the percentage increase is the highest for diamond (27%) and the lowest for GaAs (21.393%). The maximum percentage increase in critical thickness is different for various structural materials and the sequential order in which the percentage change diminishes is diamond > SiC > PolySi > Si > GaAs.Table 4
Percentage increase (%) of critical thickness of a microplate for different structural materials with size effect.
l/h0–0.50.5–10–1Boundary conditionSSCCSSCCSSCCMode-IPolySi12.3456813.0081317.391316.0377424.1758223.74101Diamond15.6069415.9090918.4931517.857142726.79739Si11.4155311.3772515.8730215.9722222.5409822.58065GaAs10.4395610.7913715.1898715.8333321.3930322.07792SiC14.0845112.8440417.3553717.204325.3086424.39024Mode-IIPolySi12.605041316.6666717.6470623.880624.77876Diamond15.7480314.9532717.5925918.8888926.5306126.82927Si11.8012411.8518515.8273416.3793122.7777823.17881GaAs10.4477610.7142915.5172415.4639221.6216221.77419SiC13.3333313.6363616.6666717.3333324.3697525


## 4.2. Effects of Boundary Conditions

The quality factor clearly increases with the clamped-clamped (CC) boundary condition compared to the simply supported (SS) case. The impact of boundary conditions on the quality factor with simply supported boundary conditions is shown in Figures2(a) and 2(b), and that with clamped-clamped is shown in Figures 2(c) and 2(d).


### 4.2.1. Quality Factor Analysis

The boundary conditions slightly change theQTED. Table 1 depicts that the quality factor is enhanced when the clamped-clamped (CC) boundary condition is selected for the microplate compared to simply supported (SS) boundary condition. In the analysis, the maximum attained quality factor QTED=148368 is for PolySi-based microplates with clamped-clamped boundary conditions and the minimum (2687.955) is obtained for simply supported microplates with SiC structural material. Table 5 shows the percentage increase in QTED with size effects when the boundary condition is changed.Table 5
Variation ofQTED (%) of a microplate for different boundary conditions.
Boundary conditionSS-CCl/h00.51Mode-IPolySi0.2906891.4841320.890208Diamond0.6098521.0754438.14786Si2.6446080.2070391.668905GaAs0.1261250.0164810.52356SiC0.44006500.4460431.36Mode-IIPolySi0.2718780.5847950.455249Diamond1.0792330.3467852.091065Si0.71688910.385356GaAs1.3098420.3033141.212788SiC0.2710031.681243.220513The maximum percentage (8.14786%) inQTED is obtained for diamond at l/h = 1 and the minimum (0.01648%) is obtained for GaAs at l/h = 0.5. Figure 3(a) shows the variation of Q−1 with the thickness of the plate for two boundary conditions at l/h = 0 for the first mode (1, 1). There is only slight variation in the quality factor when boundary conditions are different.Figure 3
Comparison of energy dissipation,Q−1 vs. thickness, h of a rectangular plate for simply supported (SS) and clamped-clamped (CC) boundary conditions vibrating in Mode-I with (a) l/h = 0, (b) l/h = 0.5, and (c) l/h = 1.
(a)(b)(c)Figures4(a)–4(c) show the variation of Q−1 with the thickness of the beam for different boundary conditions with l/h = 0, l/h = 0.5, and l/h = 1. As l/h increases, the value of QTED is enhanced, as shown in Figures 4(a) and 4(c).Figure 4
Comparison of energy dissipation,Q−1 vs. thickness, h of a rectangular plate for simply supported (SS) and clamped-clamped (CC) boundary conditions vibrating in second mode with varying dimensionless length scale parameters (l/h): (a) l/h = 0, (b) l/h = 0.5, and (c) l/h = 1.
(a)(b)(c)Figures4(a)–4(c) show the size effect variation of plate resonators for Mode-II (1, 2). The variation of the quality factor with mode switching is negligible here.


### 4.2.2.hc Analysis

The maximum critical thickness is attained for Si-based microplates with simply supported boundary condition (4.88μm), and the minimum hc (1.5 μm) is obtained for clamped-clamped, SiC-based microplates as shown in Table 2. The maximum percentage change (31.70732%) in critical thickness is obtained for PolySi-based microplate at l/h = 0.5 vibrating in first mode. The minimum percentage increase (15.67641%) in hc is found for PolySi at l/h = 0 with the second vibration mode.


## 4.2.1. Quality Factor Analysis

The boundary conditions slightly change theQTED. Table 1 depicts that the quality factor is enhanced when the clamped-clamped (CC) boundary condition is selected for the microplate compared to simply supported (SS) boundary condition. In the analysis, the maximum attained quality factor QTED=148368 is for PolySi-based microplates with clamped-clamped boundary conditions and the minimum (2687.955) is obtained for simply supported microplates with SiC structural material. Table 5 shows the percentage increase in QTED with size effects when the boundary condition is changed.Table 5
Variation ofQTED (%) of a microplate for different boundary conditions.
Boundary conditionSS-CCl/h00.51Mode-IPolySi0.2906891.4841320.890208Diamond0.6098521.0754438.14786Si2.6446080.2070391.668905GaAs0.1261250.0164810.52356SiC0.44006500.4460431.36Mode-IIPolySi0.2718780.5847950.455249Diamond1.0792330.3467852.091065Si0.71688910.385356GaAs1.3098420.3033141.212788SiC0.2710031.681243.220513The maximum percentage (8.14786%) inQTED is obtained for diamond at l/h = 1 and the minimum (0.01648%) is obtained for GaAs at l/h = 0.5. Figure 3(a) shows the variation of Q−1 with the thickness of the plate for two boundary conditions at l/h = 0 for the first mode (1, 1). There is only slight variation in the quality factor when boundary conditions are different.Figure 3
Comparison of energy dissipation,Q−1 vs. thickness, h of a rectangular plate for simply supported (SS) and clamped-clamped (CC) boundary conditions vibrating in Mode-I with (a) l/h = 0, (b) l/h = 0.5, and (c) l/h = 1.
(a)(b)(c)Figures4(a)–4(c) show the variation of Q−1 with the thickness of the beam for different boundary conditions with l/h = 0, l/h = 0.5, and l/h = 1. As l/h increases, the value of QTED is enhanced, as shown in Figures 4(a) and 4(c).Figure 4
Comparison of energy dissipation,Q−1 vs. thickness, h of a rectangular plate for simply supported (SS) and clamped-clamped (CC) boundary conditions vibrating in second mode with varying dimensionless length scale parameters (l/h): (a) l/h = 0, (b) l/h = 0.5, and (c) l/h = 1.
(a)(b)(c)Figures4(a)–4(c) show the size effect variation of plate resonators for Mode-II (1, 2). The variation of the quality factor with mode switching is negligible here.


## 4.2.2.hc Analysis

The maximum critical thickness is attained for Si-based microplates with simply supported boundary condition (4.88μm), and the minimum hc (1.5 μm) is obtained for clamped-clamped, SiC-based microplates as shown in Table 2. The maximum percentage change (31.70732%) in critical thickness is obtained for PolySi-based microplate at l/h = 0.5 vibrating in first mode. The minimum percentage increase (15.67641%) in hc is found for PolySi at l/h = 0 with the second vibration mode.


## 4.3. Effects of Mode Switching

In the case of microplate resonators, the quality factor increases with the material length scale parameter, which is applicable for all modes. The size-dependent quality factor is greater than the classical ones and can be further increased by increasing the mode number, i.e., the thermoelastic damping limited quality factor is enhanced slightly when the microplate resonators are operated at higher modes.


### 4.3.1. Quality Factor Analysis

As the vibration modes of the resonators are increased, energy dissipation decreases due to the effect of thermal relaxation time. From Table1, the maximum QTED is achieved for PolySi (148774.1) vibrating in the second mode, and the minimum quality factor (2687.955) is obtained for SiC-based microplates with Mode I.While analysing various modes, the maximum percentage increase (8.8328%) in QTED is obtained with l/h = 0.5 for diamond-based microplates with clamped-clamped boundary condition. The minimum percentage increase (0.09%) obtained in hc is for a simply supported GaAs-based microplate with l/h = 0.5 and is found to be very small. The decreasing order for the maximum percentage variation for different structural materials is diamond > SiC > Si > GaAs > PolySi.


### 4.3.2.hc Analysis

The effect of critical thickness changes for each mode is shown in Figure5 for two boundary conditions. The maximum (4.88 μm) and minimum (1.5 μm) values of hc are obtained for the first and second modes, respectively.Figure 5
Comparison of energy dissipation,Q−1 vs. thickness, h of a rectangular plate for two modes (mode I and mode II) with (a) simply supported boundary condition and varying l/h values: (A) l/h = 0, (B) l/h = 0.5, and (C) l/h = 1 and (b) clamped-clamped boundary condition and varying l/h values (D) l/h = 0, (E) l/h = 0.5, and (F) l/h = 1.
(a)(b)Maximum percentage increase (26.5896%) in hc is obtained for simply supported diamond-based microplates withl/h = 0.5 and the minimum percentage increase (18.69919%) is obtained for SiC-based microplate with clamped-clamped boundary condition and l/h = 0.A difference in boundary conditions changes critical thickness significantly, as shown in Table2. The change in quality factor is not significant, as verified by comparing Figures 5A–5C, which illustrates the energy dissipation curves corresponding to the simply supported case, with Figures 5D–5E, which illustrate the clamped-clamped case.


## 4.3.1. Quality Factor Analysis

As the vibration modes of the resonators are increased, energy dissipation decreases due to the effect of thermal relaxation time. From Table1, the maximum QTED is achieved for PolySi (148774.1) vibrating in the second mode, and the minimum quality factor (2687.955) is obtained for SiC-based microplates with Mode I.While analysing various modes, the maximum percentage increase (8.8328%) in QTED is obtained with l/h = 0.5 for diamond-based microplates with clamped-clamped boundary condition. The minimum percentage increase (0.09%) obtained in hc is for a simply supported GaAs-based microplate with l/h = 0.5 and is found to be very small. The decreasing order for the maximum percentage variation for different structural materials is diamond > SiC > Si > GaAs > PolySi.


## 4.3.2.hc Analysis

The effect of critical thickness changes for each mode is shown in Figure5 for two boundary conditions. The maximum (4.88 μm) and minimum (1.5 μm) values of hc are obtained for the first and second modes, respectively.Figure 5
Comparison of energy dissipation,Q−1 vs. thickness, h of a rectangular plate for two modes (mode I and mode II) with (a) simply supported boundary condition and varying l/h values: (A) l/h = 0, (B) l/h = 0.5, and (C) l/h = 1 and (b) clamped-clamped boundary condition and varying l/h values (D) l/h = 0, (E) l/h = 0.5, and (F) l/h = 1.
(a)(b)Maximum percentage increase (26.5896%) in hc is obtained for simply supported diamond-based microplates withl/h = 0.5 and the minimum percentage increase (18.69919%) is obtained for SiC-based microplate with clamped-clamped boundary condition and l/h = 0.A difference in boundary conditions changes critical thickness significantly, as shown in Table2. The change in quality factor is not significant, as verified by comparing Figures 5A–5C, which illustrates the energy dissipation curves corresponding to the simply supported case, with Figures 5D–5E, which illustrate the clamped-clamped case.


## 4.4. Micro/Nanoplate Resonator Modelling Applying FEM

COMSOL finite element analysis is also employed to investigate the thermoelastic damping behaviour of the plates and related displacement and temperature coupling are explored. The mean value of the displacements and temperature amplitudes at different points along the plate resonator are shown in Figures6(a) and 6(b), respectively. The results obtained from the FEM analysis are also in accordance with those attained through numerical simulations.Figure 6
(a) Temperature distribution of a rectangular microplate with thermoelastic damping and (b) the displacement of a rectangular microplate with thermoelastic damping.
(a)(b)


## 5. Conclusion

In micro/nanoplate resonators, energy dissipation due to thermoelastic damping (TED) is a significant energy loss method which is verified from the findings. When the dimensions are in micro/nanoscales, size effect is very much influential and a higher order theory like MCST is applied to incorporate scaling effects. In the current analysis, the material dependent thermoelastic coupled equations for the deflected Kirchoff’s plate have been presented using variational and Hamilton principles and the solutions forQTED are presented based on complex frequency approach with the appropriate material index parameters. The impacts of material properties, boundary conditions, mode switching, and material length scale parameter on thermoelastic damping limited quality factor (QTED) and critical thickness (hc) are investigated in the work. The most substantial conclusions of the analyses are as follows:(I)
(i)
From the investigation, thermoelastic damping limited quality factor (QTED) of rectangular micro/nanoplates is found to be related to a decisive material performance index parameter known as thermoelastic damping index (TDI). As TDI of the structural material used for making the micro/nanoplates increases, thermoelastic energy dissipation increases and thermoelastic damping limited quality factor (QTED) decreases. Hence polySi-based micro/nanoplates attains the lowest TDI and highest QTED. SiC-based plates own the lowest QTED because of its highest TDI value. According to this work, the maximum QTED is obtained for rectangular microplates with polySi as the structural material having the lowest thermoelastic damping index (TDI) which is an important observation. The lowest QTED is for SiC with the highest TDI value.(ii)
From the current analysis, critical thickness (hc) of micro/nanoplates is also found to be material dependent. The current investigation finds that hc is found to be inversely proportional to the thermal diffusion length,lT. Under the influence of lT, critical thickness of micro/nanoplates made of different structural materials is found to be different.(II)
(i)
The diminishing order in whichQTED varies is poly Si > diamond > Si > GaAs > SiC among the five different structural materials, which is substantiated to be inversely proportional to TDI.(ii)
The material order in which critical thickness (hc) drops off is Si > GaAs > diamond > polySi > SiC. Hence the relation to thermal diffusion length of structural materials is proved.(III)
(i)
When higher order theory like MCST is applied for quantifying the thermoelastic losses of rectangular plates, corresponding quality factor (QTED) is enhanced according to the material length scale parameter incorporating the small scale effect. The maximum QF is obtained when the material length scale parameter is equal to the thickness of the plates.(ii)
As the dimensionless material length scale parameter increases, hc decreases which is a significant finding from the analysis regarding the critical dimensions of micro/nanoplates.(IV)
(i)
As far as various boundary conditions such as simply supported and clamped-clamped `plates are considered,QTED of clamped-clamped plates are more for all modes and l/h values. Hence the susceptibility of micro/nanoplates to thermoelastic energy losses can be reduced by opting clamped-clamped boundary condition.(ii)
When micro/nanoplates with two different structural boundary conditions are examined, the maximum critical thickness is procured with simply supported boundary condition. The critical thickness for fully clamped plates is less compared to simply supported boundary condition.(V)
(i)
According to the increase in mode number, there is a slight enhancement inQTED for all l/h values and boundary conditions.(ii)
As the mode number increases, hc decreases which is an important observation. In the mode analysis, the maximum percentage difference in hc is obtained with the simply supported mode.The investigations of thermoelastic losses, related quality factor, and critical dimensions of microplates are highly valuable for the design and manufacture of high performance resonating devices for the state of the art technological applications in communication and sensing fields.



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*Source: 6759093-2022-04-04.xml*