Dual-phase-lag thermoelastic damping analysis of a functionally graded sandwich micro-beam resonators incorporating double nonlocal effects

Abstract

Functionally graded sandwich micro/nano-structures have attracted great attention due to the capability to resist high noise and thermal stress in a non-isothermal environment. Additionally, the design of high quality-factor micro/nano-resonators requires accurate estimation of their thermoelastic damping. However, the classical thermoelastic damping models fail at the micro/nano-scale due to the influences of the size-dependent effects related to heat transfer and elastic deformation. This work aims to investigate the influences of the size-dependent effects on the thermoelastic damping of functionally graded sandwich micro-beam resonators by combining the nonlocal dual-phase-lag heat conduction model and the nonlocal elasticity model. It is assumed that the functionally graded sandwich micro-beam resonators consist of a ceramic core and functionally graded surfaces. The energy equation and the transverse motion equation are derived. The analytical expression of thermoelastic damping is obtained by complex frequency method. Numerical results are analyzed for the effects of the thermal nonlocal parameter, the elastic nonlocal parameter, the power-law index, and the vibration modes on the thermoelastic damping of functionally graded sandwich micro-beam resonators. The results show that the thermoelastic damping of functionally graded sandwich micro-beam resonators can be adjusted by the suitably modified parameters, which strongly depends on the double nonlocal effects and the power-law index.

Introduction

Nowadays, sandwich structures have been widely used in various engineering fields such as aerospace, civil engineering and automotive field due to their high strength and excellent vibration resistance. ¹ However, the traditional sandwich structures typically exhibit delamination and stress concentration at the interfaces, which is caused by the material discontinuities. The above imperfections can be avoided by applying functionally graded (FG) sandwich structures. The Japanese scientist initially proposed the concept of functionally graded materials (FGMs) in 1984. FGMs are advanced composites with the properties varying gradually along certain direction to eliminate the sharp interface in multi-layer structures. ² Thereafter, many investigations have been carried out to study the static and dynamic responses of the FG sandwich structures.^3–12

With the rapid development of micro- and nano-technology, the concept of functionally graded sandwich has also been utilized in micro/nano-electro-mechanical systems (MEMS/NEMS) to improve their mechanical properties. For micro/nano-resonators as key components of MEMS/NEMS, experimental tests and theoretical analyses have verified that the thermoelastic damping (TED) is a crucial energy dissipation mechanism affecting their performance. ¹³ To achieve a high-quality factor, theoretical prediction of TED is essential for designing the micro/nano-resonators. Theoretically, Zener ¹⁴ originally developed the TED model for a thin beam, who used the classical Fourier law to derive an approximate inverse quality factor. Following Zener’s work, Lifshitz and Roukes ¹⁵ proposed an exact TED solution by complex frequency method. Subsequently, complex frequency method has been extensively applied to the TED analysis of microbeam or microplate. Prabhakar et al. ¹⁶ studied the TED model by considering two-dimensional heat conduction along the length and thickness directions of the microbeam. Nayfeh and Younis ¹⁷ analyzed the TED model for microplates of arbitrary shapes with different boundary conditions. Sun and Saka ¹⁸ solved the TED for a circular plate under arbitrary vibration modes.

It should be noticed that in the TED analyses mentioned above, most of them were considered in the classical thermoelasticity. For the classical thermoelasticity, there exists an inherent nature of the parabolic heat conduction model that the speed for heat transfer is infinite. ¹⁹ However, it is opposite to the experimental test. ²⁰ To address this concern, non-Fourier heat conduction models were developed, such as the Cattaneo-Vernotte (C-V) thermal wave model,^21,22 the Guyer-Krumhansl (G-K) model, ²³ the dual-phase-lag (DPL) model,^24–27 and the three-phase-lag(TPL) model. ²⁸ Compared with other models, the DPL model includes two phase-lag parameters to capture the micro-structural effect of phonon-electron interactions. Meanwhile, it is noted that a high temperature gradient at microscale causes the size-dependent effect in heat transfer progress. ²⁹ To capture the size-dependent effect in heat conduction, Guyer and Krumhansl ²³ developed the G-K model based on the eigenvalue solution of the linearized phonon Boltzmann equation. Subsequently, inspired by DPL and G-K models, Zhou et al. ³⁰ derived the nonlocal dual-phase-lag (NDPL) model and studied the analytical expression of thermoelastic damping in microbeam. Accordingly, with the aid of energy equation, the generalized thermoelastic theories such as Lord-Shulman (L-S) theory,^31–33 Green-Lindsay (G-L) theory, ³⁴ and Green-Naghdi theory ³⁵ were also developed. Later on, Chandrasekharaiah ³⁶ further proposed the DPL generalized thermoelasticity based on the DPL model to describe the micro-structural effect. In view of this, the generalized thermoelastic theories were extensively applied to studying dynamic response^37,38 and the TED analysis of FG microstructures. Based on the classical beam theory and the coupled thermoelastic dynamics, Li et al. ³⁹ studied the TED of free vibrating FGM microbeams with rectangular cross sections. Li and Ma ⁴⁰ investigated the free vibration of FGM microplates with TED by considering the continuous variation of mechanical and thermal properties of FGM plates with thickness. Azizi et al. ⁴¹ proposed a novel functionally graded piezoelectric (FGP) MEMS to investigate the TED of transverse vibration in FGP microbeam resonators.

Experimental results indicated that when the characteristic geometrical size of the structures belongs to the micro/nanoscale, the observed stiffness and strength of the micro/nano structures increase, which is called as size-dependent effect.^42,43 Nevertheless, the classical continuum theory fails in depicting such effect in the micro/nano-structures due to lacking of internal length scale parameters. ⁴⁴ To capture the size-dependent effect, three main higher-order continuum theories including the nonlocal theory, ⁴⁵ the modified strain gradient theory (MSGT), ⁴⁶ and the modified couple stress theory (MCST) ⁴⁷ were formulated. The MSGT includes three material length scale parameters. It is difficult to obtain all the material length scale parameters of microstructures through experiments. Subsequently, by ignoring the symmetric curvature tensor, the MCST with only one material length scale parameter was proposed by Yang et al. ⁴⁷ Nevertheless, the constitutive equations of the MSGT and the MCST are relatively complex. ⁴⁸ As one of the widely used theories, the nonlocal theory considers the long-range force between atoms. In this theory, the stress at a certain point depends on the strains of all points in the body instead of only the strain at this point. So far, several studies were performed to incorporate the nonlocal theory in TED analysis,^49,50 dynamic response,^51–54 and vibration^55,56 of micro/nano-structures.

According to the above literatures, although there are quite a few works concerning the TED of microstructure,^{39–41,49,50} it can be concluded that the TED analysis of the FG sandwich microbeam considering both the size-dependent effects of the heat conduction and the stress-strain relation is barely reported. In fact, with the development of micro-scale devices, the size-dependent effect on elastic deformation can’t be ignored. ³⁷ Meanwhile, the size-dependent effect and the micro-structural effect of heat conduction must be incorporated, because the micro-devices are inevitably encountered the non-uniform temperature environment or generated by themselves during operation.^23–29 On the other hand, FG sandwich microstructures can effectively reduce the delamination phenomena compared to the traditional sandwich microstructures. Enlightened by these, the present work aims to investigate a DPL thermoelastic model based on the nonlocal elasticity theory and the nonlocal heat conduction theory. Then, this model is applied to studying the TED of a FG sandwich microbeam resonator. The corresponding governing equations are derived and the analytical solution for the TED by complex frequency method is obtained. In numerical evaluation, the influences of the modified parameters on the TED are examined and discussed in detail. It is hoped that the work may provide some guidelines to accurately predict the TED for FG sandwich microstructures in a thermal environment.

Theory and formulation

Nonlocal dual-phase-lag (NDPL) heat conduction model

With the aid of the linearized Boltzmann transport equation, a nonlocal heat conduction model was proposed by Guyer and Krumhansl ²³ as follows.

$$(1+{\tau }_q\frac{\partial }{\partial t}-{\xi }^2{\nabla }^2)\mathbf{q}(\mathbf{x},t)=-\kappa \nabla \theta (\mathbf{x},t)$$

(1)

where $\mathbf{q}$ is the heat flux vector, $\mathbf{x}$ is the position vector, ∇ is the gradient operator, $\theta$ is the temperature increment, $\xi$ is the thermal nonlocal parameter, $\kappa$ is the thermal conductivity, and ${\tau }_q$ is the thermal phase lag of the heat flux.

Considering the effect of thermal phase lag of the temperature gradient, Zhou et al. ³⁰ developed a NDPL heat conduction model which can be written as

$$(1+{\tau }_q\frac{\partial }{\partial t}-{\xi }^2{\nabla }^2)\mathbf{q}(\mathbf{x},t)=-\kappa (1+{\tau }_{\theta }\frac{\partial }{\partial t})\nabla \theta (\mathbf{x},t)$$

(2)

where ${\tau }_{\theta }$ is the thermal phase lag of the temperature gradient.

Nonlocal stress-strain relations

According to the nonlocal elasticity, ⁴⁵ the constitutive relations are defined as follows

$${\sigma }_{\mathrm{ij}}(\mathbf{r})={\int }_{V}K(|\mathbf{r}-\mathbf{r}\mathbf{\prime }|,\chi ){\sigma \prime }_{\mathrm{ij}}(\mathbf{r}\mathbf{\prime })\mathrm{d}V(\mathbf{r}\mathbf{\prime })$$

(3)

$$\sigma {\prime }_{\mathrm{ij}}(\mathbf{r}\mathbf{\prime })=\lambda {\epsilon }_{\mathrm{kk}}(\mathbf{r}\mathbf{\prime }){\delta }_{\mathrm{ij}}+2\mu {\epsilon }_{\mathrm{ij}}(\mathbf{r}\mathbf{\prime })$$

(4)

$${\epsilon }_{\mathrm{ij}}(\mathbf{r}\mathbf{\prime })=\frac{1}{2}(\frac{\partial u_j(\mathbf{r}\mathbf{\prime })}{\partial {r\prime }_i}+\frac{\partial u_i(\mathbf{r}\mathbf{\prime })}{\partial {r\prime }_j})$$

(5)

where ${\sigma }_{\mathrm{ij}}\left(\mathbf{r}\right)$ and $\sigma {\prime }_{\mathrm{ij}}(\mathbf{r}\mathbf{\prime })$ are the nonlocal and the local stress components respectively, $\mathbf{r}$ is the reference position vector, $\mathbf{r}\mathbf{\prime }$ is the other position vector, $|\mathbf{r}-\mathbf{r}\mathbf{\prime }|$  is the Euclidean distance, ${\epsilon }_{\mathrm{ij}}(\mathbf{r}\mathbf{\prime })$ is the classical strain component, ${\epsilon }_{\mathrm{kk}}(\mathbf{r}\mathbf{\prime })$ is the local cubic dilation, $u_i(\mathbf{r}\mathbf{\prime })$ is the local displacement component, $\lambda$ and $\mu$ are Lamé’s constants. $K(|\mathbf{r}-\mathbf{r}\mathbf{\prime }|,\chi )$ is the kernel function, $\chi =e_{0}a/l$ is the material constant, where $a$ is the internal characteristic length, $l$ is the external characteristic length, $e_0$ is a material dependent constant, and $e_{0}a$ is the nonlocal parameter.

The integral form of equation (3) is difficult to solve. Eringen ⁴⁵ proposed that when the kernel function was represented by a Green’s function of linear differential operator $\mathcal{L}$ as

$$\mathcal{L}K(|\mathbf{r}-\mathbf{r}\mathbf{\prime }|,\chi )=\delta \left(|\mathbf{r}-\mathbf{r}\mathbf{\prime }|\right)$$

(6)

Then, equation (3) can be simplified into differential form

$$[1-{(e_{0}a)}^2{\nabla }^2]{\sigma }_{\mathrm{ij}}(\mathbf{r})=\sigma {\prime }_{\mathrm{ij}}(\mathbf{r}\mathbf{\prime })$$

(7)

Dual-phase-lag thermoelastic model with double nonlocal effects

To accurately describe the TED of microstructures, the nonlocal effects relating to both the heat conduction and the elastic deformation are further considered in generalized thermoelasticity. Inspired by the works,^30,45 the basic equations are presented below.

The energy conservation equation can be expressed as ³⁰

$$-\nabla ·\mathbf{q}=\rho c_E\frac{\partial \theta }{\partial t}+T_0\gamma \frac{\partial {\epsilon }_{\mathrm{kk}}}{\partial t}$$

(8)

where $\rho$ , $c_E$ , ${\epsilon }_{\mathrm{kk}}$ , and $T_0$ represent the material density, the specific heat at constant volume, the volumetric strain and the reference temperature, respectively. $\gamma =(3\lambda +2\mu ){\alpha }_T$ , where ${\alpha }_T$ represents the coefficient of linear thermal expansion. The Lamé’s constants can be defined as ⁵²

$$\lambda =\frac{\mathrm{Ev}}{(1+v)(1-2v)},\mu =\frac{E}{2(1+v)}$$

(9)

Combining equations (2) and (8) yields the energy equation

$$\begin{array}{c}\multicolumn{1}{c}{\kappa (1+{\tau }_{\theta }\frac{\partial }{\partial t}){\nabla }^2\theta =(1+{\tau }_q\frac{\partial }{\partial t}-{\xi }^2{\nabla }^2)}\\ \multicolumn{1}{c}{(\rho c_E\frac{\partial \theta }{\partial t}+T_0\gamma \frac{\partial {\epsilon }_{\mathrm{kk}}}{\partial t})}\end{array}$$

(10)

The constitutive equation with nonlocal effect ⁴⁵

$$[1-{\left(e_{0}a\right)}^2{\nabla }^2]{\sigma }_{\mathrm{ij}}=2\mu {\epsilon }_{\mathrm{ij}}+\lambda e_{\mathrm{kk}}{\delta }_{\mathrm{ij}}-\gamma \theta {\delta }_{\mathrm{ij}}$$

(11)

The displacement-strain relation ⁴⁵

$${\epsilon }_{\mathrm{ij}}=\frac{1}{2}(u_{i,j}+u_{j,i})$$

(12)

Substituting equation (9) into equation (11) leads to

$$\begin{array}{c}\multicolumn{1}{c}{[1-{\left(e_{0}a\right)}^2{\nabla }^2]{\sigma }_{\mathrm{ij}}=\frac{\mathrm{Ev}}{(1+v)(1-2v)}{\epsilon }_{\mathrm{kk}}{\delta }_{\mathrm{ij}}+\frac{E}{1+v}{\epsilon }_{\mathrm{ij}}}\\ \multicolumn{1}{c}{-\frac{E{\alpha }_T}{1-2v}\theta {\delta }_{\mathrm{ij}}}\end{array}$$

(13)

where E and $v$ are the elastic modulus and the Poisson’s ratio.

Formulation of the problem

Modeling of FG sandwich microbeam

As shown in Figure 1, the three-layer sandwich microbeam is made of two materials, that is, the FG-layer $h_1(h_3)$ and the homogeneous layer $h_2$ . The FG-sandwich microbeam has length $l$ , width $b$ , and thickness $h=h_1+h_2+h_3$ . Furthermore, the mechanical properties in $h_3$ and $h_1$ are symmetric about the mid-plane in the FG sandwich microbeam.

Figure 1.

Schematic drawing of the FG-sandwich microbeam.

Material properties of FG sandwich microbeam

The mechanical properties of the sandwich microbeam are assumed that the two materials are distributed functionally graded in the top and bottom layer, respectively. So, the mechanical properties $P$ of the FG sandwich microbeam follow the mixture rule ³

$$P=\{\begin{array}{c}{P}_1{V}_1+P_2{V}_{2}0\le z\le \frac{h}{4}\hfill \\ \multicolumn{1}{c}{P_2\frac{h}{4}<z<\frac{3h}{4}}\\ \multicolumn{1}{c}{P_1{V}_1+P_2{V}_2\frac{3h}{4}\le z\le h}\end{array}$$

(14)

where $P$ can be the Young’s modulus $E$ , the Poission’s ratio $\nu$ , the mass density $\rho$ , the thermal conductivity $\kappa$ , the specific heat capacity $C_E$ , or the thermal expansion ${\alpha }_T$ . The volume content fractions of two compositional materials $V_1$ and $V_2$ in the FG sandwich microbeam are defined as the power function model ⁴

$$V_1=\{\begin{array}{c}{\left(\frac{h_1-z}{h_1}\right)}^{k}0\le z\le \frac{h}{4}\hfill \\ \multicolumn{1}{c}{0\frac{h}{4}<z<\frac{3h}{4}}\\ \multicolumn{1}{c}{1-{\left[\frac{h_3-z}{h_3}\right]}^k\frac{3h}{4}\le z\le h}\end{array}$$

(15a)

$$V_2=1-V_1$$

(15b)

in which $k$ is the power-law index. The fraction $V_1$ of material 1 changing with the exponent $k$ in the FG sandwich microbeam is shown in Figure 2. At a certain thickness, the bigger the exponent is, the lower fraction of material 1 as well as the less content of material 1 is in the FG sandwich microbeam.

Figure 2.

The fraction $V_1$ of material 1 versus thickness $h$ in the FG sandwich microbeam.

Derivation of the basic governing equations

Based on the Euler-Bernoulli beam theory, the displacement components are taken as^49,50

$$u_x=-z\frac{\partial w}{\partial x},u_y=0,u_z=w(x,t)$$

(16)

Referring to equation (12), the ${\epsilon }_{\mathrm{xx}}$ can be given by ⁵⁰

$${\epsilon }_{\mathrm{xx}}=-z\frac{{\partial }^{2}w}{\partial x^2}$$

(17)

The ${\epsilon }_{\mathrm{yy}}$ and ${\epsilon }_{\mathrm{zz}}$ are given as ⁴⁹

$${\epsilon }_{\mathrm{yy}}={\epsilon }_{\mathrm{zz}}=\mathrm{vz}\frac{{\partial }^{2}w}{\partial x^2}+(1+v){\alpha }_T\theta$$

(18)

Then, the volumetric strain ${\epsilon }_{\mathrm{kk}}$ can be obtained as

$${\epsilon }_{\mathrm{kk}}={\epsilon }_{\mathrm{xx}}+{\epsilon }_{\mathrm{yy}}+{\epsilon }_{\mathrm{zz}}=(2v-1)z\frac{{\partial }^{2}w}{\partial x^2}+2(1+v){\alpha }_T\theta$$

(19)

Substituting equation (17) into equation (13), then neglecting the Poisson’s effect arrives at

$$[1-{\left(e_{0}a\right)}^2\frac{{\partial }^2}{\partial x^2}]{\sigma }_{\mathrm{xx}}=E({\epsilon }_{\mathrm{xx}}-{\alpha }_T\theta )=E(-z\frac{{\partial }^{2}w}{\partial x^2}-{\alpha }_T\theta )$$

(20)

The motion equation is written as ⁵²

$$\frac{{\partial }^{2}M}{\partial x^2}=\rho A\frac{{\partial }^{2}w}{\partial t^2}$$

(21)

where $M$ is the bending moment defined by

$$M=\int \underset{A}{\int }{\sigma }_{x}z\mathrm{d}A$$

(22)

Multiplying equation (20) by z and integrating over the cross-section area, the bending moment can be obtained as

$$M(x,t)={\left(e_{0}a\right)}^2\frac{{\partial }^{2}M}{\partial x^2}-\mathrm{EI}\frac{{\partial }^{2}w}{\partial x^2}-E{\alpha }_T{M}_T$$

(23)

where $I=b{h}^3/12$ is the inertial moment of the cross-section and $M_T$ is the thermal moment defined by

$$\begin{array}{c}\multicolumn{1}{c}{M_T=E{\alpha }_{T}b{\int }_{-h/2}^{h/2}\theta z\mathrm{d}z}\end{array}$$

(24)

Introducing equation (23) into equation (21) arrives at

$$M(x,t)={\left(e_{0}a\right)}^2\rho A\frac{{\partial }^{2}w}{\partial t^2}-\mathrm{EI}\frac{{\partial }^{2}w}{\partial x^2}-M_T$$

(25)

Combining equations (21) and (25), the motion equation is obtained as

$$\{\mathrm{EI}\frac{{\partial }^4}{\partial x^4}+\rho A\frac{{\partial }^2}{\partial t^2}[1-{\left(e_{0}a\right)}^2\frac{{\partial }^2}{\partial x^2}]\}w+\frac{{\partial }^2{M}_T}{\partial x^2}=0$$

(26)

For thin beams, the gradient of temperature along x direction is negligible, ⁴⁹ so equation (10) can be further expressed as

$$\begin{array}{c}\multicolumn{1}{c}{\kappa (1+{\tau }_{\theta }\frac{\partial }{\partial t})\frac{{\partial }^2\theta }{\partial z^2}=(1+{\tau }_q\frac{\partial }{\partial t}-{\xi }^2\frac{{\partial }^2}{\partial z^2})}\\ \multicolumn{1}{c}{(\rho c_E\frac{\partial \theta }{\partial t}+T_0\gamma \frac{\partial {\epsilon }_{\mathrm{kk}}}{\partial t})}\end{array}$$

(27)

Substituting equation (19) into equation (27) yields

$$\begin{array}{c}\multicolumn{1}{c}{\chi (\frac{{\partial }^2\theta }{\partial z^2}+{\tau }_{\theta }\frac{{\partial }^3\theta }{\partial z^2\partial t})+\frac{{\Delta }_E}{{\alpha }_T}}\\ \multicolumn{1}{c}{z(\frac{{\partial }^{3}w}{\partial x^2\partial t}+{\tau }_q\frac{{\partial }^{4}w}{\partial x^2\partial t^2}-{\xi }^2\frac{{\partial }^{5}w}{\partial x^2\partial z^2\partial t})}\\ \multicolumn{1}{c}{=(1+2{\Delta }_E\frac{(1+v)}{(1-2v)})(\frac{\partial \theta }{\partial t}+{\tau }_{\theta }\frac{{\partial }^2\theta }{\partial t^2}-{\xi }^2\frac{{\partial }^{3}w}{\partial z^2\partial t})}\end{array}$$

(28)

where $\chi =\kappa /\rho c_E$ and ${\Delta }_E=E{\alpha }_T^2{T}_0/\rho c_E$ . Due to ${\Delta }_E<<1$ , equation (28) can be reduced to

$$\begin{array}{c}\chi (\frac{{\partial }^2\theta }{\partial z^2}+{\tau }_{\theta }\frac{{\partial }^3\theta }{\partial z^2\partial t})+\frac{{\Delta }_E}{{\alpha }_T}z\hfill \\ \multicolumn{1}{c}{(\frac{{\partial }^{3}w}{\partial x^2\partial t}+{\tau }_q\frac{{\partial }^{4}w}{\partial x^2\partial t^2}-{\xi }^2\frac{{\partial }^{5}w}{\partial x^2\partial z^2\partial t})}\\ =\frac{\partial \theta }{\partial t}+{\tau }_{\theta }\frac{{\partial }^2\theta }{\partial t^2}-{\xi }^2\frac{{\partial }^{3}w}{\partial z^2\partial t}\hfill \end{array}$$

(29)

To predict the TED of microbeam resonators, the temperature increment and transverse deflection for harmonic vibration are assumed as ¹⁵

$$w(x,t)=W_0(x)e^{i\omega t},\theta (x,z,t)={\Theta }_0(x,z)e^{i\omega t}$$

(30)

where $\omega$ represents the vibration frequency of the microbeam, $W_0(x),{\Theta }_0(x,z)$  represent complex-valued functions.

Inserting equation (30) into equation (29), one obtains

$$\frac{{\partial }^2{\Theta }_0}{\partial z^2}+p^2{\Theta }_0=p^2\frac{\Delta E}{{\alpha }_T}z\frac{{\mathrm{d}}^2{W}_0}{\mathrm{d}{x}^2}$$

(31)

where $p=\frac{\zeta \eta }{h}-i\frac{\zeta a_2}{h\eta }$ .

and

$$\begin{array}{c}\multicolumn{1}{c}{\zeta =h\sqrt{\frac{\omega }{2\chi }},\eta =\sqrt{a_1+\sqrt{a_1^2+a_2^2}},}\\ \multicolumn{1}{c}{a_1=\frac{[{\tau }_q-({\tau }_{\theta }+\frac{{\xi }^2}{\chi })]\omega }{1+{({\tau }_{\theta }+\frac{{\xi }^2}{\chi })}^2{\omega }^2},a_2=\frac{1+{\tau }_q({\tau }_{\theta }+\frac{{\xi }^2}{\chi }){\omega }^2}{1+{({\tau }_{\theta }+\frac{{\xi }^2}{\chi })}^2{\omega }^2}}\end{array}$$

(31)

The general solution of equation (31) takes the following form ⁴⁹

$${\Theta }_0(x,z)=b_1\sin \left(\mathrm{pz}\right)+b_2\cos \left(\mathrm{pz}\right)+\frac{\Delta E}{{\alpha }_T}z\frac{{\mathrm{d}}^2{W}_0}{\mathrm{d}{x}^2}$$

(32)

where $b_1$ and $b_2$ are arbitrary constants. By assuming the upper and lower surfaces of the microbeam are adiabatic, that is, $\partial {\Theta }_0/\partial z=0$ at $z=\pm \frac{h}{2}$ , the ${\Theta }_0(x,z)$ is given by

$${\Theta }_0(x,z)=\frac{\Delta E}{{\alpha }_T}\frac{{\mathrm{d}}^2{W}_0}{\mathrm{d}{x}^2}(z-\frac{\sin \left(\mathrm{pz}\right)}{p\cos (\mathrm{ph}/2)})$$

(33)

Combining equations (24) and (33), the thermal moment can be further expressed as

$$M_T=\mathrm{EI}{\Delta }_E[1+f\left(\omega \right)]\frac{{\mathrm{d}}^2{W}_0}{\mathrm{d}{x}^2}{e}^{i\omega t}$$

(34)

where $f\left(\omega \right)=\frac{24}{{(\mathrm{hp})}^3}[\frac{\mathrm{hp}}{2}-\tan \left(\frac{\mathrm{hp}}{2}\right)]$ .

Substituting equation (34) into equation (26) gives

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{EI}\{1+{\Delta }_E[1+f\left(\omega \right)]\}\frac{{\mathrm{d}}^4{W}_0}{\mathrm{d}{x}^4}+{\left(e_{0}a\right)}^2\rho A{\omega }^2\frac{{\mathrm{d}}^2{W}_0}{\mathrm{d}{x}^2}}\\ \multicolumn{1}{c}{-\rho A{\omega }^2{W}_0=0}\end{array}$$

(35)

For the isothermal state of microbeam, equation (35) can be written as

$$\mathrm{EI}\frac{{\mathrm{d}}^4{W}_0}{\mathrm{d}{x}^4}+{\left(e_{0}a\right)}^2\rho A{\omega }_0^2\frac{{\mathrm{d}}^2{W}_0}{\mathrm{d}{x}^2}-\rho A{\omega }_0^2{W}_0=0$$

(36)

where ${\omega }_0$ is the isothermal frequency.

The general solution of equation (36) is given by

$$\begin{array}{c}\multicolumn{1}{c}{W_0\left(x\right)=c_1\sin \left({\beta }_{1}x\right)+c_2\cos \left({\beta }_{1}x\right)+c_3\sinh \left({\beta }_{2}x\right)}\\ \multicolumn{1}{c}{+c_4\cosh \left({\beta }_{2}x\right)}\end{array}$$

(37)

where $c_i(i=1-4)$ are unknown coefficients.

Introducing equation (37) into equation (36), one can obtain

$${\omega }_0=\frac{\mathrm{EI}{\beta }_1^4}{\rho A(1+{\left(e_{0}a\right)}^2{\beta }_1^2)}=\frac{\mathrm{EI}{\beta }_2^4}{\rho A(1-{\left(e_{0}a\right)}^2{\beta }_2^2)}$$

(38)

Assume the microbeam is simply supported at both ends, then, the boundary condition can be given as follows

$$W_0\left(0\right)=\frac{{\mathrm{d}}^2{W}_0}{\mathrm{d}{x}^2}\left(0\right)=W_0\left(L\right)=\frac{{\mathrm{d}}^2{W}_0}{\mathrm{d}{x}^2}\left(L\right)=0$$

(39)

Substituting equation (37) into equation (39), one has

$$\sin \left({\beta }_{1}L\right)=0$$

(40)

where ${\beta }_{1}L=n\pi (n=1,2,3...)$ , $n$ is the vibration modes.

By comparing equation (35) with equation (36), the relation between $\omega$ and ${\omega }_0$ can be given by

$$\omega ={\omega }_0\sqrt{1+\Delta E[1+f\left(\omega \right)]}$$

(41)

Due to $\Delta E$ is very small and the value of TED is very weak, ¹⁵ one can expand the right-hand side of equation (41) by Taylor series and the $f\left(\omega \right)$ can be replaced by $f\left({\omega }_0\right)$ as follows

$$\omega ={\omega }_0\sqrt{1+\frac{\Delta E}{2}[1+f\left({\omega }_0\right)]}$$

(42)

The real and imaginary parts of $\omega$ is expressed as

$$\mathrm{Re}\left(\omega \right)={\omega }_0\{1+\frac{{\Delta }_E}{2}+\frac{6{\Delta }_E({\eta }^2-\frac{a_2^2}{{\eta }^2})}{{\zeta }^2{({\eta }^2+\frac{a_2^2}{{\eta }^2})}^2}-\frac{12{\Delta }_E[({\eta }^3-3\frac{a_2^2}{\eta })\sin \left(\zeta \eta \right)+(3a_2\eta -\frac{a_2^3}{{\eta }^3})\sinh \left(\frac{\zeta a_2}{\eta }\right)]}{{\zeta }^3{({\eta }^2+\frac{a_2^2}{{\eta }^2})}^3[\cos \left(\zeta \eta \right)+\cosh \left(\frac{\zeta a_2}{\eta }\right)]}\}$$

(43)

$$\mathrm{Im}\left(\omega \right)=12{\omega }_0{\Delta }_E[\frac{a_2}{{\zeta }^2{({\eta }^2+\frac{a_2^2}{{\eta }^2})}^2}-\frac{(3a_2\eta -\frac{a_2^2}{{\eta }^3})\sin \left(\zeta \eta \right)-({\eta }^3-\frac{3a_2^2}{\eta })\sinh \left(\frac{\zeta a_2}{\eta }\right)}{{\zeta }^3{({\eta }^2+\frac{a_2^2}{{\eta }^2})}^3[\cos \left(\zeta \eta \right)+\cosh \left(\frac{\zeta a_2}{\eta }\right)]}]$$

(44)

The inverse of the quality factor can be given by the frequency method as ¹⁵

$$Q^{-1}=2\left|\frac{\mathrm{Im}\left(\omega \right)}{\mathrm{Re}\left(\omega \right)}\right|$$

(45)

Substituting equations (43) and (44) into equation (45), the expression of the TED is finally expressed as

$$\begin{array}{c}\multicolumn{1}{c}{Q^{-1}=2\left|\frac{\mathrm{Im}\left(\omega \right)}{\mathrm{Re}\left(\omega \right)}\right|=24{\Delta }_E[\frac{a_2}{{\zeta }^2{({\eta }^2+\frac{a_2^2}{{\eta }^2})}^2}-\frac{(3a_2\eta -\frac{a_2^3}{{\eta }^3})\sin \left(\zeta \eta \right)-({\eta }^3-\frac{3a_2^2}{\eta })\sinh \left(\frac{\zeta a_2}{\eta }\right)}{{\zeta }^3{({\eta }^2+\frac{a_2^2}{{\eta }^2})}^3[\cos \left(\zeta \eta \right)+\cosh \left(\frac{\zeta a_2}{\eta }\right)]}]}\end{array}$$

(46)

Furthermore, when the double nonlocal effects and the thermal phase lag of the temperature gradient ${\tau }_{\theta }$  are neglected, the classical TED model can be obtained as

$$Q^{-1}=\frac{6{\Delta }_E}{{\zeta }^2}[1-\frac{\sin \left(\zeta \eta \right)+\sin \left(\zeta \right)}{\zeta [\cosh \left(\zeta \right)+\cos \left(\zeta \right)]}]$$

(47)

Results and discussion

The composite microbeam is made of Si (material 1) and ZrO₂ (material 2). The parameters of the two materials at $T_0=300$ are listed in Table 1.

Table 1.

The material parameters of Si and ZrO₂. ⁵²

Materials	( $\rho$ (kg/m3)	$E$ (GPa)	$\kappa$ [W/(m·K)]	$C_E$ [J/(kg·K)]	${\alpha }_T$ (1/K)	$\nu$
Si	2300	160	150	695	2.6 × 10−6	0.22
ZrO2	3657	244.27	1.71	2.74	12.77 × 10−6	0.29

The other constants are

$$L/h=10,b/h=0.5,{\tau }_{\theta }=0.5{\tau }_q.$$

(47)

The thermal relaxation time of the heat flux ${\tau }_q$ can be given by ⁵⁷

$${\tau }_q=\frac{3\kappa }{\rho c_E{v}^2}$$

(48)

where $v^2$ is the phonon velocity, $v=\sqrt{E/\rho }$ .

Validation

The present theoretical model can be degenerated into the DPL TED model in Gu and He. ⁵⁶ when $k=0$ , $e_{0}a=0$ , and $\xi =0$ . Meanwhile, the material properties, the boundary conditions and the research object are set as the same as those used in Gu and He. ⁵⁶ In the degradation model, the TED in Figure 3 is consistent with those in Gu and He. ⁵⁶

Figure 3.

Comparison between the thermoelastic damping from the degraded model and Ref. ⁵⁶

Parametric analysis

In this section, the influences of the double nonlocal effects on the TED of the FG sandwich microbeam resonator by combining the DPL heat conduction model and the nonlocal elasticity model are studied. Based on the TED model, the effects of different parameters, such as the power-law index, the elastic nonlocal parameter, the thermal nonlocal parameter, and the different modes are discussed in detail.

Figure 4(a)–(c) show the mechanical properties of the FG sandwich microbeam at the internal surfaces of $z=h_1/16$ , $z=h_1/2$ , and $z=15h_1/16$ . The mechanical properties such as the Young’s modulus $E$ , the thermal conductivity $\kappa$ , and the heat capacity $C_E$ are selected. For $z=h_1/16$ , the mechanical properties gradually increase (for $E$ ) or decrease (for $\kappa$ and $C_E$ ) as the exponent $k$ increases, and then reach a certain value of pure ceramic. In which, the Young’s modulus arrives at the determined value earlier than the thermal conductivity and the heat capacity. For $z=h_1/2$ and $z=15h_1/16$ , the mechanical properties sharply increase (for $E$ ) or decrease (for $\kappa$ and $C_E$ ) as the exponent $k$ increases when $k<10$ , and then reach a certain value of pure ceramic in a short exponent range. Consequently, there is the property sensitive to exponent $k$ for the surface close to the homogeneous layer $h_2$ .

Figure 4.

Influence of the mechanical properties versus power-law index $k$ in FG sandwich microbeam: (a) the Young’s modulus $E$ , (b) the thermal conductivity $\kappa$ , and (c) the heat capacity $C_E$ .

Figure 5(a) and (b) illustrate the influence of the elastic nonlocal parameter $e_{0}a$ on TED (scaled by ${\Delta }_E$ ) versus length $L$ in FG sandwich microbeam by taking the thermal nonlocal parameter as $\xi =0$ . It can be found that the variation of TED with respect to length $L$ exists a peak value for different elastic nonlocal parameters $e_{0}a$ . As observed, with the increase of the elastic nonlocal parameter $e_{0}a$ , the peak values move toward the right end of the FG sandwich microbeam. Furthermore, the elastic nonlocal parameter $e_{0}a$ leads to a larger area of the axis enclosed by the curve. This means that the elastic nonlocal parameter $e_{0}a$ can increase the vibration resistance range of the FG sandwich microbeam and make the stiffness of the FG sandwich microbeam enhanced. It is consistent with the definition of nonlocal elastic model. That is, the stress at a point depends on the strains of all points in the body. ⁴⁵ Thus, it leads to the vibration resistance range increases. On another front, the peak value at $z=15h_1/16$ is higher than the peak value at $z=h_1/16$ on the TED. This phenomenon is caused by the strong vibration performance near the ceramic internal surface. Consequently, the value of the $Q^{-1}/\Delta E$ at $z=15h_1/16$ becomes larger.

Figure 5.

Influence of the elastic nonlocal parameter $e_{0}a$ on $Q^{-1}/\Delta E$ versus length $L$ in FG sandwich microbeam: (a) $z=h_1/16$ and (b) $z=15h_1/16$ .

The variation of TED (scaled by ${\Delta }_E$ ) versus elastic nonlocal parameter $e_{0}a$ for different internal surface $z$ in FG sandwich microbeam by fixing the thermal nonlocal parameter as $\xi =0$ is shown in Figure 6. For $L=100\mathrm{nm}$ , with the increase of the elastic nonlocal parameter $e_{0}a$ , the values of the TED increase. Meanwhile, a bigger internal surface $z$ acts to increase the values of the TED. Because $z=15h_1/16$ is close to the ceramic material, the ceramic internal surface exhibits strong vibrational performance, which leads to an increase in the strong vibrational performance of the FG sandwich microbeam. This view coincides with that in Figure 5. This indicates that elastic nonlocal parameter $e_{0}a$ can improve the vibration performance of FG sandwich microbeams, and the vibration performance is further enhanced with the increase of the internal surface $z$ .

Figure 6.

Influence of the internal surface z on $Q^{-1}/\Delta E$ versus elastic nonlocal parameter $e_{0}a$ in FG sandwich microbeam $(L=100\mathrm{nm})$ .

Thermal nonlocal parameter $\xi$ is employed to analyze the effect of the thermal nonlocal effect on the TED (scaled by ${\Delta }_E$ ) by taking the elastic nonlocal parameter as $e_{0}a=0$ , and the result is displayed in Figure 7. It can be observed that the values of TED decrease with the increase of the thermal nonlocal parameter $\xi$ . Thermal nonlocal parameter $\xi$ can reduce the peak value of TED in FG sandwich microbeam, resulting in an improved quality factor.

Figure 7.

Influence of the thermal nonlocal parameter $\xi$ on $Q^{-1}/\Delta E$ versus thickness $h$ in FG sandwich microbeam.

Figure 8(a) and (b) demonstrate the influence of internal surface $z$ on TED (scaled by ${\Delta }_E$ ) with thermal nonlocal parameter $\xi$ in FG sandwich microbeam by fixing the elastic nonlocal parameter as $e_{0}a=0$ . As shown, the TED values decrease with the thermal nonlocal parameter increases. It means that the thermal nonlocal parameter $\xi$ can reduce the energy dissipation of the FG sandwich microbeam. On the flip side, the values at $z=15h_1/16$ are higher than the values at $z=h_1/16$ on the TED. The phenomenon is similar to that in Figure 5.

Figure 8.

Influence of the internal surface z on $Q^{-1}/\Delta E$ versus thermal nonlocal parameter $e_{0}a$ in FG sandwich microbeam $(h=10\mu m)$ : (a) $z=h_1/16$ and (b) $z=15h_1/16$ .

Figure 9 plots the influence of the power-law index $k$ on the ratio of thermoelastic damping in ceramic material $Q_k^{-1}/Q_c^{-1}$ when different values are taken in FG material by taking the thermal nonlocal parameter as $\xi =0$ and the elastic nonlocal parameter as $e_{0}a=0$ . In the range of 0– $1.6\mu m$ for thickness $h$ , the compositions of the ceramic increase with the increase of the power-law index $k$ . This means that the vibration of the FG sandwich microbeam becomes stronger, increasing the TED. In Figure 10, the variations of TED (scaled by ${\Delta }_E$ ) with thickness $h$ for the different power-law indexes. When $h=1.6\mu m$ , the FG sandwich microbeam reaches a peak value. After the TED reaches the peak value, the compositions of silicon decrease with the increase of the power-law index $k$ . The values of the $Q_k^{-1}/Q_c^{-1}$ decrease due to small energy storage in Figure 9. It shows that the power-law index $k$ decreases the energy storage of the FG sandwich microbeam, which leads to a decrease in TED. In summary, the TED is weakened due to the existence of the power-law index $k$ .

Figure 9.

Influence of the power-law index $k$ on $Q_k^{-1}/Q_c^{-1}$ versus thickness $h$ in FG sandwich microbeam.

Figure 10.

Influence of the power-law index $k$ on $Q_k^{-1}/{\Delta }_E$ versus thickness $h$ in FG sandwich microbeam.

Figure 11 analyzes the impact of the mode on TED (scaled by ${\Delta }_E$ ) versus thickness $h$ in FG sandwich microbeam by fixing the thermal nonlocal parameter as $\xi =0$ and the elastic nonlocal parameter as $e_{0}a=0$ .With the increase of the mode of vibration $n$ , the maximum value of the TED decreases. It is seen that the peak values of the TED corresponding to third, fourth, and fifth modes occur at $h=1\mu m$ , $h=1.3\mu m$ , and $h=1.7\mu m$ , respectively. The influence of high mode shape on the structural vibration is obvious at the initial stage of vibration, and then gradually attenuates. Because the higher the mode shape is, the faster under the effect of TED attenuation is. Thus, it is obvious that the vibration mode has an influence on the peak value.

Figure 11.

Influence of the different modes $n$ on $Q_k^{-1}/{\Delta }_E$ versus thickness $h$ in FG sandwich microbeam.

Conclusions

In this work, a DPL generalized thermoelasticity with two fundamentally distinct size-dependent effect models which govern the nonlocal features of heat transport and elastic deformation is developed. Then, the model is applied to studying the TED of a FG sandwich microbeam. From the obtained results, it can be concluded

1. The nonlocal parameter in the FG sandwich microbeam shows remarkable effect on size-dependent TED, and it is also found that properly selected nonlocal parameter can adjust the vibration resistance range in the FG sandwich microbeam. This indicates that the nonlocal effect can make a certain impact on the strengthening or weakening of the FG sandwich microbeam stiffness.

2. The introduction of the thermal nonlocal parameter acts to decrease the peak value of the TED. It can be confirmed that the nonlocal heat conduction model can improve the quality factor in the FG sandwich microbeam by suitably tuning the thermal nonlocal parameter and reduce the energy dissipation of the FG sandwich microbeam.

3. As the power-law index $k$ increases, the compositions of the ceramic increase, thereby, increasing the TED value. After the TED reaches the peak value, the decrease of energy storage in FG sandwich microbeam leads to the decrease of TED. This means the existence of the power-law index $k$ causes the TED to be weakened.