Vibration of circular micro-ceramic (Si 3 N 4 ) plate resonators in the context of the generalized viscothermoelastic dual-phase-lagging theory

Abstract

In this article, the analysis and numerical results are represented for the thermoelastic of an isotropic homogeneous, thermally conducting, Kelvin–Voigt-type circular micro-plate in the context of Kirchhoff’s Love plate theory of generalized viscothermoelasticity based on the dual-phase-lagging model. The governing equations are obtained for the generalized dual-phase-lagging model and coupled viscothermoelastic plates. The scaled viscothermoelasticity has been illustrated in the case of the circular plate and the axisymmetric circular plate for an aspect ratio for clamped boundary conditions. Laplace transform has been applied, and its inversions have been calculated numerically by using the Tzou method. The results have been carried out for the ceramic (Si₃N₄). It is noted that the temperature increment and lateral deflection are significantly affected by the time, the width, the thickness, and the mechanical relaxation times of the material.

Introduction

Heat conduction has been studied using mathematical models such as dual-phase lag (DPL), which was proposed by Tzou.^1,2 The temperature gradient and heat flux were established by this model. Many scientists used this model in heat transfer problems, ³ physical systems,^4–8 and thermoelastic damping vibration.^9,10 Guo et al.^11,12 used the DPL model to analyze the thermoelastic damping theory of micro- and nanomechanical resonators; then, he investigated the dissipation in the circular micro-plate resonator.

The circular plate is a common structural in many micro-and nano-electromechanical resonators. Hao ¹³ adopted an analytical study to analyze thermoelastic damping in vibrations of micro- and nano-electromechanical circular thin-plate resonators. Sun and Tohmyoh ¹⁴ studied thermoelastic damping on axisymmetric out-of-plane vibration of circular plate resonators. Sun and Saka ¹⁵ investigated the thermoelastic damping effects on the out-of-plane vibration of circular plate resonators. They added a factor in their formula of thermoelastic damping $K=(1+\upsilon )/(1-2\upsilon )$ , which is different from that of Lifshitz and Roukes, ¹⁶ in which $\upsilon$ is Poisson’s ratio. Li et al. ¹⁷ employed an analytical study to analyze thermoelastic damping in rectangular and circular micro-plate resonators. Grover ¹⁸ established thermoelastic damping for out-of-plane vibrations of a generalized viscothermoelastic circular thin plate.

The study of viscoelastic materials or relaxation properties has become essential in mechanics. The theory of viscothermoelasticity and vibrational principles in thermodynamics has been discussed by Biot.^19,20 Drozdov ²¹ derived a constitutive model for thermoviscoelasticity behavior of polymers at finite strains. Ezzat and El-Karamany ²² used a new model of thermoviscoelasticity for isotropic media to study the relaxation effects of volume properties of viscoelastic materials in thermoelasticity. Carcione et al. ²³ used a numerical algorithm for wave simulations in an elastic medium using the Kelvin–Voigt mechanical model. Grover^18,24,25 studied transverse vibrations in micro-scale viscothermoelastic beam resonators.

Grover ²⁵ studied the closed-form expressions for the transverse vibrations of a homogeneous thermoelastic thin beam with voids in micro/nano-scale. Zhou et al. ²⁶ devolved a model based on the Kirchhoff plate theory and the continuum surface elasticity theory.

In this article, the analysis has been carried out for scaled thermoelastic damping of a homogeneous isotropic, thermally conducting, Kelvin–Voigt-type circular micro-plate based on Kirchhoff’s Love plate theory. A generalized model of the viscothermoelasticity theory of dual-phase-lagging has been used, under clamped boundary conditions for fixed aspect ratios. The analytical results have been validated for a thermoelastic circular plate as a special case, the results for which are available in the literature.

Generalized viscothermoelastic dual-phase-lagging model

We consider a homogeneous isotropic thermally conducting, Kelvin–Voigt-type thin circular plate on the basis of Kirchhoff’s Love plate theory, having origin at the center of the plate of uniform thickness “h” and radius “a” in the cylindrical coordinate system $(r,\theta ,z)$ , initially unstressed, unstrained, and at uniform room temperature “ $T_0$ ”. Assume the mechanical relaxation times ${\lambda }_1$ and ${\mu }_1$ . The time delay “ ${\tau }_q$ ” is known as the thermal relaxation time due to the fast-transient effects of thermal inertia while the time delay “ ${\tau }_T$ ” is known as the time required for thermal activation at micro-scale, caused by the micro-structural interactions such as electron–phonon interaction or phonon scattering (non-classical dual-phase-lagging model). The neutral surface is kept on the $(r,\theta )$ plane, and the z-axis taken normal to the neutral surface, as shown in Figure 1. ²⁷

Figure 1.

Circular micro-plate resonator.

Hence, the displacement components are as follows ¹⁸

$$u_r(r,\theta ,z;t)=-z\frac{\partial w(r,\theta ;t)}{\partial r},u_{\theta }(r,\theta ,z;t)=-\frac{z}{r}\frac{\partial w(r,\theta ;t)}{\partial \theta },u_z=w(r,\theta ;t)$$

(1)

The temperature increment is as follows

$$T(r,\theta ,z;t)=T_1(r,\theta ,z;t)-T_0$$

(2)

The equation of motion is as follows^27–29

$$\frac{{\partial }^2{M}_r}{\partial r^2}+\frac{2}{r}\frac{{\partial }^2{M}_{r\theta }}{\partial r\partial \theta }+\frac{2}{r}\frac{\partial M_r}{\partial r}-\frac{1}{r}\frac{\partial M_{\theta }}{\partial r}+\frac{2}{r^2}\frac{\partial M_{r\theta }}{\partial \theta }+\frac{1}{r^2}\frac{{\partial }^2{M}_{\theta }}{\partial {\theta }^2}=\rho h\stackrel{··}{w}$$

(3)

The moments of flexure and torsion are as follows

$$M_r=\underset{-h/2}{\overset{h/2}{\int }}{\sigma }_{r}z\mathrm{dz}=-D[\frac{{\partial }^{2}w}{\partial r^2}+\upsilon (\frac{1}{r}\frac{\partial w}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}w}{\partial {\theta }^2})+(1+\upsilon ){\alpha }_T{M}_T]$$

(4)

$$M_{\theta }=\underset{-h/2}{\overset{h/2}{\int }}{\sigma }_{\theta }z\mathrm{dz}=-D[\upsilon \frac{{\partial }^{2}w}{\partial r^2}+(\frac{1}{r}\frac{\partial w}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}w}{\partial {\theta }^2})+(1+\upsilon ){\alpha }_T{M}_T]$$

(5)

and

$$M_{r\theta }=\underset{-h/2}{\overset{h/2}{\int }}{\tau }_{r\theta }z\mathrm{dz}=-D\left[(1-\upsilon )\frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial w}{\partial r}\right)\right]$$

(6)

The thermal moment is as follows

$$M_T=\frac{12}{h^3}\underset{-h/2}{\overset{h/2}{\int }}Tz\mathrm{dz}$$

(7)

where

$$D=\frac{h^3{E}^{*}}{12(1-{\upsilon }^2)},E^{*}=\frac{{\mu }^{*}(3{\lambda }^{*}+2{\mu }^{*})}{({\lambda }^{*}+{\mu }^{*})}$$

(8)

and λ* and μ* are Lamè parameters, and α_T is the coefficient of thermal expansion.

The stress components are as follows^{18,27,28,30–32}

$${\sigma }_r=\frac{E}{1-{\upsilon }^2}[{\epsilon }_r+\upsilon {\epsilon }_{\theta }-(1+\upsilon ){\alpha }_{T}T]$$

(9)

$${\sigma }_{\theta }=\frac{E}{1-{\upsilon }^2}[{\epsilon }_{\theta }+\upsilon {\epsilon }_r-(1+\upsilon ){\alpha }_{T}T]$$

(10)

$${\tau }_{r\theta }=2{\mu }^{*}{\epsilon }_{r\theta }$$

(11)

The strain components are as follows

$${\epsilon }_r=\frac{\partial u_r}{\partial r}=-z\frac{{\partial }^{2}w}{\partial r^2}$$

(12)

$${\epsilon }_{\theta }=\frac{u_r}{r}+\frac{1}{r}\frac{\partial u_{\theta }}{\partial \theta }=-z(\frac{1}{r^2}\frac{{\partial }^{2}w}{\partial {\theta }^2}+\frac{1}{r}\frac{\partial w}{\partial r})$$

(13)

$${\epsilon }_{r\theta }=\frac{1}{2r}(u_{\theta }-\frac{\partial u_r}{\partial \theta })+\frac{\partial u_{\theta }}{\partial r}=-z\frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial w}{\partial \theta }\right)$$

(14)

The generalized heat conduction equation in the context of the viscothermoelastic DPL model is given by^11,27

$$K[1+{\tau }_T\frac{\partial }{\partial t}]({\nabla }^2+\frac{{\partial }^2}{\partial z^2})T=\rho C_{\upsilon }(\frac{\partial }{\partial t}+{\tau }_q\frac{{\partial }^2}{\partial t^2})T-\beta T_{0}z(\frac{\partial }{\partial t}+{\tau }_q\frac{{\partial }^2}{\partial t^2}){\nabla }^{2}w$$

(15)

where $\beta =E^{*}{\alpha }_T/1-2\nu$ , and ${\nabla }^2=({\partial }^2/\partial r^2)+(1/r)(\partial /\partial r)+(1/r^2)({\partial }^2/\partial {\theta }^2)$ .

For the viscoelastic materials, the elastic constants ${\lambda }^{*}\mathrm{and}{\mu }^{*}$ take the forms ¹⁸

$${\lambda }^{*}=\lambda (1+{\lambda }_1\frac{\partial }{\partial t})\mathrm{and}{\mu }^{*}=\mu (1+{\lambda }_1\frac{\partial }{\partial t})$$

(16)

where ${\lambda }_1\mathrm{and}{\mu }_1$ are the mechanical relaxation times.

By substituting equation (16) in the relations on equation (8), we obtain

$$E=E_0(1+E_1\frac{\partial }{\partial t})$$

(17)

and

$$\beta =\frac{{\alpha }_T{E}_0}{1-2\nu }(1+E_1\frac{\partial }{\partial t})$$

(18)

where $E_0=\mu (3\lambda +2\mu )/(\lambda +\mu ),E_1=(((3\lambda {\lambda }_1+2\mu {\mu }_1)/(2\mu +3\lambda ))+{\mu }_1)$ .

Formulation of the problem

In the absence of the body force, the equation of motion (equation (3)) takes the form

$$(1+E_1\frac{\partial }{\partial t}){\nabla }^2{\nabla }^{2}w+{\alpha }_T(1+E_1\frac{\partial }{\partial t}){\nabla }^2{M}_T+\frac{\rho A}{E_{0}I}\stackrel{··}{w}=0$$

(19)

For the axisymmetric micro-beam, the displacement components take the forms ¹⁸

$$u_r(r,\theta ,z;t)=-z\frac{\partial w(r;t)}{\partial r},u_{\theta }(r,z;t)=0,u_z=w(r;t)$$

(20)

and

$${\nabla }^2=\frac{{\partial }^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}$$

(21)

where there is no heat flow across the ends of the beam, so that

$$\frac{\partial T}{\partial z}=0\mathrm{for}z=\pm h/2$$

(22)

Thus, we can consider the temperature increment in the following form.

Because the beam is very thin and assuming the temperature varies regarding a $\sin (\mathrm{pz})$ function along the thickness direction^1–5 we have

$$T(r,z,t)=\phi (r,t)\sin \left(\mathrm{pz}\right)$$

(23)

where $p=\pi /h$ .

Hence, the equation of motion (equation (19)) takes the following form

$$(1+E_1\frac{\partial }{\partial t}){\nabla }^2{\nabla }^{2}w+\frac{12{\alpha }_T}{h^3}(1+E_1\frac{\partial }{\partial t}){\nabla }^2\phi (r,t)\underset{-h/2}{\overset{h/2}{\int }}\sin \left(\mathrm{pz}\right)z\mathrm{dz}+\frac{\rho A}{E_{0}I}\stackrel{··}{w}=0$$

(24)

After doing the integration, we obtain

$$(1+E_1\frac{\partial }{\partial t}){\nabla }^2{\nabla }^{2}w+\frac{24{\alpha }_T}{{\pi }^{2}h}(1+E_1\frac{\partial }{\partial t}){\nabla }^2\phi (r,t)+\frac{\rho A}{E_{0}I}\stackrel{··}{w}=0$$

(25)

By using equations (15) and (23), the heat conduction equation takes the form

$$[1+{\tau }_T\frac{\partial }{\partial t}]({\nabla }^2-p^2)\phi \sin \left(\mathrm{pz}\right)=(\frac{\partial }{\partial t}+{\tau }_q\frac{{\partial }^2}{\partial t^2})[\frac{\rho C_{\upsilon }}{K}\phi \sin \left(\mathrm{pz}\right)-\frac{T_0{\alpha }_T{E}_0}{K(1-2\nu )}(1+E_1\frac{\partial }{\partial t})z{\nabla }^{2}w]$$

(26)

In equation (26), we multiply both sides by z and integrating with respect to z from $-h/2$ to $h/2$ , then we obtain

$$[1+{\tau }_T\frac{\partial }{\partial t}]({\nabla }^2-p^2)\phi =(\frac{\partial }{\partial t}+{\tau }_q\frac{{\partial }^2}{\partial t^2})[\frac{\rho C_{\upsilon }}{K}\phi -\frac{{\alpha }_T{E}_0{T}_0{\pi }^{2}h}{24K(1-2\nu )}(1+E_1\frac{\partial }{\partial t}){\nabla }^{2}w]$$

(27)

where $I=\rho \pi h{a}^4/2,A=\pi a^2$ .

Finally, we have the governing equations as follows

$$(1+E_1\frac{\partial }{\partial t}){\nabla }^2{\nabla }^{2}w+\frac{24{\alpha }_T}{{\pi }^{2}h}(1+E_1\frac{\partial }{\partial t}){\nabla }^2\phi +\frac{2}{E_{0}h{a}^2}\stackrel{··}{w}=0$$

(28)

and

$$(1+{\tau }_T\frac{\partial }{\partial t}){\nabla }^2\phi =p^2\phi +(\frac{\partial }{\partial t}+{\tau }_q\frac{{\partial }^2}{\partial t^2})[\eta \phi -\frac{{\alpha }_T{E}_0{T}_0{\pi }^{2}h}{24K(1-2\nu )}(1+E_1\frac{\partial }{\partial t}){\nabla }^{2}w]$$

(29)

New dimensionless variables will be used as follows^9,33

$$(t\prime ,{\tau \prime }_0)=\eta c_0^2(t,{\tau }_0),\sigma \prime =\frac{\sigma }{E},\phi \prime =\frac{\phi }{T_0},c_0^2=\frac{E}{\rho },(r\prime ,z\prime ,w\prime ,h\prime ,a\prime ,\frac{1}{p\prime })\equiv \eta c_0(r,z,w,h,a,\frac{1}{p})$$

Hence, we obtain

$$(1+E_1\frac{\partial }{\partial t}){\nabla }^2{\nabla }^{2}w+{\alpha }_1(1+E_1\frac{\partial }{\partial t}){\nabla }^2\phi +{\alpha }_2\stackrel{··}{w}=0$$

(30)

and

$$(1+{\tau }_T\frac{\partial }{\partial t}){\nabla }^2\phi =p^2\phi +(\frac{\partial }{\partial t}+{\tau }_q\frac{{\partial }^2}{\partial t^2})[\phi -{\alpha }_3(1+E_1\frac{\partial }{\partial t}){\nabla }^{2}w]$$

(31)

where ${\alpha }_1=24{\alpha }_T{T}_0/{\pi }^{2}h$ , ${\alpha }_2=2c_0/h{a}^2$ , ${\alpha }_3=E_0{\pi }^{2}h{\alpha }_T/24K\eta (1-2\upsilon )$ , $\eta =\rho C_{\upsilon }/K$ .

The primes have been dropped for simplicity.

We will apply the Laplace transform, which is defined as

$$\overline{f}\left(s\right)=L\left[f\left(t\right)\right]=\underset{0}{\overset{\infty }{\int }}f\left(t\right)e^{-st}dt$$

(32)

Hence, we have

$$(1+E_{1}s){\nabla }^2{\nabla }^2\overline{w}+{\alpha }_1(1+E_{1}s){\nabla }^2\overline{\phi }+{\alpha }_2{s}^2\overline{w}=0$$

(33)

and

$$(1+{\tau }_{T}s){\nabla }^2\overline{\phi }=p^2\overline{\phi }+(s+{\tau }_q{s}^2)[\overline{\phi }-{\alpha }_3(1+E_{1}s){\nabla }^2\overline{w}]$$

(34)

which gives

$${\nabla }^2{\nabla }^2\overline{w}+{\alpha }_1{\nabla }^2\overline{\phi }+{\alpha }_4\overline{w}=0$$

(35)

and

$${\nabla }^2\overline{\phi }={\alpha }_5\overline{\phi }-{\alpha }_6{\nabla }^2\overline{w}$$

(36)

$${\alpha }_4=\frac{{\alpha }_2{s}^2}{(1+E_{1}s)},{\alpha }_5=\frac{p^2+(s+{\tau }_q{s}^2)}{(1+{\tau }_{T}s)},{\alpha }_6=\frac{{\alpha }_3(1+E_{1}s)(s+{\tau }_q{s}^2)}{(1+{\tau }_{T}s)}$$

While applying the Laplace transform, we assumed the initial conditions as follows

$$\phi (r,0)=0,w(r,0)=0,{\frac{\partial \phi (r,t)}{\partial t}|}_{t=0}=0,{\frac{\partial w(r,t)}{\partial t}|}_{t=0}=0$$

(37)

Elimination between equations (35) and (36) gives the following equations

$$[{\nabla }^6-({\alpha }_5+{\alpha }_6{\alpha }_1){\nabla }^4+{\alpha }_4{\nabla }^2-{\alpha }_4{\alpha }_5]\overline{w}=0$$

(38)

and

$$[{\nabla }^6-({\alpha }_5+{\alpha }_6{\alpha }_1){\nabla }^4+{\alpha }_4{\nabla }^2-{\alpha }_4{\alpha }_5]\overline{\phi }=0$$

(39)

The bounded solutions of equations (38) and (39) take the forms

$$\overline{w}(r,s)=\sum _{i=1}^3(k_i^2-{\alpha }_5)A_i{I}_0\left(k_{i}r\right)$$

(40)

and

$$\overline{\phi }(r,s)=-{\alpha }_6\sum _{i=1}^3{k}_i^2{A}_i{I}_0\left(k_{i}r\right)$$

(41)

where $I_0{(}^{*})$ is the modified Bessel function of the second kind of order zero and $\pm k_i,i=1,2,3$ are the roots of the characteristic equation

$$k^6-({\alpha }_5+{\alpha }_6{\alpha }_1)k^4+{\alpha }_4{k}^2-{\alpha }_4{\alpha }_5=0$$

(42)

We will consider the edge of the micro-beams $r=a$ is clamped and thermally loaded by thermal shock as follows ²⁷

$$w(a,t)={{\nabla }^{2}w(r,t)|}_{r=a}=0$$

(43)

and

$$\phi (a,t)={\phi }_{o}H\left(t\right)$$

(44)

Applying Laplace transform to the aforementioned boundary conditions, we have

$$\overline{w}(a,s)={{\nabla }^2\overline{w}(r,s)|}_{r=a}=0$$

(45)

and

$$\overline{\phi }(a,s)=\frac{{\phi }_0}{s}$$

(46)

where ${\phi }_0$ is the constant, and H(t) is the unit step function.

Applying the boundary conditions in equations (40) and (41), we get the following system

$$\sum _{i=1}^3(k_i^2-{\alpha }_5)A_i{I}_0\left(k_{i}a\right)=0$$

(47)

$$\sum _{i=1}^3(k_i^2-{\alpha }_5)k_i^2{A}_i{I}_0\left(k_{i}a\right)=0$$

(48)

$$\sum _{i=1}^3{k}_i^2{A}_i{I}_0\left(k_{i}a\right)=-\frac{{\phi }_0}{{\alpha }_{6}s}$$

(49)

By solving the aforementioned system, we get the solution in the Laplace transform domain as follows

$$\begin{array}{c}{A}_1=-\frac{{\phi }_0(k_2^2-{\alpha }_5)(k_3^2-{\alpha }_5)}{{\alpha }_5{\alpha }_{6}s(k_1^2-k_3^2)(k_1^2-k_2^2)I_0\left(k_{1}a\right)},\hfill \\ \multicolumn{1}{c}{A_2=-\frac{{\phi }_0(k_1^2-{\alpha }_5)(k_3^2-{\alpha }_5)}{{\alpha }_5{\alpha }_{6}s(k_2^2-k_1^2)(k_2^2-k_3^2)I_0\left(k_{2}a\right)}}\\ A_3=-\frac{{\phi }_0(k_1^2-{\alpha }_5)(k_2^2-{\alpha }_5)}{{\alpha }_5{\alpha }_{6}s(k_3^2-k_1^2)(k_3^2-k_2^2)I_0\left(k_{3}a\right)}\hfill \end{array}$$

Then, we have the temperature increment in the Laplace transform as follows

$$\overline{T}(r,s)=\frac{{\phi }_0\sin \left(pz\right)}{{\alpha }_{5}s}\left[\begin{array}{c}\frac{k_1^2(k_2^2-{\alpha }_5)(k_3^2-{\alpha }_5)}{(k_1^2-k_3^2)(k_1^2-k_2^2)I_0\left(k_{1}a\right)}{I}_0\left(k_{1}r\right)\hfill \\ \multicolumn{1}{c}{+\frac{k_2^2(k_1^2-{\alpha }_5)(k_3^2-{\alpha }_5)}{(k_2^2-k_1^2)(k_2^2-k_3^2)I_0\left(k_{2}a\right)}{I}_0\left(k_{2}r\right)}\\ \multicolumn{1}{c}{+\frac{k_3^2(k_1^2-{\alpha }_5)(k_2^2-{\alpha }_5)}{(k_3^2-k_1^2)(k_3^2-k_2^2)I_0\left(k_{1}a\right)}{I}_0\left(k_{3}r\right)}\end{array}\right]$$

(50)

Moreover, the lateral deflection $\overline{w}(r,s)$ in the Laplace transform as follows

$$\overline{w}(r,s)=-\frac{{\phi }_0}{{\alpha }_5{\alpha }_{6}s}\left[\begin{array}{c}\frac{(k_1^2-{\alpha }_5)(k_2^2-{\alpha }_5)(k_3^2-{\alpha }_5)}{(k_1^2-k_3^2)(k_1^2-k_2^2)I_0\left(k_{1}a\right)}{I}_0\left(k_{1}r\right)\hfill \\ \multicolumn{1}{c}{+\frac{(k_2^2-{\alpha }_5)(k_1^2-{\alpha }_5)(k_3^2-{\alpha }_5)}{(k_2^2-k_1^2)(k_2^2-k_3^2)I_0\left(k_{2}a\right)}{I}_0\left(k_{2}r\right)}\\ \multicolumn{1}{c}{+\frac{(k_3^2-{\alpha }_5)(k_1^2-{\alpha }_5)(k_3^2-{\alpha }_5)}{(k_3^2-k_1^2)(k_3^2-k_2^2)I_0\left(k_{3}a\right)}{I}_0\left(k_{3}r\right)}\end{array}\right]$$

(51)

To determine the solutions in the time domain, the Riemann-sum approximation method is used to obtain numerical results. In this method, any function in the Laplace domain can be inverted to the time domain as Tzou ²

$$f(t)=\frac{e^{\kappa t}}{t}[\frac{1}{2}\overline{f}\left(\kappa \right)+\mathrm{Re}\sum _{n=1}^N{(-1)}^n\overline{f}(\kappa +\frac{in\pi }{t})]$$

(52)

where Re is the real part and $i=\sqrt{-1}$ is an imaginary number unit. For faster convergence, numerous numerical experiments have shown that the value of $\kappa$ satisfies the relation $\kappa t\approx 4.7$ .

Mathematical software MAPLE 17 has been used to formulate the formula of Tzou (equation (52)) and we calculated the inverse Laplace transform of equations (51) and (52).

In the following numerical calculations, the circular micro-ceramic (Si₃N₄) plate is assumed. The thermal and mechanical properties are given by the following^18,27

$$\begin{array}{c}K=8.0\mathrm{W}/(\mathrm{m}\mathrm{K}),{\alpha }_T=3.0(10{)}^{-6}{\mathrm{K}}^{-1},\hfill \\ \multicolumn{1}{c}{\rho =3200\mathrm{kg}/{\mathrm{m}}^3,T_0=300\mathrm{K},C_{\upsilon }=937.5\mathrm{J}/(\mathrm{kg}\mathrm{K}),}\\ E_0=250\mathrm{GPa},E_1=6.7\times {10}^{-12}\mathrm{s},\mathrm{and}\upsilon =0.44.\hfill \end{array}$$

Figures 2 and 3 represent the temperature increment and the lateral deflection for the non-dimensional parameters $a/h=5,t=20,z=h/2$ , ${\tau }_q=0.02$ , ${\tau }_T=0.9{\tau }_q$ , $E_1=0.21\times {10}^3$ , and for different values of the mechanical relaxation times $E_1=0$ for the thermoelastic materials while $E_1\ne 0$ for the viscothermoelastic materials. It is noted that the effect of the parameter $E_1$ is not significant on the temperature increment distribution, while its effect is significant on the lateral deflection distribution. The values of the lateral deflection distribution decrease on the context of the viscoelastic model.

Figure 2.

The temperature increment distribution for an elastic and viscoelastic material.

Figure 3.

The lateral deflection distribution for an elastic and viscoelastic material.

Figures 4 and 5 represent the temperature increment and the lateral deflection for the non-dimensional parameters $a/h=5,t=20$ , ${\tau }_q=0.02$ , ${\tau }_T=0.9{\tau }_q$ , $E_1=0.21\times {10}^3$ , and for different values of the thickness of the plate z $(z=h/2,z=h/2)$ . It is noted that the effect of thickness of the plate is significant on the temperature increment distribution, where the farther away you are from both sides of the plate, the lower the values of temperature increment. The thickness has minimal effect on the lateral deflection distribution.

Figure 4.

The temperature increment distribution with various values of the thickness z.

Figure 5.

The lateral deflection distribution with various values of the thickness z.

Figures 6 and 7 represent the temperature increment and the lateral deflection for the non-dimensional parameters $a/h=5$ , ${\tau }_q=0.02$ , ${\tau }_T=0.9{\tau }_q$ , $E_1=0.21\times {10}^3$ , and for different values of the time t $(t=20,t=30)$ . It is noted that the effect of the time is significant on the temperature increment distribution and the lateral displacement distribution, where the increase in the value of the time leads to the increase in the values of the temperature increment and the lateral displacement distributions.

Figure 6.

The temperature increment distribution with various values of time t.

Figure 7.

The lateral deflection distribution with various values of time t.

Figures 8 and 9 represent the temperature increment and the lateral deflection for the non-dimensional parameters $t=20$ , ${\tau }_q=0.02$ , ${\tau }_T=0.9{\tau }_q$ , $E_1=0.21\times {10}^3$ , and for different values of the aspect ratio a/h $(a/h=5,a/h=10)$ . It is noted that the effect of that ratio is significant on the temperature increment and the lateral deflection distributions, where the increase in the value of that ratio leads to the increase in the values of the temperature increment and the lateral deflection distributions. The results are accurate and agree with the results in previous studies.^9,34–36

Figure 8.

The temperature increment distribution with various values of the thickness h.

Figure 9.

The lateral deflection distribution with various values of the thickness h.

Conclusion

A mathematical model of thermoelastic of an isotropic homogeneous, thermally conducting, Kelvin–Voigt-type circular micro-plate in the context of Kirchhoff’s Love plate theory of generalized viscothermoelasticity based on dual-phase-lagging has been constructed.

The mechanical relaxation times, the time, and the aspect ratio a/h have significant effects, while the thickness z has a limited effect on the lateral deflection distributions.

The time, the aspect ratio a/h, and the thickness z have significant effects, while the mechanical relaxation times has a limited effect on the temperature increment distributions.