Free vibration of a microbeam resting on Pasternak’s foundation via the Green–Naghdi thermoelasticity theory without energy dissipation

Abstract

This article investigates the effect of length-to-thickness ratio and elastic foundation parameters on the natural frequencies of a thermoelastic microbeam resonator. The generalized thermoelasticity theory of Green and Naghdi without energy dissipation is used. The governing frequency equation is given for a simply supported microbeam resting on Winkler–Pasternak elastic foundations. The influences of different parameters are all demonstrated. Natural vibration frequencies are graphically illustrated and some tabulated results are presented for future comparisons.

Introduction

Nonisothermoelastic structures are of great interest and have become increasingly important in different engineering industries. It is well known that there are many generalized thermoelasticity theories devoted to treat the errors of the classical thermoelasticity theory. In the classical theory, ¹ the heat conduction occurs at an infinite speed so that the heat wave propagation phenomena and other some behaviors cannot be successfully captured. Lord and Shulman ² used one relaxation time parameter into the Fourier heat conduction equation which turns out to be of a hyperbolic type. Green and Lindsay ³ used two relaxation time parameters in their thermoelastic model. Green and Naghdi^4,5 ignored the internal rate of production of entropy in their thermoelasticity theory without energy dissipation. Recently, Zenkour ⁶ presented a unified generalized thermoelasticity theory for the transient thermal shock problem in the context of Green–Naghdi (G–L), Lord–Shulman (L–S), and coupled thermoelasticity (CTE) theories.

The vibration analysis of micro/nano beams in the context of generalized theory of thermoelasticity has been treated by many investigators. The coupled thermoelastic problem of vibration phenomenon during pulsed laser heating of microbeams is investigated by Fang et al. ⁷ and solved using an analytical–numerical technique based on the Laplace transformation. Soh et al. ⁸ developed a generalized solution for the coupled thermoelastic vibration of a microscale beam resonator induced by pulsed laser heating. Guo et al. ⁹ investigated the coupled thermoelastic vibration characteristics of the axially moving beam. Guo and Wang ¹⁰ discussed the thermoelastic coupling vibration characteristics of the axially moving beam with frictional contact. Sharma and Grover ¹¹ derived the transverse vibrations of a homogenous isotropic, thermoelastic thin beam with voids, based on Euler–Bernoulli (E–B) theory have been. Grover ¹² derived the transverse vibrations of a homogenous isotropic, thermally conducting, Kelvin–Voigt-type viscothermoelastic thin beam, based on E–B theory. Belardinelli et al. ¹³ proposed the governing equations of a thermomechanical problem for a slender microbeam subjected to an electric actuation by making use of a unified model. The vibration of nanobeams induced by sinusoidal pulse heating via a nonlocal thermoelastic model is presented by Zenkour and Abouelregal. ¹⁴ The same authors ¹⁵ studied the nonlocal thermoelastic vibrations for variable thermal conductivity nanobeams due to harmonically varying heat. Carrera et al. ¹⁶ studied the vibrational analysis for an axially moving microbeam with two temperatures. Abouelregal and Zenkour ¹⁷ presented the generalized thermoelastic vibration of a microbeam with an axial force. Zenkour et al. ¹⁸ presented the state–space approach for the vibration of nanobeams based on the nonlocal thermoelasticity theory without energy dissipation. Zenkour and Abouelregal ¹⁹ presented the thermoelastic vibration of an axially moving microbeam subjected to sinusoidal pulse heating. Zenkour and Abouelregal ²⁰ discussed the effect of ramp-type heating on the vibration of functionally graded microbeams without energy dissipation.

However, the vibration analysis of micro/nano beams resting on elastic foundations still rare in the literature is the main subject in many investigations.^21–26 Şimşek ²³ proposed analytical and numerical solution procedures for vibration of an embedded microbeam under action of a moving microparticle based on the modified couple stress theory within the framework of E–B beam theory. Akgöz and Civalek ²⁴ investigated the vibration response of nonhomogenous and nonuniform microbeams in conjunction with E–B beam and modified couple stress theory. Şimşek ²⁵ developed a nonclassical beam theory for the static and nonlinear vibration analysis of microbeams based on a three-layered nonlinear elastic foundation within the framework of the modified couple stress theory. Vosoughi ²⁶ presented the nonlinear free vibration of functionally graded nanobeams on nonlinear elastic foundation.

In fact, the free vibration analyses of most structures are investigated using various foundation models by means of both numerical and analytical approaches. However, the vibration problems of such structures on elastic foundations with the inclusion of thermal coupling are not treated before. The present article has been proposed to the first time in its field. To the best of the author’s knowledge, the present article is not available in the literature. The inclusion of thermoelastic coupling in the free vibration of microbeam resting on elastic foundation is treated here for the first time. The present paper deals with the solution of the problem of generalized thermoelastic vibrations of a microbeam resting on two-parameter elastic foundation. The effects of the length-to-thickness ratio as well as the elastic foundation parameters are investigated. The natural frequencies are studied and graphically illustrated. Additional frequencies are tabulated for future comparisons.

The Green–Naghdi thermoelastic theory of microbeam resonators

Consider small flexural deflections of a very thin elastic beam (e.g. a nanotube, a micro/nanobeam, a microtubule) with length $L(0\le x\le L)$ , width $b(-b/2\le y\le b/2)$ , and thickness $h(-h/2\le z\le h/2)$ (Figure 1). Define the x-coordinate along the axis of the beam, with the y- and z-coordinates corresponding to the width and thickness, respectively. In equilibrium, the beam is unstrained, at zero stress and constant temperature T₀ everywhere. The beam undergoes bending vibrations of small amplitude about the x-axis such that the deflection is consistent with the linear E–B theory. That is, any plane cross-section initially perpendicular to the axis of the beam remains plane and perpendicular to the neutral surface during bending. Thus, the displacements are given by

$u=-z\frac{\partial w}{\partial x},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}v=0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}w=w(x,t)$

(1)

where w is the lateral deflection.

Figure 1.

Schematic diagram for the microbeam resting on a two-parameter elastic foundation.

The relevant constitutive equation for the axial stress ${\sigma }_x$ reads

${\sigma }_x=-E(z\frac{\partial 2w}{\partial x2}+{\alpha }_T\theta )$

(2)

where E is Young’s modulus, $\theta =T-T_0$ is the excess temperature with T₀ denoting the constant environmental temperature, ${\alpha }_T={\alpha }_t/(1-2\nu )$ in which ${\alpha }_t$ is the thermal expansion coefficient, and ν is Poisson’s ratio. The interaction between the beam and the supporting foundation follows the two-parameter Pasternak’s model as ²⁷

$R_f=K_{1}w(x,t)-K_2\frac{\partial 2w}{\partial x2}$

(3)

where R_f is the foundation reaction per unit area, K₁ is the Winkler’s foundation stiffness, and K₂ is the shear stiffness of elastic foundation. This model simply reduces to the Winkler’s type when $K_2=0$ .

For transverse vibrations, the corresponding equation of motion reads

$\frac{\partial 2M}{\partial x2}-R_f=\rho A\frac{\partial 2w}{\partial t2}$

(4)

where ρ is the material density and $A=\mathrm{bh}$ is the cross-sectional area. Accordingly, the E–B flexural moment of the cross-section is given, with aid of equation (2), by the expression

$M=-\mathrm{EI}(\frac{\partial 2w}{\partial x2}+{\alpha }_T{M}_T)$

(5)

where $I=\mathrm{bh}3/12$ is the moment of inertia, $\mathrm{EI}$ is the flexural rigidity of the beam, and M_T is the thermal moment defined by

$M_T=\frac{12}{h3}{\int }_{-h/2}^{h/2}\theta (x,z,t)z\text{d}z$

(6)

Substituting equations (3) and (5) into equation (4), one obtains the motion equation of the beam in the form

$(\frac{\partial 4}{\partial x4}-\frac{K_2}{\mathrm{EI}}\frac{\partial 2}{\partial x2}+\frac{\rho A}{\mathrm{EI}}\frac{\partial 2}{\partial t2}+\frac{K_1}{\mathrm{EI}})w+{\alpha }_T\frac{\partial 2M_T}{\partial x2}=0$

(7)

The heat conduction in the context of Green and Naghdi’s generalized thermoelasticity theory without energy dissipation is given by ²⁸

$\kappa \nabla 2\theta +(1+\frac{\partial }{\partial t})(\rho Q*)=\frac{\partial }{\partial t}(\rho C\upsilon \frac{\partial \theta }{\partial t}+\gamma T_0\frac{\partial e}{\partial t})$

(8)

where κ is the thermal conductivity (the material constant characteristic), $C\upsilon$ is the specific heat per unit mass at constant strain, $e={\varepsilon }_{\mathrm{kk}}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}$ is the volumetric strain, $Q*$ is the heat source, and $\gamma ={\alpha }_{t}E/(1-2\nu )$ is the thermoelastic coupling parameter. The corresponding thermal conduction equation for the beam under consideration without heat source $(Q*=0)$ is obtained by specializing equation (8) to present the E–B beam configuration as

$\frac{\partial 2\theta }{\partial x2}+\frac{\partial 2\theta }{\partial z2}=\frac{1}{\kappa }\frac{\partial }{\partial t}[\rho C\upsilon \frac{\partial \theta }{\partial t}-\gamma T_{0}z\frac{\partial }{\partial t}(\frac{\partial 2w}{\partial x2})]$

(9)

Multiplying the above equation by $12z/h3$ , and integrating it with respect to z through the beam thickness from $-h/2$ to $h/2$ , yields

$\frac{\partial 2M_T}{\partial x2}-\frac{12}{h3}\theta {|}_{-h/2}^{+h/2}=\frac{1}{\kappa }\frac{\partial }{\partial t}[\rho C\upsilon \frac{\partial M_T}{\partial t}-\gamma T_0\frac{\partial }{\partial t}(\frac{\partial 2w}{\partial x2})]$

(10)

For the present nanobeam, it is assumed that there is a cubic polynomial variation of temperature increment along the thickness direction. This assumption leads to ²⁹

$M_T=\frac{6}{5h}\theta {|}_{-h/2}^{+h/2}$

(11)

So, at this point equation (10) becomes

$\frac{\partial 2M_T}{\partial x2}-\frac{10}{h2}{M}_T=\frac{1}{\kappa }\frac{\partial }{\partial t}[\rho C\upsilon \frac{\partial M_T}{\partial t}-\gamma T_0\frac{\partial }{\partial t}(\frac{\partial 2w}{\partial x2})]$

(12)

Analytical solution

At this point analytical solutions are sought for the coupled system of equations (7) and (12), along with equation (5) for the bending moment. Concerning the heat conditions of the present microbeam, it is assumed that no heat flow occurs across its upper and lower surfaces (thermally insulated), that is

$\frac{\partial \theta }{\partial z}{|}_{z=\pm h/2}=0$

(13)

However, the nanobeam is subjected to simply supported mechanical conditions at its edges $x=0$ and $x=L$ as

$w=M=0$

(14)

Following the Navier-type solution, the deflection and moment that satisfy the boundary conditions maybe expressed as

$\{w(x,t),M(x,t)\}=\sum _{n=1}^N\{w_n^{*},M_n^{*}\}\sin ({\lambda }_{n}x)\text{e}i\omega t$

(15)

where $w_n^{*}$ and $M_n^{*}$ are arbitrary parameters, ${\lambda }_n=\frac{n\pi }{L}$ , n is a mode number, and ω denotes the eigen-frequency. According to equations (5) and (13), the thermal bending moment has the same behavior form as the bending moment. Substituting equation (5) into equations (7) and (12) gives

$({\lambda }_n^4+\frac{K_2}{\mathrm{EI}}{\lambda }_n^2+\frac{K_1}{\mathrm{EI}}-\frac{\rho A}{\mathrm{EI}}\omega 2)w_n^{*}-{\alpha }_T{\lambda }_n^2{M}_{\mathrm{nT}}^{*}=0$

(16)

${\lambda }_n^2\frac{\gamma T_0}{\kappa }\omega 2w_n^{*}+(\omega 2\frac{\rho C\upsilon }{\kappa }-{\lambda }_n^2-\frac{10}{h2})M_{\mathrm{nT}}^{*}=0$

(17)

where $M_{\mathrm{nT}}^{*}$ is an arbitrary thermal parameter. To get the nontrivial solution of the above equations, the parameter $w_n^{*}$ and $M_{\mathrm{nT}}^{*}$ must be nonzero. Then, the determinate of the coefficients, after integrating through the length with respect to x from 0 to 1, should be vanished. This tends to the frequency equation

$\Omega 4-A_0\Omega 2+A_1{A}_2=0$

(18)

where $\Omega =L\sqrt{\frac{\rho }{E}}\omega$ is the dimensionless frequency parameter and

$A_1=\frac{h2}{12L2}({\overline{\lambda }}_n^4+k_2{\overline{\lambda }}_n^2+k_1),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{A}_2=({\overline{\lambda }}_n^2+\frac{10L2}{h2})\frac{\kappa }{\mathrm{EC}\upsilon },A_0=A_1+A_2+{\overline{\lambda }}_n^4\frac{\gamma {\alpha }_T{T}_{0}h2}{12\rho C\upsilon L2},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{k}_1=\frac{L4K_1}{\mathrm{EI}},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{k}_2=\frac{L2K_2}{\mathrm{EI}},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\overline{\lambda }}_n=n\pi$

(19)

Numerical results and conclusions

Several numerical applications are considered here to put into evidence the influence of the length-to-thickness ratio, the foundation parameters, and the mode number of the minimum frequency. The present nanobeam is made of a silicon material with the following properties ³⁰

$E=165.9\text{GPa},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\nu =0.22,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\rho =2330\text{kg}/\text{m}3,C\upsilon =1.661\text{J}/\text{kgK},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\alpha =2.59\times 10-6/\text{K},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\kappa =156\text{W}/\text{mK}$

(20)

The reference temperature is taken as $T_0=293\text{K}$ . For the sake of completeness, sample results are presented for the dimensionless natural frequencies $\Omega$ to be used for future comparisons. The inclusion of the thermoelastic coupling effect is required to get reliable frequencies. The effects of mode number n, the foundation parameters k₁ and k₂, and the length-to-thickness ratio $L/h$ on the dimensionless natural frequency $\Omega$ are presented in Table 1. The fundamental frequency $(n=1)$ is the smallest one. The frequencies decrease as the ratio $L/h$ increases. However, the frequencies increase with the increase of the mode number n and the foundation parameters k₁ and k₂. In fact, the dimensionless vibration frequency is very sensitive to the variation of the parameters n, $L/h$ , k₁, and k₂.

Table 1.

Effect of mode number n, the foundation parameters k₁ and k₂, and the length-to-thickness ratio $L/h$ on the dimensionless natural frequency $\Omega$ .

			L/h
n	k 1	k 2	5	10	20	50
1	0	0	0.56982	0.28491	0.14246	0.05698
	100	0	0.81119	0.40560	0.20280	0.08112
	100	10	0.99349	0.49674	0.24837	0.09935
2	0	0	2.27929	1.13964	0.56982	0.22793
	100	0	2.35127	1.17564	0.58782	0.23513
	100	10	2.61619	1.30809	0.65405	0.26162
3	0	0	5.12840	2.56420	1.28210	0.51284
	100	0	5.16079	2.58040	1.29020	0.51680
	100	10	5.44010	2.72005	1.36002	0.54401
4	0	0	9.11715	4.55858	2.27929	0.91172
	100	0	9.13541	4.56771	2.28385	0.91354
	100	10	9.41911	4.70955	2.35478	0.94191
5	0	0	14.24555	7.12277	3.56139	1.42455
	100	0	14.25724	7.12862	3.56431	1.42572
	100	10	14.54282	7.27141	3.63570	1.45428

Figures 2 and 3 are prepared by using the dimensionless natural frequencies for a wide range of the length-to-thickness ratio. Figure 2 shows that the fundamental frequency $\Omega$ decreases as $L/h$ increases. However, the inclusion of the elastic foundation causes the increase of $\Omega$ . The frequencies of the microbeam resting on a two-parameter elastic foundation have the greatest values. In Figure 3, the plot the natural frequency Ω versus the length-to-thickness ratio $L/h$ is also illustrated for a microbeam resting on a two-parameter elastic foundation ( $k_1=100$ and $k_2=15$ ). Once again the natural frequency decreases with the decrease of $L/h$ and the mode number n.

Figure 2.

The fundamental frequency $\Omega$ versus the length-to-thickness ratio $L/h$ for various elastic foundations.

Figure 3.

The natural frequency $\Omega$ versus the length-to-thickness ratio $L/h$ for $k_1=100$ and $k_2=15$ .

In Figures 4 and 5, the aspect ratio of the beam is kept fixed as $L/h=10$ . Figure 4 shows that the frequency increases as the first foundation Winkler’s parameter k₁ increases. The same behavior occurs in Figure 5 that $\Omega$ increases as the second foundation Pasternak’s parameter k₂ increases. In contrast, the dimensionless natural frequency depends upon the length-to-thickness ratio and the two-parameter elastic foundation.

Figure 4.

The fundamental frequency $\Omega$ versus the Winkler’s parameter k₁ for different values of the Pasternak’s parameter k₂ $(L/h=10)$ .

Figure 5.

The fundamental frequency $\Omega$ versus the Pasternak’s parameter k₂ for different values of the Winkler’s parameter k₁ $(L/h=10)$ .

Finally, Figure 6 shows the fundamental frequency $\Omega$ versus both the Pasternak’s parameter k₂ and the length-to-thickness ratio $L/h$ for $k_1=100$ . The frequency is highly sensitive to the variation of $L/h$ and k₂. The effect of the Pasternak’s parameter k₂ is more pronounced for smaller values of the length-to-thickness ratio, especially when $L/h=5$ . However, for higher values of $L/h$ , the Pasternak’s parameter k₂ has little effect on the frequency $\Omega$ . Figure 7 shows the effect of the foundation parameters k₁ and k₂ on the fundamental frequency $\Omega$ of the microbeam. The frequency is highly sensitive to the variation of the Pasternak’s parameter k₂. The effect of the Winkler’s parameter k₁ is pronounced for all values of k₂. In general, with the increase of k₁ and k₂ the frequency $\Omega$ is increasing. The maximum frequency occurs for higher values of the foundation parameters.

Figure 6.

The fundamental frequency $\Omega$ versus the Pasternak’s parameter k₂ and length-to-thickness ratio $L/h$ for $k_1=100$ .

Figure 7.

The fundamental frequency $\Omega$ versus the two foundation parameters k₁ and k₂ for $L/h=5$ .