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

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1 Original Article Free vibration of a microbeam resting on Pasternak s foundation via the Green Naghdi thermoelasticity theory without energy dissipation Journal of Low Frequency Noise, Vibration and Active Control 2016, Vol. 35(4) ! The Author(s) 2016 Reprints and permissions: sagepub.co.uk/journalspermissions.nav DOI: / lfn.sagepub.com Ashraf M Zenkour 1,2 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. Keywords Thermoelasticity, microbeam resonator, elastic foundations 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 unieed 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 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh, Egypt Corresponding author: Ashraf M Zenkour, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. zenkourkau.edu.sa

2 304 Journal of Low Frequency Noise, Vibration and Active Control 35(4) 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 ramptype 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 Şims ek 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. Akgo z and Civalek 24 investigated the vibration response of nonhomogenous and nonuniform microbeams in conjunction with E B beam and modified couple stress theory. Şims ek 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 Lð0 x LÞ, width bð b=2 y b=2þ, and thickness hð h=2 z h=2þ (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 T 0 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 where w is the lateral deflection. u ¼ z w, v ¼ 0, w ¼ wðx, tþ ð1þ x Figure 1. Schematic diagram for the microbeam resting on a two-parameter elastic foundation.

3 Zenkour 305 The relevant constitutive equation for the axial stress x reads x ¼ E z 2 w x 2 þ T ð2þ where E is Young s modulus, ¼ T T 0 is the excess temperature with T 0 denoting the constant environmental temperature, T ¼ t =ð1 2Þ in which t is the thermal expansion coefficient, and is Poisson s ratio. The interaction between the beam and the supporting foundation follows the two-parameter Pasternak s model as 27 2 w R f ¼ K 1 wx, ð tþ K 2 x 2 ð3þ where R f is the foundation reaction per unit area, K 1 is the Winkler s foundation stiffness, and K 2 is the shear stiffness of elastic foundation. This model simply reduces to the Winkler s type when K 2 ¼ 0. For transverse vibrations, the corresponding equation of motion reads 2 M x 2 R f ¼ A 2 w t 2 ð4þ where is the material density and A ¼ 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 ¼ EI 2 w x 2 þ TM T where I ¼ bh 3 =12 is the moment of inertia, EI is the flexural rigidity of the beam, and M T is the thermal moment defined by M T ¼ 12 Z h=2 h 3 ðx, z, tþzdz ð6þ h=2 Substituting equations (3) and (5) into equation (4), one obtains the motion equation of the beam in the form 4 x 4 K 2 2 EI x 2 þ A 2 EI t 2 þ K 1 2 M T w þ T EI x 2 ¼ 0 ð7þ The heat conduction in the context of Green and Naghdi s generalized thermoelasticity theory without energy dissipation is given by 28 r 2 þ 1 þ ðq Þ ¼ t t C t þ T 0 where is the thermal conductivity (the material constant characteristic), C is the specific heat per unit mass at constant strain, e ¼ " kk ¼ u x þ v y þ w z is the volumetric strain, Q is the heat source, and ¼ t E=ð1 2Þ is the thermoelastic coupling parameter. The corresponding thermal conduction equation for the beam under consideration without heat source ðq ¼ 0Þ is obtained by specializing equation (8) to present the E B beam configuration as 2 x 2 þ 2 z 2 ¼ 1 C t t T 0z 2 w t x 2 ð9þ Multiplying the above equation by 12z=h 3, and integrating it with respect to z through the beam thickness from h=2 toh=2, yields 2 M T x 2 12 h 3 jþh=2 h=2 ¼ 1 C M T 2 w T 0 t t t x 2 ð10þ e t ð5þ ð8þ

4 306 Journal of Low Frequency Noise, Vibration and Active Control 35(4) 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 ¼ 6 5h jþh=2 h=2 So, at this point equation (10) becomes 2 M T x 2 10 h 2 M T ¼ 1 C M T t t T 0 t 2 w x 2 ð11þ ð12þ 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 z ¼ 0 ð13þ z¼h=2 However, the nanobeam is subjected to simply supported mechanical conditions at its edges x ¼ 0 and x ¼ L as w ¼ M ¼ 0 ð14þ Following the Navier-type solution, the deflection and moment that satisfy the boundary conditions maybe expressed as X N wx, ð tþ, Mx, ð tþ ¼ n¼1 w n, M n sinðn xþe i!t ð15þ where w n and M n are arbitrary parameters, n ¼ n L, n is a mode number, and! 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 4 n þ K 2 EI 2 n þ K 1 A EI EI!2 w n T 2 n M nt ¼ 0 2 T 0 n!2 w n þ C!2 2 n 10 h 2 M nt ¼ 0 where M nt is an arbitrary thermal parameter. To get the nontrivial solution of the above equations, the parameter w n and M 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 ð16þ ð17þ 4 A 0 2 þ A 1 A 2 ¼ 0 p where ¼ L ffiffi! is the dimensionless frequency parameter and E ð18þ A 1 ¼ h2 4 12L 2 n þ k 2 2 n þ k 1, A2 ¼ 2 n þ 10L2 h 2 EC, A 0 ¼ A 1 þ A 2 þ 4 T T 0 h 2 n 12C L 2, k 1 ¼ L4 K 1 EI, k 2 ¼ L2 K 2 EI, n ¼ n ð19þ

5 Zenkour 307 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 ¼ 165:9 GPa, ¼ 0:22, ¼ 2330 kg=m 3, C ¼ 1:661 J=kgK, ¼ 2: =K, ¼ 156 W=mK ð20þ Table 1. Effect of mode number n, the foundation parameters k 1 and k 2, and the length-to-thickness ratio L=h on the dimensionless natural frequency. L/h n k 1 k Figure 2. The fundamental frequency versus the length-to-thickness ratio L=h for various elastic foundations.

6 308 Journal of Low Frequency Noise, Vibration and Active Control 35(4) Figure 3. The natural frequency versus the length-to-thickness ratio L=h for k 1 ¼ 100 and k 2 ¼ 15. Figure 4. The fundamental frequency versus the Winkler s parameter k 1 for different values of the Pasternak s parameter k 2 ðl=h ¼ 10Þ. The reference temperature is taken as T 0 ¼ 293 K. For the sake of completeness, sample results are presented for the dimensionless natural frequencies 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 k 1 and k 2, and the length-to-thickness ratio L=h on the dimensionless natural frequency are presented in Table 1. The fundamental frequency ðn ¼ 1Þ 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 k 1 and k 2. In fact, the dimensionless vibration frequency is very sensitive to the variation of the parameters n, L=h, k 1, and k 2.

7 Zenkour 309 Figure 5. The fundamental frequency versus the Pasternak s parameter k 2 for different values of the Winkler s parameter k 1 ðl=h ¼ 10Þ. Figure 6. The fundamental frequency versus the Pasternak s parameter k 2 and length-to-thickness ratio L=h for k 1 ¼ 100. Figures 2 and 3 are prepared by using the dimensionless natural frequencies for a wide range of the length-tothickness ratio. Figure 2 shows that the fundamental frequency decreases as L=h increases. However, the inclusion of the elastic foundation causes the increase of. The frequencies of the microbeam resting on a two-parameter elastic foundation have the greatest values. In Figure 3, the plot the natural frequency versus the length-tothickness ratio L=h is also illustrated for a microbeam resting on a two-parameter elastic foundation (k 1 ¼ 100 and k 2 ¼ 15). Once again the natural frequency decreases with the decrease of L=h and the mode number n. In Figures 4 and 5, the aspect ratio of the beam is kept fixed as L=h ¼ 10. Figure 4 shows that the frequency increases as the first foundation Winkler s parameter k 1 increases. The same behavior occurs in Figure 5 that increases as the second foundation Pasternak s parameter k 2 increases. In contrast, the dimensionless natural frequency depends upon the length-to-thickness ratio and the two-parameter elastic foundation.

8 310 Journal of Low Frequency Noise, Vibration and Active Control 35(4) Figure 7. The fundamental frequency versus the two foundation parameters k 1 and k 2 for L=h ¼ 5. Finally, Figure 6 shows the fundamental frequency versus both the Pasternak s parameter k 2 and the lengthto-thickness ratio L=h for k 1 ¼ 100. The frequency is highly sensitive to the variation of L=h and k 2. The effect of the Pasternak s parameter k 2 is more pronounced for smaller values of the length-to-thickness ratio, especially when L=h ¼ 5. However, for higher values of L=h, the Pasternak s parameter k 2 has little effect on the frequency. Figure 7 shows the effect of the foundation parameters k 1 and k 2 on the fundamental frequency of the microbeam. The frequency is highly sensitive to the variation of the Pasternak s parameter k 2. The effect of the Winkler s parameter k 1 is pronounced for all values of k 2. In general, with the increase of k 1 and k 2 the frequency is increasing. The maximum frequency occurs for higher values of the foundation parameters. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) received no financial support for the research, authorship, and/or publication of this article. References 1. Biot MA. Thermoelasticity and irreversible thermodynamics. J Appl Phys 1956; 27: Lord HW and Shulman Y. A generalized dynamical theory of thermoelasticity. J Mech Phys Solids 1967; 15: Green AE and Lindsay KE. Thermoelasticity. J Elast 1972; 2: Green AE and Naghdi PM. On thermodynamics and the nature of the second law. Proc R Soc A 1977; 357: Green AE and Naghdi PM. A re-examination of the basic postulates of thermomechanics. Proc R Soc A 1991; 432: Zenkour AM. Three-dimensional thermal shock plate problem within the framework of different thermoelasticity theories. Compos Struct 2015; 132: Fang D-N, Sun Y-X and Soh A-K. Analysis of frequency spectrum of laser-induced vibration of microbeam resonators. Chinese Phys Lett 2006; 23: Soh A-K, Sun Y and Fang D. Vibration of microscale beam induced by laser pulse. J Sound Vib 2008; 311: Guo X-X, Wang Z-M, Wang Y, et al. Analysis of the coupled thermoelastic vibration for axially moving beam. J Sound Vib 2009; 325: Guo X-X and Wang Z-M. Thermoelastic coupling vibration characteristics of the axially moving beam with frictional contact. J Vib Acoust 2010; 132: Sharma JN and Grover D. Thermoelastic vibrations in micro-/nano-scale beam resonators with voids. J Sound Vib 2011; 330: Grover D. Transverse vibrations in micro-scale viscothermoelastic beam resonators. Arch Appl Mech 2013; 83: Belardinelli P, Lenci S and Demeio L. Vibration frequency analysis of an electrically-actuated microbeam resonator accounting for thermoelastic coupling effects. Int J Dyn Control 2015; 3:

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