Thermoelastic Vibration of Temperature-Dependent Nanobeams Due to Rectified Sine Wave Heating—A State Space Approach

Document Type: Research Paper

Authors

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh 33516, Egypt

3 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Abstract

In this study, the second type of Green and Naghdi's thermoelasticity theory is applied to present the vibration of a nanobeam subjected to rectified sine wave heating based upon the nonlocal thermoelasticity theory. Both Young's modulus and thermal conductivity are considered to be linear functions of the temperature. The Laplace transform domain is adopted to solve the governing partial differential equations using the state space approach. Numerical computations are carried out using the inverse of Laplace transforms. The effects of nonlocal parameter and angular frequency on the thermal vibration quantities are discussed. The results of all quantities are illustrated graphically and investigated.

Keywords

Main Subjects

[1] Abbas, I. A., A GN Model for Thermoelastic Interaction in a Microscale Beam Subjected to a Moving Heat Source. Acta Mechanica 226 (2015) 2527–2536.

[2] Abouelregal, A. E., and A. M. Zenkour, Generalized Thermoelastic Vibration of a Microbeam with an Axial Force. Microsystem Technologies 21 (2015) 1427–1435.

[3] Allam, M. N. M., K. A. Elsibai, and A. E. Abouelregal, Magneto-thermoelasticity for an Infinite Body with a Spherical Cavity and Variable Material Properties without Energy Dissipation. International Journal of Solids and Structures 47(20) (2010) 2631–2638.

[4] Ansari, R., M. F. Oskouie, and R. Gholami, Size-dependent Geometrically Nonlinear Free Vibration Analysis of Fractional Viscoelastic Nanobeams based on the Nonlocal Elasticity Theory. Physica E 75 (2016) 266–271.

[5] Bahar L. Y., and R. B. Hetnarski, State Space Approach to Thermoelasticity. Journal of Thermal Stresses 1(1) (1978) 135–145.

[6] Eringen A. C., Nonlocal Polar Elastic. Continua, International Journal of Engineering Science 10(1) (1972) 1–16.

[7] Eringen A. C., and D. G. B. Edelen, On Nonlocal Elasticity. International Journal of Engineering Science 10(3) (1972) 233–248.

[8] Eltaher, M. A., A. E. Alshorbagy, and F. F. Mahmoud, Vibration Analysis of Euler–Bernoulli Nanobeams by Using Finite Element Method. Applied Mathematical Modelling 37 (2013) 4787–4797.

[9] Green, A. E., and P. M. Naghdi, A Re-examination of the Basic Postulates of Thermomechanics. Proceedings of the Royal Society A, Mathematical Physical and Engineering Sciences 432 (1885) 171–194.

[10] Green, A. E., and P. M. Naghdi, On Undamped Heat Waves in an Elastic Solid. Journal of Thermal Stresses 15(2) (1992) 253–264.

[11] Green, A. E., and P. M. Naghdi, Thermoelasticity without Energy Dissipation. Journal of Elasticity 31(3) (1993) 189–208.

[12] Guo, F. L., G. Q. Wang, and G. A. Rogerson, Analysis of Thermoelastic Damping in Micro- and Nanomechanical Resonators based on Dual-phase-lagging Generalized Thermoelasticity Theory. International Journal of Engineering Science 60 (2012) 59–65.

[13] Hosseini-Hashemi, S., R. Nazemnezhad, and H. Rokni, Nonlocal Nonlinear free Vibration of Nanobeams with Surface Effects. European Journal of Mechanics A/Solids 52 (2015) 44–53.

[14] Lin, S-M., Analytical Solutions for Thermoelastic Vibrations of Beam Resonators with Viscous Damping in Non-Fourier Model. International Journal of Mechanical Sciences 87 (2014) 26–35.

[15] Lord, H. W., and Y. Shulman, A Generalized Dynamical Theory of Thermoelasticity. Journal of Mechanics and Physics of Solids 15(5) (1967) 299–309.

[16] Nejad, M. Z., and A. Hadi, Non-local Analysis of Free Vibration of Bi-directional Functionally Graded Euler–Bernoulli Nano-beams. International Journal of Engineering Science 105 (2016) 1–11.

[17] Rahmani, O., and O. Pedram, Analysis and Modeling the Size Effect on Vibration of Functionally Graded Nanobeams based on Nonlocal Timoshenko Beam Theory. International Journal of Engineering Science 77 (2015) 55–70.

[18] Rezazadeh, G., A. S. Vahdat, S, Tayefeh-rezaei, and C. Cetinkaya, Thermoelastic Damping in a Micro-beam Resonator Using Modified Couple Stress Theory. Acta Mechanica 223 (2012) 1137–1152.

[19] Sharma, J. N., and R. Kaur, Response of Anisotropic Thermoelastic Micro-beam Resonators under Dynamic Loads. Applied Mathematical Modelling 39 (2015) 2929–2941.

[20] Sharma, J. N., and D. Grover, Thermoelastic Vibrations in Micro-/nano-scale Beam Resonators with Voids. Journal of Sound and Vibration 330 (2015) 2964–2977.

[21] Taati, E., M. M. Najafabadi, and H. B. Tabrizi, Size-dependent generalized thermoelasticity model for Timoshenko microbeams. Acta Mechanica 225 (2014) 1823–1842.

[22] Tzou, D. Y., Macro-to-Microscale Heat Transfer: The Lagging Behavior. Taylor & Francis: Washington DC, 1996.

[23] Sahmani, S., and R. Ansari, Nonlocal Beam Models for Buckling of Nanobeams Using State-space Method Regarding Different Boundary Conditions. Journal of Mechanical Science and Technology 25(9) (2011) 2365–2375.

[24] Youssef, H. M., and K. A. Elsibai, Vibration of Gold Nanobeam Induced by Different Types of Thermal Loading—A State-space Approach. Nanoscale and Microscale Thermophysical Engineering 15(1) (2011) 48–69.

[25] Zenkour, A. M., Vibration analysis of generalized thermoelastic microbeams resting on visco-Pasternak’s foundations. Advances in Aircraft and Spacecraft Science 4 (2014) 269–280.

[26] Zenkour, A. M., and A. E. Abouelregal, Vibration of FG Nanobeams Induced by Sinusoidal Pulse-heating via a Nonlocal Thermoelastic Model. Acta Mechanica 225 (2014) 3409–3421.

[27] Zenkour, A. M., and A. E. Abouelregal, Nonlocal Thermoelastic Nanobeam Subjected to a Sinusoidal Pulse Heating and Temperature‑dependent Physical Properties. Microsystem Technologies 21 (2015) 1767–1776.

[28] Zenkour, A. M., A. E. Abouelregal, K. A. Alnefaie, N. H. Abu-Hamdeh, A. A. Aljinaidi and E. C. Aifantis, State Space Approach for the Vibration of Nanobeams based on the Nonlocal Thermoelasticity Theory without Energy Dissipation. Journal of Mechanical Science and Technology 29(7) (2015) 2921–2931.

[29] Zenkour, A. M., and M. Sobhy, A simplified Shear and Normal Deformations Nonlocal Theory for Bending of Nanobeams in Thermal Environment. Physica E 70 (2015) 121–128.