Effect of temperature on free vibration of functionally graded microbeams

Document Type: Research Paper


1 Department of Mechanical Engineering, Central Tehran Branch, Islamic Azad Univ., Tehran, Iran

2 Department of Mechanical Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran


Modified couple stress theory is applied to study of temperature effects on free vibration of Timoshenko functionally graded microbeams. Due to the interatomic and microstructural reactions of the structures in micro scale, the dynamic behavior of the microbeam is predicted more accurate applying the couple stress theory. Both of the simply supported and clamped boundary conditions are assumed to obtain the natural frequencies of the beam. The natural frequencies are obtained by minimization of the Lagrange function and applying the Ritz method. Finally, effects of various parameters such as; temperature change, power law index, height to length scale parameter ratio and height to length ratio on natural frequencies of the microbeam are presented and discussed in detail. The results obtained in this work is validated against numerical data given in the literature search.


Main Subjects

[1] Mindlin, R.D., “Micro-structure in Linear Elasticity”, Archive for Rational Mechanics and Analysis, Vol. 16, No. 1, pp. 51–78, (1964).

[2] Toupin, R.A., “Elastic Materials with Couple-stresses”, Archive for Rational Mechanics and Analysis, Vol. 11, No. 1, pp. 385–414, (1962).

[3] Kr¨oner, E., “Elasticity Theory of Materials with Long Range Cohesive Forces”, International Journal of Solids and Structures, Vol. 3, No. 5, pp. 731–742, (1967).

[4] Eringen, A.C., “Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves”, International Journal of Engineering Science, Vol. 10, No. 5, pp. 425 – 435, (1972).

[5] Fleck, N.A., and Hutchinson, J.W., In Hutchinson, J.W., andWu, T.Y., editors, Strain Gradient Plasticity, Volume 33 of Advances in Applied Mechanics, pages 295 – 361, Elsevier, New York, Academic Press, (1997).

[6] Gao, H., Huang, Y., Nix,W., and Hutchinson, J., “Mechanism-based Strain Gradient Plasticity I. Theory”, Journal of the Mechanics and Physics of Solids, Vol. 47, No. 6, pp. 1239 – 1263, (1999).

[7] Yang, F., Chong, A., Lam, D., and Tong, P., “Couple Stress Based Strain Gradient Theory for Elasticity”, International Journal of Solids and Structures, Vol. 39, No. 10, pp. 2731 – 2743, (2002).

[8] Yang, Y., Zhang, L., and Lim, C.W., “Wave Propagation in Double-walled Carbon Nanotubes on a Novel Analytically Nonlocal Timoshenko-beam Model”, Journal of Sound and Vibration, Vol. 330, No. 8, pp. 1704–1717, (2011).

[9] Ke, L.L., and Wang, Y.S., “Flow-induced Vibration and Instability of Embedded Doublewalled Carbon Nanotubes Based on a Modified Couple Stress Theory”, Physica E: Low- Dimensional Systems and Nanostructures, Vol. 43, No. 5, pp. 1031–1039, (2011).

[10] Wang, L., “Wave Propagation of Fluid-conveying Single-walled Carbon Nanotubes via Gradient Elasticity Theory”, Computational Materials Science, Vol. 49, No. 4, pp. 761 – 766, (2010).

[11] Ke, L.L., and Wang, Y.S., “Size Effect on Dynamic Stability of Functionally Graded Microbeams Based on a Modified Couple Stress Theory”, Composite Structures, Vol. 93, No. 2, pp. 342 – 350, (2011). 

[12] Ma, H., Gao, X.L., and Reddy, J., “A Microstructure-dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory”, Journal of the Mechanics and Physics of Solids, Vol. 56, No. 12, pp. 3379–3391, (2008).

[13] Ke, L.L.,Wang, Y.S., andWang, Z.D., “Thermal Effect on Free Vibration and Buckling of Size-dependent Microbeams”, Physica E: Low-dimensional Systems and Nanostructures, Vol. 43, No. 7, pp. 1387–1393, (2011).

[14] Asghari, M., Kahrobaiyan, M., and Ahmadian, M., “A Nonlinear Timoshenko Beam Formulation Based on the Modified Couple Stress Theory”, International Journal of Engineering Science, Vol. 48, No. 12, pp. 1749–1761, (2010).

[15] Salamat-talab, M., Nateghi, A., and Torabi, J., “Static and Dynamic Analysis of Thirdorder Shear Deformation FG Micro Beam Based on Modified Couple Stress Theory”, International Journal of Mechanical Sciences, Vol. 57, No. 1, pp. 63–73, (2012).

[16] Ebrahimi, F., and Salari, E., “Thermal Buckling and Free Vibration Analysis of Size Dependent Timoshenko FG Nanobeams in Thermal Environments”, Composite Structures, Vol. 128, pp. 363 – 380, (2015).

[17] Lee, H.L., Chu, S.S., and Chang, W.J., “Vibration Analysis of Scanning Thermal Microscope Probe Nanomachining using Timoshenko Beam Theory”, Current Applied Physics, Vol. 10, No. 2, pp. 570 – 573, (2010).

[18] Soh, A.K., Sun, Y., and Fang, D., “Vibration of Microscale Beam Induced by Laser Pulse”, Journal of Sound and Vibration, Vol. 311, No. 12, pp. 243 – 253, (2008).

[19] Park, S.K., and Gao, X.L., “BernoulliEuler Beam Model Based on a Modified Couple Stress Theory”, Journal of Micromechanics and Microengineering, Vol. 16, No. 11, pp. 2355, (2006).

[20] Kong, S., Zhou, S., Nie, Z., and Wang, K., “The Size-dependent Natural Frequency of Bernoulli-Euler Micro-beams”, International Journal of Engineering Science, Vol. 46, No. 5, pp. 427 – 437, (2008). 

[21] Xiang, H., and Yang, J., “Free and Forced Vibration of a Laminated FGM Timoshenko Beam of Variable Thickness under Heat Conduction”, Composites Part B: Engineering, Vol. 39, No. 2, pp. 292 – 303, (2008).

[22] Zenkour, A.M., Arefi, M., and Alshehri, N.A., “Size-dependent Analysis of a Sandwich Curved Nanobeam Integrated with Piezomagnetic Face-sheets”, Results in Physics, Vol. 7, pp. 2172 – 2182, (2017).

[23] Arefi, M., and Zenkour, A.M., “Vibration and Bending Analysis of a Sandwich Microbeam with Two Integrated Piezo-magnetic Face-sheets”, Composite Structures, Vol. 159, pp. 479 – 490, (2017).

[24] Arefi, M., and Zenkour, A.M., “Transient Analysis of a Three-layer Microbeam Subjected to Electric Potential”, International Journal of Smart and Nano Materials, Vol. 8, No. 1, pp. 20–40, (2017). 

[25] Ansari, R., Gholami, R., and Sahmani, S., “Free Vibration Analysis of Size-dependent Functionally Graded Microbeams Based on the Strain Gradient Timoshenko Beam Theory”, Composite Structures, Vol. 94, No. 1, pp. 221 – 228, (2011).