• Safa Hameed Majeed Department of Mechanical Engineering, College of Engineering, University of Thi-Qar, Thi-Qar,64001, Iraq
  • Talib Ehraize Elaikh Department of Mechanical Engineering, College of Engineering, University of Thi-Qar, Thi-Qar,64001, Iraq
  • Adnan Abdul-hussien Ugla Department of Mechanical Engineering, College of Engineering, University of Thi-Qar, Thi-Qar,64001, Iraq



Bidirectional FGM microbeam, RSGE, Galerkin method (GM), thermal effect


This research examines the vibrational response of a micro-scale Euler beam made from two-directional functionally graded (2D-FG) materials and subjected to thermal effects. By employing a reformulated strain gradient elasticity (RSGE) approach, the equations of motion using Hamilton’s principle for clamped -clamped and clamped-simply boundary conditions are derived and solved them using Galerkin's approach. The investigation explores the impact of temperature, gradient index, and parameters length scale materials on the bidirectional graded microbeam's dynamic characteristics. Furthermore, the normalized frequency, as based on the current reformulated strain gradient elasticity microbeam model, consistently emerges as higher than that derived from the classical model.


Şimşek, M. 2015. Bi-directional Functionally Graded Materials (BDFGMs) for Free and Forced Vibration of Timoshenko Beams with Various Boundary Conditions. Compos. Struct. 133: 968-978. Doi: 10.1016/j.compstruct.2015.08.021.

Mohammadi, M., Rajabi, M., and Ghadiri, M. 2021. Functionally Graded Materials (FGMs): A Review of Classifications, Fabrication Methods and Their Applications. Processing and Application of Ceramics. 15(4): 319-343. Doi: 10.2298/PAC2104319M.

El-Galy, I. M., Saleh, B. I., and Ahmed, M. H. 2019. Functionally Graded Materials Classifications and Development Trends from Industrial Point of View. SN Applied Sciences. 1(11). Doi: 10.1007/s42452-019-1413-4.

Li, X., Li, L., Hu, Y., Ding, Z., and Deng, W. 2017. Bending, Buckling and Vibration of Axially Functionally Graded Beams based on Nonlocal Strain Gradient Theory. Compos. Struct. 165: 250-265. Doi: 10.1016/j.compstruct.2017.01.032.

Tang, Y., Lv, X., and Yang, T. 2019. Bi-directional Functionally Graded Beams: Asymmetric Modes and Nonlinear Free Vibration. Compos. Part B Eng. 156: 319-331. Doi: 10.1016/j.compositesb.2018.08.140.

Elaikh, T. E., Abed, N. M. 2019. Stability of FG Material Micro-pipe Conveying Fluid. International Journal of Energy and Environment. 10(4): 211-222.

Ghayesh, M. H., and Farokhi, H. 2016. Coupled Nonlinear Dynamics of Geometrically Imperfect Shear Deformable Extensible Microbeams. J. Comput. Nonlinear Dyn. Doi: 10.1115/1.4031288.

Ghayesh, M. H., and Farajpour, A. 2019. A Review on the Mechanics of Functionally Graded Nanoscale and Microscale Structures. Int. J. Eng. Sci. 137: 8-36. Doi: 10.1016/j.ijengsci.2018.12.001.

Elaikh, T. E. H., Abed, N. M., and Ebrahimi-Mamaghani, A. 2020. Free Vibration and Flutter Stability of Interconnected Double Graded Micro Pipes System Conveying Fluid. IOP Conf. Ser. Mater. Sci. Eng. Doi: 10.1088/1757-899X/928/2/022128.

Şimşek, M. 2016. Buckling of Timoshenko Beams Composed of Two-Dimensional Functionally Graded Material (2D-FGM) Having Different Boundary Conditions. Compos. Struct. Doi: 10.1016/j.compstruct.2016.04.034.

Avcar, M. 2014. Free Vibration Analysis of Beams Considering Different Geometric Characteristics and Boundary Conditions. Int. J. Mech. Appl. 4(3): 94-100. Doi: 10.5923/j.mechanics.20140403.03.

Karamanli, A., and Thuc, P. Vo. 2020. Bending, Vibration, Buckling Analysis of bi-directional FG Porous Microbeams with a Variable Material Length Scale Parameter. Appl. Math. Model. Doi: 10.1016/j.apm.2020.09.058.

Şimşek, M. 2010. Dynamic Analysis of an Embedded Microbeam Carrying a Moving Microparticle based on the Modified Couple Stress Theory. Int. J. Eng. Sci. 48(12): 1721--1732. Doi: 10.1016/j.ijengsci.2010.09.027.

Asghari, M., Ahmadian, M. T., Kahrobaiyan, M. H. and Rahaeifard, M. 2010. On the Size-dependent Behavior of Functionally Graded Micro-beams. Mater. Des. 31(5): 2324-2329. Doi: 10.1016/j.matdes.2009.12.006.

Liu, H., and Zhang, Q. 2021. Nonlinear Dynamics of Two-directional Functionally Graded Microbeam with Geometrical Imperfection using Unified Shear Deformable Beam Theory. Appl. Math. Model. 98: 783-800. Doi: 10.1016/j.apm.2021.05.029.

Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J., and Tong, P. 2003. Experiments and Theory in strain Gradient Elasticity. J. Mech. Phys. Solids. 51(8): 1477-1508. Doi: 10.1016/S0022-5096(03)00053-X.

Akgöz, B., and Civalek, Ö. 2013. Buckling Analysis of Functionally Graded Microbeams based on the Strain Gradient Theory. Acta Mech. 224(9): 2185-2201. Doi: 10.1007/s00707-013-0883-5.

Ansari, R., Gholami, R., and Sahmani, S. 2014. Free Vibration of Size-dependent Functionally Graded Microbeams based on the Strain Gradient Reddy Beam Theory. Int. J. Comput. Methods Eng. Sci. Mech. 15(5): 401-412. Doi: 10.1080/15502287.2014.915249.

Attia, M. A., and Shanab, R. A. 2022. On the Dynamic Response of bi-directional Functionally Graded Nanobeams under Moving Harmonic Load Accounting for Surface Effect. Acta Mech. 233(8): 3291-3317. Doi: 10.1007/s00707-022-03243-1.

Chen, L., Liu, Y., Zhou, S., and Wang, B. 2021x. The Reformulated Micro-beam Models by Incorporating the General Strain Gradient Elasticity Theory (GSGET). Appl. Math. Model. 90: 448-465. Doi: 10.1016/j.apm.2020.08.050.

Yin, S., Xiao, Z., Liu, J., Xia, Z., and Gu, S. 2022. Variational Formulations and Isogeometric Analysis of Timoshenko–Ehrenfest Microbeam Using a Reformulated Strain Gradient Elasticity Theory. Crystals. 12(6). Doi: 10.3390/cryst12060752.

Dinachandra, M., and Alankar, A. 2021. Static and Dynamic Modeling of functionally Graded Euler–Bernoulli Microbeams based on Reformulated Strain Gradient Elasticity Theory using Isogeometric Analysis. Compos. Struct. 280: 114923. Doi: 10.1016/j.compstruct.2021.114923.

Yin, S., Xiao, Z., Zhang, G., Bui, T. Q., Wang, X., and Liu, J. 2022. Size-dependent Postbuckling for Microbeams: Analytical Solutions using a Reformulated Strain Gradient Elasticity Theory. Acta Mech. 233(12): 5045-5060. Doi: 10.1007/s00707-022-03360-x.

Yin, S., Xiao, Z., Zhang, G., Liu, J., and Gu, S. 2022. Size-Dependent Buckling Analysis of Microbeams by an Analytical Solution and Isogeometric Analysis. Crystals. Doi: 10.3390/cryst12091282.

Yin, S., Xiao, Z., Deng, Y., Zhang, G., Liu, J., and Gu, S. 2021. Isogeometric Analysis of Size-dependent Bernoulli–Euler Beam based on a Reformulated Strain Gradient Elasticity Theory. Comput. Struct. 253: 106577. Doi: 10.1016/j.compstruc.2021.106577.

Zhang, G. Y., and Gao, X. L. 2020. A New Bernoulli–Euler Beam Model based on a reformulated Strain Gradient Elasticity Theory. Math. Mech. Solids. 25(3): 630-643. Doi: 10.1177/1081286519886003.

Babaei, A., Noorani, M. R. S., and Ghanbari, A. 2017. Temperature-dependent Free Vibration Analysis of Functionally Graded Micro-beams based on the Modified Couple Stress Theory. Microsyst. Technol. 23(10): 4599-4610. Doi: 10.1007/s00542-017-3285-0.

Zanoosi, A. A. P. 2020. Size-dependent Thermo-mechanical Free Vibration Analysis of Functionally Graded Porous Microbeams based on Modified Strain Gradient Theory. J. Brazilian Soc. Mech. Sci. Eng. 42(5). Doi: 10.1007/s40430-020-02340-3.

Tang, Y., and Ding, Q. 2019. Nonlinear Vibration Analysis of a bi-directional Functionally Graded Beam under Hygro-thermal Loads. Compos. Struct. 111076. Doi: 10.1016/j.compstruct.2019.111076.

Nateghi, A., and Salamat-talab, M. 2013. Thermal Effect on Size Dependent Behavior of Functionally Graded Microbeams based on Modified Couple Stress Theory. Compos. Struct. 96: 97-110. Doi: 10.1016/j.compstruct.2012.08.048.

Liu, H., Zhang, Q., and Ma, J. 2021. Thermo-mechanical Dynamics of Two-dimensional FG Microbeam Subjected to a Moving Harmonic Load. Acta Astronaut. 178(January): 681-692. Doi: 10.1016/j.actaastro.2020.09.045.

Tang, Y., Zhong, S., Yang, T., and Ding, Q. 2019. Interaction between Thermal Field and Two-dimensional Functionally Graded Materials: A Structural Mechanical Example. Int. J. Appl. Mech. 11(10). Doi: 10.1142/S1758825119500996.

Ebrahimi-Mamaghani, A., Sotudeh-Gharebagh, R., Zarghami, R., and Mostoufi, N. 2020. Thermo-mechanical Stability of Axially Graded Rayleigh Pipes. Mech. Based Des. Struct. Mach. 1-30. Doi: 10.1080/15397734.2020.1717967.

Elaikh, T. E., and Agboola, O. O. O. 2022. Investigation of Transverse Vibration Characteristics of Cracked Axially Moving Functionally Graded Beam Under Thermal Load. Trends Sci. 19(23). Doi: 10.48048/tis.2022.1349.

Mutlak, D. A., Muhsen, S., Waleed, I., Hadrawi, S. K., Khaddour, M. H., and Ahmadi, S. 2022. Forced and Free Dynamic Responses of Functionally Graded Porous Rayleigh Small-Scale Beams on Kerr Foundation under Moving Force. Mater. Today Commun. 33: 104919. Doi: 10.1016/J.MTCOMM.2022.104919.

Nematollahi, M. S., Mohammadi, H., and Taghvaei, S. 2019. Fluttering and Divergence Instability of Functionally Graded Viscoelastic Nanotubes Conveying Fluid based on Nonlocal Strain Gradient Theory. Chaos. 29(3). Doi: 10.1063/1.5057738.

Nejad, M. Z., and Hadi, A. 2016. Non-local Analysis of Free Vibration of bi-directional Functionally Graded Euler-Bernoulli Nano-beams. Int. J. Eng. Sci. 105: 1-11. Doi: 10.1016/j.ijengsci.2016.04.011.

Yapor Genao, F., Kim, J., and Żur, K. K. 2021. Nonlinear Finite Element Analysis of Temperature-dependent Functionally Graded Porous Micro-plates under Thermal and Mechanical Loads. Compos. Struct. 256: 112931. Doi: 10.1016/j.compstruct.2020.112931.

Esen, I., Özarpa, C., and Eltaher, M. A. 2021. Free Vibration of a Cracked FG Microbeam Embedded in an Elastic Matrix and Exposed to Magnetic Field in a Thermal Environment. Compos. Struct. 261x. Doi: 10.1016/j.compstruct.2021.113552.

Shaat, M. 2018. A reduced Micromorphic Model for Multiscale Materials and Its Applications in Wave Propagation. Compos. Struct. 201: 446-454. Doi: 10.1016/j.compstruct.2018.06.057.

Zhang, G. Y., Gao, X. L., Zheng, C. Y., and Mi, C. W. 2021. A Non-classical Bernoulli-Euler Beam Model based on a Simplified Micromorphic Elasticity Theory. Mech. Mater. 161: 103967. Doi: 10.1016/j.mechmat.2021.103967.

Rao, S. S. 2007. Vibration of Continuous Systems. Doi: 10.1002/9780470117866.

Eltaher, M. A., Emam, S. A., and Mahmoud, F. F. 2012. Free Vibration Analysis of Functionally Graded Size-dependent Nanobeams. Appl. Math. Comput. 218(14): 7406-7420. Doi: 10.1016/j.amc.2011.12.090.




How to Cite




Science and Engineering