REFERENCES
[1] Koizumi, M. “FGM activities in Japan”, Composites Part B: Engineering, 28(1), pp. 1-4 (1997).
[2] Sobzak, J., Drenchev, L. “Metal based Functionally Graded Materials – Engineering and Modelling”, Bentham Science Publisher Ltd. pp. 1-24 (2009).
[3] Kapuria, M., Bhattacharya, M., Kumar, A. N. “Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation”, Composite Structures, 82(3), pp. 390-402 (2008).
[4] Yang, J., Chen, Y. Free vibration and buckling analysis of functionally graded beams with edge cracks, Composite Structures, 83(1), pp. 48-60 (2008).
[5] Alshorbagy, A. E., Eltaher, M. A., Mahmoud, F. F. “Free vibration characteristics of functionally graded beam by finite element method”, Applied mathematical modelling, 35(1), pp. 412-425 (2011).
[6] Simsek, M., Kocaturk, T. “Free and forced vibration of functionally graded beam subjected to a concentrated moving harmonic load”, composite structures, 90, pp. 465-473 (2009).
[7] Pradhan, K. K., Chakraverty, S. “Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method”, Composites: Part B, 51, pp. 175-184 (2013).
[8] Rao, R. S., Ganesan, N. “Dynamic response of non-uniform composite beams”, Journal of Sound and Vibration, 200(5), pp. 563-577 (1997).
[9] Karami, G., Malekzadeh, P., Shahpari, S. A. “A DQEM for vibration of shear deformable non-uniform beams with general boundary conditions”, Engineering structures, 25, pp. 1169-1179 (2003).
[10] Aydogdu, M., Taskin, V. “Free vibration analysis of functionally graded beams with simply supported edges”, Material and Design, 28, pp. 1651-1656 (2007).
[11] Nguyen, V. L., Quoc, T. H. “Bending and free vibration analysis of functionally graded plates using new eight unknown shear deformation theory by finite element method”, International Journal of advanced structural Engineering, 8, pp. 391-399 (2016).
[12] Huang, Y., Li, X. F. “A new approach for free vibration of axially graded beams with non-uniform cross-section”, Journal of Sound and vibration, 329, pp. 2291-2303 (2010).
[13] Cao, D., Gao, Y., Zhang, W. “Free vibration of axially functionally graded beams using the asymptotic development method”, Engineering Structures, 173, pp. 442-448 (2018).
[14] Ghayesh, M. H. “Non-linear vibration analysis of axially functionally graded shear-deformable tapered beams”, Applied mathematical modelling, 59, pp. 583-596 (2018).
[15] Huang, Y., Wang, T., Zhao, Y. et al. “Effect of axially functionally graded material on whirling frequencies and critical speeds of spinning Timoshenko beam”, Composite structures, 192, pp. 355-367 (2018).
[16] Salinic, S., Obradovic, A., Tomovic, A. “Free vibration analysis of axially functionally graded tapered, stepped and continuously segmented rods and beams”, Composites Part B, 150, pp. 135-143 (2018).
[17] Zheng, S., Chen, D., Wang, H. “Size dependent nonlinear free vibration of axially functionally graded tapered microbeams using finite element method”, Thin-Walled Structures, 139, pp. 46-52 (2019).
[18] Sahin, S., Karahan, E., Kilic, B., Ozdemir, O. “Finite element method for vibration analysis of Timoshenko beams”, 9th International Conference on Recent Advances in Space Technologies (RAST), Istanbul, Turkey, pp. 673-679 (2019).
[19] Xie, K., Wang, Y., Fu, T. “Dynamic response of axially functionally graded beam with longitudinal-transverse coupling effect”, Aerospace Science and Technology, 85, pp. 85-95 (2019).
[20] Sun, D. L., Li, X. F. “Initial value method for free vibration of axially loaded functionally graded Timoshenko beams with non-uniform cross-section”, Mechanics based design of structures and machines, 47(1), pp. 102-120 (2019).
[21] Hughes, T.J.R., Cottrell, J.A., Bzileves, Y. “Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement”, Computer Methods in Applied Mechanics and Engineering, 194(39-41), pp. 4135-4195 (2005).
[22] Nguyan, H.X., Nguyan, T., Wahab, M.A. et al. “A refined quasi-3D isogeometric analysis for functionally graded microplates based on the modified couple stress theory”, Computer Methods in Applied Mechanics and Engineering, 313, pp. 904-940 (2017).
[23] Thanh, C.L., Ferreira, A.J.M., Wahab, M.A. “A refined size-dependent couple stress theory for laminated composite micro-plates using isogeometric analysis”, Thin-Walled Structures, 145, 106427 (2019).
[24] Thanh, C.L., Tran, L.V., Bui, T.Q. et al. “Isogeometric analysis for size-dependent nonlinear thermal stability of porous FG microplates”, Composite Structures, 221, 110838 (2019).
[25] Thanh, C.L., Van, P.P., Thai, C.H. et al. “Isogeometric analysis of functionally graded carbon nanotube reinforced composite nanoplates using modified couple stress theory”, Composite Structures, 184(15), pp. 633-649 (2018).
[26] Van, P.H., Tran, L.V., Ferreira, A.J.M. et al. “Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads”, Nonlinear Dynamics, 87, pp. 879-894 (2017).
[27] Van, P.P., Thai, C.H., Xuan, H.N. et al. “Porosity-dependent nonlinear transient responses of functionally graded Nano-plates using iso-geometric analysis”, Composites Part B: Engineering, 164, pp. 215-225 (2019).
[28] Thanh, C.L., Tran, L.V., Huu, T.V. et al. “The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis”, Computer Methods in Applied Mechanics and Engineering, 350, pp. 337-361, (2019).
[29] Nguyen, T.-T., Nguyen, N.-L., Lee, J. et al. “Analysis of non-uniform polygonal cross-sections for thin-walled functionally graded straight and curved beams”, Engineering structures, 226, 111366, (2021).
[30] Nguyen, T.-T., Lee, J. “Interactive geometric interpretation and static analysis of thin-walled bi-directional functionally graded beams”, Composite Structures, 191, pp. 1-11 (2018).
[31] Nguyen, T.T., Lee, J. “Flexural-torsional vibration and buckling of thin-walled bi-directional functionally graded beams”, Composites Part B, 154, pp. 351-362 (2018).
[32] Rajasekaran, S., Khaniki, H. B. “Finite element static and dynamic analysis of axially functionally graded non-uniform small-scale beams based on nonlocal strain gradient theory”, Mechanics of Advanced Materials and Structures, 0, pp. 1-15 (2018).
[33] Voigt, W. “Ueber die beziehung zwischen den beiden elasticitätsconstanten isotroper körper”. Annalen der Physik, pp. 573-587 (1889).