[1] Newell, W.E., Miniaturization of tuning forks, Science, 161(3848), 1968, 1320-6.
[2] Craighead, H.G., Nanoelectromechanical systems. Science, 290(5496), 2000, 1532-1536.
[3] Holterman, J., Groen, P., An Introduction to piezoelectric materials and applications, Stichting Applied Piezo, 2013.
[4] Ye, Z., Handbook of Advanced Dielectric, Piezoelectric and Ferroelectric Materials: Synthesis, Properties and Applications, Woodhead Publishing, 2008.
[5] Cady, W., Piezoelectricity: an introduction to the theory and applications of electromechanical phenomena in crystals, McGraw-Hill; 1st edition, 1946.
[6] Maranganti, R., Sharma, N.D., Sharma, P., Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions. Phys. Rev. B., 74(1), 2006, 14110.
[7] Majdoub, M.S., Sharma, P., Cagin, T., Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Phys. Rev. B., 77(12), 20008, 125424.
[8] Sharma, N.D., Maranganti, R., Sharma, P., On the possibility of piezoelectric nanocomposites without using piezoelectric materials. J. Mech. Phys. Solids, 55(11), 2007, 2328-2350.
[9] Sharma, N.D., Landis, C.M., Sharma, P., Piezoelectric thin-film superlattices without using piezoelectric materials. J. Appl. Phys., 108(2), 2010, 24304.
[10] Catalan, G., Sinnamon, L.J., Gregg, J.M., The effect of flexoelectricity on the dielectric properties of inhomogeneously strained ferroelectric thin films. J. Phys. Condens. Matter., 16(13), 2004, 2253-2264.
[11] Lee, D., Yoon, A., Jang, S.Y., et al. Giant Flexoelectric Effect in Ferroelectric Epitaxial Thin Films. Phys. Rev. Lett., 107(5), 2011, 57602.
[12] Zhou, H., Hong, J., Zhang, Y., et al. Flexoelectricity induced increase of critical thickness in epitaxial ferroelectric thin films. Physica B: Condensed Matter, 407(17), 2012, 3377-3381.
[13] Eliseev, E.A., Morozovska, A.N., Glinchuk, M.D., Blinc, R., Spontaneous flexoelectric/flexomagnetic effect in nanoferroics. Phys. Rev. B., 79(16), 2009, 165433.
[14] Craciunescu, C., Wuttig, M., New Ferromagnetic and Functionally Graded Shape Memory Alloys. J. Opt. Adv. Matter., 5(39), 2003, 139-146.
[15] Fu, Y., Du, H., Zhang, S., Functionally graded TiN/TiNi shape memory alloy films. Mater. Lett., 57(20), 2003, 2995-2999.
[16] Fu, Y., Du, H., Huang, W., Zhang, S., Hu, M., TiNi-based thin films in MEMS applications: a review. Sens. Actuators A, 112(2-3), 2004, 395-408.
[17] Lee, Z., Ophus, C., Fischer, L.M., et al. Metallic NEMS components fabricated from nanocomposite Al–Mo films. Nanotechnology, 17(12), 2006, 3063-3070.
[18] Witvrouw, A., Mehta, A., The Use of Functionally Graded Poly-SiGe Layers for MEMS Applications. Mater. Sci. Forum, 492-493, 2005, 255-260.
[19] Ma, W., Cross, L.E., Observation of the flexoelectric effect in relaxor Pb(Mg1/3Nb2/3)O3 ceramics. Appl. Phys. Lett., 78(19), 2001, 2920-2921.
[20] Ma, W., Cross, L.E., Large flexoelectric polarization in ceramic lead magnesium niobate. Appl. Phys. Lett., 79(26), 2001, 4420-4422.
[21] Ma, W., Cross, L.E., Flexoelectric effect in ceramic lead zirconate titanate. Appl. Phys. Lett., 86(7), 2005, 72905.
[22] Ma, W., Cross, L.E., Flexoelectricity of barium titanate. Appl. Phys. Lett., 88(23), 2006, 232902.
[23] Maranganti, R., Sharma, P., Atomistic determination of flexoelectric properties of crystalline dielectrics. Phys. Rev. B., 80(5), 2009, 54109.
[24] Hong, J., Catalan, G., Scott, J.F., Artacho, E., The flexoelectricity of barium and strontium titanates from first principles. J. Phys. Condens. Matter., 22(11), 2010, 112201.
[25] Ponomareva, I., Tagantsev, A.K., Bellaiche, L., Finite-temperature flexoelectricity in ferroelectric thin films from first principles. Phys. Rev. B., 85(10), 2012, 104101.
[26] Kheibari, F., Beni, Y.T., Size dependent electro-mechanical vibration of single-walled piezoelectric nanotubes using thin shell model. Mater., Des., 114, 2017, 572-583.
[27] Mehralian, F., Beni, Y.T., Ansari, R., On the size dependent buckling of anisotropic piezoelectric cylindrical shells under combined axial compression and lateral pressure. Int. J. Mech. Sci., 119, 2016, 155-169.
[28] Mehralian, F., Beni, Y.T., Ansari, R., Size dependent buckling analysis of functionally graded piezoelectric cylindrical nanoshell. Compos. Struct., 152, 2016, 45-61.
[29] Yue, Y.M., Xu, K.Y., Chen, T., A micro scale Timoshenko beam model for piezoelectricity with flexoelectricity and surface effects. Compos. Struct., 136, 2016, 278-286.
[30] Kong, S., Zhou, S., Nie, Z., Wang, K., The size-dependent natural frequency of Bernoulli–Euler micro-beams. Int. J. Eng. Sci., 46(5), 2008, 427-437.
[31] Sadeghi, H., Baghani, M., Naghdabadi, R., Strain gradient elasticity solution for functionally graded micro-cylinders. Int. J. Eng. Sci., 50(1), 2012, 22-30.
[32] Yan, Z., Jiang, L., Size-dependent bending and vibration behaviour of piezoelectric nanobeams due to flexoelectricity. J. Phys. D. Appl. Phys., 46(35), 2013, 355502.
[33] Liang, X., Hu, S., Shen, S., Size-dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity. Smart. Mater. Struct., 24(10), 2015, 105012.
[34] Yan, Z., Jiang, L., Effect of flexoelectricity on the electroelastic fields of a hollow piezoelectric nanocylinder. Smart. Mater. Struct., 24(6), 2015, 65003.
[35] Kundalwal, S.I., Megui, S.A., Effect of carbon nanotube waviness on active damping of laminated hybrid composite shells. Acta Mech., 226(6), 2015, 2035-2052.
[36] Kundalwal, S.I., Suresh, Kumar, R., Ray, M.C., Smart damping of laminated fuzzy fiber reinforced composite shells using 1–3 piezoelectric composites. Smart. Mater. Struct., 22(10), 2013, 105001.
[37] Suresh Kumar, R., Kundalwal, S.I., Ray, M.C., Control of large amplitude vibrations of doubly curved sandwich shells composed of fuzzy fiber reinforced composite facings. Aerosp. Sci. Technol., 70, 2017, 10-28.
[38] Kundalwal, S.I., Shingare, K.B., Rathi, A., Effect of flexoelectricity on the electromechanical response of graphene nanocomposite beam, Int. J. Mech. Mater. Des., 2018, https://doi.org/10.1007/s10999-018-9417-6.
[39] Kundalwal, S.I., Megui, S.A., Weng, G.J., Strain gradient polarization in graphene. Carbon, 117, 2017, 462-472.
[40] Chu, L., Dui, G., Ju, C., Flexoelectric effect on the bending and vibration responses of functionally graded piezoelectric nanobeams based on general modified strain gradient theory. Compos. Struct., 186, 2018, 39-49.
[41] Wei, G., Shouwen, Y., Ganyun, H., Finite element characterization of the size-dependent mechanical behaviour in nanosystems. Nanotechnology, 17, 2016, 1118-1122.
[42] Darrall, T.B., Hadjesfandiari, A.R., Dargush G.F., Size-dependent piezoelectricity: A 2D finite element formulation for electric field-mean curvature coupling in dielectrics. Eur. J. Mech. A. Solids, 49, 2015, 308-320.
[43] Shijie, Z., Xie, Z., Wang, H., Theoretical and finite element modeling of piezoelectric nanobeams with surface and flexoelectricity effects. Mech. Adv. Mater. Struct., 2018, https://doi.org/10.1080/15376494.2018.1432799.
[44] Soleimani, I., Beni Y.T., Vibration analysis of nanotubes based on two-node size dependent axisymmetric shell element, Arch. Civ. Mech. Engg., 18, 2018, 1345–1358.
[45] Mohtashami, M., Beni Y.T., Size-Dependent Buckling and Vibrations of Piezoelectric Nanobeam with Finite Element Method. Iran. J. Sci. Tech. Trans. Civil Engg., 2018, https://doi.org/10.1007/s40996-018-00229-9.
[46] Moura, A.G., Erturk, A., Electroelastodynamics of flexoelectric energy conversion and harvesting in elastic dielectrics. J. Appl. Phys., 121, 2017, 064110.
[47] Liang, X., Zhang, R., Hu, S., Shen, S., Flexoelectric energy harvesters based on Timoshenko laminated beam theory. J. Intell. Mater. Syst. Struct., 28(15), 2017, 2064-2073.
[48] Liang, X., Hu, S., Shen, S., Nanoscale mechanical energy harvesting using piezoelectricity and flexoelectricity. Smart. Mater. Struct., 26, 2017, 035050.
[49] Deng, Q., Kammoun, M., Erturk, A., Sharma, P., Nanoscale flexoelectric energy harvesting. Int. J. Solids Struct., 51, 2014, 3218-3225.
[50] Wang, K.F., Wang, B.L., Non-linear flexoelectricity in energy harvesting. Int. J. Eng. Sci., 116, 2017, 88-103.
[51] Hu, S., Shen, S., Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Sci. China Phys. Mech. Astron., 53(8), 2010, 1497-1504.
[52] Shen, S., Hu, S., A theory of flexoelectricity with surface effect for elastic dielectrics. J. Mech. Phys. Solids, 58(5), 2010, 665-677.
[53] Li, A., Zhou, S., Qi, L., Chen, X., A reformulated flexoelectric theory for isotropic dielectrics. J. Phys. D. Appl. Phys., 48(46), 2015, 465502.
[54] Toupin, R.A., The Elastic Dielectric. J. Ration. Mech. Anal., 5, 1956, 849-915.
[55] Kuang, Z-.B., Variational principles for generalized dynamical theory of thermopiezoelectricity. Acta Mech., 203(1-2), 2009, 1-11.
[56] Beni, Y.T., Size-dependent electromechanical bending, buckling, and free vibration analysis of functionally graded piezoelectric nanobeams. J. Intell. Mater. Syst. Struct., 27(16), 2016, 2199-2215.
[57] Omidian, R., Beni, Y.T., Mehralian, F., Analysis of size-dependent smart flexoelectric nanobeams. Eur. Phys. J. Plus., 132(11), 2017, 481.
[58] Eltaher, M.A., Emam, S.A., Mahmoud, F.F., Free vibration analysis of functionally graded size-dependent nanobeams. Appl. Math. Comput., 218(14), 2012, 7406-7420.
[59] Eltaher, M.A., Emam, S.A., Mahmoud, F.F., Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct., 96, 2013, 82-88.