Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory

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A review on the mechanics of nanostructures (2018, background): “19 surface and nonlocal effects have been developed for the vibration of smart nanoscale beams such as piezoelectric and magneto-electro-elastic nanobeams [176-181] as well as nonhomogeneous nanoscale beams [144, 182-185].”
Size-dependent thermo-electrical buckling analysis of functionally graded piezoelectric nanobeams (2015, methodology): “Ke et al (2012) and Ke and Wang (2012) studied the free vibration analysis of the piezoelectric Timoshenko nanobeams under thermo-electro-mechanical field based on the Eringen’s theory.”
The size-dependent vibration of embedded magneto-electro-elastic cylindrical nanoshells (2014, positive): “Ke and Wang (2012) and Ke et al (2012) first analyzed the thermo-electro-mechanical linear and nonlinear vibration of piezoelectric nanobeams based on the nonlocal Timoshenko beam theory.”
Vibration and buckling analysis of a piezoelectric nanoplate considering surface effects and in-plane constraints (2012, background): “A (2012) strain–displacement relation (Ke & Wang 2012; Ke et al. 2012).”