This paper establishes an analytical model for the size-dependent flexural wave band structures of a piezoelectric phononic crystal (PC) Timoshenko nanobeam, considering the shear deformation and rotational inertia effects. The governing equation is derived based on the surface piezoelectricity energy and modified couple stress theory in the bulk and surface layer of the piezoelectric PC nanobeam, respectively. As a main objective of this article, the surface effects including the surface elasticity, residual surface stress and surface piezoelectricity on the frequency bandgaps in the wave propagation band structures are investigated for a piezoelectric PC Timoshenko nanobeam. Also, the effects of some geometrical parameters on the band structures are studied in details. An analytical approach is proposed for the problem. The obtained results for a piezoelectric PC Timoshenko nanobeam are compared with those of previous researches for a piezoelectric PC Euler-Bernoulli nanobeam. It is concluded that the geometrical parameters and residual surface elasticity affect the location and width of bandgaps in the frequency band structures.

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