TECHNICAL PAPER Vibration analysis of size-dependent bimorph functionally graded piezoelectric cylindrical shell based on nonlocal strain gradient theory Fahimeh Mehralian 1 • Yaghoub Tadi Beni 2 Received: 21 June 2017 / Accepted: 17 December 2017 / Published online: 15 January 2018 Ó The Brazilian Society of Mechanical Sciences and Engineering 2018 Abstract Functionally graded piezoelectric materials (FGPMs) have emerged as promising candidates for electronic nanodevices. In this paper, a study on dynamic response of a bimorph FGP cylindrical nanoshell based on nonlocal strain gradient theory is presented. The material properties are assumed to be variable across thickness direction according to power law distri- bution. The electric potential is considered to be quadratic through thickness direction. The governing equations and boundary conditions are obtained on the basis of first-order shear deformation theory using Hamilton’s principle. As case study, free vibration of simply supported bimorph FGP cylindrical nanoshell is studied and the influences of different parameters on natural frequency are illustrated. The results obtained provide detailed insights into dynamic response of bimorph FGP cylindrical nanoshell and provide evidence for its size dependency especially by increase in thickness and decrease in length, which is an important conclusion for obtaining appropriate functionality in sensors and actuators. Keywords Bimorph functionally graded piezoelectric materials  Vibration  First-order shear deformation theory  Nonlocal strain gradient theory  Nanotube 1 Introduction Recently, functionally graded piezoelectric materials (FGPMs) have been paid a lot of attention by many researchers to convert mechanical vibration energy to electrical energy and vice versa. They are widely used in transducers especially in ultrasonic sensors and actuators [1, 2]. A FGP cylindrical shell is usually as main element of transducers and their vibration characteristics should be well known. The vibration of FGP cylindrical shell was investigated by Razavi et al. and the influences of different geometrical parameters on dynamic response of FGP shell were illustrated [3]. The dynamic electromagnetoelastic responses of FGP hollow cylindrical shell under axial uniform magnetic field and subjected to mechanical loads and electric excitation was investigated by Xie et al. and the influence of gradient index was shown significant [4]. Free vibration of FG cylindrical shell with embedded piezoelectric layers was studied using three dimensional theory by Alibeigloo et al. and the effects of different parameters on frequency were shown [5]. Due to a major drawback in utilizing traditional piezo- electric sensors and actuators made of several different piezoelectric layers which results in stress concentration or interfacial debonding, functionally graded piezoelectric nanodevices are presented. FGPMs are considered attrac- tive candidates for future electronic devices. Accordingly, the comprehensive understanding of their behavior is, therefore, a major area of research for reliable design of smart structures [6]. However, little has been carried out on FGPM structures in research literature. Nonlinear static and free vibration of FGP plates under different mechanical and electrical loading were studied by Behjat et al. using finite- element method and the influences of different parameters like boundary conditions on natural frequency were illus- trated [8]. Nonlinear thermo-electro-mechanical response of FGP actuators was investigated by Komijani et al. and the effects of different parameters were presented [9]. Buckling of FGP cylindrical shell was studied by Technical Editor: Ka ´tia Lucchesi Cavalca Dedini. & Yaghoub Tadi Beni tadi@eng.sku.ac.ir 1 Mechanical Engineering Department, Shahrekord University, Shahrekord, Iran 2 Faculty of Engineering, Shahrekord University, Shahrekord, Iran 123 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018) 40:27 https://doi.org/10.1007/s40430-017-0938-y