# Analytical Electromechanical Modeling of Nanoscale Flexoelectric Energy Harvesting
## 1 Introduction  
With the development of nanotechnology, harvesting ambient waste energy into usable energy has drawn growing attention in the last decades. One of the aims in this field is to provide power for small electronic devices by harvesting ambient energy [1]. As stated by [PERSON] and [PERSON] [2] in their early work on harvesting ambient waste vibrational energy for microsystems, there are three basic vibration-to-electric energy conversion mechanisms: electromagnetic [2, 3], electrostatic [4, 5], and piezoelectric transduction [6, 7]. Using piezoelectric nanomaterials as ambient energy harvesting is considered as a promising way to supply normal microelectronic devices, such as environmental or biomedical devices, portable multimedia, distributed sensor networks, or mobile communication [8]. In addition, some important system-level and circuit-level works about micro-scale energy harvesting systems have been reported [9, 10, 11]. Piezoelectricity, which generally assumes a linear relationship between electric field and strain, exists only in non-centrosymmetric dielectric materials. Alternatively, flexoelectricity, which is the coupling of electrical polarization and the strain gradient, exists in a wide variety of dielectric materials and may lead to strong size-dependent properties at the nanoscale. A series of experimental [12; 13; 14] and theoretical works [15; 16; 17] for the flexoelectric effect have been reported. Flexoelectric effects may be exploited to enhance piezoelectric properties of materials [18; 19], or to enable new classes of electromechanically-coupled materials, such as lead-free \"pseudo-piezoelectric\" [16]. In addition, one can exploit flexoelectricity to construct materials of non-uniform shapes, which exhibit large strain gradients and can generate electricity despite being non-piezoelectric [20]. Recently, several comprehensive reviews on flexoelectricity of solid crystals, thin films, polymers, and living membranes have been published [21; 22; 23]. In the nano-electromechanical systems (NEMSs), such as resonators, sensors, actuators, and energy harvesters, the electromechanical coupling of the nanobeams and nanoplates with the flexoelectric effect has drawn a surge of interest due to the relatively large flexoelectric effect at the nanoscale. The electromechanical coupling responses of the nanobeams and nanoplates have been analyzed by numerical and analytical methods for static and dynamic problems [24; 25; 26; 27; 28; 17; 29]. Based on the linear piezoelectricity theory developed by [PERSON] [30], [PERSON] and [PERSON] [31] have established a theoretical framework by a variational principle for dielectrics including the electrostatic force, flexoelectricity, and surface effects. Based on the electric Gibbs energy, [PERSON] et al. [32] proposed the Euler-Bernoulli beam model to investigate the effect of surface and flexoelectricity on the coupling response of piezoelectric nanostructures. They found that the effective bending rigidity of the nanobeam enhances dramatically in nanoscale. [PERSON] and [PERSON] [33] discussed the flexoelectric effect on the mechanical and electrical properties of piezoelectric nanobeams under static bending and different mechanical boundary conditions based on the internal energy. Simulation results show that the flexoelectric effect is sensitive to the mechanical boundary conditions and the direction of applied electric field. [PERSON] et al. [34] investigated the effect of flexoelectricity and surface on buckling and vibration behavior of piezoelectric beams by use of the Euler-Bernoulli beam model. They found that the effects of surface and flexoelectricity influenced the resonance frequency of piezoelectric nanowires. Based on the Kirchhoff plate model and the extended linear piezoelectric theory, [PERSON] et al. [35] presented a finite difference method to solve the non-conventional governing equations of the cantilevered piezoelectric nanoplates. Recently, [PERSON] et al.