Piezoelectric nanomaterials (PNs) are attractive for applications including sensing, actuating, energy harvesting, among others in nano-electro-mechanical-systems (NEMS) because of their excellent electromechanical coupling, mechanical and physical properties. However, the properties of PNs do not coincide with their bulk counterparts and depend on the particular size. A large amount of efforts have been devoted to studying the size-dependent properties of PNs by using experimental characterization, atomistic simulation and continuum mechanics modeling with the consideration of the scale features of the nanomaterials. This paper reviews the recent progresses and achievements in the research on the continuum mechanics modeling of the size-dependent mechanical and physical properties of PNs. We start from the fundamentals of the modified continuum mechanics models for PNs, including the theories of surface piezoelectricity, flexoelectricity and non-local piezoelectricity, with the introduction of the modified piezoelectric beam and plate models particularly for nanostructured piezoelectric materials with certain configurations. Then, we give a review on the investigation of the size-dependent properties of PNs by using the modified continuum mechanics models, such as the electromechanical coupling, bending, vibration, buckling, wave propagation and dynamic characteristics. Finally, analytical modeling and analysis of nanoscale actuators and energy harvesters based on piezoelectric nanostructures are presented.

Soft active materials that show a response under external stimuli such as light, electricity, heat and magnetic fields have wide applications in the fields of soft robotics, flexible electronics, biomedicine, etc. Recently developed hard-magnetic soft (HMS) materials that undergo rapid transformations under magnetic actuations belong to a new class of soft active materials. In order to fulfill the potential applications of HMS structures, it is of great importance to investigate their mechanical responses. In this work, based on the principle of minimum potential energy, a theoretical model for HMS beams with the consideration of exact geometric nonlinearity is proposed, which is capable of characterizing the complex transformations of the HMS beam. The Galerkin method is utilized to solve the governing equation of the system. By comparing with the recent simulation and experimental data, the proposed model is validated. It is found that with the designed residual magnetic flux density in the HMS beam, the initially straight beam would deform into complex configurations under specified magnetic actuation. Interestingly, the crawling of inchworms, grasping of tongs and slithering of snakes can be well imitated by the HMS beam as a consequence. The developed theoretical model could serve as a benchmark for the analyses of HMS beams in the future, and the observed phenomena could shed light on the design of soft robotic devices used in extreme environments based on HMS structures.

The size-dependent bending and vibration behaviors of a clamped piezoelectric circular nanoplate are investigated by using a modified Kirchhoff plate model. The flexoelectricity, the surface effect and the non-local elastic effect are taken into account in the modified model by decomposing the electric Gibbs free energy into the bulk and surface parts and including the strain gradient and the electric field gradient terms into the bulk energy density function. Different from the results predicted by the classical plate model, the proposed model predicts size-dependent behaviors of the piezoelectric thin plate with nanoscale thickness. Comparisons among the models considering the flexoelectricity, the surface effect and the non-local elastic effect individually, the current model and the classical model are also given in this study. Simulation results indicate that the electromechanical coupling properties, the transverse displacements and the resonant frequencies of the plate are significantly influenced by each individual effect as well as their combined effects. It is also indicated that such effects are affected by the external applied electric potential and the plate geometries. Neglecting any individual effect may induce inaccurate characterization of the electromechanical coupling of the piezoelectric nanoplate. Therefore, the current plate model is expected to provide more accurate predictions of the electromechanical coupling and the mechanical behaviors of piezoelectric circular nanoplate-based devices in the nanoelectromechanical systems.

This work presents a two dimensional theory for a piezoelectric/piezomagnetic bilayer nanoplate in coupled extensional and flexural vibrations with both flexoelectric and surface effects. The magneto-electro-elastic (MEE) coupling equations are derived from three-dimensional equations and Kirchhoff plate theory. Based on the developed theory, a piezoelectric/piezomagnetic nano energy harvester is proposed, which can generate electricity under time-harmonic applied magnetic field. The approximate solutions for the mechanical responses and voltage of the energy harvester are obtained using the weighted residual method. Results show that the properties of the proposed energy harvester are size-dependent due to the flexoelectric and surface effects, and such effects are more pronounced when the bilayer thickness is reduced to dozens of nanometers. It is also found that the magnetoelectric coupling coefficient and power density of the energy harvester are sensitive to the load resistance, the thickness fraction of the piezoelectric or the piezomagnetic layer and damping ratios. Moreover, results indicate that the flexoelectric effect could be made use to build a dielectric/piezomagnetic nano energy harvester. This work provides modeling techniques and numerical methods for investigating the size-dependent properties of MEE nanoplate-based energy harvester and could be helpful for designing nano energy harvesters using the principle of flexoelectricity.

In this work, we present a finite element (FE) approach for analyzing flexoelectric beam energy harvesters with nonuniform cross-section. A two nodes finite element model with 10 degrees of freedom is proposed and Hermite polynomials are adopted to satisfy the higher order continuity requirement of the electric potential and displacement due to flexoelectricity. The proposed model can also be simplified to investigate a flexoelectric actuator, for which analytical solutions are obtained and thus can be used to validate the proposed FE method. Results indicate the performance of flexoelectric energy harvesters with nonuniform cross-section can exceed that of rectangular ones.