Size-dependent thermal-electro-mechanical behaviors of a piezo-flexoelectric micro-beam based on nonlocal strain gradient theory and dual-phase-lagging heat model

With the rapid development and applications of micro/nano electromechanical systems in engineering fields, piezoelectric materials have been widely used in these systems due to their excellent physical and mechanical properties. Generally, these systems work in complex thermoelastic environments, to depict the thermal-electro-mechanical behaviors of piezoelectric material structures at microscale, the existing theories need to be developed due to the arising size-dependent effect, thermal relaxation effect and flexoelectric effect etc. To this end, the present work intends to generalize the size-dependent theories by considering the thermal-electro-mechanical coupling behaviors in piezoelectric material structures at microscale. Based on the nonlocal strain gradient theory and the generalized heat conduction model, a new model is established and the corresponding constitutive equations and the governing equations of equilibrium are derived. To illustrate the application of this new model, a piezoelectric micro-beam with flexoelectricity is considered and the closed form solutions of bending deflection and the free vibration response are obtained in terms of the boundary conditions. Based on the numerical results, the size-dependent effect, the piezo-flexoelectric effect and the thermal-lagging effect on the bending and vibration responses of the microbeam are examined and illustrated graphically. This work may be potentially helpful for the efficient, optimized and adaptive designs of the NEMS/MEMS devices.