Finite element modeling of free vibration of cracked nanoplates with flexoelectric effects

This work for the first time combines the theory of high-order shear deformation and the phase-field approach to simulate the free vibration response of cracked nanoplates taking into account the flexoelectric effect, which is created by a combination of charge polarization and variation in strain. The oscillation equation is established by the finite element method, and comparisons of free vibration with analytical results, numerical methods, and experimental methods have proven the correctness of calculation theory. This paper also investigates the effects of some materials and geometry parameters on the free vibration response of the cracked nanoplates, and the numerical results demonstrate that for the crack, the thickness ratio of the plate edges has a significant effect on the value of the natural frequency of the vibration, the shape of the vibration mode, and most importantly, the polarization associated with each vibration mode shape of the plate.