Application of nonlocal modified couple stress to study of functionally graded piezoelectric plates

In this paper, we study a size-dependent functionally graded piezoelectric (FGP) plate model based on the combination of the modified couple stress and nonlocal elasticity theories, named nonlocal modified couple stress theory (NMCS). Continues variation of the material properties (chosen from PZT family) through the thickness of the plate is considered according to the power law distribution. Employing Hamilton's principle via thin plate theory, we obtain governing equations of motion for two cases: 1) using only the nonlocal parameter for stress tensor, 2) using the nonlocal parameter for both stress and couple stress tensors. Navier's approach has been used to investigate free vibration of FGP under simply supported edge conditions, analytically. In the first step, we compare our results with the obtained results for the vibration of FGP in the previous studies using the modified couple stress theory and nonlocal elasticity theory, separately. Then, the effects of the aspect ratio, the length of the plate, the side to thickness ratio, FG power index, mode numbers, and the nonlocal and length scale parameters on normalized natural frequencies are investigated.