Size-dependent electromechanical coupling behaviors of circular micro-plate due to flexoelectricity

In this paper, the flexoelectric theory is re-expressed by a set of orthogonal components of strain gradient tensor. The general formulations of flexoelectric theory in orthogonal curvilinear coordinates are derived and, then, are specified for the case of cylindrical coordinates. A flexoelectric circular micro-plate model is established based on the current formulations in cylindrical coordinates to evaluate its size-dependent static and dynamic responses. The governing equations, boundary conditions and initial conditions are obtained according to the Hamilton's principle. A static bending problem of simply supported axisymmetric circular micro-plate is solved in two cases, of which one is subjected to a distributed load and the other is subjected to a voltage across the plate thickness. And the free vibration problem of a simply supported circular microplate is also analyzed. The bending numerical results show that both the deflection and the electric potential exhibit obvious size dependency in the two cases. Both the induced electric potential in direct flexoelectric effect and the induced deflection in inverse flexoelectric effect decrease as the decrease in flexoelectric coefficient and even disappear when the flexoelectric coefficient equals zero. Moreover, the numerical results of free vibration demonstrate the dimensionless natural frequency shows obvious size effect, while the influence of flexoelectric coefficient on dimensionless natural frequency is negligible.