Strain gradient and electric field gradient effects in piezoelectric cantilever beams

A piezoelectric beam model with strain gradient and electric gradient effects is proposed. An energy variational principle with strain, strain gradient, electric field, and electric field gradient considered as independent variables is postulated to develop the governing equations and boundary conditions. Moreover, two strain gradient coefficients and one electric field gradient coefficient are introduced to account for higher order coupling effects of underlying microstructure. Then the bending problem of a cantilever beam is solved to illustrate the theory. It is found that the deflection of the cantilever may depend strongly on the so-introduced gradient effects. As a result of the electromechanical coupling, the generated electric field may also be affected by the strain gradient. However, electric field gradient has a limited effect on the electric field.