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A continuum model of piezoelectric material with strain and electric gradient effects is proposed in this work. An energy variational principle with strain, strain gradient, electric field and electric field gradient as independent variables is presented to develop the equilibrium equations and all boundary conditions. Moreover, two internal length characteristics of the underlying microstructure are introduced in the constitutive equations. This model is employed to an anti-plane problem of the piezoelectric material, and a general solution is obtained in polar coordinates. Special attention is paid to an infinite piezoelectric medium with a microvoid. It is found that electromechanical fields predicted by the gradient model not only depend on the material parameters, but also on the relative size of the microvoid with respect to the scale of microstructure. More importantly, some new interesting phenomena are observed due to the gradient effects of strain and electric fields.

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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.