Pure bending of a piezoelectric layer in second gradient electroelasticity theory

The semi-inverse analytical solution of a pure bending problem for a piezoelectric layer is developed in the framework of linear electroelasticity theory with strain gradient and electric field gradient effects. The simplified gradient theory of transversely isotropic material with a single additional length scale parameter is considered. A two-dimensional solution is derived assuming plane strain state of a layer (cylindrical bending of a plate) and low dielectric properties of the surrounding medium. The electromechanical response of a layer is found under conditions of prescribed bending moments at the end faces. Boundary conditions on the top and bottom surfaces of a layer are satisfied exactly. The analytical solution is validated based on numerical finite element modeling. It is shown that the obtained solutions can be used for the validation of size-dependent beam and plate models in the second gradient electroelasticity theory.