Piezoelasticity solutions for functionally graded piezoelectric beams

This paper considers the plane stress problem of generally anisotropic piezoelectric beams with the coefficients of elastic compliance, piezoelectric and dielectric impermeability being arbitrary functions of the thickness coordinate. Firstly, the partial differential equations for the plane problem of anisotropic functionally graded piezoelectric materials are derived, which the stress function and electric displacement function satisfy. Secondly, the stress and electric displacement functions are assumed in forms of polynomials of the longitudinal coordinate, so that the stress and electric displacement functions can be acquired through successive integrations. The analytical expressions of axial force, bending moment, shear force, displacements, electric displacements and electric potential are then deduced. Thirdly, the stress and electric displacement functions are employed to solve problems of functionally graded piezoelectric plane beams, with the integral constants completely determined from boundary conditions. Two piezoelasticity solutions are thus obtained, for cantilever beams subjected to shear force and point charge applied at the free end, for cantilever beams subjected to uniform load. These solutions can be easily degenerated into the piezoelasticity solutions for homogeneous anisotropic piezoelectric beams. Finally, a numerical example is presented to show the application of the proposed method to a specific case.