An electroelastic solution for functionally graded piezoelectric material beams with different moduli in tension and compression

In this study, we derived an electroelastic solution of functionally graded piezoelectric beams with bimodular effect under the combined action of uniformly distributed loads, concentrated loads, and bending moments. We first presented the simplified mechanical model established in previous study, and this model is based on a complete partition of tension and compression. Based on the theory of piezoelectric elasticity and functional graded property, we assumed that material parameters vary exponentially along the thickness, and the exponential functions adopt two different forms in the tensile zone and the compressive zone. Using stress function method, we obtained an analytical solution of the problem in which unknown constants are determined by boundary conditions and continuity conditions on the neutral layer. The results indicate that the introduction of exponential functions makes us be able to effectively distinguish the relative magnitude of the tensile modulus, the compressive modulus and the modulus on the neutral layer. Thus, the variation in the stress and the displacement including the electric displacement with the thickness are discussed in detail. The results obtained are helpful to refined analysis and design of functionally graded piezoelectric materials and structures with obvious bimodular effect.