On the validity of plane state assumptions in the bending of bimorph piezoelectric cantilevers

We assess the accuracy of plane state assumptions in modeling the bending of bimorph piezoelectric cantilevers. Reference solutions are obtained by means of fully three-dimensional simulations. We solve a model, compliant with the standard theoretical background, able to deal with large deflections, and aimed at estimating the electrical energy harvesting potential. We show that the two assumptions of plane stress and plane strain yield very different results in terms of strain and therefore lead to substantially different estimates of electrical power. The plane stress state is the best approximation of narrow cantilevers, while the plane strain is suited for wide ones. The validity ranges are defined for each of the two, in terms of the generated electrical power. More specifically, the plane stress assumption better approximates the reference solution in the range W/H<10. By contrast, the plane strain condition yields results closer to three-dimensional solution for ratios W/H>103 with W the width and H the total thickness of the solid. Other aspects are affected by the modeling approach, such as the estimation of maximum deformation and, to a minor extent, of resonance frequencies.