A critical review of reduced one-dimensional beam models of piezoelectric composite beams

Simplified one-dimensional models for composite beams with piezoelectric layers, which are intrinsically three-dimensional structures, are important for many applications, including piezoelectric energy harvesters. To reduce the dimensionality of the system, assumptions on the stress/strain state in the transverse direction are typically made. The most common are those of null transverse stress, used for narrow beams, null transverse deformation, used for wide beams, and continuous interface strain, suited for thin piezoelectric layers (we call this assumption thin film continuous). We show that the models based on these assumptions are often used uncritically for beam geometries for which large errors may result. In particular, null transverse stress fails even for narrow beams if the thickness is much smaller than the beam width. We give clear geometric criteria that, for any geometry, allow the selection of the most accurate model among the three. We also develop a single, unified beam equation encompassing the three models and compare the analytical results from this equation with finite element simulations over a wide range of beam lengths, widths, and layer thicknesses. The selection criteria and the unified beam equation form a valuable tool for fast and accurate design of composite piezoelectric beams.