A finite element model for the static and dynamic analysis of a piezoelectric bimorph

A finite element model for the static and dynamic analysis of a piezoelectric bimorph is proposed. It combines a 2D single-layer representation model (finite 2D isoparametric elements) for the mechanical displacement field with a layerwise-like approximation (finite sublayers) for the electric potential field to achieve the accurate prediction of both the mechanical displacement and electric potential fields. Linear through-the-thickness electric potential distribution is assumed only for each piezoelectric sublayer and the actual nonlinear distribution can be reached by using a fine thickness discretization. The unknown induced electric potentials are not included in the global governing equations of motion due to the introduction of an elementary condensation scheme based on the electric boundary conditions and thus the present model will not suffer from an excessive number of potential field variables. A PVDF bimorph beam and a PZT bimorph plate are used to verify the present model. Numerical examples show that the present model can well predict both the global and local responses such as mechanical displacements, modal frequencies as well as the through-the-thickness electric potentials, all in good agreement with those from full 3D finite element model or 3D elasticity theory solution. Furthermore, the conventional piezoelectric finite element models assuming a linear through-the-thickness electric potential field, which is a special case of the present electric potential representation, has been identified as suitable for the global response analysis of thin or moderately thick bimorph plates. The performance of 2D plane models has been evaluated as less accurate in predicting the static and dynamic responses if transversally isotropic piezoelectric materials are involved. The present study may help establish more accurate and efficient piezoelectric models based on better understanding to the piezoelectricity.