The interplay of pre-stress and higher-order basis functions in Galerkin-based postbuckling analysis of a von Karman plate

Abstract

The Galerkin method is widely used to study postbuckled von Karman plates and often yields good approximations with only a few low-order basis functions such as low-frequency vibrational modes. However, the emergence of piezoelectric bistable microelectromechanical systems (MEMS) presents new challenges. In these systems, the in-plane stress can vary over a wide range through fabrication or piezoelectric actuation. It remains unclear which basis functions are required to ensure accurate Galerkin solutions throughout this wide pre-stress range and deep into the postbuckling regime. In this work, we present a systematic study of the interplay between pre-stress and higher-order basis functions. Starting from von Karman plate theory, we derive the governing equations for both transverse deflection and in-plane stresses using an Airy stress function formulation. We show that higher-order basis functions are essential for accurately representing the in-plane stresses, even when the deflection itself is fully described by low-order basis functions. Additionally, higher-order basis functions become increasingly important for accurately capturing the static deflection and vibrational eigenmodes at elevated pre-stress levels. By analyzing linear and nonlinear coupling coefficients, we construct effective basis function sets with minimal degrees of freedom. We also introduce a distortion measure to quantify the contribution of higher-order modes and to identify the pre-stress levels beyond which a given basis set becomes insufficient for Galerkin solutions. Our results show good agreement with finite element solutions and provide practical criteria for assessing and constructing basis function sets in Galerkin methods for the study of postbuckled plates over wide pre-stress ranges. These insights enable reliable Galerkin-based modeling of the static and dynamic behavior of bistable MEMS and postbuckled plates in general.