Analytical study of large deflection of piezoelectric and elastically-bossed plate under pretension

Large deflection of a piezoelectric layered plate with an elastic boss due to lateral load is studied. von Karman´s plate theory of large deflection is utilized and extended to a symmetrically-layered case including a piezoelectric layer. Governing equations thus derived are simplified by neglecting the arising nonlinear terms, yielding a modified Bessel or Bessel equation for the lateral slope, depending on the relative magnitude of the piezoelectric load. The associated analytical solutions were developed by imposing the end support condition and the interface continuity between the center boss and the annular plate. Lateral deflection and curvature were further derived by the use of the corresponding recurrence relations. The approach was implemented with typical silicon-based materials used in miniaturized devices, for various geometrical sizes of the center boss and a wide range of pretension, as a parametric study. For a nearly monolithic plate with a narrow and shallow boss under a very low applied voltage, the results agree well with those available in literature, thus the developed approach is validated. For typically bossed plates, piezoelectric effect appears to be present only in a low pretension condition. In this case, the higher the applied voltage, the greater the normalized center deflection and lateral curvature are observed. The range of initial tension for the validity of plate behavior is lowered as well, as the applied voltage is raised.