An analytical–numerical framework for vibration and pull-in analysis of auxetic sandwich microplates with FG PZT/Ag facesheets

This paper presents a design-oriented analytical–numerical framework for vibration and electrostatic pull-in of a three-layer microplate with a re-entrant auxetic core and functionally graded (FG) PZT/Ag facesheets (PZT: lead zirconate titanate; Ag: silver). Piezoelectric coupling is not considered; the facesheets are modeled only through graded elastic properties. The aim is to provide a compact and validated workflow for predicting both resonance and pull-in of auxetic-core FG sandwich microplates under simply supported boundaries. A harmonic (series-compliance) shear-transfer correction is incorporated at the laminate level, and the transverse dynamics are solved using a Mindlin–Navier series solution. The framework is verified against an in-house Mindlin plate FEM and cross-validated using ABAQUS composite-shell simulations. Pull-in is assessed using a Mindlin-consistent single-mode ROM based on static condensation and numerical projection of the electrostatic pressure, yielding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>I</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>118.16</mml:mn> <mml:mo> </mml:mo> <mml:mi mathvariant="normal">V</mml:mi> </mml:mrow> </mml:math> for a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.52</mml:mn> <mml:mo> </mml:mo> <mml:mi>μ</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> </mml:math> gap. Size effects are examined via MCST through the normalized parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi mathvariant="script">l</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>/</mml:mo> <mml:mi>h</mml:mi> </mml:mrow> </mml:math> , reported as a sensitivity to the assumed (or calibrated) length scale.