Geometrically Nonlinear Free Vibration Analysis of Axially Loaded Super-Light Auxetic Beams Reinforced by Functionally Graded Face Sheets

Purpose
This study aims to investigate the geometrically nonlinear free vibration of axially loaded super-light auxetic beams reinforced by functionally graded (FG) face sheets using an analytical method based on the multiple-scale technique.

Methods
The FG auxetic beam is modeled with two FG top/bottom layers and a re-entrant honeycomb core. Material properties of the FG layers vary with layer thickness. The nonlinear equations of motion are derived using the Hamilton principle, first-order shear deformation theory (FSDT), and von Karman relations for moderately large deformations. These equations are solved analytically using the multiple-scale technique to obtain linear/nonlinear frequencies, mode shapes, and frequency amplitude curves under different boundary conditions. A parametric study is conducted to assess the influence of honeycomb structure parameters and the power-law index of the FG layers. The results are validated against existing references.

Results
The closed-form solution allows for the investigation of nonlinear free vibrations in FG auxetic beams. Parametric studies reveal that FG face sheets can reduce beam mass by 50% with minimal frequency change compared to homogeneous face sheets. Additionally, the maximum frequency is observed for C–C boundary conditions and under tensile axial loads. The influence of axial loads is more pronounced for positive power-law FG face sheets and lighter structures. In the studied range, both lateral and axial frequencies exhibit hardening behavior, except for the case of positive power-law FG face sheets under compressive axial loads.

Conclusion
This study proposes a closed-form solution based on the multiple-scale technique for analyzing the nonlinear free vibration of auxetic beams with FG face sheet reinforcements. The findings demonstrate the effectiveness of FG face sheets in reducing beam mass while maintaining similar frequencies. The study also highlights the influence of boundary conditions, axial loads, and material properties on the nonlinear free vibration behavior.