Secondary instability and mode jumping analysis of deep hygrothermally buckled cross-ply laminated plates

• Kinematic relations and dynamic governing PDEs are derived rigorously from VAM.
• Nondimensionalization scheme is carefully selected to ensure the proposed PDEs suitable for bifurcation analysis.
• A novel and simpler solution is developed to solve the two coupled compatibility and governing equations.
• The nonlinear PDEs are transformed to a system of nonlinear ODEs for investigation of effects of multi-mode interactions.
• The analytical approach can capture the mode shifting phenomenon in the primary post-buckling regime.

An analytical investigation is performed to study the secondary instability and dynamic aspects of the mode jumping in hygrothermally buckled cross-ply laminated plates. The governing partial differential equations (PDEs) and constitution relations are derived rigorously from an asymptotically correct, geometrically nonlinear theory. A novel and relatively simpler solution approach is developed to solve the two coupled fourth-order PDEs, namely, the compatibility equation and the dynamic governing equation. The von Kàmàn plate equation, namely, the coupled nonlinear governing equation, is reduced to a system of nonlinear ordinary differential equations (ODEs) by expressing the transverse deflection as a series of linear buckling modes. The ODEs, combined with the non-linear algebraic constraint equations obtained from in-plane boundary conditions, are then solved numerically under the parametric variation of the temperature and moisture. The comparison between the present method and the FEA shows that the secondary bifurcation point of the hygrothermally loaded plate is far beyond the primary buckling point, and the jump behavior cannot be predicted correctly without sufficient assumed modes, while the present method has the capability of exploring deeply into the post-secondary buckling realm and capturing the mode jumping phenomenon for various combinations of plate configurations boundary conditions. Furthermore, by monitoring free vibration along the stable primary postbuckling and the jumped equilibrium paths, we find that a mode shifting phenomenon (the exchange of vibration modes) exists in the primary post-buckling regime.