Keywords: hyperelastic materials, soft materials, high-order plate theories, finite elements method, free vibration, stress analysis.

This study explores higher-order 2D plate finite elements for the stress and modal analysis of soft structures. The problem is established within the domain of the Carrera Unified Formulation (CUF), integrating available hyperelastic models in a unified, fully nonlinear Finite Element (FE) approach under a Total Lagrangian formulation. The matrix form of the governing equations for the nonlinear static and free vibration analysis is carried out through the Principle of Virtual Displacements (PVD), obtaining a pure displacement-based FE model. The numerical procedure is based on a Newton-Raphson linearization approach and path-following methods. The primary objective of this research is to analyze the three-dimensional stress state of soft structures in the large strain regime and how highly nonlinear pre-stressed conditions affect natural frequencies and modal shapes. The proposed results are compared with the FE solution obtained through classical models available in commercial software. The numerical results proposed assess the efficiency of the accuracy of higher-order 2D models for displacements, strains, and modal behaviour of soft structures.

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