Buckling and post-buckling analysis of neo-Hookean plates and its correlation to a direct energetic stability analysis (Nonlinear elasticity)

The elastic stability of buckling and post-buckling deformations for incompressible neo-Hookean rectangular plate subjected to a uni-axial thrust is investigated. The buckling deformation is described by the small deformation superposed on finite homogeneous deformations. Throughout the investigation the thickness of the plate is not limited. The resulting nonlinear boundary value problem is analyzed by using the perturbation expansion method in which an associated linear problem is solved at each order.; Buckling onset is determined from the first order expansion and can occur in either flexural or barreling mode shapes with any integer number of half wavelength in the direction of thrust. The solutions from the higher order expansion correspond to post-buckling deformations. The higher order problems inherit information from problems of the previous order, both by the expansion procedure and by the application of mathematical solvability conditions. The stability criterion for post-buckling deformations is based on energy competition between the buckled deformation and unbuckled homogeneous deformation in the vicinity of buckling onset. It is formally established that the energetic favoribility correlates with the load following character of the buckled solutions (progressive buckling vs. snap buckling).; Based on the expressions obtained by these procedures, it is found that the flexural buckled deformation is energetically favored over the unbuckled homogeneous deformation when mode number is small, otherwise the homogeneous deformation is favored. The barreling buckled deformation is always energetically favored over the homogeneous deformation. This contrasts with previous results of Sawyers and Rivlin (1982), who obtain essentially opposite results for the elastic stability of homogeneous deformation. The approach to evaluate the elastic stability by using the perturbation method considered in this research gives more insights to understand the buckling phenomena and is systematically applicable to higher order analysis.; Besides the main topic of stability evaluation, several approximate schemes for the critical buckling load in neo-Hookean three-ply sandwich type plate were developed in view of practical application. The schemes are based mostly on the Rayleigh quotients approach and trial solutions. These schemes can be expanded to general multi-ply composite plates and so reduce the effort to determine the critical buckling load.