On Mixed Variational Large Deformation Theories for Corrections of Classical von Kármán Theory of Thin Plates

For thin plates, linear deformation theory is only valid, when the plate deformation is on the order of the thickness of the plate. To ensure validity of results for larger deformations, geometrically non-linear strains must be used to properly reflect the behavior of the plates deformations. One of the earliest and most widely known non-linear plate theories is von Kármán thin plate theory, which was proposed by Theodore von Kármán in 1910. This thesis reexamines the classical von Kármán thin plate theory and seeks to re-formulate the governing equilibrium equations for thin plates undergoing large geometrically nonlinear deformations in terms of the plate displacements, using the Hu-Washizu mixed field variational statement. Additionally, other mixed field and single field variational statements are examined for the same purpose in order to determine any differences between them. Governing equilibrium equations yielded from each variational statement are compared to examine differences in the terms obtained. Those governing equilibrium equations will then be compared against existing formulations for the governing equilibrium equations for von Kármán thin plate theory, and their solutions found for the cases of statically loaded simply supported, statically loaded clamped plates, and simply supported free and forced vibrations to assess the validity of the governing equilibrium equations derived. Improvements to the formulation of thin plate theory allows for improvements to the design process of ship hulls, aircraft skins, and automobile bodies that incorporate thin plates. Improvements to accurate modeling of thin plates, could lead to reductions in the weight and cost of systems that use thin plates.