On a consistent finite-strain plate theory for incompressible hyperelastic materials

In this paper, a consistent finite-strain plate theory for incompressible hyperelastic materials is formulated. Within the framework of nonlinear elasticity and through a variational approach, the three-dimensional (3D) governing system is derived. Series expansions of the independent variables in the governing system are taken about the bottom surface of the plate, which, together with some further manipulations, yield a 2D vector plate equation. Suitable position and traction boundary conditions on the edge are also proposed. The 2D plate system ensures that each term in the variations of the generalized potential energy functional attains the required asymptotic order. The associated weak formulation of the plate model is also derived, and can be simplified to accommodate distinct types of practical edge conditions. To demonstrate the validity of the derived 2D vector plate system, the pure finite-bending of a plate made of an incompressible neo-Hookean material is studied. Both the exact solutions and the plate solutions of the problem are obtained. Through some comparisons, it is found that the plate theory can provide second-order correct results.