Stress-based dimensional reduction and dual-mixed hp finite elements for elastic plates

Abstract

Starting from the linearized weak forms of the kinematic equation and the angular momentum balance equation of three-dimensional non-linear elasticity, a stressbased dimensional reduction procedure is presented for elastic plates. After expanding the three-dimensional non-symmetric stress tensor into power series with respect to the thickness coordinate, the translational equilibrium equations, written in terms of the expanded stress coefficients, are satisfied by introducing first-order stress functions. The symmetry of the stress field is satisfied in a weak sense by applying the material rotations as Lagrangian multipliers. The seven-field plate model developed in this way employs unmodified three-dimensional strain-stress relations. On the basis of the dimensionally reduced plate model derived, a new dual-mixed plate bending finite element model is developed and presented. The numerical performance of the hp-version plate elements is investigated through the solutions of standard plate bending problems. It is shown that the modeling error of the stress-based plate model in the energy norm is better than that of the displacement-based Kirchhoff- and Reissner-Mindlin plate models. The numerical solutions and their comparisons to reference solutions indicate that the dual-mixed hp elements are free from locking problems, in either the energy norm or the stress computations, both for h- and p-extensions, and the results obtained for the stresses are accurate and reliable even for extremely thin plates.