Title : Mathematical Modeling and Finite Element Computation of Cosserat Elastic Plates

Place : Monzon M201, UPRM

Time : 10:00 AM on April 17, 2013

Abstract

In this dissertation we present the mathematical modeling of Cosserat elastic plates and their Finite element computation. We develop the mathematical model for bending of Cosserat elastic plates, which assumes physically and mathematically motivated approximations over the plate thickness for stress, couple stress, displacement, and microrotation. The approximations are consistent with the three-dimensional Cosserat elasticity equilibrium equations, boundary conditions and the constitutive relationships. The Generalized Hellinger-Prange-Reissner Principle allows us to obtain the equilibrium equations, constitutive relations and optimal value for the minimization of the elastic energy with respect to the splitting parameter in the approximation of the σ33 stress component.

The comparison of the maximum vertical deflection for simply supported square plate with the analytical solution of the three-dimensional Cosserat elasticity confirms the high order of approximation of the three-dimensional (exact) solution. The computations produce a relative error of the order 1% in comparison with the exact three-dimensional solution that is stable with respect to the standard range of the plate thickness. The results are compatible with the precision of the well-known Reissner model used for bending of simple elastic plates.

For the Finite Element formulation, we present the Cosserat plate field equations as an elliptic system of nine differential equations in terms of the kinematic variables. The system includes an optimal value of the splitting parameter, which is the minimizer of the Cosserat plate stress energy. We propose the Finite Element Method for Cosserat elastic plates based on the efficient numerical algorithm for the calculation of the optimal value of the splitting parameter and the computation of the corresponding unique solution of the weak problem. We provide the numerical validation of the proposed Finite Element Method and showed that it converges to the analytical solution with optimal linear rate of convergence. The asymptotic order of the computational complexity of the proposed Finite Element algorithm is shown to be the same as of the classical Finite Element.

We provide the Finite Element modeling of the bending of clamped Cosserat elastic plates of arbitrary shapes under different loads. The numerical results are obtained for the elastic plates made of dense polyurethane foam used in structural insulated panels.