Mathematical modeling and finite element computation of cosserat elastic plates

Abstract

In this dissertation the mathematical modeling of Cosserat elastic plates and their Finite Element computation are presented. 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 is developed. 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 to obtain the equilibrium equations, constitutive relations and optimal value for the minimization of the elastic energy with respect to the splitting parameter. On of the main contributions of this dissertation is the comparison of the maximum vertical deflection for simply supported square plate with the analytical solution of the three-dimensional Cosserat elasticity. It 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, the Cosserat plate field equations are presented 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. 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 is proposed. The numerical validation of the Finite Element Method shows its convergence to the analytical solution with optimal linear rate of convergence in H1-norm. The Finite Element computation of bending of clamped Cosserat elastic plates of arbitrary shapes under different loads is provided. The numerical results are obtained for the elastic plates made of dense polyurethane foam used in structural.