Reissner's plate theory in the framework of asymmetric elasticity.

摘要

The purpose of this thesis is to develop a new mathematical model for the bending of thin elastic plates with microstructure. Our approach is based on a generalization of the classical Reissner plate theory, which takes into account the transverse microrotation of the plates. Our model assumes polynomial approximations over the plate thickness of asymmetric stress, couple stress, displacement, and microrotation, which are consistent with the elastic equilibrium, boundary conditions and the constitutive relationships. We use a Cosserat free elastic energy function which includes the energy of the transverse shear couple stress. The application of the method of Lagrange multipliers to the free elastic energy function leads to a system of 9 equations describing the bending (6 equations) and the twisting (3 equations) of the plate. Analytical solutions for the deflection of the plate are calculated for a square plate made of syntatic foam. The Fourier series method is applied. The solutions are compared with a model developed by Eringen and also with solutions obtained from the classical theory. The results illustrate the influence of transverse microrotations on the bending of the rectangular plate.