Application of Groebner bases to geometrically nonlinear analysis of axisymmetric circular isotropic plates.

摘要

This thesis demonstrates a new application of Groebner basis by finding an analytical solution to geometrically nonlinear axisymmetric isotropic circular plates. Because technology is becoming capable of creating materials that can perform materially in the linear elastic range while experiencing large deformation geometrically, more accurate models must be used to ensure the model will result in realistic representations of the structure. As a result, the governing equations have a highly nonlinear and coupled nature. Many of these nonlinear problems are solved numerically. Since analytic solutions are unavailable or limited to only a few simplified cases, their analysis has remained a challenging problem in the engineering community.;On the other hand, with the increasing computing capability in recent years, the application of Groebner basis can be seen in many areas of mathematics and science. However, its use in engineering mechanics has not been utilized to its full potential. The focus of this thesis is to introduce this methodology as a powerful and feasible tool in the analysis of geometrically nonlinear plate problems to find the closed form solutions for displacement, stress, moment, and transverse shearing force in the three cases defined in Chapter 4.;The procedure to determine the closed form solutions developed in the current study can be summarized as follows: 1) the von Karman plate theory is used to generate nonlinear governing equations, 2) the method of minimum total potential energy combined with the Ritz methodology converts the governing equations into a system of nonlinear and coupled algebraic equations, 3) and Groebner Basis is employed to decouple the algebraic equations to find analytic solutions in terms of the material and geometric parameters of the plate. Maple 13 is used to compute the Groebner basis. Some examples of Maple worksheets and ANSYS log files for the current study are documented in the thesis.;The results of the present analysis indicate that nonlinear effects for the plates subjected to larger deformation are significant for predicting the deflections and stresses in the plates and necessary compared to those based on the linear assumptions. The analysis presented in the thesis further shows the potential of the Groebner basis methodology combined with the methods of Ritz, Galerkin, and similar approximation methods of weighted residuals which may provide a useful procedure of analysis to other nonlinear problems and a basis of preliminary design in engineering practice.