In this paper, geometrically nonlinear analysis of functionally graded beams using the dual mesh finite domain method (DMFDM) and the finite element method is presented. The DMFDM makes use of a primal mesh of finite elements and associated approximation for the variables of the formulation and a dual mesh of control domains, which does not overlap the primal mesh, for the satisfaction of the governing equations. The dual variables can be postcomputed uniquely and accurately at the control domain interfaces. The method is used to obtain nonlinear (due to the von Kármán nonlinear strains) bending solutions of straight, through-thickness functionally graded beams using the Euler–Bernoulli and the Timoshenko beam theories. Mixed models, which contain displacements and the bending moment as degrees of freedom, and displacement models are developed. Numerical results of linear and nonlinear analyses are presented to illustrate the methodology and a comparison of the generalized displacements and bending moments obtained with the DMFDM and FEM models while bringing out certain interesting features of functionally graded beams.

本文采用双网格有限域方法对功能梯度梁的几何非线性分析（DMFDM）和有限元方法。DMFDM利用有限元的原始网格和公式变量的相关近似值以及控制域的双重网格（不与原始网格重叠）来满足控制方程。对偶变量可以在控制域接口上唯一且准确地进行后计算。该方法用于使用Euler–Bernoulli和Timoshenko束理论来获得直的，通过厚度的功能梯度梁的非线性（由于vonKármán非线性应变）弯曲解决方案。开发了混合模型，其中包含位移和弯曲力矩作为自由度，以及位移模型。