Isoparametric Graded Finite Elements for Nonhomogeneous Isotropic and Orthotropic Materials

Graded finite elements are presented within the framework of a generalized isoparametric formulation. Such elements possess a spatially varying material property field, e.g. Young’s modulus $E$ and Poisson’s ratio $ν$ for isotropic materials; and principal Young’s moduli $E11,E22,$ in-plane shear modulus $G12,$ and Poisson’s ratio $ν12$ for orthotropic materials. To investigate the influence of material property variation, both exponentially and linearly graded materials are considered and compared. Several boundary value problems involving continuously nonhomogeneous isotropic and orthotropic materials are solved, and the performance of graded elements is compared to that of conventional homogeneous elements with reference to analytical solutions. Such solutions are obtained for an orthotropic plate of infinite length and finite width subjected to various loading conditions. The corresponding solutions for an isotropic plate are obtained from those for the orthotropic plate. In general, graded finite elements provide more accurate local stress than conventional homogeneous elements, however, such may not be the case for four-node quadrilateral (Q4) elements. The framework described here can serve as the basis for further investigations such as thermal and dynamic problems in functionally graded materials.