Journal of the Brazilian Society of Mechanical Sciences and Engineering. Vol. XXIX, No. 1, pp. 63-81, 2007
J.H. Kim and G.H. Paulino
Department of Civil and Environmental Engineering, University of Illinois
at Urbana-Champaign, Newmark
Laboratory, 205 North Mathews Avenue, IL
61801, U.S.A.
Abstract
Functionally graded materials have an additional length scale associated to the spatial variation of the material property field. By considering the length scale of nonhomogeneity, this paper presents the weak patch test (rather than the standard one) of the graded element for nonhomogeneous materials to assess convergence of the finite element method (FEM). Both consistency (as the size of elements approach zero, the FEM approximation represents the exact solution) and stability (spurious mechanisms are avoided) conditions are addressed. The graded elements considered here are the 4-node, 8-node, and 9-node isoparametric quadrilaterals. Two dimensional plane stress and axisymmetric problems are considered. The finite element approximate solutions are compared with exact solutions available for nonhomogeneous materials, which are either available in the literature or derived in this work.
Key words: Finite element method (FEM), patch test, weak patch test, functionally graded materials, graded element, isoparametric formulation, ``isoparametric moduli''.