Marcio A. A. Cavalcante Severino P. C. Marques Center of Technology, Federal University of Alagoas, Maceio, Alagoas, Brazil Marek-Jerzy Pindera Civil Engineering Department, University of Virginia, Charlottesville, VA 22903 Parametric Formulation of the Finite-Volume Theory for Functionally Graded Materials—Part I: Analysis The recently reconstructed higher-order theory for functionally graded materials is fur- ther enhanced by incorporating arbitrary quadrilateral subcell analysis capability through a parametric formulation. This capability significantly improves the efficiency of modeling continuous inclusions with arbitrarily-shaped cross sections of a graded mate- rial’s microstructure previously approximated using discretization based on rectangular subcells, as well as modeling of structural components with curved boundaries. Part I of this paper describes the development of the local conductivity and stiffness matrices for a quadrilateral subcell which are then assembled into global matrices in an efficient manner following the finite-element assembly procedure. Part II verifies the parametric formulation through comparison with analytical solutions for homogeneous curved struc- tural components and graded components where grading is modeled using piecewise uniform thermoelastic moduli assigned to each discretized region. Results for a hetero- geneous microstructure in the form of a single inclusion embedded in a matrix phase are also generated and compared with the exact analytical solution, as well as with the results obtained using the original reconstructed theory based on rectangular discretiza- tion and finite-element analysis. DOI: 10.1115/1.2722312 Keywords: functionally graded materials, finite-volume theory, parametric formulation 1 Introduction Functionally graded materials FGMs are multiphase materials with engineered microstructures which produce property gradients aimed at optimizing structural response under different types of loads thermal, mechanical, electrical, optical, etc.. These prop- erty gradients are produced in several ways, for example by gradual variation of the content of one phase ceramic relative to other metallic used in thermal barrier coatings, or by using a sufficiently large number of constituent phases with different properties. Developed by Japanese researchers in the mid-1980s, these materials continue to evolve and to find new applications in areas other than the thermal protection/management structures for which they had been originally developed, cf. Suresh and Mortensen 1, Miyamoto et al. 2, Paulino 3, Chatzigeorgiou and Charalambakis 4. The use of graded material concepts in structural design and optimization requires the development of appropriate analysis techniques which account for the spatially variable microstruc- tures in this class of materials. Presently, there are two approaches available to analyze the response of FGMs to thermomechanical loads, called coupled and uncoupled approaches, Pindera et al. 5. In the uncoupled approach, the graded material’s microstruc- ture is replaced by equivalent homogenized properties which are either determined from micromechanics considerations or as- sumed a priori. This results in a boundary-value problem with either continuously or discretely variable elastic moduli at the scale at which the analysis is conducted, called macroscale. In the coupled approach originally proposed by Aboudi et al. 6, and summarized in a review paper by Aboudi et al. 7, the material’s microstructure is explicitly taken into account by performing the analysis at the microscale. In particular, in the original formula- tion of this so-called higher-order theory, a two-step discretization involving generic cells and subcells is employed to capture the graded material’s heterogeneous microstructure. Subsequently, thermal and displacement fields within each subcell are approxi- mated using quadratic expansions in local coordinates, and the unknown coefficients associated with the different-order terms are obtained by satisfying various moments of the field equations in a volume-averaged sense in each subcell, followed by the applica- tion of continuity conditions within each generic cell, and between adjacent cells, in a surface-average sense together with the im- posed boundary conditions. We mention that surface averaging of continuity conditions was proposed by Achenbach 8 in the con- text of the author’s cell model for unidirectional composites. This approach has recently been reconstructed by Bansal and Pindera 9 and Zhong et al. 10 based on a simplified volume discretization using subcells as the fundamental subvolumes, in place of the two-level discretization employed in the original con- struction. The use of subcells as the fundamental subvolumes, in turn, facilitated the implementation of the local/global stiffness matrix formulation, Bufler 11, Pindera 12, into the solution procedure for the unknown subcell surface-averaged interfacial displacements which became the primary unknown quantities in the reconstructed theory. The reconstruction has also revealed that the model’s theoretical framework is based on direct satisfaction of the field equations within each subcell, in contrast to the origi- nal construction wherein higher-order moments of the equilibrium equations were also satisfied, thereby erroneously suggesting this model to be a version of a micropolar continuum theory. The significantly simplified theoretical structure of this so-called higher-order theory in conjunction with the implementation of the local/global stiffness matrix approach also resulted in a substantial reduction in the final system of equations for the unknown quan- tities, thereby making it possible to analyze realistic graded mi- crostructures that required extensive discretization not possible Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 6, 2006; final manuscript received December 22, 2006. Review conducted by Robert M. McMeeking. Journal of Applied Mechanics SEPTEMBER 2007, Vol. 74 / 935 Copyright © 2007 by ASME Downloaded 14 Sep 2007 to 128.143.34.99. Redistribution subject to ASME license or copyright, see http://www.asme.org/terms/Terms_Use.cfm