Pergamon 0956-7151(95)00269-3 Acta mater. Vol. 44, No. 5, 2057-2066. 1996 pp. Elsevier ScienceLtd Copyright 0 1996 Acta MetallurgicaInc. Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00 COMPOSITES WITH FUNCTIONALLY GRADED INTERPHASES: MESOCONTINUUM CONCEPT AND EFFECTIVE TRANSVERSE CONDUCTIVITY M. OSTOJA-STARZEWSKI’, I. JASIUK’, W. WANG’ and K. ALZEBDEH* ‘Institute of Paper Science and Technology, 500 10th Street, N.W., Atlanta, GA 30318-5794, and 2Department of Materials Science and Mechanics, Michigan State University, East Lansing, MI 48824-1226, U.S.A. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ (Received 6 January 1995; in revised form 19 June 1995) Abstract-We consider a unidirectional fiber-reinforced composite with an interphase between the fiber and matrix taken as a graded zone of two randomly interpenetrating phases of these materials. In particular, we take this interphase as a functionally graded material (FGM). The objective of this paper is to present a micromechanics based method to treat FGM and to calculate the effective macroscopic properties (transverse conductivity, or, equivalently, axial shear modulus) of such a composite material. This problem requires the treatment of several length scales: the fine interphase microstructure, its mesocontinuum representation, the fiber size, and the macroscale level (of many fibers) at which the effective properties are defined. It is shown through an example that a convergent hierarchy of bounds on the effective response is obtained with systematically increasing mesoscale resolutions. 1. INTRODUCTION Interfaces in composite materials influence their local fields and effective properties [l-3]. Theoretical stud- ies in this area represent the interface as either a two dimensional bounding surface, or as a three dimen- sional region of certain microstructure, called an interphase [4]. In this paper we focus on the effective transverse conductivity of unidirectional composite materials with interphases having functionally graded properties. It can be noted that this problem is mathematically equivalent to other phenomena gov- erned locally by Laplace’s equation, such as electrical conductivity or anti-plane elasticity [S]; see Table 3 in the Appendix. The inhomogeneity of the interphase may be due, for example, to the chemical reaction(s) or diffusion. Composites with inhomogeneous interphases have been studied by several researchers recently. These works include the studies of local fields due to thermal [&lo] and mechanical [ll] loadings and the evaluation of effective elastic moduli of composites [12-171. However, in these studies the interphase region is assumed to be isotropic with one property, typically Young’s modulus, varying linearly, as a power law, or as a polynomial, and in general the Poisson’s ratio is taken as a constant. The anisotropy of the interphase is considered in [18, 191 where the anisotropic constants are assumed to vary as a power law. Also, several studies represent the inhomo- geneous interphase by a number of layers, e.g. [20]. For additional references on the subject see also [21]. Functionally graded materials (FGM in current terminology), or materials with spatially varying properties, present new theoretical challenges in mechanics and micromechanics of solids. The funda- mental problem is how to predict the effective prop- erties of such materials given the spatially inhomogeneous distribution of phases. Zuiker and Dvorak [22,23] generalized the Mori-Tanaka method to account for the variable reinforcement density and thus to linearly varying local and global fields. An alternate approach was considered by Aboudi and co-workers [24,25] who employed a modified method of cells to study an FGM model system. Recently, a study of elastoplastic phenomena in FGM has been presented in [26]. In this paper we approach this complicated prob- lem in a new way which explicitly deals with spatially random graded microstructures. We begin by consid- ering the microstructure of the interphase, which we represent as a zone of two randomly interpenetrating phases with radially dependent statistics. We admit two generic models of the microstructure: a fine- grained model with a topology of a random chess- board, and a coarser-grained model with a topology of a Voronoi tessellation, whose cells are occupied by either one of the phases. While we choose these material systems for simplicity, it has been noted that the presented model and method admit any type of FGM microstructure. Ideally, the input on details of microstructure should come from experimental observations; this however still remains a challenge [27-291.