Composites with functionally graded interphases: Mesocontinuum concept and effective transverse conductivity

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.