Effective properties of particulate composites with surface-varying interphases

Highlights

• Models based on the concept of energy equivalent inhomogeneity.

• Inhomogeneity/interphase system replaced by equivalent inhomogeneity.

• Spring layer model of interphases.

• Inhomogeneous interphase with surface varying properties.

• Closed-form formulas for overall properties of composites accounting for interphase.

Abstract

A concept of equivalent inhomogeneity is adopted to facilitate analysis of effective properties of particulate composites with surface-varying interphases. The basic idea is to replace the inhomogeneity and its interphase by a single equivalent inhomogeneity, combining properties of both. It is illustrated considering spherical inhomogeneity with spring layer model of interphases. Due to surface-varying properties of the interphase the resulting equivalent inhomogeneity is anisotropic. Thus, the Method of Conditional Moment – developed for composites with anisotropic constituents – is applied in analysis of the effective properties of the composite. For particular case of homogeneous interphase, the results obtained here are compared with those available in the literature.

Introduction

Influence of interphase properties on the effective behavior of entire composite is the focus of many recent research papers. The reported investigations deal with interphases that are homogeneous or interphases with properties varying only across their thickness, and their more detailed review is included subsequently. Although both of those models may be a good approximation of some realistic situations, one can easily envision a composite when it may be more appropriate to account for variation of the interphase properties over the inhomogeneity surface. For example, this may be the case when the inhomogeneities and/or the matrix are anisotropic, considering that the interphase properties result from the local inter-atomic interactions between the adjoined materials. Thus, the effective properties of composites with the interphases characterized by the surface-varying properties constitute the focus of this paper.

An extreme example of when it is necessary to include the surface varying properties of the interphase relates to the case of interphase debonding, often developing under loading [1,18,21,39]. This, however, lies outside of the scope of the present work as it necessitates a model to describe propagation of debonding with load.

The prior publications addressing the issue of across-the-thickness variation of interphase properties were very different from one another and quite restrictive in nature. Kanaun and Kudryavtseva [15] present a numerical solution for a single spherically layered inhomogeneity (piecewise-constant dependence of elastic moduli on radius) in an infinite medium under the assumption of Poisson's ratio in each layer and in the matrix being the same. This solution was subsequently utilized to determine the effective elastic moduli of random composites by the effective field method [19,20].

Hashin [8,9] restricted application of his composite cylinder or sphere assemblage to analysis of the effective bulk modulus with interphases possessing constant properties. He noted, however, that a single “equivalent inhomogeneity” (comprising properties of the original inhomogeneity and of its interphase) perfectly bonded to the matrix can yield the same effective properties as those resulting from independent treatment of the inhomogeneity and the interphase. This idea, in a formal setting that suits the purpose of this work, is also exploited here. Relevance of Hashin contribution [8,9] to composites with interphase properties varying across its thickness lies in included there prescription of how equivalent inhomogeneity can be defined for multilayer interphases, but still only in the context of effective bulk modulus. In Refs. [13,36] composite materials reinforced with unidirectional infinite fibers (two-dimensional case) or spherical particles (three-dimensional case) with interphase continuously varying in radial direction are analyzed. In analysis they used composite cylinder or sphere assemblage methods of Hashin [8,9] and generalized self-consistent scheme [4] to evaluate the effective properties of such composites.

Wu et al. [38] and Zhong et al. [40] employ a differential scheme based on the Hashin–Shtrikman lower bound [7], which is similar to the replacement technique used in Shen and Li [34,35] and Lombardo [22]. That idea has been subsequently followed by differential schemes of Shen and Li [34] and of Sevostianov and Kachanov [33]. In these approaches layers of infinitesimal thickness and varying properties were successively added to the original spherical inhomogeneity to form an interphase with properties varying across its thickness. With addition of each layer the properties of the system were defined either by the Mori-Tanaka scheme or Hashin–Shtrikman upper bound estimate [7].

In Sburlati et al. [31,32] particulate composites reinforced with solid spherical particles surrounded by graded interphase zone were considered. The graded interphase zone was assumed to have the shear modulus described by a power law with respect to the radial co-ordinate. The Poisson's ratio was assumed to be constant and equal to that of the matrix. Closed-form expressions for the effective bulk modulus and Hashin–Strickmann upper and lower bounds for the effective shear modulus was determined on the basis of composite sphere assemblage method of Hashin [9] and Hashin–Shtrickman bound estimate [7].

The approaches discussed above are based on the analytical solution for a single inhomogeneity in an infinite matrix [6], combined with approaches such as that of Mori-Tanaka scheme [2,3,23], composite cylinder or sphere assemblage method [8,9] or self-consistent method [4,24], Willis's closure model [37] among others. If the interphase properties vary over the inhomogeneity surface, which is of interest in this work, such a solution it is very difficult to obtain, if possible at all. The additional difficulty is related to the fact that the effective properties of the composite are then anisotropic. Thus, considering the existing state of research in the area, the aim of this work is to demonstrate that the energy-equivalent inhomogeneity approach, developed and pursued by these authors in several previous publications, provides a natural way to analyze the composites with surface-varying properties of the interphases. Unfortunately, the problem is involved and some complicated aspects of analysis are impossible to bypass.

To present the basic features of the approach we begin with a brief description of the notion of the energy equivalent inhomogeneity and its approximate definition. This is done in Section 2 for the spring layer model of the interphase without any assumptions as to the variation of its properties. The specifications related to the interphases with specific surface-varying properties are detailed in Section 3. Considering that the resulting equivalent inhomogeneity is in this case anisotropic, this section presents also the basic formulas defining effective properties of composite with orthotropic inhomogeneities; the method of choice in this regard is the Method of Conditional Moments (MCM). Numerical results are presented in Section 4 and conclusions form the content of Section 5.