Journal of Applied Mathematics

Volume 2018, Article ID 3196569, 9 pages

https://doi.org/10.1155/2018/3196569

## A Theoretical Consideration on the Estimation of Interphase Poisson’s Ratio for Fibrous Polymeric Composites

^{1}School of Applied Mathematics and Physical Sciences NTU Athens, Section of Mechanics, 5 Heroes of Polytechnion Avenue, 15773 Athens, Greece^{2}School of Applied Mathematics and Physical Sciences NTUA, Section of Mechanics, 5 Heroes of Polytechnion Avenue, 15773 Athens, Greece

Correspondence should be addressed to J. Venetis; moc.liamg@4sitenevnhoj

Received 3 July 2018; Revised 23 August 2018; Accepted 5 September 2018; Published 1 October 2018

Academic Editor: Xin-Lin Gao

Copyright © 2018 J. Venetis and E. Sideridis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An analytical approach on the evaluation of interphase Poisson’s ratio for fibrous composites, consisting of polymeric matrix and unidirectional continuous fibers, is performed. The simulation of the microstructure of the composite was carried out by means of a modified form of Hashin-Rosen cylinder assemblage model. Next, by the use of this three-phase model the authors impose some limitations to the polynomial variation laws which are commonly adopted to approximate the thermomechanical properties of the interphase layer of this type of polymeric composites and then propose an nth-degree polynomial function to approximate the Poisson’s ratio of this layer.

#### 1. Introduction

The rigorous description of a composite system consisting of a matrix in which continuous or short fibers have been dispersed is not an easy task. Indeed, a great number of parameters, geometrical, topological, mechanical, etc. are necessary, the majority of which vary in a stochastic manner or are practically unknown. Theoretical treatments usually attempt to exploit as far as possible readily available information, which, in most cases, consists of the mechanical properties of matrix and fiber and the volume fraction of the latter, whilst appropriate assumptions cover missing data. On the other hand it is well known that in lightweight structures, large specific stiffness, and strength are aiming properties for a material. By combining fibers with an appropriate polymer and controlling the production procedure it is possible to manufacture composites featuring large specific properties. Moreover, the constituents are usually cheap and easily processed, e.g., by injection molding. Hence, fibrous composites have many industrial applications where high performance per weight at a reasonable price is required. One of the most useful forms of composites for the construction of high-performance structural elements is the type of panels made from aligned fibers containing polymerized matrix. Evidently, their mechanical properties depend on the related properties and volume fractions of the constituent materials, the fiber length or aspect ratio, the degree of the alignment, the adhesion between fibers, and matrix and last but not least they are affected by the fabrication techniques [1]. As it was initially stated, the investigation of the elastic properties of unidirectional fibrous composites reinforced with long or short fibers in terms of the related properties of constituents constitutes a very difficult problem of applied mechanics. In a fundamental investigation by Hashin and Rosen [2] bounds and explicit expressions to estimate the effective elastic moduli of unidirectional fibrous composites were derived via a rigorous variational method. On the other hand, amongst the analytical models presented in the literature, some of them take into consideration the existence of an intermediate phase developed during the preparation of the polymeric composite. Evidently this new phase plays an important role in the overall thermomechanical behavior of the composite.

In the meanwhile, the existence of a boundary interphase in polymeric composites (fibrous and pariculate) was shown experimentally by Lipatov [3] who measured its thickness by means of Differential Scanning Calorimetry (DSC) experiments. Also, in this valuable investigation, it was empirically stated that the size of these heat capacity jumps for unfilled and filled polymers is expressed in terms of the thickness of this phase. Yet, the estimation of its physical properties was a remaining problem.

To this end, in a preliminary model performed by Papanicolaou et al. [4] and Theocaris et al. [5] this intermediate phase was initially assumed to be a homogeneous and isotropic material whereas in a better approximation [6] a more complex model was introduced, according to which the fiber was surrounded by a series of successive cylinders, each one of them having a different elastic modulus in a step-function variation with respect to the polar radius.

In addition, problems of obtaining closed-form solutions to many cases of elastic inclusions embedded in an elastic matrix were encountered by Muskhelishvili [7] by the use of conformal mapping techniques. In addition, numerical solution techniques such as finite difference and finite elements have been extensively used to this end [8, 9]. Another remarkable consideration of the variable modulus interphase is the so-called unfolding model, which was carried out on the basis that the boundary interphase layer constitutes a transition zone between fibers with high moduli and matrix with rather low stiffness [10]. An extension of the above fundamental model to more complex composite systems (fibrous and particulate) was exhibited by Theocaris in [11].

In addition, a remarkable study performed by Swain et al [12] has thrown light on the effect of the boundary interphase on the properties and performance of composite materials and laminates.

In the past years, there is a lot of recent research work carried out for the evaluation of elastic and thermal properties of fibrous composites and for the investigation of the influence of various parameters on these properties. A detailed study on the effect of interphase layers on local thermal displacements in fibrous composites was presented in [13]. Also, in [14], an investigation was made by the aid of strength of materials and elasticity approach, to derive explicit expressions for longitudinal, transverse, and shear moduli along with longitudinal and transverse Poisson’s ratio using the concept of boundary interphase between fiber and matrix, in the framework of a three-phase modified form of Hashin-Rosen cylinder assemblage model that was presented in [2]. Here, to evaluate the elastic properties of interphase a unified mode of variation was considered. In particular, an nth degree polynomial variation was initially adopted in terms of the radius of the introduced three-phase coaxial model and finally for facility reasons a quadratic function was taken into account. The above approach of interphase unknown properties was continued and extended to estimate the thermal conductivity of the same class of fibrous composites in [15]. In this work, to cover the whole spectrum of interphase thermal conductivity, five different laws of variation were used of which only two constitute polynomial forms (linear and parabolic). Nevertheless, it can be said that a three-layer cylindrical model to simulate the structure of unidirectional fibrous composites reinforced with continuous fibers may arise based on the theory of self-consistent models and adapting this approach to a three-layered cylinder, delineating the representative volume element for the fibrous composite [11]. Here one should elucidate that a significant variation of the self-consistent model is the three-phase model first introduced by Kerner [16], according to which the inclusion is developed by a matrix annulus which in turn is embedded in an infinite medium with the unknown macroscopic properties of the composite. Further, in [17, 18] the adoption of linear variation laws for interphase elastic constants yielded theoretical predictions of the composite moduli in a reasonable accordance with experimental values. Yet, to approach the interphase modulus and Poisson’s ratio by a quadratic (parabolic) law with respect to the radius of the adopted model of embedded cylinders is many times more convenient since many times the linear variation law cannot alleviate the fact that the transition of the elastic constants from the matrix to fiber is carried out by “jumps” in their characteristic properties [17, 19]. However, a significant problem that remains is the following: letting an interphase property be approached by a quadratic polynomial in the general form , there are three coefficients needed to be found. Unfortunately, in the context of the modified form of Hashin-Rosen model introduced in [14] there are only two boundary conditions available. Specifically, at and at . Another condition may arise by requiring the first derivative to vanish at a critical point over the interval which corresponds to a given local extremum (maximum or minimum) [14]. In particular it was assumed that the critical point coincides with one of the endpoints of this interval and thus . Nevertheless, should this critical point be outside this interval or in the interval such a condition cannot hold [20] and therefore the coefficients are unable to be calculated. To overcome this unfortunate situation, in the present work an alternative approach on the estimation of interphase elastic constants is proposed.

#### 2. Analysis

Let us simulate the microstructure of a unidirectional fibrous composite by means of a coaxial three-phase cylinder unit cell, the cross-sectional area of which can be seen in Figure 1.