Application of generalized self-consistent method to predict effective elastic properties of bristled fiber composites

Abstract

A self-consistent Eshelby method based on the three-phase model is developed to examine the behavior of bristled fiber composite material where the fibers are radially coated with micro/nanostructures such as microwhiskers, nanowires or carbon nanotubes (fuzzy fiber). The effective mechanical properties are determined by taking into account the additional bristled interphase layer that is formed between fiber and matrix due to the presence of these micro/nanofibers. The features of the proposed method are emphasized and comparative study with other methods is conducted. In addition, several parameters of the micro/nanofibers such as length, density, diameter and material affecting the effective properties of composites are examined. In general, the presence of bristled fibers can significantly improve the shear and transverse characteristics of composite materials.

Introduction

In aerospace industry, fiber reinforced plastic (FRP) composite material can be tailored to produce very strong and stiff lightweight structures. Nevertheless, its superiority is often restricted by the inevitable presence of fiber–matrix interface, which plays very important role in determining the properties of composite material [1]. A variety of techniques has been designed to improve the quality of an interface, and the most common being through improving the chemical interactions between fiber and matrix, modifying the surface of fiber, or adding a third phase material or an interphase layer to the region between fiber and matrix [2], [3], [4]. Basically, the ideas behind these methods are to improve the interfacial adhesion properties, and to increase the fiber’s surface area for more effective load transfer between fiber and matrix so that the behaviors of fiber composite material can be further improved.

Several decades ago, microwhiskers were used to increase the shear characteristics of fiber composite material. One of the techniques used is to grow them directly on top of the surface of carbon fiber. When these whiskered fibers are embedded in an epoxy matrix system, a so-called interphase layer is formed in between fiber and matrix. This layer itself is a composite layer consisting of microwhiskers and matrix material. It is reported that with the presence of these radially aligned whiskers and depending on the various combinations of carbon fibers and epoxy matrices, a three to five times increase in the interfacial shear strength of such composite system were obtained [5], [6]. However, a major drawback of this technology is that the mechanical processes in growing the microwhiskers substantially reduce the tensile strength of the underlying fiber, and hence the strength of the composite in the reinforcement direction is reduced significantly [7]. As a result, researches in this technology became dormant for quite some time. Interestingly, this former technology reappears in the present era of nanotechnology. Carbon nanotubes (CNTs) that became available since 1990s are now being designed to coat on the circumferential surface of carbon fiber. Such fiber system is known as fuzzy fiber [8], [9], [10]. Sager et al. [2] reported that the presence of fuzzy fibers could increase the interfacial shear strength of fiber composite material. Further experimental works by Agnihotri et al. [11] revealed that the length and density of CNTs grown on top of the surface of carbon fiber are critical in producing the enhanced properties of CNT-coated carbon/polyester composite material. Unfortunately, just like its predecessor, this fiber system also suffers from the significant reductions in its tensile strength and stiffness properties due to the high temperature applications used in growing the CNTs. Nevertheless, Steiner et al. [12] recently announced that they have successfully developed new techniques to produce fuzzy fibers without damaging the properties of the base fiber.

Apparently, CNTs are not the only nano-reinforcing materials used to reinforce the fiber–matrix interface in composite material. Nanoparticles and nanowires/rods are also being explored to produce similar fiber system. Such technology can be found in the work of Wang et al. [13] where silicon oxide (SiO₂) nanowires are grown perpendicularly to the surface of Cadmium Telluride (CdTe) nano-sized fiber. This type of fiber system is termed as bristled ‘nano-centipedes’ fiber due to its striking resemblance to centipedes. However, more interesting results can be found in the works of Lin et al. [3] and Galan et al. [14]. In their works, zinc oxide nanowires (ZnO NW) are radially coated to the surface of IM7 carbon fibers. It is reported that under specific configuration of nanowires and its composite, the average shear strength and shear modulus of the composite are increased by 37.5% and 38.8% respectively when compared to the uncoated carbon fiber composite specimens. In addition, it is observed that the interfacial shear strength of the composite is increased when the length or diameter of ZnO NWs increases. Interestingly, this type of coated fiber system is claimed to be no weaker after the growth of ZnO NWs [14].

With these latest technological developments in producing CNTs- or nanowires-coated carbon fibers that preserve the strength of the underlying fiber, a stronger and lighter advanced fiber composite material with enhanced fracture toughness and damage tolerance may well be achieved. For example, higher interfacial shear strength not only will improve the impact performance of the composite [15] but will also provide better resistance against fiber–matrix debonds, which are known to coalesce and form transverse matrix cracking in the composite laminate under excessive loading [16]. Furthermore, the transverse reinforcement provided by the micro/nanofibers will probably ensure better protection against plies delamination failure. It is also expected that this transverse reinforcement will address the low through-thickness characteristics inherited by fiber composite material, thus creating the possibilities of having thinner, lighter but stronger composite structure. These advantages are certainly very beneficial for the aerospace industry. Thus, herein we intend to examine the effective elastic behaviors of such promising composite materials. In addition, hereafter we will generalize all these types of micro/nanofiber-coated fiber system as bristled fibers.

Presently, there are several micromechanics models proposed by various authors in determining the effective mechanical properties of bristled fiber composite materials. Guz et al. [6], [17] studied bristled fiber composite that consists of four layers of material – base fiber, fiber-coating layer, bristled interphase layer, and matrix material. As such, they developed a four-component structural model based on the periodic unit cell. The overall effective properties are determined using a two-step homogenization procedure. The first step involves the homogenization of bristled interphase layer where rule of mixtures is employed to determine its effective properties. In the second step of homogenization procedure, the properties of all phases in the four-component structural model are averaged together. The final effective properties are then determined from the relation of average stresses and strains in the composite’s unit cell. In computing the average stresses and strains, the exact solutions for the stress and strain fields in the heterogeneous media are defined using the Muskhelishvili complex potential method. To analyze the suitability of their proposed method, the results on the effective longitudinal shear modulus with three different densities ranging from sparse, medium and dense bristlization are illustrated. It is shown that regardless of the density of bristles, the presence of bristled fibers gives significant increase in the shear modulus of composite when compared to uncoated fiber composite. Nevertheless, the difference in shear modulus between these three densities is shown to be less than 3% only.

On the effective properties of a three-phase fuzzy fiber composite material, Kundawal and Ray [9] approached the problem using mechanics of material approach and Mori–Tanaka method. Interestingly, in evaluating the properties of the composite, instead of a two-step procedure, they employed a three-step homogenization procedure. In their study, the effect of diameter of CNTs as well as the effect of an interphase between CNT and matrix on the properties of fuzzy fiber composite is investigated. Nevertheless, the effect of density (quantity) of CNT’s is not considered. The results obtained from both methods are shown to be in agreement with one another. It is also shown that the presence of CNTs improves significantly the effective transverse elastic constants of the composite while on the effective in-plane elastic constant is not affected. However, these constants increase marginally when the diameter of CNT is increased. Furthermore, the role of an interphase is found to be insignificant.

In the work of Chatzigeorgiou et al. [10], a combination approach based on the composite cylinder assemblage (CCA) method and the generalized self-consistent (GSC) method is employed to fully describe the effective properties of a three-phase fuzzy fiber composite. The use of GSC method in their analytical study is necessary since it is well known that the CCA method provides only useful upper and lower bounds when estimating the transverse shear modulus of fiber composite material [18]. In estimating the properties of bristled interphase layer, three different methods are proposed, which are CCA/GSC combination approach, Mori–Tanaka and self-consistent method [19]. Their results revealed that the transverse and shear properties of fuzzy fiber composite are strongly influenced by the presence of CNTs while the effect on the axial Young’s modulus is very marginal. The validity of their proposed method is verified by comparing its results with the effective properties obtained by a numerical method based on the asymptotic expansion homogenization method [20]. It is shown that both results are in very good agreement with one another. The effects of length and volume fractions of CNTs are also examined in their analytical study, and it is found that both parameters significantly affect the properties of fuzzy fiber composite material.

In this present work, we offer an analytical model based on the self-consistent Eshelby method or better known as the generalized self-consistent method [21], [22], [23] to examine the overall mechanical behaviors of bristled fiber composite material. The application of this method in this problem is motivated by a remark made in the work of Chatzigeorgiou et al. [10] where it is mentioned that the use of GSC method in obtaining the effective properties other than the transverse shear modulus of fiber composite material will lead to nonlinear equations, thus requiring iteration schemes when solving for the unknown constants and effective modulus. In our opinion, this condition may not be necessarily true and if so, it can be avoided. As will be shown later, it is possible to find the unknown constants linearly and as such, the effective modulus can be expressed explicitly. In addition, we will show that the values of the respective effective properties obtained from the two methods are identical as had been noted by Christensen [23] and Hashin [24]. Finally, the proposed method will be developed to accommodate different types of bristled fiber composite materials that may have different densities (quantities), length, diameter and materials of micro/nanofibers. As such, unlike in previous works reviewed earlier, the effects of all these parameters on the mechanical behaviors of bristled fiber composite material will be investigated as well.