Elsevier

Mechanics of Materials

Volume 96, May 2016, Pages 1-11
Mechanics of Materials

Overall properties of particulate composites with fractal distribution of fibers

https://doi.org/10.1016/j.mechmat.2016.01.014Get rights and content

Highlights

  • Introduction of multi-step periodic homogenization method.

  • Application of the multi-step homogenization to discrete distribution of fibers.

  • Application of the multi-step homogenization to fractal distribution of fibers.

  • Comparison with experimental data on mechanical properties of particulate composites.

Abstract

Extensive research on the micromechanical structure of materials has revealed that some of the main construction materials exhibit fractal patterns at the micro scale. Therefore, micromechanical structure of these types of materials can be viewed as periodic structures at different length scales. A new homogenization technique is proposed that is mainly based on the micromechanical averaging schemes for the determination of the mechanical properties of materials with periodic microstructures. This method can be used in determining the overall properties of particulate composites. The proposed technique is a multi-step homogenization technique in which in each step a length scale is considered until the whole reinforcing phase is taken into account. To validate the method, its results are compared with the experimental data on different composite materials with different matrices and fibers. The results show that very good estimates of the mechanical properties is reached by utilizing the proposed multi-step homogenization technique. This method can be utilized in the determination of the mechanical properties of the composites with the coated fibers, where the effect of size of particles on the mechanical properties can be investigated by the application of principles of fractal geometry.

Introduction

The interest in mixing materials to obtain new materials with desired mechanical properties is as old as human history. However, it has only been a few decades that a systematic approach for the determination of the properties of such materials is available. Starting from the seminal works of Eshelby (1957, 1959, 1961) on ellipsoidal inclusions many works have been done on various aspects of micromechanical analysis of inhomogeneous solids. One of the active research subjects in the field of micromechanics of solids is about determining the overall properties of composite materials. The early works on this subject goes back to the Voigt and Reuss bounds on elastic moduli that result directly from the rule of mixtures method. More delicate works on the subject were done later, among which one can mention the variational universal bounds of Hashin and Shtrikman (1961, 1962a, 1962b, 1963). Other significant contributions in the field are Hill (1963); Willis (1977); Talbot and Willis (1985, 1987); Weng (1990); Castañeda (1991, 1992). Another direction in the field of micromechanics of composite materials was started by the introduction of the equivalent inclusion method by Mura (1964), which was followed and extended extensively in the later works of Nemat-Nasser and Taya (1981); Nemat-Nasser et al. (1982, 1993); Nemat-Nasser and Hori (2013). Following the works of Nemat-Nasser and Taya (1981); Nemat-Nasser et al. (1982) on periodic distribution of fibers, some research studies devoted to extend the scope of periodic homogenization to the cases of non-dilute coated fiber reinforced composites (e.g. see El Mouden, Cherkaoui, Molinari, Berveiller, 1998, Shodja, Roumi, 2005).

Since the pioneering work of Mandelbrot (1983), many research studies have been carried out in various applications of fractal geometry. Fractal geometry has found many applications in various fields of science that deal with objects of irregular shapes. Introduction of this mathematical concept has extended the scope of research to objects of irregular shapes and geometries, which in the past were assumed to be mathematically intractable. The interest in the application of fractal geometry in mechanics started from the realization of the fact that the fracture surfaces of some materials exhibit fractal patterns (Saouma, Barton, 1994, Saouma, Barton, Gamaleldin, 1990). There are extensive works in the subject of the effect of fractality on the various aspects of fracture mechanics properties of materials (see Mosolov, 1991, Gol’dshtein, Mosolov, 1991, Gol’dshtein, Mosolov, 1992, Balankin, 1997, Borodich, 1997, Carpinteri, 1994, Cherepanov, Balankin, Ivanova, 1995, Xie, 1989, Xie, 1995, Yavari, Hockett, Sarkani, 2000, Yavari, Sarkani, Moyer Jr, 2002, Yavari, 2002, Wnuk, Yavari, 2003, Wnuk, Yavari, 2005, Wnuk, Yavari, 2008, Wnuk, Yavari, 2009, Yavari, Khezrzadeh, 2010, Khezrzadeh, Wnuk, Yavari, 2011, Balankin, 2015 and references therein.). Now the scope of research on fractal objects is extended to the mechanics of materials with fractal microstructure, and the continuum mechanics framework for the fractal media is being developed (see Tarasov, 2005, Tarasov, 2011, Ostoja-Starzewski, Li, Joumaa, Demmie, 2014).

The aim of this paper is to introduce a new method for characterizing overall behavior of particulate composite materials, especially those with fractal distribution of fibers. In this paper the advantages of both fractal geometry and periodic homogenization techniques are put together to develop a method for the determination of the overall properties of particulate composite materials with particles of different sizes. The presented method can be applied for the determination of size effect on the mechanical properties of particle-reinforced composites.

The paper is organized as follows. In Section 2, the geometrical model is introduced, by utilizing this geometric model, the microstructures with periodicity at different length scales can be modeled. In Section 3, at first the basics of homogenization techniques for periodic microstructures is introduced and then properties of the fundamental unit cell, which is called Unit Cell I is derived. In Section 4 the theory of multi-step homogenization is developed by implementing the results of the previous sections. In Section 5, the results from the presented method is verified against experimental results of different composite materials and it is shown that good estimations of the elastic moduli are obtained by utilizing the proposed homogenizing technique even at high concentration of fibers. Finally some concluding remarks will be given.

Section snippets

Geometrical modeling of particulate composites with fractal-like microstructure

Fractal geometry, as a mathematical tool that can describe the objects of irregular shapes has found many applications in science and engineering. In fractal geometry the irregularity is modeled by using the self-similarity and self-affinity properties of the fractal sets. Self-similarity and self-affinity indicate that the irregular object has some degree of order in different length scales. This indicates that the total object is consisted of many objects, which are identical in shape but are

Micromechanical model of particulate composites with periodic microstructures

In this section the concepts of eigenstrain homogenization method for periodic microstructures is briefly reviewed. The eigenstrain homogenization method is mainly based on the concept of equivalent eigenstrain that was first introduced by Mura (1964) and developed extensively in the later works of Nemat-Nasser et al. (1986, 1993) and Nemat-Nasser and Hori (2013). In this method a unit cell is taken as the building block of the whole structure. Suppose a cubic unit cell which is defined as

Multi-step homogenization

In this section the concept of multi-step homogenization is being introduced. This method can be used for the determination of the overall properties of materials with particles of different sizes and properties. For these materials procedure starts by the determination of the volume fraction of particles. This can simply be extended to the case of continuous size distribution of particles by defining length intervals to discretize the size distribution. Then it is required to determine the

Comparison with the experimental data

To check the validity of the proposed model its results are compared with the experimental data of different composite materials. These composites are consisted of different types of matrices and fibers. The cases with different distribution of fibers are also selected for the comparison with the current model outcomes.

In the homogenization procedure Hashin–Shtrikman homogenization scheme is being used for the determination of the overall properties of the coated fibers, and the resulting

Conclusions

A new explicit method for determining the overall properties of particulate composite materials was introduced which considers the effect of size distribution and interaction between particles on the mechanical properties. This method is mainly based on the well known periodic homogenization of the composite materials. This paper focused on the properties of particulate composites with fractal distribution of fibers. However, through comparison with some experimental data it is shown that the

Acknowledgements

The author benefited from helpful discussions with Prof. Arash Yavari of the Georgia Institute of Technology. The author wish to thank anonymous reviewer for his/her constructive comments which helped to improve the manuscript.

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