Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2021.017589

ARTICLE

Effective Elastic Properties of 3-Phase Particle Reinforced Composites with Randomly Dispersed Elastic Spherical Particles of Different Sizes

Dedicated to Professor Karl Stark Pister for his 95th birthday

1Civil Engineering and Construction Engineering Management, California State University, Long Beach, CA 90840-5101, USA

2Civil and Environmental Engineering, University of California, Los Angeles, CA 90095-1593, USA

*Corresponding Author: Yu-Fu Ko. Email: yu-fu.ko@csulb.edu

Received: 22 May 2021; Accepted: 05 August 2021

Abstract: Higher-order multiscale structures are proposed to predict the effective elastic properties of 3-phase particle reinforced composites by considering the probabilistic spherical particles spatial distribution, the particle interactions, and utilizing homogenization with ensemble volume average approach. The matrix material, spherical particles with radius

Keywords: Particle reinforced composites; micromechanics; spherical particle interactions; ensemble volume average; homogenization; probabilistic spatial distribution; higher-order bounds; multiscale

Composite materials can considerably enhance the following material properties including strength, stiffness, thermal insulation, thermal conductivity, fatigue life, acoustical insulation, wear resistance, corrosion resistance, etc. To achieve targeted engineering performance, composite materials consisting of two or more different materials to macroscopically form new materials are needed. The “inclusions” in composites can be in the forms of whiskers, fibers, and particulates. The “matrix” in composites is the binder material. The matrix material provides the support and protection to the inclusions. The matrix material also transfers stresses and strains through inclusion/matrix interfaces under complex 3-dimensional loading. Limited mechanical properties of conventional particle reinforced composites that contain a single material type of inclusions are observed such as tensile strength, compressive strength, impact resistance.

Alternatively, hybrid particle-reinforced composites, containing several different sizes and/or materials of particles into a matrix, demonstrate superior and excellent mechanical properties and have been largely employed in engineering applications, e.g., aerospace, civil engineering, automobile industries, and military equipment. The mechanical behaviors of hybrid composites, compared with conventional composites, are improved by the weighted summation of the individual particles with different material properties and distinct sizes. Therefore, enhanced engineering performance and cost could be obtained by appropriate engineering material design [1–5].

Numerous experimental research works have exhibited that mechanical behaviors of particle reinforced composites are controlled by particle sizes, micro-structural morphology, and interfacial properties between matrix material and particles [6–17]. Several theoretical methods have been developed in literatures to derive the effective elastic properties of multiphase composites such as variational methods, effective medium methods, direct micromechanical methods, and finite element methods.

Variational methods utilize linear comparison composites or variational principles to acquire mathematical upper and lower bounds for effective elastic properties of composites [18–27]. Hashin's bounds are referred to as the 2-point bounds. Furthermore, the “improved” higher-order mathematical bounds considering the statistical micro-structural information of composites were proposed [28–33]. For instance, Silnutzer [28] proposed improved bounds, referred to as 3-point bounds, on the effective in-plane shear modulus and bulk modulus. The 3-point bounds are narrower when compared with the 2-point bounds.

Effective medium methods employ effective medium to predict the effective elastic properties of composites [34–39]. Mori-Tanaka method, differential scheme, self-consistent method, and generalized self-consistent method are among the effective medium methods. These methods considered only the volume fractions and geometries of inclusions. Conversely, probabilistic distributions or spatial locations of inclusions are not considered. Therefore, the effective medium methods are best suited for low volume fractions of inclusions or some limited particular configurations.

Direct micromechanical methods consider specific geometric configurations of inclusions dispersed in the matrix and utilize approximations to determine effective elastic properties of composites with randomly located and interacting inclusions. Eshelby [40] proposed the renowned “Eshelby's equivalence principle” based on an ellipsoidal inclusion embedded in infinite matrix. Mura [41] primarily considered rigorous “local” micromechanics. Honein [42] proposed frameworks based on Kolosov-Muskhelishvili complex potentials to study circular inclusions in plane elastostatics. Moreover, the direct micromechanical methods are further utilized and studied [43]. Nonetheless, based on the above methods, “local” (not “overall”) field solutions could be achieved. To predict the effective properties of elastic multi-phase composites, Ju et al. [44,45], based on Eshelby [40], proposed a micromechanical ensemble volume average formulation, which is higher-order in volume fractions of inclusions

Finite element methods, i.e., numerical solutions, are obtained based on the “unit cell model” as well as assume particular periodic arrays of inclusions [50–53].

Finally, Ju and co-researchers, following the micromechanical frameworks [44,45], continue to study the effective elastoplastic with/without damage behaviors of advanced composite materials [54–81]. Additionally, Ko et al. [82,83] proposed new higher-order bounds on effective transverse elastic properties of 3-phase hybrid fiber-reinforced composites. The hybrid fiber-reinforced composites contain randomly located and unidirectionally aligned circular fibers featuring distinct fiber sizes and different elastic material properties.

Majority of research works on prediction of the effective elastic properties of particle reinforced composites focus on conventional composites containing one type of particle and did not consider the effects of different particle sizes. In addition, the “unit cell model,” commonly adopted within the 3-D RVE finite element models, assumes periodic arrays of particles, not randomly distributed.

The predictions of effective elastic properties of particle reinforced composites with different particle sizes are still limited in research works. The objective of this paper is to propose analytical frameworks based on the direct micromechanical methods to derive the effective elastic properties of 3-phase spherical particle reinforced composites featuring different particle sizes and same elastic material property of particle. All particles are considered randomly dispersed, non-intersection, and with perfect interfaces between matrix and particles. In addition, both “local” and “overall” ensemble volume averaged micromechanical field equations will be derived by rigorously considering the spherical particle interactions and the probabilistic spatial distribution of particles in formulations as well as utilizing homogenization technique with ensemble volume average approach.

Section 2 presents analytical local solutions of spherical particle interactions. The spherical particles are assumed to be elastic and randomly dispersed in the matrix material. 3-phase composites contain 2 inclusion phases featuring different particle sizes and same elastic properties. Consequently, Section 3 presents the derivations of the ensemble volume averaged eigenstrains based on the probabilistic particle interactions mechanism considering uniform radial distribution function (URDF). Additionally, 2 formulations with different orders, Formulation II and Formulation I, are proposed. In Section 4, the effective elastic properties of 3-phase composites consisting of spherical particles, which are randomly dispersed as well as featuring different particle sizes and same elastic material properties, are analytically obtained. In Section 5, our theoretical predictions are compared with available experimental data together with numerical examples to demonstrate the competence of the proposed frameworks. In Section 6, conclusions are presented.

2 Analytical Local Solutions of Spherical Particle Interactions

Consider a 3-phase particle reinforced composite consisting of randomly dispersed and spherical shaped elastic particles with different particle sizes embedded in an isotropic elastic matrix as shown in Fig. 1. The matrix material, spherical particles with radius

where

According to the eigenstrain concept of Eshelby [40], with the replacement of the spherical particles by the matrix material, the perturbed strain field

where

Due to the presence of the distributed eigenstrain

where V is volume of a representative volume element (RVE),

In addition, the 4th-order tensor G per Mura [41] is expressed as

where

We also define the components of the 4th-order tensor F in the Cartesian coordinates as (m = 1–6)

where

Making use of Eqs. (2) and (3),

By considering spherical particle interactions, Eq. (6) are rearranged as

where

To obtain the 1st-order solution for the eigenstrain

In addition, the 4th-order tensor S, known as interior-point Eshelby tensor of a spherical particle, is [41,58,59]

The components of S for a spherical particle are [41]

where v0 is the Poisson's ratio of the matrix material.

To obtain the effects of spherical particle interactions, Eq. (8) is subtracted by Eq. (9). Then, solving the integral equation

In addition, we define

In order to find the correction of

where the following equalities are employed.

Substitution of Eq. (14) to Eq. (12) yields

where volume of a spherical particle in the ith phase is

Moreover, the average fields shown in Eq. (16) are defined as

The 3rd-order tensor

It is noted that the leading order of

By conducting volume averaging of Eq. (16) over the

By setting i equal to 1 and j equal to 2, we obtained Eqs. (20)–(23).

in which

Similarly, by setting i equal to 2 and j equal to 1, we obtained Eqs. (24)–(27).

where

and the components of

Moreover, we define

Furthermore, Eqs. (20) and (24) are further rearranged:

Eq. (29) can be solved to obtain

It can be proved that the leading order is of

In addition,

Likewise, Eq. (31) can be expressed as

3 Averaged Eigenstrains of Spherical Particles by Ensemble Volume Average Approach

To find the solutions of averaged eigenstrains of spherical particles considering spherical particle interactions, ensemble volume average approach is adopted. The ensemble volume average process is stated as

where

In this paper, we adopt a statistically isotropic and uniform 2-point conditional probability density function known as uniform radial distribution function (URDF).

where r is the distance between the centers of the spherical particles in the ith phase and the spherical particles in the jth phase.

After substituting Eq. (32) into Eq. (35),

where

where

Additionally, we can prove that

in which,

Likewise, after substituting Eq. (33) into Eq. (36),

Similarly, we can also prove that

In the following sections, Formulation II (higher order) and Formulation I (lower order) will be presented to obtain the effective elastic properties of 3-phase particle reinforced composites. In this paper, Superscript II indicates Formulation II and Superscript I indicates Formulation I. It is noted that Formulation II represents higher-order formulation than Formulation I. In addition, with same elastic material property and same shape of spherical particles in the 1st phase and the 2nd phase,

With lengthy algebra together with employing identities Eqs. (18), (37) and (41),

where the isotropic tensor

where

Likewise,

where the isotropic tensor

where

Following similar procedures in Formulation II and dropping the higher-order components

where the isotropic tensor

Likewise,

where the isotropic tensor

4 Effective Elastic Properties of 3-Phase Particle Reinforced Composites with Randomly Dispersed Elastic Spherical Particles of Different Particle Sizes and Same Material Properties

We will employ the solutions of

The ensemble volume averaged field equations are [44]

where

With same elastic material properties of spherical particles (the 1st phase) and spherical particles (the 2nd phase), then we write

in which

Moreover, the effective shear modulus

In addition, the effective bulk modulus

where

Consequently, the effective Young's modulus

Effective properties of the composites based on Formulation I can be derived by replacing Superscript II based on Formulation II with I. It is also noted that our formulations are completely identical to Eqs. (48), (49), (55) and (56) in Lin et al. [46] when the particles featuring same particle sizes and same elastic material property.

5 Numerical Examples and Comparisons

We will present several analytical examples of 2-phase and 3-phase particle reinforced composites under consideration. In addition, the following notations will be adopted in the illustrations.

3-point upper bound of Silnutzer is SU; 2-point upper bound of Hashin is HU; 3-point lower bound of Silnutzer is SL; 2-point lower bound of Hashin is HL; Formulation II is FII; Formulation I is FI.

5.1 2-Phase Elastic Composites: Elastic Spherical Particles in Elastic Matrix

The analytical predictions of our proposed micromechanical framework are compared with the 2-point bounds [19], the 3-point bounds [30,32], and experimental data [84]. As a special case, i.e., particle size of the 1st phase and particle size of the 2nd phase are the same; the proposed formulations reduce to 2-phase formulas. The following material properties of experiments from Smith's data [84] are as follows:

The normalized effective shear modulus

Comparisons among our analytical predictions, 2-point bounds [19], and the 3-point bounds [30,32]), and experimental data [84] on the normalized effective Young's modulus

5.2 3-Phase Particle Reinforced Composites-Varying Particle Volume Fractions at Different Particle Sizes

We assumed the 3-phase composites consist of only glassy spherical particles in both the 1st phase and the 2nd phase of particles with different ratios of particle size

Fig. 4a presented the normalized effective shear modulus

The normalized effective Young's modulus

5.3 3-Phase Particle Reinforced Composites-Varying Particle Size Ratios with a Mixture of Particle Volume Fractions

To examine the effects of varying particle size ratios, various micromechanics-based predictions were conducted. We assumed the 3-phase composites consist of only glassy spherical particles in both the 1st phase and the 2nd phase of particles with varying ratios of particle size

The normalized effective shear modulus

Fig. 7 exhibits the normalized effective Young's modulus

This paper obtains effective elastic properties of 3-phase particle reinforced composites containing randomly dispersed elastic spherical particles of different sizes and the same elastic material properties by our proposed innovative new higher-order micromechanical formulations. In particular, higher-order spherical particle interaction effects together with governing field equations are considered. The randomness of dispersed spherical particles is considered within the probabilistic ensemble volume average process. The averaged eigenstrains in spherical particles are approximately derived by ensemble volume average approach through the spherical particle interactions. Consequently, we obtained compact analytical formulations.

Moreover, improved higher-order bounds on effective elastic properties of 3-phase composites are derived by proposed Formulation II and Formulation I. This paper demonstrates major improvements over the prior works by other researchers based on the same spherical particle sizes in the matrix material. As a special case, i.e., particle size of the 1st phase is the same as particle size of the 2nd phase, the proposed formulations reduce to 2-phase formulas. Our proposed micromechanical frameworks demonstrate excellent agreement with selected experimental data. Our analytical predictions for 2-phase composites also fall within the 2-point bounds [19] and the 3-point bounds [30,32]. With our proposed analytical formulations, finite element calculations and Monte Carlo simulations can be circumvented. In addition, numerical simulations as well as comparisons presented in this paper cover a wide range of 3-phase particle reinforced elastic composites consisting of different spherical particle sizes and same elastic material properties, including varying particle size ratios as well as a mixture of particle volume fractions of the 1st phase particles and the 2nd phase particles.

To further calibrate the analytical frameworks proposed in this paper, experimental validations are the key procedures. When the associated experiment data of 3-phase elastic composites become available, additional comparisons as well as experimental validations will be conducted.

Acknowledgement: This work was in part sponsored by the 2015–2016 California State University Long Beach Research, Scholarship and Creative Activity (RSCA) Award.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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