Issue 
EPJ Appl. Metamat.
Volume 9, 2022



Article Number  9  
Number of page(s)  8  
DOI  https://doi.org/10.1051/epjam/2022007  
Published online  10 June 2022 
https://doi.org/10.1051/epjam/2022007
Research Article
FFT, DA, and MoriTanaka approximation to determine the elastic moduli of threephase composites with the random inclusions
Faculty of Mechanical Engineering, Hanoi University of Industry, 298 Cau Dien Street, Bac Tu Liem District, Hanoi, Vietnam
^{*} email: nguyenvanluat@haui.edu.vn
Received:
2
May
2021
Accepted:
25
February
2022
Published online: 10 June 2022
In this work, some solutions such as MoriTanaka approximation (MTA), Differential approximations (DA), and Fast Fourier transformation method (FFT) were applied to estimate the elastic bulk and shear modulus of threephase composites in 2D. In which two different sizes of circular inclusions are arranged randomly nonoverlapping in a continuous matrix. The numerical solutions using FFT analysis were compared with DA, MTA, and HashinStrikman's bounds. The MTA and DA reasonably agreeable solution with the FFT solution shows the effectiveness of the approximation methods, which makes MTA, DA useful with simplicity and ease of application.
Key words: Elastic modulus / Fast Fourier transformation method (FFT) / MoriTanaka approximation / differential approximations / composite materials
© V.L. Nguyen, published by EDP Sciences, 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Investigations on the macroscopic properties of multiphase materials and media have been started with the classical works of Maxwell, Voigt, Reuss, Einstein [1], and continued with the bounds of Hill [2], and Hashin and Shtrikman [3], Pham & Nguyen [4, 5]. Further studies aim to construct better estimates by including more detailed information about the microstructure of the materials. Approximation schemes have been constructed based on microscopic models. Based on the classical work of Eshelby [6] on the elastic ellipsoidal inclusion problem, effective medium approximations have been developed to predict the overall behavior of the general nondilute composites such as approximation methods (Christensen [7], Mori and Tanaka [8], and the recent Polarization approximation [9]), in which construction of approximate formulae that can describe the macroscopic property accurately over ranges of volume proportions of component materials for engineering uses. Strong numerical methods such as the Finite Element and Fast Fourier ones have been developed and used effectively. Numerical homogenization techniques determining the effective properties give reliable results but challenge engineers by computational costs, especially in the case of complex microstructure. The Fast Fourier Transform has been used to compute the effective properties of periodic composites by Michel et al. [10], Moulinec [11], Bonnet [12] in the context of elasticity. However, most of these studies were applied to twophase material composites with simple periodic structures such as square or hexagonal models. In this work, some solutions are proposed to determine the macroelastic modulus for 3phase composite materials in 2D (or transverse isotropic unidirectional fiberreinforced composite) with a more complex structure than previous studies [13]. In Section 2, the MoriTanaka approximation for elastic moduli is introduced with the explicit expressions of the estimates for the threephase composite with random two different phase inclusions. In Section 3, the DA approximation for elastic bulk and shear modulus are developed, HS bounds are presented in Section 4. In Section 5, the FFT method is developed to directly calculate the effective moduli of threephase composites with the random distribution of fiber inclusions. After that (Sect. 6), numerical examples will be presented to compare the results of MTA, DA, and HashinShtrikman bounds and the FFT method.
2 MoriTanaka approximation (MTA) for elastic modulus
Consider ncomponent transversely isotropic unidirectional elastic composites of randomly oriented inclusions of type α (α = 2, …, n). The matrix phase has the volume fraction v_{M} and the αinclusion has the volume fraction v_{Iα}. The bulk modulus and shear modulus of the matrix are K_{M} and μ_{M}, respectively, those of the α inclusion phases are K_{Iα} and μ_{Iα}. The MTA, derived as an approximate solution to the field equations for the composite to compute the elastic bulk modulus K_{MTA} and shear modulus μ_{MTA} has the expressions [1,8] (1) (2)
D_{Kα}, D_{μα} are functions depending on the inclusionshape, D_{Kα}, D_{μα} with αcircular inclusion phases in 2D are specified (3) (4)
The threecomponent composite that are two circular inclusions having elastic bulk modulus K_{I1}, K_{I2} shear modulus μ_{I1}, μ_{I2} and volume fraction v_{I1}, v_{I2} in a matrix having the elastic moduli K_{M}, μ_{M} and volume fraction v_{M}. In the case of threecomponent matrix composites, the bulk modulus K and the elastic shear modulus μ formula of MoriTanaka approximation can be written as: (5) (6)
where the dilute suspension expression D_{Kα}, D_{μα} for an inclusion α has been defined in (3), (4). Equations (5) and (6) will be used to determine the elastic moduli of threephase composites.
3 Differential approximations (DA)
Consider nphase suspension of randomly oriented inclusions of type α (α = 2, …, n), with elastic bulk modulus K_{Iα}, shear modulus μ_{Iα} (volume proportion v_{Iα}) in a matrix of elastic moduli K_{M}, μ_{M} (volume proportion v_{M}). The differential scheme construction process for the suspension starts with the base matrix phase M. At each step of the procedure, we add proportionally infinitesimal volume amounts v_{Iα}Δt (Δt << 1, α = 2, …, n) of randomly oriented inclusions into already constructed composite of the previous step, which contains volume fractions v_{Iα}t of the inclusion phases (the parameter t increases from 0 to 1, as the differential scheme proceeds). The newly added particles will see an effective continuum, owing to their relative sizes, and the new composite can be considered as a dilute suspension of particles from phases α, of volume fractions (7)
where , in a matrix of elastic bulk modulus K, shear modulus μ (v_{I} is the total volume fractions of the included phases). The elastic modulus of the new composite is (8) (9)
where the dilute suspension expression D_{Kα}, D_{μα} for an inclusion α has been defined in (3), (4). Since the volume fraction of the included phase α increases by (10)
we obtain the following differential equation for the elastic bulk modulus K, shear modulus μ of the composite (11) (12) (13)
Differential equations (11) and (12) will be used to determine the elastic moduli of threephase composites. Though the above construction process of differential scheme corresponds to certain idealistic hierarchical models formed on widelyseparated scales, the approximation aims at usual multiphase suspensions of inclusions in a matrix.
4 HashinStrikman bounds
HS bounds on the effective elastic moduli of isotropic ddimensional composites can be presented as [3]
Elastic bulk modulus
Shear modulus
5 FFT simulation for threephase composites
The FFT method uses the classical expansion along with the Neuman series of the solution of the periodic elastic problem in Fourier space, based on the Green's tensor and exact expressions of the shape factors in Fourier space [11,12,14,15]. In this section, the FFT method is briefly presented for calculating the effective elastic moduli of threecomponent materials in 2D.
Behavior of the component materials is described by Hooke' law: (18)
where σ(x) and ϵ(x) are respectively the local stress and strain fields, the stress field satisfies the equilibrium condition (19)
Let x denote the position of a point in the unit cell. C(x) is the fourth order local elastic tensor of the heterogeneous medium, one is given by (20)α designates the phase (α = I_{1} ; I_{2} or M).
We shall denote the Fourier transform of a Vperiodic function F(x) of cartesian x(x_{1}, x_{2}, x_{3}) as (21)
with ξ(ξ_{1}, ξ_{2}, ξ_{3}) being the wave vector, the symbol “*” designates the product of convolution .
The Fourier transformation of elastic tensor is (22)where I_{α}(ξ) are the shape functions, defined by (23)
In the case of circleinclusion, the function I_{α}(ξ) is given by NematNasser [16]) (24)where , S_{Iα} = πR^{2}, R is the radii of circleinclusion, x_{c}(α) is the vector position of the center of the inclusion α; and ξ_{1}, ξ_{2} are the components of ξ; J_{1} is the Bessel function of first kind and first order. I_{M}(ξ) can be derived from relation (25)
For ξ = 0, one have
Substituting the Fourier transformation of the local stress, strain fields into equilibrium equation (19), the problem in a unit cell is solved by explicit recurrence process in Fourier space. That can be rewritten in the form [10,11] (26)in which and are respectively Fourier transformation of σ(x) and ϵ(x), is the Greens's tensor, the symbol “*” designates the product of convolution. The value of the Green's tensor is given for an isotropic reference medium by Mura [1]. (27) (28) (29) (30)where , , λ and μ are Lame coefficients.
Relationship between and is described by expression (31)
For the threecomponent medium considered, the expression (26) can be written as (32)
At convergence of the iterative process, one finds (33)
The numerical algorithm is given as follows
The convergence of the iterative procedure is reached when , where ϵ is a prescribed value (ϵ = 10^{−3}).
6 Applications
In this section, we use the FFT method, MTA approximation, and DA approximation to estimate the effective elastic moduli of the elasticallyisotropic composites 2D.
We consider 3 examples, with
 (A)
K_{M} = 2, μ_{M} = 1, K_{I1} = 100, μ_{I1} = 60, K_{I2} = 20, μ_{I2} = 10
 (B)
K_{M} = 30, μ_{M} = 16, K_{I1} = 10, μ_{I1} = 6, K_{I2} = 1, μ_{I2} = 0.5
 (C)
K_{M} = 30, μ_{M} = 16, K_{I1} = 4, μ_{I1} = 2, K_{I2} = 100, μ_{I2} = 60.
We consider threephase composites, in which two different size balls are arranged randomly nonoverlapping (Fig. 1). For numerical FFT illustrations, 60 circles inclusion were planted randomly in a unit cell by Matlab program such that there is no circle overlapping, in which a unit cell having the dimension L = 1 along each space direction containing inclusion (Fig. 1, left), the minimum distance of is 0.01. In our calculations, a grid 128 × 128 is considered. The FFT result simulation is obtained from the algorithm in Section 4. The FFT results compared with Differential approximation, MoriTanaka approximation, and Hashin–Shtrikman bounds over ranges of v_{I} = v_{I1} + v_{I2}, v_{I1} = 2v_{I2} (all the inclusions in one phase have the same size, the dimensionless radius R_{I1} varies from 0.01 to 0.063 and R_{I2} varies from 0.0071 to 0.0445) are reported in Figures 2–4 and Tables 1–6. Numerical FFT results and MTA, DA approximations are quite close and converge at small volume fractions of inclusion phases and diverge at large proportions of suspended particles, all the results fall inside the HashinShtrikman' bounds, as expected. The DA, MTA approximations are asymptotically exact at dilute suspensions of included particles, but become inevitably less so good at higher proportions of included phases. At large proportions of included phases, the details of particles' interactions of particular microstructures should be accounted for more accurate estimations.
Fig. 1 Unit cell containing 60 circleinclusion randomly palaced (left), three phase model of composite with two different circles inclusion (right). 
Fig. 2 Elastic bulk (left) and shear modulus (right) of threephase composites with the case (A). 
Fig. 3 Elastic bulk (left) and shear modulus (right) of threephase composites with the case (B). 
Fig. 4 Elastic bulk (left) and shear modulus (right) of threephase composites with the case (C). 
Comparison of results (K^{eff}) of FFT, MTA, DA, and HS bound for the case (A), .
Comparison of results (μ^{eff}) of FFT, MTA, DA, and HS bounds for the case (A), .
Comparison of results (K^{eff}) of FFT, MTA, DA, and HS bounds for the case (B), .
Comparison of results (μ^{eff}) of FFT, MTA, DA, and HS bound for the case (B), .
Comparison of results (K^{eff}) of FFT, MTA, DA, and HS bound for the case (C), .
Comparison of results (μ^{eff}) of FFT, MTA, DA, and HS bound for the case (C),
7 Conclusions
There have been many previous studies on the elastic moduli of twophase material composites or threephase with periodic structures such as square or hexagonal models. In practical materials, the structure of materials is often randomly distributed and has multiphases. In this work, with different asymptotic solutions, the paper has solved the problem of the elastic moduli for a threephase material model in 2D.
FFT algorithm is developed to calculate the effective elastic moduli of some complex material models such as threephase composites with an arbitrary distribution of two inclusion phases. The numerical results fall inside the HashinShtrikman bounds.
DA, MTA approaches are to solve for the threephase material model of two different sizes of circular inclusions. FFT, MTA, and DA give quite close results. All results satisfy HS bounds over all the volume proportions of the components.
MTA and DA have explicit algebraic expressions, so that easy to apply the estimates for the effective elastic moduli, hence might be more useful for engineers as first estimates of the effective elastic moduli of the composites.
References
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Cite this article as: VanLuat Nguyen, FFT, DA, and MoriTanaka approximation to determine the elastic moduli of threephase composites with the random inclusions, EPJ Appl. Metamat. 9, 9 (2022)
All Tables
Comparison of results (K^{eff}) of FFT, MTA, DA, and HS bound for the case (A), .
Comparison of results (μ^{eff}) of FFT, MTA, DA, and HS bounds for the case (A), .
Comparison of results (K^{eff}) of FFT, MTA, DA, and HS bounds for the case (B), .
Comparison of results (μ^{eff}) of FFT, MTA, DA, and HS bound for the case (B), .
Comparison of results (K^{eff}) of FFT, MTA, DA, and HS bound for the case (C), .
All Figures
Fig. 1 Unit cell containing 60 circleinclusion randomly palaced (left), three phase model of composite with two different circles inclusion (right). 

In the text 
Fig. 2 Elastic bulk (left) and shear modulus (right) of threephase composites with the case (A). 

In the text 
Fig. 3 Elastic bulk (left) and shear modulus (right) of threephase composites with the case (B). 

In the text 
Fig. 4 Elastic bulk (left) and shear modulus (right) of threephase composites with the case (C). 

In the text 
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