© V.-L. Nguyen, published by EDP Sciences, 2022

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 non-dilute 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 two-phase material composites with simple periodic structures such as square or hexagonal models. In this work, some solutions are proposed to determine the macro-elastic modulus for 3-phase composite materials in 2D (or transverse isotropic unidirectional fiber-reinforced composite) with a more complex structure than previous studies [13]. In Section 2, the Mori-Tanaka approximation for elastic moduli is introduced with the explicit expressions of the estimates for the three-phase 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 three-phase 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 Hashin-Shtrikman bounds and the FFT method.

2 Mori-Tanaka approximation (MTA) for elastic modulus

Consider n-component 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 α-inclusion 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 inclusion-shape, D_Kα, D_μα with α-circular inclusion phases in 2D are specified (3) (4)

The three-component 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 three-component matrix composites, the bulk modulus K and the elastic shear modulus μ formula of Mori-Tanaka 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 three-phase composites.

3 Differential approximations (DA)

Consider n-phase 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_IαΔt (Δt << 1, α = 2, …, n) of randomly oriented inclusions into already constructed composite of the previous step, which contains volume fractions v_Iαt 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 three-phase composites. Though the above construction process of differential scheme corresponds to certain idealistic hierarchical models formed on widely-separated scales, the approximation aims at usual multiphase suspensions of inclusions in a matrix.

4 Hashin-Strikman bounds

HS bounds on the effective elastic moduli of isotropic d-dimensional composites can be presented as [3]

• Elastic bulk modulus

(14)

where (15)

• Shear modulus

(16)

where (17)

5 FFT simulation for three-phase 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 three-component 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 = I₁ ; I₂ or M).

We shall denote the Fourier transform of a V-periodic function F(x) of cartesian x(x₁,  x₂,  x₃) as (21)

with ξ(ξ₁,  ξ₂,  ξ₃) 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 circle-inclusion, the function I_α(ξ) is given by Nemat-Nasser [16]) (24)where , S_Iα = πR², R is the radii of circle-inclusion, x_c(α) is the vector position of the center of the inclusion α; and ξ₁, ξ₂ are the components of ξ; J₁ 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 three-component 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

Iteration i=1:

Iteration i: are known

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 elastically-isotropic 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 three-phase composites, in which two different size balls are arranged randomly non-overlapping (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, Mori-Tanaka 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 Hashin-Shtrikman' 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 circle-inclusion randomly palaced (left), three phase model of composite with two different circles inclusion (right).

Fig. 2 Elastic bulk (left) and shear modulus (right) of three-phase composites with the case (A).

Fig. 3 Elastic bulk (left) and shear modulus (right) of three-phase composites with the case (B).

Fig. 4 Elastic bulk (left) and shear modulus (right) of three-phase composites with the case (C).

7 Conclusions

There have been many previous studies on the elastic moduli of two-phase material composites or three-phase with periodic structures such as square or hexagonal models. In practical materials, the structure of materials is often randomly distributed and has multi-phases. In this work, with different asymptotic solutions, the paper has solved the problem of the elastic moduli for a three-phase material model in 2D.

FFT algorithm is developed to calculate the effective elastic moduli of some complex material models such as three-phase composites with an arbitrary distribution of two inclusion phases. The numerical results fall inside the Hashin-Shtrikman bounds.

DA, MTA approaches are to solve for the three-phase 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

All Tables

All Figures

	Fig. 1 Unit cell containing 60 circle-inclusion 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 three-phase composites with the case (A).
In the text

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

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