Contents

1 Eshelby’s ellipsoidal inclusion theory

Eshelby established the basis of the micromechanics theory for isotropic body[1]. Afterward it has been extended for anisotropic cases by other researchers, and has become more sophisticated theory; for example, one can see more systematized theory in Mura’s book[2]. In this section, we introduce Eshelby’s ellipsoidal inclusion theory.

1.1 Eigenstrain

Suppose that only a small part Ω in an originally uniform substance is expanded because of thermal dilatation or a certain transformation. Then, the substance becomes in an internal-stress state. If the surroundings of Ω does not exist, the displacement due to such an intrinsic dilatation of Ω will be larger; the strain expressed by this displacement is not elastic but plastic, which is called eigenstrain or stress-free transformation strain, which is denoted as ϵ_kl^*. However, the surroundings prohibits such an intrinsic dilatation; Ω is shrinked elastically because of the constraint by the surroundings. This elastic strain is denoted as ϵ_kl. When the total displacement measured referring to the initial state is denoted as u = (u₁,u₂,u₃), the total strain can be written as γ_kl = ∂u_k∕∂x_l, which is given by sum of the eigenstrain and elastic strain;

Next suppose that a number of inclusions with an eigenstrain ϵ_kl^* exist in a substance; the spatial distribution of ϵ_kl^* is denoted as ϵ _kl^*(x). When x is located inside Ω (i.e., x ∈ Ω), ϵ_kl^*(x) = ϵ _kl^*, and when x is outside Ω (i.e., x ∋ Ω), ϵ _kl^*(x) = 0. In the case where the elastic constants C_ijkl of matrix are equal to those of inclusions, Hooke’s law can be written as

1.2 Elastic-equilibrium equation for elastic homogeneity

1.2.1 General solution

When the total displacement u or total strain γ are known, all the stress/strain fields can be analyzed. To do this, we have to solve the following elastic-equilibrium equation:

(3)

Hereafter, the summation convention is applied when the same subscripts appear. Substituting Eq. (70) into Eq. (3), we obtain

(4)

To solve this equation, we transform u_k(x) and ϵ_k′l′^*(x) into the Fourier forms: ũ_k(g) and _k′l′^*(g). Then, in the Fourier spcace, Eq. (4) is written as

(5)

The solution of Eq. (5) is given by

(6)

where K_ik ≡ C_ijklg_jg_l and the subscripts k′l′ are changed to mn. When g is denoted as gg (g is the unit vector, and g is the magnitude of the vector),

(7)

Hence, Eq. (6) is written as

(8)

and hence the expression of ∂u_k∕∂x_l in the Fourier space is

(9)

where {A(x)}_g means the Fourier transform of A(x). Therefore, the symmetric total strain γ_kl(x) is expressed as

(12)

from Eq. (8)

1.2.2 Solution for ellipsoidal inclusion

Hereafter, we consider the case wehre there is one inclusion Ω that has the eigenstrain ϵ^*(x′). Since ϵ^*(x′) = 0 for x′∋ Ω, Eq. (14) is

(28)

we obtain

(30)

we have to seek x′ in Ω so as to satisfy g ⋅ (x - x′) = 0.

We first deal with the integral ∫ _Ωϵ_mn^*(x′)δ(g ⋅ (x - x′))dx′ in Eq. (29), where we ragard that x and g are fixed, and consider the case where x = (x₁,x₂,x₃) is located in the ellipsoidal inclusion Ω with radii a₁, a₂, and a₃:

(31)

Transformations of variables are made as follows:

(34)

Since dx = a₁a₂a₃dy and g ⋅ (x - x′) = hh ⋅ (y - y′), Eq. (29) is written as

In Eq. (36), ∫ _S²ϵ_mn^*(x′)δ(hh ⋅ (y - y′))dy′ is performed under the fixed y and h. Therefore,

When x is located in the exterior matrix (i.e., x ∋ Ω), it is quite complicated to calculate the outside fields because the hatched area displayed in Fig. 2, which satisfies h ⋅ (y - y′) = 0, does not exist for any y′ when y ⋅h > 1. Taking into account this matter, Cheng and Mura has derived the outside field of ellipsoidal inclusion[3].

1.3 The Eshelby tensor

As found from Eq. (43), the total strain γ_ik in the inclusion Ω can be written in the following form:

General ellipsoidal shape As seen in Eq. (43), the most general form of the fourth-rank Eshelby tensors for the arbitrary shape of ellipsoidal inclusion are given by

where the surface integral is performed over the unit sphere |S²| that is made by unit vector g, C_jlmn denotes the elastic constants of matrix, g_p is a component of g, and (a₁, a₂, a₃) denote the radii of the ellipsoidal inclusion.

Cylindrical shape For cylindrical (needle) shape, i.e., a₃∕a₁(= a₃∕a₂) →∞, S tensors reduce to

Plate-disc shape For the plate-disc inclusion with zero aspect ratio i.e., a₃∕a₁(= a₃∕a₂) → 0, S tesors further reduce to

In general, the principle axes of ellipsoidal inclusion are not always consistent with the crystallographic axes (in which the elastic constants, strain and stress etc are defined). In such a case, it is convenient to transform the elastic constants of crystal-coordinate system into those of the inclusion-coordinate system:

Moreover, there are often cases that we want to use the Eshelby tensors of 6 × 6 matrix form, because the elastic-constant tensors can be reduced to 6 × 6 matrix form, and the tensor calculation becomes more simple. As found from Eqs. (45)-(46), there is a following symmetry S_ijkl = S_jikl = S_ijlk and S_ijkl≠S_klij. Therefore, the Eshelby-tensor matrix S can be expressed as

1.4 Equivalent inclusion method

When the elastic constants of inclusion are different from those of matrix, we have to reconsider the elastic-equilibirum equation (this will be discussed in the later section). Here, a convenient method that does not require the reconsideration of the elastic-equilibirum equation is presented. If the elastic constants of inclusion can be somehow regarded as the same as those of matrix by modifying the eigenstrain, we can deal with the same elastic-equilibrium equation as well as in the previous sections. There is a well-known method, so-called equivalent inclusion method, to solve the elastic inhomogeneity problem.

Type-I equivalent inclusion When an ellipsoidal inclusion has no eigenstrain but elastic constants C′_ijkl different from those of matrix C_ijkl, stress and strains due to an applied external stress becomes not uniform because of the elastic inhomogeneity; the stress and strain disturbances, which are denoted as σ′_ij and γ_kl, are tried to be reproduced using an equivalent inclusion.

When there is no inclusion, a uniform strain ϵ_kl⁰ is caused by external stress:

We have to seek the equivalent eigenstrain ϵ_kl^* so as to satisfy Eq. (51). Note that if the elastic constants are the same, the stress and strain associated with the external stress are uniform, even when the inclusion has an eigenstrain. Furthermore, the intrinsic (original) eigenstrain ϵ′_kl^* of the inclusion does not appear in Eq. (51). This is because the initial internal-stress state due to ϵ′_kl^* in the absence of external stress is regarded as a standard. These matters are related with Colonnetti’s theorem, which is described in Sec. 3.

Type-II equivalent inclusion We consider the case where an ellipsoidal inclusion has both eigenstrain ϵ′_kl^* and different elastic constants C′_ijkl. The internal stress due to the inhomogeneous inclusion is tried to be reproduced using an equivalent inclusion.

The stress inside the inclusion is expressed as

We have to seek the equivalent eigenstrain ϵ_kl^** so as to satisfy Eq. (52).

2 Phase-field microelasticity theory

In the previous section, we derived Eqs. (11), (16) and (14) as solutions of linear elastic-equilibrium equation (3) for elastic homogeneity, and obtained Eq. (43) for the special case where there is one ellipsoidal inclusion in the substance. However, the following two points has not been discussed yet.

1. In the elastic-equilibrium equation, we did not consider the case where the elastic constants of inclusions are different from those of matrix. As stated in previous section, there is a well-known method, so-called equivalent inclusion method, to solve the elastic inhomogeneity problem. If we do not use the equivalent inclusion method, how do we write the elastic-equilibrium equation, and how should we deal with this problem?
2. The integrand in Eq. (11), C_ijmn(G_ik^-1g _l + G_il^-1g _k)g_{j mn}^*(g) exp(ig ⋅ x), shows a singularity at g = 0, because G_ik = C_ipkqg_pg_q has no inverse matrix. In the derivation of Eq. (43) for ellipsoidal inclusion, we have avoid calculating the integrand at g = 0, by utilizing the characteristics of delta function δ(x). If we treat Eq. (11) more generally, how should we deal with this singularity?

In this section, to overcome the above problems, the phase-field microelasticity theory of Khachaturyan[5] is introduced. The theory adopts Eshelby’s concept, but the ellipsoidal inclusion method is no longer used and the discrete Fourier transformation (DFT) technique is utilized in the theory.

2.1 Macroscopic shape change and total strain

The total strain is redefined as

(53)

where γ_kl and δγ_kl(x) represent the average and deviation of the total strain, respectively. From the definition,

(54)

Thus, the macroscopic shape change of the substance is represented by γ_kl. Since the Eshelby theory treats the disturbances (of stress, strain, displacement fields) caused by the small region with eigenstrain, the shape change of the whole substance is not taken into consideration. Namely, the total strain γ_kl(x) uesd throughout the previous sections is a quantity measured referring to the sunstance without any macroscopic-shape change. According to Eq. (53), the above can be written as γ_kl = 0 and γ_kl(x) = δγ_kl(x). Also in the case where the macroscopic shape of substance changes, we have to use

(55)

where the displacement u is measured referring to the macroscopical shape with a deformation represented by γ_kl.

External strain Without macroscopic change due to the eigenstrain, when an external stress σ_kl^ext, which is given by

(56)

is applied to a substance, the macroscopic average γ_kl of the total strain is equal to the average strain ϵ_kl^ext cuased by the external stress;

(57)

Internal strain Consider a structural phase transformation, for example, the substance with crystal lattice of cubic symmetry transforms into that of tetragonal symmetry; its eigenstrain ϵ_kl^* is measured referring to the cubic lattice. Then, the macroscopic average of the total strain is

External and internal strain The strains in both cases are summed up:

2.2 Elastic-equilibrium equation for elastic inhomogeneity

We first define a new parameter Φ(x) to prescribe which material exists at a position x; Φ(x) = 0 or Φ(x) = 1 mean the matrix or inclusion, respectively. When the difference of the elastic constants between inclusion and matrix is denoted as ΔC_ijkl, the elastic constants at x are written as

(64)

Similarly, the eigenstrain ϵ_kl^* representing the inclusion is defined as

(65)

According to Eq. (70), the internal stress field is given by

Zeroth-order approximation When the effect of ΔC_ijkl is neglected (i.e., ΔC_ijkl = 0), Eq. (76) reduces to

(78)

(79)

Therefore, from Eq. (55), we obtain

First-order approximation We write Eq. (76) as

Higher-order approximation Similarly, we obtain the higher-order solution through the iterative calculations:

2.3 Application of Discrete Fourier Transformation technique

2.3.1 Definition of DFT

We try to apply the discrete Fourier transformation (DFT) technique for the calculation of Eq. (80) with Eqs. (81) [0th-order solution], (89) [1st-order solution], and (93) [Higher-order solution]. For the sake of simplicity, as an example, we here treat the solution of zeroth-order approximation, i.e., Eq. (81).

---

Table 1: Physical quantities corresponding to the indices, x_i and g_i, used in the discrete Fourier transformation. Note that 2πg_i∕NΔx in DFT corresponds to g_i = 2π∕λ_i of conventional Fourier transformation.

Index	xi	0	1	2		N - 1	(N)
L [m]	xiΔx	0	Δx	2Δx		(N - 1)Δx	(NΔx)
Index	gi	0	1	2		N - 1	(N)
λ [m]		∞	NΔx				(Δx)
[1/m]		0					()

---

In order to perform DFT, the substance is divided into N^d elements, where d denotes the dimension, and the division length is denoted as Δx [m]. Figure 3 indicates the relation between real and reciprocal spaces. With regard to physical quantities A(x) and Ã(g), the definition of DFT is given by

(97)

2.3.2 On singular point

We consider the g = 0 point in Eq. (81). The DFT forms of eigenstrain _kl^*(g) is given by

In the similar way with this DFT characteristics, the following relation

2.3.3 Constraint conditions

Next let us consider the constraint conditions with regard to the macroscopic-shape change. For this purpose, we rewrite Eq. (63) in the discrete form as

(1) Without macroscopic shape change If the substance is fixed in the rigid box, γ_kl = 0 should be used for the calculations.

(2) Without external stress If no external field is applied and the substance is elastically homogeneous, the average total strain is given by the spatial average of eigenstrain:

If the substance is elastically inhomogeneous, since

(3) With uniform strain Origin of the uniform strain (e.g., mechanical, electrical, magnetical sources) is not limited. Then, γ_kl is given as a constraint condition.

For example, when the external stress σ_ij^ext is applied to the elastically homogeneous system, the external uniform strain

(107)

is caused. Then, when calculating Eq. (103), we use as a constraint condition

(4) With external stress Since the eigenstrain effect ϵ_kl^*Φ(x) can be added later, we distinguish the effects of the external stress and eigenstress (eigenstrain), and here discuss the only effect of external stress.

This condition is the most general, but the most complicated in the case of the elastic inhomogeneity, which is totally different from the uniform strain condition. Because the uniform strain is usually not caused, when an external stress σ_kl^ext is applied to the elastically inhomogeneous substance. The relation that necessarily holds is

(109)

where

Since the eigenstrain effects can be added later, this is now ignored, and then the macroscopic average γ_kl of the total strain equals the average strain ϵ_kl^ext cuased by the external stress;

(110)

Usually, we do not know the detail of σ_ij^ext(x), and therefore this constraint condition is somewhat complicated. If you know the effective (average) elastic constants C_klij in advance, we obtain

(111)

Therefore, solving this constraint problem is converted to obtaining the effective elastic constants.

The effective elastic constants C_klij obtained by the other methods are generally not self-consistent. Thus, the following iterative calculations are needed.

1. First assume C_klij arbitrarily (with the other method), and calculate the average total strain using γ_mn = C_mnij^-1σ _ij^ext.
2. Obtain the equilibrium solution with higher-order approximation:
   
   where δγ_mn⁽⁰⁾(x) = 0 and G _ik = C_ipkqg_pg_q. Note that the eigenstrain is not related to the effective elastic constants in the regime of linear elasticity theory.
3. Check the sum of local stresses using Eq. (70), i.e.,
   
   If this sum equals the external stress σ_ij^ext, the assumed C _klij is valid, because the sum of internal stresses (eigen-stresses) is zero; see Eq. (61).
4. After obtaining the self-consistent C_klij, γ_mn is recalculated and then the effect of the eigenstrain is considered with higher-order approximation.

Remarks This method for evaluating the effective elastic constants is implicitly based on Colonnetti’s theorem, which is described next section. According to this theorem, the external stress does not interact with the elastic strain due to eigenstrain. When the substance includes the elastic inhomogeneous inclusions with an eigenstrain, the substance is in an internal-stress state, regardless of the existence of an external stress. We regard this (equilibrium) state as a standard state, and consider the case where an external stress is further applied to the equilibrium state. Colonnetti’s theorem says that the total strain energy E_str stored in the substance is given by the simple sum of the strain energy due to the eigenstrain E_str^int and that due to the external stress E_str^ext, i.e., E _str = E_str^int + E _str^ext. If the inclusion has no eigenstrain but has only different elastic constants, the strain energy is caused only by the external stress, which equals E_str^ext stated above. Namely, we may ignore the eigenstrain effect when considering the effective elastic costants. Once we obtain the accurate or self-consistent effective elastic constants, δγ_mn is to be calculated with Eq. (93) using the average total strain γ_mn = C_mnij^-1σ _ij^ext.

3 Mechanical energies

In this section, we discuss various kinds of mechanical energies. There are two types of energies to be discussed here. One is the elastic strain energy E_str, and the other is the potential energy E_pot of external stress (load). The sum of these energies is called the total mechanical energy:

3.1 Definitions and theorem

3.1.1 Strain energy

Let us consider the case where the substance has only one inclusion with eigenstrain ϵ_ij^*. The strain energy stored in the whole substance is

3.1.2 Elastic interaction energy

Let us consider the case where two inclusions exist in the substance, and we define the elastic interaction energy. The total strain energy due to the eigen-stresses (1) and (2) is given by

3.1.3 Colonnetti’s theorem

When the eigenstress/eigenstrain and the external stress/strain coexist, the elastic strain energy is expressed as

Furthermore, using the other hand, σ_ij^ext(x)ϵ _ij(x) = σ_ij^ext(x)[γ _ij(x) -ϵ_ij(x)], in the expansion,

3.1.4 Total mechanical energy and mechanical interaction

Let us consider the case where the eigenstrain and external stress coexist. Considering the potential energy due to external stress (potential energy of weight in the system consisting of spring and weight), the total mechanical energy E_mec, which corresponds to the Gibbs free energy, can be defined as

Remarks In the above arguments, there is no limitation or constraint on the elastic constants C_ijkl, i.e., the case of elastic inhomogeneity can also be dealt with, and furthermore the shape of inclusions are not specified.

3.2 For ellipsoidal inclusion

When the inclusion shape is assumed to be ellipsoidal, the calculations including volume integrals are fairly reduced, because the internal stress and strain inside the inclusion become uniform.

3.2.1 Strain energy

Elastic homogeneity From Eq. (115), the elastic strain energy is given by

Elastic inhomogeneity If an elastically inhomogeneous (C′_ijkl) inclusion has an eigenstrain ϵ′_kl^*, using the equivalent eigenstrain ϵ_kl^* (see section 1.4), the internal stress inside the inhomogeneous inclusion is given by

3.2.2 Mechanical interaction energy

From Eq. (144), the mechanical interaction energy E_pot^int is

Elastic homogeneity In this case, the above equation is straightforwardly applicable, i.e.,

Elastic inhomogeneity In the case of elastically inhomogeneous (C′_ijkl) inclusion, if it has no eigenstrain, E_pot^int is not caused. When the elastically inhomogneneous inclusion has an eigenstrain ϵ′_kl^*, E _pot^int should be considered. The applied stress to the inclusion, σ_ij^ext(x), is different from the originally uniform stress σ _ij^ext(= C _ijklϵ_kl⁰), i.e., σ_ij^ext(x) = σ _ij^ext + σ′ _ij(x) ≡ σ′_ij^ext. In order to know the stress inside inclusion, Type-I equivalent inclusion method, Eq. (51), is used:

3.3 External stress applied to elastic inhomogeneity

In the previous sections, the general definitions of the mechanical energies are presented. Here, we focus especially on the case where the external stress is applied to an elastically inhomogeneous substance. We are not concerned about the existence of eigenstrain regions; for the sake of simplicity, the substance has no eigenstrain regions.

In the case where an external stress is applied to an elastically inhomogeneity, the total mechanical energy (the residual parts of Eq. (142) after removing the eigenstrain terms) is expressed as

After all, the total mechanical energy is expressed as

Application of equivalent inclusion The above argument holds regardless of the inclusion shape. We here assume an ellopsoidal-shape inclusion. Referring to Eq. (144), we apply the Type-I equivalent inclusion method to this case:

By substituting Eq. (168) into the first term of the right-hand side of Eq. (165), we can discuss macroscopic effective elastic constants. Next section presents more systematic treatment for deriving the effective elastic constants, and this strain-energy balance will be discussed in Section 4.4.

4 Effective elastic constants

The general formulae for the effective elastic constants are to be derived based on the Eshelby’s ellipsoidal inclusion theory[1] and Mori-Tanaka’s meand-field approximation (MTMF) theory[7]. On this issue, the mathmatical treatment has been first presented by Benveniste[8]. His approach is very simple and widely applied to other relatated topics. In this section, following the procedure of Benveniste, we will present the derivation of the effective elastic constants.

4.1 Definition of effective elastic constants

Consider a composite material consisting of a matrix with the elastic constants C₀ and a type of inclusion with C₁, whose volume fractions are denoted by 1 - p and p, respectively. The average stress σ and average strain ϵ of the composite are approximated as σ = (1 - p)σ₀ + pσ₁ and ϵ = (1 - p)ϵ₀ + pϵ₁, where σ₀ = C₀ϵ₀ and σ₁ = C₁ϵ₁. The average stress σ must be equal to an external stress, because the sum of internal stresses have to be zero: ∫ _V σ_ij(x)dV = 0. Then, the average elastic constants C of the composite material can be defined as σ = Cϵ, i.e.,

(170)

When expressing ϵ₁ = Aϵ₀, the average elastic constants C of the composite can be written as

(171)

where I is the unit matrix. The matrix A is called the strain-concentration factor. In order to obtain the effective elastic constants C, we must find A.

4.2 Strain-concentration factor

The strain-concentration factor A can be derived on the basis of Eshelby’s equivalent-inclusion[1] and Mori-Tanaka’s mean-field theories[7].

4.2.1 Equivalent inclusion method

We suppose first that the external stress σ^ext is applied to an elastically uniform (elastic homogeneous) substance:

(172)

where ϵ₀ denotes the uniform strain in the absence of any inclusions. Next, we consider a substance containing, in a matrix with the elastic constants C₀, an infinitesimally small inclusion that has different elastic constants C₁ but has no intrinsic eigenstrain such as a lattice misfit, and consider the case where an external stress σ^ext is applied to such an elastically inhomogeneous substance. Then, the internal stress σ₁ inside the inclusion is given by

(173)

where σ^∞ and γ represent the disturbances of internal stress and strain. which are caused by the elastic inhomogeneity under an applied stress. Such a situation is depicted in Fig. 4. When the inclusion is ellipsoidal, according to Eshelby’s concept, Eq. (173) can be expressed using the eigenstrain ϵ^* of “the equivalent inclusion” as

(174)

where the extra strain γ is given by γ = Sϵ^* with S the Eshelby tensor. Hence, σ^∞ can be expressed as

(175)

If σ^∞ is expressed without using the Eshelby tensor S,

(176)

When ϵ₀ = 0 or σ^ext = 0, the stress disturbance is not caused i.e., σ^∞ = 0 and therefore γ = 0. When ϵ₀≠0 or σ^ext≠0, σ^∞≠0 and therefore γ = 0 because of S≠I in Eq. (175). Thus, σ^∞ and γ are functions of σ^ext or ϵ ₀.

4.2.2 Utilization of Mori-Tanaka’s mean-field theory

We deal with a composite material with a large number of inclusions. Under an external stress σ^ext′(≠σ^ext), the average stresses and strains of matrix and inclusion are denoted as σ₀,ϵ₀ and σ₁,ϵ₁, respectively. (Then, σ^ext′ = (1 - p)C ₀ϵ₀ + pC₁ϵ₁ holds.) The differences between σ₁ and σ₀ and between ϵ₁ and ϵ₀ are denoted by σ′ and γ′, i.e., σ₁ = σ₀ + σ′ and ϵ₁ = ϵ₀ + γ′.

On the other hand, Mori-Tanaka’s mean-field (MTMF) theory[7] allows us to express the average internal stress of inclusions as

(177)

Equation (177) can be understood as follows; see Fig. 4. We suppose that new one inclusion is further added into the matrix of the composite with numerous inclusions. As stated in the previous paragraph, in the case of single inclusion, the stress disturbance due to the inclusion embedded in a matrix with the external uniform strain ϵ₀ (caused by σ^ext) is given by σ^∞. Since the new inclusion is to be embedded into the arleadly stressed (or strained) matrix with σ₀ (or ϵ₀), the internal stress of new inclusion is approximately given by sum of σ^∞ and σ ₀, i.e., σ₁ ≈σ₀ + σ^∞. Moreover, as far as σ₀(x) is not so much deviated from σ₀, we may regard that the internal stress σ₁ of the newly added inclusion is virtually equal to the average internal stress σ₁ of inclusions in the pre-added original composite, i.e., σ₁ ≈ σ₁. Therefore we obtain σ₁ ≈σ₀ + σ^∞. It should be noted here that we have to use σ^∞ in the situation that ϵ ₀ equals ϵ₀. After all, we find that σ′ = σ^∞(ϵ ₀) and γ′ = γ(ϵ₀).

Then, by using Eq. (175) based on the equivalent inclusion concept, we can write Eq. (177) as

(178)

Here, expressing A as

(179)

and substituting Eq. (179) into eq. (178), we obtain

(180)

4.2.3 Multi-component composite

Essentially, the similar treatment is applicable to this case. The species of the component phases are denoted i. Then, the effective elastic constants can be expressed as

(181)

where ∑ _ip_i = 1. The average strain of i phase is denoted as ϵ_i, and we write ϵ_i in the similar way to Eq. (179) as

(182)

where A_i represents the strain-concentration factor of i phase, which is given (in the regime of Mori-Tanaka’s mean-field theory) by

(183)

Note that Eq. (183) satisfies A₀ = I.

4.3 Application to special composites

4.3.1 Porous material

We here consider the case of porous material; pore can be regarded as inclusion with C₁ = 0. Here, the slightly different way of derivation of A is presented. First, let us consider the case where the single pore exists in an infinite matrix. Since σ₁ = C₁ϵ₁ = 0 holds in Eq. (174), the following relation

(184)

must hold. Namely, according to the magnitude of an external stress σ^ext, γ is determined to be

(185)

so as to guarantee that σ₁ = 0. As mentioned abobe, one can find that γ is a function of σ^ext or ϵ ₀.

Next a porous composite is considered. Since under an external stress σ^ext′,

(186)

holds, substituting C₁ = 0 into above equation, we obtain σ^ext′ = (1 - p)C ₀ϵ₀. In order to refer to the case of an infinitesimal single inclusion, we suppose the case where the average strain ϵ₀ of the matrix is equal to ϵ₀;

(187)

i.e., σ^ext′ = (1 - p)σ^ext in the case of ϵ ₀ = ϵ₀. Referring to ϵ₀ + γ = Aϵ₀ [Eq. (179)], we obtain

(188)

or from Eqs. (179), (185) and ϵ₀ = ϵ₀, we directly obtain

(189)

Eliminating σ^ext′ or ϵ ₀ in both hand sides, after all we obtain

(190)

which is consistent with Eq. (180) in the case of C₁ = 0.

Other derivation of A From the equations,

and

we obtain

(191)

For porous materials, since σ₁ = 0,

(192)

Hence,

(193)

and from this equation we obtain

(194)

Therefore, from ϵ₀ + γ = Aϵ₀ [Eq. (179)],

(195)

which is the same as Eq. (188).

4.3.2 Liquid-inclusion composite

When the shear modulus of liquid is assumed to be zero, using the Lam constant (λ and μ = 0), the elastic constants of liquid is expressed as

4.3.3 Rigid-body-inclusion composite

Rigid body can be regarded as inclusion with C₁^-1 → 0; even when an external stress is applied to the rigid body, the external strain is not caused, i.e.,

(198)

and therefore γ = -ϵ₀. Since C₁ →∞ and ϵ₁ = 0, σ₁ = C₁ϵ₁ formally has an indeterminate form, and σ₁ represents the virtual stress applied to the rigid-body inclusions. This virtual stress is, of course, not a real stress of rigid body, but is the quantity that guarantees the average stress of matrix satisfies (1 -p)σ₀ = σ^ext′-pσ ₁ in the case of composite.

First, we consider the case where a single rigid body exists in an infinite matrix. Therefore, Eq. (174) can be written as

(199)

must hold, and therefore

(200)

Next a composite with numerous rigid-body inclusions is considered. In this case, it is convenient to consider the stress-concentration factor, not the strain-concentration factor A, because A ≡ 0 from Eq. (198). The average elastic compliances C^-1 of the composite material can be generally defined as ϵ = C^-1σ, i.e.,

(201)

When expressing σ₁ = Bσ₀, the average elastic compliances C^-1 of the composite can be written as

(203)

Comparing with Eq. (200) obtained for a rigid body, we find

(204)

where B = S^-1 when C ₀S^-1 = S^-1C ₀ holds. Since we can regard that C₁^-1 = 0 for rigid bodies, from Eq. (202), the average elastic constants for composite with rigid-body inclusions are expressed as

(205)

Thus, when p → 1, the effective elastic constants are diverged. However, since p∕(1 - p) ~ O(1) when p < 0.99, this divergence takes place steeply in the vicinity of p = 1.

4.4 Strain-energy balance

In the previous arguments, we considered the effective elastic constants on the basis of Hooke’s law, (1 - p)σ₀ + pσ₁ = C[(1 - p)ϵ₀ + pϵ₁], with the average stress and strain. Here we consider this matter in terms of the external strain energy caused by an external stress σ_ij^ext.

When the substance is uniform, the external strain energy is given by

In the matrix form, we can rewrite Eq. (212) as

into the right-hand side of Eq. (214), and taking notice of ϵ₁ = ϵ₀ + γ and (1 - p)ϵ₀ + pϵ₁ = ϵ, we obtain

4.5 Effective mean-field (EMF) theory

In general, the MTMF theory does not yield accurate elastic constants for large p, because Eq. (212) does not consider the elastic interaction between inclusions. One of the essential reasons of this problem is that the important conclusion of Mori-Tanaka’s theory, σ₁ = σ₀ + σ^∞, does not hold when the fraction of inclusions is fairly large, where σ₀ and σ₁ are the average stress fields of the matrix and inclusions, and σ^∞ denotes the stress disturbance caused by a single inclusion formed in an infinite matrix. This is because the elastic-interaction effect between inclusions cannot be neglected any longer. This means that we should avoid deriving directly the elastic constants for a composite with a high inclusion fraction from those of the non-inclusion material with the conventional MTMF theory. With this in mind, we here consider the evaluation method of effective elastic constants for large p.[9]

Equation (212) is rewritten as

As seen in the previous section, the strain-energy balance equation (209) is totally equivalent to Eq. (171) with Eq. (180) derived on the basis of Hooke’s law σ^ext = Cϵ. Therefore, Eq. (171) is still used in the EMF treatment, but is applied only in the low fraction range. Namely, C₀ in Eq. (171) is sequentially superseded by C (effective elastic constants) for a lower fraction obtained by a previous calculation step, and then the effective elastic constants of a renewed composite with a slightly higher fraction are to be calculated. When the increment of fraction with regard to the original non-porous material is given by Δp, the division number Q of the entire fraction range (0 ≤ p ≤ 1) is Q = 1∕Δp (Δp should be chosen in order for Q to be integer). Then, the following relation,

(231)

holds, where 1 - nΔp means the fraction of the previous (pre-renewed) porous sample and Δp∕(1 - nΔp) represents the virtual fraction measured referring to the previous porous sample that is regarded as a matrix. Therefore, the fraction p of a renewed porous sample is expressed as

(232)

Based on the above concept, Eq. (171) is modified as a following formula:

References