1 Eshelby’s ellipsoidal inclusion theory

1.1 Eigenstrain

1.2 Elastic-equilibrium equation for elastic homogeneity

1.2.1 General solution

1.2.2 Solution for ellipsoidal inclusion

1.3 The Eshelby tensor

1.4 Equivalent inclusion method

2 Phase-field microelasticity theory

2.1 Macroscopic shape change and total strain

2.2 Elastic-equilibrium equation for elastic inhomogeneity

2.3 Application of Discrete Fourier Transformation technique

2.3.1 Definition of DFT

2.3.2 On singular point

2.3.3 Constraint conditions

3 Mechanical energies

3.1 Definitions and theorem

3.1.1 Strain energy

3.1.2 Elastic interaction energy

3.1.3 Colonnetti’s theorem

3.1.4 Total mechanical energy and mechanical interaction

3.2 For ellipsoidal inclusion

3.2.1 Strain energy

3.2.2 Mechanical interaction energy

3.3 External stress applied to elastic inhomogeneity

4 Effective elastic constants

4.1 Definition of effective elastic constants

4.2 Strain-concentration factor

4.2.1 Equivalent inclusion method

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

4.2.3 Multi-component composite

4.3 Application to special composites

4.3.1 Porous material

4.3.2 Liquid-inclusion composite

4.3.3 Rigid-body-inclusion composite

4.4 Strain-energy balance

4.5 Effective mean-field (EMF) theory

1.1 Eigenstrain

1.2 Elastic-equilibrium equation for elastic homogeneity

1.2.1 General solution

1.2.2 Solution for ellipsoidal inclusion

1.3 The Eshelby tensor

1.4 Equivalent inclusion method

2 Phase-field microelasticity theory

2.1 Macroscopic shape change and total strain

2.2 Elastic-equilibrium equation for elastic inhomogeneity

2.3 Application of Discrete Fourier Transformation technique

2.3.1 Definition of DFT

2.3.2 On singular point

2.3.3 Constraint conditions

3 Mechanical energies

3.1 Definitions and theorem

3.1.1 Strain energy

3.1.2 Elastic interaction energy

3.1.3 Colonnetti’s theorem

3.1.4 Total mechanical energy and mechanical interaction

3.2 For ellipsoidal inclusion

3.2.1 Strain energy

3.2.2 Mechanical interaction energy

3.3 External stress applied to elastic inhomogeneity

4 Effective elastic constants

4.1 Definition of effective elastic constants

4.2 Strain-concentration factor

4.2.1 Equivalent inclusion method

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

4.2.3 Multi-component composite

4.3 Application to special composites

4.3.1 Porous material

4.3.2 Liquid-inclusion composite

4.3.3 Rigid-body-inclusion composite

4.4 Strain-energy balance

4.5 Effective mean-field (EMF) 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.

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 = (u_{1},u_{2},u_{3}), 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

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_{ijkl}g_{j}g_{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)

and from Eq. (11) Equations (11), (14) and (16) are the general solutions of the elastic-equilibrium equation (3) for an elastically homogeneous substance.

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

Since the delta function has a following characteristics(30) |

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

We first deal with the integral ∫
_{Ω}ϵ_{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 = (x_{1},x_{2},x_{3}) is located in
the ellipsoidal inclusion Ω with radii a_{1}, a_{2}, and a_{3}:

(31) |

Transformations of variables are made as follows:

where |h| = 1 and h = |h| =(34) |

Since dx = a_{1}a_{2}a_{3}dy and g ⋅ (x - x′) = hh ⋅ (y - y′), Eq. (29) is written as

In Eq. (36), ∫
_{S2}ϵ_{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].

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 |S^{2}| that is made by unit
vector g, C_{jlmn} denotes the elastic constants of matrix, g_{p} is a component of g, and (a_{1},
a_{2}, a_{3}) denote the radii of the ellipsoidal inclusion.

Cylindrical shape
For cylindrical (needle) shape, i.e., a_{3}∕a_{1}(= a_{3}∕a_{2}) →∞, S tensors reduce to

Plate-disc shape
For the plate-disc inclusion with zero aspect ratio i.e., a_{3}∕a_{1}(= a_{3}∕a_{2}) → 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

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_{kl}^{0} 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).

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.

- 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?
- The integrand in Eq. (11), C
_{ijmn}(G_{ik}^{-1}g_{ l}+ G_{il}^{-1}g_{ k})g_{j}_{mn}^{*}(g) exp(ig ⋅ x), shows a singularity at g = 0, because G_{ik}= C_{ipkq}g_{p}g_{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.

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:

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

Then, the elastic-equilibrium equation isZeroth-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

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

In the Fourier space, Eq. (83) is written as Since Eq. (85) is nonlinear, it cannot be soloved analytically. Thus, we obtain an approximate solution with the zeroth-order solution. The solution of zeroth-order equation uHigher-order approximation Similarly, we obtain the higher-order solution through the iterative calculations:

and

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).

Index | x_{i} | 0 | 1 | 2 | N - 1 | (N) | |

L [m] | x_{i}Δx | 0 | Δx | 2Δx | (N - 1)Δx | (NΔx) | |

Index | g_{i} | 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 Ă(g), the definition of DFT is given by

(97) |

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

should be satisfied from Eq. (54). By utilizing Eq. (100), we can avoid calculating the g = 0 point in Eq. (81). Equation (100) is the crucial key for all the calculations.

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

Furthermore, applying DFT formulae and using the characteristics of delta function Eq. (96), the total and elastic strains are expressed as and respectively, where note that ϵ(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.

- First assume C
_{klij}arbitrarily (with the other method), and calculate the average total strain using γ_{mn}= C_{mnij}^{-1}σ_{ ij}^{ext}. - Obtain the equilibrium solution with higher-order approximation:
_{mn}^{(0)}(x) = 0 and G_{ ik}= C_{ipkq}g_{p}g_{q}. Note that the eigenstrain is not related to the effective elastic constants in the regime of linear elasticity theory. - Check the sum of local stresses using Eq. (70), i.e.,
_{ij}^{ext}, the assumed C_{ klij}is valid, because the sum of internal stresses (eigen-stresses) is zero; see Eq. (61). - 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}^{-1}σ_{
ij}^{ext}.

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:

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

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

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,

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.

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.

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

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_{
ijkl}ϵ_{kl}^{0}), 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:

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.

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.

Consider a composite material consisting of a matrix with the elastic constants C_{0} and
a type of inclusion with C_{1}, 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)σ_{0} + pσ_{1} and ϵ = (1 - p)ϵ_{0} + pϵ_{1}, where σ_{0} = C_{0}ϵ_{0} and
σ_{1} = C_{1}ϵ_{1}. 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 ϵ_{1} = Aϵ_{0}, 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.

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].

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

(172) |

where ϵ_{0} denotes the uniform strain in the absence of any inclusions. Next, we
consider a substance containing, in a matrix with the elastic constants C_{0}, an
infinitesimally small inclusion that has different elastic constants C_{1} 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 σ_{1}
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 ϵ_{0} = 0 or σ^{ext} = 0, the stress disturbance is not caused i.e., σ^{∞} = 0 and
therefore γ = 0. When ϵ_{0}≠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}.

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 σ_{0},ϵ_{0} and σ_{1},ϵ_{1}, respectively. (Then, σ^{ext′} = (1 - p)C_{
0}ϵ_{0} + pC_{1}ϵ_{1} holds.) The
differences between σ_{1} and σ_{0} and between ϵ_{1} and ϵ_{0} are denoted by σ′ and γ′, i.e.,
σ_{1} = σ_{0} + σ′ and ϵ_{1} = ϵ_{0} + 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 ϵ_{0} (caused
by σ^{ext}) is given by σ^{∞}. Since the new inclusion is to be embedded into the
arleadly stressed (or strained) matrix with σ_{0} (or ϵ_{0}), the internal stress of new
inclusion is approximately given by sum of σ^{∞} and σ_{
0}, i.e., σ_{1} ≈σ_{0} + σ^{∞}.
Moreover, as far as σ_{0}(x) is not so much deviated from σ_{0}, we may regard that the
internal stress σ_{1} of the newly added inclusion is virtually equal to the average
internal stress σ_{1} of inclusions in the pre-added original composite, i.e., σ_{1} ≈ σ_{1}.
Therefore we obtain σ_{1} ≈σ_{0} + σ^{∞}. It should be noted here that we have to use
σ^{∞} in the situation that ϵ_{
0} equals ϵ_{0}. After all, we find that σ′ = σ^{∞}(ϵ_{
0}) and
γ′ = γ(ϵ_{0}).

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) |

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 ∑
_{i}p_{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 A_{0} = I.

We here consider the case of porous material; pore can be regarded as inclusion with
C_{1} = 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 σ_{1} = C_{1}ϵ_{1} = 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 σ_{1} = 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 C_{1} = 0 into above equation, we obtain σ^{ext′} = (1 - p)C_{
0}ϵ_{0}. In
order to refer to the case of an infinitesimal single inclusion, we suppose the case where
the average strain ϵ_{0} of the matrix is equal to ϵ_{0};

(187) |

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

(188) |

or from Eqs. (179), (185) and ϵ_{0} = ϵ_{0}, 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 C_{1} = 0.

Other derivation of A From the equations,

and

we obtain

(191) |

For porous materials, since σ_{1} = 0,

(192) |

Hence,

(193) |

and from this equation we obtain

(194) |

Therefore, from ϵ_{0} + γ = Aϵ_{0} [Eq. (179)],

(195) |

which is the same as Eq. (188).

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

Rigid body can be regarded as inclusion with C_{1}^{-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 = -ϵ_{0}. Since C_{1} →∞ and ϵ_{1} = 0, σ_{1} = C_{1}ϵ_{1} formally has an
indeterminate form, and σ_{1} 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)σ_{0} = 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^{-1}σ, i.e.,

(201) |

When expressing σ_{1} = Bσ_{0}, 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_{
0}S^{-1} = S^{-1}C_{
0} holds. Since we can regard that C_{1}^{-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.

In the previous arguments, we considered the effective elastic constants on the basis of
Hooke’s law, (1 - p)σ_{0} + pσ_{1} = C[(1 - p)ϵ_{0} + pϵ_{1}], 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

where σ

into the right-hand side of Eq. (214), and taking notice of ϵ_{1} = ϵ_{0} + γ and
(1 - p)ϵ_{0} + pϵ_{1} = ϵ, we obtain

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, = + σ^{∞}, does not hold when the fraction of inclusions is fairly large, where
and 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

where the superscript (0) means p = 0, C 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, C_{0} 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:

where C

[1] Eshelby, J.D., Proc. Roy. Soc. London, A241 (1957), 376.

[2] Mura, T., Micromechanics of Defects in Solids, Second, Revised, Edition (Martinus Nijhoff Pul., The Hegue, 1987).

[3] Mura, T. and Cheng, PC., J. Appl. Mech., 44 (1977), 591.

[4] Kinoshita N. and Mura T., Phys. Status Solidi, A5 (1971), 759.

[5] Khachaturyan AG, Theory of Structural Transformation in Solids (Wiley, New York, 1983).

[6] Hu SY. and Chen LQ., Acta Mater. 2001, 49 1879.

[7] Mori, T. and Tanaka, K., Acta Metal., 21 (1973), 573.

[8] Benveniste, Y., Mech. Mater., 6 (1987), 147.

[9] Tane, M. and Ichitsubo, T., Appl. Phys. Lett. 85 (2004), 197.