Non-interaction Approximation in the Problem of Effective Properties

Abstract

We discuss modeling of the effective properties of microstructures that contain inhomogeneities of diverse and “irregular” shapes. We focus on the effects of shapes and their diversity, in the framework of the non-interaction approximation. We also clarify the difference between the non-interaction approximation and the “dilute limit” as well as the concept of “average shape” for a mixture of inhomogeneities of diverse shapes. Further, we give an overview of the approximate schemes that utilize the non-interaction approximation as the basic building block, and discuss the key role of this approximation in establishing explicit elasticity–conductivity connection.

Appendix: Tensor Basis in the Space of Transversely Isotropic Fourth Rank Tensors. Representation of Certain Transversely Isotropic Tensors in Terms of the Tensor Basis

The operations of analytic inversion and multiplication of fourth rank tensors are conveniently done in terms of special tensor bases that are formed by combinations of unit tensor ${\delta } _{ ij}$ and one or two orthogonal unit vectors (see [55] and [50]). In the case of the transversely isotropic elastic symmetry, the following basis is most convenient (it differs slightly from the one used by Kanaun and Levin [50]):

$$\begin{aligned} T_{ ijkl}^{\left(1 \right)}&=\theta _{ ij} \theta _{kl} , \quad T_{ ijkl}^{\left(2 \right)} ={\left({\theta _{ik} \theta _{lj} +\theta _{il} \theta _{kj} -\theta _{ ij} \theta _{kl} } \right)}/2, \quad T_{ ijkl}^{\left(3 \right)} =\theta _{ ij} m_k m_l , \quad T_{ ijkl}^{\left(4 \right)} =m_i m_j \theta _{kl} \\ T_{ ijkl}^{\left(5 \right)}&={\left({\theta _{ik} m_l m_j +\theta _{il} m_k m_j +\theta _{jk} m_l m_i +\theta _{jl} m_k m_i } \right)}/4, \quad T_{ ijkl}^{\left(6 \right)} =m_i m_j m_k m_l\nonumber \end{aligned}$$

(A.1)

where $\theta _{ ij} ={\delta } _{ ij} -m_i m_j$ and ${\varvec{m}}=m_1 e_1 +m_2 e_2 +m_3 e_3$ is a unit vector along the axis of transverse symmetry.

These tensors form the closed algebra with respect to the operation of (non-commutative) multiplication (contraction over two indices):

$$\begin{aligned} \left({{\varvec{T}}^{\left({\alpha } \right)}{\varvec{T}}^{\left(\beta \right)}} \right)_{ ijkl} =T_{ ijpq}^{\left({\alpha } \right)} T_{pqkl}^{\left(\beta \right)} \end{aligned}$$

(A.2)

The table of multiplication of these tensors has the following form (the column represents the left multipliers):

	${\varvec{T}}^{(1)}$	${\varvec{T}}^{(2)}$	${\varvec{T}}^{(3)}$	${\varvec{T}}^{(4)}$	${\varvec{T}}^{(5)}$	${\varvec{T}}^{(6)}$
${\varvec{T}}^{(1)}$	$2{\varvec{T}}^{(1)}$	$0$	$2{\varvec{T}}^{(3)}$	$0$	$0$	$0$
${\varvec{T}}^{(2)}$	$0$	${\varvec{T}}^{(2)}$	$0$	$0$	$0$	$0$
${\varvec{T}}^{(3)}$	$0$	$0$	$0$	${\varvec{T}}^{(1)}$	$0$	${\varvec{T}}^{(3)}$
${\varvec{T}}^{(4)}$	$2{\varvec{T}}^{(4)}$	$0$	$2{\varvec{T}}^{(6)}$	$0$	$0$	$0$
${\varvec{T}}^{(5)}$	$0$	$0$	$0$	$0$	${\varvec{T}}^{(5)}$/2	$0$
${\varvec{T}}^{(6)}$	$0$	$0$	$0$	${\varvec{T}}^{(4)}$	$0$	${\varvec{T}}^{(6)}$

Then the inverse of any fourth rank tensor ${\varvec{X}}$, as well as the product ${\varvec{X}}{:}{\varvec{Y}}$ of two such tensors are readily found in the closed form, as soon as the representation in the basis

$$\begin{aligned} {\varvec{X}}=\sum _{k=1}^6 {X_k {\varvec{T}}^{\left(k \right)}} , \quad {\varvec{Y}}=\sum _{k=1}^6 {Y_k {\varvec{T}}^{\left(k \right)}} \end{aligned}$$

(A.3)

are established. Indeed:

(a) inverse tensor ${\varvec{X}}^{-1}$ defined by $X_{ ijmn}^{-1} X_{mnkl} =\left({X_{ ijmn} X_{mnkl}^{-1}} \right)=J_{ ijkl}$ is given by

$$\begin{aligned} {\varvec{X}}^{-1}=\frac{X_6 }{2\Delta }{\varvec{T}}^{\left(1 \right)}+\frac{1}{X_2 }{\varvec{T}}^{\left(2 \right)}-\frac{X_3 }{\Delta }{\varvec{T}}^{\left(3 \right)}-\frac{X_4 }{\Delta }{\varvec{T}}^{\left(4 \right)}+\frac{4}{X_5 }{\varvec{T}}^{\left(5 \right)}+\frac{2X_1 }{\Delta }{\varvec{T}}^{\left(6 \right)} \end{aligned}$$

(A.4)

where $\Delta =2\left({X_1 X_6 -X_3 X_4 } \right)$.

(b) product of two tensors ${\varvec{X}}{:}{\varvec{Y}}$ (tensor with $ijkl$ components equal to $X_{ ijmn} Y_{mnkl} )$ is

$$\begin{aligned} {\varvec{X}}{:}{\varvec{Y}}&=\left({2X_1 Y_1 +X_3 Y_4 } \right){\varvec{T}}^{\left(1 \right)}+X_2 Y_2 {\varvec{T}}^{\left(2 \right)}+\left({2X_1 Y_3 +X_3 Y_6 } \right){\varvec{T}}^{\left(3 \right)} \nonumber \\&\quad \;+\left({2X_4 Y_1 +X_6 Y_4 } \right){\varvec{T}}^{\left(4 \right)}+\frac{1}{2}X_5 Y_5 {\varvec{T}}^{\left(5 \right)}+\left({X_6 Y_6 +2X_4 Y_3 } \right){\varvec{T}}^{\left(6 \right)} \\ \end{aligned}$$

(A.5)

If $x_3$ is the axis of transverse symmetry, tensors ${\varvec{T}}^{\left(1 \right)}, \ldots ,{\varvec{T}}^{\left(6 \right)}$given by (A.1) have the following non-zero components:

$$\begin{aligned} {\varvec{T}}_{1111}^{\left(1 \right)}&={\varvec{T}}_{2222}^{\left(1 \right)} ={\varvec{T}}_{1122}^{\left(1 \right)} ={\varvec{T}}_{2211}^{\left(1 \right)} =1 \nonumber \\ {\varvec{T}}_{1212}^{\left(2 \right)}&={\varvec{T}}_{2121}^{\left(2 \right)} ={\varvec{T}}_{1221}^{\left(2 \right)} ={\varvec{T}}_{2112}^{\left(2 \right)} ={\varvec{T}}_{1111}^{\left(2 \right)} ={\varvec{T}}_{2222}^{\left(2 \right)} =\frac{1}{2},\nonumber \\ {\varvec{T}}_{1122}^{\left(2 \right)}&={\varvec{T}}_{2211}^{\left(2 \right)}=-\frac{1}{2} \nonumber \\ {\varvec{T}}_{1133}^{\left(3 \right)}&={\varvec{T}}_{2233}^{\left(3 \right)} =1; \nonumber \\ {\varvec{T}}_{3311}^{\left(4 \right)}&={\varvec{T}}_{3322}^{\left(4 \right)}=1 \nonumber \\ {\varvec{T}}_{1313}^{\left(5 \right)}&={\varvec{T}}_{2323}^{\left(5 \right)} ={\varvec{T}}_{1331}^{\left(5 \right)} ={\varvec{T}}_{2332}^{\left(5 \right)} ={\varvec{T}}_{3113}^{\left(5 \right)} ={\varvec{T}}_{3223}^{\left(5 \right)} ={\varvec{T}}_{3131}^{\left(5 \right)} ={\varvec{T}}_{3232}^{\left(5 \right)} =\frac{1}{4} \nonumber \\ {\varvec{T}}_{3333}^{\left(6 \right)}&=1 \nonumber \\ \end{aligned}$$

(A.6)

General transversely isotropic fourth-rank tensor, being represented in this basis

$$\begin{aligned} \varPsi _{ ijkl} =\sum {\psi _m T_{ ijkl}^m } \end{aligned}$$

has the following components:

$$\begin{aligned} \psi _1&={\left({\varPsi _{1111} +\varPsi _{1122} } \right)}/2; \quad \psi _2 =2\varPsi _{1212} ; \quad \psi _3 =\varPsi _{1133} ; \quad \psi _4 =\varPsi _{3311} ; \\ \psi _5&=4\varPsi _{1313} ; \quad \psi _6 =4\varPsi _{3333} \nonumber \end{aligned}$$

(A.7)

Utilizing (A.7) one obtains the following representations:

• Tensor of elastic compliances of the isotropic material $S_{ ijkl} =\sum {s_m T_{ ijkl}^m }$ has the following components

  $$\begin{aligned} s_1 =\frac{1-\nu }{4G\left({1+\nu } \right)}; \quad s_2 =\frac{1}{2G}; \quad s_3 =s_4 =\frac{-\nu }{2G\left({1+\nu } \right)}; \quad s_5 =\frac{1}{G}; \quad s_6 =\frac{1}{2G\left({1+\nu } \right)}. \end{aligned}$$

  (A.8)

• Tensor of elastic stiffness of the isotropic material by $C_{ ijkl} =\sum {c_m T_{ ijkl}^m }$ has components

  $$\begin{aligned} c_1 =\lambda +G; \quad c_2 =2G; \quad c_3 =c_4 =\lambda ; \quad c_5 =4G; \quad c_6 =\lambda +2G. \end{aligned}$$

  (A.9)

where $\lambda ={2G\nu }/{\left({1-2\nu } \right)}$.

• Unit fourth rank tensors are represented in the form

  $$\begin{aligned} J_{ ijkl}^{\left(1 \right)} ={\left({{\delta } _{ik} {\delta } _{lj} +{\delta } _{il} {\delta } _{kj} } \right)}/2=\frac{1}{2}T_{ ijkl}^1 +T_{ ijkl}^2 +2T_{ ijkl}^5 +T_{ ijkl}^6 \end{aligned}$$

  (A.10)
  $$\begin{aligned} J_{ ijkl}^{\left(2 \right)} ={\delta } _{ ij} {\delta } _{kl} =T_{ ijkl}^1 +T_{ ijkl}^3 +T_{ ijkl}^4 +T_{ ijkl}^6 \end{aligned}$$

  (A.11)