# Hashin–Shtrikman bounds on the effective thermal conductivity of a transversely isotropic two-phase composite material

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## Abstract

This paper is concerned with the estimation of the effective thermal conductivity of a transversely isotropic two phase composite. We describe the general construction of the Hashin–Shtrikman bounds from first principles in the conductivity setting. Of specific interest in composite design is the fact that the shape of the inclusions and their distribution can be specified independently. This case covers a multitude of composites used in applications by taking various limits of the spheroid aspect ratio, including both layered media and unidirectional composites. Furthermore the expressions derived are equally valid for a number of other effective properties due to the fact that Laplace’s equation governs a significant range of applications, e.g. electrical conductivity and permittivity, magnetic permeability and many more. We illustrate the implementation of the scheme with several examples.

### Keywords

Hashin–Shtrikman bounds Conductivity Transport problem Hill tensor### Mathematics Subject Classification

74Q20 74Q15## 1 Introduction

The determination of effective physical properties of heterogeneous materials obtained by mixing different phases, usually on a very small scale denoted by \(\eta >0\) is a widely studied problem in the physical sciences. Motivation comes from a number of areas, e.g. prediction of the overall behavior of ceramics or superconducting fibre-reinforced materials. Predicting exact fields is difficult due to the size of the microstructure. It could also be argued that such a precise solution is unnecessary if one is only interested in overall behaviour on the macroscale. A number of different techniques have been devised to determine effective properties. One such method is asymptotic homogenization theory, which involves taking the limit \(\eta \rightarrow 0\), providing a homogenized governing boundary value problem with constant coefficients, see e.g. Tartar [1, 2] for the \(N\)-dimensional case, or in a two dimensional setting by Lurie and Cherkaev [3].

*micromechanics*community, who are often concerned with determining approximations and bounds on effective properties. This information is often very useful from a practical viewpoint. Bounds are determined via variational principles and have been studied extensively by many authors, e.g. [4, 5, 6, 7, 8]. The quasi-static

*transport problem*(e.g. electrical and thermal conductivity, etc) is of great interest in many applications. In the conductivity setting, the Maxwell principle for the conductivity of a host material containing a suspension of spheres, is very well-known [9]. When the only information known regarding the microstructure is the volume fraction \(\phi _r\) and the conductivity tensor \(\mathbf {K}^r\) (with Cartesian components \(K_{ij}^r\)) associated with the \(r\)th phase, \(r=0,\,1,\ldots ,n\), the effective conductivity tensor \(\mathbf {K}^*\) associated with an arbitrary medium (isotropic or anisotropic) can be estimated by the Wiener bounds for the transport problem [10], as follows

In general, bounds that appear in the literature are almost always merely “stated” (not derived) and it is often unclear how to construct them when the material is not of simple type (e.g. isotropic spheres inside an isotropic host phase). Furthermore, discussion of how the distribution tensor affects the Hashin Shtrikman bounds for the transport problem does not appear to have been studied in any detail, in contrast to elastostatics where some studies have taken place [7]. For this reason, the objective of this work is to illustrate a direct way of constructing the HS bounds for the thermal conductivity of transversely isotropic (TI) composites from first principles incorporating information about the, perhaps non-isotropic, distributions of inclusions. That is, given the phase volume fractions and thermal conductivity, the shapes of the inclusion phases and their spatial distribution, we construct a procedure by which the HS bounds can be obtained in a straightforward manner defining the correct tensor basis set and the appropriate expressions for the Hill tensors. In particular in this respect, assuming homogeneous temperature conditions in the far field and by using the associated Green’s tensor, we exploit the uniformity of the Hill tensor and the known explicit expressions for spheroidal inclusions and distributions. This formulation should be of a great utility for engineers and material scientists who may wish to construct these kinds of expressions for a variety of such media.

The paper is organized as follows. In Sect. 2 we introduce the basic formulation of the two-phase problem. Following this, in order to obtain explicit expressions for the tensors that appear in the general scheme, we make use of the so-called single inclusion problem related to the Eshelby conjecture regarding isolated inclusions. Then, in Sect. 4 we describe the general formulation of the HS bounds, initially for the general multiphase case before restricting attention to two phases. We specialize in Sect. 5 to the case of macroscopically TI materials. This specialization therefore motivates the definition of a TI second order tensor basis set. In Sect. 6 we illustrate the implementation of the construction with some examples where we analyze the influence on the effective conductivity of the different characteristics of the microstructure. We conclude in Sect. 7.

## 2 Context of the problem

*macroscopic effective properties*, represented by a second order conductivity tensor \({\mathbf K}^*\) giving the linear relationship between the body averages of heat flux and the thermal gradient as follows

**Definition 1**

A composite is statistically homogeneous if \(p^r\) the probability density of finding an inclusion of type \(r\) centered at a point \({\mathbf x}\), is a constant equal to \(p^r={n^r}/{|\varOmega |}\) with \(n^r\) equal to the number of inclusions of type \(r\).

**Definition 2**

A composite has ellipsoidal symmetry [6] for the distribution of the inclusions if the conditional probability density function \(p^{s|r}({\mathbf x'},{\mathbf x})\) for finding an inclusion of type \(s\) centered at \({\mathbf x'}\) given that there exists an inclusion of type \(r\) centered at point \({\mathbf x}\), depends on \({\mathbf x''}={\mathbf x'}-{\mathbf x}\) only through the expression \(|A_d^{(rs)}{\mathbf x''}|\), for some matrix \(A_d^{(rs)}\) which defines an ellipsoid \(\omega _d^{(rs)}=\{{\mathbf z}:\quad |{\mathbf A}_d^{(rs)}{\mathbf z}|<1\}\).

## 3 The single inclusion problem: the Hill tensor for TI isotropic materials

The so-called single inclusion problems are well known and utilized in micromechanics because they allow one to derive very useful explicit expressions for the effective properties of heterogeneous materials [12]. Consider the domain \(\varOmega \) infinitely extended in \(\mathbb R^3\). Assuming the inclusion is an ellipsoidal inhomogeneity \(\omega \) embedded in the host matrix \(\varOmega \) (with thermal conductivity tensors \({\mathbf K}^1\) and \({\mathbf K}^0\) respectively), there exists the fundamental property of uniformity of thermal gradients interior to the ellipsoid under homogeneous far-field conditions (see [9, 13]). An analogous property for the elasticity setting was obtained by Eshelby [14] in the case of isotropic host phases. Effective properties of inhomogeneous media can often be written rather conveniently in terms of the Eshelby or Hill tensors as a result of these useful uniformity properties.

*spheroidal*inclusion and the function \({{\fancyscript{S}}}\) is given by the expression

*spherical*shape (\(\varepsilon =1\)), (3.8) leads to

*spherical*inclusions embedded inside an

*isotropic*phase (\({\mathbf K}^0=\kappa _0 I\), \({\mathbf K}^1=\kappa _1 I\)), the Hill-tensor simplifies to the form \({\mathbf P}=\displaystyle \frac{1}{3\kappa _0} I\).

## 4 The Hashin–Shtrikman variational principle

Hashin and Shtrikman [4] derived a variational principle to bound the overall conductivity of a heterogeneous composite with *statistically isotropic* microstructure. It was based on an alternative representation of the effective energy of the heterogeneous media that makes use of a proper homogeneous comparison material (see also [5]). An extension of the Hashin–Shtrikman formulation was studied by several authors, additionally in the elasticity context (see [16]). In particular, Willis [6] developed a variational structure to obtain these classical bounds on an anisotropic composite comprising a matrix and \(n\) different types of inclusion phase. In this work, a polarization field is introduced relative to the comparison material, and the distribution tensor of inclusions is assumed the same as their shape. (given by integrals of the associated two-point correlation functions). Later, Ponte Castañeda and Willis generalized the previous structure given by Willis [6] to the case where the influence of the shape and the distribution of the different phases are taken into account independently.

In this section, following Ponte Casteñeda and Willis’ [7] derivation of the variational principle for the elasticity context, we will derive explicit bounds for the effective conductive energy of an anisotropic composite, first for the multiphase-case before restricting attention to the two-phase scenario. In this formulation the hypotheses of *ellipsoidal symmetry* for the distribution of the inclusions (possibly with a different ellipsoidal shape to the shape of the inclusion) is assumed. Therefore, the explicit expressions (3.8–3.10) for the P-tensor can be used, thanks to the linearity of the problem, leading to explicit bounds on the effective conductivity \({\mathbf K}^*\).

Assume at first that the composite occupies the domain \(\varOmega \) comprising \(n\) different types of inclusion phases that could be selected independently of their spatial distribution depending on two parameters \(\varepsilon ,\,\rho >0\) respectively. These parameters are the aspect ratio of the spheroidal inclusion (\(\varepsilon \)) and the statistics associated with the spheroidal distribution function (\(\rho \)). We denote the conductivity tensor of the \(r\)th phase by \({\mathbf K}^r\), \(r=1,\ldots ,n\) and by \({\mathbf K}^0\) the conductivity of the matrix within which the inclusion phases are embedded. \(\varOmega _r\) represents the total domain of the \(r\)th phase, i.e. it is the collection of \(n^r\) ellipsoidal inclusions (each aligned and having the shape defined by a domain \(\omega ^r\)) with total volume fraction equal to \(\phi _r=|\varOmega _r|/|\varOmega |\). The volume fraction of the host phase is therefore \(\phi _0=1-\sum _{r=1}^n\phi _r\). Under these conditions, we have the following theorem [7]

**Theorem 1**

*optimal polarizations*\(\varvec{\tau }_k^*\), which satisfy the relations

*Proof*

*optimal polarizations*\(\varvec{\tau }_k^*\) which satisfy the relationships given by (4.12).

Following [7], in order to simplify the above expression for the microstructure tensor given by (4.18), we make use of the hypotheses of *statistical homogeneity* and *spheroidal symmetry* of the composite. Namely, if the inclusions are assumed to be spheroidally distributed (but maybe with a different ellipsoid from one that defines the inclusion shapes), the tensor (4.18) can be expressed as (4.13). In (4.13) \({\mathbf P}_s^{\varepsilon ,k}\) and \({\mathbf P}_d^{\rho ,(k,\ell )}\) are uniform P-tensors given by (3.7) with \(\omega \) replaced by the \(r\)th phase ellipsoidal inclusion \(\omega ^r\) and by the ellipsoid \(\omega ^{(k\ell )}\) given by Definition 2 respectively, and whose shapes depend on \(\varepsilon \) and \(\rho \). \(\square \)

*Remark 1*

Recall that the subscripts \(s\) and \(d\) refer to the *shape* and *distribution* of inclusions depending on \(\varepsilon \) and \(\rho \) respectively. By definition, we have \({\mathbf P}_d^{\rho ,(k\ell )}={\mathbf P}_d^{\rho ,(\ell k)}\), and for conciseness we will write \({\mathbf P}_d^{\rho ,k}\), to denote \({\mathbf P}_d^{\rho ,(kk)}\), \(k=0, \ldots ,n.\)

**Corollary 1**

**Corollary 2**

### 4.1 The Hashin–Strikman bounds for a two-phase composite

In this work we shall restrict attention to two-phase particulate media so that there is a single inclusion phase. For simplicity, we suppose that the inclusion phase is a distribution of (possibly different sized) aligned spheroids but where each spheroid has the same aspect ratio \(\varepsilon =a_3/a\) and where the long/short axis (same direction as the semi-axis \(a_3\) of the spheroid) is aligned with \(x_3\). Note that this spheroidal shape is taken into account thanks to the P-tensor \({\mathbf P}_s^\varepsilon \). Its distribution is accounted for by virtue of the P-tensor \({\mathbf P}_d^\rho \) which we shall also consider to be governed by spheroidal statistics, of aspect ratio \(\rho \).

Various alternative expressions for the HS bounds can be obtained. Let us first consider the case when the comparison phase can be chosen as either the host or inclusion phase. Note that this may not always be possible however.

#### 4.1.1 Comparison phase can be identified as either host or inclusion phase

*Remark 2*

#### 4.1.2 Comparison material cannot be identified as either host or inclusion phase

#### 4.1.3 Inclusions have the same shape as their distribution

### 4.2 Distribution P-tensor

*security*spheroid, containing a single spheroidal inclusion, which is not intersected by any other security spheroid.

Assuming that \({\mathbf P}_s^\varepsilon \) and \({\mathbf P}_d^\rho \) are given, this construction of the composite means that there exists a maximal volume fraction associated with how much of the inclusion can fit into the security spheroid. This depends on whether \(\varepsilon >\rho \) or \(\varepsilon <\rho \) (see Fig. 1). Simple calculations show that when \(\varepsilon >\rho \) we have \( 0 \le \phi _1 \le {\rho ^2}/{\varepsilon ^2}, \) whereas if \(\varepsilon <\rho \) we have \( 0 \le \phi _1 \le {\varepsilon }/{\rho }. \)

Alternatively, suppose that the inclusion aspect ratio \(\varepsilon \) is fixed in addition to the volume fraction. This then gives a condition on the maximum \(\rho \) permitted. In particular when \(\rho <\varepsilon \) we can determine that \(0\le \rho \le \varepsilon \sqrt{\phi _1}\le \varepsilon \) whereas when \(\rho >\varepsilon \) we have \(\varepsilon \le \rho \le \varepsilon /\phi _1\). A special case is when inclusions are spherical, so that \(\varepsilon =1\).

## 5 Construction of the bounds for transversely isotropic tensors

*explicit*general construction of the HS bounds that take into account not only the inclusion shapes but also their spatial distribution, in terms of the conductive modulus tensors \(\mathbf {K}^r\) and the P-tensors \({\mathbf P}_d^\rho \) and \({\mathbf P}_s^\varepsilon \) However the implementation can cause great difficulty, particularly for anisotropic phases. To simplify this issue, we shall proceed as follows. First, we observe that a second order TI tensor can be defined with respect to the tensor basis set

**Definition 3**

Therefore, given the explicit expressions for the HS bounds derived in Sect. 4, one can straightforwardly construct them using (5.25) and introducing the tensors coded as functions with arguments as 2-vectors.

## 6 Implementation and examples

In this section we illustrate the above scheme for a heterogeneous media containing aligned spheroidal inclusions of alumina embedded in a host matrix made up of aluminum. Aluminum (Al) is probably the most common matrix for metal-host composites for several reasons. It is very light, which makes it of great interest for the aerospace industry and other applications such as laptop computers. Its low cost due and low melting temperature also makes composite fabrication very economical. However, the high heat dissipation of this material is a serious problem in applications that use this material as a thermal conductor. An effective way to resolve this problem is the addition of a low filler to form a metal matrix composite. The most common filler used is silicon carbide (\(\mathrm {SiC}\)) inclusions due to its low cost and low coefficient of thermal expansion (CTE=\(3.7\,10^{-6/^oC}\)). In this process [17], \(\mathrm {SiC}\) reacts with aluminum through the chemical reaction \(\mathrm {3 SiC+4Al\rightarrow 3Si+Al_4C_3}\) and the produced silicon (\(\mathrm {Si}\)) weakens the interface between the filler and the host of the composite. Alternative, alumina (\(\mathrm {Al_2O_3}\)) and aluminium nitride (\(\mathrm {AlN }\)) are fillers that do not react with aluminium. We refer to [18] for more details related to the chemical reaction between these materials. Particularly, particles of alumina (\(\mathrm {Al_2O_3}\)) are the second most popular type that are employed. Therefore, in this section we illustrate the above bounding scheme for a heterogeneous media containing aligned spheroidal inclusions of alumina embedded in a host matrix made up of aluminum. According to [17] and [18] the numerical values of thermal conductivities for aluminum and alumina are, respectively \(\kappa _0= 247 \,W/Km\) (CTE= \(23\)\(10^{-6}/^oC\)) and \(\kappa _1= 20\,W/Km\) (CTE= \(7\)\(10^{-6}/^oC\)). For simplicity we compute only the transversal effective component \(\kappa _*\) of the composite. The HS bounds are plotted as a function of the volume fraction \(\phi =\phi _1\) of the alumina inclusion phase.

*spherical*alumina inclusions uniformly distributed with

*spherical*symmetry in the aluminum host phase (\(\rho =\varepsilon =1\)). Therefore the effective material is isotropic. We plot the Wiener bounds (dashed lines) on these effective properties together with the HS bounds (solid lines), noting the improvement of the HS bounds in particular. On the right, we consider the case of

*spheroidal*alumina inclusions with general aspect ratio \(\varepsilon =1\) (solid line), \(\varepsilon =0.1\) (dashed lines) distributed with

*spheroidal*symmetry. So that the effective material is TI. Limiting cases \(\varepsilon \rightarrow \infty \) (dotted line) and \(\varepsilon \rightarrow 0\) (dot-dashed line) corresponding to long fibre-reinforced and layered materials are also plotted.

In Fig. 3 we consider the case when the distribution spheroid has different shape to that of the inclusion shape (\({\mathbf P}_s^\varepsilon \ne {\mathbf P}_d^\rho \)). On the left, we take *spherical* alumina inclusions (\(\varepsilon =1\)) and consider both oblate with \(\rho =\sqrt{\phi }\) (dot-dash lines) and \(\rho =\frac{1}{\phi }\) for the prolate (dashed lines) distributions. On the right, we consider the effect when the inclusion becomes a *spheroid* with (oblate and prolate) aspect ratio \(\varepsilon =\phi ,\, {1}/{\sqrt{\phi }}\) (dot-dash and dashed lines resp.) and the distribution is *spherical* (\(\rho =1\)). It is worth observing the fact that according to (4.22), the effect of the aspect ratio of the inclusions on the effective properties of the material is larger that the corresponding one due to the distribution of spherical inclusions. Of course, each point on these curves represents a different type of composite in the sense that the inclusion and the distribution has different spheroidal statistics. Wiener bounds (solid lines) are also represented. On the right of Fig. 4, we plot the influence of *spheroidal* inclusions with aspect ratio \(\varepsilon \), possibly distinct from the aspect ratio \(\rho \) of the *spheroidal* distribution for a fixed \(\phi =0.3\). We plot two set of curves, one set corresponding to a *spheroidal* distribution with aspect ratio \(\rho =\varepsilon \sqrt{0.3}\) (solid lines) and another set corresponding to the same aspect ratio as that of the inclusion \(\rho =\varepsilon \) (dashed lines). Note that the plot has log-linear scaling and as should be expected the most significant effect is felt away from the limiting cases when \(\varepsilon \rightarrow 0\) and \(\varepsilon \rightarrow \infty \) corresponding to the layer and long fibre limits.

## 7 Conclusions

We have presented a straightforward mechanism for the *construction* of the HS bounds for TI composites focusing in particular on the two-phase case in the conductivity setting. The scheme takes into account microstructural information of the media through the shape and distribution of the inclusions. The explicit form of the Hill tensors for spheroidal inclusions, which is often derived from its integral form- and the definition of an appropriate TI tensor basis set is used. The associated vector notation described in Sect. 5 leads to an clear way to develop a mathematical theory that generalizes some existing formulas in the literature. We implement different constructions for a specific composite material, showing the improvement of the HS bounds over the Wiener bounds. Analogous schemes may be developed for several phases and materials of arbitrary anisotropy, although in general the corresponding Green tensor, and therefore the Hill tensor cannot be derived analytically. The mechanism proposed can be extended to the more general elasticity context by deriving the corresponding Hill tensors and the appropriate basis tensor. In this sense, future work will try to consider the construction of HS bounds for multi-phase composites, taking into account enough microstructural information to derive accurate property predictions.

## Notes

### Acknowledgments

This work has been partially supported by the project MTM 2011-24457 of the “Ministerio de Ciencia e Innovación” of Spain and the research group FQM-309 of the “Junta de Andalucía”.

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