# On the computation of the Hashin–Shtrikman bounds for transversely isotropic two-phase linear elastic fibre-reinforced composites

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## Abstract

Although various forms for the Hashin–Shtrikman bounds on the effective elastic properties of inhomogeneous materials have been written down over the last few decades, it is often unclear how to construct and compute such bounds when the material is not of simple type (e.g. isotropic spheres inside an isotropic host phase). Here, we show how to construct, in a straightforward manner, the Hashin–Shtrikman bounds for generally transversely isotropic two-phase particulate composites where the inclusion phase is spheroidal, and its distribution is governed by spheroidal statistics. Note that this case covers a multitude of composites used in applications by taking various limits of the spheroid, including both layered media and long unidirectional composites. Of specific interest in this case is the fact that the corresponding Eshelby and Hill tensors can be derived analytically. That the shape of the inclusions and their distribution can be specified independently is of great utility in composite design. We exhibit the implementation of the computations with several examples.

### Keywords

Fibre-reinforced materials Hashin–Shtrikman bounds Linear elasticity## 1 Introduction

Fibre-reinforced composites (FRCs) are commonly employed in numerous applications in science and engineering, one of their main uses being to provide improved tensile strength. It is therefore of great interest to possess knowledge of the overall (effective) properties of such materials; of specific interest in this article will be their effective linear *elastic* properties and the ability to determine useful bounds on these for a variety of fibre microstructures. FRCs typically occur as one of two types: long unidirectional fibre materials where the fibres are aligned (parallel) to some common axis and they are so long that end effects can be neglected, or short fibre materials where the short fibres may or may not be aligned, depending upon the application. Such FRCs are frequently then used as *ply* phases in order to build up layered composites, see e.g. Tsai [1]. Furthermore, such layered and FRC materials have the potential to be of great use in modern metamaterial applications as was described by Torrent and Sanchez-Dehesa [2] and Amirkhizi et al. [3] for example. The main objective of this article is to pull together ideas relating to FRCs from a variety of diverse publications, placing particular emphasis on the explicit construction and computation of the Hashin–Shtrikman bounds on effective elastic constants, including information regarding the two-point correlation functions associated with the distribution of inclusion phases.

*microstructure*(i.e. a lengthscale that is much smaller than the characteristic lengthscale of loading, e.g. a propagating wavelength) is difficult, because the partial differential equations that arise possess rapidly varying coefficients (in space). Before the early 1960s, there was a paucity of theoretical work regarding the prediction of the effective behaviour of elastic composite materials, i.e. elastic materials composed of two or more so-called

*phases*that are mixed together in order to improve and/or optimize the overall material behaviour in some sense. In particular, very limited information was available regarding the prediction of the fourth-order effective linear elastic modulus tensor \(\mathbf {C}^*\). Key results were those of Voigt [4] and Reuss [5] in 1889 and 1929, respectively, whose approximations of uniform strain and stress (respectively) throughout the composite are straightforwardly identified as upper and lower

*bounds*(respectively) on the effective elastic properties. For a composite with phases labelled \(r=0,\ldots ,n\) and with associated elastic modulus tensors \(\mathbf {C}^r\), these bounds are written

Numerous important contributions to the subject occurred in the early 1960s, the main motivation being the understanding of the overall behaviour of FRCs. At that time, such materials were used in a variety of applications but principally within the aerospace industry because of their high strength-to-weight ratio compared with more conventional structural materials. In particular, Hashin and Shtrikman [7] established a variational principle for elastostatics which they subsequently applied to multiphase (macroscopically isotropic) composites [8]. The resulting bounds on the effective bulk and shear moduli are the celebrated Hashin–Shtrikman bounds. Hashin [9] extended the principle in order to derive bounds for long FRCs where the macroscopic anisotropy is that of *transverse isotropy* and phases are isotropic. Hashin [10] also derived results for fibre reinforced composites where phases are transversely isotropic but not via the classical Hashin–Shtrikman variational scheme. Derivations of the Hashin–Shtrikman bounds have been improved and revised by many authors since they were originally devised. In particular, we note the works [11] and [12, 13, 14, 15, 16]. The analogous problem for nonlinear heterogeneous media has been treated less frequently. In this regard, we can refer to Willis [17] where the approach of Hill [18] was applied to nonlinear dielectrics, and Talbot and Willis [19] where theoretical aspects of the extension of the Hashin–Shtrikman variational principles to *nonlinear* heterogeneous systems were formulated. Other variational structures for nonlinear media, based on comparisons with *linear homogeneous media* that allow the estimation of the effective energy function of nonlinear composites, in terms of the corresponding properties for linear problems with the same microscopic distribution, were obtained by Ponte Castañeda [20, 21].

The articles just described were instrumental in enabling general forms of the Hashin–Shtrikman bounds to be written down for arbitrarily anisotropic composites.

*homogeneous*linear elastic

*comparison material*(the choice of which will be discussed shortly), and \(\varvec{\tau }\) is the so-called

*polarization stress*, which is necessarily spatially dependent. The equations of elastostatics thus reduce to

Optimizing with respect to \(\varvec{\tau }^r\) and *choosing the distribution function to have the same spheroidal statistics as the shape of the corresponding inclusion phase*, we are led to the classical (and much quoted) form of the Hashin–Shtrikman bounds ((3.13) below) derived by Willis [12]. This form can also be deduced from the more general formulation (3.8) developed by Ponte Castañeda and Willis [16] which allows for more general distributions. The key advantage of the Hashin–Shtrikman bounds over the Reuss–Voigt bounds is that the former uses information about the macroscopic anisotropy; this permits an improvement over the Reuss–Voigt bounds in almost all cases.

It is worth noting at this point that improved bounds, using more general polarizations leading to higher-order statistics, can also be derived [22, 23, 24]. However, it is often the case that such higher-order statistical correlation information for a given material is not known or difficult to determine accurately.

Although the general form for the Hashin–Shtrikman bounds applicable to arbitrarily anisotropic composites can be derived in a straightforward manner in some cases, only very recently have *explicit* bounds for generally transversely isotropic materials been written down [25]. Furthermore, as far as the authors are aware, works concerning the *construction* and *computation* of such bounds for a given material, using tensor bases (thus permitting fast implementation), are not available. Indeed it appears that the Hashin–Shtrikman bounds have often taken on a mysterious air in the literature. These bounds always appear to be merely stated (not derived constructively) and almost no information can be found in the literature regarding bounds on the effective properties of composites that are not simply either macroscopically isotropic or transversely isotropic of the long FRC variety. Therefore, the step taken in [25] was a useful and important one since explicit bounds were stated and compared with numerous homogenization and micromechanical methods. However, in [25], very little discussion of the required tensor-basis for transverse isotropy was given, and furthermore the uniformity of the so-called \(\mathbf {P}\)-tensor (directly related to the Eshelby tensor and sometimes known as the *Hill* tensor in the literature) was not exploited fully since expressions for the \(\mathbf {P}\)-tensor were given in integral form, rather than explicit expressions which have been previously derived for spheroidal inclusions and distributions.

For all of the reasons above, the construction and computation of the Hashin–Shtrikman bounds for those “not in the know” are far from straightforward. This is important specifically for engineers, materials scientists and industrialists who may wish to construct the Hashin–Shtrikman bounds for a variety of such media. They are often restricted from doing so by lack of information regarding the detail of the construction. It would be convenient to have a systematic and prescriptive way of constructing the Hashin–Shtrikman bounds from the first principles. That is, given the volume fractions, elastic properties, shapes of phases of the composite and their spatial distribution, it would be convenient to have a mechanism by which the Hashin–Shtrikman bounds could be constructed in a straightforward manner using the correct tensor basis set and the appropriate expressions for the Eshelby (\(\mathbf {S}\)) tensors and/or Hill (\(\mathbf {P}\)) tensors. In particular in this respect, it appears that although knowledge of explicit bounds is very useful, we would argue that it is less important than being able to construct the bounds from the first principles.

The principal objective of this article is therefore the discussion of such a construction and computation of the Hashin–Shtrikman bounds for transversely isotropic composites restricting attention to two-phase fibre-reinforced media. We consider cases that incorporate separately the influence of the fibre (or “inclusion”) phase and the distribution of these inclusions. Initially, we shall consider the general case of an arbitrary number of phases, but considering general statistics makes the study of media with more then one inclusion phase more complicated, and therefore that case shall be considered in a follow-up article.

The paper is organized as follows. In Sect. 2, we introduce notation and the basics concerning Hill’s tensor. In Sect. 3, we describe the general formulation of the Hashin–Shtrikman bounds, initially for the general multiphase case before restricting attention to two phases, the primary concern of this article. We specialize in Sect. 4 to the case of macroscopically transversely isotropic materials. This specialization therefore motivates the discussion of transversely isotropic fourth-order tensors in Sect. 5 and in particular we describe a convenient basis set for such tensors which was introduced in this context by Hill [26]. In Sect. 6, we describe how such a basis set, together with knowledge of the appropriate Hill and Eshelby tensors (stated in the appendices), allows us to construct the Hashin–Shtrikman bounds in a straightforward, procedural manner. We discuss the specific limiting cases of layered and long cylindrical fibre-reinforced media in Sect. 7 before illustrating the implementation of the construction via some examples in Sect. 8. The main focus is the illustration of the simple implementation via the basis set employed as well as the study of distributions of inclusions that have a different spheroidal spread to their shape, the latter being associated with the appropriate Hill tensor. We conclude in Sect. 9 indicating required areas for future study.

## 2 Notation and preliminaries

Let us first define some language: we shall say that a tensor is *uniform* if all of its components are constant (and these components can be *different* constants of course). In this article, we speak generally at first with regard to multi-phase particulate composite materials occupying the domain \(\varOmega \) with closed boundary \(\partial \varOmega \) and with \(n\) types of inclusion phase that could be chosen independently of their spatial distribution [16]. We denote the elastic modulus tensor of the \(r\)th inclusion phase by \(\mathbf{C}^r\!, r=1,\ldots ,n\). Inclusion phases are embedded inside a host phase with elastic modulus tensor \(\mathbf{C}^0\). Although results can in principle be obtained for ellipsoidal inclusions, let us restrict attention to the case of *spheroidal* inclusions; this is convenient for reasons that will become clear shortly. We note that this case is extremely useful since many composites are of this class (including the limiting cases of long FRCs and layered media). Let us denote \(\varOmega ^r\) as the total domain of the \(r\)th phase, i.e. it is the collection of all spheroidal inclusions which constitute that phase, each aligned and having the shape defined by a domain \(V^r\). Each spheroid constituting the \(r\)th phase is located at a different point in space; in terms of the computation of the Hashin–Shtrikman bounds, only their *shape* and *distribution* is important. The total volume fraction of this phase is therefore \(\phi _r=|\varOmega ^r|/|\varOmega |\) where \(|\cdot |\) denotes a volume. The volume fraction of the host phase is therefore \(\phi _0=1-\sum _{r=1}^n\phi _r\).

*Hill’s polarization tensor*, has components defined by

*comparison*phase having elastic modulus tensor \(\mathbf {C}^c\). We note that the integral in (2.1) is over the domain of a spheroid in the \(r\)th phase, i.e. \(V_r\) (hence the superscript \(r\) on \(\mathbf{P}\)) and dependent only on the ratio of semi-axes, and not the size of the spheroid. It is also independent of the material properties of the inclusion so that in the particular case where all inclusion phases are of the same shape, the \(\mathbf {P}\)-tensor considered here is identical for any phase. The notation \(|_{(ij),(k\ell )}\) indicates symmetrization with respect to these indices, i.e. for a tensor \(Q_{ijk\ell }\) such a symmetrization would be

*Eshelby tensor*\(S^c_{ijk\ell }\) via the expression

*shape*, to distinguish it from an analogous quantity (with subscript \(d\)) that we will encounter shortly associated with

*distributions*.

## 3 The Hashin–Shtrikman variational principle

In his paper of 1977, Willis [12] developed a variational structure to derive the Hashin–Shtrikman bounds. Introducing a polarization stress relative to a so-called comparison material, both the inclusion shape and their distribution (from integrals of the associated two-point correlation functions) are incorporated through appropriate influence tensors. The distribution tensors were assumed to be the same as the associated shape tensor for simplicity. Later, Ponte Castañeda and Willis [16] generalized the structure of Willis [12, 14], whereby the derived bounds contain general distribution tensors. It makes use of the alternative variational representation for the effective energy function given by Talbot and Willis [19], which also applies for nonlinear composites.

*optimal polarizations*\(\varvec{\tau }_k^*\).

*shape*and

*distribution*of inclusions, respectively. By definition, we have \(\mathbf {P}_d^{(k\ell )}=\mathbf {P}_d^{(\ell k)}\), and for conciseness, we will write \(\mathbf {P}_d^{k}\), to denote \(\mathbf {P}_d^{(kk)}\), \(k=0, \ldots ,n.\)

Given the above, let us therefore recall the Hashin–Shtrikman variational principle which states the following regarding the choice of the elastic modulus tensor \(\mathbf {C}^c\) of the comparison material and the way that this leads to bounds, denoted by \(\mathbf {C}^B=\mathbf {C}^+\) for the upper and \(\mathbf {C}^B=\mathbf {C}^-\) for the lower bounds. The theorem that we state has been stated in a number of different ways depending upon the context and application. See e.g. [16].

**Theorem 1**

Therefore, the lower bound \(\mathbf {C}^B\) in (3.7) is given whenever \(\mathbf {C}^r-\mathbf {C}^c\) is positive semi-definite for all \(r=0,\ldots ,n\). If \(\mathbf {C}^r-\mathbf {C}^c\) is negative semi-definite, the upper bound is given.

In some instances, instead of obtaining bounds, authors have chosen to obtain an *approximation* to the effective properties by choosing \(\mathbf {C}^c=\mathbf {C}^0\), the host elastic modulus tensor, regardless of its relationship to other moduli [16]. Let us consider this case first.

### 3.1 The Hashin–Shtrikman estimates

*estimate*(note that it is

*not*a bound unless \(\mathbf {C}^0\) is either minimal or maximal over all phase modulus tensors).

### 3.2 The Hashin–Shtrikman bounds

As the title suggests we shall restrict attention to two-phase particulate media so that there is a single inclusion phase. The bounding technique for greater than one inclusion phase is not well established. Suppose that the inclusion phase is a distribution of (possibly different sized) aligned spheroids but where each spheroid has the same aspect ratio \(\delta =a_3/a\) 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 tensor \(\mathbf {P}_s\). Their distribution is accounted for by virtue of the tensor \(\mathbf {P}_d\) which we shall consider to be governed by spheroidal statistics, of aspect ratio \(\epsilon \). We shall consider this notion further shortly. 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.

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

Of course, it may not always be the case that the comparison material can be identified as exactly one of the inclusion phases. As such, it is instructive to determine general bounds in terms of the comparison modulus tensor.

#### 3.2.2 Comparison phase cannot be identified as either host or inclusion phase

#### 3.2.3 The distribution \(\mathbf {P}\)-tensor is identical to the shape \(\mathbf {P}\)-tensor

### 3.3 Distribution \(\mathbf {P}\)-tensor

*safety*spheroid, containing a single spheroidal inclusion and which is not intersected by any other security spheroid (hence the terminology). The details of the distribution tensors for more than one inclusion phase become quite complex and will be discussed in a future article. Here we focus on a single inclusion phase.

## 4 Transversely isotropic composites

*symmetric*TI tensors, a concept that we will discuss shortly. Therefore, in the \(r\)th phase, the Cauchy stress \(\varvec{\sigma }^r\) is linked to the linear elastic strain \(\mathbf {e}^r\) via \(\varvec{\sigma }^r=\mathbf {C}^r\mathbf {e}^r\) where \(\mathbf {C}^r\) is (symmetrically) transversely isostropic. We use the following notation (following Hill [26]) in component form:

*in-plane*bulk and shear moduli of phase \(r=1,0\), respectively, and \(p_r\) is the

*anti-plane*(or longitudinal) shear modulus. We note here that \(C^r_{1111}=k_r+m_r, C^r_{1133}=\ell _r, C^r_{3333}=n_r, C_{1122}^r=k_r-m_r\), \(C^r_{1313}=p_r\) and \(C^r_{1212}=(C^r_{1111}-C^r_{1122})/2 = m_r\). Finally, we note that the engineering notation for the elastic modulus tensor is often used, which expresses stress in terms of strain via multiplication by a six by six matrix \(C^r_{ij}\). The coefficients of this matrix are related to the elastic modulus tensor by the expressions \(C^r_{1111}=C^r_{11}, C^r_{1133}=C^r_{13}, C^r_{3333}=C^r_{33}, C^r_{1212}=C^r_{66}, C^r_{1313}=C^r_{55}\).

*engineering constants*

As detailed above, we restrict attention to inclusions (and distributions) that are *spheroidal* and with the axis of TI of the phases being the \(x_3\) axis. This yields macroscopically TI materials. Note that the limits \(\delta \rightarrow \infty \) and \(\delta \rightarrow 0\) correspond to the *long cylindrical fibre* and *layer* (or *penny-shape*) limits, respectively.

The macroscopic stress–strain law associated with the effective material will be that in (4.1)–(4.3) with \(r\) replaced by \(*\) everywhere (denoting the effective material). In what follows then we shall describe how to construct the Hashin–Shtrikman bounds on the effective elastic properties \(k_*, \ell _*, n_*, m_*\) and \(p_*\) and subsequently on the Engineering moduli \(E^\mathrm{T}*, E^A_*, \nu ^\mathrm{T}_*\) and \(\nu ^A_*\).

## 5 Transversely isotropic tensors

*isotropic*tensor can be defined with respect to the tensor basis set \(\{I_{ijk\ell }^{(1)},I_{ijk\ell }^{(2)}\}\) where

*transverse*isotropy, where the axis of transverse symmetry is the \(x_3\) axis. In this case, a transversely isotropic (TI) tensor \(H_{ijk\ell }\) can also be defined with respect to a tensor basis set. Several of these have been proposed; all slight variants of each other, see for example [30, 31]. However we shall use the basis associated with the notation introduced for TI tensors by Hill [26] since this is a frequently used notation in the composite materials community, particularly when the Hashin–Shtrikman bounds are discussed. Although it does not appear that Hill wrote down this specific basis set, it is clear that this was what his notation referred to. This basis set enables a TI tensor \(H_{ijk\ell }\) to be written in the form:

*Hill*basis.

The set of *symmetric TI tensors* (\(H_{ijk\ell }=H_{k\ell ij}\)) is defined by setting \(X_2=X_3\) (e.g. a TI elastic modulus tensor falls into this class). The set of *symmetric* TI tensors is not closed under multiplication in the sense that if we take two symmetric TI tensors \(A_{ijk\ell }\) and \(B_{ijk\ell }\) and perform the double contraction \(A_{ijmn}B_{nmk\ell }=D_{ijk\ell }\), the resulting tensor \(D_{ijk\ell }\) is *not* symmetric TI because \(D_{3311}\ne D_{1133}\), and so \(X_2\ne X_3\) in this case; see Eq. (5.8) below. Hence in general, when considering transversely isotropic tensors, we must consider them with respect to the *six* basis tensors \(\mathcal {H}_{ijk\ell }^{(n)}\) defined above; this basis set *is* closed with respect to double contraction. Note this important difference between *transversely isotropic tensors* and *symmetric transversely isotropic tensors*. For more discussion of this point, see [32], although the terminology *symmetric* TI tensors is specific to this article.

The contractions of the basis tensors \(\mathcal {H}_{ijk\ell }^{(n)}\)

\(\mathcal {H}^{(1)}\) | \(\mathcal {H}^{(2)}\) | \(\mathcal {H}^{(3)}\) | \(\mathcal {H}^{(4)}\) | \(\mathcal {H}^{(5)}\) | \(\mathcal {H}^{(6)}\) | |
---|---|---|---|---|---|---|

\(\mathcal {H}^{(1)}\) | \(\mathcal {H}^{(1)}\) | \(\mathcal {H}^{(2)}\) | 0 | 0 | 0 | 0 |

\(\mathcal {H}^{(2)}\) | 0 | 0 | 2\(\mathcal {H}^{(1)}\) | \(\mathcal {H}^{(2)}\) | 0 | 0 |

\(\mathcal {H}^{(3)}\) | \(\mathcal {H}^{(3)}\) | 2\(\mathcal {H}^{(4)}\) | 0 | 0 | 0 | 0 |

\(\mathcal {H}^{(4)}\) | 0 | 0 | \(\mathcal {H}^{(3)}\) | \(\mathcal {H}^{(4)}\) | 0 | 0 |

\(\mathcal {H}^{(5)}\) | 0 | 0 | 0 | 0 | \(\mathcal {H}^{(5)}\) | 0 |

\(\mathcal {H}^{(6)}\) | 0 | 0 | 0 | 0 | 0 | \(\mathcal {H}^{(6)}\) |

*not*equal in general, i.e. contraction between two symmetric TI tensors produces a non-symmetric TI tensor in general. This is why we must write out the theory for the evaluation of the Hashin–Shtrikman bounds for TI materials by using the general TI tensor basis set. We note that (5.8) is a correction to the typographical error in the corresponding expression in Appendix A of [25].

## 6 Construction and computation of the bounds

Here we describe the procedure for *constructing* the Hashin–Shtrikman bounds for TI composites of the type considered above. This procedure can be easily coded in any standard mathematical package or alternatively in widely used programming languages, in a straightforward manner. In particular, we have described the construction in a way that should be simple to implement in such languages, by defining functions that represent the operations of taking the inverse and double contraction of TI tensors.

The construction is achieved by using the short-hand notation for TI tensors described in the previous section. In particular, we specify a TI tensor by writing this short-hand list of six constants as a 6-vector \((2k,\ell ,\ell ',n,2m,2p)\), with symmetric transverse isotropy if \(\ell '=\ell \). We then exploit the properties of the Hill basis defined above by defining the appropriate tensor operations of contraction and inversion as in (5.8) and (5.10). This vector notation is particularly convenient for computational implementation of the computation of the HS bounds.

*Two*TI comparison materials are defined (as 6-vectors) as in Remark 1:

*symmetric*transversely isotropic tensor. We also recall that the \(\mathbf{P}\) tensor is associated with the

*comparison*material.

Let us now consider the different computations, according to the choice of comparison material. Suppose first that we are able to choose the comparison material as either the host or inclusion as in Sect. 3.2.1.

### 6.1 Construction when comparison phase is host or inclusion phase

If we have \(\mathbf {P}_d=\mathbf {P}_s=\mathbf {P}\), then one can simply use this in the procedure above, or alternatively one could use the contraction and inversion procedures to code up the approach written down in Sect. 3.2.3. In this two phase case (a single inclusion phase), re-writing in this manner is no more advantageous. However, in the multi-phase case, this alternative approach does make the construction more simple because we may exploit the form (3.13). We will describe this further in future work.

### 6.2 Construction in the general case

## 7 Limiting cases of transversely isotropic composites

It is straightforward (but time consuming) to derive the results (7.1)–(7.3) analytically using the construction of the Hashin–Shtrikman bounds in the form (3.13). In this specific limiting case, the Hashin–Shtrikman bounds coincide for all five effective properties of course, thus giving the exact results (7.1)–(7.3) for a layered medium (or a medium consisting of aligned uniformly distributed penny-shaped inclusions).

*not*coincide; they become the form of bounds given in [9] on long FRCs for \(k_*, m_*\) and \(p_*\). Bounds on \(n_*\) and \(\ell _*\) were not given in [9] but they may be determined by appealing to the so-called universal properties of composites [18]). In [9], Hashin dealt with only

*isotropic*phases, although since he wrote the paper in terms of (de-coupled) in-plane and anti-plane problems and wrote the expressions in terms of shear moduli and in-plane bulk moduli, it is straightforward to derive the bounds for TI phases by simple replacement of the corresponding moduli with their TI counterparts. As in the layered medium case above, it is straightforward to derive these bounds analytically in this limiting case using the details regarding transversely isotropic tensors in Sect. 5 (once again noting the use of (5.8) and (5.10) in particular) in the construction of the Hashin–Shtrikman bounds described above, together with the

*exact*form of the Eshelby tensor in this limit (given in 1). These results for an \(n\) phase composite of the type considered in this article are the following, in terms of \(k^B, \ell ^B, n^B, m^B\) and \(p^B\), specified in terms of the comparison phase:

## 8 Implementation

Elastic material properties of the phases in GPa

Material | \(C_{11}=k+m\) | \(C_{12} = k-m\) | \(C_{13} =\ell \) | \(C_{33}=n\) | \(C_{44}=p\) |
---|---|---|---|---|---|

Glass | 77.77 | 25.77 | 25.77 | 77.77 | 26 |

Epoxy | 6.73 | 4.19 | 4.19 | 6.73 | 1.27 |

PZT-7A | 157 | 85.4 | 73 | 175 | 47.2 |

Epoxy II | 8 | 4.4 | 4.4 | 8 | 1.4 |

### 8.1 Example 1: Isotropic glass inclusions in the isotropic epoxy host phase

*spherical*glass inclusions, and therefore the effective material is isotropic. In Fig. 2, we plot bounds on the effective bulk modulus \(\kappa _*\) and shear modulus \(\mu _*\) of the composite material. In particular, we plot the Reuss (lower) and Voigt (upper) bounds on these effective properties together with the Hashin–Shtrikman bounds. Note the improvement of the Hashin–Shtrikman bounds over the Reuss/Voigt bounds.

*spheroidal*glass inclusions with general aspect ratio \(\delta \) so that the effective material is transversely isotropic. In Fig. 3, we plot the Hashin–Shtrikman bounds on the effective shear moduli \(m_*/m_0\) and \(p_*/p_0\). Note that the bounds coincide in the (layered) limit when \(\delta \rightarrow 0\). In the case of the in-plane shear modulus, the bounds become equal to the Voigt bound, whereas in the anti-plane shear modulus they become equal to the Reuss bound; this can be seen from the exact results in (7.3). Note also that the anti-plane shear modulus does not change significantly when we increase \(\delta \) from \(\delta =1\) to the \(\delta \rightarrow \infty \) limit. This can also be seen on the right of Fig. 6 where we plot the bounds on the effective shear moduli as a function of \(\delta \) for \(\phi =0.3\). In Fig. 4, we plot the effective properties \(k_*/k_0\) and \(n_*/n_0\), noting that the layered limit bounds coincide. We also note that in this instance, the bounds on \(n_*/n_0\) for \(\delta \rightarrow \infty \) are very tight. The different regimes of \(\ell _*/\ell _0\) are plotted in Fig. 5.

In Fig. 6, we illustrate the dependence of the bounds as a function of \(\delta \) for fixed volume fraction \(\phi =0.5\). In particular, we note the recovery of the (exact) layered expressions (7.1)–(7.3) as \(\delta \rightarrow 0\) (denoted by disks) and also the recovery of the classical Hashin–Shtrikman bounds (7.4)–(7.8) for long FRCs when \(\delta \rightarrow \infty \) (also denoted by disks). Note in particular that the fibres do not have to be particularly long before they reach this limit: \(\delta = \mathcal{O}(10)\) suffices for this limit to be reached, with the exception of \(n_*\) which requires a larger \(\delta \), perhaps \(\delta = \mathcal{O}(100)\).

### 8.2 Example 2: Transversely isotropic PZT-7A inclusions in the isotropic Epoxy II host phase

In Fig. 7, we plot the bounds on the effective shear moduli of this composite. Note in particular the high contrast achieved by addition of the inclusion phase (particularly the in-plane shear modulus). As in example 1, the effective shear moduli are not affected greatly by increasing \(\delta \) from unity, particularly the anti-plane shear modulus. This is also seen in the plot on the right of Fig. 10. In Fig. 8, we plot corresponding bounds on \(k_*/k_0\) and \(n_*/n_0\). For \(k_*/k_0\), we plot only the upper bound for \(\delta =1\) since this does not change significantly until \(\delta \) is very small, when the upper and lower bounds coincide in the layered limit as can be seen in the plot on the left. This can also be seen in Fig. 10 (noting the logarithmic scale for \(\delta \)). As in the isotropic case, bounds on \(n_*/n_0\) become relatively tight in the long cylindrical fibre limit, \(\delta \rightarrow \infty \). In Fig. 9, we plot \(\ell _*/\ell _0\), on the left as we decrease \(\delta \) below unity and on the right as we increase \(\delta \) above unity. On the left, we see that bounds become progressively tighter as \(\delta \) is decreased until the layered limit is reached. On the right, we see that the bounds do not become modified greatly by increasing \(\delta \) above unity.

For all properties, as \(\delta \rightarrow 0\), the layered limit is achieved, as should be expected. We note from Fig. 10 that this limit is not reached as quickly as in the isotropic phase case in example 1. In particular even for \(\delta =10^{-3}\), this limit has not yet been reached for the in-plane shear modulus \(m_*/m_0\).

### 8.3 Example 3: Spheroidal glass inclusions with spheroidal distribution in the epoxy host phase

Let us now consider the effect when the inclusion becomes a spheroid with aspect ratio \(\delta \) possibly distinct from \(\epsilon \) the aspect ratio of the spheroidal distribution. In particular, we refer to Fig. 14 where we consider the modification to the plot in Fig. 6 when the spheroidal distribution is \(\epsilon =\delta \sqrt{\phi }=\delta /\sqrt{2}<\delta \). 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 \(\delta \rightarrow 0\) and \(\delta \rightarrow \infty \) corresponding to the layer and long fibre limits.

Finally with reference to Fig. 15, we study the glass/epoxy composite with fixed volume fraction \(\phi =0.2\) of glass inclusions. We then fix the inclusion aspect ratio and study the dependence of the shear moduli \(m_*/m_0\) and \(p_*/p_0\) on the spheroidal distribution aspect ratio \(\epsilon \). In particular, we present results for the four values \(\delta =1.5, 1\) (spheres), \(\delta =0.6\) and \(0.2\). In plotting the range of \(\epsilon \), we take into account the inequalities given in Sect. 3.3. In particular from subfigure to subfigure, we see the shift in bounds as \(\delta \) is decreased. Note in particular that there is little variation in the bounds as \(\epsilon \) increases above unity. More variation is seen for \(\epsilon <1\).

## 9 Concluding remarks

We have shown how to construct, in a straightforward manner, the Hashin–Shtrikman bounds for transversely isotropic composites focusing in particular on the case of two-phase FRCs. The shape of the inclusion phase and their corresponding distributions can be chosen independently by use of the appropriate Hill and Eshelby tensors and TI tensor basis set. Note in particular that for TI materials, the Eshelby tensor can be derived analytically, and we summarized its various forms for spheroids in the Appendix. We implemented different computations for two specific composite materials, exhibiting the improvement of the Hashin–Shtrikman bounds over the Reuss–Voigt bounds, also showing how bounds behave for different inclusion (\(\delta \)) and distribution (\(\epsilon \)) aspect ratios, including the limiting cases of layered media (\(\delta \rightarrow 0\)) where bounds coincide, agreeing with the Backus expressions [33] and also long fibre-reinforced media (\(\delta \rightarrow \infty \)) where the bounds are then the classical Hashin–Shtrikman bounds (some of which were derived by Hashin [9]) for such media. Entirely analogous approaches may be followed for phases and materials of arbitrary anisotropy, although in general, the corresponding Green’s tensor (and hence \(\mathbf {P}\)-tensor) cannot be derived analytically.

Future work needs to consider the construction and computation of the Hashin–Shtrikman bounds for multi-phase composites, a case that has not been studied sufficiently in the literature.

## Footnotes

- 1.
This latter point does not appear to have been recognized in the original papers on this subject, e.g. [34]. An example of a transversely isotropic material for which \(v_2=\overline{v_1}\in \mathbb {C}\) is zinc with (all in GPa) \(k=80, \ell =33, n=50, m=63, p=40\), for which \(v_1\) = 1.1284 + 0.6465i to four digits of precision.

## Notes

### Acknowledgments

The authors thank Prof. I.D. Abrahams (University of Manchester, UK) for helpful discussions, Prof. P.A. Martin (Colorado School of Mines, USA) for reading through an early draft, and Prof. A.N. Norris (Rutgers University, USA) for providing useful references on tensor bases. 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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