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

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.

From a mathematical viewpoint, modelling inhomogeneous materials such as FRCs, which possess a 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 writtenHere the bounds are interpreted component-wise and the Reuss (lower) bound \(\mathbf {C}^R\) is defined bywhere \(\phi _r\) is the volume fraction of the \(r\)th phase, and \(\mathbf {I}\) is the fourth-order identity tensor with components \(I_{ijk\ell }=(\delta _{ik}\delta _{j\ell }+\delta _{i\ell }\delta _{jk})/2\). We have used \(\mathbf{A}^{-1}\) to denote the inverse of the tensor \(\mathbf{A}\). Later, we discuss technical aspects for the inversion of transversely isotropic tensors. Finally, the Voigt (upper) bound \(\mathbf {C}^V\) is defined byIn addition to these bounds, there also existed Eshelby’s dilute estimate for particulate composites derived in his seminal paper of 1957 [6]. Such dilute estimates are only valid for small volume fractions \(\phi _r\) (asymptotically correct to \({\mathcal O}(\phi _r)\)) and furthermore, although the establishment of the Reuss/Voigt bounds was an important step, in many cases, the distance between these bounds can be large, meaning that little can be inferred as regards the effective elastic properties themselves.

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 [12‐16]. 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.

Working in Cartesian coordinates \(x_1, x_2, x_3\), let us denote differentiation of a second-order tensor \(\varvec{\sigma }=\varvec{\sigma }(x_1,x_2,x_3)\) with respect to \(x_k\) by \(\varvec{\sigma }_{\!,k}\), and we adopt the Einstein summation convention for summation over repeated indices. We also introduce the notation \(\mathbf{x}=(x_1,x_2,x_3)\). The key step in a most succinct derivation of the Hashin–Shtrikman bounds in the structure developed by Willis [12] is to write the standard governing equations of elastostatics for an inhomogeneous material in an alternative form. Upon neglecting body forces and supposing that the elastic modulus tensor of the inhomogeneous material is \(\mathbf {C}=\mathbf {C}(\mathbf{x})\), the equations of elastostatics arewhere \(\varvec{\sigma }\) is the Cauchy stress tensor, \(\mathrm{div}\) is the divergence operator, \(\mathbf {e}\) is the linear strain tensor, and the superscript \(\mathrm{T}\) here denotes transpose. In common with much of the existing literature, we use notation \(\mathbf{A}\mathbf{B}\) to denote contraction over the final two and first two indices of the tensors \(\mathbf{A}\) and \(\mathbf{B}\), respectively. This is rather convenient as it can be applied to tensors of arbitrary order. Here the \(ij\)th component of \(\mathbf{C}\mathbf{e}\) is therefore \(C_{ijk\ell }e_{\ell k}=C_{ijk\ell }e_{k\ell }\) and also later, we will use this notation extensively for the contraction of two fourth-order tensors \(\mathbf{A}\) and \(\mathbf{B}\) so that the \(ijk\ell \)th component of \(\mathbf{A}\mathbf{B}\) is \(A_{ijmn}B_{nmk\ell }\).

The alternative form of (1.2) used by Willis is derived by writing Hooke’s law as \(\varvec{\sigma } = \mathbf {C}^c\mathbf {e} + \varvec{\tau }(\mathbf {x})\) where \(\mathbf {C}^c\) is a 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 towhere \(\varvec{\sigma }^c=\mathbf {C}^c\mathbf {e}\). The divergence of the polarization stress can therefore be interpreted as a spatial distribution of body forces in a homogeneous medium with elastic modulus tensor \(\mathbf{C}^c\). Upon introducing the second-order elastostatic Green’s tensor \(\mathbf{G}^c\) associated with the (anisotropic) comparison material, a variational principle on the strain energy is then found in terms of the polarization stress \(\varvec{\tau }(\mathbf{x})\) [12, 16]. Next, we choose the polarization stress to be piecewise uniform (uniform in each phase), usually written as \(\varvec{\tau }(\mathbf{x})=\sum _{k=0}^N\chi _k(\mathbf{x})\varvec{\tau }^r\) where \(\chi _k(\mathbf{x})\) is unity when \(\mathbf{x}\) resides in the \(k\)th phase and is zero otherwise, and each component of \({\varvec{\tau }}^r\) is constant.

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 [22‐24]. 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.

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 \(\phi _0=1-\sum _{r=1}^n\phi _r\).

In micromechanics and homogenization, a specific tensor known as the \(\mathbf {P}\)-tensor frequently arises [13]. This fourth-order tensor, also often known as Hill’s polarization tensor, has components defined bybeing linked with the free-space Green’s tensor having components \(G^c_{ij}(\mathbf{x}-\mathbf{y})\), i.e. that associated with the linear 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 beIf the domain \(V_r\) is spheroidal as here (ellipsoidal even) then the \(\mathbf {P}\)-tensor defined in (2.1) is uniform (i.e. each component is constant) [6, 27, 28]. Note that \(P_{ijk\ell }^c\), Hill’s tensor associated with the comparison phase, is related to the Eshelby tensor \(S^c_{ijk\ell }\) via the expressionwhere the second equality is due to the inclusions being ellipsoids so that \(S^r_{ijmn}=S^r_{jimn}\).

The \(\mathbf {P}\)-tensor associated with an ellipsoidal inclusion of the \(k\)th phase, say \(V_{k}= \{\mathbf {x}:\, |\mathbf {A}^{(k)}_s \mathbf {x}|^2<1\}\), for some second-order tensor \(\mathbf{A}_s^{(k)}\) (which is usually diagonal with diagonal components corresponding to the semi-axes of the ellipsoid) may be computed from the expressionwhere \(S^2\) is the unit sphere and \(\bar{\varvec{\xi }}\) is a unit vector pointing from the centre of \(S^2\) to a point on its surface. Furthermore,For spheroidal inclusions, explicit forms may be obtained as is detailed in Appendix B. The subscript ‘s’ is associated with shape, to distinguish it from an analogous quantity (with subscript \(d\)) that we will encounter shortly associated with distributions.

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.

Let us now consider the linear composite described above whose stress–strain constitutive relation is given byUnder homogeneous boundary conditions, the fourth-order-effective moduli tensor \(\mathbf {C}^*\) can be defined asand the average strain energy density (see [29]) is determined asFollowing [7, 11], a comparison material is introduced with uniform modulus \( \mathbf {C}^c\) and a polarization field is defined by \(\varvec{\tau }(\mathbf {x})=(\mathbf {C}(\mathbf {x})-\mathbf {C}^c)\mathbf {e}\). Next assume that \( \mathbf {C}^c\le \min _{0\le r\le n} \mathbf {C}^r (\ge \max _{0\le r\le n} \mathbf {C}^r)\), in the sense that \(\mathbf {e} ( \mathbf {C}^c- \mathbf {C}^r) \mathbf {e}\le (\ge ) \,0\), \(r = 0,\ldots , n\). Certainly, choosing \(\mathbf {C}^c- \mathbf {C}^r\le (\ge ) \,0\) componentwise achieves this. Finally select a piecewise constant polarization stressSuch an assumption subsequently yields at most two-point statistics regarding phase distribution. Under the hypothesis of ellipsoidal symmetry for the distribution of the inclusions (possibly with a different ellipsoidal shape to the shape of the inclusion), the following bounds on the effective energy are derived in [16]:whereis the average of the so-called optimal polarizations \(\varvec{\tau }_k^*\).

By linearity, using (3.1), the following expression for the optimal bounds is derived from (3.2):where \(\le \) is here interpreted componentwise, and where \(\mathbf {C}^B=\mathbf {C}^+\) refers to an upper bound and \(\mathbf {C}^B=\mathbf {C}^-\) to a lower bound. These bounds depend on the choice of comparison modulus tensor and the optimal polarizations which satisfy the relations:The tensor parameters \(\mathbf{M}^{(k\ell )}\) in (3.4) depend on \(\mathbf {C}^0\) and on the microstructure. They can be shown to be symmetric and to have the form (see [16]):Here \(\mathbf {P}^k_s\) is the \(\mathbf {P}\)-tensor associated with the shape of the inclusion in the \(k\)th phase as introduced in (3.6). On the other hand, the notation \(\mathbf {P}_d^{(k\ell )}\) is the (uniform) \(\mathbf {P}\)-tensor associated with the ellipsoidal distribution tensor concerning interaction between the \(k\)th and \(\ell \)th phases. It is associated with the ellipsoid \(V_d^{(k\ell )}= \{\mathbf {x}:|\mathbf {A}^{(k\ell )}_d \mathbf {x}|^2<1\}\) for some symmetric matrix \(\mathbf {A}^{(k\ell )}_d\) and may be computed from the expression:Recall that the subscripts \(s\) and \(d\) refer to the 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].

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.

We have spoken of a general construction of bounds in terms of the elastic modulus tensors \(\mathbf{C}^r\). Of course, above when it comes to implementation, one must consider a specific form and in general, this can cause great difficulty, particularly for anisotropic phases. Let us now restrict attention to two-phase composites where phases are (at most) transversely isotropic (TI), all with \(x_3\) as the axis of transverse symmetry so that the \(x_1x_2\) plane is the plane of isotropy. This is a very commonly occurring case in practice. Many fibre-reinforced media are of this type with fibres being TI in order to provide high tensile strength. Elastic modulus tensors that are TI are in fact 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: Here \(k_r\) and \(m_r\) are known as the 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 \(C^r_{1111}=C^r_{11}, C^r_{1133}=C^r_{13}, C^r_{3333}=C^r_{33}, C^r_{1212}=C^r_{66}, C^r_{1313}=C^r_{55}\).

Note that in the above, we have worked with the elastic modulus tensor and the Hill notation for this. We also note the following strain–stress relationship in terms of the so-called engineering constants where \(E^\mathrm{T}\) and \(E^A\) are the transverse and axial Young’s moduli and \(\nu ^\mathrm{T}\) and \(\nu ^A\) are the transverse and axial Poisson ratios, respectively. In particular, \(\nu ^A=\nu ^{31}\) corresponds to the lateral contraction when a force is applied in the axial direction, and we note the relationship \(\nu ^A_{31}/E_A=\nu ^A_{13}/E_T\). Note also that there are still only five independent constants due to the relationship \(E^\mathrm{T}_r = 2m_r(1+\nu ^\mathrm{T}_r)\). The relationships between the Engineering moduli and the constants appearing in the Hill notation of the elastic modulus tensor are given in appendix.

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 \(\nu ^A_*\).

A fourth-order isotropic tensor can be defined with respect to the tensor basis set \(\{I_{ijk\ell }^{(1)},I_{ijk\ell }^{(2)}\}\) whereTherefore, a fourth-order isotropic tensor \(H_{ijk\ell }\) can be written asGiven the basis set (5.1), an isotropic tensor \(H_{ijk\ell }\) may therefore be specified by a list of two parameters, in short-hand notation: \(\mathbf {H}=(X_1,X_2)\). In order to specify an isotropic elastic modulus tensor \(C_{ijk\ell }\), frequently the short-hand notation \(\mathbf {C}=(3\kappa ,2\mu )=(X_1,X_2)\) is adopted where \(\kappa =\lambda +2\mu /3\) corresponds to the bulk modulus and \(\mu \) to the shear modulus. The parameters \(\lambda \) and \(\mu \) are the Lamé constants of the material.

Analogously, suppose now that instead of isotropy we consider the case of 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:where \(X_n\) are constants and the basis tensors \(\mathcal {H}^{(n)}_{ijk\ell }\) are defined by where \(\varTheta _{ij} = \delta _{ij}-\delta _{i3}\delta _{j3}\). The notation \(\mathcal {H}\) has been used to signify the 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.

Analogous to the short-hand notation introduced for isotropic tensors above, in order to specify a TI tensor \(H_{ijk\ell }\), the shorthand notation \(\mathbf {H}=(2k_H,\ell _H,\ell '_H,n_H,2m_H,2p_H)=(X_1,X_2,X_3,X_4,X_5,X_6)\) is adopted, i.e. the tensor is defined by these six constants (a symmetric TI tensor corresponds to \(\ell _H'=\ell _H\)). This notation is motivated by the fact that Hooke’s law with respect to this basis (4.1)–(4.3) links stress to strain via the (symmetric) transversely isotropic tensor \(C_{ijk\ell }\) which in short-hand notation is written \(\mathbf {C}=(2k,\ell ,\ell ',n,2m,2p)\) with \(\ell '=\ell \). Before we proceed, let us define the shorthand notation:for contraction between the basis tensors defined in (5.3)–(5.5). The contractions defined in (5.6) are summarized in Table 1.

With the notation above, using the contractions in Table 1, we can therefore write down the operations of addition and contraction (in short-hand notation) on the two TI tensors \(H_{ijk\ell }^1\) and \(H_{ijk\ell }^2\) defined in short-hand notation by \(\mathbf {H}^1=(2k_1,\ell _1,\ell _1',n_1,2m_1,2p_1)\) and \(\mathbf {H}^2=(2k_2,\ell _2,\ell _2',n_2,2m_2,2p_2)\), respectively. We define the operation of addition \(H_{ijk\ell }=H_{ijk\ell }^1+H_{ijk\ell }^2\) in short-hand notation byand the operation of double contraction \(H_{ijk\ell }=H^1_{ijmn}H^2_{nmk\ell }\) byNote that even if \(\ell '_1=\ell _1\) and \(\ell '_2=\ell _2\) (corresponding to symmetric TI tensors \(\mathbf {H}^1\) and \(\mathbf {H}^2\)), the second and third elements on the right-hand side of (5.8) are 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].

The fourth-order identity tensor \(I_{ijk\ell }\) and isotropic basis tensors \(I^{(1)}_{ijk\ell }\) and \(I_{ijk\ell }^{(2)}\), written with respect to this TI basis areFinally the inverse of the tensor \(\mathbf {H}^1=(2k_1,\ell _1,\ell _1',n_1,2m_1,2p_1)\) is

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.

First we define the (symmetric TI) elastic modulus tensor:We also define the fourth-order identity tensor and isotropic basis tensors in 6-vector forms as in (5.9). Two TI comparison materials are defined (as 6-vectors) as in Remark 1:This means that we take the minimal and maximal elements over all of the phases. For a multi-phase material therefore, it could be that the comparison phase does not correspond to any of the specific phases constituting the composite, since, for example, the modulus \(m_1\) of phase 1 may be maximal but the modulus \(n_2\) of phase 2 could be maximal.

We now define the operations of double contraction and inversion in terms of the 6-vector notation introduced here. With reference to (5.8) and (5.10), we introduce the functions \(\mathbb {C}[\mathbf {H}^1,\mathbf {H}^0]:\mathbb {R}^6\times \mathbb {R}^6\rightarrow \mathbb {R}^6\) and \(\mathcal {\mathbb {I}}[\mathbf {H}^1]:\mathbb {R}^6\rightarrow \mathbb {R}^6\) operating on the 6-vectors \(\mathbf {H}^1=(2k_1,\ell _1,\ell _1',n_1,2m_1,2p_1)\) and \(\mathbf {H}^0=(2k_0,\ell _0,\ell _0',n_0,2m_0,2p_0)\) in the following way: Such operations can therefore be straightforwardly coded by introducing such functions with arguments as 6-vectors.

Therefore, given the (non-symmetric) TI Eshelby tensor (defined in Appendix A) corresponding to the appropriate comparison phase, written in 6-vector form as \(\mathbf {S}^{\delta }=(2k_{\delta },\ell _{\delta },\ell _{\delta }^{\prime },n_{\delta },2m_{\delta },2p_{\delta })\) the super or subscript \(\delta \) (or \(\epsilon \)) refers to the aspect ratio of the spheroid in question involved in the Eshelby tensor. The \(\mathbf {P}\)-tensor (also written as a 6-vector) can thus be defined via (2.2) asand we note that \(\mathbf {P}^c\) is a 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.

We note that the two limits \(\delta =\epsilon \rightarrow 0\) and \(\delta =\epsilon \rightarrow \infty \) correspond to a layered medium and a long fibre-reinforced medium, respectively. As regards the former, the effective properties of such media are known exactly. They are the well-known [33] results, which for a material with TI phases, of the type considered here take the form: where the superscript \(L\) denotes a layered medium.

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).

In the limit \(\delta \rightarrow \infty \), the Hashin–Shtrikman bounds do 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: and therefore lower (upper) bounds follow from setting the comparison material equal to the smallest (largest) moduli, as described above. Therefore in this limit we are not as fortunate as the layered case where the bounds coincide. Many theories have been proposed to model the effective elastic properties of long FRC materials and a first check on the efficacy such theories is that the prediction sits within the bounds (7.4)–(7.8).

We now implement the construction above in various scenarios. Table 2 states the phase properties (taken from [25]). In particular, note that traditionally the Hashin–Shtrikman bounds are plotted as a function of volume fraction \(\phi =\phi _1\) of the inclusion phase, for a fixed aspect ratio \(\delta \) of inclusion. Here we not only plot this case, but also consider the bounds plotted as a function of \(\delta \) for a fixed volume fraction of inclusion. This was also done in [25] but note that here the \(\mathbf {P}\)-tensor is computed analytically, and therefore it is easy to plot such results for a wide range of \(\delta \) (we plot results for a logarithmic scale in \(\delta \)). Moreover, here we also study the dependence of the bounds on the distribution aspect ratio (\(\epsilon \)). Such plots are very informative particularly since the Reuss–Voigt bounds are independent of \(\delta \) and \(\epsilon \) and so they cannot give this information.

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.