Hashin–Shtrikman bounds of periodic linear elastic media with cubic symmetry

Abstract

Based on the Hashin–Shtrikman variational principle, novel bounds on the effective shear moduli of a two-phase periodic composite are derived. The composite constituents are assumed to be isotropic, while the microstructure is assumed to exhibit cubic symmetry. A solution of the subsidiary boundary value problem is expressed as a double contraction of a fourth-order cubic tensor with the applied macroscopic strain. The bounds for cubic shear moduli are new, while the bounds for the bulk modulus are equal to the classical ones. The new bounds are verified for composites with the cubic, frame, octet and cubic + octet structures. It is shown that they are nearly attained for the cubic, octet and cubic + octet structures.

Keywords

Hashin–Shtrikman bounds cubic symmetry cubic shear moduli

1. Introduction

Hashin–Shtrikman bounds are one of the cornerstones of homogenization theory of composite materials. They are based on the Hashin–Shtrikman variational principle [1] and are obtained by evaluating the strain energy of the perturbation displacement field. To achieve this, certain simplifying assumptions are invoked, usually that the material is statistically homogeneous and isotropic. In this way, the original Hashin–Shtrikman bounds [2] were obtained and further extensively elaborated by the works of Walpole [3], Willis [4] and others. The original classical Hashin–Shtrikman bounds are bounds on isotropic moduli and are thus also called isotropic bounds.

First steps to the extension to statistically transversely isotropic composites were made by Hill [5] and Hashin [6] and later completed by Willis [7], Rodríguez-Ramos et al. [8] and Parnell and Calvo-Jurado [9]. The paper of Parnell and Calvo-Jurado [9] contains a relevant discussion of the perspective evolution of the Hashin–Shtrikman bounds. For the development of Hashin–Shtrikman bounds for polycrystalline materials, see Fernández and Böhlke [10] and references therein. The main step in extending the isotropic bounds to the anisotropic ones is to solve the corresponding subsidiary boundary value problem for the equivalent homogeneous solid. This is formally done by introducing [8, 9] the Hill tensor or so-called P tensor, which is closely related to the Eshelby tensor. Since an explicit expression for the Hill tensor is known only for special geometrical shapes, this approach is restricted in the literature only to examples with ellipsoidal inclusions.

An alternative to the assumption of statistical homogeneity used in the previously cited literature is the assumption of the periodicity of microstructures. Since a symmetry group of a material with periodic microstructure is not the whole orthogonal group, its effective tensor is, in general, not isotropic. In this paper, we consider the case where the unit cell has geometrical and material cubic symmetry. Then the effective tensor is cubic and has three moduli: the bulk modulus and two shear moduli that need their own bounds.

Assuming cubic symmetry of the unit cell, the Hill tensor can be expressed using the Fourier series [11]; hence, in conjunction with a piecewise constant representation of the polarization tensor, the action of the Hill tensor can be evaluated in a closed form. This, using eigenstrains instead of polarization, was utilized in Mejak [12] to find closed-form approximations of effective properties of certain periodic structures with cubic symmetry. Although it was noted that, in the case of the isotropy of the effective elasticity tensor, the approximation coincides with one of the Hashin–Shtrikman bounds, a true nature of the approximation was not investigated. This is rectified in this paper. Using the Hashin–Shtrikman variational principle for problems with periodic boundary conditions, it is shown that a closed-form action of the Hill tensor on piecewise constant polarization tensors gives bounds on effective cubic moduli of the composite. While the bounds on the bulk modulus are well known – they coincide with the classical Hashin–Shtrikman bounds – the bounds on the cubic shear moduli are new. The new bounds on shear moduli are structure-dependent; they rest on a function $β$ that depends on Fourier coefficients of the characteristic functions of the material phases. Based on these bounds, new structure-independent bounds are derived by the estimation of the structural function $β$ .

The paper is organized as follows. In Section 2, Hashin–Shtrikman variational principle in a form suitable for our purposes is presented. First, a subsidiary boundary value problem is formulated. Its solution is given by the Hill operator. The action of the Hill operator on piecewise constant polarization tensors is represented by the Hill tensor, whose Fourier series representation is given. Section 3 is devoted to derivation of the new Hashin–Shtrikman bounds. The admissible variation of the polarization tensor is restricted in a usual way to the space of piecewise constant tensors that are nonzero on only one of the phases. Then, exploiting isotropy of material phases, periodicity and cubic geometric symmetry of the problem, the Hill tensor in a closed form is derived. It is shown that the Hill tensor is a function of Fourier coefficients of the structure. Putting the Hill tensor into variational bounds, the bounds on effective cubic moduli are derived; a well-known bound for the bulk modulus and two new bounds for cubic shear moduli. Since derivation of the Fourier coefficients for certain structures is rather cumbersome, structure-independent bonds are also derived. Bounds on the Zener ratio are given, too. In Section 4, the bounds are compared with the effective moduli of four cubic structures: cubic, frame, octet and cubic + octet. It is observed that the bounds on shear moduli can nearly be achieved and it is thus plausible that they are the best possible bounds. The paper is concluded by final remarks in Section 5. Two appendices are given. In the first, the Hashin–Shtrikman variational principle for problems with periodic boundary conditions is derived. In the second, cubic symmetry of the Hill operator and the Hill tensor is proved.

1.1. Notation

Throughout the paper, compact tensorial notation is used. Vectors are denoted by an underlined lower letter $(\underline{a})$ , second-order tensors by a double underlined lower letter $(\underline{\underline{a}})$ and fourth-order tensors by a double underlined capital letter $(\underline{\underline{A}})$ . The second-order identity tensor and the fourth-order symmetric identity tensor are denoted $\underline{\underline{i}}$ and $\underline{\underline{I}}$ , respectively. A double contraction is denoted by a semicolon. Thus, in the index notation ${(\underline{\underline{A}} : \underline{\underline{a}})}_{ij} = A_{ijkl} a_{kl}$ . Symmetrization of a tensor $\underline{\underline{A}}$ with respect to the pair of indices $i, j$ is denoted ${sym}_{ij} \underline{\underline{A}}$ . Function spaces of tensors are denoted by underlined bold letters and spaces with periodic functions by the subscript $#$ . For example, ${\underline{\underline{l}}}_{#}^{2} (Y)$ is a space of Y-periodic second-order tensors defined on Y that are square integrable.

2. Variational principle

Let Y be a unit cell of a two-phase composite with periodic microstructure. Its material phases are denoted Y₁ and Y₂. It is assumed that Y is an open cube in $ℝ^{3}$ and that Y₁ and Y₂ are disjoint open subsets of Y. Then $\bar{Y} = {\bar{Y}}_{1} \cup {\bar{Y}}_{2}$ and $Y_{1} \cap Y_{2} = \emptyset$ . Here, the bar over a set denotes its closure. Since cubic symmetry is considered in the paper, it is assumed that Y has cubic symmetry, that is, $QY = Y$ for any octahedral rotation Q. Restrictions of the elasticity tensor $\underline{\underline{C}}$ on $Y_{i}$ , $i = 1, 2$ , are denoted ${\underline{\underline{C}}}_{i}$ . Composites usually consist of one stiff and one soft phase. Accordingly, we assume that ${\underline{\underline{C}}}_{i}$ are well ordered; ${\underline{\underline{C}}}_{2} \geq {\underline{\underline{C}}}_{1} > 0$ . The assumption allows us to bound uniformly all effective moduli. A general case where not all eigenvalues of ${\underline{\underline{C}}}_{2}$ are greater or equal to the corresponding eigenvalues of ${\underline{\underline{C}}}_{1}$ is not considered in this paper. In the following phases, Y₁ and Y₂ are used interchangeably. To simplify presentation, a symbol $i +$ with the meaning $1 + = 2$ and $2 + = 1$ is introduced.

The bounds rest on the Hashin–Shtrikman variational principle [11]; see Berdichevsky [13] for a modern exposition. In the literature, the principle is usually stated for problems with homogeneous boundary conditions. Since in this paper effective properties of periodic microstructures are studied, we need the Hashin–Shtrikman variational principle for problems with periodic boundary conditions. To make the paper self-contained, we give its proof in Appendix A.

To formulate the principle, a subsidiary boundary value problem is associated: for any Y-periodic symmetric second-order tensor $\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y)$ with the support in $Y_{i +}$ , find $\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)$ such that

\begin{matrix} div ({\underline{\underline{C}}}_{i} : \underline{\underline{e}} (\underline{u}) + \underline{\underline{p}}) = \underline{0} in Y, \\ {〈 \underline{\underline{e}} (\underline{u}) 〉}_{Y} = \underline{\underline{0}}, \\ 〚 ({\underline{\underline{C}}}_{i} : \underline{\underline{e}} (\underline{u}) + \underline{\underline{p}}) \cdot \underline{n} 〛 = \underline{0} on Γ = \partial Y_{i +} \ \partial Y . \end{matrix}

(1)

Here, ${\underline{h}}_{# 0}^{1} (Y)$ is a Sobolev space of the first order, consisting of Y-periodic displacements $\underline{u}$ with zero average strains $\underline{\underline{e}} (\underline{u}) = (\nabla \underline{u} + \nabla {\underline{u}}^{T}) ∕ 2$ over Y, ${〈 • 〉}_{Y}$ is the average of • over Y and ⟦•⟧ is the jump of its argument over $Γ$ . Note that equation (1) has the variational formulation

\int_{Y} (\underline{\underline{e}} (\underline{u}) : {\underline{\underline{C}}}_{i} + \underline{\underline{p}}) : \underline{\underline{e}} (\underline{w}) d \underline{y} = 0 for all \underline{w} \in {\underline{h}}_{# 0}^{1} (Y) .

(2)

Tensor $\underline{\underline{p}}$ is called the polarization tensor. The problem is linear in $\underline{\underline{p}}$ and has a unique solution [14, 15] $\underline{u}$ . Therefore, its deformation tensor $\underline{\underline{e}} (\underline{u})$ is a bounded linear function of $\underline{\underline{p}}$ , denoted $\underline{\underline{e}} (\underline{u}) = P_{i} \underline{\underline{p}}$ . Note that the subscript i of $P$ refers to the fact that ${\underline{\underline{C}}}_{i}$ is used (equation (1)). Operator $P_{i}$ is called the Hill operator. It is shown in Appendix B that $P_{i}$ inherits the intersection of geometrical symmetry of Y and material symmetry of ${\underline{\underline{C}}}_{i}$ .

It is well known, see for example Nemat-Nasser and Hori [11], that $P_{i}$ has a form

P_{i} \underline{\underline{p}} (\underline{y}) = \sum_{\underline{m} \neq \underline{0}} {\underline{\underline{K}}}_{i} (\underline{m}) : \underline{\underline{c}} (\underline{\underline{p}}; \underline{m}) e^{i 2 π \underline{m} \cdot \underline{y}} {| Y |}^{- 1 ∕ 3},

(3)

where $\underline{m} \in ℤ^{3}$ ,

{\underline{\underline{K}}}_{i} (\underline{m}) = - {sym}_{12} {sym}_{34} \underline{m} \otimes {(\underline{m} \cdot {\underline{\underline{C}}}_{i} \cdot \underline{m})}^{- 1} \otimes \underline{m},

(4)

and $\underline{\underline{c}} (\underline{\underline{p}}; \underline{m})$ is the $\underline{m}$ th Fourier coefficient of $\underline{\underline{p}}$ .

If $\underline{\underline{p}}$ is a piecewise constant function, zero in $Y_{i}$ and constant ${\underline{\underline{p}}}_{i}$ in $Y_{i +}$ , then

\underline{\underline{c}} (\underline{\underline{p}}; \underline{m}) = \frac{1}{| Y |} \int_{Y_{i} +} e^{- i 2 π \underline{m} \cdot \underline{y}} {| Y |}^{- 1 ∕ 3} d \underline{y} {\underline{\underline{p}}}_{i} = c_{i +} (\underline{m}) {\underline{\underline{p}}}_{i} .

(5)

Here, $c_{i +} (\underline{m})$ is the $\underline{m}$ th Fourier coefficient of the characteristic function $χ_{i +}$ of $Y_{i +}$ with values 1 in $Y_{i +}$ and 0 in $Y_{i}$ . Using equations (3) to (5), it follows that for such $\underline{\underline{p}}$

P_{i} \underline{\underline{p}} = \sum_{\underline{m} \neq \underline{0}} {\underline{\underline{K}}}_{i} (\underline{m}) c_{i +} (\underline{m}) e^{i 2 π \underline{m} \cdot \underline{y}} {| Y |}^{- 1 ∕ 3} : {\underline{\underline{p}}}_{i} = : {\underline{\underline{P}}}_{i} : {\underline{\underline{p}}}_{i} .

(6)

Tensor ${\underline{\underline{P}}}_{i}$ is called the Hill tensor. In the following, by the abuse of notation, a polarization tensor with a constant value ${\underline{\underline{p}}}_{i}$ on $Y_{i +}$ and a zero value on $Y_{i}$ is denoted ${\underline{\underline{p}}}_{i}$ . Action of the Hill operator on ${\underline{\underline{p}}}_{i}$ is thus expressed in equation (6) as a double contraction of the Hill tensor with the constant tensor ${\underline{\underline{p}}}_{i}$ .

Since ${\underline{\underline{p}}}_{i}$ is nonzero only in one material phase, the problem of equation (1) is often referred to as the inclusion problem. Frequently [12, 16], instead of the polarization tensor, the inclusion problem is stated using the eigenstrain tensor $\underline{\underline{ϵ}} *$ , defined as $\underline{\underline{ϵ}} * = - {\underline{\underline{C}}}_{i}^{- 1} : {\underline{\underline{p}}}_{i}$ . Then, instead of the Hill operator and tensor, Eshelby’s operator $S_{i} = P_{i} \circ {\underline{\underline{C}}}_{i}$ and tensor ${\underline{\underline{S}}}_{i} = {\underline{\underline{P}}}_{i} : {\underline{\underline{C}}}_{i}$ are used [17].

Let ${\underline{\underline{e}}}_{b}$ be an arbitrary homogeneous symmetric second-order tensor, and let us define functionals

J_{1} (\underline{\underline{p}}) = \frac{1}{| Y |} \int_{Y_{2}} (2 \underline{\underline{p}} : {\underline{\underline{e}}}_{b} - \underline{\underline{p}} : {({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} : \underline{\underline{p}} + \underline{\underline{p}} : P_{1} \underline{\underline{p}}) d \underline{y}

(7)

and

J_{2} (\underline{\underline{p}}) = \frac{1}{| Y |} \int_{Y_{1}} (- 2 \underline{\underline{p}} : {\underline{\underline{e}}}_{b} - \underline{\underline{p}} : {({\underline{\underline{C}}}_{1} - {\underline{\underline{C}}}_{2})}^{- 1} : \underline{\underline{p}} + \underline{\underline{p}} : P_{2} \underline{\underline{p}}) d \underline{y} .

(8)

The Hashin–Shtrikman variational principle, under the assumption ${\underline{\underline{C}}}_{2} \geq {\underline{\underline{C}}}_{1}$ , states that

\min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} \frac{1}{| Y |} \int_{Y} ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) : \underline{\underline{C}} : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) d \underline{y} = {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{1} : {\underline{\underline{e}}}_{b} + \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} J_{1} (\underline{\underline{p}}),

(9)

\min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} \frac{1}{| Y |} \int_{Y} ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) : \underline{\underline{C}} : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) d \underline{y} = {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{2} : {\underline{\underline{e}}}_{b} + \min_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{1})} J_{2} (\underline{\underline{p}}),

(10)

It is well known [13, 18] that the minimum of the strain energy in equations (9) and (10) is attained at the statically admissible stress field $\tilde{\underline{\underline{t}}} = \underline{\underline{C}} : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\tilde{\underline{u}}))$ , where $\tilde{\underline{u}}$ is the minimizer of the strain energy. Then, by Clapeyron’s theorem

\min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} \frac{1}{2 | Y |} \int_{Y} ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) : \underline{\underline{C}} : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) d \underline{y} = \frac{1}{2} {\underline{\underline{e}}}_{b} : {〈 \tilde{\underline{\underline{t}}} 〉}_{Y} = : \frac{1}{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{e} : {\underline{\underline{e}}}_{b} .

(11)

Here, ${\underline{\underline{C}}}_{e}$ is the effective elasticity tensor defined by ${〈 \tilde{\underline{\underline{t}}} 〉}_{Y} = {\underline{\underline{C}}}_{e} : {〈 {\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\tilde{\underline{u}}) 〉}_{Y} = {\underline{\underline{C}}}_{e} : {\underline{\underline{e}}}_{b}$ . It is a classical result [19 –21] that equation (11) defines the effective elasticity tensor, which is independent of the prescribed macroscopic boundary conditions for composites with periodic microstructure.

Using equation (11) in equations (9) and (10), it follows that

{\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{1} : {\underline{\underline{e}}}_{b} + J_{1} ({\underline{\underline{p}}}_{1}) \leq {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{e} : {\underline{\underline{e}}}_{b} \leq {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{2} : {\underline{\underline{e}}}_{b} + J_{2} ({\underline{\underline{p}}}_{2}),

(12)

for arbitrary ${\underline{\underline{p}}}_{1}, {\underline{\underline{p}}}_{2} \in {\underline{\underline{l}}}_{#}^{2} (Y)$ , such that the support of ${\underline{\underline{p}}}_{i}$ is within $Y_{i +}$ . The bounds of equation (12) are of special form of the bounds first obtained by Hashin and Shtrikman [2]. In particular, in our case, the comparison elasticity tensor is either ${\underline{\underline{C}}}_{1}$ or ${\underline{\underline{C}}}_{2}$ , while in the general case the comparison tensor can be an arbitrary isotropic tensor ${\underline{\underline{C}}}_{0}$ bounded from above or below by ${\underline{\underline{C}}}_{1}$ or ${\underline{\underline{C}}}_{2}$ , respectively.

3. Cubic Hashin–Shtrikman bounds

To compute the bounds (equation (12)), one needs to evaluate functionals $J_{i} ({\underline{\underline{p}}}_{i})$ ; this amounts to solving the inclusion problem (equation (1)) in a closed form. This is possible only under further assumptions that the comparison tensor is an isotropic tensor and that $\underline{\underline{p}}$ has a simple form. Hashin and Shtrikman [2], in the context of isotropic two-point statistics, assumed an isotropic distribution of phases $Y_{i}$ and, in accordance with this, restricted their variations of $\underline{\underline{p}} = \underline{\underline{p}} (\underline{y})$ only to isotropic functions, $\underline{\underline{p}} (\underline{\underline{q}} \cdot \underline{y}) = \underline{\underline{q}} \cdot \underline{\underline{p}} (\underline{y}) \cdot {\underline{\underline{q}}}^{T}$ for all $\underline{\underline{q}} \in O (3)$ . In this way, they obtained the well-known bounds on the effective isotropic moduli, bulk and shear moduli. Their assumption of the isotropic distribution of phases $Y_{i}$ made the composite isotropic and precluded them from obtaining bounds for anisotropic composites. However, as they did not restrict the comparison tensor to the elasticity tensor of one of the phases, they managed to obtain isotropic bounds on composites made of anisotropic phases; see Hashin and Shtrikman [22] for cubic polycrystals and Watt [23] and references therein for other symmetry classes.

Henceforth, it is assumed that ${\underline{\underline{C}}}_{i}$ , $i = 1, 2$ , are isotropic tensors. Note that in our paper the unit cell Y has a well-defined microstructure with cubic symmetry and is thus neither statistically homogeneous nor statistically isotropic. The elasticity moduli of ${\underline{\underline{C}}}_{i}$ , the bulk modulus, the shear modulus and the Poisson ratio are denoted $κ_{i}$ , $μ_{i}$ and $ν_{i}$ , respectively,

{\underline{\underline{C}}}_{i} = 2 μ_{i} (\frac{ν_{i}}{1 - 2 ν_{i}} \underline{\underline{i}} \otimes \underline{\underline{i}} + \underline{\underline{I}}) .

Assuming that ${\underline{\underline{p}}}_{1}$ is constant on $Y_{2}$ it follows from equations (6) and (7) that

J_{1} ({\underline{\underline{p}}}_{1}) = f_{2} (2 {\underline{\underline{p}}}_{1} : {\underline{\underline{e}}}_{b} - {\underline{\underline{p}}}_{1} : {({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} : {\underline{\underline{p}}}_{1} + {\underline{\underline{p}}}_{1} : {〈 {\underline{\underline{P}}}_{1} 〉}_{2} : {\underline{\underline{p}}}_{1}),

(13)

where $f_{2} = | Y_{2} | ∕ | Y |$ and ${〈 • 〉}_{2}$ is the average of • over Y₂ The functional is stationary at the solution of

{\underline{\underline{e}}}_{b} = ({({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} - {〈 {\underline{\underline{P}}}_{1} 〉}_{2}) : {\underline{\underline{p}}}_{1} .

(14)

In terms of the eigenstrain $\underline{\underline{ϵ}} * = - {\underline{\underline{C}}}_{1}^{- 1} : {\underline{\underline{p}}}_{1}$ , equation (14) is known as the equivalent eigenstrain equation [16, 24] and ${\underline{\underline{S}}}_{1} = - {\underline{\underline{P}}}_{1} : {\underline{\underline{C}}}_{1}$ is called the Eshelby tensor. Expressing a solution of equation (14) by ${\underline{\underline{p}}}_{1} = {\underline{\underline{Z}}}_{1} : {\underline{\underline{e}}}_{b}$ and putting it into equation (13), it follows that

f_{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{Z}}}_{1} : {\underline{\underline{e}}}_{b} \leq \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} J_{1} (\underline{\underline{p}}) .

(15)

Similarly,

- f_{1} {\underline{\underline{e}}}_{b} : {\underline{\underline{Z}}}_{2} : {\underline{\underline{e}}}_{b} \geq \min_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{1})} J_{2} (\underline{\underline{p}}),

(16)

where ${\underline{\underline{p}}}_{2} = {\underline{\underline{Z}}}_{2} : {\underline{\underline{e}}}_{b}$ is a solution of

{\underline{\underline{e}}}_{b} = ({({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} + {〈 {\underline{\underline{P}}}_{2} 〉}_{1}) : {\underline{\underline{p}}}_{2} .

(17)

Introducing equations (15) and (16) into equation (12), bounds on the effective tensor

{\underline{\underline{C}}}_{1} + f_{2} {\underline{\underline{Z}}}_{1} \leq {\underline{\underline{C}}}_{e} \leq {\underline{\underline{C}}}_{2} - f_{1} {\underline{\underline{Z}}}_{2}

(18)

are obtained.

To evaluate the bounds, closed forms of ${\underline{\underline{Z}}}_{i}$ and ${\bar{\underline{\underline{P}}}}_{i} = {〈 {\underline{\underline{P}}}_{i} 〉}_{i +}$ are needed. We begin with the derivation of a closed form of ${\bar{\underline{\underline{P}}}}_{i}$ . Once it is found, a closed form of ${\underline{\underline{Z}}}_{i}$ readily follows from equations (14) and (17).

Using equation (6), it follows that

{\bar{\underline{\underline{P}}}}_{i} = \frac{1}{f_{i +}} \sum_{\underline{m} \neq \underline{0}} {\underline{\underline{K}}}_{i} (\underline{m}) c_{i +} (\underline{m}) c_{i +} (- \underline{m}) .

(19)

As shown in Appendix B, tensor ${\bar{\underline{\underline{P}}}}_{i}$ is a cubic tensor and thus has three independent components. The easiest way to determine them is to evaluate its Cartesian components ${\bar{\underline{\underline{P}}}}_{i 1111} (= {\bar{\underline{\underline{P}}}}_{i 2222} = {\bar{\underline{\underline{P}}}}_{i 3333})$ , ${\bar{\underline{\underline{P}}}}_{i 1122} (= {\bar{\underline{\underline{P}}}}_{i 1133} = {\bar{\underline{\underline{P}}}}_{i 2233})$ and ${\bar{\underline{\underline{P}}}}_{i 1212} (= {\bar{\underline{\underline{P}}}}_{i 1313} = {\bar{\underline{\underline{P}}}}_{i 2323})$ . To this end, we compute

\begin{matrix} {\hat{\underline{\underline{K}}}}_{1} (\underline{m}) = \frac{1}{3} ({\underline{\underline{K}}}_{i 1111} (\underline{m}) + {\underline{\underline{K}}}_{i 2222} (\underline{m}) + {\underline{\underline{K}}}_{i 3333} (\underline{m})), \\ {\hat{\underline{\underline{K}}}}_{2} (\underline{m}) = \frac{1}{3} ({\underline{\underline{K}}}_{i 1122} (\underline{m}) + {\underline{\underline{K}}}_{i 1133} (\underline{m}) + {\underline{\underline{K}}}_{i 2233} (\underline{m})), \\ {\hat{\underline{\underline{K}}}}_{3} (\underline{m}) = \frac{1}{3} ({\underline{\underline{K}}}_{i 1212} (\underline{m}) + {\underline{\underline{K}}}_{i 1313} (\underline{m}) + {\underline{\underline{K}}}_{i 2323} (\underline{m})) . \end{matrix}

(20)

It is at this step that the assumption of the isotropy of ${\underline{\underline{C}}}_{i}$ is needed in order to find a closed form of ${\bar{\underline{\underline{P}}}}_{i}$ . Indeed, if

{\underline{\underline{C}}}_{i} = 2 μ_{i} (\frac{ν_{i}}{1 - 2 ν_{i}} \underline{\underline{i}} \otimes \underline{\underline{i}} + \underline{\underline{I}}),

then equation (4) simplifies to

{\underline{\underline{K}}}_{i} (\underline{m}) = \frac{1}{2 μ_{i} (1 - ν_{i}) {| \underline{m} |}^{4}} \underline{m} \otimes \underline{m} \otimes \underline{m} \otimes \underline{m} - \frac{1}{μ_{i} {| \underline{m} |}^{2}} \underline{m} \hat{\otimes} \underline{\underline{i}} \hat{\otimes} \underline{m},

(21)

where $\hat{\otimes} is$ the symmetric dyadic product. Note that ${\underline{\underline{K}}}_{i} (\underline{m})$ is not a cubic tensor. Direct evaluation of equations (20) and (21) gives

\begin{matrix} {\hat{\underline{\underline{K}}}}_{1} (\underline{m}) = - \frac{1 - 2 ν_{i}}{6 μ_{i} (1 - ν_{i})} - \frac{b (\underline{m})}{3 μ_{i} (1 - ν_{i})}, \\ {\hat{\underline{\underline{K}}}}_{2} (\underline{m}) = \frac{b (\underline{m})}{6 μ_{i} (1 - ν_{i})}, \\ {\hat{\underline{\underline{K}}}}_{3} (\underline{m}) = - \frac{1}{6 μ_{i}} + \frac{b (\underline{m})}{6 μ_{i} (1 - ν_{i})}, \end{matrix}

(22)

where

b (\underline{m}) = \frac{m_{1}^{2} m_{2}^{2} + m_{1}^{2} m_{3}^{2} + m_{2}^{2} m_{3}^{2}}{{| \underline{m} |}^{4}} .

Now we put equation (22) into equation (19) and use Parseval’s identity, $\sum_{\underline{m} \neq \underline{0}} {| c_{i +} (\underline{m}) |}^{2} = f_{i +} (1 - f_{i +})$ . After some simplification, we get

\begin{matrix} {\bar{\underline{\underline{P}}}}_{i 1111} = - \frac{1}{3 μ_{i} (1 - ν_{i})} (\frac{1}{2} (1 - 2 ν_{i}) (1 - f_{i +}) + \frac{1}{f_{i +}} β (Y_{i +})), \\ {\bar{\underline{\underline{P}}}}_{i 1122} = \frac{1}{6 μ_{i} (1 - ν_{i}) f_{i +}} β (Y_{i +}), \\ {\bar{\underline{\underline{P}}}}_{i 1212} = - \frac{1}{6 μ_{i}} (1 - f_{i +} - \frac{1}{(1 - ν_{i}) f_{i +}} β (Y_{i +})), \end{matrix}

(23)

where

β (Y_{i +}) = \sum_{\underline{m} \neq \underline{0}} b (\underline{m}) {| c_{i +} (\underline{m}) |}^{2} = \sum_{\underline{m} \neq \underline{0}} \frac{m_{1}^{2} m_{2}^{2} + m_{1}^{2} m_{3}^{2} + m_{2}^{2} m_{3}^{2}}{{| \underline{m} |}^{4}} {| c_{i +} (\underline{m}) |}^{2} .

(24)

Since $0 < b (\underline{m}) \leq 1 ∕ 3$ for $\underline{m} \neq \underline{0}$ , the maximum is attained at $| m_{1} | = | m_{2} | = | m_{3} |$ , and we get by Parseval’s identity, that

0 < β (Y_{i +}) \leq \frac{1}{3} f_{i +} (1 - f_{i +}) .

(25)

Function $β (Y_{i})$ is similarly defined. However, noting that $0 = c_{i} (\underline{m}) + c_{+ i} (\underline{m})$ for $\underline{m} \neq \underline{0}$ , it follows from equation (24) that $β (Y_{i}) = β (Y_{i + 1})$ .

To proceed, let us recall that a cubic tensor has a unique spectral representation [26]. A cubic tensor $\underline{\underline{A}}$ is identified with a triple $a = (a_{1}, a_{2}, a_{3})$ of its eigenvalues

a_{1} = {\underline{\underline{A}}}_{1111} + 2 {\underline{\underline{A}}}_{1122}, a_{2} = {\underline{\underline{A}}}_{1111} - {\underline{\underline{A}}}_{1122}, a_{3} = 2 {\underline{\underline{A}}}_{1212} .

(26)

In particular, the identity $\underline{\underline{I}}$ is identified with $i = (1, 1, 1)$ and ${\underline{\underline{C}}}_{i} \equiv c_{i} = 2 μ_{i} ((1 + ν_{i}) ∕ (1 - 2 ν_{i}), 1, 1)$ . The double contraction is identified as $\underline{\underline{A}} : \underline{\underline{B}} \equiv a ⋆ b = (a_{1} b_{1}, a_{2} b_{2}, a_{3} b_{3})$ and, if $\underline{\underline{A}}$ is invertible, ${\underline{\underline{A}}}^{- 1} \equiv a^{- 1} = (1 ∕ a_{1}, 1 ∕ a_{2}, 1 ∕ a_{3})$ . The ring of symmetric cubic tensors with operations of addition and double contraction is thus isomorphic to $(ℝ^{3}, +, ⋆)$ . In the following, this identification will be used to evaluate the bounds (equation (18)).

Straightforward evaluation of equations (23) and (26) gives the eigenvalues of ${\bar{\underline{\underline{P}}}}_{i}$

{\bar{\underline{\underline{P}}}}_{i} \equiv p_{i} = (1 - f_{i +}) a_{i} + \frac{β (Y_{i +})}{f_{i +}} b_{i},

(27)

where

a_{i} = - \frac{1}{6 μ_{i}} (\frac{1 - 2 ν_{i}}{1 - ν_{i}}, \frac{1 - 2 ν_{i}}{1 - ν_{i}}, 2) and b_{i} = \frac{1}{6 μ_{i} (1 - ν_{i})} (0, - 3, 2) .

(28)

Using equations (14) and (17), it follows that

\begin{matrix} {\underline{\underline{Z}}}_{1} = {({({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} - {\bar{\underline{\underline{P}}}}_{1})}^{- 1} \equiv z_{1} = {({(c_{2} - c_{1})}^{- 1} - p_{1})}^{- 1}, \\ {\underline{\underline{Z}}}_{2} = {({({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} + {\bar{\underline{\underline{P}}}}_{2})}^{- 1} \equiv z_{2} = {({(c_{2} - c_{1})}^{- 1} + p_{2})}^{- 1} . \end{matrix}

(29)

Combining equation (18) with equations (27) and (29), we arrive at

c^{l} = c_{1} + f_{2} {({(c_{2} - c_{1})}^{- 1} - (1 - f_{2}) a_{1} - \frac{β (Y_{2})}{f_{2}} b_{1})}^{- 1}

(30)

and

c^{u} = c_{2} + f_{1} {({(c_{1} - c_{2})}^{- 1} - (1 - f_{1}) a_{2} - \frac{β (Y_{1})}{f_{1}} b_{2})}^{- 1} .

(31)

The Hashin–Shtrikman bounds for two-phase periodic composites with cubic symmetry

c^{l} \leq c^{eff} \leq c^{u}

(32)

are established. They depend on elasticity moduli of material phases and a function $β (Y_{i})$ , which is determined by microstructural geometry. Hashin–Shtrikman bounds are usually given in terms of elasticity moduli $κ_{i}$ and $μ_{i}$ . Using equation (28), it follows from equations (30) to (32), after some algebra, that the effective moduli $κ_{e}$ and $μ_{e}^{k}$ , $k = 1, 2$ have bounds

\begin{matrix} κ_{1} + \frac{f_{2}}{\frac{1}{κ_{2} - κ_{1}} + \frac{1 - f_{2}}{κ_{1} + 4 μ_{1} ∕ 3}} \leq κ_{e} \leq κ_{2} + \frac{f_{1}}{\frac{1}{κ_{1} - κ_{2}} + \frac{1 - f_{1}}{κ_{2} + 4 μ_{2} ∕ 3}}, \\ μ_{1} + \frac{f_{2}}{\frac{1}{μ_{2} - μ_{1}} + \frac{6 (1 - f_{2}) (κ_{1} + 2 μ_{1})}{5 μ_{1} (3 κ_{1} + 4 μ_{1})} + G_{1}^{k}} \leq μ_{e}^{k} \leq μ_{2} + \frac{f_{1}}{\frac{1}{μ_{1} - μ_{2}} + \frac{6 (1 - f_{1}) (κ_{2} + 2 μ_{2})}{5 μ_{2} (3 κ_{2} + 4 μ_{2})} + G_{2}^{k}} . \end{matrix}

(33)

Here, $G_{i}^{k}$ , $i, k \in {1, 2}$ are given by

\begin{matrix} G_{i}^{1} = - \frac{2 (3 κ_{i} + μ_{i}) (1 - f_{i +} - \frac{5 β (Y_{i})}{f_{i +}})}{5 μ_{i} (3 κ_{i} + 4 μ_{i})}, \\ G_{i}^{2} = - \frac{2}{3} G_{i}^{1} . \end{matrix}

(34)

Bounds on the effective bulk modulus are the same as the original Hashin–Shtrikman bounds, while bounds on the shear modulus are different. Let us denote them by $μ_{L}^{k}$ and $μ_{U}^{k}$ ,

\begin{matrix} μ_{L}^{k} = μ_{1} + \frac{f_{2}}{\frac{1}{μ_{2} - μ_{1}} + \frac{6 (1 - f_{2}) (κ_{1} + 2 μ_{1})}{5 μ_{1} (3 κ_{1} + 4 μ_{1})} + G_{1}^{k}}, \\ μ_{U}^{k} = μ_{2} + \frac{f_{1}}{\frac{1}{μ_{1} - μ_{2}} + \frac{6 (1 - f_{1}) (κ_{2} + 2 μ_{2})}{5 μ_{2} (3 κ_{2} + 4 μ_{2})} + G_{2}^{k}} . \end{matrix}

(35)

They have the same appearance but with an additional term $G_{i}^{k}$ in the denominators. The bounds are valid for all $μ_{1} \leq μ_{2}$ and $κ_{1} \leq κ_{2}$ . In the limits $(μ_{1} \to 0, κ_{1} \to 0)$ and $(μ_{2} \to \infty, κ_{2} \to \infty)$ , the lower bound and the upper bound are trivial, respectively. Functions $G_{i}^{k}$ depend on the structural function $β$ ; thus, the bounds are not universal as they depend on its structure. It is seen from equation (34) that for $β (Y_{i}) = f_{i} (1 - f_{i}) ∕ 5$ functions $G_{i}^{k}$ are identical equal to zero. In this case, the bounds of equation (35) coincide with the original Hashin–Shtrikman bounds, which we denote $μ_{L}^{0}$ and $μ_{U}^{0}$ . Note that $β (Y_{1}) = f_{1} (1 - f_{1}) ∕ 5$ and $β (Y_{2}) = f_{2} (1 - f_{2}) ∕ 5$ hold simultaneously. For $β (Y_{i}) > f_{i} (1 - f_{i}) ∕ 5$ , it follows that $G_{i}^{1} > 0$ and $G_{i}^{2} < 0$ . This means that, compared with the original Hashin–Shtrikman bounds, bounds on $μ_{e}^{1}$ are shifted downwards while bounds on $μ_{e}^{2}$ are shifted upwards. For $β (Y_{i}) < f_{i} (1 - f_{i}) ∕ 5$ , the shift is reversed; bounds on $μ_{e}^{1}$ are shifted upwards and bounds on $μ_{e}^{2}$ are shifted downwards. Therefore,

\begin{matrix} β (Y_{i}) \geq \frac{1}{5} f_{i} (1 - f_{i}) & \Rightarrow μ_{L}^{1} \leq μ_{L}^{0} \leq μ_{L}^{2} \leq μ_{e}^{2} and μ_{e}^{1} \leq μ_{U}^{1} \leq μ_{U}^{0} \leq μ_{U}^{2}, \\ β (Y_{i}) \leq \frac{1}{5} f_{i} (1 - f_{i}) & \Rightarrow μ_{e}^{2} \leq μ_{U}^{2} \leq μ_{U}^{0} \leq μ_{U}^{1} and μ_{L}^{2} \leq μ_{L}^{0} \leq μ_{L}^{1} \leq μ_{e}^{1} . \end{matrix}

(36)

The bounds of equation (36) exceed the original Hashin–Shtrikman bounds. This is not inconsistent, since in their derivation we assume that the effective solid is isotropic; in particular, we assume that the effective strain energy density is of the form

U = \frac{1}{2} (9 κ_{e} {(tr {\underline{\underline{e}}}_{b})}^{2} + 2 μ_{e} {\underline{\underline{e}}}_{b}^{'} : {\underline{\underline{e}}}_{b}^{'}),

(37)

where ${\underline{\underline{e}}}_{b}^{'}$ is the deviatoric part of the applied macroscopic strain ${\underline{\underline{e}}}_{b}$ .

The obtained bounds depend on the structural function $β (Y_{i})$ . Using equation (25), it is easy to obtain universal bounds. Of course, they are weaker but they usually circumvent a rather complex evaluation of the Fourier coefficients $c_{i} (\underline{m})$ . It follows from equation (25) that

\begin{matrix} - \frac{2 (3 κ_{i} + μ_{i}) (1 - 4 f_{i +})}{5 μ_{i} (3 κ_{i} + 4 μ_{i})} < G_{i}^{1} < \frac{4 (3 κ_{i} + μ_{i}) (1 - f_{i +})}{15 μ_{i} (3 κ_{i} + 4 μ_{i})}, \\ - \frac{8 (3 κ_{i} + μ_{i}) (1 - 4 f_{i +})}{45 μ_{i} (3 κ_{i} + 4 μ_{i})} < G_{i}^{2} < \frac{4 (3 κ_{i} + μ_{i}) (1 - f_{i +})}{15 μ_{i} (3 κ_{i} + 4 μ_{i})} . \end{matrix}

(38)

It is interesting that the upper bounds on $G_{i}^{k}$ are the same. Introducing equation (38) into equation (35), we obtain the universal bounds

\begin{matrix} {\tilde{μ}}_{L}^{1} = μ_{1} + \frac{f_{2}}{\frac{1}{μ_{2} - μ_{1}} + \frac{2 (1 - f_{2})}{3 μ_{1}}} \leq μ_{e}^{1} \leq μ_{2} + \frac{f_{1}}{\frac{1}{μ_{1} - μ_{2}} + \frac{2 (1 - f_{1})}{3 κ_{2} + 4 μ_{2}}} = {\tilde{μ}}_{U}^{1}, \\ {\tilde{μ}}_{L}^{2} = μ_{1} + \frac{f_{2}}{\frac{1}{μ_{2} - μ_{1}} + \frac{2 (1 - f_{2})}{3 μ_{1}}} \leq μ_{e}^{2} \leq μ_{2} + \frac{f_{1}}{\frac{1}{μ_{1} - μ_{2}} - \frac{2 (f_{1} - 1) (3 κ_{2} + 10 μ_{2})}{9 μ_{2} (3 κ_{2} + 4 μ_{2})}} = {\tilde{μ}}_{U}^{2} . \end{matrix}

(39)

Note that ${\tilde{μ}}_{L}^{1} = {\tilde{μ}}_{L}^{2}$ .

Anisotropy of cubic materials is characterized by the Zener ratio $A_{e} = μ_{e}^{2} ∕ μ_{e}^{1}$ . Using equations (35) and (39), bounds

{\tilde{A}}_{L} = \frac{{\tilde{μ}}_{L}^{2}}{{\tilde{μ}}_{U}^{1}} \leq A_{L} = \frac{μ_{L}^{2}}{μ_{U}^{1}} \leq A_{e} \leq A_{U} = \frac{μ_{U}^{2}}{μ_{L}^{1}} \leq {\tilde{A}}_{U} = \frac{{\tilde{μ}}_{U}^{2}}{{\tilde{μ}}_{L}^{1}}

(40)

can be established. However, they are rather conservative. We shall see in the next section that the bounds of equation (35) are attainable for certain structures. In these examples, the bounds are attained simultaneously for both shear moduli, either as the lower or upper bounds. Then $μ_{L}^{2} ∕ μ_{L}^{1}$ or $μ_{U}^{2} ∕ μ_{U}^{1}$ is a good estimate of $A_{e}$ . Conversely, the bounds of equation (40) are weak.

Figures 1 and 2 illustrate the bounds for $β (Y_{i}) = f_{i} (1 - f_{i}) ∕ 4$ and for $β (Y_{i}) = f_{i} (1 - f_{i}) ∕ 8$ for material parameters $κ_{2} = 5 κ_{1}$ , $μ_{2} = 10 μ_{1}$ and $ν_{1} = 1 ∕ 4$ . Note that the inequalities of equation (36) are confirmed.

Figure 1.

New Hashin–Shtrikman bounds of $μ_{e}^{k}$ . Dashed curves: the original Hashin–Shtrikman bounds, dot-dashed curves: universal bounds ${\tilde{μ}}_{L,U}^{k}$ , solid curves: structure-dependent bounds $μ_{L,U}^{k}$ . Left $β (Y_{i}) = f_{i} (1 - f_{i}) ∕ 4$ , right $β (Y_{i}) = f_{i} (1 - f_{i}) ∕ 8$ .

Figure 2.

Bounds on the Zener ratio. Solid curves: bounds $A_{L}$ and $A_{U}$ , dashed curves ${\tilde{A}}_{L}$ and ${\tilde{A}}_{U}$ . Left $β (Y_{i}) = f_{i} (1 - f_{i}) ∕ 4$ , right $β (Y_{i}) = f_{i} (1 - f_{i}) ∕ 8$ .

4. Examples

The validity of the bounds of equation (35) is demonstrated for four periodic microstructures: cubic, frame, octet and cubic + octet. Their unit cells are shown schematically in Figure 3. For a comprehensive description of the octet and cubic + octet structure, we refer to Mejak [12]. Note that the inclusions in the octet unit cell are all congruent, while the congruency set of inclusions of the cubic + octet unit cells has two simplices.

Figure 3.

Unit cells of cubic, frame, octet and cubic + octet microstructures.

The Fourier coefficients (equation (5)) of the cubic and frame structure can be readily evaluated. For the cubic inclusion, they are

c_{1} (\underline{m}) = \frac{sin (m_{1} δπ) sin (m_{2} δπ) sin (m_{3} δπ)}{π^{3} m_{1} m_{2} m_{3}} .

(41)

Here, $δ$ is the ratio of the length of the side of the embedded cube to the length of the side of the unit cell cube ${[- π, π]}^{3}$ . The volume ratio of the cubic inclusion is $f_{1} = δ^{3}$ . Since $c_{1} (\underline{m})$ is a function of $δ$ and hence of f₁, $β (Y_{1})$ is expressed as a function of f₁, $β (Y_{1}) = β (f_{1})$ .

Fourier coefficients for the frame structure are

\begin{matrix} c_{1} (m_{1} \neq 0, m_{2} \neq 0, m_{3} \neq 0) = - \frac{2 sin (m_{1} δπ) sin (m_{2} δπ) sin (m_{3} δπ)}{π^{3} m_{1} m_{2} m_{3}}, \\ c_{1} (m_{1} \neq 0, m_{2} \neq 0, 0) = \frac{(3 - 2 δ) sin (m_{1} δπ) sin (m_{2} δπ)}{π^{2} m_{1} m_{2}}, \end{matrix}

(42)

where $δ$ is the ratio of the thickness of the frame beam to the length of the side of the unit cell. Values of the coefficients (equations (41) and (42)) for $m_{j} = 0$ are evaluated as the limit values $m_{j} \to 0$ . The volume ratio of the frame structure is $f_{1} = (3 - 2 δ) δ^{2}$ . Again, $β (Y_{1}) = β (f_{1})$ . Fourier coefficients for the octet and cubic + octet structures are much more involved and are given in Mejak [12].

With given values of the Fourier coefficients, functions $β (Y_{1}) = : β (f_{1})$ are computed by the numerical summation of equation (24). Their values are tabulated and approximated by the best fit of a form $\hat{β} (f_{1}) = f_{1} (1 - f_{1}) \sum_{j = 0}^{5} a_{j} f_{1}^{j}$ . Values of the coefficients $a_{j}$ are given in Table 1. For the frame structure relation, $β (f_{1}) = β (1 - f_{1}) = β (f_{2})$ holds. Graphs of $β (f)$ are shown in Figure 4. In this paper, only the cubic + octet structure with the ratio $3 \sqrt{3} ∕ 8$ of the thicknesses of the octet to the cubic walls is considered. As first demonstrated by Berger et al. [27], the effective elasticity tensor of the cubic + octet foam with this ratio is isotropic. Figure 4 shows that $β (f)$ of the cubic + octet structure is also almost equal to $f (1 - f) ∕ 5$ for all values of f and not only in the limiting case. Therefore, the shear moduli $μ_{e}^{k}$ are almost equal and thus ${\underline{\underline{C}}}_{e}$ is nearly isotropic. This generalizes the results of Berger et al. [27] to composites with the cubic + octet microstructure. Note in Figure 4 that $β (f) \leq f (1 - f) ∕ 5$ for the cubic and frame structures and $β (f) \geq f (1 - f) ∕ 5$ for the octet structure. It is interesting that for the octet structure $β (f)$ is very close to $f (1 - f) ∕ 3$ for $f > 0.8$ .

Table 1.

Coefficients of $\hat{β}$

Structure	$a_{0}$	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$
Cubic	0.1656	−0.5242	0.8888	−1.0275	0.7029	−0.2062
Frame	0.1004	−0.2293	0.4514	−0.4440	0.2220	0.0000
Octet	0.2195	−0.0343	1.3713	−3.2377	3.1061	−1.0962
Cubic + octet	0.1817	0.1504	−0.2925	0.1240	0.1545	−0.1211

Figure 4.

Plots of $β (f)$ for cubic (C), frame (F), octet (O) and cubic + octet (I) structures. The dotted curves are s(1–f) ∕3 and f (1–f) ∕5.

After obtaining the bounds, the question is how accurate they are. This issue is addressed in Figures 5 to 7 by comparison of the bounds with the computed values of the effective shear moduli. The moduli are computed by a standard numerical homogenization. The unit cell problem is solved for prescribed macroscopic strains by the finite-element method and the effective moduli are then determined by the linear relation between the averages of the computed stresses and the prescribed macroscopic strains. For computational details, see, for example Bornert et al. [28] or Mejak [29]. Values of the computed moduli are denoted by plot marks in Figures 5 to 7. Results are given for the glass epoxy composite with $E_{2} = 70$ GPa, $ν_{2} = 0.2$ , $E_{1} = 3$ GPa, $ν_{1} = 0.35$ . The corresponding ratios of the phase moduli are $κ_{2} ∕ κ_{1} = 11.67$ and $μ_{2} ∕ μ_{1} = 26.25$ .

Figure 5.

Comparison of the bounds $μ_{L,U}^{k}$ and ${\tilde{μ}}_{L,U}^{k}$ , with the exact values $μ_{e}^{1} (□)$ and $μ_{e}^{2} (♢)$ , with respect to the concentration ratio $f_{1}$ . Left: soft cubical inclusion with phase $Y_{1}$ , right: stiff cubical inclusion with phase $Y_{2}$ .

Figure 6.

Comparison of the bounds with computed vales of the effective shear moduli $μ_{e}^{1} (□)$ , $μ_{e}^{2} (♢)$ and Zener ratio $A (□)$ , with respect to the concentration ratio $f_{1}$ for the frame structure.

Figure 7.

Comparison of the bounds with computed vales of the effective shear moduli $μ_{e}^{1} (□)$ and $μ_{e}^{2} (♢)$ with respect to the concentration ratio $f_{1}$ for the octet and cubic + octet structures.

Figure 5 gives a comparison for the cubical inclusion. Two cases are considered; soft and stiff cubical inclusion. In the first case, the inclusion is phase Y₁ and in the second it is Y₂. Note that for the stiff inclusion $β (Y_{1}) = β (1 - f_{1})$ , where $β (f)$ is given in Table 1 and plotted in Figure 4. As can be seen, the new bounds $μ_{L,U}^{k}$ are nearly attained, the upper bounds for the soft inclusion and the lower bounds for the stiff inclusion. The match is particularly good for high concentrations $f_{1} > 0.8$ . Note in Figure 5 that ${\tilde{μ}}_{L}^{2} \approx μ_{L}^{2}$ and ${\tilde{μ}}_{U}^{1} \approx μ_{U}^{1}$ and thus the universal bounds are tight bounds. Figure 6 gives results for the frame structure. The effective moduli obey the bounds but they are not achieved except at very high or very low concentration ratios. Note, however, that, curiously, $μ_{e}^{1} \approx μ_{U}^{0}$ for all values of f₁. By contrast with the cubical inclusion, where both bounds are attained simultaneously, either as the lower or upper bounds, bounds for the frame structure at f₁ close to 1 are attained oppositely; $μ_{e}^{1} \approx μ_{U}^{1}$ and $μ_{e}^{2} \approx μ_{L}^{2}$ . Therefore, the lower bound on the Zener ratio $A_{L}$ is a tight bound near $f_{1} = 1$ , as shown in Figure 6. Comparisons for the octet and cubic + octet structures are presented in Figure 7. For the octet structure, the walls are made of stiff material phase Y₂, while for the cubic + octet structure, the walls are soft. For the octet structure, the upper bounds give a very good approximation of the effective shear moduli for $f_{1} > 0.7$ . The agreement between the lower bounds and effective moduli is also very good for the cubic + octet structure. As already noted, $μ_{L}^{1} = μ_{L}^{2}$ and $μ_{U}^{1} = μ_{U}^{2}$ for $β (f) = f (1 - f) ∕ 5$ and hence the cubic + octet structure is indeed effectively isotropic. The agreement of the bounds with the effective moduli is universal; it does not hold only for the given values of phases moduli, but holds for all possible values, even for a void or a rigid material phase, see Mejak [12]. If the material phases are interchanged, with a soft inclusion within a strong matrix, the upper attained bounds become the lower attained bounds and vice versa. In particular, as first observed by Berger et al. [27], a thin walled cubic + octet foam achieves the upper Hashin–Shtrikman bounds and is thus capable of storing maximal strain energy. With regards to bounds, cubic, octet and cubic + octet structures are special; for a general cubic structure, one cannot expect that the bounds are nearly achieved. This confirms the frame structure in Figure 6.

5. Final remarks

The bounds in Figures 5 to 7 are quite wide. This is due to the large mismatch of the moduli of the phases. For smaller differences between phases, the bounds are narrower. It is of some interest whether the bounds on cubic moduli are narrower than the classical bounds. This depends in a rather complicated way on the material moduli, concentration f and a structural function $β (f)$ . In general, for $β (f) \neq f (1 - f) ∕ 5$ the cubic bounds are noticeably narrower for f over a substantial part of the interval $(0, 1)$ and only slightly worse in the remaining part of the interval.

Examples in Figures 5 to 7 confirm the bounds of equations (35), (39) and (40). For certain structures, they are nearly achieved and they can be used as closed-form approximations of the effective shear moduli. Their close agreement with the effective moduli suggests that they are the best obtainable bounds without knowing further geometrical details of phases. Since the bounds of equation (33) on the bulk modulus are equal to the classical Hashin–Shtrikman bounds, they are not considered in this paper. However, we remark that they are nearly achieved for the cubic, octet and cubic + octet structures. The reason that the bounds on the bulk modulus are equal to the classical ones can be inferred from Mejak [12]. Indeed, it can be shown that the restriction of the variation of $J_{i} (\underline{\underline{p}})$ (see equation (13)) to the space of octahedral invariant tensors $\underline{\underline{p}} (\underline{\underline{q}} \cdot \underline{y}) = \underline{\underline{p}} (\underline{y})$ , where $\underline{y} \in Y$ and $\underline{\underline{q}}$ is any element of the octahedral group, gives the classical bounds on the bulk modulus.

It is possible to obtain higher-order bounds by considering the variation of $J_{i} (\underline{\underline{p}})$ for a larger set of admissible tensors. In particular, instead of the constant approximation, a piecewise constant approximation can be considered. This results in improved bounds. However, in this case, the bounds depend on the way in which the piecewise constant approximation is constructed and thus the bounds are not universal. Another interesting possibility is to use the polynomial approximation.

Generalization to other symmetry classes is possible. The cubic algebra must be replaced with an appropriate algebra. All expressions become more complex and the bounds have a more involved form. Using a cuboid unit cell instead of the cubic unit cell, the rhombic symmetry must be used. In particular, we report that for the tetragonal symmetry the bounds involves six functions $β_{i} (f)$ , which are given as series with coefficients that are rational functions of $c_{i} (\underline{m})$ and m₁, m₂ and m₃.

Footnotes

A. Hashin–Shtrikman variational principle for periodic boundary conditions

The Hashin–Shtrikman variational principle for problems with homogeneous boundary conditions is well known. It is shown in this appendix that it can be generalized almost ad verbatim to the periodic boundary conditions.

Let ${\underline{\underline{e}}}_{b}$ be an arbitrary homogeneous symmetric second-order tensor. It is well known that the minimum of the average strain energy

(43)

F (\underline{u}) = \frac{1}{2 | Y |} \int_{Y} ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) : \underline{\underline{C}} : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) d \underline{y}

on ${\underline{h}}_{# 0}^{1} (Y)$ is attained at $\tilde{\underline{u}}$ , which is a solution of the stationary equation $\underline{0} = div \tilde{\underline{\underline{t}}}$ , where

(44)

\tilde{\underline{\underline{t}}} = \underline{\underline{C}} : (\underline{\underline{e}} (\tilde{\underline{u}}) + {\underline{\underline{e}}}_{b})

is the equilibrium stress field. Then, by Clapeyron’s theorem,

(45)

\begin{matrix} \min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} F (\underline{u}) = F (\tilde{\underline{u}}) = \frac{1}{2 | Y |} {\underline{\underline{e}}}_{b} : \int_{Y} \underline{\underline{C}} : (\underline{\underline{e}} (\tilde{\underline{u}}) + {\underline{\underline{e}}}_{b}) dy \\ = \frac{1}{2} {\underline{\underline{e}}}_{b} : {〈 \tilde{\underline{\underline{t}}} 〉}_{Y} = \frac{1}{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{e} : {\underline{\underline{e}}}_{b} . \end{matrix}

Here, ${\underline{\underline{C}}}_{e}$ is the effective stress tensor defined by ${〈 \tilde{\underline{\underline{t}}} 〉}_{Y} = {\underline{\underline{C}}}_{e} : {〈 \underline{\underline{e}} (\tilde{\underline{u}}) + {\underline{\underline{e}}}_{b} 〉}_{Y} = {\underline{\underline{C}}}_{e} : {\underline{\underline{e}}}_{b}$ .

To obtain the Hashin–Shtrikman variational principle, $F (\underline{u})$ is rewritten as a maximization problem. First, we note that

(46)

F (\underline{u}) = \frac{1}{2 | Y |} \int_{Y} ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) : {\underline{\underline{C}}}_{1} : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) d \underline{y} + \frac{1}{2 | Y |} \int_{Y_{2}} ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) : ({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1}) : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) d \underline{y} .

Owing to the periodicity of $\underline{u}$ , it follows that the first integral in equation (46) is equal to

(47)

\frac{1}{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{1} : {\underline{\underline{e}}}_{b} + \frac{1}{2 | Y |} \int_{Y} \underline{\underline{e}} (\underline{u}) : {\underline{\underline{C}}}_{1} : \underline{\underline{e}} (\underline{u}) d \underline{y} .

The integrand of the second integral in equation (46) is transformed using the Young–Fenchel transformation [13]

(48)

\frac{1}{2} ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) : ({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1}) : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) = \max_{\underline{\underline{p}}} (\underline{\underline{p}} : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) - \frac{1}{2} \underline{\underline{p}} : {({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} : \underline{\underline{p}}) .

Note that, in equation (48), the assumption that ${\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1}$ is positive definite is required. The Young–Fenchel transformation is applied for all $\underline{y} \in Y_{2}$ ; thus, $\underline{\underline{p}}$ is considered a function of $\underline{y} \in Y_{2}$ . Note that ${\underline{\underline{l}}}_{#}^{2} (Y_{2}) = {\underline{\underline{l}}}^{2} (Y_{2})$ if $Y_{2} \subset Y$ . Since the solution of the maximization problem is $\underline{\underline{p}} = ({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1}) : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u}))$ , a space of admissible functions $\underline{\underline{p}} (\underline{y})$ can be restricted to ${\underline{\underline{l}}}_{#}^{2} (Y_{2})$ . Using equation (48) in the second integral in equation (46), it is easy to show that the maximum can be moved in front of the integral. Then, combining this with equations (45) to (47), we arrive at

(49)

\frac{1}{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{e} : {\underline{\underline{e}}}_{b} = \frac{1}{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{1} : {\underline{\underline{e}}}_{b} + \frac{1}{| Y |} \min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} I (\underline{u}, \underline{\underline{p}}),

where

(50)

I (\underline{u}, \underline{\underline{p}}) = \frac{1}{2} \int_{Y} \underline{\underline{e}} (\underline{u}) : {\underline{\underline{C}}}_{1} : \underline{\underline{e}} (\underline{u}) d \underline{y} + \int_{Y_{2}} (\underline{\underline{p}} : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) - \frac{1}{2} \underline{\underline{p}} : {({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} : \underline{\underline{p}}) d \underline{y} .

To proceed, we would like to change the order of minimization and maximization in equation (49). It is easy to show that

(51)

\min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} I (\underline{u}, \underline{\underline{p}}) \geq \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} \min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} I (\underline{u}, \underline{\underline{p}}) .

To reverse the inequality, one can refer to the minimax theorem [30]. However, to make the paper self-contained, we prove this next by elementary means.

To this end, let us consider for an arbitrary $\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})$ a minimization problem

(52)

\min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} G (\underline{u}),

where

(53)

G (\underline{u}) = \frac{1}{2} \int_{Y} \underline{\underline{e}} (\underline{u}) : {\underline{\underline{C}}}_{1} : \underline{\underline{e}} (\underline{u}) d \underline{y} + \int_{Y_{2}} \underline{\underline{p}} : \underline{\underline{e}} (\underline{u}) d \underline{y} .

The problem is equivalent to the boundary value problem: find $\hat{\underline{u}} \in {\underline{h}}_{# 0}^{1} (Y)$ such that

(54)

\begin{matrix} div ({\underline{\underline{C}}}_{1} : \underline{\underline{e}} (\hat{\underline{u}}) + \underline{\underline{p}}) = \underline{0} in Y, \\ 〚 ({\underline{\underline{C}}}_{1} : \underline{\underline{e}} (\hat{\underline{u}}) + \underline{\underline{p}}) \cdot \underline{n} 〛 = \underline{0} on Γ = \partial Y_{2} \ \partial Y . \end{matrix}

Then the minimum of equation (52) is achieved at $\hat{\underline{u}}$ , and by a general form¹ of the Clapeyron’s theorem, the minimum is

(55)

G (\hat{\underline{u}}) = \frac{1}{2} \int_{Y_{2}} \underline{\underline{p}} : \underline{\underline{e}} (\hat{\underline{u}}) d \underline{y} = - \frac{1}{2} \int_{Y} \underline{\underline{e}} (\hat{\underline{u}}) : {\underline{\underline{C}}}_{1} : \underline{\underline{e}} (\hat{\underline{u}}) d \underline{y} .

Solution $\hat{\underline{u}}$ of equation (54) depends linearly on $\underline{\underline{p}}$ . Hence, $\underline{\underline{e}} (\hat{\underline{u}}) = P_{1} \underline{\underline{p}}$ , where $P_{1}$ is a linear operator from ${\underline{\underline{l}}}_{#}^{2} (Y_{2})$ to ${\underline{\underline{l}}}_{#}^{2} (Y)$ . Summarizing equations (49) to (53) and equation (55), we arrive at

(56)

\frac{1}{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{e} : {\underline{\underline{e}}}_{b} \geq \frac{1}{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{1} : {\underline{\underline{e}}}_{b} + \frac{1}{| Y |} \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} [\int_{Y_{2}} (\frac{1}{2} \underline{\underline{p}} : P_{1} \underline{\underline{p}} + \underline{\underline{p}} : {\underline{\underline{e}}}_{b} - \frac{1}{2} \underline{\underline{p}} : {({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} : \underline{\underline{p}}) d \underline{y}] .

To conclude the derivation of the Hashin–Shtrikman variational principle, it remains to show that the equality sign applies in equation (56).

First, we note that, according to Clapeyron’s theorem,

(57)

\max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} [\int_{Y_{2}} (\frac{1}{2} \underline{\underline{p}} : P_{1} \underline{\underline{p}} + \underline{\underline{p}} : {\underline{\underline{e}}}_{b} - \frac{1}{2} \underline{\underline{p}} : {({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} : \underline{\underline{p}}) d \underline{y}] = \frac{1}{2} \int_{Y_{2}} \hat{\underline{\underline{p}}} : {\underline{\underline{e}}}_{b} d \underline{y},

where $\hat{\underline{\underline{p}}}$ is a solution of the corresponding stationary equation

(58)

{\underline{\underline{e}}}_{b} + P_{1} \hat{\underline{\underline{p}}} = {({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} \hat{\underline{\underline{p}}} in Y_{2} .

Therefore,

(59)

\max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} \min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} I (\underline{u}, \underline{\underline{p}}) = \frac{1}{2} \int_{Y_{2}} \hat{\underline{\underline{p}}} : {\underline{\underline{e}}}_{b} d \underline{y} .

Let us now evaluate $\min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} I (\underline{u}, \underline{\underline{p}})$ . Using Clapeyron’s theorem again, we have

(60)

\max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} I (\underline{u}, \underline{\underline{p}}) = \frac{1}{2} \int_{Y} \underline{\underline{e}} (\underline{u}) : {\underline{\underline{C}}}_{1} : \underline{\underline{e}} (\underline{u}) d \underline{y} + \frac{1}{2} \int_{Y_{2}} \tilde{\underline{\underline{p}}} : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) d \underline{y},

where

(61)

{\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u}) = {({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} \tilde{\underline{\underline{p}}} in Y_{2} .

Evidently,

(62)

\min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} I (\underline{u}, \underline{\underline{p}}) \leq \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} I (\hat{\underline{u}}, \underline{\underline{p}}),

where $\underline{\underline{e}} (\hat{\underline{u}}) = P_{1} \hat{\underline{\underline{p}}}$ and $\hat{\underline{\underline{p}}}$ is a solution of the maximization problem (equation (57)). For such $\underline{u} = \hat{\underline{u}}$ , it follows from equations (58) and (61) that $\tilde{\underline{\underline{p}}} = \hat{\underline{\underline{p}}}$ and then, by equation (55),

(63)

\max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} I (\underline{u}, \underline{\underline{p}}) = - \frac{1}{2} \int_{Y} \hat{\underline{\underline{p}}} : P_{1} \hat{\underline{\underline{p}}} d \underline{y} + \frac{1}{2} \int_{Y_{2}} \hat{\underline{\underline{p}}} : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) d \underline{y} = \frac{1}{2} \int_{Y_{2}} \hat{\underline{\underline{p}}} : {\underline{\underline{e}}}_{b} d \underline{y} .

Using equations (59), (62) and (63), we obtain

(64)

\min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} I (\underline{u}, \underline{\underline{p}}) \leq \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} \min_{\underline{u} \in {\underline{h}}_{# 0}^{1} (Y)} I (\underline{u}, \underline{\underline{p}}) .

Therefore, by equation (51), the equality sign holds in equation (64) and hence also in equation (56); thus, the Hashin–Shtrikman variational principle

(65)

\frac{1}{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{e} : {\underline{\underline{e}}}_{b} = \frac{1}{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{1} : {\underline{\underline{e}}}_{b} + \frac{1}{| Y |} \max_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{2})} [\int_{Y_{2}} (\frac{1}{2} \underline{\underline{p}} : P_{1} \underline{\underline{p}} + \underline{\underline{p}} : {\underline{\underline{e}}}_{b} - \frac{1}{2} \underline{\underline{p}} : {({\underline{\underline{C}}}_{2} - {\underline{\underline{C}}}_{1})}^{- 1} : \underline{\underline{p}}) d \underline{y}]

is established.

Let us denote by ${\tilde{\underline{\underline{p}}}}_{1}$ extension of $\tilde{\underline{\underline{p}}}$ by zero to Y, ${\tilde{\underline{\underline{p}}}}_{1} = \underline{\underline{0}}$ on Y₁ and ${\tilde{\underline{\underline{p}}}}_{1} = \tilde{\underline{\underline{p}}}$ on Y₂. Then, combining equations (44) and (61), it follows that

(66)

\tilde{\underline{\underline{t}}} = {\underline{\underline{C}}}_{1} : (\underline{\underline{e}} (\tilde{\underline{u}}) + {\underline{\underline{e}}}_{b}) + {\tilde{\underline{\underline{p}}}}_{1} in Y .

Tensor ${\tilde{\underline{\underline{p}}}}_{1}$ is called the stress polarization tensor.

To prove a dual form of the Hashin–Shtrikman variational principle

(67)

\frac{1}{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{e} : {\underline{\underline{e}}}_{b} = \frac{1}{2} {\underline{\underline{e}}}_{b} : {\underline{\underline{C}}}_{2} : {\underline{\underline{e}}}_{b} + \frac{1}{| Y |} \min_{\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y_{1})} [\int_{Y_{1}} (\frac{1}{2} \underline{\underline{p}} : P_{2} \underline{\underline{p}} - \underline{\underline{p}} : {\underline{\underline{e}}}_{b} - \frac{1}{2} \underline{\underline{p}} : {({\underline{\underline{C}}}_{1} - {\underline{\underline{C}}}_{2})}^{- 1} : \underline{\underline{p}}) d \underline{y}],

one proceeds in a similar manner, where instead of equation (46) a decomposition

(68)

F (\underline{u}) = \frac{1}{2 | Y |} \int_{Y} ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) : {\underline{\underline{C}}}_{2} : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) d \underline{y} + \frac{1}{2 | Y |} \int_{Y_{1}} ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) : ({\underline{\underline{C}}}_{1} - {\underline{\underline{C}}}_{2}) : ({\underline{\underline{e}}}_{b} + \underline{\underline{e}} (\underline{u})) d \underline{y}

is used. Similarly to equation (66), we have

(69)

\tilde{\underline{\underline{t}}} = {\underline{\underline{C}}}_{2} : (\underline{\underline{e}} (\tilde{\underline{u}}) + {\underline{\underline{e}}}_{b}) + {\tilde{\underline{\underline{p}}}}_{2} in Y .

B. Symmetry of the Hill operator and the Hill tensor

In this appendix, it is shown that the Hill operator $P_{i}$ , tensor ${\underline{\underline{P}}}_{i}$ and ${\bar{\underline{\underline{P}}}}_{i} = {〈 {\underline{\underline{P}}}_{i} 〉}_{i +}$ have cubic symmetry. This will be proved using the variational formulation of the inclusion problem (equation (2)). An alternative approach is to use the Fourier representation (equation (3)).

Our approach will be slightly more general. To begin with, we first recall relevant definitions of symmetry. Let $G$ be a symmetry group. A set A is called $G$ -symmetric if $A = \underline{\underline{g}} (A) = {\underline{\underline{g}} \cdot \underline{a} | \underline{a} \in A}$ for all $\underline{\underline{g}} \in G$ . A tensor $\underline{\underline{A}}$ is called a $G$ -symmetric tensor if $\underline{\underline{g}} * \underline{\underline{A}} = \underline{\underline{A}}$ for all $\underline{\underline{g}} \in G$ . Here, $\underline{\underline{g}} * \underline{\underline{A}}$ is the Rayleigh product [31]. A tensorial function $\underline{\underline{A}} : \underline{a} \mapsto \underline{\underline{A}} (\underline{a})$ is called $G$ -symmetric if $\underline{\underline{A}} (\underline{\underline{g}} \cdot \underline{a}) = \underline{\underline{g}} * \underline{\underline{A}} (\underline{a})$ for all $\underline{\underline{g}} \in G$ . A tensorial operator $A : \underline{\underline{a}} (\cdot) \mapsto (A \underline{\underline{a}}) (\cdot) = \underline{\underline{A}} (\cdot)$ , which maps a tensorial function $\underline{\underline{a}} : \underline{a} \mapsto \underline{\underline{a}} (\underline{a})$ into a tensorial function $\underline{\underline{A}} : \underline{a} \mapsto \underline{\underline{A}} (\underline{a})$ is called $G$ -symmetric if

(70)

(A \underline{\underline{a}}) (\underline{\underline{g}} \cdot) = \underline{\underline{g}} * A ({\underline{\underline{g}}}^{- 1} * \underline{\underline{a}} (\underline{\underline{g}} \cdot)) (\cdot) .

If $G$ is a group of octahedral rotations $O$ , we call $\underline{\underline{A}}$ , $\underline{\underline{A}} (\cdot)$ and $A$ a cubic tensor, a cubic function and a cubic operator, respectively.

Let $G_{1} \subset O (3)$ be a geometrical symmetry group of the problem, that is $\underline{\underline{g}} (Y) = Y$ for all $\underline{\underline{g}} \in G_{1}$ and let $G_{2} \subset O (3)$ be a material symmetry group of the problem, $\underline{\underline{g}} * {\underline{\underline{C}}}_{i} = {\underline{\underline{C}}}_{i}$ for all $\underline{\underline{g}} \in G_{1}$ . Then, as shown next, the Hill operator $P_{i}$ is $G$ -symmetric for $G = G_{1} \cap G_{2}$ . For an arbitrary $\underline{\underline{g}} \in G$ , $\underline{v} \in {\underline{h}}_{# 0}^{1} (Y)$ and $\underline{\underline{p}} \in {\underline{\underline{l}}}_{#}^{2} (Y)$ , we define $\underline{v}' (\underline{y}') = {\underline{\underline{g}}}^{T} \cdot \underline{v} (\underline{y})$ and $\underline{\underline{p}}' (\underline{y}') = {\underline{\underline{g}}}^{T} * \underline{\underline{p}} (\underline{y})$ , where $\underline{y}' = {\underline{\underline{g}}}^{T} \cdot \underline{y}$ . Using variational formulation of the inclusion problem (equation (2)), for $\underline{\underline{p}}'$ we have

(71)

\int_{Y} (P_{i} \underline{\underline{p}}' : {\underline{\underline{C}}}_{i} + \underline{\underline{p}}') : \frac{\partial \underline{v}'}{\partial \underline{y}'} d \underline{y}' = 0 .

Since

\frac{\partial \underline{v}'}{\partial \underline{y}'} (\underline{y}') = {\underline{\underline{g}}}^{T} * \frac{\partial \underline{v}}{\partial \underline{y}} (\underline{y}),

$\underline{\underline{g}} (Y) = Y$ and ${\underline{\underline{g}}}^{T} * {\underline{\underline{C}}}_{i} = {\underline{\underline{C}}}_{i}$ , it follows from equation (71) that

(72)

\int_{Y} (\underline{\underline{g}} * (P_{i} \underline{\underline{p}}') ({\underline{\underline{g}}}^{T} \cdot \underline{y}) : {\underline{\underline{C}}}_{i} + \underline{\underline{p}}) : \frac{\partial \underline{v}}{\partial \underline{y}} d \underline{y} = 0 .

This equation holds for an arbitrary $\underline{v} \in {\underline{h}}_{# 0}^{1} (Y)$ and hence $P_{i} \underline{\underline{p}} (\underline{y}) = \underline{\underline{g}} * (P_{i} \underline{\underline{p}}') ({\underline{\underline{g}}}^{T} \cdot \underline{y})$ , since the inclusion problem is uniquely solvable. Therefore, by the definition of $\underline{\underline{p}}'$

(73)

(P_{i} \underline{\underline{p}}) (\underline{\underline{g}} \cdot) = \underline{\underline{g}} * P_{i} ({\underline{\underline{g}}}^{T} * \underline{\underline{p}} (\underline{\underline{g}} \cdot)) (\cdot) .

and $P_{i}$ is $G$ -symmetric. If $\underline{\underline{p}} = {\underline{\underline{p}}}_{i}$ is constant and has the support in $Y_{i +}$ , then $(P_{i} \underline{\underline{p}}) (\cdot) = {\underline{\underline{P}}}_{i} (\cdot) : {\underline{\underline{p}}}_{i}$ and

(74)

{\underline{\underline{P}}}_{i} (\underline{\underline{g}} \cdot) : {\underline{\underline{p}}}_{i} = \underline{\underline{g}} * {\underline{\underline{P}}}_{i} (\cdot) : {\underline{\underline{p}}}_{i}

and hence ${\underline{\underline{P}}}_{i} (\cdot)$ is a $G$ -symmetric tensorial function. It readily follows from equation (74) and $G$ -symmetry of $Y_{i +}$ that ${\bar{\underline{\underline{P}}}}_{i}$ is $G$ -symmetric tensor.

In particular, if Y symmetric for all octahedral rotations, and if ${\underline{\underline{C}}}_{i}$ is a cubic tensor, then ${\bar{\underline{\underline{P}}}}_{i}$ is a cubic tensor.

Acknowledgements

The author thanks anonymous reviewers for their careful reading and valuable comments and suggestions.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

ORCID iD

George Mejak

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