Periodic homogenization and material symmetry in linear elasticity

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demon- strates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.


Introduction
Composite materials are of great interest as their properties can be better than those of their constituents. If the properties of these constituents are known, then homogenization theory is a way to obtain the properties of the composite. Here we are interested in the case where the composite consists of a periodic arrangement of linearly elastic materials. In this instance, homogenization theory says that the behavior of the composite is approximately linearly elastic. The elasticity tensor obtained by homogenization is called the macroscopic or effective elasticity tensor. See, for example, Cioranescu and Donato [3], Jikov, Kozlov, and Oleinik [8], and Oleinik, Shamaev, and Yosifian [11] for results on the homogenization of the equations of elasticity.
It is possible for the response of a material (not necessarily a composite) to be unaffected by a change in reference configuration. This leads to the notion of material symmetry. Roughly speaking, the material symmetry group consists of all transformations that leave the response of the material unchanged. A connection between material symmetry and microstructure is well-known as different material symmetry groups are connected with different crystalline structures. For a list of which material symmetry group is appropriate to assume for a given crystalline structure see, for example, Coleman and Noll [4] or Gurtin [6]. Thus, it is reasonable to conjecture that in the context of homogenization theory, the properties of a material on the microscopic scale yields information about the material symmetry group on the macroscale. In this paper we show that if the elasticity tensor describing the properties of the microstructure satisfies an invariance condition, see (4), involving an affine transformation that preserves volume, then the gradient of this transformation is a material symmetry of the macroscopic elasticity tensor. Previous results of this type were established by Jikov, Kozlov, and Oleinik [8] and Alexanderian, Rathinam, and Rostamian [1], but these authors only considered transformations that are rotations or reflections about a fixed point. Examples of transformations not considered before include a translation together with a rotation or a unimodular transformation that is not a rotation. A detailed discussion of how previous results compare with what is established in this paper is given after Theorem 3.4.
One of the advantages of knowing the material symmetry group of the macroscopic elasticity tensor is related to numerics. The components of the macroscopic tensor are obtained by solving unit cell problems, which are systems of elliptic partial differential equations. Taking into account the major and minor symmetries of the elasticity tensor, the 21 elasticities are determined by solving six unit cell problems. Hence, information that reduces the number of unknown components can decrease the number of unit cell problems needed to be solved and, thus, save considerable computational time. Knowledge of the material symmetry group yields this kind of information. For example, if the macroscopic elasticity tensor possesses orthotropic symmetry, the nine unknown elasticities can be computed by solving four unit cell problems.
The outline of the paper is as follows. In Section 2, the concept of a periodic elastic structure and an appropriate notion of symmetry are introduced. Section 3 contains a review of the homogenization of the equations of linear elasticity in a domain with periodic microstructure and the main result connecting symmetries of the periodic elastic structure with the material symmetry group of the macroscopic elasticity tensor. In Section 4, several examples of periodic elastic structures are given and the corresponding material symmetry groups are mentioned. Through these examples it is shown that it is possible for the constituents of a composite to be completely anisotropic and yet the macroscopic elasticity tensor can have a nontrivial symmetry group. Other examples show that transverse isotropy is not always the appropriate symmetry assumption for materials having a bundled or fibered structure. Finally, Section 5 contains some concluding remarks.

Periodic linearly elastic structures
Consider a parallelepiped Y in a Euclidean point space E with vector space V. One can think of E as R n . The cases where E is two-or three-dimensional are of primary interest, however here it is assumed that E is n-dimensional, with n ∈ N. The set Y can be used to form a periodic tessellation of E. Let b i , i = 1, . . . , n, be the linearly independent vectors in V that are the length of and parallel to the edges of Y ; see Figure 1(a). For all z ∈ E, there are unique integers k i , i = 1, . . . , n, such that Here we are interested in linearly elastic materials that have a periodic structure. To this end, consider a position dependent elasticity tensor C defined on Y and extended Y -periodically to all of E so that The pair (Y, C) is referred to as a periodic (linearly) elastic structure. In such a construction, Y is called the unit cell. It is possible for two periodic elastic microstructures to be the same even when they are specified by different unit cells. To make this precise, we say that the periodic elastic structures (Y, C) and (Ŷ ,Ĉ) are equivalent if the periodic extensions of C andĈ to all of E, relative to their respective unit cells, are equal. Consider a periodic elastic structure (Y, C) and let a be a vector. Extend C as in (2) and define C a to be its restriction to the set a + Y . The structure (a + Y, C a ) is equivalent to the original.
Here we consider symmetries of a periodic elastic structure generated by affine transformations that preserve volume. Such a transformation h of E has the form where z • is a point in E, a is a vector, and H is a unimodular linear mapping, so that |det H| = 1. Unless otherwise stated, when speaking of a symmetry h, it is assumed to have the form (3). The motivation for considering mappings of this form is that they are used when discussing material symmetry. Let Sym denote the set of all symmetric linear mappings from V to itself. It is easily seen that the collection of symmetries of a periodic elastic structure form a group under function composition. For any unimodular linear mapping H, define If C is periodic, then so is C H . Using the notation in (5), h is a symmetry of the periodic elastic structure if and only if Notice that translations by integer linear combination of b i , where i = 1, . . . , n, are symmetries of the periodic elastic structure.
Proposition 2.2. If h be a symmetry of the periodic elastic structure (Y, C), then C is periodic inb i := Hb i .
Proof. Considerẑ ∈ E and set z := h −1 (ẑ). From (6) and the periodicity of C in b i , we have and so C isb i -periodic.
The next result says that periodic elastic structures related by symmetries are equivalent. Proposition 2.3. Let (Y, C) be a periodic elastic structure with symmetry h, and setŶ := h(Y ) andb i := Hb i , for i = 1, . . . , n. IfĈ is defined bŷ then the periodic elastic structures (Y, C) and (Ŷ ,Ĉ) are equivalent.
Proof. It must be shown that the periodic extensions of C andĈ relative to Y andŶ are equal. Letẑ ∈ E be given. Define {ẑ}Ŷ analogous to (1), so that there are integersk i , i = 1, . . . , n, such that By (7) and the periodicity of C andĈ, see (2), we havê Applying h −1 to (8) yields from whence it follows that Using this last equation in (9), the periodicity of C H , and (6) results in It follows that C =Ĉ and, hence, the elastic structures are equivalent.
A consequence of Proposition 2.3 is that C andĈ are equal, and hence (Y, C) and (Ŷ , C) are equivalent elastic structures. It is this formulation of the previous result that will be used in the next section.
To illustrate the previous two results, consider the tessellation of the plane depicted in Figure 1(b). Define C so that it equals one isotropic tensor on the white triangles and another isotropic tensor on the grey triangles. This elastic structure can be generated by the unit cells Y andŶ shown in the figure. The transformation h that takes Y toŶ is a symmetry of the periodic elastic structure. Moreover, the vectorsb 1 andb 2 associated with the periodicity ofŶ are related to the vectors b 1 and b 2 associated with the periodicity of Figure 2: A depiction of a periodic elastic structure together with a scaling of it. The unit cell Y generates the initial structure, and the grey and white regions represent different elastic materials. The smaller, transparent structure generated by the unit cell Y ε is the result of scaling the original structure about the point q by ε = 1/2.

Macroscopic elasticity tensor
Homogenization of the equations of linear elasticity is classical [3,8,11]. However, for the sake of completeness, a sketch of the derivation of the macroscopic equations of linear elasticity in a domain with periodic microstructure is given here.
The starting point is to consider a linearly elastic material with a periodic microstructure. We consider the periodic elastic microstructure given by where q is an arbitrary, but fixed, point, ε is a small parameter associated with the length scale of the microstructure, and (Y, C) is a periodic elastic structure. If Y ε is the result of scaling Y by ε about the point q, then (Y ε , C ε ) is a periodic elastic structure, which can be viewed as a microstructure since ε is small; see Figure 2.
From here on we assume that · CB for all linear mappings A and B, and Let Ω be a reference configuration of the elastic material with elasticity tensor C ε , and assume that Ω is open and bounded with Lipschitz boundary. Let Γ 1 and Γ 2 be disjoint subsets of ∂Ω, and assume a zero displacement condition on Γ 1 and the traction is given by t on Γ 2 . The resulting mixed boundary-value problem in elastostatics with body force b is in Ω, where e(u ε ) = 1 2 (∇u ε + (∇u ε ) ) is the symmetrized gradient and n is the exterior unitnormal to Ω. By the Lax-Milgram theorem a unique solution of (11) exists in the space provided that b ∈ L 2 (Ω, V) and t ∈ L 2 (Γ 2 , V). Moreover, the solutions u ε of (11) are bounded in the H 1 -norm independent of ε and, hence, there is a u ∈ H 1 (Ω, V) such that up to a subsequence u ε converges weakly to u in H 1 (Ω, V); see [11]. The goal of homogenization is to find an equation that characterizes u.
To derive the macroscopic equations associated with (11), the notion of two-scale convergence first introduced by Nguetseng [10] and further developed by Allaire [2] is applied using the following function spaces.
is the set of all functions ψ from E to V that are Y -periodic and Y |ψ(y)| 2 dy is finite.
is the completion with respect to the H 1 -norm of the space of smooth functions from E to V that are Y -periodic.
where two functions are equivalent if they differ by a constant vector.
is finite.
• C(Ω, L 2 per (Y, V)) consists of functions ψ : Ω × E → V such that for all x ∈ Ω the function y → ψ(x, y) is in L 2 per (Y, V) and the function x → ψ(x, ·) from Ω to L 2 per (Y, V) is continuous.
Two-scale convergence, in the context presented here, is defined as follows.
where the symbol − denotes the average integral.
By using the change of variables x → x − q, one can see that this definition is equivalent to the standard definition [10,2] involving a vector space rather than a Euclidean point space. Associated with two-scale convergence is the following compactness result [2,10].
is a sequence of functions that weakly converges to v 0 ∈ H 1 (Ω, V), then there is a v 1 ∈ L 2 (Ω, W per (Y, V)) such that up to a subsequence ∇v ε converges two-scale to ∇v 0 + ∇ y v 1 .
Following the tradition in homogenization theory, the symbol ∇ y denotes the gradient taken with respect to the microscopic variable y. It follows from Theorem 3.2 that (13) implies there is a w ∈ L 2 (Ω, W per (Y, V)) such that up to a subsequence ∇u ε converges two-scale to ∇u + ∇ y w. Define Multiply (11) 1 by v ε , integrate over Ω, and then integrate by parts to obtain where is the symmetric tensor product and the symmetry of C ε has been used. Since the mapping is a suitable test function in the definition of two-scale convergence, we can take the limit as ε → 0 in (17) using (15), (16), and the fact that Taking v 0 = 0 in (18) and using the arbitrariness of ψ ∈ C ∞ 0 (Ω) yields It can be shown that given e(u), there is exactly one w ∈ L 2 (Ω, W per (Y, V)) that satisfies (19) [11]. Also, w only depends on x through e(u)(x) and, moreover, w depends linearly on e(u)(x). Thus, (19) can be reformulated as follows: for any E ∈ Sym, there is a unique which is called the unit cell problem. The solutions of (20) can be used to define which is called the macroscopic elasticity tensor. Given an orthonormal bases e 1 , . . . , e n of V, (21) can be written in components: C ijpq (y)∂ yq w e k e l p (y)) dy.
Next, take ψ = 0 in (18) and use (21) to obtain which is equivalent to A unique solution of (24) exists in H by the Lax-Milgram theorem [11]. Numerically, it is more efficient to solve the system (24) than (11) since the elasticity tensor in (24) is constant, while in (11) the elasticity tensor can have rapid oscillations. An analogous homogenization argument can be carried out for elasticity problems involving different boundary conditions or including an inertial term. However, the resulting formula (21) for the macroscopic elasticity tensor, which is of primary interest here, would remain unchanged.
The macroscopic elasticity tensor C 0 can be given in terms of an equivalent periodic elastic structure with a different unit cell. In particular, if h is a symmetry of the structure andŶ := h(Y ), then the remark made immediately after Proposition 2.3 say that (Ŷ , C) is an equivalent elastic structure. Hence, the above homogenization procedure can be carried out usingŶ instead of Y to find that C 0 is also given by The following result relates the solutions of the unit cell problems (20) and (26).
Notice that v ∈ W per (Y, V) and, hence, using (20) with E replaced by H EH results in Y C(y) H EH + e y (w H EH )(y) · e y (v)(y) dy = 0.
Using the change of variablesŷ = h(y) and (28) yields Since h is a symmetry of the elastic structure and, hence, (4) holds, (29) becomes Since (30) holds for allv ∈ W per (Ŷ , V) and the solution of (26) is unique, the lemma holds.
The following is the main result.
Theorem 3.4. If h is a symmetry of the periodic elastic structure, then Proof. Let E ∈ Sym be given and setŶ := h(Y ). By (21), the change of variablesŷ = h(y), Lemma 3.3, (4), and (25), we have Equation (31) is the condition for H to be in the material symmetry group of C 0 . Thus, Theorem 3.4 implies that the gradient of every symmetry of the periodic elastic structure is a material symmetry of the macroscopic elasticity tensor. For a detailed discussion of material symmetry see, for example, Gurtin, Fried, and Anand [7]. Although the above analysis was carried out in the context of linear elasticity, it could of been done for the diffusion equation. In that case, a diffusion tensor D would be defined on a unit cell and (4) would be replaced by Notice that Theorem 3.4 does not say that every element of the material symmetry group of C 0 is generated by a symmetry of the periodic elastic structure. It is unknown if given a material symmetry H of C 0 , there is a symmetry of the elastic structure whose gradient is H. Results similar to Theorem 3.4 were established by Jikov, Kozlov, and Oleinik [8] and by Alexanderian, Rathinam, and Rostamian [1]. To accurately compare these results with Theorem 3.4, take the Euclidean space E to be R n . Jikov, Kozlov, and Oleinik proved Theorem 3.4 for symmetries of the form where Q is an orthogonal linear mapping and Alexanderian, Rathinam, and Rostamian also considered symmetries of this form in the context of the homogenization of diffusive random Figure 3: Examples of periodic elastic structures containing symmetries that are not covered by previous results similar to Theorem 3.4. In these examples, the white and grey regions should be viewed as different isotropic materials. (a) An example of a periodic elastic structure having a symmetry consisting of the translation a together with a reflection in which neither the translation nor the reflection by themselves symmetries of the elastic structure. (b) This periodic elastic structure has amongst its symmetries reflections about the vertical and horizontal lines passing through p and rotations about the point q by π/2 radians. media. Thus, these previous results only considered symmetries that are rotations or reflections about a single point-in particular, the origin. Whereas Theorem 3.4 also includes symmetries about different points, symmetries consisting of an orthogonal linear mappping together with a translation, and symmetries involving unimodular linear mappings.
A periodic elastic structure with a symmetry consisting of a translation and reflection is depicted in Figure 3(a). In this figure consider the white and grey regions to consist of different isotropic elastic materials. Notice that a reflection about any vertical line is not a symmetry of the elastic structure, however the transformation consisting of the translation a together with the reflection about the vertical line aligned with a is a symmetry of the periodic elastic structure. Figure 3(b) is an example of a periodic elastic structure that contains symmetries of the form for different H associated with different points z • . Namely, there are reflection symmetries about point p and a rotation about the point q by π/2 radians is a symmetry. An example of a periodic elastic structure that has a symmetry where H is not orthogonal is described at the end of the next section.
When C takes only a finite number of values, the elastic structure is made up of a finite number of materially uniform constituents. In this case, two factors determine the symmetries of such an elastic structure and, hence, the material symmetry of C 0 : the material symmetries of the constituents and their arrangement. This is illustrated in the next section. Parnell and Abrahams [12] realized this in the context of a three dimensional elastic structure made of two constituents.

Examples
In this section several examples of periodic elastic structures are given. For each example, the symmetries of the elastic structure are mentioned together with the resulting material symmetries of the macroscopic elasticity tensor guaranteed by Theorem 3.4.
For simplicity, consider E to be the three-dimensional space R 3 and use the standard basis e 1 , e 2 , e 3 . Given a unit vector a and an angle φ, let R φ a be the right-handed rotation through the angle φ about the axis in the direction of e. Notice that −R π a is the reflection with respect to the plane with normal a.
Since the macroscopic elasticity tensor C 0 is given by an integral expression, if C is changed on a set of measure zero, then the macroscopic elasticity tensor remains unchanged. For this reason, in these examples the microscopic elasticity tensor C is only specified up to a set of measure zero. Similarly, the unit cell Y which defines the periodic elastic structure only tiles R 3 up to a set of measure zero.
Orthotropic symmetry: It is possible for the elasticity tensor C associated with the elastic structure to be fully anisotropic at each point and yet the macroscopic elasticity tensor can have orthotropic symmetry, meaning that To obtain this, the anisotropies of the elastic structure are arranged in a particular way. Begin by considering an anisotropic elasticity tensor C an and the unit cell Y = (−1, 1) 3 .
The same symmetry persists if there are several fibers appearing in the unit cell arranged as depicted in Figure 4(b). Moreover, as long as the material symmetry groups of the matrix and fibers contain R π e 1 and R π/2 e 3 , the result still holds. Thus, if the fibers are transversely isotropic and the matrix is isotropic, then the macroscopic elasticity tensor has tetragonal symmetry. This type of periodic elastic structure was used by Ptashnyk and Seguin [13] in modeling plant cell walls. The macroscopic elasticity tensor also has tetragonal symmetry when the fibers are oriented in orthogonal directions, as depicted in Figure 4(c). To see this, let and z → e 1 + R π/2 e 1 z are symmetries of the elastic structure and, thus, the macroscopic elasticity has tetragonal symmetry.
Hexagonal symmetry: Let R be a rhombus in R 2 whose sides have length 2 and whose centroid is at the origin. Consider the unit cell Y = (−1, 1) × R. As in the example involving tetragonal symmetry, assume that Y consists of a matrix and fibers and that the fibers occupy the domain and the rest of Y is occupied by the matrix. A cross-section of Y for a constant y 1 value is shown in Figure 5(a). The elastic structure resulting from tiling R 3 with this unit cell has fibers arranged in a hexagonal bundle structure; see Figure 5(b). When the matrix and the fibers are isotropic, then the elastic structure generated by this unit cell has the symmetry It follows that the corresponding macroscopic elasticity tensor has hexagonal symmetry.
It is often claimed that when a material has a bundled or fibered structure it is appropriate to assume that it is transversely isotropic. However, the examples above involving tetragonal and hexagonal symmetry indicate that the arrangement of the bundled fibers on the microscale dictates the symmetry of the macroscopic elasticity tensor and that transverse isotropy is not the appropriate material symmetry assumption, at least when the fibers are arranged periodically. However, if the axes of all of the fibers are parallel but arranged randomly in the orthogonal plane, then it may be appropriate to assume transverse isotropy on the macroscale.
This symmetry group is useful in the modeling of lipid bilayers; see, for example, Deseri, Piccioni, and Zurlo [5] and Maleki, Seguin, and Fried [9]. It is readily verified that for all H ∈ G, the transformation y → Hy is a symmetry of the periodic elastic structure and, hence, Theorem 3.4 implies that the macroscopic elasticity tensor also has in-plane fluidity.

Conclusion
In this paper we showed a connection between the symmetries of a periodic linearly elastic structure and the material symmetry of the macroscopic elasticity tensor using homogenization theory. Our result generalizes previous work by considering a larger class of symmetries on the microscopic scale. Since the material symmetry group is a subgroup of the unimodular group, we conjecture that symmetries of the form (3) include all possible symmetries of the periodic elastic structure that lead to material symmetries of the macroscopic elasticity tensor. However, it remains an open question as to whether all material symmetries are generated in this way. The proof of Theorem 3.4 relies on the formula (21) for the macroscopic elasticity tensor. Since homogenization of locally-periodic or random microstructures also results in explicit formulas for the macroscopic elasticity tensor, it is possible that with suitable modifications the arguments used here could yield a result similar to Theorem 3.4 for these kinds of microstructures. It would be interesting to investigate whether there is an analog of Theorem 3.4 in the case of nonlinear elasticity.