Periodic rhomboidal cells for symmetry-preserving homogenization and isotropic metamaterials

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Periodic rhomboidal cells for symmetry-preserving homogenization and isotropic

metamaterials

Giulio G. Giusteria,∗, Raimondo Pentab

aDipartimento di Matematica“Tullio Levi-Civita”, Università degli Studi di Padova, via Trieste 63, 35121, Padova, Italy

bSchool of Mathematics and Statistics, University of Glasgow, University Place, G128QQ, Glasgow, UK

Abstract

In the design and analysis of composite materials based on periodic arrangements of sub-units it is of paramount

importance to control the emergent material symmetry in relation to the elastic response. The target material symmetry

plays also an important role in additive manufacturing. In numerous applications it would be useful to obtain eﬀectively

isotropic materials. While these typically emerge from a random microstructure, it is not obvious how to achieve isotropy

with a periodic order. We prove that arrangements of inclusions based on a rhomboidal cell that generates the Face-

Centered Cubic lattice do in fact preserve any material symmetry of the constituents, so that spherical inclusions of

isotropic materials in an isotropic matrix produce eﬀectively isotropic composites.

Keywords: Material symmetry, Periodic composite, Homogenization, Isotropic metamaterial

1. Introduction

Composite or microstructured materials have been long

since considered as important means to engineer and op-

timize mechanical properties for speciﬁc applications [1,

2, 3]. With the advent of additive manufacturing (also

known as 3D-printing), production of artiﬁcial constructs

conceived to possess speciﬁc optimal properties is now be-

coming possible. The design of the mechanical behaviour

of composites is increasingly relevant in a large variety of

scenarios of practical interests, ranging from construction

[4] to biomimetic materials [5].

The architecture of such materials is typically based on

designing features at a small scale, that lead to the desired

large-scale behavior of structural elements. In light of this,

theoretical studies of composites often involve asymptotic

(periodic) homogenization or alternative upscaling tech-

niques based on average ﬁeld theories (see for example the

review [6] where the two approaches are compared) to ob-

tain suitable predictions of the eﬀective material behavior.

Examples can be found in Refs. [7], [8, 9], and [10] concern-

ing poroelastic composites, biophysical applications (such

as bone, tendons, tumors, and organs) and metamaterials,

respectively.

On the one hand, obtaining detailed quantitative infor-

mation on material parameters typically requires employ-

ment of sophisticated combinations of analytical and/or

computational techniques [11, 12, 13, 14, 15]. On the other

hand, some qualitative information can be deduced by sim-

ple symmetry arguments and this will be the focus of the

∗Corresponding author

present note. One of the most important qualitative prop-

erties of elastic solids is the material symmetry, and the

target symmetry for a composite is always considered in

the design process. Luckily, one can draw some conclusion

on material symmetry by considering how the response to

simple homogeneous deformations interacts with the ge-

ometric properties of the assembly of microscopic units

that form a composite. This is especially true when the

large-scale specimen is built by a periodic reproduction of

identical units.

Several studies dealt with material symmetries in the

context of linearized elasticity [16, 17] and the minimal

symmetry induced by a periodic arrangement of inclusions

in a binary composite has been repeatedly investigated by

diﬀerent methods [18, 19, 20]. Nevertheless, it is not clear

whether it be possible to obtain an eﬀectively isotropic re-

sponse by a periodic arrangement of inclusions in a three-

dimensional body, while it is well known that a hexagonal

lattice would suﬃce to get such a maximal symmetry in a

two-dimensional context. In particular, a periodic arrange-

ment of uniaxially aligned ﬁbers leads to either tetragonal

or transverse isotropic response depending on weather the

planar projections of such ﬁbers are encoded in a square

or hexagonal periodic cell [21, 22]. Topological optimiza-

tion procedures oﬀer a way to approach an isotropic re-

sponse by resorting to nontrivial geometries. While in rare

cases the lack of isotropy is negligible [12, 13], a residual

anisotropy is usually found in three-dimensional compos-

ites in the presence periodicity. Introducing randomness

in the microstructure remains the most accepted way to

reach an isotropic response.

Based on geometric symmetry considerations, it is well

known how to achieve a large-scale cubic symmetry with a

Preprint submitted to Elsevier October 11, 2022

arXiv:2210.04463v1 [math-ph] 10 Oct 2022

periodic inclusion lattice. By considering a standard cubic

periodic cell, then the existence of three planes of sym-

metry guarantees the material orthotropy as long as no

additional degree of anisotropy is induced by the material

symmetry of the individual phases in the composite. This

is true when the individual phases are geometrically ar-

ranged in the host in order to guarantee the existence of

such planes of symmetries, and they are individually at

most orthotropic. When assuming, in addition, that the

individual phases are either all isotropic or at most cubic,

and that the resulting geometry is invariant with respect to

rotation of the three orthogonal axis (this can be achieved

for example by considering either a cubic or a spherical

inclusion in three dimensions), the resulting material re-

sponse is then in general cubic. This is also shown in the

works [23] and [24], where the authors deﬁne a suitable

function which measures the deviation from isotropy for

composites in the context of asymptotic homogenisation.

Common lore suggests that, with a periodic inclusion

lattice, one cannot achieve isotropy without additional con-

straining strategies [12, 13], even with isotropic compo-

nents. Should this be the case, then it would be somewhat

unpleasant because, on the one hand, periodic arrange-

ments are by far the most convenient approach in both

additive manufacturing and computational material design

and, on the other hand, isotropic elasticity is very often

assumed in practical applications, especially in the context

of simple models used to validate experimental results.

In this work, we show that the maximal symmetry that

can be achieved for three-dimensional periodic composites

is not the cubic one. We give a rigorous and yet sim-

ple proof of the fact that a periodic arrangement on a

Face-Centered Cubic (FCC) lattice of spherical inclusions

of an isotropic solid within an isotropic matrix gives rise

to a large-scale isotropic response. In doing so, we also

show that any rhomboidal computational cell that gen-

erates such a lattice can be used to successfully design

homogenized solids in which the material symmetry is not

aﬀected by the periodicity of the construction, since the

latter would preserve even the largest possible symmetry

group. It is signiﬁcant to observe that the geometric sym-

metry group of such rhomboidal cells is strictly smaller

than the symmetry group of the lattice they generate, but

the lattice and not the cell is the geometrically relevant

structure when analyzing large-scale properties.

Incidentally, our result shows that the small lack of

isotropy computationally found for several periodic arrange-

ments based on a FCC lattice is to be ascribed to (un-

avoidable) numerical approximations rather than to real

geometric obstructions. This leads to important changes

in perspective for the interpretation of numerical results

and towards the design of isotropically elastic metamate-

rials, with important consequences on several applications.

We frame our discussion in the context of linear elas-

ticity by introducing, in Section 2, a normalized Voigt

representation of the elasticity tensor which is very con-

venient for the identiﬁcation of material parameters and

symmetries. We then discuss the link between lattice sym-

metries and material symmetries for periodic composites

in Section 3 and, ﬁnally, present our main result and a

symmetry-preserving rhomboidal cell in Section 4.

2. Normalized Voigt representation

We are interested in describing the eﬀective linear elas-

tic response of a composite that consists of two isotropi-

cally elastic phases. One is the matrix and the other one

occupies spherical inclusions with centers distributed on

a periodic lattice. Due to the spherical shape of the in-

clusions and the isotropic nature of the two materials, the

only source of anisotropy in the homogenized material re-

sponse can be the geometry of the inclusion lattice. It

is thus convenient to represent the linearized measure of

strain and the Cauchy stress tensor on a basis for the space

of symmetric tensors that is adapted to the geometry of

the inclusion lattice. This leads to the construction of a

normalized Voigt representation.

We denote by (a1,a2,a3)the generators of the lattice,

namely linearly independent vectors such that the centers

of the spherical inclusions are obtained as combinations of

a1,a2, and a3with integer coeﬃcients. The set of lat-

tice sites is then denoted by L=ha1,a2,a3iZ. A set of

directors of the lattice can be constructed building an or-

thonormal basis for R3out of the generators. For instance,

we may choose

l1=a1

ka1k,kl2kl2=a2−(a2·l1)l1,

kl3kl3=a3−(a3·l1)l1−(a3·l2)l2.

We now introduce an orthogonal basis for the linear

space of symmetric tensors built upon the lattice directors.

The basis Z= (Z1,Z2,Z3,Z4,Z5,Z6)is given in terms of

dyadic products by

Z1=l1⊗l1,Z2=l2⊗l2,Z3=l3⊗l3,

Z4=l2⊗l3+l3⊗l2

√2,Z5=l1⊗l3+l3⊗l1

√2,

Z6=l1⊗l2+l2⊗l1

√2.

Such a basis is orthonormal with respect to the tensor

scalar product deﬁned by A:B:= tr(ATB).

As customary in linear elasticity, we decompose the

deformation gradient tensor as F=J+∇u, with uthe

displacement ﬁeld and Jthe isometry that maps spatial

vectors to material ones. The standard linearized strain

measure is then E=1

2(∇uJ+JT∇uT). In the inﬁnitesimal-

displacement regime considered in linear elasticity, Jis

constant and homogeneous and can be taken as the iden-

tity. Eis a symmetric tensor, characterized by six degrees

of freedom. A possible choice of objective quantities to

represent them are the eigenvalues of Eand the orienta-

tion of its eigenvectors with respect to the lattice directors.

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摘要：

Periodicrhomboidalcellsforsymmetry-preservinghomogenizationandisotropicmetamaterialsGiulioG.Giusteria,,RaimondoPentabaDipartimentodiMatematicaTullioLevi-Civita,UniversitàdegliStudidiPadova,viaTrieste63,35121,Padova,ItalybSchoolofMathematicsandStatistics,UniversityofGlasgow,UniversityPlace,G128QQ,...

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Periodic rhomboidal cells for symmetry-preserving homogenization and isotropic metamaterials

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