# Scale-size dependent multi-continuum homogenization of complex bodies

Grigor Nika

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Grigor Nika. Scale-size dependent multi-continuum homogenization of complex bodies. Quarterly of Applied Mathematics, 2024. hal-04378065v2

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Distributed under a Creative Commons Attribution 4.0 International License
Scale-size dependent
multi-continuum homogenization of complex bodies

Grigor Nika*

Mathematics & Computer Science
Karlstad University
Universitetsgatan 2
65188 Karlstad, Sweden
grigor.nika@kau.se

May 9, 2024

**Abstract**

We derive effective equations of a periodically heterogeneous Cosserat material encom-
passing intrinsic lengths modelling scale-size effects. The resultant homogenized material
supports internal body torques and leads to an asymmetric effective stress providing a con-
nection to the theory of *odd elasticity*. Furthermore, a link to the classical *Cauchy* stress is
given. Moreover, the corresponding local problem exhibits asymmetry as well, due to the
micropolar couple modulus inherited from the original microscopic Cosserat problem. We
validate our results by conducting numerical simulations using the finite element method on
circularly perforated square and rectangular unit cells, highlighting the impact, of not only
volume fraction but also of internal body torques on effective coefficients. Additionally, we
numerically quantify the “amount” that the body can torque internally.

MSC 2020: 74Q05, 35B27, 35J58, 35Q74

**Keywords:** multi-continuum homogenization, Cosserat continuum, scale-size effects, odd elasticity

# 1 Introduction

In recent years, there has been a prevalent observation in materials science regarding the reliance of
mechanical properties on specific characteristic lengths. These lengths are intricately determined
either by the inherent structure itself or by the nuanced attributes of an underlying microstructure,
or often, by a synergistic interplay of both factors (see, e.g., [32], [49], and [33]). The implica-
tions of these phenomena extend beyond the microscale, manifesting as macroscopic properties

<footnote>*Email: grigor.nika@kau.se</footnote>
of the composite material that may starkly deviate from the intrinsic properties of its constituents. Particularly intriguing is the scenario wherein these characteristic lengths align closely with the dimensions of the microstructure. In such cases, the classical framework of classical continuum mechanics<fnref n="1" /> loses its accuracy in characterizing the mechanical behavior of these materials. These phenomena, encapsulated by the term *scale-size effects*, prompt the exploration of alternative frameworks, and one way involves modeling them through generalized continuum theories (see, e.g., [34], [1], [18], [48], [40]).

Generalized continuum theories<fnref n="2" /> find applications in modeling materials with intricate microstructures [11, 26, 53, 54, 31, 37, 15, 16, 38, 14, 39, 46]. Among the possible generalized continuum theories two seem to be the most utilized: higher-grade theories, that introduce higher gradients of the displacement field, and higher-order theories, that introduce additional degrees of freedom. Among the pioneering contributions to generalized continuum mechanics theories, the Cosserat brothers' formulation of continuum mechanics stands out (see [11]). Notably, Cosserat elasticity can model scale-size effects through intrinsic length scales<fnref n="3" />, a phenomenon that classical elasticity theory lacks the mechanism to model (see, e.g., [13], [27], [48]).

Simultaneously with the resurgence of generalized continua, homogenization theory emerged independently as a particularly apt method for analyzing highly heterogeneous and microstructured materials (see, e.g., [4, 51, 3, 7, 36]). Homogenization seems to be a natural bridging mechanism for exploring connections between generalized continuum theories and classical theories as they pertain to heterogeneous media. The combination of these theories enables a more comprehensive understanding of the mechanical behavior of complex materials. A series of investigations starting in the late eighties onwards starting applying the homogenization method within the context of Cosserat continua (see, e.g., [5, 55, 19, 20, 18]). The latter inquiries focused on estimating effective properties of heterogeneous Cosserat materials while accounting for scale-size effects (see, e.g., [18]).

Our objectives in this article are the following: Starting from foundational non-classical thermo-dynamic principles and balance laws, we deduce a set of constitutive laws for a linear Cosserat continuum, grounded in the tenets of the principle of virtual work (see, e.g., [23], [35], [17]). In systems characterized by isothermal conditions and time stationarity, we enhance the aforementioned constitutive laws through the application of the homogenization method. Concurrently, within this homogenization framework, we incorporate intrinsic lengths linked to scale-size phenomena inherent in the mechanics of the Cosserat continuum. In contrast to previous studies, the resulting homogenized stress is asymmetric. Moreover, our approach involves solving only one local problem to determine the homogenized coefficients, which is also locally asymmetric due to the micropolar couple modulus. Asymmetry of the homogenized stress implies that the resultant homogenized body is capable of supporting internal torques, precisely as the theory of odd elasticity<fnref n="4" /> dictates (see, e.g., [52], [21]).

<footnote n="1">¹The Cauchy postulate loses its validity. See the edited volume of articles in [34] and references therein for an excellent exposition to the topic.</footnote>

<footnote n="2">²Following the example of G. Maugin in [34], I would like to apologize in advance if I have neglected to cite someone that I should have cited. I have done my best to comprehend the many contributions of all the authors I have cited but, almost surely, I have missed others.</footnote>

<footnote n="3">³Furthermore, Cosserat elasticity proves versatile enough to model chirality. The reader can consult the work in [27] for a precise definition of chirality in the context of Cosserat rods.</footnote>

<footnote n="4">⁴Odd elasticity assumes that elastic stresses arise solely from displacement gradients, ensuring linear momentum conservation but not angular momentum conservation. This means that odd-elastic materials can experience internal</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: Schmatic of the approach proposed in the article and how it can lead to a stress that is asymmetric. These type of phenomena are often referred as *odd elasticity* and usually occur in complex bodies that are capable of supporting internal torques.</figcaption></figure>

Asymmetric stresses, also, occur quite commonly in field responsive materials. For instance, the stress induced by an external magnetic field in a hard-magnetorheological elastomer (hMREs) (see, e.g., [56], [22], [41]). These type of materials support internal torques and in order to ensure equilibrium an additional equation is required to balance angular momentum<fnref n="5" /> since the stress is not symmetric. Usually, the above effect is distinguished from non-classical continua, such as Cosserat continua, which admit an asymmetric stress. However, as this work shows, the two concepts are intimately related with homogenization being the connecting lynch pin.

The structure of the article is as follows: In Section 2 we develop the model from fundamental laws of non-classical thermodynamics and derive the constitutive laws based on the celebrated Coleman-Noll procedure (see [45]). In Section 3 we perform the analysis of the corresponding isothermal and time stationary system using the method of periodic unfolding, derive the local problem, and show the relation between the *Cauchy stress* and the derived effective stress. Subsequently, in Section 4 we carry out simulations of the derived local problem and ascertain the influence of internal torques on the effective coefficients as well as on the shearing resistance or lack thereof of the material. Finally, in Section 5 we describe some conclusions.

## 82 Notation

We denote by bold capital letters tensors of order four and by blackboard font letters tensors of order two. Their corresponding elements will be denoted by a scalar quantity with the appropriate number of indices corresponding to the order of the tensor. To expedite the presentation of our results, here-onwards we will make use of the following notation:

87 - Throughout the article we employ the Einstein summation notation of repeated indices unless otherwise stated.

88

torque density even without stress induced by solid body rotations, unlike classical elasticity which relies on an elastic potential to conserve energy and momentum (see, e.g., [52], [21]).

<footnote n="5">⁵While sometimes the term *Cauchy stress* is used to also refer to the overall stress in hMREs, strictly speaking, it doesn't meet the criteria of Cauchy stress because it violates the third axiom (see, e.g., [6]).</footnote>
98 - The third order tensor $\epsilon_{ijk}$ is the Levi-Civita symbol that is equal to 1 if $(i, j, k)$ is an even permutation of $(1, 2, 3)$, -1 if it is an odd permutation, and zero if any index is repeated.

91 - **Tensor contractions:** Let $\mathbf{v}$ be a vector, $\mathbf{A}$ and $\mathbf{B}$ be second order tensors. We define the following contractions of tensor-tensor or tensor-vector contractions:

93 1. $\mathbf{A}:\mathbf{B}=\mathbf{A}_{ij}\mathbf{B}_{ij}$ (Frobenius inner product)

94 2. $\epsilon.\mathbf{v}=\epsilon_{ijk}\mathbf{v}_i$ (contraction of the Levi-Civita tensor with a vector leads to second order tensor)

96 2. $\epsilon:\mathbf{A}=\epsilon_{ijk}\mathbf{A}_{ij}$ (contraction of the Levi-Civita tensor with a second order tensor leads to a vector)

## 2 Modeling and thermodynamic restrictions

### 2.1 Balance laws

1. *Balance of linear and angular momentum.* Denote by $s_{ij}:=\partial_{x_j}u_i - \sum_{k=1}^3 \epsilon_{kji}\varphi_k$ and $\kappa_{ji}=\partial_{x_j}\varphi_i$ for $i, j \in \{1, 2, 3\}$ where $\mathbf{u}$ is the displacement and $\mathbf{\varphi}$ is the rotation. The balance of linear and angular momentum<fnref n="6" /> is written in the form of the principle of virtual work. Define,

$$ 
\begin{aligned}
W_{\text{int}}(\Omega, \dot{\mathbf{u}}, \dot{\mathbf{\varphi}}) &= - \int_{\Omega} \sigma : \nabla \dot{\mathbf{u}}^* \, dv - \int_{\Omega} \mu : \nabla \dot{\mathbf{\varphi}}^* \, dv + \int_{\Omega} \epsilon.\dot{\mathbf{\varphi}}^*:\sigma \, dv \\
&= - \int_{\Omega} \sigma : \mathbf{s}^* \, dv - \int_{\Omega} \mu : \mathbf{i}^* \, dv
\end{aligned}
\quad (2.1) $$

$$ W_{\text{ext}}(\Omega, \dot{\mathbf{u}}, \dot{\mathbf{\varphi}}) = \int_{\partial\Omega} \mathbf{t}.\dot{\mathbf{u}}^* \, da + \int_{\partial\Omega} \mathbf{m}.\dot{\mathbf{\varphi}}^* \, da + \int_{\Omega} \rho \mathbf{f}.\dot{\mathbf{u}}^* \, dv + \int_{\Omega} \rho \mathbf{l}.\dot{\mathbf{\varphi}}^* \, dv \quad (2.2) $$

$$ W_{\text{acc}}(\Omega, \dot{\mathbf{u}}, \dot{\mathbf{\varphi}}) = \int_{\Omega} \rho \dot{\mathbf{u}}.\dot{\mathbf{u}}^* \, dv + \int_{\Omega} \rho \dot{\mathbf{\varphi}}.\dot{\mathbf{\varphi}}^* \, dv \quad (2.3) $$

103 The sum of the virtual work of internal, external, and acceleration forces acting on any subdomain D of the body for any virtual set of vector fields $(\dot{\mathbf{u}}^*, \dot{\mathbf{\varphi}}^*)$ vanishes at each instant $t$ of the evolution.

$$ W_{\text{int}}(\Omega, \dot{\mathbf{u}}, \dot{\mathbf{\varphi}}) + W_{\text{ext}}(\Omega, \dot{\mathbf{u}}, \dot{\mathbf{\varphi}}) = W_{\text{acc}}(\Omega, \dot{\mathbf{u}}, \dot{\mathbf{\varphi}}), \quad (2.4) $$

105 for all $\Omega$ and for all admissible $(\dot{\mathbf{u}}^*, \dot{\mathbf{\varphi}}^*)$. Consequences of the virtual power principle are,

<footnote n="6">⁶In the linear Cosserat context where we assume infinitesimal displacements, the mass density per unit volume $\rho \approx \rho_0$, where $\rho_0$ is the Lagrangian constant mass density. Similarly, under infinitesimal rotations, the microinertia term reduces to $\rho t$ due to the spherical symmetry assumption we make. In general, the microinertia term, is a second order tensor $\mathbf{I}_{ij}$ that under the spherical symmetry assumption becomes $\mathbf{I}_{ij} = \rho t \delta_{ij}$. The reader can consult the works of, e.g., [15] and [15] (this where the concert of microinertia was introduced), [13], [24], [25]. In the nonlinear Cosserat context, in addition to having conservation of mass, we also need to have conservation of microinertia.</footnote>
$$
\begin{align}
\operatorname{div} \sigma + \rho f &= \rho \ddot{u} && \text{in } \Omega, \nonumber \\
\operatorname{div} \mu + \epsilon : \sigma + \rho \iota \ell &= \rho \iota \dot{\phi} && \text{in } \Omega, \nonumber \\
\sigma \cdot \mathbf{n} &= \mathbf{t} && \text{on } \partial\Omega, \nonumber \\
\mu \cdot \mathbf{n} &= \mathbf{m} && \text{on } \partial\Omega. \tag{2.5}
\end{align}
$$

106 The tensor $\sigma$ is the non-symmetric stress tensor, $\mu$ the couple-stress tensor, $\epsilon$ the Levi-Civita tensor
107 with $\epsilon:\sigma = \sum_{j,k=1}^{3} \epsilon_{ijk}\sigma_{jk}$ for $i \in \{1, 2, 3\}$, $f$ the body forces, and $\ell$ the body couples.

108 2. The first law of thermodynamics. The balance law for the energy is,

$$
\frac{d}{dt} \int_{\Omega} \rho e \, dv + W_{\text{acc}}(\Omega, \dot{\mathbf{u}}, \dot{\mathbf{\phi}}) = W_{\text{ext}}(\Omega, \dot{\mathbf{u}}, \dot{\mathbf{\phi}}) - \int_{\partial\Omega} \mathbf{q} \cdot \mathbf{n} \, da + \int_{\Omega} \rho r \, da, \quad (2.6)
$$

or, equivalently after taking the principle of virtual work into account,

$$
\frac{d}{dt} \int_{\Omega} \rho e \, dv = - \int_{\partial \Omega} \mathbf{q} \cdot \mathbf{n} \, da + \int_{\Omega} \rho r \, da. \quad (\text{balance of energy})
$$

110 The local equations then take the form,

$$
\rho \dot{e} = \sigma : \dot{s} + \mu : \dot{\kappa} - \operatorname{div} \mathbf{q} + \rho r. \quad (2.7)
$$

111 3. *The second law of thermodynamics.* The second law of thermodynamics keeps its classical
112 global form for any material subdomain D,

$$
\frac{d}{dt} \int_{\Omega} \rho \eta \, dv \geq - \int_{\partial \Omega} \frac{\mathbf{q}}{T} \cdot \mathbf{n} \, da + \int_{\Omega} \rho \frac{r}{T} \, dv. \quad (\text{growth of entropy})
$$

113 Combining the **growth of entropy** and the **balance of energy** together with the Helmholtz free en-
114 ergy $\psi:=e - T\eta$ we obtain the Clausius-Duhem inequality,

$$
\rho(\dot{\psi} + \dot{T}\eta) - \sigma : \dot{s} - \mu : \dot{\kappa} + \frac{\mathbf{q} \cdot \text{grad } T}{T} \le 0. \quad (\text{Clausius-Duhem ineq.})
$$

115 **2.2 Constitutive laws based on thermodynamic restrictions**

116 Following the celebrated Coleman-Noll procedure, constitutive functions depend on the state vari-
117 ables (S, K, T):

$$
\psi = \hat{\psi}(s, K, T). \tag{2.8}
$$

118 We require that the Helmholtz free energy satisfy the *principle of material objectivity*, namely,
$$ \hat{\psi}(s, \kappa, T) = \hat{\psi}(Qs, Q\kappa, T) \qquad (2.9) $$

119 for all $Q \in SO(3)$. Hence, the classical chain rule gives,

$$ \psi = \frac{\partial \hat{\psi}}{\partial s} : \dot{s} + \frac{\partial \hat{\psi}}{\partial \kappa} : \dot{\kappa} + \frac{\partial \hat{\psi}}{\partial T} : \dot{T} \qquad (2.10) $$

120 Replacing the above expression in the Clausius-Duhem inequality, we obtain,

$$ \left( \rho \frac{\partial \hat{\psi}}{\partial s} - \sigma \right) : \dot{s} + \left( \rho \frac{\partial \hat{\psi}}{\partial \kappa} - \mu \right) : \dot{\kappa} + \left( \rho \frac{\partial \hat{\psi}}{\partial T} + \eta \right) : \dot{T} + \frac{1}{T} \mathbf{q} \cdot \nabla T \le 0. \qquad (2.11) $$

121 Thermodynamic restrictions give the following constitutive relations:

$$ \sigma = \rho \frac{\partial \hat{\psi}}{\partial s}, \quad \mu = \rho \frac{\partial \hat{\psi}}{\partial \kappa}, \quad \eta = -\rho \frac{\partial \hat{\psi}}{\partial T}. \qquad (2.12) $$

122 The above relations reduce the dissipative part of the system to the purely thermal component, namely,

$$ \frac{1}{T} \mathbf{q} \cdot \nabla T \le 0. \qquad (2.13) $$

## 3 Analysis of an isothermal and quasi-stationnary system

125 Henceforth, we will assume that the material is in its natural configuration, meaning the pre-

126 stress and the couple pre-stress are both zero. Moreover, we will consider a system that is quasi-

127 stationary and thermally isolated where heat exchange does not occur. A reasonable free energy

128 could have the following form,

$$ \rho \hat{\psi} = \frac{1}{2} E s : s + \frac{1}{2} M_{\kappa : \kappa} + C_s : \kappa, \qquad (3.1) $$

129 and the ensuing constitutive laws from (2.12) become,

$$ \begin{bmatrix} \sigma \\ \mu \end{bmatrix} = \begin{bmatrix} E & C \\ C^T & M \end{bmatrix} \begin{bmatrix} s \\ \kappa \end{bmatrix} \qquad (3.2) $$

130 The term $s$ is the Cosserat strain tensor and $\kappa$ is the torsion tensor (or curvature-twist tensor or

131 curvature tensor or curvature). Additionally, thermodynamic relations impose the following sym-

132 symmetry on the tensors $E_{jikl}$, $M_{jikl}$, and $C_{jikl}$:

<footnote n="7">⁷Within this context, should one contemplate a quadratic free energy formulation for a material in its inherent state—wherein both prestress and couple prestress parameters assume a value of zero—a comprehensive thermomechanical system for a Cosserat continuum can be deduced. However, the exploration of this avenue diverges from the primary focus of this article. Consequently, we opt to defer this exploration to a subsequent undertaking, as it falls outside the immediate scope of our present objectives.</footnote>
$$E_{jikl} = E_{klji}, M_{jikl} = M_{klji}, \text{ and } C_{jikl} = C_{klji} \qquad (3.3)$$

The pseudo-tensor $C_{jikl}$, expresses the chirality$^8$ effects. If $C_{jikl} \equiv 0$ then the material is called centrosymmetric (see, e.g., [46]) and the problem has been addressed formally in [19], [18] and rigorously via unfolding in [42]. Regarding non-centrosymmetric material, one can consult [5] as well as recent work in [43], [44].

## 3.1 Dimensional analysis

Let L be the characteristic length of the body $\Omega$ and $\ell$ the characteristic length of the periodic cell. We normalize all the parameters involved, including material parameters, as follows:

$$\mathbf{x}^* = \frac{\mathbf{x}}{L}, \quad \mathbf{u}^*(\mathbf{x}^*) = \frac{\mathbf{u}(\mathbf{x})}{L}, \quad \varphi^*(\mathbf{x}^*) = \varphi(\mathbf{x}). \qquad (3.4)$$

In non-centrosymmetric Cosserat media there are two additional and independent length scale parameters that are of importance, namely, the Cosserat intrinsic length $\ell_{coss}$ of the constituents (see, e.g., [18], [19]), and the chirality coefficient $\ell_{chiral}$. The Cosserat intrinsic length and the chirality coefficient are defined as follows,

$$\mathcal{M} = \varepsilon \ell_{coss}^2, \quad \mathcal{C} = \varepsilon \ell_{coss}^{1/p} \ell_{chiral}^{1/p'} \quad \text{with} \quad \frac{1}{p} + \frac{1}{p'} = 1, \quad p, p' \in (1, +\infty), \qquad (3.5)$$

where $E = \max_{z \in Y_{\ell}} |E_{jikl}(z)|$, $C = \max_{z \in Y_{\ell}} |C_{jikl}(z)|$, $M = \max_{z \in Y_{\ell}} |M_{jikl}(z)|$ for i, j, k, l $\in \{1, 2, 3\}$, and $Y_{\ell} = (-\ell/2, \ell/2]^3$ is periodic cell characterizing the body $\Omega$. Additionally, the non-symmetric strain and curvature scale as,

$$\mathcal{S}^* = \mathcal{S} \text{ and } \mathcal{K}^* = \mathcal{L} \mathcal{K}. \qquad (3.6)$$

Moreover, we define the scaled stress, couple-stress, and fourth order material tensors as follows,

$$\sigma^* = \varepsilon^{-1} \sigma, \quad \mu^* = (\varepsilon L)^{-1} \mu, \qquad (3.7)$$

$$\varepsilon^*(\mathbf{x}^*) = \varepsilon^{-1} \mathbf{E}(\mathbf{x}), \quad \mathcal{C}^*(\mathbf{x}^*) = \mathcal{C}^{-1} \mathcal{C}(\mathbf{x}), \quad \mathcal{M}^*(\mathbf{x}^*) = \mathcal{M}^{-1} \mathcal{M}(\mathbf{x}). \qquad (3.8)$$

We remark that the fourth order material tensors $E^*(\mathbf{x}^*)$, $C^*(\mathbf{x}^*)$, and $M^*(\mathbf{x}^*)$ are $Y^*$ periodic where,

<footnote>$^8$Chiral materials exhibit a distinctive quality wherein they lack invariance under a coordinate transformation in-volving inversion. This implies an inherent orientation within the body. In the realm of three dimensions, this orientation is defined by adopting either a right-hand rule or a left-hand rule convention to determine the positioning of the axes. Frequently, in scholarly literature, chirality and non-centrosymmetry have been used interchangeably. However, this interchangeability holds true exclusively in spatial dimensions of three or more. In two dimensions, these two concepts diverge. For clarity, our focus in this current study will adhere to three dimensions, thereby sidestepping potential confusion. Readers seeking further insight are encouraged to refer to the recent work in [2].</footnote>
$$Y^* = \frac{\ell}{L} Y, \quad Y := \left(-\frac{1}{2}, \frac{1}{2}\right]^3. \qquad (3.9)$$

150 Hence, the system of equations in (2.5) scales as:

$$\begin{aligned} -\operatorname{div} \sigma^* &= \mathbf{f}^* && \text{in } \Omega, \\ -\operatorname{div} \mu^* - \mathcal{E}:\sigma^* &= \ell^* && \text{in } \Omega, \end{aligned} \qquad (3.10)$$

151 where $\mathbf{f}^*$, and $\ell^*$ are the appropriately scaled body forces and body couples<fnref n="9" /> (see [18, Eq. (14), pg. 4589]) and with constitutive laws, for i, j ∈ {1, 2, 3}:

$$\sigma_{ji}^* = \sum_{k, \ell=1}^{3} \left( E_{jikl}^* s_{kl}^* + \left(\frac{\ell_{\text{coss}}}{L}\right)^{1/p} \left(\frac{\ell_{\text{chiral}}}{L}\right)^{1/p'} C_{ijk\ell}^* \kappa_{k\ell}^* \right), \qquad (3.11)$$

$$\mu_{ji}^* = \sum_{k, \ell=1}^{3} \left( \left( \frac{\ell_{\text{coss}}}{L} \right)^{1/p} \left( \frac{\ell_{\text{chiral}}}{L} \right)^{1/p'} C_{k\ell ij}^* s_{k\ell}^* + \left( \frac{\ell_{\text{coss}}}{L} \right)^2 M_{jik\ell}^* \kappa_{k\ell}^* \right). \qquad (3.12)$$

153 Thus, one can generate an $\varepsilon$ periodic problem by defining the nondimensional number $\varepsilon$ as the ratio of $\ell$/L and let $\varepsilon \to 0$ to obtain an effective medium. However, different cases ought to be considered depending on how $\ell_{\text{coss}}$ and $\ell_{\text{chiral}}$ scale with $\ell$ and L, respectively. Here we consider the case where $p = p' = 2$,

$$\ell_{\text{coss}}/L \sim \varepsilon^{1/2}, \quad \ell_{\text{chiral}}/L \sim \varepsilon^{3/2}. \qquad \text{(scaling scheme)}$$

### 3.1.1 Scaling of the stress and couple-stress

158 The stress and couple-stress equations scale<fnref n="10" /> then as follows,

$$\begin{aligned} \sigma_{ij}^{\varepsilon} &= \sum_{k, \ell=1}^{3} \left( E_{ijkl} \left(\frac{\mathbf{x}}{\varepsilon}\right) s_{kl}^{\varepsilon} + \varepsilon C_{ijkl} \left(\frac{\mathbf{x}}{\varepsilon}\right) \mathcal{L}_{kl}^{\varepsilon} \right), \\ \mu_{ij}^{\varepsilon} &= \sum_{k, \ell=1}^{3} \left( \varepsilon C_{klij} \left(\frac{\mathbf{x}}{\varepsilon}\right) \mathcal{L}_{kl}^{\varepsilon} + \varepsilon M_{ijkl} \left(\frac{\mathbf{x}}{\varepsilon}\right) s_{kl}^{\varepsilon} \right). \end{aligned} \qquad (3.13)$$

<footnote n="9">⁹For expediting the presentation, we have also absorbed into the body forces and body couples the mass density, $\rho$, and the microinertia density, $\rho_t$, respectively.</footnote>

<footnote n="10">¹⁰This scaling in this work is quite different than the scaling introduced in [18]. As we will see in the preceding section where we make the homogenization rigorous through the periodic unfolding technique, some more effort is required to pass to the limit for the rotations under this scheme. The reader is invited to compare this result with the work in [42], [44].</footnote>
## 3.2 The microscopic mathematical model

We consider an elastic solid with periodic microstructure of period $\epsilon$ occupying a region $\Omega$. The region $\Omega$, that the body occupies, is assumed to be a connected (possibly non-simple), open set of $\mathbb{R}^3$ with a Lipschitz boundary. $Y = (-1/2, 1/2]^3$ is the unit cube in $\mathbb{R}^3$ and $Z^3$ is the set of all vectors with integer components. For every positive $\epsilon$, let $N_\epsilon$ be the set of all points $m \in \mathbb{Z}^3$ such that $\epsilon(m + Y)$ is strictly included in $\Omega$. Let $T \subset \text{interior}(Y)$ be a Lipschitz domain such that $T \subset Y$ with $T_m^\epsilon := \epsilon(m + T)$ will represent the region containing one of the material phases (see Fig. 2). Hence, we can define the following subsets of $\Omega^{11}$:

$$ \Omega_{1\epsilon} := \bigcup_{m \in N_{\epsilon}} T_m^{\epsilon}, \quad \Omega_{2\epsilon} := \Omega \setminus \Omega_{1\epsilon}, \quad \Omega := \Omega_{1\epsilon} \cup \Omega_{2\epsilon}. $$

<figure><img src="image_3.png" /><figcaption>Figure 2: Schematic of the heterogeneous Cosserat medium with perfect transmission conditions. The red arrows are there to indicate that the body can also independently rotate in addition to displacing.</figcaption></figure>

<footnote n="11">¹¹In this work, we assume perfect contact between the two different phases of the body.</footnote>
$$
\begin{align}
-\operatorname{div} \sigma^{\epsilon} &= \mathbf{f} && \text{in } \Omega, \nonumber \\
-\operatorname{div} \mu^{\epsilon} - \epsilon\sigma^{\epsilon} &= \boldsymbol{\ell} && \text{in } \Omega, \nonumber \\
\sigma \cdot \mathbf{n} &= \mathbf{0} && \text{on } \Sigma_N, \nonumber \\
\mu \cdot \mathbf{n} &= \mathbf{0} && \text{on } \Sigma_N, \nonumber \\
\mathbf{u}^{\epsilon} &= \mathbf{0} && \text{on } \Sigma_D, \nonumber \\
\varphi^{\epsilon} &= \mathbf{0} && \text{on } \Sigma_D. \tag{3.14}
\end{align}
$$

$$
\begin{bmatrix} \sigma^{\epsilon} \\ \mu^{\epsilon} \end{bmatrix} = \begin{bmatrix} E^{\epsilon} & C^{\epsilon} \\ C^{\epsilon \top} & M^{\epsilon} \end{bmatrix} \begin{bmatrix} \mathfrak{s}^{\epsilon} \\ \mathfrak{k}^{\epsilon} \end{bmatrix} \qquad (3.15)
$$

$$
\begin{equation}
\begin{split}
E_{jikl}^{\epsilon}(\mathbf{x}) := E_{jikl}(\frac{\mathbf{x}}{\epsilon}) ={}& (\vartheta(\frac{\mathbf{x}}{\epsilon}) + \alpha(\frac{\mathbf{x}}{\epsilon})) \delta_{jk} \delta_{il} \\
& + (\vartheta(\frac{\mathbf{x}}{\epsilon}) - \alpha(\frac{\mathbf{x}}{\epsilon})) \delta_{jl} \delta_{ik} + \bar{\omega}(\frac{\mathbf{x}}{\epsilon}) \delta_{ij} \delta_{kl},
\end{split}
\tag{3.16}
\end{equation}
$$

$$
\begin{equation}
\begin{split}
\mathbf{M}_{jikl}^{\epsilon}(\mathbf{x}) := {}& \mathbf{M}_{jikl}(\frac{\mathbf{x}}{\epsilon}) = (\theta(\frac{\mathbf{x}}{\epsilon}) + \beta(\frac{\mathbf{x}}{\epsilon})) \delta_{jk} \delta_{il} \\
& + (\theta(\frac{\mathbf{x}}{\epsilon}) - \beta(\frac{\mathbf{x}}{\epsilon})) \delta_{jl} \delta_{ik} + \omega(\frac{\mathbf{x}}{\epsilon}) \delta_{ij} \delta_{kl},
\end{split}
\tag{3.17}
\end{equation}
$$

$$
\begin{align}
C_{jikl}^{\epsilon}(\mathbf{x}) :=& C_{jikl}(\frac{\mathbf{x}}{\epsilon}) \nonumber \\
&= c_1(\frac{\mathbf{x}}{\epsilon}) \delta_{kl} \delta_{ij} + c_2(\frac{\mathbf{x}}{\epsilon}) \delta_{ik} \delta_{jl} + c_3(\frac{\mathbf{x}}{\epsilon}) \delta_{il} \delta_{jk}, \tag{3.18}
\end{align}
$$

<footnote n="167">167 Moreover, we denote by $\Sigma = \partial\Omega$ the boundary of $\Omega$ such that $\Sigma = \Sigma_D \cup \Sigma_N$ with $\Sigma_D \cap \Sigma_N = \emptyset^{12}$.</footnote>

<footnote n="168">168 The vector **n** will denote the unit normal on Σ pointing in the outward direction. Hence, the</footnote>

<footnote n="169">169 heterogeneous Cosserat continuum is characterized by the following coupled system:</footnote>

<footnote n="170">170 Here $\sigma^\epsilon$ is the stress, $\mu^\epsilon$ is the couple-stress, $\mathbf{u}^\epsilon$ is the displacement, and $\varphi^\epsilon$ is the rotation. The
171 system of equations (3.14) characterizes the mechanical deformation that the body undergoes. The
172 equations are fully coupled and the system is closed with homogeneous Dirichlet boundary con-
173 ditions on $\Sigma$. In addition to computing the displacement we must also compute the rigid rotations
174 which makes for a fully coupled system of partial differential equations.</footnote>

<footnote n="175">175 **Constitutive relations.** The stress and couple-stress are related to the strain and curvature-twist, respectively, though the fourth order material tensors $E^{\epsilon}(\mathbf{x}) := E(\frac{\mathbf{x}}{\epsilon})$, $C^{\epsilon}(\mathbf{x}) := C(\frac{\mathbf{x}}{\epsilon})$, and $M^{\epsilon}(\mathbf{x}) := M(\frac{\mathbf{x}}{\epsilon})$ by the relation (see [46]),</footnote>

<footnote n="178">178 The tensors $E^{\epsilon}$, $M^{\epsilon}$, and $C^{\epsilon}$ are linear isotropic tensors in each phase and have the following form,</footnote>

<footnote n="12">¹²The component $\Sigma_D$ ($\Sigma_N$) represents the part of the boundary where we are going to apply a Dirichlet (Neumman / traction) boundary condition.</footnote>
respectively. The coefficients $\vartheta, \theta, \alpha, \beta, \omega, \omega$, and $c_t$ for $t = 1, 2, 3$ are material parameters that are piece-wise constant in each phase. The restrictions on the coefficients are the following<fnref n="13" /> (see [46], [47]),

$$ \begin{array}{l} \vartheta > 0, \quad 3\omega + 2\vartheta > 0, \\ \vartheta + \alpha > 0, \quad \alpha > 0, \end{array} \qquad \begin{array}{l} \vartheta > 0, \quad 3\omega + 2\vartheta > 0, \\ \vartheta + \beta > 0, \quad \beta > 0. \end{array} \qquad (3.19) $$

### 3.2.1 Assumptions

We frame model (3.14) under the following general assumptions:

* The body force $f$ and body couple $\ell$ are such that $f \in L^2(\Omega; \mathbb{R}^3)$ and $\ell \in H(\text{curl}, \Omega)$.

* The fourth order material characterization tensors $E^\epsilon, M^\epsilon, C^\epsilon$ belong in $L^\infty$. Moreover, $E^\epsilon$ and $M^\epsilon$ satisfy the Legendre condition, i.e for any fourth rank tensor $A$,

$$ A_{\alpha\beta ij} \xi_i^\alpha \xi_j^\beta \geq \lambda |\xi|^2, \quad \forall \xi \in \mathbb{R}^{3 \times 3} \qquad (3.20) $$

where $\lambda > 0$.

**Existence and uniqueness.** Denote by,

$$ J_{\epsilon}[\mathbf{v}, \mathbf{\psi}] = \frac{1}{2} \int_{\Omega} [\mathbf{s} \otimes \mathbf{k}] \begin{bmatrix} \mathbf{E}^{\epsilon} & \epsilon \mathbf{C}^{\epsilon} \\ \epsilon \mathbf{C}^{\epsilon}^{\top} & \epsilon \mathbf{M}^{\epsilon} \end{bmatrix} [\mathbf{s} \otimes \mathbf{k}] \, d\mathbf{x} \qquad (3.21) $$

where $\mathbf{s} := \nabla \mathbf{v} - \mathbf{\epsilon} \cdot \mathbf{\phi}$ and $\mathbf{k} := \nabla \mathbf{\psi}$. Then the total energy of the system described in (3.14),

$$ \mathcal{E}_{\epsilon}[\mathbf{v}, \mathbf{\psi}] = J_{\epsilon}[\mathbf{v}, \mathbf{\psi}] - \int_{\Omega} (\mathbf{f} \cdot \mathbf{v} + \mathbf{l} \cdot \mathbf{\psi}) \, d\mathbf{x}. \qquad (3.22) $$

Computing the Euler-Lagrange equations and grouping terms by test function we can recover system (3.14) in the sense of distributions. Moreover, we readily verify that the above energy is sequentially weakly lower semicontinuous in $H_{\Sigma_D}^1(\Omega, \mathbb{R}^3) \times H_{\Sigma_D}^1(\Omega, \mathbb{R}^3)$, where $H_{\Sigma_D}^1(\Omega, \mathbb{R}^3) = \{\mathbf{g} \in H^1(\Omega, \mathbb{R}^3) | \mathbf{g}|_{\Sigma_D} = 0\}$ and $\mathbf{g}|_{\Sigma_D}$ is the usual trace operator and coercive. Then the direct method will guarantee the existence of a unique minimizer. Furthermore, one has the estimate<fnref n="14" />,

$$ \begin{align} \|\mathbf{u}^{\epsilon}\|_{H_{\Sigma_D}^1(\Omega; \mathbb{R}^3)}^2 &+ (\|\boldsymbol{\varphi}^{\epsilon}\|_{L^2(\Omega; \mathbb{R}^3)}^2 + \epsilon \|\nabla \boldsymbol{\varphi}^{\epsilon}\|_{L^2(\Omega; \mathbb{R}^{3 \times 3})}^2) \nonumber \\ &\leq c (\|\mathbf{f}\|_{L^2(\Omega; \mathbb{R}^3)}^2 + \|\mathbf{l}\|_{L^2(\Omega; \mathbb{R}^3)}^2), \tag{3.23} \end{align} $$

for some generic constant $c$ independent of $\epsilon$.

<footnote n="13">¹³The constants $c_1, c_2$, and $c_3$ in the pseudo tensor $C^\epsilon$ are the chirality constants (see, e.g., [30]). The tensor changes sign under an inversion of coordinates transformation. No restrictions, as the ones mentioned for the other coefficients in the case of isotropy, seem to exist in the literature for the coefficients $c_1, c_2$, and $c_3$.</footnote>

<footnote n="14">¹⁴All of the above can be proven in a similar fashion as in [42] by simply making the necessary adjustments to the coefficients.</footnote>
### 3.2.2 The periodic unfolding operator

We define the following domain decompositions (see [8, 12, 9, 10]):

$$ K_{\varepsilon}^{-} := \{ \ell \in \mathbb{Z}^3 \mid \varepsilon(\ell + Y) \subset \bar{\Omega} \}, \quad \Omega_{\varepsilon}^{-} := \text{int}(\cup_{\ell \in K_{\varepsilon}^{-}} \varepsilon(\ell + Y)), \quad \Lambda_{\varepsilon}^{-} := \Omega \setminus \Omega_{\varepsilon}^{-}. \quad (3.24) $$

<figure><img src="image_4.png" /><figcaption>Figure 3: Unfolding operator on a periodic grid.</figcaption></figure>

Let $[\mathbf{z}]_Y = ([z_1], [z_2], [z_3])$ denote the integer part of $\mathbf{z} \in \mathbb{R}^3$ and denote by $\{\mathbf{z}\}_Y$ the difference $\mathbf{z} - [\mathbf{z}]_Y$ which belongs to $Y$. Regarding our multiscale problem that depends on a small length parameter $\varepsilon > 0$, we can decompose any $\mathbf{x} \in \mathbb{R}^3$ using the maps $[\cdot]_Y : \mathbb{R}^3 \to \mathbb{Z}^3$ and $\{\cdot\}_Y : \mathbb{R}^3 \to Y$ the following way (see Fig. 3 (right)),

$$ \mathbf{x} = \varepsilon \left( \left[ \frac{\mathbf{x}}{\varepsilon} \right]_Y + \left\{ \frac{\mathbf{x}}{\varepsilon} \right\}_Y \right). \quad (3.25) $$

For any Lebesgue measurable function $\phi$ on $\Omega$ we define the periodic unfolding operator by,

$$ \mathcal{T}_{\varepsilon}(\phi)(\mathbf{x}, \mathbf{y}) = \begin{cases} \phi(\varepsilon[\frac{\mathbf{x}}{\varepsilon}]_Y + \varepsilon\mathbf{y}) & \text{for a.e. } (\mathbf{x}, \mathbf{y}) \in \Omega_{\varepsilon}^{-} \times Y \\ 0 & \text{for a.e. } (\mathbf{x}, \mathbf{y}) \in \Lambda_{\varepsilon}^{-} \times Y. \end{cases} \quad (3.26) $$

In a similar fashion we define the local average operator $\mathcal{M}_{\varepsilon} : L^p(\Omega) \to L^p(\Omega)$ for $p \in [1, +\infty]$,

$$ \mathcal{M}_{\varepsilon}(\phi)(\mathbf{x}) = \begin{cases} \frac{1}{|Y|} \int_Y \phi(\varepsilon[\frac{\mathbf{x}}{\varepsilon}] + \varepsilon\mathbf{z}) \, d\mathbf{z} & \text{for a.e. } \mathbf{x} \in \Omega_{\varepsilon}^{-} \\ 0 & \text{for a.e. } \mathbf{x} \in \Lambda_{\varepsilon}^{-}. \end{cases} \quad (3.27) $$

Note that the local average $\mathcal{M}_{\varepsilon}$ is connected to the unfolding operator $\mathcal{T}_{\varepsilon}$ via the average over $Y$, namely,

$$ \mathcal{M}_{\varepsilon}(\phi) = \frac{1}{|Y|} \int_{Y} \mathcal{T}_{\varepsilon}(\phi)(\cdot, \mathbf{y}) \, d\mathbf{y} \quad (3.28) $$
**Proposition 3.1.** For any $p \in [1, +\infty)$ the unfolding operator $\mathcal{T}_\varepsilon : L^p(\Omega) \mapsto L^p(\Omega \times Y)$ is linear, continuous, and has the following properties:

I. $\mathcal{T}_\varepsilon(\phi \psi) = \mathcal{T}_\varepsilon(\varphi) \mathcal{T}_\varepsilon(\psi)$ for every pair of Lebesgue measurable functions $\varphi, \psi$ on $\Omega$

II. For every $\phi \in L^1(\Omega)$ we have,

$$ \frac{1}{|Y|} \int_{\Omega \times Y} \mathcal{T}_\varepsilon(\phi)(\mathbf{x}, \mathbf{y}) \, d\mathbf{x} \, d\mathbf{y} = \int_{\Omega_\varepsilon} \phi(\mathbf{x}) \, d\mathbf{x} = \int_{\Omega_\varepsilon} \phi(\mathbf{x}) \, d\mathbf{x} - \int_{\Lambda_\varepsilon} \phi(\mathbf{x}) \, d\mathbf{x} \quad (3.29) $$

III. $\|\mathcal{T}_\varepsilon(\phi)\|_{L^p(\Omega \times Y)} \le |Y|^{1/p} \|\phi\|_{L^p(\Omega)}$ for every $\phi \in L^p(\Omega)$

IV. $\mathcal{T}_\varepsilon(\phi) \to \phi$ strongly in $L^p(\Omega \times Y)$ for $\phi \in L^p(\Omega)$ as $\varepsilon \to 0$

V. If $\{\phi_\varepsilon\}_\varepsilon$ is a sequence in $L^p(\Omega)$ such that $\phi_\varepsilon \to \phi$ strongly in $L^p(\Omega)$, then $\mathcal{T}_\varepsilon(\phi_\varepsilon) \to \phi$ strongly in $L^p(\Omega \times Y)$

VI. If $\phi \in L^p(Y)$ is $Y$-periodic and $\phi_\varepsilon(\mathbf{x}) = \phi(\frac{\mathbf{x}}{\varepsilon})$ then $\mathcal{T}_\varepsilon(\phi_\varepsilon) \to \phi$ strongly in $L^p(\Omega \times Y)$ as $\varepsilon \to 0$

VII. If $\phi_\varepsilon \to \phi$ in $H^1(\Omega)$ then there exists an non-relabeled subsequence and a $\hat{\phi} \in L^2(\Omega; H_{\text{per}}^1(Y))$ such that

a. $\mathcal{T}_\varepsilon(\phi_\varepsilon) \to \phi$ in $L^2(\Omega; H^1(Y))$

b. $\mathcal{T}_\varepsilon(\nabla\phi_\varepsilon) \to \nabla_x\phi + \nabla_y\hat{\phi}$ in $L^2(\Omega \times Y)$

VIII. Let $\phi_\varepsilon \in H^1(\Omega)$ and assume that $\{\phi_\varepsilon\}_\varepsilon$ is converging weakly to some $\phi$ in $L^2(\Omega)$ and satisfying for some strictly positive sequence $\{\delta(\varepsilon)\}_\varepsilon$ converging to zero,

$\delta(\varepsilon) \|\nabla\phi_\varepsilon\|_{L^2(\Omega)}$ is bounded as $\varepsilon$ tends to zero.

If $\lim_{\varepsilon \to 0} \delta(\varepsilon)/\varepsilon = +\infty$ then,

a. $\mathcal{T}_\varepsilon(\phi_\varepsilon) \to \phi$ in $L^2(\Omega; H^1(Y))$,

b. $\nabla_y \mathcal{T}_\varepsilon(\phi_\varepsilon) \to 0$ in $L^2(\Omega \times Y; \mathbb{R}^3)$.

Furthermore, there exists a $\hat{\phi} \in L^2(\Omega; H_{\text{per}}^1(Y))$ with zero average over $Y$, such that, up to a sub-sequence$^{15}$,

c. $\frac{\delta(\varepsilon)}{\varepsilon} \mathcal{T}_\varepsilon(\phi_\varepsilon - \mathcal{M}_\varepsilon(\phi_\varepsilon)) \to \hat{\phi}$ in $L^2(\Omega; H^1(Y))$,

d. $\delta(\varepsilon) \mathcal{T}_\varepsilon(\nabla\phi_\varepsilon) \to \nabla_y\hat{\phi}$ in $L^2(\Omega \times Y; \mathbb{R}^3)$.

<footnote>$^{15}$ Property VIII. is not as well-known in classical homogenization literature. Nevertheless, this result, along with its proof, can be found in [10, Lemma 1.44 (ii), pg. 32].</footnote>
## 3.3 Presentation and discussion of the results

**Theorem 3.1.** If $(\mathbf{u}^\epsilon, \boldsymbol{\phi}^\epsilon)$ is the solution set to (3.14) under the scaling scheme with $\delta(\epsilon):=\epsilon^{1/2}$ then there exist $\mathbf{u}^0 \in H_{\Sigma_D}^1(\Omega; \mathbb{R}^3)$, $\mathbf{u}^1 \in L^2(\Omega; H_{\text{per}}^1(Y; \mathbb{R}^3))$, $\boldsymbol{\phi}^0 \in L^2(\Omega; H_{\text{per}}^1(Y; \mathbb{R}^3))$ such that,

$$ \mathcal{T}_{\epsilon}(\mathbf{u}^{\epsilon}) \to \mathbf{u}^{0} \text{ in } L^{2}(\Omega; H^{1}(Y; \mathbb{R}^{3})), \quad (3.30) $$

$$ \mathcal{T}_{\epsilon}(\nabla \mathbf{u}^{\epsilon}) \to \nabla_{\mathbf{x}} \mathbf{u}^{0} + \nabla_{\mathbf{y}} \mathbf{u}^{1} \text{ in } L^{2}(\Omega \times Y; \mathbb{R}^{3} \times 3), \quad (3.31) $$

$$ \mathcal{T}_{\epsilon}(\boldsymbol{\phi}^{\epsilon}) \to \boldsymbol{\phi}^{0} \text{ in } L^{2}(\Omega; H^{1}(Y; \mathbb{R}^{3})), \quad (3.32) $$

$$ \frac{\delta(\epsilon)}{\epsilon} \mathcal{T}_{\epsilon}(\boldsymbol{\phi}_{\epsilon} - \mathcal{M}_{\epsilon}(\boldsymbol{\phi}_{\epsilon})) \to \boldsymbol{\phi} \text{ in } L^{2}(\Omega; H^{1}(Y; \mathbb{R}^{3})), \quad (3.33) $$

$$ \delta(\epsilon) \mathcal{T}_{\epsilon}(\nabla \boldsymbol{\phi}_{\epsilon}) \to \nabla_{\mathbf{y}} \boldsymbol{\phi} \text{ in } L^{2}(\Omega \times Y; \mathbb{R}^{3}) \quad (3.34) $$

and $(\mathbf{u}^0, \boldsymbol{\phi}^0, \mathbf{u}^1)$ is the unique solution set of,

$$ \frac{1}{|Y|} \int_{\Omega \times Y} \left[ \mathbf{E}(\mathbf{y})(\nabla_{\mathbf{x}}\mathbf{u}^0 + \nabla_{\mathbf{y}}\mathbf{u}^1 - \epsilon.\boldsymbol{\phi}^0) \right] : (\nabla_{\mathbf{x}}\mathbf{V} + \nabla_{\mathbf{y}}\overline{\mathbf{W}} - \epsilon.\mathbf{\Psi}) \, d\mathbf{y} \, d\mathbf{x} \quad (3.35) \\ - \frac{1}{|Y|} \int_{\Omega \times Y} (\mathbf{f} \cdot \mathbf{V} + \ell \cdot \mathbf{\Psi}) \, d\mathbf{y} \, d\mathbf{x} = 0. $$

for all $(\mathbf{V}, \mathbf{\Psi}) \in (H_{\Sigma}^1(\Omega; \mathbb{R}^3), L^2(\Omega, \mathbb{R}^3))$ and $\overline{\mathbf{W}} \in L^2(\Omega; H^1(Y; \mathbb{R}^3))$.

Equation (3.35) can be written in the sense of distributions as the following system of odd elasticity,

$$ \begin{gathered} \operatorname{div} \sigma^{\text{eff}} := \mathbf{f} \text{ in } \mathcal{D}(\Omega), \\ \epsilon: \sigma^{\text{eff}} + \underbrace{\epsilon:(1/2\epsilon \cdot \mathbf{l})}_{\text{body torques}} = \mathbf{0} \text{ in } \mathcal{D}(\Omega), \quad (3.36) \end{gathered} $$

with the effective stress<fnref n="16" /> $\sigma^{\text{eff}}:=\mathbf{E}^{\text{eff}}:\nabla_{\mathbf{x}}\mathbf{u}^0$, the homogenized coefficients,

$$ \mathbf{E}_{\text{jipq}}^{\text{eff}} := \frac{1}{|Y|} \int_{Y} \mathbf{E}_{\text{jikl}}(\mathbf{y}) \left( \delta_{\text{kp}} \delta_{\text{ql}} + \partial_{y_l} \mathbf{w}_{\text{k}}^{\text{pq}} - \epsilon_{\text{vkl}} \epsilon_{\text{v\alpha\beta}} \partial_{y_\beta} \mathbf{w}_{\text{\alpha}}^{\text{pq}} \right) d\mathbf{y}, \quad (3.37) $$

and $\mathbf{w}^{\text{pq}}$ the local solution<fnref n="17" /> satisfying the problem,

$$ \begin{gathered} \mathbf{w}^{\text{pq}} \in H_{\text{per}}^{1}(Y; \mathbb{R}^{3}), \quad \int_{Y} \mathbf{w}^{\text{pq}} \, d\mathbf{y} = 0, \quad (3.38) \\ \int_{Y} \mathbf{E}(\mathbf{y}): (\mathbf{e}_{\text{p}} \otimes \mathbf{e}_{\text{q}} + \nabla_{\mathbf{y}} \mathbf{w}^{\text{pq}} - \epsilon \cdot \text{curl}_{\mathbf{y}} \mathbf{w}^{\text{pq}}) : \nabla_{\mathbf{y}} \mathbf{v} \, d\mathbf{y} = 0 \quad \text{for all } \mathbf{v} \in H_{\text{per}}^{1}(Y; \mathbb{R}^{3}). \end{gathered} $$

<footnote n="16">¹⁶This is not the Cauchy stress per se. For a precise definition of the Cauchy stress the reader can consult the book by P. Ciarlet [6].</footnote>

<footnote n="17">¹⁷We remark here that there is only one local solution. By contrast, in the works of [18], [42], [44] there are two local problems. The approach herein is both mathematically more consistent as well as computationally more efficient since there is only one local problem.</footnote>
239 **Lemma 3.1.** The homogenized system in (3.36) can be written, equivalently,

$$ -\operatorname{div} \mathfrak{T}^{\text{Cauchy}} = \frac{1}{2} \operatorname{curl}_x \boldsymbol{\ell} + \mathbf{f} \text{ in } \mathcal{D}(\Omega), \qquad (3.39) \\ \mathfrak{T}^{\text{Cauchy}} = (\mathfrak{T}^{\text{Cauchy}})^{\top}, $$

240 with $\mathfrak{T}^{\text{Cauchy}}:=\varphi^{\text{eff}}+\frac{1}{2}\boldsymbol{\epsilon}.\boldsymbol{\ell}$ the Cauchy stress$^{18}$.

241 **Remark 3.1.** Theorem 3.1 provides an important link between the theory of odd elasticity and generalized continua through multi-continuum homogenization. Essential our argument says that the theory of odd elasticity can been seen as an approximating limit of a Cosserat theory under the scaling assumption in our outlined scaling scheme. Hence, in the context of periodic homogenization we can provide even more information regarding the stiffness tensor based on morphology, volume fraction, and absolute size of the constituents. This latter property is a novelty from the point of view of homogenization. Thus, within the above context, the homogenized system an effective odd elastic system.

249 **Remark 3.2.** Lemma 3.1 establishes a crucial connection between the stress encountered in the theory of odd elasticity (which is asymmetric) and the conventional Cauchy stress (which is symmetric). The lemma addresses a natural question: Given that the quantity $\varphi^{\text{eff}}$ in Theorem 3.1 does not conserve angular momentum, what quantity does then conserve angular momentum and how is it related to $\varphi^{\text{eff}}$?

254 **3.4 Proofs of the main results**

255 **3.4.1 Proof of Theorem 3.1**

256 **Theorem 3.1.** If $(\mathbf{u}^{\varepsilon}, \boldsymbol{\varphi}^{\varepsilon})$ is the solution set to (3.14) under the scaling scheme with $\delta(\varepsilon):=\varepsilon^{1/2}$ then there exist $\mathbf{u}^{0} \in H_{\Sigma_D}^{1}(\Omega; \mathbb{R}^3)$, $\mathbf{u}^{1} \in L^2(\Omega; H_{\text{per}}^{1}(Y; \mathbb{R}^3))$, $\boldsymbol{\varphi}^{0} \in L^2(\Omega; H_{\text{per}}^{1}(Y; \mathbb{R}^3))$ such that,

$$ \mathcal{T}_{\varepsilon}(\mathbf{u}^{\varepsilon}) \to \mathbf{u}^{0} \text{ in } L^2(\Omega; H^1(Y; \mathbb{R}^3)), \qquad (3.30) $$

$$ \mathcal{T}_{\varepsilon}(\nabla \mathbf{u}^{\varepsilon}) \to \nabla_{\mathbf{x}} \mathbf{u}^{0} + \nabla_{\mathbf{y}} \mathbf{u}^{1} \text{ in } L^2(\Omega \times Y; \mathbb{R}^{3 \times 3}), \qquad (3.31) $$

$$ \mathcal{T}_{\varepsilon}(\boldsymbol{\varphi}^{\varepsilon}) \to \boldsymbol{\varphi}^{0} \text{ in } L^2(\Omega; H^1(Y; \mathbb{R}^3)), \qquad (3.32) $$

$$ \frac{\delta(\varepsilon)}{\varepsilon} \mathcal{T}_{\varepsilon}(\boldsymbol{\varphi}_{\varepsilon} - \mathcal{M}_{\varepsilon}(\boldsymbol{\varphi}_{\varepsilon})) \to \boldsymbol{\Phi} \text{ in } L^2(\Omega; H^1(Y; \mathbb{R}^3)), \qquad (3.33) $$

$$ \delta(\varepsilon) \mathcal{T}_{\varepsilon}(\nabla \boldsymbol{\varphi}_{\varepsilon}) \to \nabla_{\mathbf{y}} \boldsymbol{\Phi} \text{ in } L^2(\Omega \times Y; \mathbb{R}^3) \qquad (3.34) $$

258 and $(\mathbf{u}^0, \boldsymbol{\varphi}^0, \mathbf{u}^1)$ is the unique solution set of,

<footnote>$^{18}$The reader should remark that the relationship between effective stress and effective strain is not linear but rather affine. Nevertheless, in the proof of the lemma we will show that the effective stress is symmetric. Hence, the body-couples play an important role in stress equilibrium.</footnote>
$$ \frac{1}{|Y|} \int_{\Omega \times Y} \left[ \mathbf{E}(\mathbf{y})(\nabla_{\mathbf{x}}\mathbf{u}^0 + \nabla_{\mathbf{y}}\mathbf{u}^1 - \epsilon.\boldsymbol{\varphi}^0) \right] : (\nabla_{\mathbf{x}}\mathbf{V} + \nabla_{\mathbf{y}}\overline{\mathbf{W}} - \epsilon.\mathbf{\Psi}) \, d\mathbf{y} \, d\mathbf{x} \tag{3.35} \\ - \frac{1}{|Y|} \int_{\Omega \times Y} (\mathbf{f} \cdot \mathbf{V} + \boldsymbol{\ell} \cdot \mathbf{\Psi}) \, d\mathbf{y} \, d\mathbf{x} = 0. $$

for all $(\mathbf{V}, \mathbf{\Psi}) \in (H_{\Sigma}^{1}(\Omega; \mathbb{R}^{3}), L^{2}(\Omega, \mathbb{R}^{3}))$ and $\overline{\mathbf{W}} \in L^{2}(\Omega; H^{1}(Y; \mathbb{R}^{3}))$.

Equation (3.35) can be written in the sense of distributions as the following system of odd elasticity,

$$ \operatorname{div} \sigma^{\text{eff}} := \mathbf{f} \text{ in } \mathcal{D}(\Omega), \\ \epsilon: \sigma^{\text{eff}} + \underbrace{\epsilon:(1/2\epsilon \cdot \boldsymbol{\ell})}_{\text{body torques}} = \mathbf{0} \text{ in } \mathcal{D}(\Omega), \tag{3.36} $$

with the effective stress<fnref n="19" /> $\sigma^{\text{eff}}:=\mathbf{E}^{\text{eff}}:\nabla_{\mathbf{x}}\mathbf{u}^0$, the homogenized coefficients,

$$ \mathbf{E}_{\text{jipq}}^{\text{eff}} := \frac{1}{|Y|} \int_{Y} \mathbf{E}_{\text{jikl}}(\mathbf{y}) (\delta_{\text{kp}} \delta_{\text{ql}} + \partial_{y_l} w_k^{\text{pq}} - \epsilon_{\text{vkl}} \epsilon_{\text{v\alpha\beta}} \partial_{y_\beta} w_\alpha^{\text{pq}}) \, d\mathbf{y}, \tag{3.37} $$

and $\mathbf{w}^{\text{pq}}$ the local solution<fnref n="20" /> satisfying the problem,

$$ \mathbf{w}^{\text{pq}} \in H_{\text{per}}^{1}(Y; \mathbb{R}^{3}), \quad \int_{Y} \mathbf{w}^{\text{pq}} \, d\mathbf{y} = 0, \tag{3.38} \\ \int_{Y} \mathbf{E}(\mathbf{y}) : (\mathbf{e}_{\text{p}} \otimes \mathbf{e}_{\text{q}} + \nabla_{\mathbf{y}} \mathbf{w}^{\text{pq}} - \epsilon.\text{curl}_{\mathbf{y}} \mathbf{w}^{\text{pq}}) : \nabla_{\mathbf{y}} \mathbf{v} \, d\mathbf{y} = 0 \text{ for all } \mathbf{v} \in H_{\text{per}}^{1}(Y; \mathbb{R}^{3}). $$

Proof. Using the estimates from (3.23) and the results from Proposition 3.1 VII. and VIII. we obtain (3.30)–(3.34) with $\delta(\epsilon):=\epsilon^{1/2}$. Moreover, the weak form (3.14) of the Cosserat continuum reads:

Find $(\mathbf{u}^{\epsilon}, \boldsymbol{\varphi}^{\epsilon}) \in H_{\Sigma_D}^1(\Omega; \mathbb{R}^3) \times H_{\Sigma_D}^1(\Omega; \mathbb{R}^3)$ such that,

$$ \int_{\Omega} \mathbf{E}(\frac{\mathbf{x}}{\epsilon})(\nabla \mathbf{u}^{\epsilon} - \epsilon.\boldsymbol{\varphi}^{\epsilon})(\nabla \mathbf{v} - \epsilon.\mathbf{\psi}) \, d\mathbf{x} - \int_{\Omega} \epsilon \mathbf{C}(\frac{\mathbf{x}}{\epsilon}) \nabla \boldsymbol{\varphi}^{\epsilon} : (\nabla \mathbf{v} - \epsilon.\mathbf{\psi}) \, d\mathbf{x} \\ - \int_{\Omega} \epsilon \mathbf{C}(\frac{\mathbf{x}}{\epsilon})(\nabla \mathbf{u}^{\epsilon} - \epsilon.\boldsymbol{\varphi}^{\epsilon}) : \nabla \mathbf{\psi} \, d\mathbf{x} - \int_{\Omega} \epsilon \mathbf{M}(\frac{\mathbf{x}}{\epsilon}) \nabla \boldsymbol{\varphi}^{\epsilon} : \nabla \mathbf{\psi} \, d\mathbf{x} \tag{3.40} \\ - \int_{\Omega} (\mathbf{f} \cdot \mathbf{v} + \mathbf{g} \cdot \mathbf{\psi}) \, d\mathbf{x} = 0. $$

<footnote n="19">¹⁹This is not the Cauchy stress per se. For a precise definition of the Cauchy stress the reader can consult the book by P. Ciarlet [6].</footnote>

<footnote n="20">²⁰We remark here that there is only **one** local solution. By contrast, in the works of [18], [42], [44] there are two local problems. The approach herein is both mathematically more consistent as well as computationally more efficient since there is only one local problem.</footnote>
268 Unfold the above expression using Proposition 3.1 properties I., II., and IV. and obtain,

$$
\begin{align}
& \frac{1}{|Y|} \int_{\Omega \times Y} \mathcal{E}(\mathbf{y}) (\mathcal{T}_{\varepsilon}(\nabla \mathbf{u}^{\varepsilon}) - \varepsilon \cdot \mathcal{T}_{\varepsilon}(\boldsymbol{\varphi}^{\varepsilon})): (\mathcal{T}_{\varepsilon}(\nabla \mathbf{v}) - \varepsilon \cdot \mathcal{T}_{\varepsilon}(\boldsymbol{\psi})) \, d\mathbf{y} \, d\mathbf{x} \nonumber \\
& - \frac{1}{|Y|} \int_{\Omega \times Y} \varepsilon C(\mathbf{y}) \mathcal{T}_{\varepsilon}(\nabla \boldsymbol{\varphi}^{\varepsilon}): (\mathcal{T}_{\varepsilon}(\nabla \mathbf{v}) - \varepsilon \cdot \mathcal{T}_{\varepsilon}(\boldsymbol{\psi})) \, d\mathbf{x} \, d\mathbf{y} \nonumber \\
& - \frac{1}{|Y|} \int_{\Omega \times Y} \varepsilon C(\mathbf{y}) (\mathcal{T}_{\varepsilon}(\nabla \mathbf{u}^{\varepsilon}) - \varepsilon \cdot \mathcal{T}_{\varepsilon}(\boldsymbol{\varphi}^{\varepsilon})): \mathcal{T}_{\varepsilon}(\nabla \boldsymbol{\psi}) \, d\mathbf{x} \, d\mathbf{y} \tag{3.41} \\
& - \frac{1}{|Y|} \int_{\Omega \times Y} \varepsilon \mathcal{M}(\mathbf{y}) \mathcal{T}_{\varepsilon}(\nabla \boldsymbol{\varphi}^{\varepsilon}): \mathcal{T}_{\varepsilon}(\nabla \boldsymbol{\psi}) \, d\mathbf{y} \, d\mathbf{x} \nonumber \\
& - \frac{1}{|Y|} \int_{\Omega \times Y} (\mathcal{T}_{\varepsilon}(\mathbf{f}) \cdot \mathcal{T}_{\varepsilon}(\mathbf{v}) + \mathcal{T}_{\varepsilon}(\mathbf{g}) \cdot \mathcal{T}_{\varepsilon}(\boldsymbol{\psi})) \, d\mathbf{y} \, d\mathbf{x} = 0. \nonumber
\end{align}
$$

269 Set $(\mathbf{v}, \boldsymbol{\psi}) := (\mathbf{V}(\mathbf{x}), \boldsymbol{\Psi}(\mathbf{x}))$, where $(\mathbf{V}, \boldsymbol{\Psi}) \in \{C^{\infty}(\Omega; \mathbb{R}^3) \mid \mathbf{V}=0 \text{ on } \Sigma_D\} \times \{C^{\infty}(\Omega; \mathbb{R}^3) \mid \boldsymbol{\Psi}=0 \text{ on } \Sigma_D\}$
270 are arbitrary test functions, in (3.41) and let $\varepsilon \to 0$ to obtain,

$$
\frac{1}{|Y|} \int_{\Omega \times Y} \mathbf{E}(\mathbf{y}) (\nabla_{\mathbf{x}} \mathbf{u}_k^0 + \nabla_{\mathbf{y}} \mathbf{u}^1 - \varepsilon \cdot \mathbf{\varphi}^0) : (\nabla_{\mathbf{x}} \mathbf{V} - \varepsilon \cdot \mathbf{\Psi}) \, d\mathbf{y} \, d\mathbf{x} \\
- \frac{1}{|Y|} \int_{\Omega \times Y} (\mathbf{f} \cdot \mathbf{V} + \mathbf{g} \cdot \mathbf{\Psi}) \, d\mathbf{x} = 0. \tag{3.42}
$$

271 Select now a test function of the form $(\mathbf{v}, \boldsymbol{\psi})=(\mathbf{v}^{\varepsilon}, \boldsymbol{\psi}^{\varepsilon}) := (\varepsilon \mathbf{U}(\mathbf{x}) \mathbf{W}(\frac{\mathbf{x}}{\varepsilon}), \varepsilon \Phi(\mathbf{x}) \Xi(\frac{\mathbf{x}}{\varepsilon}))$ where $(\mathbf{U}, \Phi) \in C_0^{\infty}(\Omega) \times C_0^{\infty}(\Omega)$ and $(\mathbf{W}, \Xi) \in H_{\text{per}}^1(Y; \mathbb{R}^3) \times H_{\text{per}}^1(Y; \mathbb{R}^3)$. It is immediately clear that $(\mathbf{v}^{\varepsilon}, \boldsymbol{\psi}^{\varepsilon}) \to (\mathbf{0}, \mathbf{0})$ in $L^2(\Omega; \mathbb{R}^3) \times L^2(\Omega; \mathbb{R}^3)$. Moreover, for $\mathbf{v}^{\varepsilon}$ we have

$$
\partial_{x_j} v_i^{\epsilon}(\mathbf{x}) = \epsilon \partial_{x_j} \mathbf{U}(\mathbf{x}) \mathbf{W}_i(\mathbf{x}/\epsilon) + \mathbf{U}(\mathbf{x}) \partial_{y_j} \mathbf{W}_i(\mathbf{x}/\epsilon) \quad (3.43)
$$

274 which implies $\mathcal{T}_{\varepsilon}(\partial_{x_j} v_i^{\varepsilon}) \to \partial_{y_j} \overline{W}_i(\mathbf{x}, \mathbf{y})$ in $L^2(\Omega \times Y)$ as $\varepsilon \to 0$ where $\overline{W}_i(\mathbf{x}, \mathbf{y}) := \mathbf{U}(\mathbf{x}) W_i(\mathbf{y})$.
275 Likewise, for $\psi v^{\varepsilon}$, we have

$$
\partial_{x_j} \psi_i^\epsilon(\mathbf{x}) = \epsilon \partial_{x_j} \Phi(\mathbf{x}) \Xi_i(\mathbf{x}/\epsilon) + \Phi(\mathbf{x}) \partial_{y_j} \Xi_i(\mathbf{x}/\epsilon) \quad (3.44)
$$

276 which implies $\mathcal{T}_{\varepsilon}(\partial_{x_j} \psi_i^{\varepsilon}) \to \partial_{y_j} \Xi_i(\mathbf{x}, \mathbf{y})$ in $L^2(\Omega \times Y)$ as $\varepsilon \to 0$ where $\Xi_i(\mathbf{x}, \mathbf{y}) := \Phi(\mathbf{x}) \Xi_i(\mathbf{y})$.
277 Hence, unfolding (3.41) with the test function $(\mathbf{v}^{\varepsilon}, \boldsymbol{\psi}^{\varepsilon})$ and letting $\varepsilon$ tend to zero we obtain,

$$
\frac{1}{|Y|} \int_{\Omega \times Y} \mathbf{E}(\mathbf{y}) (\nabla_{\mathbf{x}} \mathbf{u}^0 + \nabla_{\mathbf{y}} \mathbf{u}^1 - \epsilon \cdot \mathbf{\varphi}^0) : \nabla_{\mathbf{y}} \overline{\mathbf{W}}(\mathbf{x}, \mathbf{y}) \, d\mathbf{y} \, d\mathbf{x} = 0. \quad (3.45)
$$

278 Adding (3.42) and (3.45) we arrive at,
$$ \frac{1}{|Y|} \int_{\Omega \times Y} \mathbf{E}(\mathbf{y})(\nabla_{\mathbf{x}} \mathbf{u}^0 + \nabla_{\mathbf{y}} \mathbf{u}^1 - \epsilon \cdot \boldsymbol{\varphi}^0):(\nabla_{\mathbf{x}} \mathbf{V} + \nabla_{\mathbf{y}} \bar{\mathbf{W}} - \epsilon \cdot \boldsymbol{\Psi}) \, d\mathbf{y} \, d\mathbf{x} = \frac{1}{|Y|} \int_{\Omega \times Y} (\mathbf{f} \cdot \mathbf{V} + \boldsymbol{\ell} \cdot \boldsymbol{\Psi}) \, d\mathbf{y} \, d\mathbf{x}. \quad (3.46) $$

By the density of $C_0^\infty(\Omega) \otimes H_{\text{per}}^1(Y; \mathbb{R}^3)$ in $L^2(\Omega; H_{\text{per}}^1(Y; \mathbb{R}^3))$ the above result holds for all $(\bar{\mathbf{W}}(\mathbf{x}, \mathbf{y}), \bar{\boldsymbol{\Xi}}(\mathbf{x}, \mathbf{y})) \in L^2(\Omega; H_{\text{per}}^1(Y; \mathbb{R}^3)) \times L^2(\Omega; H_{\text{per}}^1(Y; \mathbb{R}^3))$. From here we can obtain two sets of equations. By considering, first, that $(\mathbf{V}, \bar{\mathbf{W}}, \boldsymbol{\Psi}) = (\mathbf{V}, \bar{\mathbf{W}}, \mathbf{0})$ we have,

$$ \frac{1}{|Y|} \int_{\Omega \times Y} \mathbf{E}(\mathbf{y})(\nabla_{\mathbf{x}} \mathbf{u}^0 + \nabla_{\mathbf{y}} \mathbf{u}^1 - \epsilon \cdot \boldsymbol{\varphi}^0):(\nabla_{\mathbf{x}} \mathbf{V} + \nabla_{\mathbf{y}} \bar{\mathbf{W}}) \, d\mathbf{y} \, d\mathbf{x} = \frac{1}{|Y|} \int_{\Omega \times Y} \mathbf{f} \cdot \mathbf{V} \, d\mathbf{y} \, d\mathbf{x} = 0. \quad (\text{linear momentum}) $$

By considering, $(\mathbf{V}, \bar{\mathbf{W}}, \boldsymbol{\Psi}) = (\mathbf{0}, \mathbf{0}, \boldsymbol{\Psi})$ we have,

$$ -\frac{1}{|Y|} \int_{\Omega \times Y} \mathbf{E}(\mathbf{y})(\nabla_{\mathbf{x}} \mathbf{u}^0 + \nabla_{\mathbf{y}} \mathbf{u}^1 - \epsilon \cdot \boldsymbol{\varphi}^0):\epsilon \cdot \boldsymbol{\Psi} \, d\mathbf{y} \, d\mathbf{x} = -\frac{1}{|Y|} \int_{\Omega \times Y} \boldsymbol{\ell} \cdot \boldsymbol{\Psi} \, d\mathbf{y} \, d\mathbf{x} = 0. \quad (\text{angular momentum}) $$

Shifting our focus to the linear momentum equation, when we opt for $\mathbf{V} \equiv \mathbf{0}$, it becomes evident that both $\mathbf{u}^1$ and $\boldsymbol{\varphi}^0$ exhibit a linear dependence on $\nabla_{\mathbf{x}} \mathbf{u}^0$. Consequently, there are two outcomes for the local problem: it either lacks a solution OR the vector fields $\mathbf{u}^1$ and $\boldsymbol{\varphi}^0$ are dependent<fnref n="21" /> at the local level<fnref n="22" />. In the case where the displacement and rotation are dependent at the local level then a possible way to construct local solutions is as follows,

$$ \mathbf{u}^1(\mathbf{x}, \mathbf{y}) = \mathbf{w}^{pq}(\mathbf{y}) \partial_{x_q} \mathbf{u}_p^0(\mathbf{x}) + \mathbf{c}(\mathbf{x}), \quad (3.47) $$

$$ \boldsymbol{\varphi}^0(\mathbf{x}, \mathbf{y}) = \partial_{x_q} \mathbf{u}_p^0(\mathbf{x}) \operatorname{curl}_y (\mathbf{w}^{pq}), \quad (3.48) $$

Hence, the corrector $\mathbf{w}^{pq}$ satisfies the following problem<fnref n="23" />,

<footnote n="21">²¹The dependence at the local level is simply $\boldsymbol{\varphi}^0 = \operatorname{curl}_y \mathbf{u}^1$, which is quite natural. It says that when we “collapse” the Cosserat continuum to a classical continuum through multi-continuum homogenization, the rotations cannot be independent of the displacement.</footnote>

<footnote n="22">²²In the work by [42], [44], it was posited that the expressions for $\mathbf{u}^1$ and $\boldsymbol{\varphi}^0$ are as follows: $\mathbf{u}^1(\mathbf{x}, \mathbf{y}) = \zeta^{k\ell}(\mathbf{y})$, $\partial_{x_\ell} \mathbf{u}_k^0(\mathbf{x}) + \mathbf{c}(\mathbf{x})$ and $\boldsymbol{\varphi}^0(\mathbf{x}, \mathbf{y}) = \xi^{k\ell}(\mathbf{y})$, $\partial_{x_\ell} \mathbf{u}_k^0(\mathbf{x})$. Here, two distinct local solutions, denoted as $(\zeta^{k\ell}, \xi^{k\ell})$, were assumed, each satisfying disparate local problems. It is noteworthy that a similar construction of two local independent solutions was also presented in [18]. Our aforementioned methodology stands out for its efficiency, characterized by the consolidation of considerations into a singular local problem. This streamlined approach not only simplifies potential computations but also enhances overall computational efficiency.</footnote>

<footnote n="23">²³The local problem is not a classical elasticity problem. The material parameter, $\alpha$ usually referred to as the Cosserat angle, provides a rotational component. Absent the Cosserat angle, the local problem reduces to the usual local problem we encounter in the homogenization of classical elasticity.</footnote>
$$ 
\begin{array}{ll}
\boldsymbol{w}^{pq} \in H_{\text{per}}^{1}(Y; \mathbb{R}^{3}), & \displaystyle\int_{Y} \boldsymbol{w}^{pq} \, d\boldsymbol{y} = 0, \\
& \text{(local problem)} \\
\displaystyle\int_{Y} \mathbf{E}(\boldsymbol{y}) (\boldsymbol{e}_{p} \otimes \boldsymbol{e}_{q} + \nabla_{y} \boldsymbol{w}^{pq} - \boldsymbol{\epsilon}.\text{curl}_{y}(\boldsymbol{w}^{pq})) : \nabla_{y} \boldsymbol{v} \, d\boldsymbol{y} = 0,
\end{array}
$$

289 for all $\boldsymbol{v} \in H_{\text{per}}^{1}(Y; \mathbb{R}^{3})$.

290 If we now opt for $\bar{\mathbf{W}}=\mathbf{0}$ and substitute $\mathbf{u}^1$ and $\boldsymbol{\phi}^0$ from (3.47) and (3.48), respectively, in the equation of linear momentum we obtain,

$$ \int_{\Omega} \sigma^{\text{eff}} : \nabla_{x} \mathbf{V} \, d\mathbf{x} = \int_{\Omega} \mathbf{f} \cdot \mathbf{V} \, d\mathbf{x}, \qquad (3.49) $$

292 where $\sigma^{\text{eff}}=\mathbf{E}^{\text{eff}}:\nabla_{x}\mathbf{u}^0$ and $\mathbf{E}_{\text{ipq}}^{\text{eff}} = \int_Y \mathbf{E}_{\text{jkl}}(\mathbf{y}) (\delta_{kp} \delta_{q\ell} + \partial_{y\ell} w_k^{pq} - \epsilon_{\nu k \ell} \epsilon_{\nu \alpha \beta} \partial_{y\beta} w_{\alpha}^{pq}) \, d\mathbf{y}$. From 293 which one can derive the balance of linear momentum equation in the sense of distributions.

294 Returning to the equation of angular momentum and substituting $\mathbf{u}^1$ and $\boldsymbol{\phi}^0$ from (3.47) and (3.48) respectively, and using the formula $\ell_i = \frac{1}{2}\epsilon_{ipq} \epsilon_{jpq} \ell_j$, $i = 1, 2, 3$. we obtain,

$$ -\int_{\Omega} \sigma^{\text{eff}} : \boldsymbol{\epsilon} \cdot \mathbf{\Psi} \, d\mathbf{x} - \int_{\Omega} \frac{1}{2} [\boldsymbol{\epsilon} \cdot \mathbf{l}] : [\boldsymbol{\epsilon} \cdot \mathbf{\Psi}] \, d\mathbf{x} = 0. \qquad (3.50) $$

296 Hence, locally we have $\boldsymbol{\epsilon} : \sigma^{\text{eff}} + \boldsymbol{\epsilon} : \frac{1}{2} [\boldsymbol{\epsilon} \cdot \mathbf{l}] = \mathbf{0}$. We remark that the effective stress is not symmetric 297 and, therefore, not a Cauchy stress by the strict definition of the word. Thus, it seems that when 298 we “collapse” a microscopic Cosserat continuum to an equivalent homogeneous medium through 299 homogenization the resulting effective system displays odd elasticity. From the physical point 300 of view, the result says that the system governing the equivalent homogeneous medium cannot 301 be derived from conservative forces. That is to say, we are dealing with forces that break the 302 conservation of energy but not the conservation of linear momentum (see, e.g., [52], [21]). $\square$

303 **Lemma 3.1.** The homogenized system in (3.36) can be written, equivalently,

$$ 
\begin{array}{ll}
-\operatorname{div} \mathcal{T}^{\text{Cauchy}} = \frac{1}{2} \operatorname{curl}_x \boldsymbol{l} + \mathbf{f} \text{ in } \mathcal{D}(\Omega), & (3.39) \\
\mathcal{T}^{\text{Cauchy}} = (\mathcal{T}^{\text{Cauchy}})^{\top}, &
\end{array}
$$

304 with $\mathcal{T}^{\text{Cauchy}} := \sigma^{\text{eff}} + \frac{1}{2} \boldsymbol{\epsilon} \cdot \boldsymbol{l}$ the Cauchy stress$^{24}$.

305 **Proof.** In order to better understand what happens to the balance of linear momentum and angular momentum and how they relate to the Cauchy stress, we re-write the body couple force as follows: 306 $\ell_i = \frac{1}{2} \epsilon_{ipq} \epsilon_{jpq} \ell_j$, $i = 1, 2, 3$. Then, grouping the terms in (3.46) by test function we obtain,

<footnote>$^{24}$The reader should remark that the relationship between effective stress and effective strain is not linear but rather affine. Nevertheless, in the proof of the lemma we will show that the effective stress is symmetric. Hence, the body-couples play an important role in stress equilibrium.</footnote>
$$ 
\begin{aligned}
& \frac{1}{|Y|} \int_{\Omega \times Y} \left[ E(\mathbf{y})(\nabla_x \mathbf{u}^0 + \nabla_y \mathbf{u}^1 - \epsilon.\boldsymbol{\varphi}^0) + \frac{1}{2} \epsilon.\boldsymbol{\ell} \right] : (\nabla_x \mathbf{V} + \nabla_y \bar{\mathbf{W}}) \, dy \, dx \\
& - \frac{1}{|Y|} \int_{\Omega \times Y} \left[ E(\mathbf{y})(\nabla_x \mathbf{u}^0 + \nabla_y \mathbf{u}^1 - \epsilon.\boldsymbol{\varphi}^0) + \frac{1}{2} \epsilon.\boldsymbol{\ell} \right] : \epsilon.\boldsymbol{\Psi} \, dy \, dx \\
& - \frac{1}{|Y|} \int_{\Omega \times Y} \left( \mathbf{f} \cdot \mathbf{V} + \frac{1}{2} \epsilon.\boldsymbol{\ell} (\nabla_x \mathbf{V} + \nabla_y \bar{\mathbf{W}}) \, dy \, dx \right] = 0.
\end{aligned} 
\quad (3.51) $$

308 From here, again, we can obtain two sets of equations. By considering, first, that $(\mathbf{V}, \bar{\mathbf{W}}, \boldsymbol{\Psi}) = (\mathbf{V}, \bar{\mathbf{W}}, \mathbf{0})$ we have,

$$ 
\begin{aligned}
& \frac{1}{|Y|} \int_{\Omega \times Y} \left[ E(\mathbf{y})(\nabla_x \mathbf{u}^0 + \nabla_y \mathbf{u}^1 - \epsilon.\boldsymbol{\varphi}^0) + \frac{1}{2} \epsilon.\boldsymbol{\ell} \right] : (\nabla_x \mathbf{V} + \nabla_y \bar{\mathbf{W}}) \, dy \, dx \\
& - \frac{1}{|Y|} \int_{\Omega \times Y} \left( \mathbf{f} \cdot \mathbf{V} + \frac{1}{2} \epsilon.\boldsymbol{\ell} (\nabla_x \mathbf{V} + \nabla_y \bar{\mathbf{W}}) \, dy \, dx \right] = 0.
\end{aligned} 
\quad (3.52) $$

310 This is precisely the balance of linear momentum. By considering, $(\mathbf{V}, \bar{\mathbf{W}}, \boldsymbol{\Psi}) = (\mathbf{0}, \mathbf{0}, \boldsymbol{\Psi})$ we have,

$$ -\frac{1}{|Y|} \int_{\Omega \times Y} \left[ E(\mathbf{y})(\nabla_x \mathbf{u}^0 + \nabla_y \mathbf{u}^1 - \epsilon.\boldsymbol{\varphi}^0) + \frac{1}{2} \epsilon.\boldsymbol{\ell} \right] : \epsilon.\boldsymbol{\Psi} \, dy \, dx = 0. \quad (3.53) $$

312 Hence, the balance of angular momentum is satisfied for the homogenized equation in the unfolding space. Utilizing the formulas derived in (3.47) and (3.48) we obtain from (3.52),

$$ \int_{\Omega} \mathcal{T}^{\text{Cauchy}} : \nabla_x \mathbf{V} \, d\mathbf{x} = \frac{1}{2} \int_{\Omega} \epsilon.\boldsymbol{\ell} : \nabla_x \mathbf{V} \, d\mathbf{x} + \int_{\Omega} \mathbf{f} \cdot \mathbf{V} \, d\mathbf{x}. \quad (3.54) $$

314 Moreover, the stress, $\mathcal{T}^{\text{Cauchy}}$, is symmetric. To see this, select $\mathbf{u}^1$, $\boldsymbol{\varphi}^0$ as in (3.47), (3.48), respectively with $\boldsymbol{\Psi}$ an arbitrary test function, and substitute them in (3.53) to obtain,

$$ -\int_{\Omega} \mathcal{T}^{\text{Cauchy}} : \epsilon.\boldsymbol{\Psi} \, d\mathbf{x} = -\int_{\Omega} (\epsilon : \mathcal{T}^{\text{Cauchy}}).\boldsymbol{\Psi} \, d\mathbf{x} = 0. \quad (3.55) $$

316 Hence, locally we have $\epsilon : \mathcal{T}^{\text{Cauchy}} = \mathbf{0}$ (or in components $\epsilon_{ijk} \mathcal{T}_{jk}^{\text{Cauchy}} = \mathbf{0}$, $i = 1, 2, 3$) which implies that $\mathcal{T}^{\text{Cauchy}} = (\mathcal{T}^{\text{Cauchy}})^{\top}$ and therefore angular momentum is conserved. $\square$
### 3.4.2 The local problem

The local problem that arises from the above multi-continuum homogenization scheme is,

$$ 
\begin{gathered}
\mathbf{w}^{\mathrm{pq}} \in H_{\mathrm{per}}^{1}(Y; \mathbb{R}^{3}), \quad \int_{Y} \mathbf{w}^{\mathrm{pq}} \, d\mathbf{y} = 0, \\
\int_{Y} \mathrm{E}(\mathbf{y}) (\mathbf{e}_{\mathrm{p}} \otimes \mathbf{e}_{\mathrm{q}} + \nabla_{\mathrm{y}} \mathbf{w}^{\mathrm{pq}} - \epsilon \cdot \mathrm{curl}_{\mathrm{y}}(\mathbf{w}^{\mathrm{pq}})) : \nabla_{\mathrm{y}} \mathbf{v} \, d\mathbf{y} = 0,
\end{gathered}
\quad (3.56) $$

for all $\mathbf{v} \in H_{\mathrm{per}}^{1}(Y; \mathbb{R}^{3})$. The above cell/local problem differs fundamentally from the classical cell problem that one finds in the homogenization literature in two ways: First, the material tensor $\mathrm{E}$ is assumed to be isotropic in the Cosserat mechanics context and has the following form,

$$ 
\begin{align}
\mathrm{E}_{\mathrm{jikl}}^{\epsilon}(\mathbf{x}) := & \mathrm{E}_{\mathrm{jikl}}(\frac{\mathbf{x}}{\epsilon}) \nonumber \\
& = (\vartheta(\frac{\mathbf{x}}{\epsilon}) + \alpha(\frac{\mathbf{x}}{\epsilon})) \delta_{jk} \delta_{il} \tag{3.57} \\
& + (\vartheta(\frac{\mathbf{x}}{\epsilon}) - \alpha(\frac{\mathbf{x}}{\epsilon})) \delta_{jl} \delta_{ik} + \omega(\frac{\mathbf{x}}{\epsilon}) \delta_{ij} \delta_{kl}, \nonumber
\end{align}
$$

One can remark immediately the parameter $\alpha$, often referred to as the Cosserat angle, that gives the tensor an anti-symmetric component.

The second way the cell problem differs, is through the extra term that appears and it is a remnant of the rotations,

$$ \mathrm{E}(\mathbf{y}) \epsilon \cdot \mathrm{curl}_{\mathrm{y}}(\mathbf{w}^{\mathrm{pq}}). $$

Nevertheless the cell problem does have a unique solution.

**Proposition 3.2.** Assume that the material tensor $\mathrm{E}$ is isotropic as given by (3.16) and $\vartheta \ge \alpha$ then the local problem has a unique solution up to an additive constant.

*Proof.* Define the bi-linear form,

$$ \mathrm{B}[\mathbf{w}^{\mathrm{pq}}, \mathbf{v}] = \int_{Y} \mathrm{E}(\mathbf{y}) (\nabla_{\mathrm{y}} \mathbf{w}^{\mathrm{pq}} - \epsilon \cdot \mathrm{curl}_{\mathrm{y}}(\mathbf{w}^{\mathrm{pq}})) : \nabla_{\mathrm{y}} \mathbf{v} \, d\mathbf{y}. \quad (3.58) $$

The form is indeed bi-linear given the isotropy of the material tensor $\mathrm{E}$ in (3.16). By using tensor identities and straightforward calculations we see,

$$ \mathrm{E}(\mathbf{y}) \nabla_{\mathrm{y}} \mathbf{w}^{\mathrm{pq}} : \nabla_{\mathrm{y}} \mathbf{v} = 2\vartheta \epsilon_{ij}(\mathbf{w}^{\mathrm{pq}}) \partial_{y_i} v_j + 2\alpha \omega_{ij}(\mathbf{w}^{\mathrm{pq}}) \partial_{y_i} v_j + \omega \mathrm{div}_{\mathrm{y}}(\mathbf{w}^{\mathrm{pq}}) \mathrm{div}_{\mathrm{y}}(\mathbf{v}), \quad (3.59) $$

$$ \mathrm{E}(\mathbf{y}) \epsilon \cdot \mathrm{curl}_{\mathrm{y}}(\mathbf{w}^{\mathrm{pq}}) : \nabla_{\mathrm{y}} \mathbf{v} = \mathrm{E}_{\mathrm{jikl}} \epsilon_{rkl} \epsilon_{rmn} \partial_{y_n} \mathbf{w}_{\mathrm{m}}^{\mathrm{pq}} \partial_{y_i} v_j = -4\alpha \omega_{ij}(\mathbf{w}^{\mathrm{pq}}) \partial_{y_i} v_j, \quad (3.60) $$
where $e_{ij}(f) = 1/2(\partial_{y_i}f_j + \partial_{y_j}f_i)$ and $\omega_{ij}(f) = 1/2(\partial_{y_i}f_j - \partial_{y_j}f_i)$ denote the symmetric and anti-symmetric part of a second order tensor. Using the above calculation we arrive at the bi-linear form$^{25}$,

$$B[\mathbf{w}^{pq}, \mathbf{v}] = \int_{\mathcal{Y}} \hat{\mathbf{E}}(\mathbf{y}) \nabla_{\mathbf{y}} \mathbf{w}^{pq} : \nabla_{\mathbf{y}} \mathbf{v} \, d\mathbf{y}, \qquad (3.61)$$

336 where

$$\hat{\mathbf{E}}_{jikl}(\mathbf{y}) = (\vartheta(\mathbf{y}) - \alpha(\mathbf{y})) \delta_{jk} \delta_{il} \\ + (\vartheta(\mathbf{y}) + \alpha(\mathbf{y})) \delta_{jl} \delta_{ik} + \omega(\mathbf{y}) \delta_{ij} \delta_{kl}. \qquad (3.62)$$

337 Now the problem,

$$\int_{\mathcal{Y}} \mathbf{E}(\mathbf{y}) (\mathbf{e}_p \otimes \mathbf{e}_q + \nabla_{\mathbf{y}} \mathbf{w}^{pq} - \epsilon \cdot \mathrm{curl}_y(\mathbf{w}^{pq})) : \nabla_{\mathbf{y}} \mathbf{v} \, d\mathbf{y} = 0 \qquad (3.63)$$

$$\mathbf{B}[\mathbf{w}^{pq}, \mathbf{v}] + \int_{\mathcal{Y}} \mathbf{E}(\mathbf{y}) \mathbf{e}_p \otimes \mathbf{e}_q : \nabla_{\mathbf{y}} \mathbf{v} \, d\mathbf{y} = 0.$$

The form $\mathbf{B}[\mathbf{w}^{pq}, \mathbf{v}]$ is coercive, in the space of functions $H_{\mathrm{per}}^{1}(\mathcal{Y}, \mathbb{R}^{3})$ with zero average, due to Poincaré's inequality and continuous. Moreover, the remaining term is linear and continuous and thus, Lax-Milgram provides a unique solution$^{26}$. □

## 4 Numerical computation of the homogenized tensor $E^{\mathrm{eff}}$

In this section, we carry out two-dimensional computations of the shear components of the homogenized tensor $E^{\mathrm{eff}}$ defined in (3.37). We employ the finite element method to determine first the local solution $\mathbf{w}^{pq}$ defined in the **local problem** and, subsequently, using (3.37) to compute certain shear values. It's noteworthy that, in contrast to conventional methodologies, we examine two scenarios: first, when the **micropolar couple modulus**, denoted as $\alpha$, approaches zero, and subsequently, when the micropolar couple modulus closely aligns with the Lamé parameter $\theta$.

In the latter scenario, the rotational effects introduce internal body torques. Thus, the homogenized tensor has non-zero value entries that are classically forbidden, and thus the tensor aligns with the

<footnote>$^{25}$It is imperative for the reader to discern the nuanced yet significant distinction between $E$ and $\hat{\mathbf{E}}$. Notably, the positions of the material parameters $\vartheta(\mathbf{y}) - \alpha(\mathbf{y})$ and $\vartheta(\mathbf{y}) + \alpha(\mathbf{y})$ have undergone a reversal compared to the original isotropic definition of the material tensor $E$. In instances where $\alpha \equiv 0$, "the classical cell problem encountered in the periodic homogenization of linear elasticity is recovered. In such cases, the meticulous tracking of index orders becomes inconsequential, owing to the symmetry that naturally emerges.</footnote>

<footnote>$^{26}$As a clarifying remark, we point out that the bilinear form, $\mathbf{B}[\mathbf{w}^{pq}, \mathbf{v}]$, is constructed with the tensor $\hat{\mathbf{E}}$ while the continuous and linear form contains the tensor $E$.</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 4: A square (1 × 1) unit cell with a circular “hole”. The “hole” is represented a very weak material with a Young’s modulus of 10⁻³GPa.</figcaption></figure>

theory of odd elasticity. In the former scenario, the outcome aligns with those anticipated in the classical theory of linear elasticity.

A secondary objective is to ascertain the influence exerted by the micropolar couple modulus on the shear coefficients (essentially we are interested in understaind the effects of internal torques on the effective coefficients). Consequently, we graphically represent the shear coefficients over a spectrum of micropolar couple moduli spanning the interval [0, 1], numerically pinpointing a critical micropolar couple modulus. This critical micropolar couple modulus marks the juncture at which the body torque becomes prominent, rendering the material more resistant to shear forces. Lastly, we graphically represent the deformed unit cell for a 1 × 1 unit cell under two distinct conditions: first, when the micropolar couple modulus approaches zero, and subsequently, when the micropolar couple modulus closely aligns with the parameter θ. This graphical demonstra- tion is then replicated for a 1/2 × 2 unit cell, enhancing our ability to elucidate the impact of the micropolar couple modulus on the deformation of the unit cell.

The computational domain is set within the confines of the unit square and expands thereafter to encompass the unit rectangle. Both computational domains adhere to periodic boundary con- ditions imposed on opposing faces and are permeated with a ”weak” material configured in the form of a circle (refer to Fig. 4). The focal point of these simulations lies in understanding the impact that local rotations impart upon the elastic coefficients, enabling a comparative analysis with their classical counterparts without the rotational influences. In all computational analyses, we employed a conventional triangular mesh. Specifically, we utilized 200 P2 elements to mesh the circular inclusion, complemented by an additional 20 P2 elements for the remainder of the unit cell. Throughout the entirety of our investigations, we assigned a Poisson’s ratio of ν = 0.3 to each phase. The Young’s modulus for the “strong” phase was set at ℘¹ = 0.91 GPa, while the “weak” phase was set at ℘⁰ = 10⁻³ GPa. All computations were carried out using the software, FreeFem++ [29]. In all computations, the material parameters were defined as follows:

$$ \vartheta = \frac{\mathfrak{E}}{2(1 + \nu)}, \quad \omega = \frac{\mathfrak{E}\nu}{(1 + \nu)(1 - 2\nu)}, \quad \alpha = \eta\vartheta, \qquad (4.1) $$

with
$$ \mathcal{E} = \begin{cases} \mathcal{E}^0, & \text{for the weak material,} \\ \mathcal{E}^1, & \text{for the strong material} \end{cases} \quad \text{and} \quad \alpha = \begin{cases} 10^{-3}, & \text{for the weak material,} \\ \eta\theta, & \text{for the strong material,} \end{cases} \qquad (4.2) $$

376 were $\eta$ is a dimensionless number within the closed interval [0,1], the symbols $\vartheta$ and $\rho$ represent the classical Lamé constants, with $\alpha$ commonly denoted as the 'micropolar couple modulus' (see, 377 e.g., [28]). Moreover, in the two-dimensional setting, the tensor, 378

$$ E = \begin{pmatrix} E_{1111} & E_{1122} & E_{1112} & E_{1121} \\ E_{2211} & E_{2222} & E_{2212} & E_{2221} \\ E_{1211} & E_{1222} & E_{1212} & E_{1221} \\ E_{2111} & E_{2122} & E_{2112} & E_{2121} \end{pmatrix} = \begin{pmatrix} 2\vartheta + \omega & \omega & 0 & 0 \\ \omega & 2\vartheta + \omega & 0 & 0 \\ 0 & 0 & \vartheta + \alpha & \vartheta - \alpha \\ 0 & 0 & \vartheta - \alpha & \vartheta + \alpha \end{pmatrix} \quad (4.3) $$

379 with the tensor $\hat{E}$ defined in (3.62) having almost the same form as the one above with the exception that the signs in the lower right 2 × 2 block matrix in red being reversed. 380

## 4.1 Effect of volume fraction on effective coefficients

382 In Fig. 5 we plot the effective coefficients $E_{1212}^{\text{eff}}$ and $E_{1221}^{\text{eff}}$ as a function of the volume fraction. We 383 have only computed the shear coefficients since they are the only ones (up to symmetry) where the 384 rotations make a contribution. From the analysis of Fig. 5, it is evident when internal torques are 385 present (i.e. for non-zero values of $\alpha$) there is discernible distinction in the coefficients, notably 386 $E_{1212}^{\text{eff}} \neq E_{1221}^{\text{eff}}$ just as in the theory of odd elasticity. This “discrepancy” is not observed when 387 $\alpha=0$, where the coefficients are equal just like classical theory of linear elasticity dictates.

388 Additionally, in accordance with expectations, an augmentation in the volume fraction of the 389 weaker phase renders the material more amenable to shear. Intriguingly, the computations reveal 390 a substantial discrepancy in the value of $E_{1212}^{\text{eff}}$ between the scenarios with and without internal 391 torques present, showcasing that the presence of microstructural elements allowing for displacement and (constrained) rotation leads to a pronounced ease of shear along one direction (21- 392 direction) and a corresponding difficulty in the direction (12-direction). This asymmetry aligns 393 with the findings from numerical computations of elastic moduli in [50] for a two-phase centrosymmetric laminated composite, as well as the investigations in [44] focusing on a two-phase 394 hexagonal unit cell within the realm of compact bone biomechanics$^{27}$. 395

396 As a last comment, we remark that the size of the difference between $E_{1212}^{\text{eff}}$ ($\alpha > 0$) and $E_{1212}^{\text{eff}}$ ($\alpha \approx 0$) remains relatively constant as the volume fraction increases. However, this is not the case 397 between $E_{1221}^{\text{eff}}$ ($\alpha > 0$) and $E_{1212}^{\text{eff}}$ ($\alpha \approx 0$) where they values diverge in as the volume fraction 398 increases. 400

<footnote>$^{27}$In the study conducted in [44], the methodology relies on unrestricted rotations, formulated under a somewhat idealized assumption and constructed via a distinct local problem. Our current approach represents a refinement of this methodology in that it streamlines the process to a singular local problem for computation. Notably, the rotations are constrained, introducing a more realistic depiction within the context of a homogenized Cauchy problem.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 5: The effective coefficients $E_{1212}^{\text{eff}}$ and $E_{1221}^{\text{eff}}$ are graphically illustrated across varying volume fractions with and without internal body torques present (i.e. $\alpha > 0$ OR $\alpha \approx 0$, respectively). The plotted lines, distinguished by diamond and triangle markers, correspond to the values of $E^{\text{eff}}_{1212}$ and $E^{\text{eff}}_{1221}$ respectively, with $\eta = 0.999$ ($\alpha \approx \theta$). Conversely, the line adorned with crossed diamonds signifies $E^{\text{eff}}_{1212} = E^{\text{eff}}_{1221}$ for $\eta = 0.0001$ ($\alpha \approx 0$).</figcaption></figure>

## 4.2 Effect of internal torques on effective coefficients

An inherent question that arises is the exploration of the influence exerted by the internal torques on the effective coefficients. This question is crucial for comprehending the intricate dynamics at play within the material system. Equally vital is the question of identifying a value that quantifies the “amount” of internal torque present in the body, past which one recovers results consistent within the theory of odd elasticity.

Understanding the displacement patterns and structural alterations incurred when a material supports internal torques is integral for gaining a comprehensive insight into the nuanced behavior of the material at a microscale. This line of inquiry delves into the finer details of how the material structure adapts and deforms in the presence and absence of internal torques, respectively. In the plot in Fig. 6, the variation of the coefficient $\eta$ with respect to shear coefficients for a unit cell exhibiting a volumetric fraction of 15% is illustrated.

For lower values of $\eta$, the influence that internal body torques exert on the shear coefficients is negligible, resulting in homogenized coefficients that are indistinguishable from the coefficients obtained within the classical linear elasticity framework (i.e. $E_{1212}^{\text{eff}}$ equals $E_{1221}^{\text{eff}}$). However, as $\eta$ approaches the value one, a noteworthy transformation in behavior is observed. Computations reveal a pivotal threshold at approximately $\eta \approx 0.42$, denoting the point at which internal body torques become influential, consequently giving different values for the coefficients $E_{1212}^{\text{eff}}$ and $E_{1221}^{\text{eff}}$. This pivotal transition is visually demarcated in Fig. 6 by a black, dotted vertical line. Beyond this critical value, the homogenized coefficients exhibit a discernible divergence, progressively amplifying until $\eta$ attains the value one. At this upper limit, the discrepancy between the homogenized coefficients reaches its maximum. To further elucidate the impact of the micropolar couple modulus on cellular displacement, we employ a visual representation in Fig. 7.
<figure><img src="image_7.png" /><figcaption>Figure 6: The effective shear coefficients $E_{1212}^{\text{eff}}$ and $E_{1221}^{\text{eff}}$ plotted against different values of $\eta$ for a unit cell with 15% volume fraction. Numerically there seems to exist a pivotal threshold of $\eta \approx 0.42$ at which the internal torques become influential, consequently leading to the divergence in values of the coefficients $E_{1212}^{\text{eff}}$ and $E_{1221}^{\text{eff}}$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: Comparison of the displacement of the unit cell under different values of the micropolar couple modulus. The image on Fig. 7b shows that when $\eta \approx 0$, the weak material undergoes displacement, assuming an elliptical configuration in a clockwise trajectory. Conversely, as $\eta$ traverses beyond the critical threshold, indicating internal body torques the cellular microstructure undergoes displacement in a counterclockwise direction.</figcaption></figure>

Fig. 7 illustrates the morphological changes undergone by the mesh under the influence of the local solution $w^{12}$ computed from the local problem. The examination is conducted first under conditions when internal body torques are negligible, i.e., $\eta \approx 0$, and subsequently when internal body torques are present, i.e., $\eta \approx 1$. Upon inspection, the following transformation unfolds: when $\eta \approx 0$, the compliant material undergoes displacement, assuming an elliptical configuration in a clockwise trajectory. Intriguingly, as $\eta$ traverses beyond the critical threshold the cellular microstructure undergoes displacement in a counterclockwise direction. Notably, the resulting elliptical form appears to exhibit a more expansive area, suggesting a nuanced interplay between internal body torques and the intricate deformation of the material.

In a conceptual exploration aimed at further illuminating the ramifications of the internal body torques on the local displacement, we present the same visual exposition as before, this time utilizing a 1/2 × 2 unit cell, thus introducing a measure of anisotropy. The visual representation
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in Fig. 8, akin to its precursor in Fig. 7, delineates the displacement transformations exhibited by
the mesh. These alterations are due to the local solution $\mathbf{w}^{12}$ derived from the solution to the
**local problem**, albeit with heightened visualization due the specific geometric arrangement. As
the critical threshold is surpassed and internal body torques are activated, the microstructural
framework of the cellular unit undergoes displacement in a counterclockwise trajectory—an ob-
servable consistency with the earlier explicated scenario. Nevertheless, in contrast to the 1 × 1
unit cell, when $\eta \approx 0$, the displaced configuration of the weak material takes on a non-elliptical
form. However, when $\eta \approx 1$ then the displaced configuration of the weak material deforms to an
elliptical shape.

<figure><img src="image_9.png" /><figcaption>(a) Undeformed unit cell.</figcaption></figure>

<caption>(b) Deformed unit cell ($\eta \approx 0$).</caption>

<caption>(c) Deformed unit cell ($\eta \approx 1$).</caption>

**Figure 8:** Comparison of the displacement of the 1/2 × 2 unit cell in the presence and absence of internal body torques. The visual representation in Fig. 8a elucidates that for $\eta \approx 0$, the weak material experiences displacement, adopting a non-elliptical configuration, for this type of cell, in a clockwise trajectory. Conversely, when $\eta$ surpasses the critical threshold so that internal body torques become active, the cellular microstructure undergoes displacement in a counterclockwise direction consistent with the previous example. The anisotropy of the cell vividly illustrates the pronounced impact of body torques, providing a more discernible manifestation in this particular cell type.

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# 5 Conclusions

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We used multi-continuum homogenization to derive effective equations for a heterogeneous Cosserat continuum taking into account scale-size effects. The homogenized system is that of odd elasticity where the effective stress is asymmetric implying that the system supports internal body torques. Moreover, we showed how the derived effective stress is related to the classical *Cauchy* stress. Additionally, our approach to decomposing the micro-macro solutions in equation (3.48), leads to a streamlined representation of a single local problem with constraint rotations. This decomposition results in more efficient computations compared to existing methodologies (see, e.g., [18], [42], [44]).
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Furthermore, we support our findings using numerical computations, showcasing the pronounced influence of the internal body torques on both the effective coefficients and the displacement of the unit cell. The numerical computations revealed a critical value that quantifies the “amount” of internal body torque present beyond which there is a transformative shift in the displacing behavior of the cell and consequently the macroscopic material properties.

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## Acknowledgements

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The author gratefully acknowledges the financial support by the Knowledge Foundation (project nr. KK 2020-0152). Part of this work was completed while the author was visiting the group of Prof. Dr. Michael Stingl at the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU). Their hospitality is gratefully appreciated.

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## References

465

466

[1] E. Aifantis. On the microstructural origin of certain inelastic models. *J. Eng. Mater. Technol.*, 106(4):326-330, 1984.

467

468

[2] N. Auffray, S. El Ouafa, G. Rosi, and B. Desmorat. Anisotropic structure of two-dimensional linear Cosserat elasticity. *Math. Mech. Complex Syst.*, 10(4):321-356, 2023.

469

470

471

[3] N. Bakhvalov and G. Panasenko. *Homogenisation: averaging processes in periodic media: mathematical problems in the mechanics of composite materials*. Kluwer Academic Publishers, 1989.

472

473

[4] A. Bensoussan, J.-L. Lions, and G. Papanicolaou. *Asymptotic Analysis for Periodic Structures*. AMS Chelsea Publishing, Providence, RI, 1978.

474

475

[5] S. Bytner and B. Gambin. Homogenization of Cosserat continuum. *Archiwum Mechaniki Stosowanej*, 38(3):289-299, 1986.

476

477

478

[6] P.G. Ciarlet. *MATHEMATICAL ELASTICITY: VOLUME I: THREE-DIMENSIONAL ELASTICITY*. Studies in Mathematics and Its Applications, vol. 20, North-Holland, Amsterdam, 1988.

479

480

[7] C. Ciorănescu and P. Donato. *An Introduction to Homogenization*. Oxford University Press, Oxford, UK, 2000.

481

482

[8] D. Ciorănescu, A. Damlamian, and G. Griso. *Éclatement périodique et homogénéisation*. *C. R. Acad. Sci. Paris, Sér. I Math.*, 335:99-104, 2002.

483

484

[9] D. Ciorănescu, A. Damlamian, and G. Griso. The periodic unfolding method in homogenization. *SIAM J. Math. Anal.*, 40(4):1585-1620, 2008.

485

486

487

[10] D. Ciorăncescu, A. Damlamian, and G. Griso. *The Periodic Unfolding Method. Theory and Applications to Partial Differential Problems*. Series in Contemporary Mathematics 3. Springer, 1st edition, 2018.
488

489

[11] E. Cosserat and F. Cosserat. *Théorie des Corps Déformables*. Librairie Scientifique A. Hermann et Fils, 6, Rue de la Sorbonne, 1909.

490

491

[12] A. Damlamian. An elementary introduction to periodic unfolding. *Gakuto Int. Series, Math. Sci. Appl.*, 24:1651–1684, 2005.

492

493

[13] A. C. Eringen. *Microcontinuum Field Theories: I. Foundations and Solids*, volume 1. Springer Verlag, 1999.

494

[14] A.C. Eringen. Linear theory of micropolar elasticity. *J. Math. Mech.*, 15(6):909–923, 1966.

495

496

[15] A.C. Eringen and E.S. Suhubi. Nonlinear theory of simple microelastic solids-I. *Int. J. Eng. Sci.*, 2(2):189–203, 1964.

497

498

[16] A.C. Eringen and E.S. Suhubi. Nonlinear theory of simple microelastic solids-II. *Int. J. Eng. Sci.*, 2(4):389–404, 1964.

499

[17] S. Forest. *Milieux continus généralisés et matériaux hétérogènes*. Presses des MINES, 2006.

500

501

[18] S. Forest, F. Pradel, and K. Sab. Asymptotic analysis of heterogeneous Cosserat media. *Int. J. Solids Structures*, 38(26-27):4585–4608, 2001.

502

503

[19] S. Forest and K. Sab. Cosserat overall modeling of heterogeneous material. *Mech. Res. Commun.*, 25(4):449–454, 1998.

504

505

[20] S. Forest and K. Sab. Estimating the overall properties of heterogeneous Cosserat materials. *Modelling Simul. Mater. Sci. Eng.*, 7(5):829–840, 1999.

506

507

[21] M. Fruchart, C. Scheibner, and V. Vitelli. Odd viscosity and odd elasticity. *Annu. Rev. Condens. Matter Phys.*, 14:471–510, 2023.

508

509

[22] D. Garcia-Gonzalez. Magneto-visco-hyperelasticity for hard-magnetic soft materials: theory and numerical applications. *Smart Mater. Struct.*, 28(8):085020, 2019.

510

511

[23] P. Germain. La méthode des puissances virtuelles en mécanique des milieux continus, I: *Théorie du second gradient*. *J. Mécanique*, 12(2):235–274, 1973.

512

513

[24] P. Germain. The method of virtual power in continuum mechanics. Part 2: Microstructure. *SIAM J. Appl. Math.*, 25(3):556–575, 1973.

514

515

[25] E. Grekova and P. Zhilin. Basic equations of kelvin's medium and analogy with ferromagnets. *J. Elast.*, 64(1):29–70, 2001.

516

[26] G. Grioli. Elasticita asimmetrica. *Ann. Mat. Pura Appl.*, 50:389–417, 1960.

517

518

[27] C. S. Ha, M. E. Plesha, and R.S. Lakes. Chiral three-dimensional lattices with tunable Poisson's ratio. *Smart Mater. Struct.*, 25:6pp, 2016.

519

520

521

[28] S. Hassanpour and G.R. Heppler. Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations. *Math. Mech. Solids*, 22(2):224–242, 2017.
522

[29] F. Hecht. New development in FreeFem++. *J. Numer. Math.*, 20:251–265, 2012.

523

524

525

[30] H. Joumaa and M. Ostoja-Starzewski. Stress and couple-stress invariance in non-centrosymmetric micropolar planar elasticity. *Proc. R. Soc. A: Math. Phys. Eng. Sci.*, 467(2134):2896–2911, 2011.

526

527

[31] W.T. Koiter. Couple-stress in the theory of elasticity. In *Proc. K. Ned. Akad. Wet*, volume 67, pages 17–44. North Holland Pub, 1964.

528

529

[32] R.S. Lakes. Size effects and micromechanics of porous solids. *J. Mat. Scien.*, 18:2572–2581, 1983.

530

531

[33] R.S. Lakes. Strongly Cosserat elastic lattice and foam materials for enhanced toughness. *Cell. Polym.*, 12:17–30, 1993.

532

[34] G. Maugin and A. Metrikine (eds.). *Mechanics of generalized continua*. Springer, 2010.

533

534

[35] GA Maugin. The method of virtual power in continuum mechanics: application to coupled fields. *Acta Mech.*, 35(1-2):1–70, 1980.

535

536

[36] C. C. Mei and B. Vernescu. *Homogenization Methods for Multiscale Mechanics*. World Scientific, 2010.

537

[37] R.D. Mindlin. Micro-structure in linear elasticity. *Arch. Rat. Mech. Anal.*, 16:51–78, 1964.

538

539

[38] R.D. Mindlin. On the equations of elastic materials with micro-structure. *Int. J. Solids Structures*, 1(1):73–78, 1965.

540

541

[39] R.D. Mindlin and N.N. Eshel. On first strain-gradient theories in linear elasticity. *Int. J. Solids Structures*, 4(1):109–124, 1968.

542

543

544

[40] D. Molavitabrizi, S. Khakalo, R. Bengtsson, and S.M. Mousavi. Second-order homogenization of 3-D lattice materials towards strain gradient media: numerical modelling and experimental verification. *Cont. Mechanics & Thermodyn.*, pages 1–20, 2023.

545

546

547

[41] M. A. Moreno-Mateos, M. Hossain, P. Steinmann, and D. Garcia-Gonzalez. Hard magnetics in ultra-soft magnetorheological elastomers enhance fracture toughness and delay crack propagation. *J. Mech. Phys. Solids*, 173:105232, 2023.

548

549

[42] G. Nika. Derivation of effective models from heterogenous Cosserat media via periodic unfolding. *Ricerche Mat.*, pages 1–26, 2021.

550

551

[43] G. Nika. Cosserat continuum modelling of chiral scale-size effects and their influence on effective constitutive laws. *Forc. Mechanics*, 9:100140, 2022.

552

553

[44] G. Nika. On a hierarchy of effective models for the biomechanics of human compact bone tissue. *IMA J. Appl. Math.*, 88(2):282–307, 2023.

554

555

556

[45] W. Noll and B. Coleman. The thermodynamics of elastic materials with heat conduction and viscosity. In W. Noll, editor, *The Foundations of Mechanics and Thermodynamics: Selected Papers*, pages 145–156. Springer, 1974.
557

[46] W. Nowacki. *The Theory of Micropolar Elasticity*. Springer, 1972.

558

559

[47] W. Nowacki. *Theory of asymmetric elasticity*. Pergamon Press/Warsaw: PWN-Polish Scientific Publ., 1986. (Translated from the polish version by H. Zorski).

560

561

[48] M. Ostoja-Starzewski. *Microstructural Randomness and Scaling in Mechanics of Materials*. Chapman and Hall/CRC, 2007.

562

563

[49] H.C. Park and R.S. Lakes. Cosserat micromechanics of human bone: strain redistribution by a hydration-sensitive constituent. *J. Biomech.*, 19:385-397, 1986.

564

565

566

567

[50] R. Rodríguez-Ramos, V. Yanes, Y. Espinosa-Almeyda, J.A. Otero, F.J. Sabina, C.F. Sánchez-Valdés, and F. Lebon. Micro–macro asymptotic approach applied to heterogeneous elastic micropolar media. analysis of some examples. *Int. J. Solids Structures*, 239-240:111444, 2022.

568

569

[51] E. Sanchez-Palencia. *Non-homogeneous media and vibration theory*. Lecture Notes in Physics, Springer-Verlag Berlin Heidelberg, 1980.

570

571

[52] C. Scheibner, A. Souslov, D. Banerjee, P. Surówka, W.T.M. Irvine, and V. Vitelli. Odd elasticity. *Nat. Phys.*, 16(4):475–480, 2020.

572

573

[53] R. Toupin. Elastic materials with couple-stresses. *Arch. Rat. Mech. Anal.*, 11(1):385–414, 1962.

574

[54] R. Toupin. Theory of elasticity with couple-stress. *Arch. Rat. Mech. Anal.*, 17:85–112, 1964.

575

576

577

[55] N. Triantafyllidis and S. Bardenhagen. The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models. *J. Mech. Phys. Solids*, 44(11):1891–1928, 1996.

578

579

[56] R. Zhao, Y. Kim, S. Chester, P. Sharma, and X. Zhao. Mechanics of hard-magnetic soft materials. *J. Mech. Phys. Solids*, 124:244–263, 2019.