# Homogenisation of periodic Cosserat media

K. Sab (a) and F. Pradel (b)

**Abstract:** The present work deals with the development of homogenization procedures for Cosserat materials with periodic micro-structure. The procedure developed by (Forest et al., 2001) for continuum heterogeneous linear elastic Cosserat media is recalled and extended to discrete Cosserat media. The nature of the homogeneous equivalent medium depends on the hierarchy of three characteristic lengths: the size *l* of the heterogeneities, the Cosserat intrinsic length *l*<sub>c</sub> and the typical size *L* of the considered structure. When *l* and *l*<sub>c</sub> are comparable and much smaller than *L*, the effective medium is a Cauchy one, whereas the case *l*<sub>c* ~*L* leads to a Cosserat effective medium. Applications to square, hexagonal, cubic and tetrakaidecahedral cell micro-structures are provided. We critically address the other existing homogenization procedures.

**Keywords:** Homogenization; micropolar theory; periodic lattices; cellular materials.

## 1 INTRODUCTION

Granular materials can be considered at the microscopic level as an assembly of interacting particles. Each particle is endowed with six degrees of freedom: three translations and three rotations. The interactions are governed by contact laws. Similarly, a honeycomb material can be seen as a lattice of beams which are rigidly connected in vertices. If the beams are loaded only at their ends, the elastic deformation can be completely determined in terms of translations and rotations of the vertices. In this case, the interactions between two particles (vertices) depend on the geometry and the material properties of the beam connecting these particles.

Consider an infinite body consisting of a periodic lattice of interacting particles. The so-called *n*-type lattice is generated by periodicity with *n* particles, i.e. a periodic unit cell containing *n* particles can be identified. A possible effective equivalent homogeneous medium can be described, a priori, either by a Cauchy continuum or by a generalized continuum like Cosserat representation [1]. The questions are the following: does a homogeneous substitute medium exists? Under which conditions does it still have a non-local character? What is the relation between the effective characteristic length and that of the constituents? Reasons for introducing generalized continuum models may be twofold. Firstly, it is a natural way to obtain an explicit dependence of the effective properties of composites or multiphase materials on the absolute size of the constituents with a continuum model and to account for size effects observed for instance in cellular materials [2]. Indeed, a homogenization method leading to a Cauchy continuum is unable to predict size effects in the mechanical behavior of heterogeneous materials. On the other hand, generalized continua can be used to limit strain localization phenomena that may occur when the constituents exhibit a strain softening behavior, as advocated by Mühlhaus and Vardoulakis [3]. These authors used Cosserat theory for the pre-

diction of the shear band thickness in granular materials. Various applications of Cosserat theory exist in the literature: for granular materials, the works of Mühlhaus and Oka [4], Sab [5], Kruyt [6] and Ehrlers et al. [7] can be mentioned among others. Lakes [8], Pradel and Sab [9], Onck [10], Diebels and Steeb [11], Kumar and McDowell [12] studied foam materials. Sulem and Mühlhaus [13] and Trovalusci and Masiani [14] considered masonry-like materials in the framework of linear and non-linear Cosserat theory...etc. The main purpose of this paper is to substitute an effective homogeneous continuum for a periodic lattice of interacting particles which are endowed with three translations and three rotations. Non-linear interaction constitutive laws are out of the scope of this paper. See for instance Laroussi et al.,[15] for geometrical nonlinearities, and [16-17] for plastic behavior.

The homogenization of elastic periodic discrete materials was previously investigated by many authors. In the sixties, using Taylor series expansion method, Banks and Sokolowski [18] developed an analogy between rectangular lattice and couple stress theory. The couple stress theory is equivalent to the reduced Cosserat theory, in which rotations are obtained from the skew-symmetric part of the displacement gradient tensor. Askar and Cakmak [19] obtained full Cosserat theory representation of a rectangular lattice with diagonals. Kim [20] and Pradel and Sab [21] used the strain energy approach to identify the Cosserat overall elastic constants of periodic lattices. Much more recently, using analogous energy approach Kumar and McDowell [12] obtained the linear elastic Cosserat constants for various 1-type lattice topologies. Sab [5] and Pradel [22] used asymptotic expansion methods and other related homogenization techniques to show that the overall elastic properties of a general *n*-type lattice can be derived by solving a discrete unit cell problem involving a finite number of degrees of freedom. The same methods have been used by Forest et al.[23] for the homogenization of continuum heterogeneous linear elastic Cosserat media. It

<footnote>(a) Institut Navier, LAMI (ENPC/LCPC), Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France</footnote>

<footnote>(b) Applied Mechanics Division, Institut Français du Pétrole, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, France</footnote>
has been rigorously proved in these works that the nature of the homogeneous equivalent medium depends on the hierarch of three characteristic lengths: the size $l$ of the heterogeneousities, the Cosserat intrinsic length $l_c$ and the typical size $L$ of the considered structure. When $l$ and $l_c$ are comparable and much smaller than $L$, the effective medium is a Cauchy one, whereas the case $l_c \sim L$ leads to a Cosserat effective medium. In the present paper, the main results of Pradel [22] and Forest et al. [23] are presented and used to critically address the other existing homogenization procedures Ebinger et al [24]. It will be explained why the Cosserat theory is unable to capture the size effects and the edge effects in usual n-type ($n \ge 2$) periodic lattice microstructures as it has been numerically shown by Tekoglu and Onck [25] for hexagonal honeycomb materials.

The paper is organized as follows: The Cosserat medium is described in Section 2. In Section 3, the homogenization procedure for periodic heterogeneous continuum Cosserat media is presented. This procedure is extended to periodic lattices (discrete materials) in Section 4. Section 5 is dedicated to the applications of the proposed homogenization procedure for square, hexagonal, cubic and tetrakaidecahedral cell microstructures. The paper is ended by a conclusion which summarizes the main results.

## 2 THE COSSERAT CONTINUUM

A wide use of the nabla operator $\nabla$ is made in the sequel. The notation used for the gradient and divergence operators are the following:

$$a \underline{\nabla} = a_{,i} \underline{\mathbf{e}}_i, \quad \underline{\mathbf{a}} \otimes \underline{\nabla} = a_{i,j} \underline{\mathbf{e}}_i \otimes \underline{\mathbf{e}}_j, \quad \underline{\mathbf{a}} \cdot \underline{\nabla} = a_{ij,j} \underline{\mathbf{e}}_i,$$

where $a$, $\underline{\mathbf{a}}$ and $\underline{\mathbf{a}}$ respectively, denote a scalar, first- and second-rank tensors. The $(\underline{\mathbf{e}}_i)_{i=1,2,3}$ are the vectors of an orthonormal basis of space and the associated Cartesian coordinates have been used. Third- and fourth-rank tensors are, respectively, denoted by $\underline{\mathbf{a}}$ and $\underline{\mathbf{a}}$. Indices can be contracted as follows:

$$\underline{\mathbf{a}} : \underline{\mathbf{b}} = a_{ij} b_{ji}, \quad \underline{\mathbf{a}} : \underline{\underline{\mathbf{A}}} : \underline{\mathbf{b}} = a_{ji} A_{ijkl} b_{lk}, \\ \underline{\underline{\mathbf{a}}} : \underline{\mathbf{b}} = a_{ijk} b_{kj} \underline{\mathbf{e}}_i, \quad \underline{\underline{\mathbf{a}}} : \underline{\mathbf{b}} = a_{ijkl} b_{lk} \underline{\mathbf{e}}_i \otimes \underline{\mathbf{e}}_j.$$

The third-rank permutation tensor reads $\underline{\underline{\epsilon}}$, its component $\epsilon_{ijk}$ is the signature of permutation $(i, j, k)$: $\epsilon_{ijk} = +1$ for an even permutation of $(1, 2, 3)$, $-1$ for an odd permutation and 0 otherwise. A skew-symmetric tensor $\underline{\underline{a}}$ can be represented by the (pseudo-) vector $\underline{\underline{a}}$:

$$\underline{\underline{a}} = -\underline{\underline{\epsilon}} : \underline{\underline{a}}, \quad \underline{\underline{a}} = -\underline{\underline{\epsilon}} : \underline{\underline{a}}.$$

The following relations then hold for the vector product:

$$\underline{\underline{a}} \times \underline{\underline{b}} = -\underline{\underline{\epsilon}} : (\underline{\underline{a}} \otimes \underline{\underline{b}}) = \underline{\underline{a}} \cdot \underline{\underline{b}} = -\underline{\underline{b}} \cdot \underline{\underline{a}}.$$

If $\underline{\underline{a}}$ and $\underline{\underline{b}}$ are skew-symmetric:

$$\underline{\underline{a}} : \underline{\underline{b}} = -2\underline{\underline{a}} \cdot \underline{\underline{b}}.$$

A rotation tensor $\underline{\underline{R}}$ can be represented by a rotation vector $\underline{\underline{\Phi}}$ such that

$$\underline{\underline{R}} = \underline{\underline{1}} - \underline{\underline{\epsilon}} \cdot \underline{\underline{\Phi}} \quad \text{or} \quad \underline{\underline{\Phi}} = -\underline{\underline{\epsilon}} : \underline{\underline{R}}$$

in the case of small rotations.

The motion of a Cosserat body $\Omega$ is described by two independent sets of degrees of freedom: the displacement $\underline{\underline{u}}$ and the micro-rotation $\underline{\underline{\phi}}$ attributed to each material points. The micro-rotation accounts for the rotation of a triad associated with underlying micro-structure. The associated deformation fields are the Cosserat deformation tensor $\underline{\underline{e}}$ and the torsion-curvature tensor $\underline{\underline{\kappa}}$ defined by:

$$\underline{\underline{e}} = \underline{\underline{u}} \otimes \underline{\underline{\cdot}} + \underline{\underline{\epsilon}} \cdot \underline{\underline{\phi}}, \quad \underline{\underline{\kappa}} = \underline{\underline{\phi}} \otimes \underline{\underline{\cdot}}. \qquad (1)$$

The symmetric part of $\underline{\underline{e}}$ corresponds to the usual strain tensor, whereas its skew-symmetric part accounts for the relative rotation of the material with respect to micro-structure. In this paper, the analysis is restricted to small deformations, small micro-rotations and small curvatures. The statistics of the Cosserat continuum is described by the generally non-symmetric force-stress tensor $\underline{\underline{\sigma}}$ and the generally non-symmetric couple-stress tensor $\underline{\underline{\mu}}$. These tensors must fulfill the local form of the balance equations in the static case for given body forces $\underline{\underline{f}}_{\text{ext}}$ and body couples $\underline{\underline{c}}_{\text{ext}}$:

$$\underline{\underline{\sigma}} \cdot \underline{\underline{\nabla}} + \underline{\underline{f}}_{\text{ext}} = 0, \quad \underline{\underline{\mu}} \cdot \underline{\underline{\nabla}} + \underline{\underline{\epsilon}} : \underline{\underline{\sigma}} + \underline{\underline{c}}_{\text{ext}} = 0. \qquad (2)$$

The constitutive equations for linear elastic centro-symmetric Cosserat materials read:

$$\underline{\underline{\sigma}} = \underline{\underline{\mathbf{a}}} : \underline{\underline{\mathbf{e}}}, \quad \underline{\underline{\mu}} = \underline{\underline{\mathbf{e}}} : \underline{\underline{\mathbf{\kappa}}}. \qquad (3)$$

The elasticity tensors display the major symmetries:

$$a_{ijkl} = a_{klij}, \quad c_{ijkl} = c_{klij}.$$

When the material does not admit point symmetry in its symmetry group, a coupling term arises between deformation and curvature in the previous constitutive equations. This case is considered in [22]. This is excluded here for the sake of simplicity. The setting of the boundary value problem on body $\Omega$ is then closed by the boundary conditions. Dirichlet boundary conditions are of the form:

$$\underline{\underline{u}}(\underline{\underline{x}}) = 0, \quad \underline{\underline{\phi}}(\underline{\underline{x}}) = 0 \quad \forall \underline{\underline{x}} \in \partial\Omega, \qquad (4)$$

where $\partial\Omega$ denotes the boundary of $\Omega$. Equations (1-4) define the boundary value problem $\mathcal{P}$ on $\Omega$. The diameter $L$ of the body $\Omega$ is defined as the maximum distance between two points. It is the typical size of the considered structure.

## 3 HOMOGENIZATION OF CONTINUUM COSSERAT MEDIA

This section is restricted to Cosserat materials with periodic micro-structure. The heterogeneous material is then obtained
by space tesselation with cells translated from a single cell $Y^l$. The period of the micro-structure is described by three dimensionless vectors ($\underline{\mathbf{a}}_1, \underline{\mathbf{a}}_2, \underline{\mathbf{a}}_3$) such that

$$l = lY, \quad Y = \left\{ \underline{\mathbf{y}} = y_i \underline{\mathbf{a}}_i, \quad |y_i| \le \frac{1}{2} \right\}$$

where $Y$ is the dimensionless unit cell and $l$ is the characteristic size of the cell $Y^l$. We call $\underline{\mathbf{a}}^l$ and $\underline{\mathbf{c}}^l$ the elasticity tensor fields of the periodic Cosserat material. They are such that

$$\begin{aligned}
\underline{\mathbf{a}}^l(\underline{\mathbf{x}}) &= \underline{\mathbf{a}}^l(l^{-1}\underline{\mathbf{x}}), \quad \underline{\mathbf{c}}^l(\underline{\mathbf{x}}) = \underline{\mathbf{c}}^l(l^{-1}\underline{\mathbf{x}}), \\
\forall \underline{\mathbf{y}} &\in \mathbb{R}^3, \quad \forall (n_1, n_2, n_3) \in \mathbb{Z}^3, \\
\underline{\mathbf{a}}(\underline{\mathbf{y}}) &= \underline{\mathbf{a}}(\underline{\mathbf{y}} + n_1\underline{\mathbf{a}}_1 + n_2\underline{\mathbf{a}}_2 + n_3\underline{\mathbf{a}}_3), \\
\underline{\mathbf{c}}(\underline{\mathbf{y}}) &= \underline{\mathbf{c}}(\underline{\mathbf{y}} + n_1\underline{\mathbf{a}}_1 + n_2\underline{\mathbf{a}}_2 + n_3\underline{\mathbf{a}}_3).
\end{aligned}$$

It is necessary to introduce next a norm of the elasticity tensors:

$$A = \max_{\underline{\mathbf{y}} \in Y} |a_{ijkl}(\underline{\mathbf{y}})|, \quad C = \max_{\underline{\mathbf{y}} \in Y} |c_{ijkl}(\underline{\mathbf{y}})|$$

whereby a characteristic length $l_c$ can be defined

$$C = A_c^2, \quad \underline{\mathbf{c}} = \underline{\mathbf{c}}^2 \underline{\mathbf{\tilde{c}}}.$$

Hence, the constitutive equations (3) read:

$$\underline{\mathbf{\tilde{c}}}(\underline{\mathbf{x}}) = \underline{\mathbf{a}}^l(l^{-1}\underline{\mathbf{x}}) : \underline{\mathbf{e}}(\underline{\mathbf{x}}), \quad (5)$$

$$\underline{\mathbf{\mu}}(\underline{\mathbf{x}}) = \underline{\mathbf{c}}^2 \underline{\mathbf{\tilde{c}}}^l(l^{-1}\underline{\mathbf{x}}) : \underline{\mathbf{\kappa}}(\underline{\mathbf{x}}). \quad (6)$$

Using the well-known asymptotic expansion method, Forest, Sab and Pradel [22-23][26] have shown that the nature of the homogeneous equivalent medium depends on the hierarchy of the three characteristic lengths: the size $l$ of the heterogeneities, the Cosserat intrinsic length $l_c$ and the typical size $L$ of the considered structure: for given $\Omega$, $Y$-periodic elasticity tensor fields $\underline{\mathbf{a}}$ and $\underline{\mathbf{c}}$, body forces $\underline{\mathbf{f}}$ and body couples $\underline{\mathbf{c}}$, two cases are relevant for the asymptotic analysis. The first case corresponds to a limiting process for which $l_c/l$ remains constant when $l/L$ goes to zero. The second case corresponds to the situation for which $l_c/L$ remains constant when $L$ goes to zero. These assumptions lead to two different homogenization schemes labelled HS1 and HS2 in the sequel. HS1 (resp. 2) will be relevant when the ratio $l/L$ is small enough and when $l_c$ and $l$ (resp. $L$) have the same order of magnitude.

## 3.1 First homogenization scheme
(HS1: $l \sim l_c \ll L$)

The effective medium turns out to be a Cauchy continuum with volume couples, described by the displacement field $\underline{\mathbf{U}}$ and governed by the following effective equations:

$$\left\{
\begin{array}{l}
\underline{\mathbf{\Sigma}} \cdot \underline{\mathbf{\nabla}} + \underline{\mathbf{f}}_{\text{ext}} = 0, \\
\underline{\mathbf{\Sigma}} = \underline{\mathbf{A}}^{\text{HS1}} : (\underline{\mathbf{U}} \otimes \underline{\mathbf{s}}) - \frac{1}{2} \underline{\mathbf{\xi}} \underline{\mathbf{\Sigma}} \underline{\mathbf{c}}_{\text{ext}}, \\
\underline{\mathbf{U}} = 0 \text{ on } \partial \Omega
\end{array}
\right.
\quad (7)$$

where $\underline{\mathbf{U}} \otimes \underline{\mathbf{s}} \underline{\mathbf{\nabla}}$ is the symmetric part of $\underline{\mathbf{U}} \otimes \underline{\mathbf{\nabla}}$. The tensor of effective moduli $\underline{\mathbf{A}}^{\text{HS1}}$ possesses all symmetries of classical elastic moduli for a Cauchy medium. It is obtained by solving the following boundary value problem on $l$:

$$\left\{
\begin{array}{l}
\underline{\mathbf{\Sigma}} \cdot \underline{\mathbf{\nabla}} = 0, \quad \underline{\mathbf{\mu}} \cdot \underline{\mathbf{\nabla}} + \underline{\mathbf{\xi}} : \underline{\mathbf{\Sigma}} = 0, \\
\underline{\mathbf{\Sigma}} = \underline{\mathbf{a}}^l : (\underline{\mathbf{u}}^{\text{per}} \otimes \underline{\mathbf{\xi}} + \underline{\mathbf{\xi}} \cdot \underline{\mathbf{\phi}}^{\text{per}} + \underline{\mathbf{E}}^s), \\
\underline{\mathbf{\mu}} = \underline{\mathbf{c}}^l : (\underline{\mathbf{\phi}}^{\text{per}} \otimes \underline{\mathbf{\nabla}}), \\
\underline{\mathbf{u}}^{\text{per}}, \underline{\mathbf{\phi}}^{\text{per}} \quad \begin{array}{l} \text{Y}^l\text{-periodic,} \\ \underline{\mathbf{\xi}} \cdot \underline{\mathbf{n}}, \underline{\mathbf{\mu}} \cdot \underline{\mathbf{n}} \end{array} \\
\underline{\mathbf{\Sigma}} \cdot \underline{\mathbf{n}}, \underline{\mathbf{\mu}} \cdot \underline{\mathbf{n}} \quad \text{skew-periodic}
\end{array}
\right.
$$

where $\underline{\mathbf{n}}$ is the outer normal to the boundary of $Y^l$. Introducing $\langle \cdot \rangle_{Y^l}$, the volume-average operator over the unit cell $l$, we have

$$\underline{\mathbf{\Sigma}}^s = \langle \underline{\mathbf{\sigma}} \rangle_{Y^l} = \underline{\mathbf{A}}^{\text{HS1}} : \underline{\mathbf{E}}^s$$

for any given symmetric tensor $\underline{\mathbf{E}}^s$.

## 3.2 Second homogenization scheme
(HS2: $l \ll L \sim l_c$)

The effective medium turns out to be a full Cosserat continuum, described by the displacement field $\underline{\mathbf{U}}$ and the rotation field $\underline{\mathbf{\Phi}}$. It is governed by the following effective equations:

$$\left\{
\begin{array}{l}
\underline{\mathbf{\Sigma}} \cdot \underline{\mathbf{\nabla}} + \underline{\mathbf{f}}_{\text{ext}} = 0, \\
\underline{\mathbf{M}} \cdot \underline{\mathbf{\nabla}} + \underline{\mathbf{\xi}} : \underline{\mathbf{\Sigma}} + \underline{\mathbf{c}}_{\text{ext}} = 0, \\
\underline{\mathbf{\Sigma}} = \underline{\mathbf{A}}^{\text{HS2}} : (\underline{\mathbf{U}} \otimes \underline{\mathbf{\nabla}} + \underline{\mathbf{\xi}} \cdot \underline{\mathbf{\Phi}}), \\
\underline{\mathbf{M}} = \underline{\mathbf{C}}^{\text{HS2}} : (\underline{\mathbf{\Phi}} \otimes \underline{\mathbf{\nabla}}), \\
\underline{\mathbf{U}} = 0, \quad \underline{\mathbf{\Phi}} = 0 \text{ on } \partial \Omega.
\end{array}
\right.
\quad (8)$$

The tensor of effective moduli $\underline{\mathbf{A}}^{\text{HS2}}$ possesses the major symmetry properties: $A_{ijkl}^{\text{HS2}} = A_{klij}^{\text{HS2}}$. It is obtained by solving the following boundary value problem on $l$:

$$\left\{
\begin{array}{l}
\underline{\mathbf{\Sigma}} \cdot \underline{\mathbf{\nabla}} = 0, \\
\underline{\mathbf{\xi}} = \underline{\mathbf{a}}^l : (\underline{\mathbf{u}}^{\text{per}} \otimes \underline{\mathbf{\nabla}} + \underline{\mathbf{E}}), \\
\underline{\mathbf{u}}^{\text{per}} \quad \begin{array}{l} \text{Y}^l\text{-periodic,} \\ \underline{\mathbf{\xi}} \cdot \underline{\mathbf{n}} \end{array} \\
\underline{\mathbf{\xi}} \cdot \underline{\mathbf{n}} \quad \text{skew-periodic,}
\end{array}
\right.
$$

and

$$\underline{\mathbf{\Sigma}} = \langle \underline{\mathbf{\sigma}} \rangle_{Y^l} = \underline{\mathbf{A}}^{\text{HS2}} : \underline{\mathbf{E}}$$

for any given second-rank tensor $\underline{\mathbf{E}}$.

The tensor of effective moduli $\underline{\mathbf{C}}^{\text{HS2}}$ possesses the major symmetry properties: $C_{ijkl}^{\text{HS2}} = C_{klij}^{\text{HS2}}$. It is obtained by solving
the following boundary value problem on $l$:

$$ \left\{ \begin{array}{l} \underline{\underline{\mu}} = \underline{\underline{c}}^l : (\underline{\underline{\phi}}^{\text{per}} \otimes \underline{\underline{\cdot}} + \underline{\underline{K}}), \\ \underline{\underline{\phi}}^{\text{per}} Y^l\text{-periodic}, \\ \underline{\underline{\mu}}.\underline{\underline{n}} \text{ skew-periodic.} \end{array} \right. $$

and

$$ \underline{\underline{M}} = \langle \underline{\underline{\mu}} \rangle_{Y^l} = \underline{\underline{C}}^{\text{HS2}} := $$

for any given second-rank tensor $\underline{\underline{K}}$.

### 3.3 Third homogenization scheme (HS3: $l < l_c < L$)

In HS1, the constitutive equations for deformation and curvature are coupled at the local level via the micro-rotation $\underline{\underline{\phi}}$ but the resulting effective continuum is a Cauchy one. In contrast, in HS2, the problems in displacement and micro-rotation are fully decoupled. In the non-linear case, the secant or tangent stiffness associated with $l_c$ may vary during the deformation process. It may become stiffer or undergo some softening, so that $l_c$ may lie between $l$ and $L$. An intermediate scheme, labelled HS3 in the sequel, resorting to coupled strain measures and allowing to prescribe a mean curvature and therefore leading to a Cosserat effective medium, may be useful for a heuristic extension to non-linear behaviors. This scheme is not deduced from rigorous asymptotic methods but rather combines features of both schemes HS1 and HS2. Actually, HS3 has been constructed such that to give the same "classical" moduli as HS1. According to HS3, the tensors of effective Cosserat moduli are obtained by solving the following boundary value problem on $l$:

$$ \left\{ \begin{array}{l} \underline{\underline{\sigma}}.\underline{\underline{\nabla}} = 0, \quad \underline{\underline{\mu}}.\underline{\underline{\nabla}} + \underline{\underline{\epsilon}} : \underline{\underline{\sigma}} = \text{constant}, \\ \langle \underline{\underline{\phi}}^{\text{per}} \rangle_{Y^l} = 0, \\ \underline{\underline{\sigma}} = \underline{\underline{a}}^l : (\underline{\underline{u}}^{\text{per}} \otimes \underline{\underline{\nabla}} + \underline{\underline{\epsilon}}.\underline{\underline{\phi}}^{\text{per}} + \underline{\underline{E}}), \\ \underline{\underline{\mu}} = \underline{\underline{c}}^l : (\underline{\underline{\phi}}^{\text{per}} \otimes \underline{\underline{\cdot}} + \underline{\underline{K}}), \\ \underline{\underline{u}}^{\text{per}}, \underline{\underline{\phi}}^{\text{per}} Y^l\text{-periodic}, \\ \underline{\underline{\sigma}}.\underline{\underline{n}}, \underline{\underline{\mu}}.\underline{\underline{n}} \text{ skew-periodic,} \end{array} \right. $$

and

$$ \langle \underline{\underline{\sigma}} \rangle_{Y^l} = \underline{\underline{A}}^{\text{HS3}} : \underline{\underline{E}}, \quad \langle \underline{\underline{\mu}} \rangle_{Y^l} = \underline{\underline{C}}^{\text{HS3}} : = $$

for any given second-rank tensors $\underline{\underline{E}}$ and $\underline{\underline{K}}$. A coupling term between $\underline{\underline{E}}$ and $\underline{\underline{K}}$ may in general arise in the effective behavior above but, if the unit cell admits a symmetry point, it can be shown to vanish.

### 4 HOMOGENIZATION OF PERIODIC DISCRETE MEDIA

In this Section, the above described homogenization procedure is extended to periodic discrete media. Consider an infinite body consisting of a periodic lattice of interacting particles. Particles $P^{\alpha,i}$, $\alpha = (\alpha_1, \alpha_2, \alpha_3) \in \mathbb{Z}^3, i = , \dots, n$, of

the so-called n-type lattice are generated by periodicity with n particles as follows:

$$ \underline{\underline{X}}^{\alpha,i} = \underline{\underline{X}}^{0,i} + l (\alpha_1 \underline{\underline{a}}_1 + \alpha_2 \underline{\underline{a}}_2 + \alpha_3 \underline{\underline{a}}_3) $$

where $\underline{\underline{X}}^{\alpha,i}$ is the coordinates vector of $P^{\alpha,i}$ in the reference configuration, and $(\underline{\underline{a}}_1, \underline{\underline{a}}_2, \underline{\underline{a}}_3)$ are three dimensionless independent vectors. $Y$ is the parallelepipedic cell constructed from this base, and $\mathcal{P}$ is the set of all particles of the lattice. It is assumed that for every particle $P$ in $\mathcal{P}$ there is a unique couple of $\alpha \in \mathbb{Z}^3$ and $i = 1, \dots, n$ such that $= P^{\alpha,i}$.

In the small strain regime, each particle $^{,i}$ is endowed with a translation vector, $\underline{\underline{u}}^{,i}$, and a rotation vector, $\underline{\underline{\phi}}^{,i}$. The discrete deformation field is noted as follows:

$$ (\underline{\underline{u}}, \underline{\underline{\phi}}) = \{ (\underline{\underline{u}}^{\alpha,i}, \underline{\underline{\phi}}^{\alpha,i}), i = 1, \dots, n, \alpha \in \mathbb{Z}^3 \} $$

In this paper, the analysis is restricted to interactions between couples of particles. This is the case for a granular material seen as a periodic assembly of rigid particles interacting at contact points. For these materials, $^{\alpha,i}$ is the center of particle $^{,i}$. This is also the case for cellular materials seen as periodic lattices of beams that are rigidly connected in vertices. Here, particles are vertices of the lattice and interacting particles are couples of vertices which are connected by a beam element. It will be shown in the sequel how the overall mechanical behaviour of periodic discrete media can be determined in terms of the constitutive laws that govern the interactions between connected particles.

So, let $c = \{P^{\beta,j}, P^{\alpha,i}\}$ be an interacting couple of particles: $\underline{\underline{f}}_c^{\alpha,i}$ (resp. $\underline{\underline{f}}_c^{\beta,j}$) is the force exerted by particle $P^{\beta,j}$ (resp. $P^{\alpha,i}$) on particle $P^{\alpha,i}$ (resp. $P^{\beta,j}$), and $\underline{\underline{m}}_c^{\alpha,i}$ (resp. $\underline{\underline{m}}_c^{\beta,j}$) is the moment at point $\underline{\underline{X}}_c^{\alpha,i}$ (resp. $\underline{\underline{X}}_c^{\beta,j}$) exerted by particle $P^{\beta,j}$ (resp. $^{,i}$) on particle $^{,i}$ (resp. $P^{\beta,j}$). The set of interaction forces and moments of the couple $c$, noted

$$ I_c = \{ \underline{\underline{f}}_c^{\beta,j}, \underline{\underline{m}}_c^{\beta,j}, \underline{\underline{f}}_c^{,i}, \underline{\underline{m}}_c^{\alpha,i} \} $$

is self-balanced:

$$ \underline{\underline{f}}_c^{\alpha,i} + \underline{\underline{f}}_c^{\beta,j} = 0, \quad \underline{\underline{m}}_c^{,i} + \underline{\underline{m}}_c^{\beta,j} - \underline{\underline{f}}_c^{\alpha,i} \times \underline{\underline{l}}_c^{\alpha,i} = 0, \quad (9) $$

where

$$ \underline{\underline{l}}_c^{\alpha,i} = \underline{\underline{X}}^{\beta,j} - \underline{\underline{X}}^{\alpha,i}, \quad \underline{\underline{l}}_c^{\beta,j} = \underline{\underline{X}}^{\alpha,i} - \underline{\underline{X}}^{\beta,j} = -\underline{\underline{l}}_c^{\alpha,i}. $$

The balance equations at particle $^{,i}$ read:

$$ \underline{\underline{f}}_c^{,i} = \underline{\underline{f}}_{\text{ext}}^{,i}, \quad \underline{\underline{m}}_c^{,i} = \underline{\underline{c}}_{\text{ext}}^{,i}, \quad (10) $$

where $\mathcal{C}$ is the set of interacting couples of particles and $\underline{\underline{f}}_{\text{ext}}^{\alpha,i}$ (resp. $\underline{\underline{m}}_{\text{ext}}^{,i}$) is the external force (resp. moment) applied to particle $^{,i}$. In the above summations, $\underline{\underline{f}}_c^{\alpha,i}$ and $\underline{\underline{m}}_c^{,i}$ are zero if particle $P^{\alpha,i}$ is not one of the two particles of $c$. It is assumed that there are two smooth fields $\underline{\underline{f}}_{\text{ext}}(\underline{\underline{x}})$ and $\underline{\underline{c}}_{\text{ext}}(\underline{\underline{x}})$ which, respectively, describe the external forces and couples by:
$$ \underline{\mathbf{f}}_{\text{ext}}^{\alpha, i} = \frac{l^3 |\underline{Y}|}{n} \underline{\mathbf{f}}_{\text{ext}}(\underline{\mathbf{X}}^{\alpha, i}), \quad \underline{\mathbf{c}}_{\text{ext}}^{\alpha, i} = \frac{l^3 |\underline{Y}|}{n} \underline{\mathbf{c}}_{\text{ext}}(\underline{\mathbf{X}}^{\alpha, i}). $$

The internal work done by $I_c$ is:

$$ c = \underline{\mathbf{f}}_{c}^{\alpha, i} \cdot \underline{\mathbf{u}}_{c}^{\alpha, i} + \underline{\mathbf{m}}_{c}^{\alpha, i} \cdot \underline{\mathbf{\phi}}_{c}^{\alpha, i} + \underline{\mathbf{f}}_{c}^{\beta, j} \cdot \underline{\mathbf{u}}_{c}^{\beta, j} + \underline{\mathbf{m}}_{c}^{\beta, j} \cdot \underline{\mathbf{\phi}}_{c}^{\beta, j}. $$

Introducing the generalized strains $(\underline{\mathbf{d}}_c^{\alpha, i}, \underline{\mathbf{\delta}}_c^{\alpha, i})$ and $(\underline{\mathbf{d}}_{c}^{\beta, j}, \underline{\mathbf{\delta}}_{c}^{\beta, j}) = -(\underline{\mathbf{d}}_{c}^{\alpha, i}, \underline{\mathbf{\delta}}_{c}^{\alpha, i})$ with

$$ \begin{aligned} \underline{\mathbf{d}}_{c}^{\alpha, i} &= \underline{\mathbf{u}}^{\beta, j} - \underline{\mathbf{u}}_{c}^{\alpha, i} - \underline{\mathbf{l}}_{c}^{\alpha, i} \times \frac{\underline{\mathbf{\phi}}_{c}^{\alpha, i} + \underline{\mathbf{\phi}}^{\beta, j}}{\underline{\mathbf{\phi}}_{c}^{\beta, j}}, && (11) \\ \underline{\mathbf{\delta}}_{c}^{\alpha, i} &= \underline{\mathbf{\phi}}^{\beta, j} - \underline{\mathbf{\phi}}_{c}^{\alpha, i}, \end{aligned} $$

and using equation (9), it is found that

$$ \begin{aligned} c &= - \left( \underline{\mathbf{f}}_{c}^{\alpha, i} \cdot \underline{\mathbf{d}}_{c}^{\alpha, i} + \underline{\mathbf{m}}_{c}^{\alpha, i} \cdot \underline{\mathbf{\delta}}_{c}^{\alpha, i} \right) && (12) \\ &= - \left( \underline{\mathbf{f}}_{c}^{\beta, j} \cdot \underline{\mathbf{d}}_{c}^{\beta, j} + \underline{\mathbf{m}}_{c}^{\beta, j} \cdot \underline{\mathbf{\delta}}_{c}^{\beta, j} \right) \\ &= - (\underline{\mathbf{f}}_{c} \cdot \underline{\mathbf{d}}_{c} + \underline{\mathbf{m}}_{c} \cdot \underline{\mathbf{\delta}}_{c}), \end{aligned} $$

where $\underline{\mathbf{m}}_{c}^{\alpha, i}$ (resp. $\underline{\mathbf{m}}_{c}^{\beta, j} = -\underline{\mathbf{m}}_{c}^{\alpha, i}$) is the moment at the mid-point of $P^{\alpha, i}$ and $P^{\beta, j}$ which is exerted by particle $P^{\beta, j}$ (resp. $^{i)}$ on particle $P^{\alpha, i}$ (resp. $P^{\beta, j}$):

$$ \underline{\mathbf{m}}_{c}^{\alpha, i} = \underline{\mathbf{m}}_{c}^{\alpha, i} - \frac{1}{2} \underline{\mathbf{f}}_{c}^{\alpha, i} \times \underline{\mathbf{l}}_{c}^{\alpha, i}. \quad (13) $$

Depending on the orientation of the couple $c$, which must be fixed once for all, the branch vector $\underline{l}_c$ and the generalized stress $(\underline{\mathbf{f}}_c, \underline{\mathbf{m}}_c)$ and strain $\underline{\mathbf{d}}_c, \underline{\mathbf{\delta}}_c)$ associated to $c$ can be equivalently defined as

$$ \underline{l}_c^{\alpha, i} = \underline{l}_c, \quad \begin{pmatrix} \underline{\mathbf{f}}_c^{\alpha, i} \\ \underline{\mathbf{m}}_c^{\alpha, i} \end{pmatrix} = \begin{pmatrix} \underline{\mathbf{f}}_c \\ \underline{\mathbf{m}}_c \end{pmatrix}, \quad \begin{pmatrix} \underline{\mathbf{d}}_c^{\alpha, i} \\ \underline{\mathbf{\delta}}_c^{\alpha, i} \end{pmatrix} = \begin{pmatrix} \underline{\mathbf{d}}_c \\ \underline{\mathbf{\delta}}_c \end{pmatrix}, $$

or as

$$ \underline{l}_c^{\beta, j} = \underline{l}_c, \quad \begin{pmatrix} \underline{\mathbf{f}}_c^{\beta, j} \\ \underline{\mathbf{m}}_c^{\beta, j} \end{pmatrix} = \begin{pmatrix} \underline{\mathbf{f}}_c \\ \underline{\mathbf{m}}_c \end{pmatrix}, \quad \begin{pmatrix} \underline{\mathbf{d}}_c^{\beta, j} \\ \underline{\mathbf{\delta}}_c^{\beta, j} \end{pmatrix} = \begin{pmatrix} \underline{\mathbf{d}}_c \\ \underline{\mathbf{\delta}}_c \end{pmatrix}. $$

The constitutive equations for linear elastic symmetric interactions read:

$$ \underline{\mathbf{f}}_c = \underline{\mathbf{a}}^c \cdot \underline{\mathbf{d}}_c, \quad \underline{\mathbf{m}}_c = \underline{\mathbf{a}}^c \cdot \underline{\mathbf{\delta}}_c, \quad (14) $$

where the generalized elasticity tensors are second-rank symmetric and positive tensors. When the interacting couple of particle does not admit point symmetry, a coupling term arises between relative displacements and relative rotations in the previous constitutive equations. This case is considered in [22] and it is excluded here for the sake of simplicity.

For $\gamma \in \mathbb{Z}^3$ and $c = \{P^{\beta, j}, P^{\alpha, i}\}$, let $c^\gamma = \{P^{\beta+\gamma, j}, +\gamma, i\}$ denote the interacting couple of particles obtained by $\gamma$-translation of $c$. Then, the set $C$ of interacting

couples of particles is generated by periodicity with a set of $r$ couples as:

$$ C = \{c_k^\gamma, k = 1, \dots, r, \gamma \in \mathbb{Z}^3\} $$

This set is choosen such that $c_k$ in $\{c_1, c_2, \dots, c_r\}$ cannot be obtained by translating another $c_l$ in $\{c_1, c_2, \dots, c_r\}$:

$$ c_k \neq c_l \text{ for all } l \neq k \text{ and all } \in \mathbb{Z}^3. $$

We have:

$$ \underline{\mathbf{a}}_{k}^{c_{k}^{\gamma}} = \underline{\mathbf{a}}^{k}, \quad \underline{\mathbf{c}}_{k}^{c_{k}^{\gamma}} = \underline{\mathbf{c}}^{k} \quad \forall k, \quad \dots $$

The interaction forces and moments in the infinite lattice are periodic if

$$ I_{c_k}^{\gamma} = I_{c_k} \text{ for all } k \text{ and all } \dots \quad (15) $$

Let us introduce next a norm of the elasticity tensors:

$$ A = \max_k |a_{ij}^k|, \quad C = \max_k |c_{ij}^k| $$

whereby a characteristic length $l_c$ can be defined

$$ C = A l_c^2, \quad \underline{\underline{\mathbf{c}}}^k = l_c^2 \underline{\underline{\mathbf{\delta}}}^k. \quad (16) $$

The discrete problem to solve is the following: set to zero all the displacements and the rotations of the particles which are situated outside a given domain $\Omega$ and find the displacements and the rotations of the particles which are situated inside $\Omega$ such that equations (10,11,14) are satisfied. For given $\Omega$, $\underline{\underline{\mathbf{a}}}^k$, $\underline{\underline{\mathbf{\delta}}}^k$, $\underline{\underline{\mathbf{f}}}_{\text{ext}}$ and $\underline{\underline{\mathbf{c}}}_{\text{ext}}$, Sab [5], Pradel and Sab[9][21] and Pradel [22] showed that it is possible to replace the periodic lattice by an homogeneous continuum "effective" material, as the size $l$ of the heterogeneities goes to zero. As for the continuous case, the nature of the homogeneous equivalent medium is a Cauchy continuum with volume couples if $l_c$ remains constant in this limiting process, while it is a Cosserat one if $l_c$ remains constant. These assumptions lead to two different homogenization schemes: HS1 (resp. 2) will be relevant when the ratio $l_c$ is small enough and when $l_c$ and $l$ (resp. $L$) have the same order of magnitude.

## 4.1 First homogenization scheme (HS1: $l \sim l_c \ll L$)

The effective Cauchy continuum with volume couples is described by the displacement field $\underline{\underline{\mathbf{U}}}$ and it is governed by the same effective equations (7) where a third-rank tensor, denoted $\underline{\underline{\mathbf{B}}}$, is substituted for the permutation tensor $\underline{\underline{\mathbf{\epsilon}}}$ Here, the tensor of effective moduli $\underline{\underline{\mathbf{A}}}^{\text{HS1}}$ also possesses all symmetries of classical elastic moduli for a Cauchy medium. It is obtained by solving the following discrete auxiliary problem on the infinite periodic lattice:

$$ \left\{ \begin{array}{l} (\underline{\underline{\mathbf{u}}}, \underline{\underline{\mathbf{\phi}}}) \in \mathcal{K} A^1 (\underline{\underline{\mathbf{E}}})^s, \quad (I_c)_{c \in C} \in SA^1 (\underline{\underline{\mathbf{\Sigma}}})^s, \\ \underline{\underline{\mathbf{f}}}_{c_k} = \underline{\underline{\mathbf{a}}}^k \cdot \underline{\underline{\mathbf{d}}}_{c_k}, \quad \underline{\underline{\mathbf{m}}}_{c_k} = \underline{\underline{\mathbf{c}}}^k \cdot \underline{\underline{\mathbf{\delta}}}_{c_k} \quad \forall k, \end{array} \right. $$
and,

$$ \underline{\underline{\Sigma}}^s = \underline{\underline{A}}^{\text{HS1}} : \underline{\underline{E}}^s $$

for any given symmetric tensors $\underline{\underline{E}}^s$ and $\underline{\underline{\Sigma}}^s$.

In this problem,

$\mathcal{K}A^1(\underline{\underline{E}}^s)$ is the set of the kinematically admissible deformation fields associated to $\underline{\underline{E}}^s$: $(\underline{\underline{u}}, \underline{\underline{\phi}})$ is in $\mathcal{K}A^1(\underline{\underline{E}}^s)$ means that it is obtained by superposing the linear field $(\underline{\underline{u}}^{\alpha,i} = \underline{\underline{E}}^s \cdot \underline{\underline{X}}^{\alpha,i}, \underline{\underline{\phi}}^{\alpha,i} = 0)$ to a periodic deformation field $(\underline{\underline{u}}^{\alpha,i} = \underline{\underline{u}}^{\text{per},i}, \underline{\underline{\phi}}^{\alpha,i} = \underline{\underline{\phi}}^{\text{per},i})$.

The set $\mathcal{S}A^1(\underline{\underline{\Sigma}})$ of statically admissible periodic interactions forces and moments associated to the symmetric stress tensor $\underline{\underline{\Sigma}}^s$ is the set of $(I_c)_{c \in C}$ verifying (9), (10) with $\underline{\underline{f}}_{\text{ext}}^{\alpha,i} = \underline{\underline{c}}_{\text{ext}}^{\alpha,i} = 0$ and (15) and such that

$$ \underline{\underline{\Sigma}}^s = \frac{1}{l^3 |\underline{\underline{Y}}|} \sum_{k=1, \dots, r} \underline{\underline{f}}_{c_k} \otimes \underline{\underline{l}}_{c_k}^s. $$

The tensor $\underline{\underline{B}}$, which links the skew-symmetric part of $\underline{\underline{\Sigma}}$ to the applied volume couples, is determined by solving a similar discrete auxiliary problem on the infinite periodic lattice [22].

## 4.2 Second homogenization scheme

(HS2: $l \ll L \sim l_c$)

The effective Cosserat continuum is described by the displacement field $\underline{\underline{U}}$ and the rotation field $\underline{\underline{\Phi}}$ and it is governed by equations (8) where a coupling term between strains and curvaures may in general arise in the effective behavior but, if the unit cell admits a symmetry point, it can be shown to vanish. The tensor of effective moduli $\underline{\underline{A}}^{\text{HS2}}$ and $\underline{\underline{C}}^{\text{HS2}}$ are obtained by solving the following discrete auxiliary problem on the infinite periodic lattice:

$$ \left\{ \begin{array}{l} (\underline{\underline{u}}, \underline{\underline{\phi}}) \in \mathcal{K}A^2(\underline{\underline{E}}, \underline{\underline{K}}), \quad (I_c)_{c \in C} \in \mathcal{S}A^2(\underline{\underline{\Sigma}}, \underline{\underline{M}}), \\ \underline{\underline{f}}_{c_k} = \underline{\underline{a}}^k \cdot \underline{\underline{d}}_{c_k}, \quad \hat{\underline{\underline{m}}}_{c_k} = \underline{\underline{c}}^k \cdot \underline{\underline{\delta}}_{c_k} \quad \forall k, \end{array} \right. $$

and,

$$ \underline{\underline{\Sigma}} = \underline{\underline{A}}^{\text{HS2}} : \underline{\underline{E}}, \quad \underline{\underline{M}} = \underline{\underline{C}}^{\text{HS2}} : \underline{\underline{K}} $$

for any given tensors $\underline{\underline{E}}, \underline{\underline{K}}, \underline{\underline{\Sigma}}$ and $\underline{\underline{M}}$.

In this problem,

$\mathcal{K}A^2(\underline{\underline{E}}, \underline{\underline{K}})$ is the set of the kinematically admissible deformation fields associated to $(\underline{\underline{E}}, \underline{\underline{K}}): (\underline{\underline{u}}, \underline{\underline{\phi}})$ is in $\mathcal{K}A^2(\underline{\underline{E}}, \underline{\underline{K}})$ means that it is obtained by superposing the linear field $(\underline{\underline{u}}^{\alpha,i} = \underline{\underline{E}} \cdot \underline{\underline{X}}^{\alpha,i}, \underline{\underline{\phi}}^{\alpha,i} = \underline{\underline{K}} \cdot \underline{\underline{X}}^{\alpha,i})$ to a periodic deformation field $(\underline{\underline{u}}^{\alpha,i} = \underline{\underline{u}}^{\text{per},i}, \underline{\underline{\phi}}^{\alpha,i} = \underline{\underline{\phi}}^{\text{per},i})$.

The coupling term in the expression (11) for the relative displacement of a pair of interacting particles is omitted:

$$ \check{\underline{\underline{d}}}_{c}^{,i} = \underline{\underline{u}}^{\beta,j} - \underline{\underline{u}}^{,i}. $$

The set $\mathcal{S}A^2(\underline{\underline{\Sigma}}, \underline{\underline{M}})$ of statically admissible periodic interactions forces and moments associated to $(\underline{\underline{\Sigma}}, \underline{\underline{M}})$ is the set of $(I_c)_{c \in C}$ verifying the self-balance equation (9) where the coupling term $\underline{\underline{f}}_{c}^{,i} \times \underline{\underline{l}}_{c}^{\alpha,i}$ is omitted (and also in (13)), (10) with $\underline{\underline{f}}_{\text{ext}}^{,i} = \underline{\underline{c}}_{\text{ext}}^{,i} = 0$, and (15), and such that

$$ \underline{\underline{\Sigma}} = \frac{1}{l^3 |\underline{\underline{Y}}|} \sum_{k=1, \dots, r} \underline{\underline{f}}_{c_k} \otimes \underline{\underline{l}}_{c_k}, \quad (17) $$

$$ \underline{\underline{M}} = \frac{1}{l^3 |\underline{\underline{Y}}|} \sum_{k=1, \dots, r} \hat{\underline{\underline{m}}}_{c_k} \otimes \underline{\underline{l}}_{c_k}. \quad (18) $$

## 4.3 Third homogenization scheme

(HS3: $l < l_c < L$)

As for the continuous case, in HS1, the constitutive equations for deformation and curvature are coupled at the local level via the micro-rotation $\underline{\underline{\phi}}$ (11) but the resulting effective continuum is a Cauchy one. In contrast, in HS2, the resulting effective continuum is a Cosserat one but the problems in displacement and micro-rotation are fully decoupled. The aim of the intermediate scheme (HS3) is to allow for coupling between deformation and curvature in the identification of effective Cosserat moduli. According to HS3, the previous identification procedure is modified as follows:

$$ \left\{ \begin{array}{l} (\underline{\underline{u}}, \underline{\underline{\phi}}) \in \mathcal{K}A^3(\underline{\underline{E}}, \underline{\underline{K}}), \quad (I_c)_{c \in C} \in \mathcal{S}A^3(\underline{\underline{\Sigma}}, \underline{\underline{M}}), \\ \underline{\underline{f}}_{c_k} = \underline{\underline{a}}^k \cdot \underline{\underline{d}}_{c_k}, \quad \hat{\underline{\underline{m}}}_{c_k} = \underline{\underline{c}}^k \cdot \underline{\underline{\delta}}_{c_k} \quad \forall k, \end{array} \right. $$

and,

$$ \underline{\underline{\Sigma}} = \underline{\underline{A}}^{\text{HS2}} : \underline{\underline{E}}, \quad \underline{\underline{M}} = \underline{\underline{C}}^{\text{HS2}} : \underline{\underline{K}} $$

for any given tensors $\underline{\underline{E}}, \underline{\underline{K}}, \underline{\underline{\Sigma}}$ and $\underline{\underline{M}}$.

In this problem,

$(\underline{\underline{u}}, \underline{\underline{\phi}})$ is in $\mathcal{K}A^3(\underline{\underline{E}}, \underline{\underline{K}})$ means that it is in $\mathcal{K}A^2(\underline{\underline{E}}, \underline{\underline{K}})$ and $\sum_{i=1, \dots, n} \underline{\underline{\phi}}^{\text{per},i} = 0$

The expression (11) for the relative displacement of a pair of interacting particles is used.

$\mathcal{S}A^3(\underline{\underline{\Sigma}}, \underline{\underline{M}})$ is the set of $(I_c)_{c \in C}$ verifying (9) (the coupling term $\underline{\underline{f}}_{c}^{\alpha,i} \times \underline{\underline{l}}_{c}^{\alpha,i}$ is not omitted), (10) with $\underline{\underline{f}}_{\text{ext}}^{\alpha,i} = 0$ and $\underline{\underline{c}}_{\text{ext}}^{\alpha,i} = \text{Constant}$, (15), (17) and (18).

A coupling term between $\underline{\underline{E}}$ and $\underline{\underline{K}}$ may in general arise in the effective behavior above but, if the unit cell admits a symmetry point, it can be shown to vanish.

## 5 APPLICATIONS

In this Section, the above described homogenization procedure is applied for the determination of the overall elastic moduli of various periodic lattices of Euler-Bernoulli beams
that are rigidly connected in vertices. Square, hexagonal, cubic and tetrakaidecahedral cell micro-structures are considered. The elastic energy of a beam is:

$$ \psi_c = \frac{1}{2} (\underline{\underline{d}}_c \cdot \underline{\underline{a}}^c \cdot \underline{\underline{d}}_c + \underline{\underline{\delta}}_c \cdot \underline{\underline{c}}^c \cdot \underline{\underline{\delta}}_c) = \frac{1}{2} (k_t d_t^2 + k_{n_1} d_{n_1}^2 + k_{n_2} d_{n_2}^2 + k_\theta \delta_t^2 + k_\phi \delta_{n_1}^2 + k_\psi \delta_{n_2}^2) $$

where $(d_t, d_{n_1}, d_{n_2})$ and $(\delta_t, \delta_{n_1}, \delta_{n_2})$ are the components of $\underline{\underline{d}}_c$ and $\underline{\underline{\delta}}_c$ in the local basis of the beam ($\underline{\underline{t}}, \underline{\underline{n}}_1, \underline{\underline{n}}_2$), respectively; $\underline{\underline{t}}$ is unit vector connecting the two ends of the beam. The generalized elastic moduli are:

$$ \begin{aligned} k_t &= \frac{E_s S}{G_s I_1} & k_{n_1} &= \frac{12 E_s I_1}{l^3} & k_{n_2} &= \frac{12 E_s I_2}{l^3} \\ k &= \frac{I_1}{l} & k &= \frac{l^2}{12} k_{n_1} & k_\psi &= \frac{l^2}{12} k_{n_2} \end{aligned} $$

where $E_s$ and $G_s$ are, respectively, the Young modulus and the shear modulus of the constitutive material of the beam. $l$ is the length, $I_1$ and $I_2$ are the inertia in directions $\underline{n}_1$ and $\underline{n}_2$, respectively, and $S$ is the section of the beam. For 2D applications, the section of the beam is rectangular, so for further purposes, we introduce $h$, the thickness of the beam. Hence, a characteristic length similar to (16) is $l_c \sim \frac{l}{\sqrt{12}}$. This means that only HS1 scheme leading to a Cauchy continuum is justified for general n-type periodic lattice of Euler-Bernoulli beams. Nevertheless, one can always determine the overall elastic moduli for the three proposed schemes by solving the auxiliary problem on the unit cell. For instance, the first scheme HS1 can be rewritten in the following variational form as:

$$ \Psi^{\text{HS1}}(\underline{\underline{E}}^s) = \frac{1}{2} \underline{\underline{E}}^s : \underline{\underline{A}}^{\text{HS1}} : \underline{\underline{E}}^s = \text{Min}_{(\underline{\underline{u}}, \underline{\underline{\phi}}) \in \mathcal{K}A^1}(\underline{\underline{E}}^s) \Psi $$

where the normalized strain energy is:

$$ \Psi = \frac{1}{l^3 |Y|} \sum_{k=1, \dots, r} c_k $$

Actually, one can take advantage of the material symmetries of the unit cell to reduce the number of degrees of freedom to be optimized. Like a crystal, the discrete structure of the material may be invariant for a finite number of orthogonal point transformations. These transformations form a finite group which must impact the mechanical properties. As stated by Neumann's principle, the overall elastic mechanical properties (represented by a fourth order tensor) must cope with the same finite group of transformations. For a further reading, Schouten's work [27] is a good reference to start with. Hereafter, for each considered micro-structure, the finite symmetry group, $\mathcal{K}$, will be delivered. Hence by combining the intrinsic symmetries of the mechanical loading which frame a group, $G$, and the one from the micro-structure, the complexity of the variational problem to solve may be reduced by substituting $\mathcal{K}A_{\text{sym}}^1$ for $\mathcal{K}A^1(\underline{\underline{E}})$. Here, $\mathcal{K}A_{\text{sym}}^1$ is the set of elements of $\mathcal{K}A^1(\underline{\underline{E}})$ which are globally invariant by applying the finite transformation group.

<figure><img src="image_1.png" /><figcaption>Figure 1: square cell micro-structure</figcaption></figure>

## 5.1 2-D lattices

### 5.1.1 Square cell micro-structure

In this section, a square cell micro-structure is considered. This is a 1-type lattice. As a consequence, in all the homogenization schemes, there is only one degree of freedom to optimize (the rotation). It is worth noting that the results of the homogenization process for HS1 and HS2 schemes do not depend on the choice of the unit cell. For instance, the choice of cell composed by one particle with four half-beams (middle cell on figure 1) delivers the same macroscopic constitutive law as the choosen one made of one type of particle with two beams according a L assembly (top right cell on figure 1). The generating set of the symmetry group is $\{C, \underline{\underline{F}}, \underline{\underline{E}}, \underline{\underline{x}}_1\}^1$, which leads to a cubic material symmetry for the overall elastic behavior and the fact that the unknown degree of freedom is null. Hence, all the homogenization schemes amount to a Voigt scheme. According to this scheme, the microscopic discrete displacements and rotations coincide with their macroscopic expressions. The macroscopic Cauchy constitutive equations are the same for all the schemes:

$$ \Sigma_{ii} = E_s \frac{h}{E_{ii}}, \Sigma_{ij} = E_s \frac{h^3}{l^3} E_{ij}, \quad i \neq j. $$

For the HS1 scheme which delivers a pure Cauchy medium, the macroscopic strain is symmetrical (i.e: $E_{12} = E_{21}$). The HS2 and HS3 schemes lead to a homogenized Cosserat medium: $E_{12} \neq E_{21}$, and the above constitutive equations are completed by:

$$ M_{31} = \frac{E_s h^3}{2l} K_{31}, \quad M_{32} = \frac{E_s h^3}{12l} K_{32} $$

### 5.1.2 Hexagonal cell micro-structure

This is a 2-type lattice. A minimum periodic cell is reported on figure 2. It contains two different types of particle and

<footnote n="1">¹C refers to the reflection at the origin of the cell, $\underline{\underline{x}}_1$ stands for the reflections at the $\underline{x}_2 \underline{x}_3$ plane and $\underline{\underline{F}}$ is the reflection at the line A (see figure 1).</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 2: Unit cell for hexagonal cell micro-structure</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 3: Hexagonal cell micro-structure</figcaption></figure>

three beams. On figure 3, one can see how this minimum cell can generate the overall lattice.

Here, with the notations on the figure 3, the symmetry group is generated by the set on the hexagonal pattern : $\{C, F, E\}_{x_1}$. This case is very special since the material ends with a classification which is a extension of the cristallo-graphic system : the material is presently an isotropic medium which is fully determined, for the Cauchy part, with two parameters.

Thanks to the symmetry group, the Cauchy part of the constitutive law happens to be the same for all the schemes. It is expressed by the equation :

$$ \Sigma_{11} = (K_{hex} + G_{hex}) E_{11} + (K_{hex} - G_{hex}) E_{22} $$

$$ \Sigma_{22} = (K_{hex} - G_{hex}) E_{11} + (K_{hex} + G_{hex}) E_{22} $$

$$ \Sigma_{12} = (G_{hex} + c_{hex}) E_{12} + (G_{hex} - c_{hex}) E_{21} $$

$$ \Sigma_{21} = (G_{hex} - c_{hex}) E_{12} + (G_{hex} + c_{hex}) E_{21} $$

where $E_{12} = E_{21}$ only for HS1 scheme. The parameters are the 2D bulk modulus, $K_{hex}$ :

$$ K_{hex} = \frac{\sqrt{3}E_s h}{6}, $$

the 2D shear modulus, $G_{hex}$ :

$$ h_{hex} = \frac{\sqrt{3}E_s h^3}{3(l^2 + h^2)} $$

<figure><img src="image_4.png" /><figcaption>Figure 4: Cubic lattice</figcaption></figure>

and the coupling modulus, $c_{hex}$ :

$$ c_{hex} = \frac{\sqrt{3}E_s h^3}{6l^3} $$

The HS2 and HS3 Schemes, lead to the same homogenized Cosserat medium: $E_{12} \neq E_{21}$, and the above constitutive equations are completed by:

$$ M_{31} = \frac{E_s h^3}{\sqrt{3}} K_{31}, \quad M_{32} = \frac{E_s h^3}{12\sqrt{3}l} K_{32}. $$

It should be noted that if the homogenization procedure proposed by Ebinger et al [24] is applied, then the Cosserat part of the constitutive law would be different for both square and hexagonal lattices. Indeed, these authors proposed to apply the following uniform strain conditions $(\underline{u}^{-i} = \underline{E} \underline{X}^{-i}, \underline{\phi}^{-i} = \underline{K} \underline{X}^{-i})$ on the boundary of a selected unit cell, and then to identify the volume average overall strain energy of the cell with the Cosserat strain energy density. Unlike for HS2 scheme which is based on periodic boundary conditions, the Cosserat elastic moduli determined with this scheme depend on the choice of the unit cell.

## 5.2 3-D lattices

Two lattices are studied: cubic lattice and tetrakaidecahedral (Kelvin) lattice.

### 5.2.1 Cube cell micro-structure

A sketch of the unit cell of the cubic lattice is reported on figure 4. The unit cell is composed of one type of particle with eight beams. The inertia principal frame is set to comply with the overall cubic symmetry of the material. Here, we restrict the study to beam elements with the same inertia in each normal direction (i.e. : $I_1 = I_2$).

Here the symmetry analysis makes naturally appear that overall medium has the cubic material symmetries. The Cauchy constitutive law is the same for the three schemes ($E_{12} = E_{21}$):
<figure><img src="image_5.png" /><figcaption>Figure 5: Tetrakaidecahedral unit cell</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 6: Tetrakaidecahedral lattice</figcaption></figure>

$$ \Sigma_{ii} = \frac{E_s S}{l^2} E_{ii}, $$

$$ \Sigma_{ij} = \frac{12E_s I_1}{l^4} E_{ij}, \quad i \neq j. $$

For the HS2 and HS3 Scheme, the Cosserat part of the constitutive law is given by $E_{12} \neq E_{21}$ and:

$$ M_{ii} = \frac{s I_t}{l^2} K_{ii}, $$

$$ M_{ij} = \frac{E_s I_1}{l^2} K_{ij}, \quad i \neq j. $$

### 5.2.2 Kelvin's cell micro-structure

The last selected micro-structure is a beam assembly according to a tetrakaidecahedral pattern. In this case described on the figure 5 with the associated periodic frame {$a_1, a_2, a_3$}, a minimum unit cell which is compatible with the periodicity is composed of six types of particles (black on the figure 5) and twelve beams.

Here, on the full tetrakaihedron pattern plotted on the figure 6, the symmetry group is generated by the set : $\{C, x_1, x_2\}$.

It is worth noting that the generating set must be applied for the overall lattice and not on the unit cell. The inertia axes are oriented to respect the cubic symmetries of the material. For instance, in the present paper, the first normal vector $n_1$ of each beam is oriented towards the center of the adjacent square. The material is also classified in the cubic system. Here the obtained constitutive law is described by :

$$ \begin{aligned} \Sigma_{11}^s &= \lambda + 2\mu) E_{11} + \lambda E_{22} + \lambda E_{33} \\ \Sigma_{22}^s &= \lambda + 2\mu) E_{22} + \lambda E_{11} + \lambda E_{33} \\ \Sigma_{33}^s &= \lambda + 2\mu) E_{33} + \lambda E_{22} + \lambda E_{11} \\ \Sigma_{ij}^s &= G_f E_{ij}^s, \quad i \neq j, \\ \Sigma_{ij}^{as} &= d E_{ij}^{as}, \quad i \neq j, \\ M_{ij} + M_{ji} &= (h + g) (G_{ij} + G_{ji}), \quad i \neq j, \\ M_{ij} - M_{ji} &= (g - h) (G_{ij} - G_{ji}), \quad i \neq j, \\ M_{ii} &= \begin{cases} ii + f (G_{jj} + kk), & i \neq j, j \neq k, k \neq i \end{cases} \end{aligned} $$

Here the Lamé coefficient are usually defined by :

$$ \lambda = E_f \frac{\nu_f}{(1 + \nu_f)(1 - \nu_f)}, \quad \mu = \frac{E_f}{(1 + \nu_f)}. $$

As in the previous cases, the constitutive law for the Cauchy medium is the same for the HS1 and HS2 scheme. We find the same Cauchy elastic constitutive law as the one in [28].

$$ \begin{aligned} E_f &= 6 \frac{S I_1 E_s \sqrt{2}}{l^2 (12 I_1 + l^2 S)} \\ f &= -1/2 \frac{12 I_1 - l^2 S}{12 I_1 + l^2 S} \\ f &= \frac{3 \sqrt{S} E_s I_2 \left( I_2 + \frac{G_s}{E_s} I_t + 4 I_1 \right)}{\alpha + \beta} \\ \alpha &= l^2 \left( 12 I_2^2 + 48 I_2 I_1 + 12 I_2 \frac{G_s}{E_s} I_t \right) \\ \beta &= 2 \left( l^2 \frac{G_s}{E_s} I_t S + 4 l^2 S I_2 + 4 l^2 I_1 S \right) \end{aligned} $$

$E_f$, $\nu_f$, and $G_f$ are respectively the Young modulus, the Poisson's ratio and the shear modulus for the homogeneous medium. In the case of the HS2 scheme, the Cosserat part of the constitutive law is given by the coefficients :

$$ \begin{aligned} d &= \frac{3\sqrt{E_s S}}{l^2} \\ &= \frac{(E_s I_2 + G_s I_t)\sqrt{}}{8l^2} \\ f &= \frac{(5 s I_t - E_s I_2)\sqrt{}}{16l^2} \\ g &= E_s \frac{\sqrt{\left(3G_s I_t I_1 - G_s I_t I_2 - E_s I_1^2 - E_s I_1 I_2\right)}}{8l^2 (E_s I_1 + G_s I_t)} \\ h &= E_s \frac{\sqrt{2}\left(G_s I_t I_2 + 5G_s I_t I_1 + E_s I_1^2 + E_s I_1 I_2\right)}{8l^2 (E_s I_1 + G_s I_t)} \end{aligned} $$
# 6 CONCLUSION

Classical continuum theory, laking a length scale, cannot explain size dependent mechanical behaviours of materials with discrete micro-structure like granular materials or cellular materials. One way to account for these size effects is to explicitly model the discrete micro-structure morphology. Even though discrete models give a very good agreement with the experiments, they are computationally expensive for complex micro-structures, especially in three dimensions. To overcome this, one can use a generalized continuum theory, such as Cosserat continuum theory, which incorporates a material length scale. The problem is to link the Cosserat continuum to the discrete micro-structure morphology. It is shown in this paper that this is possible for 1-type periodic lattices. Indeed, for these lattices a Voight scheme can be used to determine the elastic moduli. Comparing the continuum predictions with the exact discrete simulations of 1-type periodic lattices, the utility of the Cosserat modeling has been numerically demonstrated by Kumar and McDowell [12]. On the other hand, for n-type periodic lattices, the determination of the overall elastic Cosserat is obtained by solving an auxiliary problem on the unit cell. This procedure (HS2 scheme) is justified when the length $l_c$ that describes the coupling between the rotations and the translations of a pair of interacting particles is much larger than the size $l$ of the heterogeneities. Unfortunately, this is not true for most usual discrete microstructures. For Euler-Bernoulli beams, we have $l_c \sim l/\sqrt{12}$. This explains why the Cosserat theory is unable to capture the size effects and the edge effects in hexagonal honeycomb materials as it has been numerically shown by Tekoglu and Onck [25].

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