## 1 Introduction and modeling
### The constitutive model  
Concerning the constitutive model, the purpose of this section is to present the basic equations governing micro-fractured media modeled within the framework of multifield continua. Reference is made to [8] for an exhaustive treatment on this subject and to [9] for the micro-fractured model investigated herein. For simplicity sake, we adopt the hypothesis of small displacements and strains.  
Within this assumption, the macro and micro equilibrium equations may be respectively written as  
\[\underline{\mbox{div}}\ \underline{\underline{T}}\ +\ \underline{b}\ =\ \underline{0},\ \ - \underline{z}+\underline{\mbox{div}}\ \underline{\underline{S}}\ =\ \underline{0}, \tag{1}\]  
where \(\underline{\underline{T}}\) and \(\underline{\underline{S}}\) are the (usual) Cauchy and micro stress tensors, \(\underline{b}\) is the load per unit area and \(\underline{z}\) is the _self-force_, the static dual of the micro-displacement vector \(\underline{d}\). Coming to the compatibility equations, an additive decomposition of the displacement field is introduced as  
\[\underline{U}=\underline{u}+\underline{d}, \tag{2}\]  
along with the linear kinematics hypothesis  
\[\left\  
where  
\[\mathbb{A}=\frac{\mathcal{W}\ell_{M}^{2}}{RVE^{m}}\left[\begin{array}{cccc}2+1/ \sqrt{2}&0&0&1/\sqrt{2}\\ &1/\sqrt{2}&1/\sqrt{2}&0\\ &&1/\sqrt{2}&0\\ &&&2+1/\sqrt{2}\end{array}\right]+\frac{\frac{1}{2}\ell_{M}^{2}\mathcal{H}}{| RVE^{M}-RVE^{m}|}\left[\begin{array}{cccc}1&0&0&1\\ &1&1&0\\ &&1&0\\ &&&1\end{array}\right],\]  
\[\mathbb{G}=\frac{\frac{1}{2}\mathcal{Q}\ell_{m}^{2}}{RVE^{m}}\left[\begin{array} []{cccc}1&0&0&1\\ &1&1&0\\ &&1&0\\ &&&1\end{array}\right]+\frac{\frac{1}{2}\ell_{m}^{2}\mathcal{H}}{|RVE^{M}-RVE^{ m}|}\left[\begin{array}{cccc}1&0&0&1\\ &1&1&0\\ &&1&0\\ &&&1\end{array}\right],\]  
\[\mathbb{B}=\frac{\frac{1}{2}\ell_{m}^{2}\ell_{M}^{2}\mathcal{H}}{|RVE^{M}-RVE ^{m}|}\left[\begin{array}{cccc}1&0&0&1\\ &1&1&0\\ &&1&0\\ &&&1\end{array}\right],\ \ \mathbb{C}=\frac{2\mathcal{Q}}{RVE^{m}}\left[\begin{array}{ cc}1&0\\ 0&1\end{array}\right].\]  
The phisical parameters entering the equations above are defined as follows (see also Figure 1 for the definitions of the caracteristic lengths \(\ell_{M}\), \(\ell_{m}\) and \(\ell_{C}\)):  
\begin{tabular}{l l l} \(t\) & = & specimen thickness \\ \(RVE^{M}\) & = & material reference volume element = \(\ell_{M}^{2}t\) \\ \(RVE^{m}\) & = & micro-fracture reference volume element = \(\ell_{m}^{2}t\) \\ \(E\) & = & Young modulus of the bonds between macro-spheres \\ \(E^{*}\) & = & Young modulus of the bonds between macro-spheres and micro-holes \\ \(\mathcal{W}\) & = & \(EA/\ell_{M}\) = macro-lattice stiffness \\ \(\mathcal{Q}\) & = & \(EA/(\pi\ell_{c})\) = mean stiffness of the ellipsoidal micro-holes \\ \(\mathcal{H}\) & = & \(2 EA/[\sqrt{2}(\ell_{m}-\ell_{M})]\) = stiffness of the bonds between macro and meso lattices \\ \(\hat{A}\) & = & cross section of rods between adjacent micro-cracks. \\ \end{tabular}  
In view of the derivation of mixed variational principles of Hellinger-Reissner type, generalized to cope with multifield problems, the formal inverse of the constitutive law (4) is written as  
\[\left\{\begin{array}{ccccl}\underline{\underline{\
abla}}^{s}\underline{u}& =&\mathbb{E}\ \underline{\underline{T}}&+&\mathbb{H}\ \underline{\underline{S}}\\ \underline{\underline{\
abla}}^{s}\underline{d}&=&\mathbb{H}\ \underline{\underline{T}}&+&\mathbb{M}\ \underline{\underline{S}}\\ \underline{\underline{z}}&=&\mathbb{C}\ \underline{\underline{d}}\end{array}\right., \tag{5}\]  
where  
\[\left[\begin{array}{cc}\mathbb{E}&-\mathbb{H}\\ -\mathbb{H}&\mathbb{M}\end{array}\right]\]  
is the inverse of the constitutive matrix derived from its direct form, i.e.  
\[\left[\begin{array}{cc}\mathbb{A}&-\mathbb{B}\\ -\mathbb{B}&\mathbb{G}\end{array}\right].\]