## 2 Macroscopic Unit-Cell  
### Theoretical Scope  
The macro-element has a theoretical background on the works of [PERSON] [79; 80; 81] and was implemented in the ABAQUS software. The strategy is especially suited for the study of planar elements, i.e., for a three-dimensional body, \(\Omega_{M}\subseteq\Re^{3}\), with one dimension much smaller than the other two. Such an assumption is also postulated when studying a given structure with a plate or flat shell FE's. Therefore, the modelling using the macro-element tries to describe the structure using the mid-surface of the body, \(\Omega_{M}\). Given the cartesian space and for the undeformed configuration, it is assumed that the thickness \(t\) of the structure develops in the z-direction (\(z=[-t/2,+t/2]\)) and that the two planar coordinates are defined through global x and y coordinates (see Figure 1). Such an assumption is important to recall bearing the integration with a multi-scale framework, as it should be linked with the Cartesian system adopted at a meso/micro-scale. A uniformisation between the allowable macro-deformation modes and the respective meso-scale ones is convenient when material orthotropy exists. The macroscopic unit-cell is composed of the assemblage of discrete quadrilateral rigid plate elements connected through a set of rigid and deformable FE trusses in its interfaces. Rigid plates are modelled as four linear quadrilateral elements within a finite-membrane strain formulation (S4 element in ABAQUS). Full integration is adopted to prevent hour-glass modes. The rigid linear elements are defined as three-dimensional Timoshenko FE beams (B31 in ABAQUS). In this scope, the deformation and damage of the structure are restricted to three-dimensional two-node FE trusses (T3D2 in ABAQUS). These are comparable with spring elements since they have just one DOF and are directly integrated. Their stiffness matrix is defined only by axial stiffness terms, given by \(diag\big{(}\frac{EA}{L},\frac{EA}{L}\big{)}\), in which E is the Young's modulus, \(A\) is its cross-section area and \(L\) is its axial length. The macro-deformation is thus governed by the trusses, in which in- and out-of-plane failure modes are considered within a decoupled approach. The FE trusses incorporate the material information, in which both orthotropy and full softening behaviour can be provided by defining the properties of the different interface sides. Regarding the out-of-plane (OOP) behaviour, the macroscopic cell is composed of a set of four flexural and torsional trusses. The flexural trusses are placed at the mid-centre of each interface and the torsional trusses are placed at each node of the squared rigid plate. Mid-span hinges on interfaces allow the fixing of the axis of rotation for torsion movements without compromising the deformed shape. Regarding the in-plane (IP) modes, trusses have been assumed for axial and shear behaviours. For the former, a total of three axial trusses are placed per-interface (two in the edges and one in the centre), and for the latter, a total of two shear trusses (defining the shear for each half-length of the interface) have been placed per interface. Inertial forces can be either modelled via a direct lumped or consistent mass matrix strategy. The former requires the computation of the representative mass for each rigid plate and its introduction through nodal mass elements on each rigid plate node. The latter allows it to be more straightforward, as the mass of the system is embodied by the quadrilateral rigid plates using an equivalent material density. By following a linear displacement interpolation assumption, one achieves a consistent mass matrix. Still, a comparison between both strategies will be given in Section 3.1.2. The computation of the system density \(\rho_{system}\) is given as \(\rho_{real}\cdot t/t_{eq}\), \(t\) being the real thickness of the structural element, \(\rho_{real}\) the real density of the material, and \(t_{eq}\) the thickness attributed to the rigid plates (defined as 10 mm).