## 2 Macroscopic Unit-Cell
### Material Constitutive Law and Damage Model
This parameter should be treated with care as it can misrepresent the obtained results. For the present macro-element, the default value of 0 (zero) has been adopted, even though it has been noticed that within a quasi-static pushover analysis a value of \(1\times 10^{-4}\) can decrease the computational cost without apparently affecting the results. Since truss beams define the material behaviour of the interfaces, the system will undergo only uniaxial loading conditions and, therefore, the plastic strain rates in tension, \(\dot{\bar{\varepsilon}}^{pl}_{t}\), and compression, \(\dot{\bar{\varepsilon}}^{pl}_{c}\), are a function of the uniaxial plastic strain rate, \(\dot{\bar{\varepsilon}}^{pl}_{11}\), and read as:  
\[\dot{\bar{\varepsilon}}^{pl}_{t}=\dot{\bar{\varepsilon}}^{pl}_{11}\;\text{ and}\;\dot{\bar{\varepsilon}}^{pl}_{c}=\;-\;\dot{\bar{\varepsilon}}^{pl}_{11} \tag{18}\]  
Likewise, even if the CDP has been extended for the general multiaxial conditions, the uniaxial character of the system may also be adopted for the cyclic loading analysis, which simplifies, to a great extent, its follow-up validation. In such cases, the recovery of the elastic stiffness when the sign of the imposed load changes is an important aspect to consider. The so-called 'unilateral effect' holds both in tensile and compressive sides of the cycle, in which, for uniaxial conditions, the damage parameter variable d is given as:  
\[(1-d)=(1-s_{t}d_{c})(1-s_{c}d_{t}),\;s_{t}\;\geq\;0,\;s_{c}\;\leq\;1 \tag{19}\]  
Here, \(s_{t}\) and \(s_{c}\) are functions of the stress state which represents the referred stiffness recovery with the related stress reversals and are defined as:  
\[\left\{ \begin{array}{c}s_{t}\;=\;1-\omega_{t}r^{*}(\overline{\sigma}_{11}) \\ \\ s_{c}\;=\;1-\omega_{c}(1-r^{*}(\overline{\sigma}_{11}))\end{array} \right. \tag{20}\]  
Here, \(\omega_{t}\) and \(\omega_{c}\) are weight factors and assumed as input material parameters for the CDP model and limited by \(0\leq\omega_{t}\leq 1\) and \(0\leq\omega_{c}\leq 1\), and \(r^{*}(\overline{\sigma}_{11})=1\) if \(\overline{\sigma}_{11}>0\) or \(r^{*}(\overline{\sigma}_{11})=0\) if \(\overline{\sigma}_{11}<\;0\). In concrete-like materials, the effect is more marked when the material is in a compression regime \((\sigma_{11}<0)\), because tensile cracks tend to close [86]. Nevertheless, the aim here is to fully reproduce the homogenised behaviour in both regimes and so the [PERSON] effect is not reproduced. Therefore, the tensile and compressive elastic stiffnesses have a full recovery effect, which is the same as defining \(\omega_{t}=\;1\) and \(\omega_{c}=1\). To fulfil the input requirements of the CDP model in ABAQUS, information regarding the post-failure behaviour may be introduced for each element that features material nonlinearity, i.e., the truss beams, in terms of stress and inelastic strain \(\bar{\varepsilon}^{ck}\) values. The latter must be obtained for each point of the post-peak homogenised curve by Equation (21):  
\[\bar{\varepsilon}^{ck}=\bar{\varepsilon}-\bar{\varepsilon}^{cl}_{0} \tag{21}\]in which \(\varepsilon_{o}^{l}\) is the elastic strain corresponding to the undamaged material and \(\varepsilon\) is the total axial strain of the multi-linear stress envelope. If the damage parameters, \(d\), are introduced, the plasticity model is thus coupled with a damage description and is suitable for the cyclic behaviour description of the material. The plastic strain values, \(\varepsilon^{p\prime}\), are calculated, for each point of the curve, as:  
\[\varepsilon^{p\prime}=\varepsilon^{cr}-\frac{d}{(1-d)}\frac{\sigma^{P}}{E_{0} ^{d}} \tag{22}\]  
Since the permanent plastic strain can be just positive or null, the latter can constitute a good checkpoint to foresee if the damage parameters have been properly computed. Lastly, it is important to recall that to increase the robustness of the problem, and as given in Figure 4, multi-linear curves (e.g., with 5-nodes) are defined.