## 2 Theoretical Background: Material Damage Model  
Among several criteria for the prediction of the initiation and evolution of intra-laminar damage for structures made of unidirectional fibre-reinforced composite materials, the Hashin criteria are probably the most adopted ones. These criteria, based on the separation of the failure modes by introducing separated equations, allow to clearly identify the damage of the matrix and of the fibre under tension or compression stress conditions. Indeed, according to Hashin failure criteria [44], in Equations (1)-(4), the limit values for the onset of the damage for the fibre traction (\(F_{fl}\)), fibre compression (\(F_{fc}\)), matrix traction (\(F_{mt}\)) and matrix compression (\(F_{mc}\)) are, respectively, introduced. \[F_{fl}=\left(\frac{\sigma_{11}}{X_{T}}\right)^{2}+\left(\frac{\sigma_{12}}{S_{ L}}\right)^{2}=1 \tag{1}\]  
\[F_{fc}=\left(\frac{\sigma_{11}}{X_{\text{C}}}\right)^{2}=1 \tag{2}\]  
\[F_{mt}=\left(\frac{\sigma_{22}}{Y_{T}}\right)^{2}+\left(\frac{\sigma_{12}}{S_ {L}}\right)^{2}=1 \tag{3}\]  
\[F_{mc}=\left(\frac{\sigma_{22}}{2S_{T}}\right)^{2}+\left[\left(\frac{Y_{\text {C}}}{2S_{T}}\right)^{2}-1\right]\cdot\frac{\sigma_{22}}{Y_{\text{C}}}+\left( \frac{\sigma_{12}}{S_{L}}\right)^{2}=1 \tag{4}\]  
In Equations (1)-(4) \(\sigma_{11}\), \(\sigma_{22}\), \(\sigma_{12}\) are the components of the effective stress tensor, respectively, along fibre direction, matrix direction and shear while \(\text{X}_{\text{T}}\), \(\text{X}_{\text{C}}\), \(\text{Y}_{\text{T}}\), \(\text{Y}_{\text{C}}\), \(\text{S}_{\text{L}}\) and \(\text{S}_{\text{T}}\) are, respectively, the fibre tensile, fibre compressive, matrix tensile, matrix compressive and shear strengths in longitudinal and transversal directions. The evolution of the damage for each failure mode is modelled according to the bilinear law schematically represented in Figure 1.  
According to Figure 1, it is possible to identify the onset and evolution damage phases for each damage mode. In particular, point A of Figure 1 identifies the condition of Hashin's criterion satisfaction. At this condition, the element has absorbed an amount of energy represented by the yellow triangle area. Beyond this condition, the evolution of the damage and the corresponding loss of stiffness of the material is simulated by introducing a parameter allowing to reduce linearly the characteristics of the undamaged material. The phase of the damage evolution is represented by the segment in Figure 1 connecting point A to point C representing the completely damaged element condition. Actually, the element is considered completely damaged when it has absorbed an energy represented by the sum of the yellow and blue triangles areas. Indeed, this global area is representative of the fracture toughness of simulated material system. Figure 1: Constitutive relation (damage onset and evolution). In order to evaluate the degradation status of the element, a material degradation coefficient \(di\) is introduced which is evaluated independently for matrix and fibre under traction and compression conditions. The degradation coefficient \(di\) definition is introduced in Equation (5). \[d_{i}=\frac{\delta_{i,eq}^{t}\Big{(}\delta_{i,eq}-\delta_{i,eq}^{0}\Big{)}}{ \delta_{i,eq}\Big{(}\delta_{i,eq}^{t}-\delta_{i,eq}^{0}\Big{)}};\quad\delta_{i, eq}^{0}\leq\delta_{i,eq}\leq\delta_{i,eq}^{t};\quad i\in(f_{c},f_{t},m_{c},m_{t}) \tag{5}\]  
where \(\delta_{i,eq}\) is equivalent displacement; \(\delta_{i,eq}^{0}\) is equivalent displacement on set damage; \(\delta_{i,eq}^{t}\) is equivalent displacement on full damage (Point C in Figure 1). Moreover, \(f_{c},f_{t},m_{c},m_{t}\) are fibre failure compression, fibre failure tension, matrix failure compression and matrix failure tension. In Equation (6), the definition of the maximum equivalent displacement, reached at point C of Figure 1, is introduced.