## 2 Theoretical background
### Composites Damage Criteria  
In order to account for the damage behavior of the laminate, a User Material was developed in the ABAQUS FEM environment capable to consider the fiber and matrix damage mechanisms in tension and compression [4; 19; 29]. Each failure mode consisted of two different phases. Initially, a linear mechanical behavior is considered up to the damage initiation threshold. Then, the damage evolution up to the complete damage is evaluated. The criteria used for evaluate the damage initiation are based on [PERSON]'s (fiber and matrix tension and fiber compression) and Puck-Shurmann [38] (matrix compression) failure criteria, as reported in Equations (7)-(10). \[F_{ft}=\left(\frac{\sigma_{1}}{X_{T}}\right)\geq 1\quad if\quad\sigma_{1}\geq 0, \tag{7}\]  
\[F_{fc}=\left(\frac{\sigma_{1}}{X_{C}}\right)\geq 1\quad if\quad\sigma_{1}<0, \tag{8}\]  
\[F_{mt}=\left(\frac{\sigma_{2}}{Y_{T}}\right)^{2}+\left(\frac{\tau_{12}}{S_{12 }}\right)^{2}+\left(\frac{\tau_{23}}{S_{23}}\right)^{2}\geq 1\quad if\quad \sigma_{2}\geq 0, \tag{9}\]  
\[F_{mc}=\left(\frac{\tau_{nt}}{S_{23}^{A}-\mu_{nt}\sigma_{m}}\right)^{2}+\left( \frac{\tau_{nl}}{S_{12}-\mu_{nl}\sigma_{n}}\right)^{2}\geq 1\quad if\quad \sigma_{2}<0, \tag{10}\]  
where \(X_{T}\) and \(X_{C}\) are the fiber tensile and compressive strengths, \(Y_{T}\) is the matrix tensile strength, and \(S_{12}\) and \(S_{23}\) are respectively the longitudinal and transverse shear strengths. Moreover, the parameters introduced in Equation (10) have been defined with respect to the potential fracture plane; in particular, \(S^{A}\)\({}_{23}\) is the transverse shear strength in the potential fracture plane, \(\mu_{nt}\) and \(\mu_{nl}\) are respectively the friction coefficients in the transverse and longitudinal directions, \(\sigma_{m}\) is the normal stress, and \(\tau_{nt}\) and \(\tau_{nl}\) are respectively the shear stresses in the transverse and longitudinal directions. Indeed, experiments have demonstrated that unidirectional laminates experience shear fail [39] when subjected to transverse compressive loads, with a fracture plane oriented with an angle \(\theta_{f}=53^{\circ}\pm 2^{\circ}\)[21]. Hence, to correctly evaluate the damage status of the laminate, the stress/strain components in the \(\theta_{f}\) oriented fracture plane have been considered, instead of the nominal \(0^{\circ}\) oriented nominal plane. Therefore, the stress components in the general fracture plane L-N-T oriented with a fracture angle \(\theta_{f}\), as in Figure 1, can be obtained as a function of the stress components defined in the lamina coordinate system 1-2-3:  
\[\left\{ \begin{array}{l}\sigma_{n}=\sigma_{2}\cos^{2}\theta_{f}+\sigma_{3}\sin ^{2}\theta_{f}+2\tau_{23}\cos\theta_{f}\sin\theta_{f}\\ \tau_{nl}=\tau_{23}\cos\theta_{f}+\tau_{31}\cos\theta_{f}\\ \tau_{nt}=(\sigma_{3}-\sigma_{2})\cos\theta_{f}\sin\theta_{f}+\tau_{23}\Big{(} \cos^{2}\theta_{f}-\sin^{2}\theta_{f}\Big{)}\end{array} \right.. \tag{11}\]  
Hence, the parameters related to the fracture angle introduced in Equation (10) can be defined [40]:  
\[\phi=2\cdot\theta_{f}-90^{\circ}, \tag{12}\]  
\[\mu_{nt}=\tan\phi, \tag{13}\]  
\[S_{23}^{A}=\frac{Y_{C}(1-\sin\phi)}{2\cos\phi}, \tag{14}\]  
\[\mu_{nl}=\mu_{nt}\cdot\frac{S_{12}}{S_{23}^{A}}, \tag{15}\]  
where \(Y_{T}\) is the matrix compressive strength. Each mode is characterized by the failure behavior reported in Figure 2. In particular, a linear elastic behavior with an initial stiffness \(K\) can be observed up to location A, which identifies the damage onset. Then, each location B on the segment AC identifies a partial damage condition characterized by a degraded stiffness \(K_{d}\), up to location C, where the complete failure can be observed. Figure 1: Compressive matrix failure and fracture plane.