## 3 3D Orthotropic Stiffness Formulation
### Hashin 3D Failure Criterion
 \tag{22}\]  
The fourth simplification is acknowledging that the tensile matrix failure criterion only depends on quadratic terms (19), which is in accordance with some cohesive models [46], and promotes a closed failure surface function (24), (25). \(Y_{t}\) and \(Y_{c}\) are the matrix tensile and compressive strength, respectively. \[A_{mt}=0;B_{mt}=\frac{1}{Y_{t}^{2}};C_{mt}=\frac{1}{S_{t}^{2}};D_{mt}=D_{m}= \frac{1}{S_{L}^{2}} \tag{24}\]  
\[F_{mt}=\left(\frac{\hat{\sigma}_{2}+\hat{\sigma}_{3}}{Y_{t}}\right)^{2}+\frac {\sharp_{23}^{2}-\flat_{2}\flat_{3}}{S_{T}^{2}}+\frac{\sharp_{21}^{2}+\sharp_{ 23}^{2}}{S_{L}^{2}}\leq 1.0\qquad\text{if }\hat{\sigma}_{2}+\hat{ \sigma}_{3}\geq 0 \tag{25}\]  
The fifth simplification pertains to the matrix's resistance to compression, in which the collapse during biaxial behavior is linked to crushing when \(\sigma_{2}=\sigma_{3}=-\sigma\gg Y_{c}\). However, it must verify the collapse during uniform compression when \(\sigma_{2}=-Y_{c}\). In these conditions, if the shear stress components are null, and we are on the onset of matrix compressive failure, the following systems of Equation (26) are assembled. \[\left\{ \begin{array}{l}A_{mc}(\hat{\sigma}_{2}+\hat{\sigma}_{3})+B_{mc}(\hat {\sigma}_{2}+\hat{\sigma}_{3})^{2}-\frac{\phi_{2}\flat_{3}}{S_{T}^{2}}=1.0\\ A_{mc}(\hat{\sigma}_{2})+B_{mc}(\hat{\sigma}_{2})^{2}=1.0\end{array} \right. \Leftrightarrow\left\{ \begin{array}{l}-2\partial A_{mc}+4\partial^{2}B_{mc }-\frac{\hat{\sigma}^{2}}{S_{T}^{2}}=1.0\\ -Y_{c}A_{mc}+Y_{c}^{2}B_{mc}=1.0\end{array} \right. \tag{26}\]  
Since the system of equations has three unknowns and only two equations, these cannot be solved directly but rather by using the assumption that in biaxial compression \(\sigma_{2}=\sigma_{3}=-\sigma\gg Y_{c}\), the following limit (27) can be admitted. It is possible to determine the remaining unknown matrix compressive parameters for a closed-envelope surface using this limit, as long as \(Y_{c}>2S_{T}\). As a result, the compressive matrix failure criterion can be put together (28). \[\lim_{\sigma\rightarrow\infty}\Rightarrow B_{mc}=\frac{1}{4S_{T}^{2}} \Leftrightarrow A_{mc}=\frac{\frac{Y_{c}^{2}}{4S_{T}^{2}}-1}{Y_{c}} \tag{27}\]  
\[F_{mc}=\left[\left(\frac{Y_{c}}{2S_{T}}\right)^{2}-1\right]\left(\frac{\partial _{2}+\partial_{2}}{Y_{c}}\right)+\left(\frac{\partial_{2}+\partial_{2}}{2S_{ T}}\right)^{2}+\frac{\sharp_{2}^{2}-\flat_{2}\flat_{3}\flat_{3}}{S_{T}^{2}}+\frac{ \sharp_{21}^{2}+\sharp_{23}^{2}}{S_{L}^{2}}\leq 1.0\qquad\text{if }\hat{\sigma}_{2}+\hat{ \sigma}_{3}<0 \tag{28}\]  
The condition of matrix tensile and compressive behavior is demonstrated in the work of [43], and it uses proof by contradiction, in which it is the only condition that does not violate the principal of \(\hat{\sigma}_{2}\geq 0;\hat{\sigma}_{3}\geq 0\) in al quadrants of the space \(\hat{\sigma}_{2}-\hat{\sigma}_{3}\). In the case of the interlaminar failure criterion, a cohesive model (29) is used for the tensile interlaminar failure criterion with the influence of the shear stresses in the interlaminar plane (30), which is based on the initial work of [21]. For the interlaminar compressive failure, the influence of the shear is minor, therefore, in this case, only the  
The evolution law is then revised, finally taking the form embodied in Figure 1, in which the area \(\Delta\text{BC}\Rightarrow\left(\varepsilon_{eq}(d=0)\right)\) under the curve is linked to the fracture energy \(G_{f,m}\). At the moment, new experimental methodologies are emerging to calculate fracture energy [51], but for orthotropic materials, the classical Compact Tension test is still suggested. The initial values \(\sigma_{eq}^{0}\) and \(\delta_{eq}^{0}\) are calculated in the onset of damage, by analytically solving Equations (23), (25), (28), and (30).