## 3 3D Orthotropic Stiffness Formulation
### Hashin 3D Failure Criterion  
The Hashin 3D failure criterion in this work is based on the fusion of formulations published by several academics [21,30,43]. These continue to acknowledge a primary plane behavior that is consistent with the material's primary resistance stress [6], in which its failure criteria are assembled. Since it is challenging to accurately describe the real behavior of composites in the 3D space, the original failure criteria make various material and tensor field simplifications. In this work, to extend to a fully 3D Hashin damage model, these simplifications will also be applied and explained throughout the manuscript. The first simplification admits that all unidirectional fibers are transversely isotropic, in which the failure criterion depends on effective stresses (18), with the plane 2-3 rotating around axis 1 [43]. Consequently, the failure criteria for fiber breaking are established \(F_{f}\) in axis-1 and matrix cracking \(F_{m}\) for plane 2-3, using their respective tensorial plane stresses (18). For this work, all the tensors are expressed in terms of effective stresses because these failure criteria are later employed in damage models. \[ \begin{split} F_{f}(\partial_{1},\tau_{12},\tau_{13})=1.0\\ F_{m}(\partial_{2},\partial_{3},\tau_{21},\tau_{31},\tau_{23})=1.0 \end{split}  \tag{18}\]  
Despite the fact that the Hashin criterion is always positioned in the material principal directions, it is preferable to rewrite the failure criterion functions in terms of stress invariants using the envelope quadratic Equation (19), in order to support a more reliable failure formulation similar to Von-Mises. \[ \begin{split} F_{f}(I_{1},I_{4})=A_{f}I_{1}+B_{f}I_{1}^{2}+D_{f}I_{ 4}=1.0\\ F_{m}(I_{2},I_{3},I_{4})=A_{m}I_{2}+B_{m}I_{2}^{2}+C_{m}I_{3}+D_{m}I_{ 4}=1.0\end{split}  \tag{19}\]  
with:  
\[\left\{ \begin{array}{l}I_{1}=\partial_{1}\\ I_{2}=\hat{\sigma}_{2}+\hat{\sigma}_{3}\\ I_{3}=\hat{\tau}_{23}^{2}-\hat{\sigma}_{2}\hat{\sigma}_{3}\\ I_{4}=\hat{\tau}_{12}^{2}+\hat{\tau}_{13}^{2}\end{array} \right. \tag{20}\]  
It is possible to simplify in the case of pure shear stress \(\left(\tau_{ij},\tau_{jl}=0,\sigma=0\right)\), for each of the respective shear directions, and compute the values of \(D_{f}\), \(C_{m}\), and \(D_{m}\) (21), for which \(S_{L}\) and \(S_{T}\) are the longitudinal and shear stress, respectively. \[\left\{ \begin{array}{l}C_{m}=\frac{1}{S_{T}^{2}}\\ D_{f}=D_{m}=\frac{1}{S_{L}^{2}}\end{array} \right. \tag{21}\]  
The second simplification is that the tensile and compressive failure criteria may present different failure modes for both the fiber and the matrix, resulting in different failure surface functions. For the third simplification, it concedes that for the fiber buckling in compression (22), there is no influence of shear stresses. Other authors have used the fiber kinking mode to contest this last fiber simplification for 3D model compression [28]. Previous studies have observed the deterioration of matrix shear modulus in fiber cycle compression [44,45], but with small influence and still some uncertainties. For this reason, simplification is still applied in this work. For the third simplification, only quadratic terms are admitted in the fiber failure criterion to preserve the envelope ecliptic surface \(\left(A_{ft}=0;B_{ft}=1/X_{t}^{2}\right)\). In any case, the possibility of a fully closed surface is eliminated by the simplification of the compressive fiber failure criterion. (23). \(X_{t}\) and \(X_{c}\) are the fiber tensile and compressive strength, respectively. \[ \begin{array}{l}A_{ft}=0;B_{ft}=\frac{1}{X_{t}^{2}};D_{ft}=D_{f}=\frac{1}{S_{ L}^{2}}\\ A_{fc}=0;B_{ft}=\frac{1}{X_{c}^{2}};D_{ft}=0\\ \\ F_{ft}=\left(\frac{\phi_{1}}{X_{t}}\right)^{2}+\frac{\sharp_{2}^{2}+\sharp_{2 }^{2}}{S_{L}^{2}}\leq 1.0\qquad\text{if }\hat{\sigma}_{1}\geq 0\\ \\ F_{fc}=\left(\frac{\phi_{1}}{X_{c}}\right)^{2}\leq 1.0\qquad\qquad\qquad\text{if }\hat{ \sigma}_{1}<0\\ \end{array}