## 4 Failure Criteria  
### Microlevel Constituent Failure Criteria  
Microlevel failure based on GMC was implemented using proper failure criterion within the constituents of the composite. Maximum stress criteria were utilized to detect the failure of fiber, which was deemed to be brittle before it broke.  
\[\left(\frac{\sigma_{11}^{(\beta_{f}\gamma_{f})}}{X_{ft}}\right)^{2}=d_{f}^{2},\ \overline{\sigma}_{11}>0 \tag{16}\]  
\[\left(\frac{\sigma_{11}^{(\beta_{f}\gamma_{f})}}{X_{fc}}\right)^{2}=d_{f}^{2}, \ \overline{\sigma}_{11}<0 \tag{17}\]  
where \(\beta_{f}\) and \(\gamma_{f}\) denoted the fiber subcell indices, and \(X_{ft}\) and \(X_{fc}\) represented the tensile and compressive fiber strengths, finally \(\sigma_{11}\) was the applied axial stress.  
A 3D Tsai-Hill criterion was chosen to predict the failure arising in the matrix subcells, which was shown as below [31].  
\[\frac{\left(\sigma_{11}^{(\beta_{\text{m}}\gamma_{\text{m}})}\right)^{2}+ \left(\sigma_{22}^{(\beta_{\text{m}}\gamma_{\text{m}})}\right)^{2}+\left( \sigma_{33}^{(\beta_{\text{m}}\gamma_{\text{m}})}\right)^{2}}{Y_{\text{m}}^{2 }}+\frac{-\sigma_{11}^{(\beta_{\text{m}}\gamma_{\text{m}})}\left(\sigma_{23} ^{(\beta_{\text{m}}\gamma_{\text{m}})}\right)^{2}-\sigma_{11}^{(\beta_{\text{ m}}\gamma_{\text{m}})}\left(\sigma_{33}^{(\beta_{\text{m}}\gamma_{\text{m}})} \right)^{2}-\sigma_{22}^{(\beta_{\text{m}}\gamma_{\text{m}})}\sigma_{33}^{( \beta_{\text{m}}\gamma_{\text{m}})}}}{Y_{\text{m}}^{2}} \tag{18}\]  
\[\frac{\left(\sigma_{11}^{(\beta_{\text{m}}\gamma_{\text{m}})}\right)^{2}+ \left(\sigma_{22}^{(\beta_{\text{m}}\gamma_{\text{m}})}\right)^{2}+\left( \sigma_{33}^{(\beta_{\text{m}}\gamma_{\text{m}})}\right)^{2}}{Y_{\text{m}}^{2 }}+\frac{-\sigma_{11}^{(\beta_{\text{m}}\gamma_{\text{m}})}\left(\sigma_{23} ^{(\beta_{\text{m}}\gamma_{\text{m}})}\right)^{2}}{Y_{\text{m}}^{2}}=d_{m}^{2}, \ \overline{\sigma}_{22}>0 \tag{19}\]  
where \(Y_{mt}\), \(Y_{mc}\), and \(T_{m}\) belonging to the properties of the matrix subcells represented the transverse tensile strengths, transverse compressive and shear strength, respectively.  
Once \(d_{m}\) was greater than or equal to 1 in a matrix subcell or \(d_{f}\) was less than or equal to in a fiber subcell, failure occurred, and then all the elastic properties of those failed subcells were degraded appropriately. All the stiffness matrices of the constituents were reduced by multiplying 1-D\({}_{\text{max}}\), and D\({}_{\text{max}}\) was set to 99.9% herein.  
At each material point, a micromechanical analysis was performed using GMC by calling the UMAT subroutine, in which maximum stress criteria and Tsai-Hill failure criteria were implemented to predict the failure of subcells. The square-packed RUC consisting of 3 matrix subcells colored green and 1 fiber subcells colored blue was chosen to analyze the properties of composite materials, though arbitrary architecture defined by the users was available. The number of subcells embedded in the RUCs is shown in Figure 3. The flow chart showing the hierarchy of this multiscale algorithm is shown in Figure 4.  
ABAQUS/standard provided the total strain tensor fields at every element integration point of the FEM model, and the global strains were imposed as basic known variables on the RUC to calculate the global stress at each iteration based on the GMC. Additionally, the criteria applied on the constituent material reduced and updated the global stiffness calculated with the GMC, by which the multiscale method of analysis was implemented.