# Crack Growth and Delamination Analysis in GFR Composite Materials
## 4 Results and Discussion
### Delaminations and Shear Lag Zone  
Cohesive zone modeling (CZM) provides a simulation pathway to model cohesion for interfaces with zero thickness. In the present study, it is used to simulate matrix cracking and delamination. As aforementioned, prior to damage, a linear elastic behavior is assumed  
Figure 4: Modeling of the possible transverse cracks with a cohesive interface.  
Figure 3: Geometry of the cross-ply composite model.  
in this traction separation model, which is valid. The aforementioned behavior is given by the following equation:  
\[\begin{bmatrix}t_{n}\\ t_{s}\\ t_{t}\end{bmatrix}=\left[\begin{array}{ccc}K_{nn}&K_{ns}&K_{nt}\\ K_{ns}&K_{ss}&K_{st}\\ K_{nt}&K_{st}&K_{tt}\end{array}\right]\begin{bmatrix}\delta_{n}\\ \delta_{s}\\ \delta_{t}\end{bmatrix} \tag{14}\]  
The quantities involved are the normal (\(t_{n}\)) and the two (\(t_{s},t_{t}\)) shear tractions, respectively. The corresponding separations are \(\delta_{n},\delta_{s},\delta_{t}\). It is crucial to underline that the cohesive contact is an interaction and not a material property, which is thus interpreted differently from cohesive elements. Additionally, CZM necessitates a geometrical nonlinear analysis.  
In Figure 5, the calculated quantity \(CSQUADSCRT=(t_{n}/t_{0}^{n})^{2}+(t_{s}/t_{0}^{5})^{2}+(t_{t}/t_{0}^{t})^{2}\) denotes the criterion for damage initiation for the quadratic contact stress. If \(CSQUADSCRT\) = 1, the criterion is fulfilled. It should be stressed that the \(CSQUADSCRT\) value cannot exceed 1 so as to specify damage evolution. For CZM, damage progression is manifested by the reduction of stiffness of the cohesive contacts, whereby, in the case of cohesive elements, damage propagation corresponds to stiffness degradation for the material itself. The present work is adopted based on energy criteria, following the analytical forms proposed by [PERSON] and [PERSON] [31]. The aforementioned forms are useful in cases where the critical energies become equal during separation along the two shear directions (\(G_{s}^{C}=G_{t}^{C}\)) and  
\[G_{n}^{C}+(G_{shear}/G_{T})^{\eta}(G_{s}^{C}-G_{n}^{C})=G^{C} \tag{15}\]  
\[G_{shear}=G_{s}+G_{t} \tag{16}\]  
\[G_{T}=G_{n}+G_{shear} \tag{17}\]  
with the quantities involved being the cohesive property parameter \(\eta\), and \(G_{n}^{C},G_{s}^{C},G_{t}^{C}\) are respectively the required critical fracture energies for failure at the normal, the first and the second direction.  
As the first cracks appear and the loading continues, almost immediately delaminations [32] are initiated too. At the area over and under the tip of the cracks, stress concentration is observed. The cracked composite material and the extension of the delamination area around the cracks are depicted in Figure 5.  
Figure 5: Modeling of the possible transverse cracks with cohesive interface. The existence of the shear lag zone is apparent. The upper layer is hidden to get a more clear image of the cracks and the delamination area. Values of \(CSQUADSCRT\) quantity are given in the colorbar.  
The shear lag zone is apparent. In this zone, the normal tensile stresses are almost zero, while the shear stresses at the interface are high, and this constitutes the main reason for delamination appearance. As we move away from that zone, the exact opposite seems to happen; as the normal stresses are high, the next cracks of the composite will be far enough from an existing crack. The zone's size strongly depends on the: (i) layer thickness, (ii) stiffness of the outer layers and the (iii) value of the interlaminar shear strength.