# Crack Growth and Delamination Analysis in GFR Composite Materials
## 3 Modeling a Cross Ply Composite Material [0/90]s  
In the relevant literature, several micromechanical [18; 19] or mesoscale [20; 21; 22; 23; 24; 25; 26] approaches have been proposed. In this study, the material at each ply is a transversely isotropic material whose elastic properties were calculated with a Mori-Tanaka micromechanical formula based on [PERSON]'s solution [13], and the structural simulation follows a mesoscale approach. The material properties' values used were the Young's modulus of epoxy resin (5.5 GPa), glass fiber (73.1 GPa) and the respective Poisson's ratio values were 0.395 and 0.18. The shape of the glass fibers having \(V_{f}=0.63\) is described by their aspect ratio, which was 10,000.  
The total composite elastic properties after homogenization procedure are reported in Table 1. The viscous effects' contribution of the matrix material to the whole composite with respect to time can be seen in Figure 2. The strength values, including Interlaminar Shear Strength (ILSS) for the homogenized GFRP material, are taken from the literature [27].  
In order to model the interfaces between the layers, cohesive contacts are used in a nonlinear analysis framework. The adoption of cohesive contacts dominated the alternative options of cohesive elements or XFEM. The cohesive elements approach has two undesirable effects. The first one is the significant increase of the total finite elements number, and the second one has to do with the meshing procedure of the different layers, which should discretized in a rather restrictive similar manner. As should be stressed, the XFEM approach cannot be used in the dynamic analysis framework with explicit integration. In the traction-separation model of ABAQUS, a linear elastic behavior is initially assumed followed by the initiation and evolution of damage. The elastic behavior is expressed in terms of an elastic constitutive matrix that relates nominal stresses to nominal strains across  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **Instantaneous** & **270 s** & **Quasi-Static** \\ \hline Longitudinal modulus \(E_{1}\) (GPa) & 48.1621 & 46.396 & 46.386 \\ Transverse modulus \(E_{2}\) (GPa) & 18.7977 & 3.930 & 3.7541 \\ In plane Poisson ratio \(\
u_{12}\) & 0.252155 & 0.25029 & 0.25038 \\ Transverse Poisson ratio \(\
u_{23}\) & 0.485318 & 0.5658 & 0.56396 \\ In plane shear modulus \(G_{12}\) (GPa) & 6.32782 & 1.255 & 1.2002 \\ Transverse shear modulus \(G_{23}\) (GPa) & 6.88076 & 1.415 & 1.3505 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Elastic Properties after elastic and viscoelastic Homogenization.  
Figure 2: Time dependent homogenized axial Young’s modulus \(E11\) from viscoelastic homogenization.  
the interface. The maximum shear stress criteria are employed to model the failure at the contacts.  
The model material geometry is represented by a domain between two adjacent cracks that is undamaged and has a 30 mm length. Every layer has a 0.25 mm thickness value and the composite is loaded uniaxially. Tabular data are employed for the incremental application of the strain up to a total value of 1.5% using a loading duration of 270 s. The geometry of the model is depicted in Figure 3.  
For the geometry mesh generation procedure, four rows of continuum shell elements along the thickness direction of each laminate were placed for higher accuracy, resulting in a total number of 96,000 continuum SC8R type shell elements.  
The placement of a high number of potential cracks permits the successful prediction of random matrix cracking (Figure 4). For every potential crack location, different strength values are attributed as non-uniform stresses are developed, as the latter would lead to the simultaneous failure for all interfaces at the cohesion zone.  
The contacts' strength follows a Gaussian distribution with a mean equal to the strength of the matrix and 15% standard deviation.