# Crack Growth and Delamination Analysis in GFR Composite Materials
## 2 Homogenization Procedure  
### Homogenization Method for Elastic Materials  
The elastic properties of the material are obtained using a two-step micromechanics homogenization algorithm. Firstly, all of the inclusion families were homogenized separately as a composite material with two phases following the Mori-Tanaka homogenization scheme [12]. Secondly, all pseudo grains were homogenized to a single composite by adopting a Voigt homogenization approach, which is schematically depicted in Figure 1 and in mathematical form can be described by \(C_{eff}=\sum_{i=1}^{n}V_{i}C_{i}\), where \(C_{eff}\) is the homogenized stiffness tensor, and \(C_{i}\) is the stiffness tensor of the pseudo grain \(i\) with volume fraction \(V_{i}\). In case of a two-phase composite material such as a glass fiber reinforced polymer (GFRP) with unidirectional long fibers, the two-step method coincides with the simple Mori-Tanaka method. The homogenized stiffness tensor is given from the Mori-Tanaka formula:  
\[C_{MT}=C_{m}+[V_{f}<(C_{f}-C_{m})A_{Eshelby}>][V_{m}\mathcal{I}+V_{f}<A_{Eshelby }>]^{-1} \tag{1}\]  
where \(<>\) stands for orientation averaging of fiber orientations, which can be ignored for aligned fibers. The quantities \(C_{f}\), \(C_{m}\) are the stiffness matrices of the epoxy and the glass fibers, respectively. The volume fractions of the fiber and the matrix are denoted with \(V_{f}\) and \(V_{m}\). In addition, \(\mathcal{I}\) is the fourth order identity tensor, and \(A_{Eshelby}\) is the strain concentration tensor of the dilute solution defined as:  
\[A_{Eshelby}=[\mathcal{I}+EC_{m}^{-1}(C_{f}-C_{m})]^{-1} \tag{2}\]  
where \(E\) is the Eshelby tensor that depends on the aspect ratio of the inclusion and the properties of the matrix material. The Eshelby tensor can be computed by using analytical solutions that have been developed for certain inclusion geometries and matrix materials. The general solution for inclusions of an arbitrary shape embedded in an isotropic matrix is described as follows:  
\[E_{ijkl} =\frac{1}{8\pi(1-v)}\{\delta_{ij}\delta_{kl}(2 vI_{i}+I_{ik})+( \delta_{ik}\delta_{jl}+\delta_{jk}\delta_{il}) \tag{3}\] \[[(1-v)(I_{k}+I_{i})+J_{ij}]\}\]  
\[I_{i}=I(\lambda)=2\pi\alpha_{1}\alpha_{2}\alpha_{3}\int_{\lambda}^{\infty} \frac{(A_{i}+s)^{-1}}{\Delta s}ds \tag{4}\]  
\[J_{ij}=J_{ij}(\lambda)=A_{i}I_{ij}(\lambda)-I_{i}(\lambda) \tag{5}\]  
\[I_{ij}=I_{ij}(\lambda)=2\pi\alpha_{1}\alpha_{2}\alpha_{3}\int_{\lambda}^{ \infty}\frac{(A_{i}+s)^{-1}(A_{j}+s)^{-1}}{\Delta s}ds \tag{6}\]  
\[\Delta s^{2}=(A_{1}+s)(A_{2}+s)(A_{3}+s) \tag{7}\]  
where  
\[A_{i}=\alpha_{i}^{2} \tag{8}\]  
The quantities involved are: the Poisson ratio \(\
u\) of the matrix, \(\alpha_{1},\alpha_{2},\alpha_{3}\) are the three ellipsoid axes dimensions of the inclusions geometry, and \(\delta\) is the Kronecker's delta and the numerical quantities \(I_{i},I_{ij}\) that are defined in terms of standard elliptic integrals and details can be found in [12; 13].  
The homogenization algorithm deals with the random orientations by performing orientational averaging using a fiber orientation tensor after retrieving a fourth order tensor using hybrid closure approximation [14]. To calculate the effective properties of a continuous fiber reinforced composite, the ellipsoid principal dimension that corresponds to the fiber's axis is much bigger than the other two axes' dimensions: \(\alpha_{1}>>\alpha_{2}\) and \(\alpha_{1}>>\alpha_{3}\).