# Study on the Properties and Fatigue Characteristics of Glass Fiber Composites Due to Porosity
## 3 Materials and Methods
### Numerical Models
#### 3.2.2 Damage Propagation  
In this paper, progressive damage analysis was conducted using the Matzenmiller-Lubliner-Taylor (MLT) model. The MLT model is primarily applied to long fiber composite materials and requires the definition of Hashin 2D or Hashin-Rotem 2D failure indicators. In the MLT model, the relationship between effective strain and stress is expressed by the following matrix, representing the matrix indicating the extent of damage by the damage variable (\(d\)). \[\overline{S}= \begin{pmatrix}\frac{1}{(1-d_{f})E_{11}}&\frac{-v_{12}}{E_{11}}& 0\\ \frac{-v_{12}}{E_{11}}&\frac{1}{\left(1-\left[\sigma_{22}\right]_{+}d_{m} \right)E_{22}}&0\\ 0&0&\frac{1}{\sigma_{12}(1-d_{s})}\end{pmatrix}  \tag{5}\]  
\(\sigma_{11}\), \(\sigma_{22}\), and \(\sigma_{12}\) are components of the stress tensor. \(E_{11}\), \(E_{22}\), and \(G_{12}\) are Young's and shear moduli, respectively. \(v_{12}\) is the Poisson's ratio, \(d_{f}\) is the fiber-related longitudinal damage index, \(d_{m}\) is matrix-related longitudinal damage index, and \(d_{s}\) is the in-plane shear damage index (combinative). \(d_{s}\) can be expressed as follows. \[d_{s}=1-\left(1-d_{f}\right)(1-d_{m}) \tag{6}\]  
These variables are often obtained from failure criteria. When the damage variable is 0 it indicates that no damage has occurred in the composite material. When it is 1 it signifies that the damage in the composite material has fully occurred, resulting in stiffness becoming 0. The damage variable changes according to the extent of damage from the point where the damage initiation criteria are met. Consequently, the stiffness of the material decreases, and the damage progresses gradually. Damage initiation and progressive damage can be expressed as follows:  
\[\frac{1}{\left(1-d_{f}\right)\overline{E}_{11}}=\mathcal{F}_{A}(\sigma)= \frac{\sigma_{11}}{X_{t}}\;if\;\sigma_{11}\geq 0,\;0\;otherwise\;(Warp\;failure) \tag{7}\]  
\[\frac{1}{\left(1-d_{f}\right)\overline{E}_{11}}=\mathcal{F}_{B}(\sigma)=-\frac {\sigma_{11}}{X_{c}}\;if\;\sigma_{11}<0,\;0\;otherwise\;(Warp\;failure) \tag{8}\]  
\[\frac{1}{\left(1-\left[\sigma_{22}\right]_{+}d_{m}\right)\overline{E}_{22}}= \mathcal{F}_{C}(\sigma)=\frac{\sigma_{22}}{Y_{t}}\;if\;\sigma_{22}\geq 0,\;0\; otherwise\;(Wleft\;failure) \tag{9}\]  
\[\frac{1}{\left(1-\left[\sigma_{22}\right]_{+}d_{m}\right)\overline{E}_{22}}= \mathcal{F}_{D}(\sigma)=-\frac{\sigma_{22}}{Y_{c}}\;if\;\sigma_{22}\geq 0,\;0\; otherwise\;(Wleft\;failure) \tag{10}\]\[\frac{1}{\overline{\hat{G}_{12}}(1-d_{s})}=\mathcal{F}_{E}(\sigma)=\frac{|\sigma_{1 2}|}{S}\ (\mathit{Inplane\ shear\ failure}) \tag{11}\]  
When considering compliance (Equation (5) using the MLT method, \(d_{f}\) becomes a function of Equation (7) for tension in one direction (fiber direction) and a function of Equation (8) for compression. \(d_{m}\) becomes a function of Equation (9) for tension in two directions and a function of Equation (10) for compression. For \(d_{s}\), it becomes a function of shear, as in Equation (11). Progressive damage analysis uses the damage progression as the criterion for damage initiation, which is based on the specific fracture energy per unit area. The fracture energy per unit area is calculated using equivalent stress and equivalent displacement. The point at which the stiffness of the element begins to decrease due to damage initiation is when the equivalent stress is at its maximum, and at this point, the damage variable is 0. After damage initiation, the point where the equivalent stress is 0 and the equivalent displacement is maximum corresponds to a damage variable of 1, indicating the point where stiffness becomes 0. In other words, according to the criteria for damage initiation in composite materials, damage initiation occurs when the damage variable is 0 and the equivalent stress is maximized.