## 3 Boundary Conditions and Material Models
### Multiscale Failure Analysis
#### 3.3.3 Failure Model for Composites Tows  
A 3D progressive failure model [21] is used to predict the final failure of composites tows. This failure model connects the material elastic properties with internal state variables that functions of the type of damage. Before the local structural failures develop, the composite tows typically behave are considered as linear elastic manner, and the constitutive relations for undamaged are given by Equation (4).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Tensile modulus, GPa & \(E_{\text{f1}}=230\), \(E_{\text{f2}}=28\) \\ Poisson’s ratio & \(\
u_{\text{f12}}=0.23\) \\ Shear modulus, Gpa & \(G_{\text{f12}}=50\) \\ Tensile strength, Mpa & \(X_{\text{fT}}\), \(Y_{\text{fT}}\), \(Z_{\text{fT}}=4900\) \\ Compressive strength, Mpa & \(X_{\text{KC}}\), \(Y_{\text{KC}}\), \(Z_{\text{KC}}=3920\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The mechanical properties of fiber carbon.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Tensile Modulus** & **Poisson’s Ratio** & **Tensile Failure** & **Compressive Yield** \\ \hline \(E_{\text{m}}=2.73\) GPa & \(v_{\text{m}}=0.4\) & \(\begin{array}{c}\sigma=85.25\text{MPa}\\ \varepsilon=3.62\%\end{array}\) & \(\begin{array}{c}\sigma=133\text{MPa}\\ \varepsilon=6.5\%\end{array}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mechanical properties of Matrix.  
To detect the onset 3D failure including fiber direction failure and transverse direction failure, the modified Hashin failure criterion is used. In each direction, the tensile and compressive failures are handled separately. The failure modes are modified for the case of 3D Stress as follow.  
Fiber failure mode [25],  
\[\sigma_{11}>0\to f_{\text{ft}}=\left(\frac{\sigma_{11}}{X_{\text{T}}} \right)^{2}+\left(\frac{\tau_{12}}{S_{12}}\right)^{2}+\left(\frac{\tau_{13}}{S _{13}}\right)^{2};\quad f_{\text{ft}}\geq 1 \tag{16}\]  
\[\sigma_{11}<0\to f_{\text{fc}}=\left(\frac{\sigma_{11}}{X_{\text{C}}} \right)^{2};\quad f_{\text{fc}}\geq 1 \tag{17}\]  
Matrix failure mode,  
\[\sigma_{22}>0\to f_{\text{mt}}=\left(\frac{\sigma_{22}}{Y_{\text{T}}} \right)^{2}+\left(\frac{\tau_{12}}{S_{12}}\right)^{2}+\left(\frac{\tau_{23}}{S _{23}}\right)^{2};\quad f_{\text{mt}}\geq 1 \tag{18}\]  
\[\sigma_{22}<0\to f_{\text{mc}}=\left(\frac{\sigma_{22}}{Y_{\text{C}}} \right)^{2}+\left(\frac{\tau_{12}}{S_{12}}\right)^{2}+\left(\frac{\tau_{23}}{S _{23}}\right)^{2};\quad f_{\text{mc}}\geq 1 \tag{19}\]  
where \(S_{12}\), \(S_{13}\) and \(S_{23}\) are the strength for shear in 1-2 plane, shear in 1-3 plane and shear in 2-3 plane, respectively. \(\tau_{12}\), \(\tau_{13}\) and \(\tau_{23}\) are the shear stress in 1-2 plane, shear in 1-3 plane and shear in 2-3 plane, respectively. \(\sigma_{11}\) and \(\sigma_{22}\) are normal stress in 1-direction and 2-direction, respectively. \(f_{\text{ft}}\) and \(f_{\text{fc}}\), are the failure indices for fiber tension and fiber compression, respectively. \(f_{\text{mt}}\) is the failure indices for matrix tension or shear cracking. \(f_{\text{mc}}\) is the failure indices for matrix compression or shear cracking.  
The material stiffness changes, after local failures within the tows. The effects of damage on the stiffness of the tows are represented using internal state variables. These state variables associated with crack density under loading, a more detailed description is given in [21].  
The progressive failure model is implemented as a user-defined material model using Abaqus user interface UMAT (user subroutine to define a material's mechanical behavior). In this procedure, a nonlinear analysis is performed until a converged solution is obtained.