# Modelling of Failure Behaviour of 3D-Printed Composite Parts
## 2 Materials and Methods
### Computational Modelling  
In this work, we applied numerical homogenization techniques for linear material modelling and for nonlinear material modelling. Computational models for modelling linear material behaviour were recently developed by authors [38]. In the present work, we focused on developing the computational procedure for nonlinear material modelling of the printed parts. A multiscale designer tool by Altair Engineering Inc is utilized for computational modelling. In this work, two different computational methodologies are developed for damage modelling. In the computational modelling of linear material behaviour of 3D-printed parts, a unit cell is taken from the single layers' microstructure of the printed parts. The unit cell is also referred to as the representative volume element (RVE), and is shown in Figure 2 for a composite 3D part. The layers of printed parts are considered as orthotropic and \begin{table}
\begin{tabular}{l l l} \hline \hline
**Printing Strategy** & **Materials** & \\ \hline  & **ABS Polymer** & **ABS + sCF Composite** \\ \hline Cross-ply & Cross-ply\_ABS & Cross-ply\_ABS + sCF \\ Angle-ply & Angle-ply\_ABS & Angle-ply\_ABS + sCF \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Materials and printing strategies of 3D-printed parts. Figure 1: 3D-printed parts for tensile testing (**a**) cross\(-\)ply (**b**) angle\(-\)ply printed laminates. therefore, the RVE is treated as an orthotropic material. The stress-strain relationship for an orthotropic material is  
\[ \begin{bmatrix}\mathbb{C}_{11}&\mathbb{C}_{12}&\mathbb{C}_{13}&0&0&0\\ \mathbb{C}_{12}&\mathbb{C}_{22}&\mathbb{C}_{23}&0&0&0\\ \mathbb{C}_{13}&\mathbb{C}_{23}&\mathbb{C}_{33}&0&0&0\\ 0&0&0&\mathbb{C}_{44}&0&0\\ 0&0&0&0&\mathbb{C}_{55}&0\\ 0&0&0&0&0&\mathbb{C}_{66}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{33}\\ \gamma_{23}\\ \gamma_{13}\\ \gamma_{12}\end{bmatrix} \text{; or }\{\sigma\}=[C]\{\varepsilon\} \tag{1}\]  
where \(C\) is a constitutive matrix. The compliance matrix (\(\mathbf{S}\)) is obtained by inverting the above equation. \[ \begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{33}\\ \gamma_{23}\\ \gamma_{13}\\ \gamma_{12}\end{bmatrix} = \begin{bmatrix}S_{11}&S_{12}&S_{13}&0&0&0\\ S_{12}&S_{22}&S_{23}&0&0&0\\ S_{13}&S_{23}&S_{33}&0&0&0\\ 0&0&0&S_{44}&0&0\\ 0&0&0&S_{55}&0\\ 0&0&0&0&S_{66}\end{bmatrix}\begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \tau_{23}\\ \tau_{13}\\ \tau_{12}\end{bmatrix} \text{; or }\{\varepsilon\}=[S]\{\sigma\} \tag{2}\]  
the coefficients \(S\) matrix are  
\[S_{11}=\frac{1}{E_{1}}\text{, }S_{12}=-\frac{\
u_{12}}{E_{1}}\text{, }S_{13}=-\frac{\
u_{13}}{E_{1}}\text{, }S_{22}=\frac{1}{E_{2}}\text{, }S_{23}=-\frac{\
u_{23}}{E_{2}}\text{, }S_{33}=\frac{1}{E_{3}}\text{, }S_{44}=\frac{1}{G_{23}}\text{, }S_{55}=\frac{1}{G_{13}}\text{, }S_{66}=\frac{1}{G_{12}} \tag{3}\]  
The RVE is considered macroscopically homogeneous in the homogenization technique, and the average stresses \(\sigma_{ij}\) and average strains \(\varepsilon_{ij}\) are calculated using the following equations  
\[\sigma_{ij}=\frac{1}{V_{\text{RVE}}}\int_{V}\sigma_{ij}(x_{1},x_{2},x_{3})dV \text{, }\varepsilon_{ij}=\frac{1}{V_{\text{RVE}}}\int_{V}\varepsilon_{ij}(x_{1},x_{ 2},x_{3})dV \tag{4}\]  
The average stress and strain fields for fibre and matrix in an RVE are written as  
\[\varepsilon_{ij}=v_{f}\varepsilon_{ij}^{f}+v_{m}\varepsilon_{ij}^{m} \tag{5}\]  
\[\sigma=v_{f}\sigma_{ij}^{f}+v_{m}\sigma_{ij}^{m} \tag{6}\]  
Figure 2: Unit cell or representative volume element (RVE) of 3D-printed composite part [38]. where \(f\) denotes fibre, and \(m\) denotes matrix material, respectively. The elastic constitutive relation for homogenized RVE is given as  
\[\{\sigma\}=[C]\{\epsilon\} \tag{7}\]  
From Equation (7) the constitutive matrix \(C\) is calculated by following the procedure as described in [39].