## 4 Implementation of the Pre-Structured Material Behavior
### Transverse Isotropic Law Identification
#### 4.1.2 The Out-of-Plane Poison's Ratio \(\mathbf{\
u}_{TT}\)  
The compliance matrix \(\underline{\underline{\underline{\underline{\underline{\underline{\underline{ \underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{ \underline{\underline{ \underline{ \underline{     }}}}}}}}}}}}}\) of the transverse isotropic material shows that the out-of-plane Poisson's ratio \(\mathbf{\
u}_{TT}\) is not activated in the case of the plane stress assumption unless the out-of-plane displacement is calculated. However, the DIC analysis employed here provides only the displacement field in 2D ROI. Therefore, the out-of-plane Poisson's ratio \(\mathbf{\
u}_{TT}\) cannot be identified using the available data. Otherwise, it was proven that its value would not influence the 2D material behavior because it only affects the out-of-plane strain component. For the developed model, \(\mathbf{\
u}_{TT}\) was taken as the average value between \(\mathbf{\
u}_{TL}\) and \(\mathbf{\
u}_{LT}\). A value of \(\mathbf{\
u}_{TT}\) =0.3416 will be employed.  
1.3 Stiffness Matrix \(\underline{\underline{\underline{\underline{\underline{\underline{ \underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{ \underline{\underline{\underline{\underline{\leftleftleftleftleft({ \mathbf{\underline{\underline{\underline{ \underline{     }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}\  
To access the in-plane shear yield strength, results from the 45\({}^{\circ}\) specimen are used. The following relation (see Equation (5)) is then established by taking \(\mathbf{\theta}\) = 45\({}^{\circ}\) and replacing \(\mathbf{\sigma_{11}}(\mathbf{\theta})\) by the experimental yield strength of the 45\({}^{\circ}\)spceicmen.  
\[\mathbf{\sigma_{LT}^{0}}=\left(\frac{4}{\mathbf{\sigma_{45}^{5}}}-\frac{1}{\mathbf{\sigma _{T}^{2}}}\right)^{-\frac{1}{2}} \tag{5}\]  
A value of 11.65 MPa was found similar to the value found by [31] 12 \(\pm\) 0.7 MPa.  
For the out-of-plane shear yield strength, the torsion test is commonly used to identify its value. The experimental results reported in [37] by [PERSON] et al. show that the out-of-plane shear yield is about 25 MPa. However, this value cannot be utilized herein since the printing parameters are substantially different. Therefore, both in-plane and out-of-plane shear yield strengths were assumed equal (\(\mathbf{\sigma_{TT}^{0}}=\mathbf{\sigma_{LT}^{0}}\)).  
The Hill criterion was implemented in ABAQUS software using the user-defined Hill's potential function in ABAQUS [38]. It implies the definition of six different anisotropic yield stress ratios. As \(\mathbf{\sigma_{TT}^{0}}=\mathbf{\sigma_{LT}^{0}}\), three independent ratios must be introduced as inputs. Those ratios (see Table 4) are expressed using a reference tensile yield stress \(\mathbf{\sigma^{0}}\), which was chosen as the tensile yield strength in the longitudinal direction \(\mathbf{\sigma_{L}^{0}}\), and a reference shear yield stress \(\mathbf{\tau^{0}}\) where \(\mathbf{\tau^{0}}=\frac{\mathbf{\sigma^{0}}}{\sqrt{3}}\).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Yield Strength Designation** & **Loading State on the Filaments** \\ \hline Tensile yield strength in the longitudinal direction: \(\mathbf{\sigma_{L}^{0}}\) & \\ Tensile yield strength in the transversal direction: \(\mathbf{\sigma_{T}^{0}}\) & \\ In-plane shear yield strength: \(\mathbf{\sigma_{LT}^{0}}\) & \\ Out-of-plane shear yield strength: \(\mathbf{\sigma_{TT}^{0}}\) & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Yield strengths designation for the transverse isotropic material.