## 2 Materials and Methods  
This research focuses on studying the manufacturability and mechanical properties of 3D re-entrant auxetic structures. Therefore, 24 half-factorial design of experiments (DoE) were designed, as shown in Table 1, to analyse the manufacturability of different design configurations by PBF and determine the influence of design parameters on their mechanical properties. Four unit cell design parameters, namely, length of vertical strut \"H\", length of re-entrant strut \"L\", re-entrant angle \"\(\theta\)\" and strut thickness \"t\" were taken into consideration, as shown in Figure 1. The values selected for H, L and \(\theta\) were based on an initial virtual analysis of the buildability of the structures within the CAD environment. The values selected would, theoretically, enable all samples to be manufactured successfully (without failure and with no use of supporting structures), subsequently allowing subsequent ANOVA to be performed. \begin{table}
\begin{tabular}{c c c c c} \hline
**Sample** & **H (mm)** & **L (mm)** & \(\theta\) (\(\diamond\)) & **t (mm)** \\ \hline
1 & 6.5 & 2.5 & 45 & 0.8 \\ \hline
2 & 7 & 2.5 & 45 & 1 \\ \hline
3 & 6.5 & 3.5 & 45 & 1 \\ \hline
4 & 7 & 3.5 & 45 & 0.8 \\ \hline
5 & 6.5 & 2.5 & 70 & 1 \\ \hline
6 & 7 & 2.5 & 70 & 0.8 \\ \hline
7 & 6.5 & 3.5 & 70 & 0.8 \\ \hline
8 & 7 & 3.5 & 70 & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: Design of experiment for auxetic structure design. Figure 1: Re-entrant unit cell parameters. From the research carried out by [PERSON] el al., it could be seen that there is no variation in the mechanical properties once the repetition of the unit cell is greater than 3, and the structures behave as a continuum [36]. Hence, samples with \(4\times 4\times 4\) unit cells were designed for manufacture. The eight corresponding structures are given in Figure 2. The structures were designed using SolidWorks CAD software (Dassault Systemes, Coventry, UK). All samples were manufactured in maraging steel (MS1) from EOS GmbH (EOS GmbH, Eschweiler, Germany) using an EOS M280 PBF system, with a 400W laser at 40\(\upmu\)m layers under a nitrogen atmosphere. The CAD designs were orientated at 35\({}^{\circ}\) to the XY (horizontal build plane) so as to exclude the use of internal support structures, requiring only external support structures. A solid base support structure was added to the designs to act as a heat-sink to prevent over-heating of the structures. An example part set-up is shown in Figure 3. It was necessary to angle sample 8 at 10\({}^{\circ}\) to the XY plane in order to compensate for the angle of build of the long struts. This is below the accepted angle for this material for an unsupported surface (35\({}^{\circ}\)), however, as the length of the re-entrant strut is short, the machine is capable of building the strut at the shallow 10\({}^{\circ}\) angle without failure. Default build parameters for MS1 [37] were modified to reduce energy levels to prevent over-heating. Non-standard laser power, scanning speed and hatch spacing was used for up skin, in skin and down skin (Figure 4a), using the parameters given in Table 2. Stripes scanning configuration was used, as shown in Figure 4b. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Layer** & **Laser Power (W)** & **Scan Speed (mm/s)** & **Hatch Spacing (\(\upmu\)m)** \\ \hline Up Skin & 165 & 600 & 110 \\ \hline In Skin & 305 & 1010 & 90 \\ \hline Down Skin & 150 & 2000 & 60 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Scan parameters employed for sample manufacture. Figure 2: CAD models of the 8 design variations. (1–8) refer to samples 1–8. The values selected for H, L and \(\uptheta\) were based on an initial virtual analysis of the buildability of the structures within the CAD environment. The values selected would, theoretically, enable all samples to be manufactured successfully (without failure and with no use of supporting structures), subsequently allowing subsequent ANOVA to be performed. Four sets of each of the 8 designs (Table 1) were manufactured.