## 4 Conclusions  
This paper has presented a manufacturability study on 3D re-entrant structures using PBF and analysis of their mechanical properties. Twenty-four half-factorial design of experiments (DOE) were designed to study the influence of unit cell parameters, namely, the length of the vertical strut 'H', the length of the re-entrant strut 'L', the re-entrant angle '\(\theta\)' and the strut thickness 't' on the mechanical properties of the structure. It has been demonstrated that auxetic 3D structures can be successfully built using laser powder bed fusion (PBF) with a density \(>\) 99.3% for all geometries. Porosity in the struts is found to be dependent on the orientation of the strut to the vertical z-axis. Lowest porosity is achieved for struts aligned nearest to the z-axis, and highest porosity was found on struts aligned furthest away from the z-axis. Porosity levels are higher nearer the strut surfaces and the become less circular. The geometry of the structure has a significant effect on the accuracy to which it can be manufactured. Struts with alignment nearer to the vertical (Z) axis have the best accuracy (\(<\)15 \(\upmu\)m thickness deviation from nominal). Those built at low angle to the vertical exhibit the largest deviation from nominal (\(>\)300 \(\upmu\)m thicker than nominal). The strut length accuracy depends on type of strut within the structure. Vertical struts were built with greatest accuracy (\(<\)2.5% deviation from nominal), and re-entrant struts showed the greatest levels of length inaccuracy (7-18%). The level of inaccuracy was lowest for struts built aligned nearer to the Z-axis. The re-entrant angle was sensitive to geometry  
Figure 20: (**a**) Struts measured for dimensional change under compression in the y direction, (**b**) points analysed to assess degree of strut bending, (**c**) illustration of strut bending. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\Delta\)**L1 (mm)** & \(\Delta\)**L1 (mm)** & \(\Delta\)**H (mm)** & \(\Delta\)**\(\theta\) (\({}^{\circ}\))** \\ \hline Dimension Change & \(-\)0.06 & \(-\)0.06 & \(+\)0.08 & 4.77 \\ \hline  & A1 (mm) & A2 (mm) & B1 (mm) & B2 (mm) \\ \hline Dimension Change & \(-\)0.004 & \(-\)0.001 & \(+\)0.025 & \(+\)0.046 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Calculated strut length changes and re-entrant angle change for compression in the y direction. (and thus to alignment to the Z-axis), with an average of 5% deviation across all structures, but was 4x higher on average for the samples with 45\({}^{\rm o}\) re-entrant angles than those with 70\({}^{\rm o}\) re-entrant angles. There is minimal variation (\(1.02\pm 0.07\) GPa) in the elastic modulus (E\({}_{\rm x}\)) measured in the x direction (perpendicular to the horizontal strut) across all auxetic structures, whereas there was considerable variation (\(4.4\pm 0.1\) GPa) observed for E\({}_{y}\) (measured parallel to the vertical strut). There is considerable variation in the yield stresses measured in both x and y directions (\(\sigma_{x}\) and \(\sigma_{y}\)), each varying similarly with structure type. Averages for yield stress are within errors equal for both test directions (\(\overline{\sigma}_{x}=44\pm 6\), \(\overline{\sigma}_{y}=52\pm 10\)). Poisson's ratio for compression in the x direction, \(\
u_{xy}\), was found to not vary considerably with geometry type, whereas for compression in the y direction, \(\
u_{y}\) was found to vary considerably with geometry. [PERSON]'s ratio of the unit cell at the centre cannot represent the whole structure due to the presence of end effects. \(\
u_{xy}\) decreases as it proceeds to the outside. However, the influence of the end effects on \(\
u_{xy}\) can be reduced by fabricating samples with large arrays of unit cells. There was no correlation found between relative density and Poisson's ratio.