# Low Thermal Expansion Machine Frame Designs Using Lattice Structures
## 3 Results
### Pattern Selection for Low-CTE Lattice Design  
When a (20 \(\times\) 20 \(\times\) 20) mm single-unit cell was examined using the FE method described in Section 2.2, there were differences between CTEs in the X- and Z-axes. This anisotropic thermal expansion would be undesirable for the end-application of the precision machine frame, where predictable response is paramount. One solution to decrease the differences between CTEs in the X- and Z-axes was to compose the unit cells in an arrangement that compensated for the anisotropy. The in-plane CTE (in the XZ plane) was investigated for five unique orientation patterns (see Figure 4). The 1 st pattern lattice provided the lowest in-plane CTE (5.4 \(\times\) 10\({}^{-8}\) K\({}^{-1}\)), while the 4 th pattern gave the highest in-plane CTE (29.0 \(\times\) 10\({}^{-8}\) K\({}^{-1}\)), as shown in Table 2.  
\begin{table}
\begin{tabular}{c c} \hline
**Pattern** & **In-Plane CTE (10\({}^{-8}\) K\({}^{-1}\))** \\ \hline
1 & 5.4 \\
2 & 12.8 \\
3 & 8.9 \\
4 & 29.0 \\
5 & 9.5 \\ \hline \end{tabular}
\end{table}
Table 2: The in-plane coefficient of thermal expansion of designs of the 2 \(\times\) 1 \(\times\) 2 lattice.  
Figure 9: The low planar CTE lattice results and its surface fit, where \(L\) is the original length of the lattice and \(D\) is the original diameter of the cylindrical inner part, and \(\alpha_{2}/\alpha_{1}\) is the ratio of CTEs of the cylindrical inner part material and the lattice outer part material.  
Table 2 shows the in-plane CTE results of five unique orientation patterns. The 1 st pattern was the lattice that was composed of the \(0^{\circ}\), \(90^{\circ}\), \(180^{\circ}\), and \(270^{\circ}\) counterclockwise orientated unit cells in the top left, top right, bottom right, and bottom left cells, respectively. The 1 st pattern provided the lowest in-plane CTE because it was the only pattern with connecting layers in a single plane (the XZ plane), see Figure 10c, so it means that the displacement would occur only in the XZ plane. The deformation of the lattice due to the expansion of the inner cylindrical part was, therefore, constrained to a single plane. In Patterns 2 to 5, they provided higher in-plane CTEs because the arrangement of the connecting layers were out of the XZ plane, resulting in displacements coming out of the XZ plane (see Figure 10d). Moreover, when the 1 st pattern lattice was tessellated over a greater numbers of cells, it was observed that even numbers of unit cells provided low CTE anisotropy (\(3\times 10^{-8}\) K\({}^{-1}\) for the \(2\times 1\times 2\) lattice and \(1\times 10^{-8}\) K\({}^{-1}\) for the \(4\times 1\times 4\) lattice), as shown in Table 3, because they consisted of \(2\times 1\times 2\) lattices which constrained the deformation to a single plane, e.g., \(4\times 1\times 4\) lattice was composed of four sets of \(2\times 1\times 2\) lattice. However, this paper focuses on design concepts, and results from larger lattices with more cells require considerably greater computational resources than could be employed here.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Lattice** & **In-Plane CTE (\(10^{-8}\) K\({}^{-1}\))** & **In-Plane CTE Anisotropy (\(10^{-8}\) K\({}^{-1}\))** \\ \hline \(1\times 1\times 1\) & 0.1 & 1379 \\ \(2\times 1\times 2\) & 5.4 & 3 \\ \(3\times 1\times 3\) & 8.7 & 149 \\ \(4\times 1\times 4\) & 0.03 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The in-plane coefficient of thermal expansion of designs.  
Figure 10: The effect of layer alignment on the deformation of the \(2\times 1\times 2\) lattice outer part. (**a**) shows an example of the 1 st pattern; (**b**) shows an example of the cross-section of the 1 st pattern; (**c**) shows an example of the cross-section of the 1 st pattern at Layer 2; and (**d**) shows any of the other patterns’ connecting layers.