# Low Thermal Expansion Machine Frame Designs Using Lattice Structures
## 1 Introduction
### 2. Methodology  
#### Motivation and Lattice Design Concept  
The motivating application for this investigation is a machine frame to support precision measurement equipment (e.g., cameras and optical projection systems). Such a frame would ultimately be used in a temperature-controlled metrology laboratory, with the temperature range around 19-23 \({}^{\circ}\)C. Figure 1 shows an example of the use of the (40 \(\times\) 20 \(\times\) 30) cm lattice with the precision measurement equipment. The thermal expansion of the frame over this temperature range should not be greater than the resolution of the precision measurement equipment [30], so it does not dominate measurement uncertainty for the instrument. To address the need of low planar CTE frames, lattice designs were developed and examined for their ability to provide a minimal, or zero, planar thermal expansion. A single-material lattice will generally possess similar CTE to a solid structure composed of the same material [31]. Thus, we focused on multi-material design solutions to obtain tailorable \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Paper Reference** & **Structure Type** & **Material(s)** & **Effective CTE (10\({}^{-6}\) K\({}^{-1}\))** \\ \hline
[26] & 2D metamaterials using bi-material re-entrant planar lattice structures & Stainless steel 431 or AI/075, and Invar & \(-\)3 to 2.5 \\ \hline
[27] & 3D metamaterials using bi-material re-entrant planar lattice structures & Stainless steel 431 or AI/075, and Invar & \(-\)8.69 to \(-\)5.22 \\ \hline
[28] & Micro-lattice composite structure & Two different CTE materials & Negative or zero \\ \hline
[20] & Stretch-dominated planar lattices with the low CTE with high stiffness & AI/075 and Ti–6 Al–4V & Zero with high stiffness \\ \hline
[23] & Stretch-dominated planar lattices in the micro-scale (thin film) & Aluminium and titanium & \(-\)0.6 \\ \hline
[18] & 1D to 3D multi-stable architected materials with zero Poisson’s ratio and controllable CTE & Polyamide 12 and glass beads reinforced polyamide 12 & Large positive, zero, and large negative \\ \hline
[29] & Lattice cylindrical shells with tailorable axial and radial CTE & Stainless steel 431 or AI/075, and Invar & \(-\)64.6 to 88.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(Cont.\)  
Figure 1: Illustration of precision measurement equipment with the 8 \(\times\) 4 \(\times\) 6 lattice; an optical vertical dilatometer. thermal expansion. The principal challenge was how the internal lattice geometry could be created to minimise the structure's CTE using two different positive CTE materials with void spaces. Initial concept lattice designs were modelled using computer-aided design (CAD) before being meshed and analysed using a finite element (FE) method. Mesh convergence analyses were carried out to ensure that the mesh density was sufficient to accurately predict the performance of the lattices. Using modelling and numerical analysis in this way provides a relatively inexpensive tool for the examination (and elimination) of concept designs in comparison with manufacture and physical testing. Finally, the most promising candidate design was examined in detail using computational methods prior to its manufacture and assembly being demonstrated. The low-CTE lattice design was created as a unit cell which was then tessellated into a lattice structure (Figure 2). The unit cell was composed of a lattice outer part (marked 'Low-CTE' in Figure 2) and a cylindrical inner part (marked 'High-CTE'). The lattice outer part contained the CTE-minimising mechanism, and was intended to be made using an AM process, particularly polymer laser powder bed fusion (LPBF). The cylindrical inner part was selected from a range of conventional industrial materials and was, therefore, not restricted by the limited materials selection associated with LPBF.