# Low Thermal Expansion Machine Frame Designs Using Lattice Structures

^{1}

^{2}

^{*}

## Abstract

**:**

^{−9}K

^{−1}(cf. 109 × 10

^{−6}K

^{−1}for solid Nylon 12). This paper showed that the combination of design optimisation and additive manufacturing can be used to achieve low CTE structures and, therefore, low thermal expansion machine frames of a few tens of centimetres in height.

## 1. Introduction

^{−5}K

^{−1}and 1.85 × 10

^{−5}K

^{−1}, respectively), resulting in an effective CTE of −68.1 × 10

^{−6}K

^{−1}for the lattice structure.

^{−4}K

^{−1}to 1 × 10

^{−3}K

^{−1}.

Paper Reference | Structure Type | Material(s) | Effective CTE (10^{−6} K^{−1}) |
---|---|---|---|

[8] | Lightweight cellular metal composites | Aluminium and Invar | −14 to 17.1 |

[11] | Composites with extremal CTEs using topology optimisation | Invar and nickel | −4.97 to 35.0 |

[12] | A porous material with planar negative CTE | VeroWhitePlus and TangoBlack Plus | −434 to 396 |

[13] | Porous composites with tunable CTE | VeroWhitePlus and TangoBlack Plus | −300 to 1000 |

[14] | A repeating hexagonal lattice with bi-material ribs | Two materials with CTE difference of 10^{−5} K^{−1} | Large positive, zero, and large negative |

[24] | A honeycomb lattice with bi-material ribs | Invar and steel | Zero |

[15] | 2D and 3D lattices with bi-material elements | Aluminium and copper | −68.1 |

[25] | Planar chiral lattices and cylindrical shells | Stainless steel 431 or Al7075, and Invar | −65.77 to 91.64 |

[16] | A continuous honeycomb structure with inserts | Two different CTE materials | Near-zero |

[17] | An Octet bi-materials | Al6061 and Ti–6Al–4V | 0.17 |

[26] | 2D metamaterials using bi-material re-entrant planar lattice structures | Stainless steel 431 or Al7075, and Invar | −3 to 2.5 |

[27] | 3D metamaterials using bi-material re-entrant planar lattice structures | Stainless steel 431 or Al7075, and Invar | −8.69 to −5.22 |

[28] | Micro-lattice composite structure | Two different CTE materials | Negative or zero |

[20] | Stretch-dominated planar lattices with the low CTE with high stiffness | Al7075 and Ti–6Al–4V | Zero with high stiffness |

[23] | Stretch-dominated planar lattices in the micro-scale (thin film) | Aluminium and titanium | −0.6 |

[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 |

[29] | Lattice cylindrical shells with tailorable axial and radial CTE | Stainless steel 431 or Al7075, and Invar | −64.6 to 88.0 |

## 2. Methodology

#### 2.1. Motivation and Lattice Design Concept

#### 2.2. The Finite Element Method

## 3. Results

#### 3.1. Low Planar CTE Lattice Results

^{−6}K

^{−1}to 96 × 10

^{−6}K

^{−1}by varying the design parameters. ${\alpha}_{X,Z}$ data were fitted with a first order polynomial surface function of the form

^{2}= 0.994) over the examined range of $D/L$ and ${\alpha}_{2}/{\alpha}_{1}$.

^{−6}K

^{−1}, b = (−132 ± 5) × 10

^{−6}K

^{−1}, and c = (−20 ± 10) × 10

^{−6}K

^{−1}, which can henceforth be used to determine the $D/L$ and ${\alpha}_{2}/{\alpha}_{1}$ parameters to achieve a pre-defined CTE.

^{−6}K

^{−1}and 200 × 10

^{−6}K

^{−1}[36], respectively. The ratio of ${\alpha}_{2}/{\alpha}_{1}$ for our chosen materials was, therefore, approximately 1.83, meaning that a unit cell made of such materials can provide CTEs between 7.8 × 10

^{−6}K

^{−1}and 1 × 10

^{−9}K

^{−1}.

#### 3.2. Pattern Selection for Low-CTE Lattice Design

^{−8}K

^{−1}), while the 4th pattern gave the highest in-plane CTE (29.0 × 10

^{−8}K

^{−1}), as shown in Table 2.

^{−8}K

^{−1}for the 2 × 1 × 2 lattice and 1 × 10

^{−8}K

^{−1}for the 4 × 1 × 4 lattice), as shown in Table 3, because they consisted of 2 × 1 × 2 lattices which constrained the deformation to a single plane, e.g., 4 × 1 × 4 lattice was composed of four sets of 2 × 1 × 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.

## 4. Discussion

^{−6}K

^{−1}to 396 × 10

^{−6}K

^{−1}from Akihiro et al. [12] and −300 × 10

^{−6}K

^{−1}to 1000 × 10

^{−6}K

^{−1}from Talezawa and Kobashi [13]), the internal geometries of those structures were more complex, required additional topology optimisation software to model the structures, and still needed an advanced multi-material AM process to fabricate. The structure with the chosen parameters could provide a CTE of 1 × 10

^{−9}K

^{−1}, which was a much lower CTE than many commercial instruments, such as 23 × 10

^{−6}K

^{−1}for aluminium 7075’s CTE and 10 × 10

^{−6}K

^{−1}for stainless steel 431′s CTE [25] of the working surface of commercial optical breadboards.

## 5. Conclusions

^{−9}K

^{−1}, which was the nearest to the near-zero CTE that could be achieved from the design.

## 6. Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Illustration of precision measurement equipment with the 8 × 4 × 6 lattice; an optical vertical dilatometer.

**Figure 3.**The concept of low-CTE design. (

**a**) Top view of a drafted layer, where L is the original length of the lattice and D is the original diameter of the cylindrical inner part; (

**b**) Isometric view of a unit cell (the two-way arrows indicate the deformation movement of the lattice).

**Figure 4.**Annotation and patterns of the lattices. (

**a**) Top view and front view of the unit cell; (

**b**) Patterns of 2 × 1 × 2 lattices.

**Figure 6.**Representative example of the converged mesh of the low-CTE concept lattice unit cell and its boundary conditions applied on a lattice unit cell design.

**Figure 8.**Illustration of the top view and the side view of (

**a**) the $(D/L)=0.30$ model; (

**b**) the $(D/L)=0.50$ model; and (

**c**) the $(D/L)=0.73$ model.

**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.

**Figure 10.**The effect of layer alignment on the deformation of the 2 × 1 × 2 lattice outer part. (

**a**) shows an example of the 1st pattern; (

**b**) shows an example of the cross-section of the 1st pattern; (

**c**) shows an example of the cross-section of the 1st pattern at Layer 2; and (

**d**) shows any of the other patterns’ connecting layers.

**Figure 11.**The total volume of the (10 × 10 × 10) mm unit cell by varying the ratio of $D/L$ between 0.3 and 0.73.

Pattern | In-Plane CTE (10^{−8} K^{−1}) |
---|---|

1 | 5.4 |

2 | 12.8 |

3 | 8.9 |

4 | 29.0 |

5 | 9.5 |

Lattice | In-Plane CTE (10^{−8} K^{−1}) | In-Plane CTE Anisotropy (10^{−8} K^{−1}) |
---|---|---|

1 × 1 × 1 | 0.1 | 1379 |

2 × 1 × 2 | 5.4 | 3 |

3 × 1 × 3 | 8.7 | 149 |

4 × 1 × 4 | 0.03 | 1 |

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**MDPI and ACS Style**

Juasiripukdee, P.; Maskery, I.; Ashcroft, I.; Leach, R. Low Thermal Expansion Machine Frame Designs Using Lattice Structures. *Appl. Sci.* **2021**, *11*, 9135.
https://doi.org/10.3390/app11199135

**AMA Style**

Juasiripukdee P, Maskery I, Ashcroft I, Leach R. Low Thermal Expansion Machine Frame Designs Using Lattice Structures. *Applied Sciences*. 2021; 11(19):9135.
https://doi.org/10.3390/app11199135

**Chicago/Turabian Style**

Juasiripukdee, Poom, Ian Maskery, Ian Ashcroft, and Richard Leach. 2021. "Low Thermal Expansion Machine Frame Designs Using Lattice Structures" *Applied Sciences* 11, no. 19: 9135.
https://doi.org/10.3390/app11199135