# Comparative Study on the Uniaxial Behaviour of Topology-Optimised and Crystal-Inspired Lattice Materials

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Cell Architecture Design

#### 2.1. Topology Optimisation Problem

^{TM}by integrating with Ls-dyna

^{®}(LSTC, Livermore, CA, USA). The implicit algorithm is employed. The optimisation problem can be mathematically stated as follows:

**x**). The design variable x, with 0 < x

_{min}≤ x ≤ 1 is known as relative density. $n$ denotes the total number of design variables. $f$ and $u$ are, respectively, the vectors of nodal force and displacement, while $K\left(\mathit{x}\right)$ is the structure stiffness matrix. ${M}_{0}$ and $M\left(\mathit{x}\right)$ represent the initial mass and the current mass of the design domain. A constraint is imposed on the mass fraction ${M}_{f}$. The base material is a high-strength steel referred to Yang et al. [41], with the key parameters presented in Table 1. Three different loading conditions, i.e., face-centred loadings, edge-centred loadings and vertexes loadings, represented by arrows, are illustrated in Table 2. The magnitude of loadings (1 N) is well selected to make sure that the design domain only suffers elastic deformation and the loading points are fixed during the optimisation process. Increasing the value of loading may lead to different optimised cells and it may also cause a problem in designing structures with a high level of porosity [1]. The topology optimisation method is a hybrid cellular automata (HCA) algorithm [42], and the solid isotropic material with penalization (SIMP) model is adopted. Two termination conditions are used to stop the optimisation process: (i) the number of iterations exceeds the maximum number of iterations, or (ii) the change in the topology is smaller than the tolerance [43]. Corresponding optimal cells are also presented in Table 2.

#### 2.2. Analogy with Crystal Structures

^{3}) is composed of six faces, twelve edges, and eight vertexes. Therefore, three lattice materials can be generated by, respectively, connecting the face-centre points, edge-centre points, and vertexes. The obtained lattice materials are referred to as the face-centred cube (FCC), edge-centred cube (ECC), and vertex cube (VC), which is consistent with Xiao et al. [1].

_{1}is cube length, and d/l

_{1}denotes the aspect ratio.

_{1}given by Equations (3) and (4). Computer-aided design (CAD) predictions are also given in the same figure to properly validate the responses of both equations. As can be seen from Figure 3, good consistency is observed between theoretical and CAD predictions.

#### 2.3. Numerical Results

## 3. Finite Element Modelling

^{®}to predict the uniaxial compressive behaviour of the lattice materials. The geometry models of lattice materials are meshed into tetrahedral elements, which are constant stress solid elements. Instead of using periodic boundary conditions on a single unit cell, a 3 × 3 × 3 cell model with dimension of 22.5 × 22.5 × 22.5 mm

^{3}is built. The influence of the number of cells on the elastic modulus of lattice material was numerically studied by Maskery et al. [47], and they found that the converged modulus of the 3 × 3 × 3 cell diamond lattice was just 1% below the upper bound of the theoretical elastic modulus. The FE model of FCC-TO is shown in Figure 4. The lattice materials are placed between two parallel plates, which are modelled as rigid bodies by *Mat.020_Mat_Rigid [48]. The bottom plate is fixed while the top plate impacts the specimen at a constant speed (10 m/s). “*Mat.024_Mat_Piecewise_Linear_Plasticity” [48] is selected to bilinearly approximate the stress–strain curve of elastic-plastic material in Table 1. The impact force is captured by using the *Automatic_Surface_To_Surface contact algorithm [48] applied among specimens and boundaries. In this algorithm, the stiffness of contact elements is penalised by a scale factor [48]. Mesh size is determined, as a result of a sensitivity analysis, as a compromise between accuracy and low computing time. Figure 5 plots the stress–strain curves of the FCC-TO sample calculated by various element sizes, i.e., 0.10~0.15 mm, 0.125~0.200 mm, 0.15~0.25 mm, 0.20~0.30 mm, and 0.25~0.35 mm. In Figure 5, convergence can be obtained when element size is equal to 0.125~0.200 mm. Therefore, this element size is employed for computation throughout this study, which means the main parts of the lattice have an element size of 0.200 mm, while other parts (e.g., joints) are characterised by a smaller element size, i.e., 0.125 mm. The finite element model has been validated by means of experimental results obtained for a body-centred-cubic (BCC) lattice made of 316L stainless steel, taken from [49]. As shown in Figure 5b, a good agreement is observed between experimental and numerical data. It should be noted that the simulation curve in Figure 5b is filtered by using Butterworth Filter with a frequency of 20,000 Hz. The frequency is carefully selected to filter the numerical perturbation while maintaining the characteristics of the stress–strain curve.

## 4. Results and Discussion

#### 4.1. Deformation Modes

#### 4.2. Stress-Strain Curves

_{L}), which is driven by the bending or stretching for the inclined or vertical cell struts/walls, respectively [53]. The plateau regime due to the plastic hinges at sections or joints can be measured by plateau stress (σ

_{pl}). The densification regime starts from a densification strain (ε

_{cd}), where the individual cell strut/wall comes into contact with each other, and exhibits dramatically increasing strength [53].

_{L}is defined as the slope of the initial linear elastic region; the initial highest peak stress is defined as the strength σ

_{b}; and the plateau stress σ

_{pl}is calculated as the arithmetical mean of the stress at a strain interval between 20% and 40% according to ISO 13314: 2011 [54].

_{cd}is identified by using the energy absorption efficiency (η) method.

_{cd}(Figure 7). Comparing the two lattice generation methods, topology-guided lattices generally produce higher σ-ε curves than manually generated structures. Specifically, the gap between TO- and CI- lattices is larger at a higher relative density, although the difference is not obvious at a lower relative density. This is because of the cell walls formed in topology-optimised lattices at high mass fraction, as shown in Table 2. In general, σ

_{pl}of optimised structures is higher than that of manually designed structures, especially for FCC (see Table 5). However, three cases (i.e., ECC with $\overline{\rho}\approx 0.15$, VC with $\overline{\rho}\approx 0.15$ and $0.20$) are excepted, which may be attributed to the slightly lower relative density of topology-optimised lattices comparing with the corresponding crystal-inspired lattices. For example, the $\overline{\rho}$ of VC-TO is 0.147, while that of corresponding VC-CI is 0.150 (see Figure 6).

#### 4.3. Mechanical Properties and Energy Absorption

_{L}, and collapse strength, σ

_{b}, scale with relative density,$\overline{\rho}$, according to the relationships:

_{1}and n

_{2}represent the structural bending/stretching dominated mode. For bending-dominated structures (e.g., body-centred lattice), n

_{1}= 1.5, and n

_{2}= 2; for stretching-dominated structures (e.g., Octet-truss lattice), both n

_{1}and n

_{2}are equal to 1. C

_{1}and C

_{2}are constants related to the lattice’s architecture as well as the base material properties.

_{1}and C

_{2}) and exponent n (n

_{1}and n

_{2}) are tabulated in Table 6. It is noticed that the exponents, n

_{1}and n

_{2}, of the as-designed lattice materials are respectively close to 1.50 and 1.70, which indicates a bending-dominated behaviour mixed with a light stretching mode. Montemayor and Greer [46] pointed out that the FCC lattice behaves as a bending-dominated structure due to the rotation between and within unit cells, although the unit cell is a stretching-dominated structure. This phenomenon can be observed in Table 3. Gross et al. [13] made it clear that the ECC lattice is also a bending-dominated structure. Compression tests were carried out on the VC-CI lattices by Mei et al. [2], where the power law with n

_{1}= 1.5 and n

_{2}= 2 was used to fit the experimental data.

_{T}) is defined as the amount of energy per unit volume up to the strain of 0.25 [50].

_{T}and W

_{V}are also be fitted by the power law.

_{V}is plotted with respect to the stress (σ) to simplify the relationships of different compressive stages. The maximum allowable stress (σ

_{max}) under a certain energy absorption ability can also be obtained. This curve helps to find a cellular material that bears the required σ

_{max}by maximising the energy absorption capability [56]. An energy-efficient structure gives a high envelope. In Figure 9, VC-TO lattice is able to absorb more energy than others with the same allowable stress. For instance, if the σ

_{max}is 300 MPa, the values of W

_{V}for FCC-TO, ECC-TO, and VC-TO with $\overline{\rho}=0.196$ are 64 MJ/m

^{3}, 47 MJ/m

^{3}, and 80 MJ/m

^{3}, respectively.

## 5. Conclusions

- a)
- Topology optimisation-guided lattice materials are highly similar to the corresponding crystal-inspired lattice materials, especially at a low relative density. The topology optimisation-guided lattice materials are generally non-uniform in terms of strut thickness and joints shape, while the crystal-inspired cells are uniform.
- b)
- Formulae relating the relative density ($\overline{\rho}$) and aspect ratio (d/l
_{1}) of crystal-inspired lattices are presented, which has been well validated by CAD predictions. - c)
- Comparing the topology-guided and manually generated structures, FCC-TO and ECC-TO exhibit a highly similar bending-dominated deformation mode to FCC-CI and ECC-CI, respectively. However, differences are found between VC-TO and VC-CI lattices. Shear band is observed in VC-CI structures at a low relative density while the VC-TO lattice deforms stably.
- d)
- In terms of collapse strength and elastic modulus, the VC lattice is stronger than the FCC and ECC lattices because its struts are arranged along the loading direction. On the other hand, topology-generated lattices outperform the corresponding crystal-guided lattices in aspects of toughness and energy absorption per unit volume.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic illustration of manually designed lattice materials: (

**a**) FCC (face-centre cube); (

**b**) edge-centre cube (ECC); (

**c**) vertex cube (VC).

**Figure 5.**(

**a**) Effects of element size on the stress-strain curve of the FCC-TO lattice material; (

**b**) validation of the modelling method by using experimental data reproduced from [49], copyright permission: Elsevier, 2020.

**Figure 6.**Stress-strain curves of lattice materials generated from two methods with various relative densities: (

**a**) FCC; (

**b**) ECC; (

**c**) VC.

**Figure 7.**The $\eta \left(\epsilon \right)-\epsilon $ curves of FCC-TO lattices with various relative densities.

**Figure 8.**The relationship between the mechanical properties/energy absorption and relative density of lattice materials: (

**a**) normalised collapse strength; (

**b**) normalised elastic modulus; (

**c**) toughness; (

**d**) energy absorption per unit volume.

**Table 1.**Material properties of the base material, data from [41].

Density, ρ_{0}/(kg·m^{−3)} | Young’s Modulus, E_{0}/GPa | Poisson’s Ratio, μ | Tangent Modulus, E_{tan}/MPa | Yield Strength, σ_{y}/MPa | Ultimate Strength σ_{u}/MPa |
---|---|---|---|---|---|

7850 | 206 | 0.26 | 517 | 382 | 482 |

**Table 2.**Lattice materials with various relative densities generated from topology optimisation and crystal inspiration.

Lattice Material | $\overline{\mathit{\rho}}=0.10$ | $\overline{\mathit{\rho}}=0.15$ | $\overline{\mathit{\rho}}=0.20$ | $\overline{\mathit{\rho}}=0.25$ | $\overline{\mathit{\rho}}=0.30$ |
---|---|---|---|---|---|

FCC-TO | |||||

ECC-TO | |||||

VC-TO | |||||

FCC-CI | |||||

ECC-CI | |||||

VC-CI |

**Table 3.**Deformation features of lattice materials with $\overline{\rho}=0.10$ at different compressive strains.

Lattice Materials | $\mathit{\epsilon}=0.015$ | $\mathit{\epsilon}=0.15$ | $\mathit{\epsilon}=0.35$ | $\mathit{\epsilon}=0.50$ |
---|---|---|---|---|

FCC-TO | ||||

FCC-CI | ||||

ECC-TO | ||||

ECC-CI | ||||

VC-TO | ||||

VC-CI |

**Table 4.**Deformation features of lattice materials with $\overline{\rho}$ = 0.20 at different compressive strains.

Lattice Materials | $\mathit{\epsilon}=0.015$ | $\mathit{\epsilon}=0.15$ | $\mathit{\epsilon}=0.35$ | $\mathit{\epsilon}=0.50$ |
---|---|---|---|---|

FCC-TO | ||||

FCC-CI | ||||

ECC-TO | ||||

ECC-CI | ||||

VC-TO | ||||

VC-CI |

**Table 5.**Plateau stress, σ

_{pl}/[MPa], of the as-designed lattice materials with various relative densities.

$\mathbf{Relative}\text{}\mathbf{Density},\text{}\overline{\mathit{\rho}}$ [-] | FCC-TO | FCC-CI | Difference | ECC-TO | ECC-CI | Difference | VC-TO | VC-CI | Difference |
---|---|---|---|---|---|---|---|---|---|

0.10 | 20.63 | 15.59 | 32.33% | 11.43 | 10.70 | 6.78% | 32.19 | 28.36 | 13.49% |

0.15 | 33.69 | 28.08 | 19.98% | 18.45 | 21.24 | −13.13% | 52.96 | 60.08 | −11.84% |

0.20 | 65.43 | 45.05 | 45.24% | 36.76 | 34.16 | 7.63% | 90.26 | 93.16 | −3.12% |

0.25 | 110.53 | 69.13 | 59.89% | 61.00 | 52.64 | 15.89% | 127.66 | 124.92 | 2.19% |

0.30 | 156.75 | 100.09 | 56.60% | 86.66 | 80.37 | 7.82% | 190.38 | 158.42 | 20.18% |

**Table 6.**Values of the parameters of the power laws used in fitting mechanical properties and energy absorption.

Lattice Material | Collapse Strength, σ_{b} [MPa] | Elastic Modulus, E_{L} [GPa] | Toughness, U_{T} [MJ/m^{3}] | Strain Energy, W_{V} [MJ/m^{3}] | ||||
---|---|---|---|---|---|---|---|---|

C_{1} | n_{1} | C_{2} | n_{2} | C_{3} | n_{3} | C_{4} | n_{4} | |

FCC-TO | 1.048 | 1.548 | 2.369 | 1.707 | 205.554 | 1.773 | 566.801 | 1.782 |

ECC-TO | 1.165 | 1.482 | 2.807 | 1.611 | 154.456 | 1.849 | 354.951 | 1.849 |

VC-TO | 1.242 | 1.453 | 3.512 | 1.691 | 147.402 | 1.438 | 884.025 | 1.719 |

FCC-CI | 0.830 | 1.418 | 2.106 | 1.705 | 107.533 | 1.502 | 183.828 | 1.241 |

ECC-CI | 0.935 | 1.397 | 2.613 | 1.603 | 103.767 | 1.625 | 126.143 | 1.230 |

VC-CI | 1.250 | 1.353 | 3.614 | 1.608 | 105.342 | 1.292 | 499.610 | 1.449 |

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**MDPI and ACS Style**

Yang, C.; Xu, K.; Xie, S. Comparative Study on the Uniaxial Behaviour of Topology-Optimised and Crystal-Inspired Lattice Materials. *Metals* **2020**, *10*, 491.
https://doi.org/10.3390/met10040491

**AMA Style**

Yang C, Xu K, Xie S. Comparative Study on the Uniaxial Behaviour of Topology-Optimised and Crystal-Inspired Lattice Materials. *Metals*. 2020; 10(4):491.
https://doi.org/10.3390/met10040491

**Chicago/Turabian Style**

Yang, Chengxing, Kai Xu, and Suchao Xie. 2020. "Comparative Study on the Uniaxial Behaviour of Topology-Optimised and Crystal-Inspired Lattice Materials" *Metals* 10, no. 4: 491.
https://doi.org/10.3390/met10040491