# 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

## References

- Xiao, Z.; Yang, Y.; Xiao, R.; Bai, Y.; Song, C.; Wang, D. Evaluation of topology-optimized lattice structures manufactured via selective laser melting. Mater. Des.
**2018**, 143, 27–37. [Google Scholar] [CrossRef] - Mei, H.; Zhao, R.; Xia, Y.; Du, J.; Wang, X.; Cheng, L. Ultrahigh strength printed ceramic lattices. J. Alloys Compd.
**2019**, 797, 786–796. [Google Scholar] [CrossRef] - Kadic, M.; Milton, G.; van Hecke, M.; Wegener, M. 3D metamaterials. Nat. Rev. Phys.
**2019**, 1, 198–210. [Google Scholar] [CrossRef] - Peng, C.; Tran, P.; Nguyen-Xuan, H.; Ferreira, A.J.M. Mechanical performance and fatigue life prediction of lattice structures: Parametric computational approach. Compos. Struct.
**2020**, 235, 111821. [Google Scholar] [CrossRef] - Helou, M.; Kara, S. Design, analysis and manufacturing of lattice structures: An overview. Int. J. Comput. Integr. Manuf.
**2017**, 31, 243–261. [Google Scholar] [CrossRef] - Daynes, S.; Feih, S.; Lu, W.; Wei, J. Design concepts for generating optimised lattice structures aligned with strain trajectories. Comput. Methods Appl. Mech. Eng.
**2019**, 354, 689–705. [Google Scholar] [CrossRef] - Hazeli, K.; Babamiri, B.; Indeck, J.; Minor, A.; Askari, H. Microstructure-topology relationship effects on the quasi-static and dynamic behavior of additively manufactured lattice structures. Mater. Des.
**2019**, 176, 107826. [Google Scholar] [CrossRef] - Alaña, M.; Lopez-Arancibia, A.; Pradera-Mallabiabarrena, A.; Ruiz de Galarreta, S. Analytical model of the elastic behavior of a modified face-centered cubic lattice structure. J. Mech. Behav. Biomed. Mater.
**2019**, 98, 357–368. [Google Scholar] [CrossRef] - Leary, M.; Mazur, M.; Elambasseril, J.; McMillan, M.; Chirent, T.; Sun, Y.; Qian, M.; Easton, M.; Brandt, M. Selective laser melting (SLM) of AlSi12Mg lattice structures. Mater. Des.
**2016**, 98, 344–357. [Google Scholar] [CrossRef] - Sienkiewicz, J.; Płatek, P.; Jiang, F.; Sun, X.; Rusinek, A. Investigations on the Mechanical Response of Gradient Lattice Structures Manufactured via SLM. Metals
**2020**, 10, 213. [Google Scholar] [CrossRef] [Green Version] - Gautam, R.; Idapalapati, S. Compressive Properties of Additively Manufactured Functionally Graded Kagome Lattice Structure. Metals
**2019**, 9, 517. [Google Scholar] [CrossRef] [Green Version] - Al-Saedi, D.; Masood, S.; Faizan-Ur-Rab, M.; Alomarah, A.; Ponnusamy, P. Mechanical properties and energy absorption capability of functionally graded F2BCC lattice fabricated by SLM. Mater. Des.
**2018**, 144, 32–44. [Google Scholar] [CrossRef] - Gross, A.; Pantidis, P.; Bertoldi, K.; Gerasimidis, S. Correlation between topology and elastic properties of imperfect truss-lattice materials. J. Mech. Phys. Solids
**2019**, 124, 577–598. [Google Scholar] [CrossRef] - Deshpande, V.; Fleck, N.; Ashby, M. Effective properties of the octet-truss lattice material. J. Mech. Phys. Solids
**2001**, 49, 1747–1769. [Google Scholar] [CrossRef] [Green Version] - Nguyen, D.S.; Tran, T.T.; Le, D.K.; Le, V.T. Creation of Lattice Structures for Additive Manufacturing in CAD Environment. In Proceedings of the IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), Bangkok, Thailand, 16–19 December 2018; pp. 396–400. [Google Scholar]
- Hedayati, R.; Sadighi, M.; Mohammadi-Aghdam, M.; Zadpoor, A. Analytical relationships for the mechanical properties of additively manufactured porous biomaterials based on octahedral unit cells. Appl. Math. Model.
**2017**, 46, 408–422. [Google Scholar] [CrossRef] [Green Version] - Bonatti, C.; Mohr, D. Mechanical performance of additively-manufactured anisotropic and isotropic smooth shell-lattice materials: Simulations & experiments. J. Mech. Phys. Solids
**2019**, 122, 1–26. [Google Scholar] - Ling, C.; Cernicchi, A.; Gilchrist, M.; Cardiff, P. Mechanical behaviour of additively-manufactured polymeric octet-truss lattice structures under quasi-static and dynamic compressive loading. Mater. Des.
**2019**, 162, 106–118. [Google Scholar] [CrossRef] - Lozanovski, B.; Downing, D.; Tran, P.; Shidid, D.; Qian, M.; Choong, P.; Brandt, M.; Leary, M. A Monte Carlo simulation-based approach to realistic modelling of additively manufactured lattice structures. Addit. Manuf.
**2020**, 32, 101092. [Google Scholar] [CrossRef] - Lozanovski, B.; Leary, M.; Tran, P.; Shidid, D.; Qian, M.; Choong, P.; Brandt, M. Computational modelling of strut defects in SLM manufactured lattice structures. Mater. Des.
**2019**, 171, 107671. [Google Scholar] [CrossRef] - Peng, C.; Tran, P. Bioinspired functionally graded gyroid sandwich panel subjected to impulsive loadings. Compos. Part B Eng.
**2020**, 188, 107773. [Google Scholar] [CrossRef] - Tran, P.; Peng, C. Triply periodic minimal surfaces sandwich structures subjected to shock impact. J. Sandw. Struct. Mater.
**2020**, 1–30. [Google Scholar] [CrossRef] - Bai, L.; Zhang, J.; Chen, X.; Yi, C.; Chen, R.; Zhang, Z. Configuration optimization design of Ti6Al4V lattice structure formed by SLM. Materials
**2018**, 11, 1856. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hu, Z.; Gadipudi, V.; Salem, D. Topology Optimization of Lightweight Lattice Structural Composites Inspired by Cuttlefish Bone. Appl. Compos. Mater.
**2018**, 26, 15–27. [Google Scholar] [CrossRef] - Chiandussi, G. On the solution of a minimum compliance topology optimisation problem by optimality criteria without a priori volume constraint specification. Comput. Mech.
**2005**, 38, 77–99. [Google Scholar] [CrossRef] - Zhang, W.; Li, D.; Kang, P.; Guo, X.; Youn, S.-K. Explicit topology optimization using IGA-based moving morphable void (MMV) approach. Comput. Methods Appl. Mech. Eng.
**2020**, 360, 112685. [Google Scholar] [CrossRef] - Bendsoe, M.P.; Sigmund, O. Topology Optimization: Theory, Methods, and Applications, 2nd ed.; Springer Science & Business Media: Berlin, Germany, 2004. [Google Scholar]
- Sigmund, O.; Maute, K. Topology optimization approaches. Struct. Multidiscip. Optim.
**2013**, 48, 1031–1055. [Google Scholar] [CrossRef] - Kentli, A. Topology optimization applications on engineering structures. In Truss and Frames-Recent Advances and New Perspectives; IntechOpen: London, UK, 2019. [Google Scholar]
- Zhang, X.; Ramos, A.S.; Paulino, G.H. Material nonlinear topology optimization using the ground structure method with a discrete filtering scheme. Struct. Multidiscip. Optim.
**2017**, 55, 2045–2072. [Google Scholar] [CrossRef] - Costa, G.; Montemurro, M.; Pailhès, J. A 2D topology optimisation algorithm in NURBS framework with geometric constraints. Int. J. Mech. Mater. Des.
**2018**, 14, 669–696. [Google Scholar] [CrossRef] [Green Version] - Costa, G.; Montemurro, M.; Pailhès, J. Minimum length scale control in a NURBS-based SIMP method. Comput. Methods Appl. Mech. Eng.
**2019**, 354, 963–989. [Google Scholar] [CrossRef] - Costa, G.; Montemurro, M.; Pailhès, J. NURBS hyper-surfaces for 3D topology optimization problems. Mech. Adv. Mater. Struct.
**2019**, 1582826, 1–20. [Google Scholar] [CrossRef] - Lazarov, B.S.; Wang, F.; Sigmund, O. Length scale and manufacturability in density-based topology optimization. Arch. Appl. Mech.
**2016**, 86, 189–218. [Google Scholar] [CrossRef] [Green Version] - Geoffroy-Donders, P.; Allaire, G.; Pantz, O. 3-d topology optimization of modulated and oriented periodic microstructures by the homogenization method. J. Comput. Phys.
**2020**, 401, 108994. [Google Scholar] [CrossRef] [Green Version] - Da, D.; Xia, L.; Li, G.; Huang, X. Evolutionary topology optimization of continuum structures with smooth boundary representation. Struct. Multidiscip. Optim.
**2017**, 57, 2143–2159. [Google Scholar] [CrossRef] - Allaire, G.; Jouve, F.; Toader, A.-M. Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys.
**2004**, 194, 363–393. [Google Scholar] [CrossRef] [Green Version] - Yamada, T.; Izui, K.; Nishiwaki, S.; Takezawa, A. A topology optimization method based on the level set method incorporating a fictitious interface energy. Comput. Methods Appl. Mech. Eng.
**2010**, 199, 2876–2891. [Google Scholar] [CrossRef] [Green Version] - Afrousheh, M.; Marzbanrad, J.; Göhlich, D. Topology optimization of energy absorbers under crashworthiness using modified hybrid cellular automata (MHCA) algorithm. Struct. Multidiscip. Optim.
**2019**, 60, 1021–1034. [Google Scholar] [CrossRef] - Yang, C.; Xu, P.; Xie, S.; Yao, S. Mechanical performances of four lattice materials guided by topology optimisation. Scr. Mater.
**2020**, 178, 339–345. [Google Scholar] [CrossRef] - Yang, C.; Xu, P.; Yao, S.; Xie, S.; Li, Q.; Peng, Y. Optimization of honeycomb strength assignment for a composite energy-absorbing structure. Thin-Walled Struct.
**2018**, 127, 741–755. [Google Scholar] [CrossRef] - Tovar, A. Bone Remodelling as a Hybrid Cellular Automaton Optimization Process. Ph.D. Thesis, University of Notre Dame, Cotabato City, Philippines, 2004. [Google Scholar]
- LS-TaSC™. Theory Manual (Version 4.0) of Topology and Shape Computations; Livermore Software Technology Corporation: Livermore, CA, USA, 2018. [Google Scholar]
- Tro, N.J. Chemistry: A Molecular Approach, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2010. [Google Scholar]
- Yuan, S.; Chua, C.; Zhou, K. 3D-Printed Mechanical Metamaterials with High Energy Absorption. Adv. Mater. Technol.
**2019**, 4, 1800419. [Google Scholar] [CrossRef] - Montemayor, L.; Greer, J. Mechanical Response of Hollow Metallic Nanolattices: Combining Structural and Material Size Effects. J. Appl. Mech.
**2015**, 82, 071012. [Google Scholar] [CrossRef] [Green Version] - Maskery, I.; Aremu, A.O.; Parry, L.; Wildman, R.D.; Tuck, C.J.; Ashcroft, I.A. Effective design and simulation of surface-based lattice structures featuring volume fraction and cell type grading. Mater. Des.
**2018**, 155, 220–232. [Google Scholar] [CrossRef] - Ls-dyna. Keyword User’s Manual Volume I (Version 971 R8.0); Livermore Software Technology Corporation: Livermore, CA, USA, 2015. [Google Scholar]
- Zhang, L.; Feih, S.; Daynes, S.; Chang, S.; Wang, M.; Wei, J.; Lu, W. Energy absorption characteristics of metallic triply periodic minimal surface sheet structures under compressive loading. Addit. Manuf.
**2018**, 23, 505–515. [Google Scholar] [CrossRef] - Al-Ketan, O.; Rowshan, R.; Abu Al-Rub, R. Topology-mechanical property relationship of 3D printed strut, skeletal, and sheet based periodic metallic cellular materials. Addit. Manuf.
**2018**, 19, 167–183. [Google Scholar] [CrossRef] - Liu, Y.; Li, S.; Zhang, L.; Hao, Y.; Sercombe, T. Early plastic deformation behaviour and energy absorption in porous β-type biomedical titanium produced by selective laser melting. Scr. Mater.
**2018**, 153, 99–103. [Google Scholar] [CrossRef] - Sun, Y.; Li, Q.M. Dynamic compressive behaviour of cellular materials: A review of phenomenon, mechanism and modelling. Int. J. Impact Eng.
**2018**, 112, 74–115. [Google Scholar] [CrossRef] [Green Version] - Yang, L.; Mertens, R.; Ferrucci, M.; Yan, C.; Shi, Y.; Yang, S. Continuous graded Gyroid cellular structures fabricated by selective laser melting: Design, manufacturing and mechanical properties. Mater. Des.
**2019**, 162, 394–404. [Google Scholar] [CrossRef] - ISO 13314. Mechanical Testing of Metals-Ductility Testing-Compression Test for Porous and Cellular Metals; International Organization for Standardization: Geneva, Switzerland, 2011. [Google Scholar]
- Gibson, L.J.; Ashby, M.F. Cellular Solids: Structure and Properties, 2nd ed.; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Yu, S.; Sun, J.; Bai, J. Investigation of functionally graded TPMS structures fabricated by additive manufacturing. Mater. Des.
**2019**, 182, 108021. [Google Scholar] [CrossRef]

**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