# Mechanical Properties of Lattice Structures with a Central Cube: Experiments and Simulations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experiments

#### 2.1. Design of Lattice Structures

#### 2.2. Additive Manufacturing of Lattice Structures

^{TM}PLA with a diameter of 2.85 mm. The UltimakerS5 3D printer has dual printheads: the AA printhead and the BB printhead. In order to print lattice structures with better precision, the BB nozzle is employed to fill the structure with 2.85 mm diameter PolyMax polyvinyl (Polymaker, Changshu, China) alcohol (PVA) water-soluble material. The PVA is used as the substrate in the suspended part of the structure. During the printing of lattice specimens, the temperature of the glass plate is set to 60 °C, the printing layer height is set to 0.1 mm, and the printing temperature is 220 °C. Considering the fabrication quality of the specimens as well as the fabrication time, the nozzle moving speed is set to 55 mm/s, and the pumping distance is 10 mm. The external size of all 4 × 4 × 4 specimens is 60 mm and the diameter of struts is 2 mm.

#### 2.3. Mechanical Tests

^{−1}.

#### 2.4. Preliminary Analysis

#### 2.4.1. Energy Absorption and Specific Strength

_{A}is the apparent volume of the lattice structure, which is the solid volume plus the pore volume. It is obvious that the structures with higher specific strength exhibit better load-bearing capacity.

#### 2.4.2. Bending Strength of Struts

#### 2.4.3. Deflection of Vertical Central Struts

_{2}represents the length of the cubic truss at the center of the TLC structure, EI represents the bending rigidity, and I represents the second moment of area. Since the diameter of struts and the elastic modulus of the material are the same, the factor affecting strut deflection is only related to the length of the central cubic truss, L

_{2}. Hence, the selection of the length L

_{2}, the size of the central cubic truss, is crucial to its mechanical properties.

## 3. Finite Element Modeling

#### 3.1. Material Properties

^{TM}PLA materials [54]. In Figure 4a, the dimensions of the ASTM D638 Type I dog-bone specimens are given. Figure 4b shows the 3D image of the dog-bone specimen for printing. The tensile properties of three dog-bone specimens are tested at 3.41 mm/min using the universal testing machine (ZQ-990LA). The true stress–strain curve is shown in Figure 5. The elastic and plastic material parameters are given in Table 2 and Table 3, respectively.

#### 3.2. Finite Element Model

^{−1}. In the simulation, the large deformation of lattice structure induces complex self-contact between structural struts. The contact between the loading plate and the lattice structure adopts “general contact”. The friction formula for the tangential behavior is “penalty”, and the friction coefficient is set to be 0.2 [59]. For FE simulations of complex structures, especially for nonlinear compression with large deformations, it is usually calculated using ABAQUS/Explicit [60,61]. Compared to ABAQUS/Standard, explicit solvers allow for reductions in computational resources and time and also avoid convergence problems that may be encountered with implicit solvers. To further reduce the computational time of the simulation, the mass scaling function of ABAQUS is used. Keeping the ratio of kinetic energy to internal energy below 5% and ensuring that the artificially introduced inertial effects are minimized, the simulation can be considered quasi-static compression [60,61]. Based on many attempts, the time period is set to 0.01 on the premise of ensuring the accuracy of the simulation results.

## 4. Results and Discussion

#### 4.1. The Optimal Size of the Central Cubic Truss in the TLC Structure

_{2}of the veritical strut, the more easily deflection will occur, which leads to the destruction of the lattice structure at lower strains.

_{2}represents the length of the cubic truss and L

_{1}the size of the single cell. L

_{1}has a fixed value of 15 mm for all structures. The optimal length of L

_{2}is investigated in this study. L

_{2}takes values of 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, and 7.5 mm, respectively, as shown in Table 4. The TLC structures with six values of central cubic trusses are analyzed by experimental tests and numerical simulations in the following.

_{2}) is, the earlier deflection happens. Furthermore, this deflection results in earlier destruction of the single cell. Therefore, TLC-e and TLC-f are not the best choices for the central truss. In contrast, L

_{2}of TLC-a is the shortest and the deflection of the vertical struts is more difficult. However, the increased length of the inclined struts will cause decreased yield strength. As shown in Figure 11, the stress–strain curve of TLC-a almost coincides with that of the BCC structure, and its mechanical properties are not obviously improved compared to the BCC structure. Similarly, the stress level of the TLC-b structure is not significantly elevated compared to the BCC structure. Therefore, both TLC-a and TLC-b should not be regarded as the optimal sizes for the TLC structure.

_{2}is determined to be 5 mm. Therefore, the size of the central truss is fixed at 5 mm in the following study.

#### 4.2. Experimental Results for 4 × 4 × 4 Structures

#### 4.3. Finite Element Modeling for Lattice Structures with Defects

#### 4.3.1. Establishment of the Finite Element Model for Lattice Structures with Microcracks

#### 4.3.2. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Ashby, M.; Evans, A.; Fleck, N.; Gibson, L.; Hutchinson, J.; Wadley, H.; Delale, F. Reviewer Metal Foams: A Design Guide. Appl. Mech. Rev.
**2001**, 54, B105–B106. [Google Scholar] [CrossRef] - Wadley, H.N. Multifunctional Periodic Cellular Metals. Philos. Transact. A Math. Phys. Eng. Sci.
**2006**, 364, 31–68. [Google Scholar] [CrossRef] [PubMed] - Gibson, L.J.; Ashby, M.F.; Harley, B.A. Cellular Materials in Nature and Medicine; Cambridge University Press: Cambridge, UK, 2010; ISBN 978-0-521-19544-7. [Google Scholar]
- Craster, R.; Guenneau, S. Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013. [Google Scholar] [CrossRef]
- Khelif, A.; Adibi, A. (Eds.) Phononic Crystals: Fundamentals and Applications; Springer: New York, NY, USA, 2016; ISBN 978-1-4614-9392-1. [Google Scholar]
- Babaee, S.; Viard, N.; Wang, P.; Fang, N.X.; Bertoldi, K. Harnessing Deformation to Switch on and off the Propagation of Sound. Adv. Mater. Deerfield Beach Fla
**2016**, 28, 1631–1635. [Google Scholar] [CrossRef] [PubMed] - Reyes, R.L.; Ghim, M.-S.; Kang, N.-U.; Park, J.-W.; Gwak, S.-J.; Cho, Y.-S. Development and Assessment of Modified-Honeycomb-Structure Scaffold for Bone Tissue Engineering. Addit. Manuf.
**2022**, 54, 102740. [Google Scholar] [CrossRef] - Hajare, D.M.; Gajbhiye, T.S. Additive Manufacturing (3D Printing): Recent Progress on Advancement of Materials and Challenges. Mater. Today Proc.
**2022**, 58, 736–743. [Google Scholar] [CrossRef] - Song, S.; Xiong, C.; Zheng, J.; Yin, J.; Zou, Y.; Zhu, X. Compression, Bending, Energy Absorption Properties, and Failure Modes of Composite Kagome Honeycomb Sandwich Structure Reinforced by PMI Foams. Compos. Struct.
**2021**, 277, 114611. [Google Scholar] [CrossRef] - Landi, D.; Zefinetti, F.C.; Spreafico, C.; Regazzoni, D. Comparative Life Cycle Assessment of Two Different Manufacturing Technologies: Laser Additive Manufacturing and Traditional Technique. Procedia CIRP
**2022**, 105, 700–705. [Google Scholar] [CrossRef] - Zhong, R.; Ren, X.; Yu Zhang, X.; Luo, C.; Zhang, Y.; Min Xie, Y. Mechanical Properties of Concrete Composites with Auxetic Single and Layered Honeycomb Structures. Constr. Build. Mater.
**2022**, 322, 126453. [Google Scholar] [CrossRef] - Wei, Y.-L.; Yang, Q.-S.; Liu, X.; Tao, R. Multi-Bionic Mechanical Metamaterials: A Composite of FCC Lattice and Bone Structures. Int. J. Mech. Sci.
**2022**, 213, 106857. [Google Scholar] [CrossRef] - Yang, J.; Chen, X.; Sun, Y.; Zhang, J.; Feng, C.; Wang, Y.; Wang, K.; Bai, L. Compressive Properties of Bidirectionally Graded Lattice Structures. Mater. Des.
**2022**, 218, 110683. [Google Scholar] [CrossRef] - Feng, G.; Li, S.; Xiao, L.; Song, W. Energy Absorption Performance of Honeycombs with Curved Cell Walls under Quasi-Static Compression. Int. J. Mech. Sci.
**2021**, 210, 106746. [Google Scholar] [CrossRef] - Zhao, S.; Zhang, Y.; Fan, S.; Yang, N.; Wu, N. Design and Optimization of Graded Lattice Structures with Load Path-Oriented Reinforcement. Mater. Des.
**2023**, 227, 111776. [Google Scholar] [CrossRef] - Wang, S.; Wang, J.; Xu, Y.; Zhang, W.; Zhu, J. Compressive Behavior and Energy Absorption of Polymeric Lattice Structures Made by Additive Manufacturing. Front. Mech. Eng.
**2020**, 15, 319–327. [Google Scholar] [CrossRef] - Zhang, J.; Hong, R.; Wang, H. 3D-Printed Functionally-Graded Lattice Structure with Tunable Removal Characteristics for Precision Polishing. Addit. Manuf.
**2022**, 59, 103152. [Google Scholar] [CrossRef] - Caiazzo, F.; Alfieri, V.; Campanelli, S.L.; Errico, V. Additive Manufacturing and Mechanical Testing of Functionally-Graded Steel Strut-Based Lattice Structures. J. Manuf. Process.
**2022**, 83, 717–728. [Google Scholar] [CrossRef] - Plocher, J.; Panesar, A. Effect of Density and Unit Cell Size Grading on the Stiffness and Energy Absorption of Short Fibre-Reinforced Functionally Graded Lattice Structures. Addit. Manuf.
**2020**, 33, 101171. [Google Scholar] [CrossRef] - Zhao, M.; Li, X.; Zhang, D.Z.; Zhai, W. TPMS-Based Interpenetrating Lattice Structures: Design, Mechanical Properties and Multiscale Optimization. Int. J. Mech. Sci.
**2023**, 244, 108092. [Google Scholar] [CrossRef] - Dong, L. Mechanical Responses of Ti-6Al-4V Truss Lattices Having a Combined Simple-Cubic and Body-Centered-Cubic (SC-BCC) Topology. Aerosp. Sci. Technol.
**2021**, 116, 106852. [Google Scholar] [CrossRef] - Yang, X.; Gong, Y.; Zhao, L.; Zhang, J.; Hu, N. Compressive Mechanical Properties of Layer Hybrid Lattice Structures Fabricated by Laser Powder Bed Fusion Technique. J. Mater. Res. Technol.
**2023**, 22, 1800–1811. [Google Scholar] [CrossRef] - Chen, X.; Ji, Q.; Wei, J.; Tan, H.; Yu, J.; Zhang, P.; Laude, V.; Kadic, M. Light-Weight Shell-Lattice Metamaterials for Mechanical Shock Absorption. Int. J. Mech. Sci.
**2020**, 169, 105288. [Google Scholar] [CrossRef] - Zhao, M.; Li, X.; Zhang, D.Z.; Zhai, W. Design, Mechanical Properties and Optimization of Lattice Structures with Hollow Prismatic Struts. Int. J. Mech. Sci.
**2023**, 238, 107842. [Google Scholar] [CrossRef] - Zhong, H.Z.; Song, T.; Li, C.W.; Das, R.; Gu, J.F.; Qian, M. Understanding the Superior Mechanical Properties of Hollow-Strut Metal Lattice Materials. Scr. Mater.
**2023**, 228, 115341. [Google Scholar] [CrossRef] - Xiao, L.; Feng, G.; Li, S.; Mu, K.; Qin, Q.; Song, W. Mechanical Characterization of Additively-Manufactured Metallic Lattice Structures with Hollow Struts under Static and Dynamic Loadings. Int. J. Impact Eng.
**2022**, 169, 104333. [Google Scholar] [CrossRef] - Červinek, O.; Pettermann, H.; Todt, M.; Koutný, D.; Vaverka, O. Non-Linear Dynamic Finite Element Analysis of Micro-Strut Lattice Structures Made by Laser Powder Bed Fusion. J. Mater. Res. Technol.
**2022**, 18, 3684–3699. [Google Scholar] [CrossRef] - Liu, X.; Wada, T.; Suzuki, A.; Takata, N.; Kobashi, M.; Kato, M. Understanding and Suppressing Shear Band Formation in Strut-Based Lattice Structures Manufactured by Laser Powder Bed Fusion. Mater. Des.
**2021**, 199, 109416. [Google Scholar] [CrossRef] - Bai, L.; Xu, Y.; Chen, X.; Xin, L.; Zhang, J.; Li, K.; Sun, Y. Improved Mechanical Properties and Energy Absorption of Ti6Al4V Laser Powder Bed Fusion Lattice Structures Using Curving Lattice Struts. Mater. Des.
**2021**, 211, 110140. [Google Scholar] [CrossRef] - Zhu, S.; Ma, L.; Wang, B.; Hu, J.; Zhou, Z. Lattice Materials Composed by Curved Struts Exhibit Adjustable Macroscopic Stress-Strain Curves. Mater. Today Commun.
**2018**, 14, 273–281. [Google Scholar] [CrossRef] - Tancogne-Dejean, T.; Mohr, D. Stiffness and Specific Energy Absorption of Additively-Manufactured Metallic BCC Metamaterials Composed of Tapered Beams. Int. J. Mech. Sci.
**2018**, 141, 101–116. [Google Scholar] [CrossRef] - Liu, Y.; Zhang, J.; Gu, X.; Zhou, Y.; Yin, Y.; Tan, Q.; Li, M.; Zhang, M.-X. Mechanical Performance of a Node Reinforced Body-Centred Cubic Lattice Structure Manufactured via Selective Laser Melting. Scr. Mater.
**2020**, 189, 95–100. [Google Scholar] [CrossRef] - Topologically Optimized Lattice Structures with Superior Fatigue Performance. Int. J. Fatigue
**2022**, 165, 107188. [CrossRef] - Ma, X.; Zhang, D.Z.; Luo, T.; Zheng, X.; Zhou, H. Novel Optimal Design-Making Model of Additive Manufactured Lattice Structures Based on Mechanical Physical Model. Mater. Today Commun.
**2024**, 38, 107823. [Google Scholar] [CrossRef] - Sun, Z.; Gong, Y.; Bian, Z.; Zhang, J.; Zhao, L.; Hu, N. Mechanical Properties of Bionic Lattice and Its Hybrid Structures Based on the Microstructural Design of Pomelo Peel. Thin-Walled Struct.
**2024**, 198, 111715. [Google Scholar] [CrossRef] - Deshpande, V.S.; Ashby, M.F.; Fleck, N.A. Foam Topology: Bending versus Stretching Dominated Architectures. Acta Mater.
**2001**, 49, 1035–1040. [Google Scholar] [CrossRef] - Karamooz Ravari, M.R.; Kadkhodaei, M.; Badrossamay, M.; Rezaei, R. Numerical Investigation on Mechanical Properties of Cellular Lattice Structures Fabricated by Fused Deposition Modeling. Int. J. Mech. Sci.
**2014**, 88, 154–161. [Google Scholar] [CrossRef] - Kaur, M.; Yun, T.G.; Han, S.M.; Thomas, E.L.; Kim, W.S. 3D Printed Stretching-Dominated Micro-Trusses. Mater. Des.
**2017**, 134, 272–280. [Google Scholar] [CrossRef] - Carlton, H.D.; Volkoff-Shoemaker, N.A.; Messner, M.C.; Barton, N.R.; Kumar, M. Incorporating Defects into Model Predictions of Metal Lattice-Structured Materials. Mater. Sci. Eng. A
**2022**, 832, 142427. [Google Scholar] [CrossRef] - Gautam, R.; Idapalapati, S.; Feih, S. Printing and Characterisation of Kagome Lattice Structures by Fused Deposition Modelling. Mater. Des.
**2018**, 137, 266–275. [Google Scholar] [CrossRef] - Li, D.; Qin, R.; Chen, B.; Zhou, J. Analysis of Mechanical Properties of Lattice Structures with Stochastic Geometric Defects in Additive Manufacturing. Mater. Sci. Eng. A
**2021**, 822, 141666. [Google Scholar] [CrossRef] - Computational Modelling of Strut Defects in SLM Manufactured Lattice Structures. Mater. Des.
**2019**, 171, 107671. [CrossRef] - Sun, Z.P.; Guo, Y.B.; Shim, V.P.W. Influence of Printing Direction on the Dynamic Response of Additively-Manufactured Polymeric Materials and Lattices. Int. J. Impact Eng.
**2022**, 167, 104263. [Google Scholar] [CrossRef] - Alghamdi, A.; Maconachie, T.; Downing, D.; Brandt, M.; Qian, M.; Leary, M. Effect of Additive Manufactured Lattice Defects on Mechanical Properties: An Automated Method for the Enhancement of Lattice Geometry. Int. J. Adv. Manuf. Technol.
**2020**, 108, 957–971. [Google Scholar] [CrossRef] - Bacciaglia, A.; Ceruti, A.; Liverani, A. Structural Analysis of Voxel-Based Lattices Using 1D Approach. 3D Print. Addit. Manuf.
**2022**, 9, 365–379. [Google Scholar] [CrossRef] - Amirpour, M.; Battley, M. Study of Manufacturing Defects on Compressive Deformation of 3D-Printed Polymeric Lattices. Int. J. Adv. Manuf. Technol.
**2022**, 122, 2561–2576. [Google Scholar] [CrossRef] - Cao, X.; Jiang, Y.; Zhao, T.; Wang, P.; Wang, Y.; Chen, Z.; Li, Y.; Xiao, D.; Fang, D. Compression Experiment and Numerical Evaluation on Mechanical Responses of the Lattice Structures with Stochastic Geometric Defects Originated from Additive-Manufacturing. Compos. Part B Eng.
**2020**, 194, 108030. [Google Scholar] [CrossRef] - Yang, H.; Wang, W.; Li, C.; Qi, J.; Wang, P.; Lei, H.; Fang, D. Deep Learning-Based X-Ray Computed Tomography Image Reconstruction and Prediction of Compression Behavior of 3D Printed Lattice Structures. Addit. Manuf.
**2022**, 54, 102774. [Google Scholar] [CrossRef] - Lei, H.; Li, C.; Meng, J.; Zhou, H.; Liu, Y.; Zhang, X.; Wang, P.; Fang, D. Evaluation of Compressive Properties of SLM-Fabricated Multi-Layer Lattice Structures by Experimental Test and μ-CT-Based Finite Element Analysis. Mater. Des.
**2019**, 169, 107685. [Google Scholar] [CrossRef] - Zanini, F.; Sorgato, M.; Savio, E.; Carmignato, S. Dimensional Verification of Metal Additively Manufactured Lattice Structures by X-Ray Computed Tomography: Use of a Newly Developed Calibrated Artefact to Achieve Metrological Traceability. Addit. Manuf.
**2021**, 47, 102229. [Google Scholar] [CrossRef] - Geng, L.; Wu, W.; Sun, L.; Fang, D. Damage Characterizations and Simulation of Selective Laser Melting Fabricated 3D Re-Entrant Lattices Based on in-Situ CT Testing and Geometric Reconstruction. Int. J. Mech. Sci.
**2019**, 157–158, 231–242. [Google Scholar] [CrossRef] - Tkac, J.; Toth, T.; Molnar, V.; Dovica, M.; Fedorko, G. Observation of Porous Structure’s Deformation Wear after Axial Loading with the Use of Industrial Computed Tomography (CT). Measurement
**2022**, 200, 111631. [Google Scholar] [CrossRef] - Dallago, M.; Winiarski, B.; Zanini, F.; Carmignato, S.; Benedetti, M. On the Effect of Geometrical Imperfections and Defects on the Fatigue Strength of Cellular Lattice Structures Additively Manufactured via Selective Laser Melting. Int. J. Fatigue
**2019**, 124, 348–360. [Google Scholar] [CrossRef] - Sun, Z.P.; Guo, Y.B.; Shim, V.P.W. Characterisation and Modeling of Additively-Manufactured Polymeric Hybrid Lattice Structures for Energy Absorption. Int. J. Mech. Sci.
**2021**, 191, 106101. [Google Scholar] [CrossRef] - Anisotropic Compression Behaviors of Bio-Inspired Modified Body-Centered Cubic Lattices Validated by Additive Manufacturing—ScienceDirect. Available online: https://www.sciencedirect.com/science/article/pii/S1359836822001093 (accessed on 8 October 2022).
- Wang, P.; Yang, F.; Li, P.; Zheng, B.; Fan, H. Design and Additive Manufacturing of a Modified Face-Centered Cubic Lattice with Enhanced Energy Absorption Capability. Extreme Mech. Lett.
**2021**, 47, 101358. [Google Scholar] [CrossRef] - Yang, X.; Ma, W.; Zhang, Z.; Liu, S.; Tang, H. Ultra-High Specific Strength Ti6Al4V Alloy Lattice Material Manufactured via Selective Laser Melting. Mater. Sci. Eng. A
**2022**, 840, 142956. [Google Scholar] [CrossRef] - ASTM D638; Standard Test Method for Tensile Properties of Plastics. ASTM International: West Conshohocken, PA, USA, 2022.
- Wang, Z. Recent Advances in Novel Metallic Honeycomb Structure. Compos. Part B
**2019**, 166, 731–741. [Google Scholar] [CrossRef] - Gümrük, R.; Mines, R.A.W. Compressive Behaviour of Stainless Steel Micro-Lattice Structures. Int. J. Mech. Sci.
**2013**, 68, 125–139. [Google Scholar] [CrossRef] - Chen, Y.; Ye, L.; Zhang, Y.X.; Fu, K. Compression Behaviours of 3D-Printed CF/PA Metamaterials: Experiment and Modelling. Int. J. Mech. Sci.
**2021**, 206, 106634. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagrams of (

**a**) BCC cell topology, (

**b**) a cubic truss, and (

**c**) TLC cell topology (the length of a single cell is 15 mm and the diameter for all struts is 2 mm).

**Figure 3.**Experimental setup for quasi-static compression tests of lattice structures (the downward blue arrow indicates the direction of compression).

**Figure 4.**(

**a**) Dimensions of ASTM D638 Type I dog-bone specimens; (

**b**) 3D image of ASTM D638 Type I dog-bone specimen for printing (the blue arrow represents the direction of FDM-based 3D printing).

**Figure 6.**Finite element model with boundary conditions and the red square defining a single cell of TLC structure meshed with C3D4 tetrahedral elements.

**Figure 7.**Deformation characteristics of BCC and TLC single-cell lattice structures with different strains.

**Figure 8.**Schematic diagram of force analysis of inclined struts in (

**a**) the BCC lattice structure and (

**b**) the TLC lattice structure.

**Figure 9.**Deflection of the vertical strut in the center of the TLC lattice structure due to lateral force (assuming that the lower end of the vertical strut is fixed).

**Figure 11.**Compressive stress–strain curves of single cells obtained from experiments (a–f denote different lengths of the central cubic truss, as given in Table 4).

**Figure 12.**Compressive stress–strain curves for TLC-e and TLC-f; the insets show the deformed specimens at the highest stress points.

**Figure 13.**Specific energy absorption corresponding to various strains for TLC and BCC single cells.

**Figure 15.**(

**a**) Engineering stress–strain curves of the 2 × 2 × 2 TLC-c and TLC-d obtained from experimental tests; (

**b**) energy absorption efficiency curves (the green dashed lines represent the highest energy absorption); (

**c**,

**d**) deformation characteristics of TLC-c and TLC-d at a strain of 0.53, respectively.

**Figure 16.**(

**a**) Engineering stress–strain curves of 4 × 4 × 4 BCC and TLC; (

**b**) deformation characteristics of TLC specimens with different strains.

**Figure 18.**(

**a**) Energy absorption and (

**b**) specific energy absorption of BCC and TLC lattice structures with strain.

**Figure 19.**(

**a**) The stress–strain curve of the 4 × 4 × 4 TLC structure; (

**b**) images of TLC specimens at points I and II (the dashed red lines in the images indicate the status of the central vertical struts).

**Figure 22.**Engineering stress–strain curves of experimental and simulation results for BCC and TLC 4 × 4 × 4 lattice structures.

**Figure 23.**TLC lattice structure model with microcracks located at (

**a**) Type I nodes; (

**b**) Type II nodes; and (

**c**) both Type I and II nodes.

**Figure 24.**(

**a**) Comparison of stress–strain curves between simulations of ideal models, Model A, and experiments; (

**b**) comparison of stress–strain curves between simulations of ideal models, Model B, and experiments; (

**c**) comparison of stress–strain curves between simulations of ideal models, Model C, and experiments (the dashed lines for Model A and B are added in (

**c**) for reference).

**Figure 25.**Comparison of deformation characteristics between experiments and simulations of Model C at various strains (The red boxes and lines highlight the locations with similar characteristics of damage/failure between experiments and simulations).

**Table 1.**Geometrical parameters of the PolyMax

^{TM}(Polymaker, Changshu, China) PLA lattice structures.

Structure | Volume | Theoretical Mass | Actual Mass | Relative Density | Apparent Density |
---|---|---|---|---|---|

BCC | 18.37 cm^{3} | 21.70 g | 21.55 g | 0.0850 | 0.0998 |

TLC | 22.11 cm^{3} | 26.09 g | 26.11 g | 0.1024 | 0.1209 |

Elastic Modulus (GPa) | Poisson’s Ratio | Density (kg/m^{3}) | Yield Strength (MPa) |
---|---|---|---|

1.85 | 0.35 | 1200 | 42 |

Plastic strain | 0 | 0.0045 | 0.012 | 0.021 | 0.034 | 0.132 | 0.2 | 0.3 |

Plastic stress (MPa) | 40 | 36 | 32 | 29 | 26 | 28 | 30 | 34 |

${\mathit{L}}_{2}/{\mathit{L}}_{1}$ | 3/15 a | 4/15 b | 5/15 c | 6/15 d | 7/15 e | 7.5/15 f |
---|---|---|---|---|---|---|

${L}_{1}$ | 15 mm | 15 mm | 15 mm | 15 mm | 15 mm | 15 mm |

${L}_{2}$ | 3 mm | 4 mm | 5 mm | 6 mm | 7 mm | 7.5 mm |

Initial Flow Stress at a Strain of 0.1 | Maximum Stress | |
---|---|---|

Experiment | 0.14 | 1.64 |

Ideal structure | 0.31(121% ↑) | 3.00 (83% ↑) |

Model A | 0.20 (43% ↑) | 2.15 (31% ↑) |

Model B | 0.26 (86% ↑) | 1.87 (14% ↑) |

Model C | 0.19 (36% ↑) | 1.55 (6% ↓) |

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## Share and Cite

**MDPI and ACS Style**

Guo, S.; Ma, Y.; Liu, P.; Chen, Y.
Mechanical Properties of Lattice Structures with a Central Cube: Experiments and Simulations. *Materials* **2024**, *17*, 1329.
https://doi.org/10.3390/ma17061329

**AMA Style**

Guo S, Ma Y, Liu P, Chen Y.
Mechanical Properties of Lattice Structures with a Central Cube: Experiments and Simulations. *Materials*. 2024; 17(6):1329.
https://doi.org/10.3390/ma17061329

**Chicago/Turabian Style**

Guo, Shuai, Yuwei Ma, Peng Liu, and Yang Chen.
2024. "Mechanical Properties of Lattice Structures with a Central Cube: Experiments and Simulations" *Materials* 17, no. 6: 1329.
https://doi.org/10.3390/ma17061329