# 3D Printed Voronoi Structures Inspired by Paracentrotus lividus Shells

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Biomimetic Design Strategy

#### 2.2. Design Parameters and 3D Printing

#### 2.3. Compression Testing Supported by FEA

^{®}Academic Research Mechanical, Release 23.1, ANSYS, Inc., Canonsburg, PA, USA) was used to study the mechanical behavior of all the Voronoi lattice structures. An explicit dynamic analysis was conducted to accurately simulate the mechanical response of the lattices which was necessary to capture their large deformations and bi-linear material behavior.

^{3}and ρ

_{s}is the density of PLA (1.24 g/cm

^{3}) [42]. Stress σ (MPa) was calculated as the ratio between force F (N) and the apparent cross-sectional area A (mm

^{2}) of the specimens [43]:

_{D}must first be determined. Densification strain is the effective strain when the cells of the Voronoi structure have entirely collapsed, and further strain would compress the bulk PLA material. The densification strain ε

_{D}of porous materials is derived based on its energy absorption efficiency [45], which is calculated with the following equation:

_{D}is the strain that corresponds to the maximum value of the η(ε) curve [32,45]. After this point, the stress increases rapidly, as the bulk material is compressed, resulting in a substantial drop in the efficiency of the structure [32]. A typical energy efficiency curve is illustrated in Figure 5 for Model 1 of the designed and fabricated specimens. The energy absorption efficiency η of the model is plotted against the strain ε. The value of strain corresponding to the maximum value of the efficiency curve is the densification strain ε

_{D}of Model 1 [45].

_{pl}is a significant parameter used to assess the compressive performances of porous materials as it describes the plateau region of the stress–strain curve of cellular solids and is calculated by the following equation [46]:

_{y}is the yield strain, which corresponds to the onset of plastic deformation.

_{v}(MJ/m

^{3}) of the foams is estimated by the energy absorbed per unit of volume up to the densification strain ε

_{D}. [45]:

_{m}(KJ/kg) as the energy absorbed per unit of mass [47]:

## 3. Results and Discussion

#### 3.1. Characterization of the Biomimetic Voronoi Structure

#### 3.2. Compression Results of the Biomimetic Voronoi Structures

#### 3.2.1. Compressive Behavior, Strength, and Modulus

_{y}, of Model 1 (baseline) is 6.26 ± 0.12 MPa, 8.80 ± 0.12 MPa for Model 2, and 10.84 ± 0.10 MPa for Model 3. The compressive modulus E of the porous structure is also improved as the rods become thicker. Model 1 has a compressive modulus of 310.29 ± 15.38 MPa, Model 2 has 443.64 ± 1.10 MPa, and Model 3 has 603.66 ± 13.93 MPa. It should be noted that the strength and modulus of the structures are significantly lower than those of PLA because of their porous geometry and anisotropic layer bonding in the rods [49].

#### 3.2.2. Densification Strain, Plateau Stress, and Energy Absorption

_{D}, of Model 1 is 49.39 ± 1.28%. Model 2 has a densification strain of 47.61 ± 1.14%, and Model 3, 47.43 ± 0.47%. The densification strain of Model 4 is 48.68 ± 1.30% and Model 5, 46.57 ± 0.42%. Models 6 and 7 have a densification strain of 48.88 ± 1.59% and 45.07 ± 0.83%, respectively. A reverse trend can be identified in these results. As the thickness of the rods, the number of nodes, and the smoothness of the structure are raised, the densification strain gradually decreases. This can be attributed to the additional PLA material that increases its overall compressible volume, allowing for smaller densification strain values and the observed decline. The plateau stress, σ

_{pl}, of Model 1 is 6.15 ± 0.09 MPa, while the plateau stress for Models 2 and 3 were calculated at 9.38 ± 0.14 MPa and 12.64 ± 0.22 MPa, indicating a rising trend as the struts become thicker. A similar trend is noticed as the node count is raised according to the values of Models’ 4 and 5 plateau stress which are 10.31 ± 0.02 MPa and 13.55 ± 0.55 MPa, respectively. The same can be said for enhancing the smoothness of the structure, since Models 6 and 7 demonstrate plateau stresses of 7.42 ± 0.08 MPa and 8.94 ± 0.41 MPa, respectively. A correlation between the strength of the structure and plateau stress can be traced. Thicker rods, more nodes, and smoother edges increase not only the compressive strength of the Voronoi structure but also the sustained stress at which the structure progressively collapses up until the densification point is reached.

_{v}of 2.94 ± 0.11 MJ/m

^{3}and specific energy capacity W

_{m}of 6.01 ± 0.30 KJ/kg. The calculated energy capacity and specific energy capacity for Model 2 are 4.32 ± 0.14 MJ/m

^{3}and 7.26 ± 0.31 KJ/kg and Model 3 are 5.78 ± 0.09 MJ/m

^{3}and 8.38 ± 0.20 KJ/kg, respectively. Similarly, to yield strength and plateau stress, the energy absorption capability of the structure increases as the radius of the rods is raised. Model 4 is characterized by an energy capacity of 4.85 ± 0.14 MJ/m

^{3}and a specific capacity of 7.98 ± 0.19 KJ/kg while Model 5 shows 6.07 ± 0.19 MJ/m

^{3}and 8.85 ± 0.27 KJ/kg, respectively. Thus, it becomes obvious that the Voronoi structure can absorb more energy when the count of its nodes is increased. Lastly, the capacities of Models 6 and 7 are 3.51 ± 0.14 MJ/m

^{3}and 3.89 ± 0.19 MJ/m

^{3}and their specific capacities are 6.75 ± 0.37 KJ/kg and 6.85 ± 0.31 KJ/kg; therefore, smoother Voronoi geometries have superior energy absorption. Overall, enhanced geometric parameters result in higher energy dissipation through rod buckling and collapse at higher constant stress rates.

#### 3.2.3. Correlation of Mechanical Properties to Relative Density

^{2}= 0.9596), as illustrated in Figure 9a. The same relation can be observed when the energy absorption capacity (R

^{2}= 0.9783) of the samples is plotted against their respective relative densities, as shown in Figure 9b. A second-degree polynomial expression (R

^{2}= 0.9804) also describes the relation between plateau stress and relative density in Figure 9c. This trend can be translated as an accelerated increase in the mechanical properties of the structure as more material is added to it and its relative density is raised. It becomes obvious that such an improvement can be achieved either by increasing the thickness of the rods, or by raising the number of the nodes, or by smoothing the geometry, depending on the design requirements or technical and fabrication constraints.

#### 3.3. FEA Validation of Experimental Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A fresh sample of the sea urchin Paracentrotus lividus; (

**b**) the porous structure of its shell.

**Figure 2.**The stages “Research and Analysis”, “Abstraction and Emulation”, and “Technical Evaluation” of the novel biomimetic strategy are interconnected via bi-directional feedback loops. The Technical Evaluation stage consists of prototyping and mechanical characterization of the biomimetic model.

**Figure 3.**The updated definition in the Grasshopper environment. A volume Boolean union is added to ensure constant rod thickness.

**Figure 4.**Designed and fabricated samples: (

**a**) Model 1; (

**b**) Model 2; (

**c**) Model 3; (

**d**) Model 4; (

**e**) Model 5; (

**f**) Model 6; and (

**g**) Model 7.

**Figure 5.**A typical energy efficiency curve η(ε) for Model 1 of the biomimetic Voronoi structure and the respective densification strain.

**Figure 6.**Compressive behavior up until the densification point of: (

**a**) Model 1; (

**b**) Model 2; (

**c**) Model 3; (

**d**) Model 4; (

**e**) Model 5; (

**f**) Model 6; and (

**g**) Model 7.

**Figure 7.**Microscopic images in (

**α**,

**β**) show typical failure points of the rods as they buckle under compressive load and the bonded 3D printed layers are separated.

**Figure 9.**A second-degree polynomial correlation is discernible when the following factors are plotted against their relative densities: (

**a**) Elastic modulus; (

**b**) energy capacity; and (

**c**) plateau stress of the Voronoi structures.

**Figure 10.**(

**a**) Experimental load–displacement response for the Voronoi lattice structures curve-fitted by FEA generated data; (

**b**) vertical deformation and (

**c**) stress distribution of the Voronoi structure under compression load, utilizing the PLA material properties in the FE model.

Model | Node Count | Rod Radius (mm) | Smoothness Scale |
---|---|---|---|

1 | 60 | 1.8 | 1 |

2 | 60 | 2 | 1 |

3 | 60 | 2.2 | 1 |

4 | 80 | 1.8 | 1 |

5 | 100 | 1.8 | 1 |

6 | 60 | 1.8 | 4 |

7 | 60 | 1.8 | 7 |

Printer Parameter | Value |
---|---|

Nozzle size | 0.4 mm |

Materials | PLA |

Layer Thickness | 0.2 mm |

Wall Thickness | 0.8 mm |

Infill Pattern | Lines |

Infill Density | 100% |

Outer Wall Speed | 15 mm/s |

Inner Wall Speed | 30 mm/s |

Infill Speed | 30 mm/s |

Printing Temp. | 205 °C |

Build Plate Temp. | 55 °C |

Support | No |

Print Time | 6–7 h |

**Table 3.**Designed relative density of the models compared to the calculated values of the 3D printed specimens.

Model | Designed Relative Density | Calculated Relative Density | Discrepancy (%) |
---|---|---|---|

1 | 0.44 | 0.40 ± 0.004 | 9.09 |

2 | 0.52 | 0.48 ± 0.005 | 7.69 |

3 | 0.61 | 0.56 ± 0.005 | 8.2 |

4 | 0.53 | 0.49 ± 0.03 | 7.55 |

5 | 0.6 | 0.55 ± 0.03 | 8.33 |

6 | 0.46 | 0.42 ± 0.08 | 8.7 |

7 | 0.5 | 0.46 ± 0.02 | 8 |

Model | Porosity (%) | Yield Strength (MPa) | Compressive Modulus (MPa) | Densification Strain (%) | Plateau Stress (MPa) | Energy Capacity (MJ/m^{3}) | Specific Energy Capacity (KJ/kg) |
---|---|---|---|---|---|---|---|

1 | 60.43 ± 0.43 | 6.26 ± 0.12 | 310.29 ± 15.30 | 49.39 ± 1.28 | 6.15 ± 0.09 | 2.94 ± 0.11 | 6.01 ± 0.30 |

2 | 52.03 ± 0.47 | 8.80 ± 0.12 | 443.64 ± 1.10 | 47.61 ± 1.14 | 9.38 ± 0.14 | 4.32 ± 0.14 | 7.26 ± 0.31 |

3 | 44.25 ± 0.50 | 10.84 ± 0.10 | 603.66 ± 13.93 | 47.43 ± 0.47 | 12.64 ± 0.22 | 5.78 ± 0.09 | 8.38 ± 0.20 |

4 | 50.99 ± 0.28 | 9.47 ± 0.13 | 498.31 ± 8.83 | 48.68 ± 1.30 | 10.31 ± 0.02 | 4.85 ± 0.14 | 7.98 ± 0.19 |

5 | 44.69 ± 0.32 | 11.55 ± 0.40 | 658.07 ± 80.29 | 46.57 ± 0.42 | 13.55 ± 0.55 | 6.07 ± 0.19 | 8.85 ± 0.27 |

6 | 58.04 ± 0.85 | 7.40 ± 0.06 | 381.33 ± 11.40 | 48.88 ± 1.59 | 7.42 ± 0.08 | 3.51 ± 0.14 | 6.75 ± 0.37 |

7 | 54.23 ± 0.21 | 8.57 ± 0.45 | 443.01 ± 14.37 | 45.07 ± 0.83 | 8.94 ± 0.40 | 3.89 ± 0.19 | 6.85 ± 0.31 |

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Efstathiadis, A.; Symeonidou, I.; Tsongas, K.; Tzimtzimis, E.K.; Tzetzis, D.
3D Printed Voronoi Structures Inspired by *Paracentrotus lividus* Shells. *Designs* **2023**, *7*, 113.
https://doi.org/10.3390/designs7050113

**AMA Style**

Efstathiadis A, Symeonidou I, Tsongas K, Tzimtzimis EK, Tzetzis D.
3D Printed Voronoi Structures Inspired by *Paracentrotus lividus* Shells. *Designs*. 2023; 7(5):113.
https://doi.org/10.3390/designs7050113

**Chicago/Turabian Style**

Efstathiadis, Alexandros, Ioanna Symeonidou, Konstantinos Tsongas, Emmanouil K. Tzimtzimis, and Dimitrios Tzetzis.
2023. "3D Printed Voronoi Structures Inspired by *Paracentrotus lividus* Shells" *Designs* 7, no. 5: 113.
https://doi.org/10.3390/designs7050113