# Optimization of Environment-Friendly and Sustainable Polylactic Acid (PLA)-Constructed Triply Periodic Minimal Surface (TPMS)-Based Gyroid Structures

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Design and Methodology

#### Design of Experiment

## 3. Grey Relational Analysis

#### 3.1. Phase 1—Data Processing

#### 3.2. Phase 2—Normalization of Data

_{ij}represents the corresponding data points for the three output parameters (compressive modulus (Y), compressive strength (U), and toughness (E)) in Table 3, while x

_{ij}represents the resulting normalized data. The output data for the three studied parameters in Table 3 were normalized using Equation (1), and Table 4 displays the obtained results.

#### 3.3. Phase 3—Determining the Deviation Sequence

#### 3.4. Phase 4—Determining the Grey Relational Coefficient

#### 3.5. Phase 5—Determining the Grey Relational Grade

## 4. Conclusions

- It was observed that a lower value (20%) of relative density provided the lowest value of yield strength. It can be concluded that relative density significantly controls the ability of the TPMS gyroid structure to resist plastic deformation.
- The ninth gyroid sample demonstrated the best yield strength, compressive modulus, and energy absorption. The ninth gyroid sample was printed with a printing speed of 40 mm/s, a relative density of 60%, and a cell size of 3.17 mm. The ninth experiment for the above-mentioned condition improved yield strength by 16.9%, the compression modulus by 34.8%, and energy absorption by 29.5% when compared with the second-best performer, the third experiment. The mechanism of toughness is linked with the plastic deformation and buckling of the cell wall. The optimal printing speed provides better material bonding because of the more precise alignment of layers. The optimal condition provided an improvement of 0.2424 in grey relational grade (GRG) when compared with the initial condition.
- To conclude, scanning electron micrographs were examined for the best- and worst-ranked samples. The micrographs confirmed the significant impact of geometry, wall thickness, and stress distribution on fracture behavior. The gyroid geometries generally involve thin walls, sharp corners, and thickness variations within structures that are regarded as potential sites of fracture development and crack formation due to stress concentrations. Larger cell sizes were observed to reduce stress concentration and enhance fracture strength. The SEM image of the first specimen (worst case), shows fractured walls and minimal material flow, exemplifying the brittle nature of the fracture arising from stress concentration because of its smaller cell size. This observation aligns with the lowest toughness value recorded for this specimen, further highlighting the critical role of design parameters in optimizing fracture resistance and mechanical performance in gyroid TPMS structures.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AM | additive manufacturing |

CFRP | carbon fiber-reinforced polymer |

CFRTPC | continuous fiber-reinforced thermoplastic composite |

DLP | digital light processing |

FDM | fused deposition Modeling |

FFF | Fused Filament Fabrication |

GRC | grey relational coefficient |

GRG | grey relational grade |

HUs | Hounsfield units |

LPBF | laser powder bed fusion |

MJF | Multi Jet Fusion |

PACs | periodic auxetic cellular structures |

PLA | polylactic acid |

PBF | powder bed fusion |

RCA | re-entrant chiral auxetic |

SEM | scanning electron microscopy |

STL | standard triangulation language |

TPMS | Triply Periodical Minimal Surface |

TCP/BG | tricalcium phosphate/bio glass composite |

µCT | micro-computed tomography |

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**Figure 1.**Deformation in TPMS cell structures under compressive testing: (

**a**) gyroid, (

**b**) primitive, (

**c**) IWP, (

**d**) Neovius, (

**e**) deformed gyroid, (

**f**) deformed primitive, (

**g**) deformed IWP, and (

**h**) deformed Neovius [adopted from 23 with kind permission from Elsevier].

**Figure 4.**(

**a**) TOBECA 3D printer used to fabricate the 3D-printed specimens. (

**b**) Cross section of PLA filament.

**Figure 6.**(

**a**) Using the Universal Testing Machine to compress each 3D-printed specimen. (

**b**) Stress–strain curve obtained during the run of Sample # 4 (printing speed = 33 mm/s, relative density = 20%, and cell size = 3.17 mm).

**Figure 9.**Scanning electron micrographs of (

**a**) Sample # 1, ranked 9th, and (

**b**) Sample # 9, ranked 1st.

Printer Parameters | Set Values |
---|---|

Printing temperature | 210 °C |

Printer bed temperature | 60 °C |

Infill density | 100% |

Nozzle diameter | 0.4 mm |

Run | A: Printing Speed (mm/s) | B: Relative Density (%) | C: Unit Cell Size (mm) | Weight (g) |
---|---|---|---|---|

1 | 22 | 20 | 1.58 | 5.7 |

2 | 22 | 40 | 3.17 | 12.6 |

3 | 22 | 60 | 6.35 | 19.7 |

4 | 33 | 20 | 3.17 | 5.86 |

5 | 33 | 40 | 6.35 | 13.13 |

6 | 33 | 60 | 1.58 | 19.2 |

7 | 40 | 20 | 6.35 | 6.07 |

8 | 40 | 40 | 1.58 | 11.59 |

9 | 40 | 60 | 3.17 | 19.9 |

Run | A | B | C | Yield Strength (MPa) | Compression Modulus (MPa) | Densification Strain (%) | Energy Absorption (MJ/m^{3}) |
---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1.3 | 53.22 | 32.5 | 0.75 |

2 | 1 | 2 | 2 | 63.47 | 1999.6 | 37.5 | 24.34 |

3 | 1 | 3 | 3 | 122.89 | 4386.2 | 30.2 | 38.28 |

4 | 2 | 1 | 2 | 3.49 | 557.1 | 37.1 | 2.06 |

5 | 2 | 2 | 3 | 50 | 1500.7 | 36 | 11.46 |

6 | 2 | 3 | 1 | 52.1 | 4028.5 | 33 | 38.98 |

7 | 3 | 1 | 3 | 9.29 | 441.97 | 41 | 2.45 |

8 | 3 | 2 | 1 | 14.53 | 856 | 37.3 | 13.47 |

9 | 3 | 3 | 2 | 143.7 | 5913 | 30.8 | 49.58 |

Normalization | ||||
---|---|---|---|---|

Run | Yield Strength (MPa) | Compression Modulus (MPa) | Densification Strain (%) | Energy Absorption (MJ/m^{3}) |

1 | 0 | 0 | 0.2129 | 0 |

2 | 0.4365 | 0.3321 | 0.6759 | 0.4830 |

3 | 0.8538 | 0.7394 | 0 | 0.7685 |

4 | 0.0153 | 0.0859 | 0.6388 | 0.0267 |

5 | 0.3419 | 0.2470 | 0.5370 | 0.2192 |

6 | 0.3567 | 0.6784 | 0.2592 | 0.7829 |

7 | 0.0561 | 0.0663 | 1 | 0.0347 |

8 | 0.0929 | 0.1369 | 0.6574 | 0.2604 |

9 | 1 | 1 | 0.0555 | 1 |

Deviation Sequence | ||||
---|---|---|---|---|

Run | Yield Strength (MPa) | Compression Modulus (MPa) | Densification Strain (%) | Energy Absorption (MJ/m^{3}) |

1 | 1 | 1 | 0.7870 | 1 |

2 | 0.5634 | 0.6678 | 0.3240 | 0.5169 |

3 | 0.1461 | 0.2605 | 1 | 0.2314 |

4 | 0.9846 | 0.9140 | 0.3611 | 0.9732 |

5 | 0.6580 | 0.7529 | 0.4629 | 0.7807 |

6 | 0.6432 | 0.3215 | 0.7407 | 0.2170 |

7 | 0.9438 | 0.9336 | 0 | 0.9652 |

8 | 0.9070 | 0.8630 | 0.3425 | 0.7395 |

9 | 0 | 0 | 0.9444 | 0 |

Grey Relational Coefficient | ||||
---|---|---|---|---|

Run | Yield Strength (MPa) | Compression Modulus (MPa) | Densification Strain (%) | Energy Absorption (MJ/m^{3}) |

1 | 0.3333 | 0.3333 | 0.3884 | 0.3333 |

2 | 0.4701 | 0.4281 | 0.6067 | 0.4916 |

3 | 0.7738 | 0.6574 | 0.3333 | 0.6835 |

4 | 0.3367 | 0.3536 | 0.5806 | 0.3393 |

5 | 0.4317 | 0.3990 | 0.5192 | 0.3904 |

6 | 0.4373 | 0.6085 | 0.4029 | 0.6972 |

7 | 0.3462 | 0.3487 | 1 | 0.3412 |

8 | 0.3553 | 0.3668 | 0.5934 | 0.4033 |

9 | 1 | 1 | 0.3461 | 1 |

Run | Grade | Rank |
---|---|---|

1 | 0.3471 | 9 |

2 | 0.4991 | 5 |

3 | 0.6120 | 2 |

4 | 0.4026 | 8 |

5 | 0.4351 | 6 |

6 | 0.5365 | 3 |

7 | 0.5090 | 4 |

8 | 0.4297 | 7 |

9 | 0.8365 | 1 |

Response Table for Grey Relational Grade | ||||
---|---|---|---|---|

1 | 2 | 3 | Rank (Max–Min) | |

A | 0.4861 | 0.4580 | 0.5917 | 0.1336 (3rd) |

B | 0.4196 | 0.4546 | 0.6617 | 0.2421 (1st) |

C | 0.4378 | 0.5794 | 0.5187 | 0.1416 (2nd) |

Levels | Yield Strength (MPa) | Compression Modulus (MPa) | Densification Strain (%) | Energy Absorption (MJ/m^{3}) | Grey Relational Grade | |
---|---|---|---|---|---|---|

Initial controllable parameters | A1B1C1 | 1.3 | 53.22 | 32.5 | 0.7536 | 0.3471 |

Optimal controllable parameters | A3B3C2 | |||||

Average experimental readings (02 replications) | 128 | 2987.3 | 34.25 | 33.84 | 0.5895 | |

Improvement in GRG = 0.2424 |

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## Share and Cite

**MDPI and ACS Style**

Razi, S.S.; Pervaiz, S.; Susantyoko, R.A.; Alyammahi, M.
Optimization of Environment-Friendly and Sustainable Polylactic Acid (PLA)-Constructed Triply Periodic Minimal Surface (TPMS)-Based Gyroid Structures. *Polymers* **2024**, *16*, 1175.
https://doi.org/10.3390/polym16081175

**AMA Style**

Razi SS, Pervaiz S, Susantyoko RA, Alyammahi M.
Optimization of Environment-Friendly and Sustainable Polylactic Acid (PLA)-Constructed Triply Periodic Minimal Surface (TPMS)-Based Gyroid Structures. *Polymers*. 2024; 16(8):1175.
https://doi.org/10.3390/polym16081175

**Chicago/Turabian Style**

Razi, Syed Saarim, Salman Pervaiz, Rahmat Agung Susantyoko, and Mozah Alyammahi.
2024. "Optimization of Environment-Friendly and Sustainable Polylactic Acid (PLA)-Constructed Triply Periodic Minimal Surface (TPMS)-Based Gyroid Structures" *Polymers* 16, no. 8: 1175.
https://doi.org/10.3390/polym16081175