# Comparison of Mechanical Properties and Energy Absorption of Sheet-Based and Strut-Based Gyroid Cellular Structures with Graded Densities

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Gibson-Ashby Model of Cellular Structures

_{L}, ρ

_{L}, and σ

_{L}are the elastic modulus, density, and yield strength of cellular structures, respectively; E

_{S}, ρ

_{S}, and σ

_{S}are the elastic modulus, density, and yield strength of base solid material, respectively. ε

_{D}is the value of onset densification strain of the cellular structure where the densification region starts. Moreover, ε

_{D}is considered to be the practical limit for energy absorption applications using cellular structures. After this strain limit, the cellular structure will continue to absorb energy, but at the expense of transferring stress throughout the cellular structure. This can result in damage to the objects or injury to the human body. Moreover, the constant parameters C

_{1}, C

_{2}, n, m, and α are calculated by fitting the result of the compression test. In this work, numerical integration is used to determine the energy absorption capacity of the cellular structure, giving the area under the compressive stress-strain curves [27]. Then, the energy absorption per unit volume W

_{v}is expressed as:

#### 2.2. Design of Gyroid-Based Cellular Structures

#### 2.3. Manufacturing the Cellular Specimens

#### 2.4. Numerical Homogenization

_{11}= C

_{22}= C

_{33}, C

_{12}= C

_{13}= C

_{23}, C

_{44}= C

_{55}= C

_{66}and other constants are zero. Then, the expression form of the stiffness matrix is:

^{H}, (axial) Poisson’s ratio υ

^{H}, and (axial) shear modulus G

^{H}can be calculated using tensor components. Moreover, we can use Zener [34] ration A

^{H}to measure the isotropy of cellular material, and if the value is close to 1, it means the cellular structure is isotropic.

#### 2.5. Mechanical Testing

_{S}= 850.1 ± 15.2 MPa, yield stress σ

_{S}= 132.5 ± 12.8 MPa Poisson’s ratio is 0.33.

## 3. Results and Discussion

#### 3.1. Formatting of Mathematical Components

#### 3.2. Deformation of Uniform Gyroid-Based Cellular Structure

_{1}, C

_{2,}and α of the Gibson-Ashby model, where n = 2 and m = 3/2 are assumed here. The estimation results are recorded in Table 2. For the Strut-Gyroid-U and Sheet-Gyroid-U cellular types, the determined value of C

_{1}is in the range of 0.1 to 4.0 previously given by Gibson and Ashby [27]. However, the determined values of parameters C

_{2}and α are lower than the ranges given by Gibson and Ashby from 0.25 to 0.35 and 1.4 to 2.0. This means that the plateau strength is lower than the predictable strength of the Gibson-Ashby model. Moreover, the densification strain observed here is higher than might be predicted. It is predicted that the coefficients of the Gibson-Ashby model should be calculated using cellular structures of different densities, but this is beyond the scope of this paper.

#### 3.3. Deformation of Graded Gyroid-Based Cellular Structures

#### 3.4. Energy Absorption Capability of the Cellular Structures

_{v}of the cellular structure is obtained by Equation (4), that is, the integral value of the stress in the strain range is the energy absorption per unit volume when it reaches the corresponding strain value. Moreover, Figure 10 shows the relationship between strain and energy absorption value per unit volume. The total energy absorption values of the respective samples when in the densified state are recorded in Table 1.

_{v}and σ

_{L}, and they are normalized by the modulus of the base material, E

_{S}. This expression is useful in allowing a designer to select a cellular structure that minimizes the stress while the required energy is absorbed [27]. We divide the curve in Figure 11 into three regions, I, II, and III. First, the region I corresponds to the elastic deformation region of the uniform structure and the collapse deformation region of the low-density region of the graded structure. In this area, they all absorb less total energy. Second, in region II, the uniform structure starts to be plastically deformed, so that the absorbed energy sharply increases, and the stress increases less. However, the graded structure absorbs energy at a lower rate in this region. Third, the uniform structure enters the densified state in the III region, and a significant turning point and a subsequent rapid increase in the stress appear in the curves. In contrast, the strut-based and sheet-based gyroid structures with gradient-2 do not show a significant turning point, but gradually enters the densification state, and the rate of energy absorption generally remains unchanged. For the gradient-1 structures, there is a slight turning point, but the Strut-Gyroid-G1 is relatively more obvious than the Sheet-Gyroid-G1. Furthermore, according to the features described above, the sheet-based gyroid structure has a more gradual energy absorption process than the strut-based gyroid structure, and there is no point of sharp turning.

## 4. Conclusions

- (1)
- Anisotropic analysis of strut-based gyroid cellular structure and sheet-based gyroid cellular structure was performed by numerical homogenization method. It is found that the sheet-based gyroid cellular structure tends to be isotropic over the entire density interval and theoretically more suited for energy absorption. In addition, the strut-based gyroid structure exhibits anisotropy, and its Young’s modulus is also smaller than the sheet-based gyroid structure at the same density.
- (2)
- It is found from the video recorded in the experiment that the uniform structure exhibits a global collapse deformation mode during the compression process, and the graded structure exhibits a layer-by-layer collapse deformation mode from a low-density. Besides, the modulus and yield strength of the two structures are calculated according to the values of the linear elastic phase. It can be seen from the results that the modulus and the yield strength of the sheet-based gyroid structure are higher than the corresponding strut-based gyroid structure.
- (3)
- The energy absorption per unit volume of each sample was calculated separately based on the data recorded in the experiment. It can be seen from the results that the energy absorption capacity of the graded structure is better than that of the uniform structure, and the graded structure exhibits a smoother energy absorption process until the fully dense strain. However, the uniform structure will suddenly increase sharply when it is fully dense. Additionally, the sheet-based gyroid structure has better energy absorption than the strut-based gyroid structure. Besides, the gradient levels also have an effect on energy absorption and deformation. From the results of this work, it is known that cellular structures with large gradient exhibit better energy absorption capacity than those with a small gradient.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**(

**a**) Gyroid structure derived from bionic butterfly wings; gyroid surface (

**b**) close to be strut-based gyroid structure (

**c**) and double gyroid surface (

**d**) close to be sheet-based gyroid structure (

**e**); (

**f**) is the relationship between relative density and parameter t of the two types gyroid cellular structures.

**Figure 3.**3D models of the (

**a**) and (

**b**) are the uniform gyroid-based cellular structures; (

**c**) and (

**d**) are the graded gyroid-based cellular structure structures with gradient-1 and (

**e**) and (

**f**) with gradient-2.

**Figure 4.**The fabricated gyroid uniform and graded cellular structures with different design parameters.

**Figure 5.**Anisotropy of strut-based gyroid (

**a**) and sheet-based gyroid (

**b**). (If A

^{H}is close to unity, the structure could be treated as isotropic).

**Figure 6.**Deformation stages of Strut-Gyroid-U (

**a**) and Sheet-Gyroid-U; (

**b**) cellular structures in the presence of 0%, 15%, 30%, 45%, and 60% compression strain from the video capture.

**Figure 7.**Compressive stress-strain curves of strut-based (

**a**,

**c**) and sheet-based (

**b**,

**d**) gyroid with different gradient.

**Figure 8.**Deformation stages of layer-by-layer collapses of Strut-Gyroid-G1 (

**a**) and Sheet-Gyroid-G1 (

**b**) cellular structures from the video frames.

**Figure 9.**Deformation stages of layer-by-layer collapses of Strut-Gyroid-G2 (

**a**) and Sheet-Gyroid-G2 (

**b**) cellular structures from the video frames.

**Figure 10.**Energy absorption per unit volume for strut-based (

**a**) and sheet-based (

**b**) gyroid uniform and graded cellular structures.

**Figure 11.**Normalized energy absorption of strut-based and sheet-based gyroid uniform and graded cellular structures.

**Table 1.**Mechanical properties and energy absorption of uniform and graded gyroid-based cellular structures under compressive testing.

Properties | Strut- Gyroid-U | Sheet- Gyroid-U | Strut- Gyroid-G1 | Sheet- Gyroid-G1 | Strut- Gyroid-G2 | Sheet- Gyroid-G2 |
---|---|---|---|---|---|---|

E^{H} (MPa) | 40.56 | 63.67 | — | — | — | — |

E_{L} (MPa) | 39.21 ± 0.23 | 62.51 ± 0.44 | 28.25 ± 0.31 | 36.67 ± 0.18 | 23.96 ± 0.16 | 32.21 ± 0.31 |

E* (10^{−3}) | 46.12 ± 0.02 | 73.54 ± 0.02 | 33.23 ± 0.05 | 43.14 ± 0.03 | 28.18 ± 0.01 | 37.89 ± 0.03 |

σ_{L} (MPa) | 2.92 ± 0.08 | 3.89 ± 0.12 | 1.98 ± 0.04 | 2.46 ± 0.16 | 0.69 ± 0.02 | 0.87 ± 0.05 |

σ* (10^{−3}) | 22.12 ± 0.02 | 29.46 ± 0.11 | 15.00 ± 0.12 | 18.63 ± 0.09 | 5.23 ± 0.10 | 6.59 ± 0.06 |

ε_{D} (%) | 62.3 ± 0.12 | 66.4 ± 0.01 | 68.5 ± 0.06 | 69.8 ± 0.13 | 70.1 ± 0.02 | 72.2 ± 0.03 |

W_{v} (KJ/m^{3}) | 1734 ± 4 | 2276 ± 2 | 2116 ± 6 | 2623 ± 1 | 2341 ± 5 | 2781 ± 4 |

Coefficients | C_{1} | C_{2} | α |
---|---|---|---|

Strut-Gyroid-U | 0.512 | 0.135 | 1.257 |

Sheet-Gyroid-U | 0.817 | 0.179 | 1.120 |

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**MDPI and ACS Style**

Li, D.; Liao, W.; Dai, N.; Xie, Y.M.
Comparison of Mechanical Properties and Energy Absorption of Sheet-Based and Strut-Based Gyroid Cellular Structures with Graded Densities. *Materials* **2019**, *12*, 2183.
https://doi.org/10.3390/ma12132183

**AMA Style**

Li D, Liao W, Dai N, Xie YM.
Comparison of Mechanical Properties and Energy Absorption of Sheet-Based and Strut-Based Gyroid Cellular Structures with Graded Densities. *Materials*. 2019; 12(13):2183.
https://doi.org/10.3390/ma12132183

**Chicago/Turabian Style**

Li, Dawei, Wenhe Liao, Ning Dai, and Yi Min Xie.
2019. "Comparison of Mechanical Properties and Energy Absorption of Sheet-Based and Strut-Based Gyroid Cellular Structures with Graded Densities" *Materials* 12, no. 13: 2183.
https://doi.org/10.3390/ma12132183