# A Numerical Study of Geometry’s Impact on the Thermal and Mechanical Properties of Periodic Surface Structures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Numerical Simulations Results and Discussion

^{3}, c—specific heat capacity 1000 J/(kg·K). The material properties are similar to those of aluminum–copper alloys—materials that are significant and often used in the industry. Particularly, aluminum alloys (AlSi, AlCu, AlSiMg, and others) are widely used in additive manufacturing. Authors considered the properties of sample materials Al2%Cu alloy because of their previous research of thermo-mechanics response during solidifying. On one of the boundaries (on the left side), the Neumann boundary condition with a fixed value of 10 kW/m

^{2}was prescribed. All other five sides were implicitly set to a no-flux boundary condition (perfect insulation). It was assumed that there is no heat exchange with the environment, and the initial temperature was equal to 300 K. Mechanical properties are as follows: Young’s modulus E = 6.9 × 10

^{10}MPa, Poisson’s ratio υ = 0.33, density ρ = 2700 kg/m

^{3}. All cubes had fixed bottom faces (all degrees of freedom were removed), and all cubes were subject to a load of value 10 MPa on the top face. With this value of the load, all investigated domains remained in the elastic region.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The results of heat simulation after 250 s. Columns from left to right: diamond (D) surface, gyroid (G) surface, primitive (P) surface. Rows from top to bottom: 2 cells, thickness δ = 0.125; 2 cells, thickness δ = 0.2; 4 cells, thickness δ = 0.125; 4 cells, thickness δ = 0.2

**Figure 2.**Results for a reference cube. The following result from the upper-left clockwise results: (

**a**) heat distribution after 250 s, (

**b**) total displacement from stress analysis, (

**c**) mesh deformation with scale ×250, and (

**d**) von Mises stress distribution in MPa.

**Figure 3.**The results of total displacement from stress analysis. Columns from left to right: diamond (D) surface, gyroid (G) surface, primitive (P) surface. Rows from top to bottom: 2 cells, relative thickness δ = 0.125; 2 cells, relative thickness δ = 0.2; 4 cells, relative thickness δ = 0.125; 4 cells, relative thickness δ = 0.2.

**Figure 4.**Von Mises stress distribution in MPa. Columns from left to right: diamond (D) surface, gyroid (G) surface, primitive (P) surface. Rows from top to bottom: 2 cells, relative thickness δ = 0.125; 2 cells, relative thickness δ = 0.2; 4 cells, relative thickness δ = 0.125; 4 cells, relative thickness δ = 0.2.

**Figure 5.**Mesh deformation. Columns from left to right: diamond (D) surface, gyroid (G) surface, primitive (P) surface. Rows from top to bottom: 2 cells, relative thickness δ = 0.125; 2 cells, relative thickness δ = 0.2; 4 cells, relative thickness δ = 0.125; 4 cells, relative thickness δ = 0.2.

**Table 1.**Parameters used for generating different periodic surface structures, together with their impact on volume and the side area of a specific structure.

Name | Relative Thickness δ [.] | Thickness [m] | Volume [m^{3}] | Volume Fraction [.] | Side Area [m^{2}] | Side Area Fraction [.] |
---|---|---|---|---|---|---|

cube | - | - | 6.400 × 10^{−5} | 1.0000 | 16.000 × 10^{−4} | 1.0000 |

diamond (D) 2 cells | 0.125 | 0.0023 | 2.633 × 10^{−5} | 0.4114 | 4.385 × 10^{−4} | 0.2741 |

diamond (D) 2 cells | 0.2 | 0.0041 | 4.254 × 10^{−5} | 0.6647 | 8.389 × 10^{-4} | 0.5243 |

diamond (D) 4 cells | 0.125 | 0.0011 | 2.638 × 10^{−5} | 0.4122 | 3.555 × 10^{−4} | 0.2222 |

diamond (D) 4 cells | 0.2 | 0.0020 | 4.266 × 10^{−5} | 0.6666 | 7.536 × 10^{−4} | 0.4710 |

gyroid (G) 2 cells | 0.125 | 0.0023 | 2.069 × 10^{−5} | 0.3233 | 3.603 × 10^{−4} | 0.2252 |

gyroid (G) 2 cells | 0.2 | 0.0040 | 3.343 × 10^{−5} | 0.5223 | 6.603 × 10^{−4} | 0.4127 |

gyroid (G) 4 cells | 0.125 | 0.0012 | 2.071 × 10^{−5} | 0.3235 | 2.892 × 10^{−4} | 0.1808 |

gyroid (G) 4 cells | 0.2 | 0.0020 | 3.347 × 10^{−5} | 0.5230 | 5.876 × 10^{−4} | 0.3673 |

primitive (P) 2 cells | 0.125 | 0.0028 | 1.828 × 10^{−5} | 0.2856 | 3.138 × 10^{−4} | 0.1961 |

Primitive (P) 2 cells | 0.2 | 0.0045 | 2.930 × 10^{−5} | 0.4579 | 5.545 × 10^{−4} | 0.3466 |

primitive (P) 4 cells | 0.125 | 0.0012 | 1.829 × 10^{−5} | 0.2857 | 2.733 × 10^{−4} | 0.1708 |

primitive (P) 4 cells | 0.2 | 0.0024 | 2.933 × 10^{−5} | 0.4584 | 5.096 × 10^{−4} | 0.3185 |

Name | Relative Thickness [.] | Number of Nodes | Number of Elements |
---|---|---|---|

cube | - | 45,142 | 245,979 |

diamond (D) 2 cells | 0.125 | 325,910 | 1,087,687 |

diamond (D) 2 cells | 0.2 | 322,198 | 1,122,255 |

diamond (D) 4 cells | 0.125 | 551,898 | 1,731,035 |

diamond (D) 4 cells | 0.2 | 544,391 | 1,911,853 |

gyroid (G) 2 cells | 0.125 | 269,511 | 890,282 |

gyroid (G) 2 cells | 0.2 | 283,175 | 967,104 |

gyroid (G) 4 cells | 0.125 | 457,834 | 1,410,877 |

gyroid (G) 4 cells | 0.2 | 479,653 | 1,610,759 |

primitive (P) 2 cells | 0.125 | 211,320 | 706,578 |

Primitive (P) 2 cells | 0.2 | 224,711 | 775,721 |

primitive (P) 4 cells | 0.125 | 355,299 | 1,096,937 |

primitive (P) 4 cells | 0.2 | 378,416 | 1,274,439 |

Name | Relative Thickness [.] | Minimum Temperature [K] | Maximum Temperature [K] | Temperature Difference ΔT [K] |
---|---|---|---|---|

cube | - | 322.1 | 322.9 | 0.8 |

diamond (D) 2 cells | 0.125 | 313.7 | 314.5 | 0.8 |

diamond (D) 2 cells | 0.2 | 316.7 | 317.6 | 0.9 |

diamond (D) 4 cells | 0.125 | 311.6 | 312.2 | 0.6 |

diamond (D) 4 cells | 0.2 | 314.7 | 315.4 | 0.7 |

gyroid (G) 2 cells | 0.125 | 314.2 | 315.2 | 1.0 |

gyroid (G) 2 cells | 0.2 | 316.7 | 317.7 | 1.0 |

gyroid (G) 4 cells | 0.125 | 312.5 | 313.2 | 0.7 |

gyroid (G) 4 cells | 0.2 | 314.5 | 315.2 | 0.7 |

primitive (P) 2 cells | 0.125 | 314.7 | 315.5 | 0.8 |

Primitive (P) 2 cells | 0.2 | 316.1 | 316.9 | 0.8 |

primitive (P) 4 cells | 0.125 | 312.2 | 312.9 | 0.7 |

primitive (P) 4 cells | 0.2 | 314.8 | 315.5 | 0.7 |

**Table 4.**Maximum total displacement, maximum and minimum von Mises stress from elastic stress analysis.

Name | Relative Thickness [.] | Maximum Total Displacement [m] | Minimum von Mises Stress [MPa] | Maximum von Mises Stress [MPa] |
---|---|---|---|---|

cube | - | 5.803 × 10^{6} | 6.359 × 10^{6} | 2.710 × 10^{7} |

diamond (D) 2 cells | 0.125 | 1.062 × 10^{5} | 340.3 | 5.161 × 10^{7} |

diamond (D) 2 cells | 0.2 | 1.007 × 10^{5} | 119.5 | 6.727 × 10^{7} |

diamond (D) 4 cells | 0.125 | 6.316 × 10^{6} | 2216 | 3.747 × 10^{7} |

diamond (D) 4 cells | 0.2 | 6.536 × 10^{6} | 330.9 | 4.276 × 10^{7} |

gyroid (G) 2 cells | 0.125 | 3.747 × 10^{5} | 224.5 | 1.019 × 10^{8} |

gyroid (G) 2 cells | 0.2 | 2.585 × 10^{5} | 334.5 | 9.635 × 10^{7} |

gyroid (G) 4 cells | 0.125 | 1.618 × 10^{5} | 2702 | 7.621 × 10^{7} |

gyroid (G) 4 cells | 0.2 | 1.313 × 10^{5} | 928.5 | 7.318 × 10^{7} |

primitive (P) 2 cells | 0.125 | 1.642 × 10^{5} | 9.146 × 10^{5} | 5.612 × 10^{7} |

Primitive (P) 2 cells | 0.2 | 1.254 × 10^{5} | 2.723 × 10^{5} | 6.460 × 10^{7} |

primitive (P) 4 cells | 0.125 | 1.200 × 10^{5} | 2.247 × 10^{5} | 5.222 × 10^{7} |

primitive (P) 4 cells | 0.2 | 9.722 × 10^{6} | 3.891 × 10^{5} | 5.847 × 10^{7} |

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**MDPI and ACS Style**

Gawronska, E.; Dyja, R.
A Numerical Study of Geometry’s Impact on the Thermal and Mechanical Properties of Periodic Surface Structures. *Materials* **2021**, *14*, 427.
https://doi.org/10.3390/ma14020427

**AMA Style**

Gawronska E, Dyja R.
A Numerical Study of Geometry’s Impact on the Thermal and Mechanical Properties of Periodic Surface Structures. *Materials*. 2021; 14(2):427.
https://doi.org/10.3390/ma14020427

**Chicago/Turabian Style**

Gawronska, Elzbieta, and Robert Dyja.
2021. "A Numerical Study of Geometry’s Impact on the Thermal and Mechanical Properties of Periodic Surface Structures" *Materials* 14, no. 2: 427.
https://doi.org/10.3390/ma14020427