# Specific Yielding of Selective Laser-Melted Ti6Al4V Open-Porous Scaffolds as a Function of Unit Cell Design and Dimensions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Unit Cell Designs and Configuration

- a cubic design (C),
- a truncated pyramidal design (P), and
- a twisted design with crossing struts (T).

#### 2.2. Fabrication of the Scaffolds via an SLM Process

^{3}). The energy density (E) is defined by the following equation [52]:

_{0}, a fully dense part (10 mm diameter and 6 mm height) was manufactured.

#### 2.3. Calculating the Scaffold Porosity

_{str}is the volume of the CAD scaffold struts and V

_{cyl}is the overall volume enclosed by the outer periphery.

_{0}is the density of non-porous Ti6Al4V (i.e., 4.43 g/cm

^{3}) and ρ

_{sc}is the density of the manufactured scaffolds, calculated by weight and volume of the scaffolds. The weight is determined by means of scales (Mettler Toledo XS 105, exactness 0.1 mg, Gießen, Germany). The volume was determined by means of measurement in a digital microscope (VHX 2000, Keyence, Neu-Isenburg, Germany).

#### 2.4. Determination of the Mechanical Properties with Compression Testing

## 3. Results

#### 3.1. Porosities of the Scaffolds

#### 3.2. Compressive Modulus, Strength and Strain of the Scaffolds

#### 3.3. Correlations between Geometrical and Mechanical Properties

^{2}= 1) and Yavari et al. (R

^{2}= 0.9998) have a high correlation. The power law is also valid if the orientation of a unit cell varies at constant scaffold orientation within the building space. These results are shown by Weißmann et al. [53] for the twisted design in different unit cell orientations with a correlation of 100%.

## 4. Discussion

_{S}, where L is the cell size and L

_{S}is the size of the specimen. Results of this study show that the design with the smaller λ achieves higher results in the elastic modulus. This relationship does not apply universally and to all tested design types. However, in general, this knowledge can be confirmed. Accordingly, the results for the cubic design range from 0.188 to 0.133 and from 0.32 to 0.3 for the pyramidal design. The E moduli for the cubic design are between 7.2 and 21.6 MPa, and those for the pyramidal design are between 3.4 and 6.9 MPa.

## 5. Conclusions

- In our work, we could demonstrate the possibility of fabricating different scaffold structures made of Ti6Al4V with high geometrical accuracy by using the SLM process. Porosity of the fabricated scaffolds differed less than 3.2% from the idealized porosity of the scaffolds as calculated by CAD.
- Based on tests under uniaxial loading conditions, the influence of unit cell dimension and of porosity on the mechanical properties was demonstrated.
- The size variation in unit cell dimension results in a change of their mechanical properties, this can be attributed to
- ○
- the geometric conditions of the unit cell, in particular the length ratio;
- ○
- the position of the struts towards the affected force

- A functional correlation between elastic modulus and compressive strain or compressive strength for all three geometric designs could be established.
- Nevertheless, a functional correlation with regard to the ratio of width of unit structure (a) and strut diameter (d) could be established.
- There is a relationship between the porosity and the elastic modulus of open-porous structures. This relationship is valid for all tested structures and describable with the power law (5). The relationship is consistent with that found in other open-cell structures fabricated from Ti6AL4V by SLM. A direct comparison with human bone is possible. Since specific yielding is directly related to the volume of structures, it is possible to selectively decide which geometries, i.e., which structures are to be preferred.
- All scaffolds were strong enough to bear the impacting loads and achieved an elastic modulus within the specified range of human cortical bone depending on the geometrical parameter variations.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Structural composition of the three investigated designs: cubic (

**left**), pyramidal (

**middle**) and twisted (

**right**). The geometrical parameters for the structures are height (c), width (a) and depth (b) of each unit structure. Struts from all designs had a circular cross-section with the strut diameter (d). The pyramidal design had additional parameters a1 and b1 for width and depth of the smaller base and top surface, respectively.

**Figure 2.**Unit cell (twisted design) and a large scaffold of diameter, D, and height, H, for mechanical testing—here as an example for the representation of the test body design.

**Figure 3.**Porosity of the three investigated designs, i.e., cubic, pyramidal and twisted, and all of their configurations. The results for CAD porosity are shown in comparison to the experimentally determined porosity.

**Figure 4.**Representation of the roughness of struts as a result of their individual orientations; (

**Left)**—slanted struts show at the bottom (the powder bed side) a build-up of molten powder particles; (

**Centre)**—image vertical struts, as well as the top of horizontal struts (the powder bed opposite) show no build-up of powder particles; (

**Right)**—horizontal struts show at the bottom (of the powder bed facing side) a build-up of melted powder particles.

**Figure 5.**Macroscopic view—overview of tested samples. Failure courses are identified in these figures by a red line for the macroscopically visible line of fracture. Twisted and pyramidal designs showed shear deformation at an angle of ca. 45°. The cubic design showed a layer-by-layer failure mechanism.

**Figure 6.**Elastic modulus and compressive strength for the three investigated designs (cube, pyramid and twisted) and the different configurations as calculated from axial compression testing. Results are shown as mean values with the corresponding standard deviation (n = 5 for each design).

**Figure 7.**Correlation between porosity and elastic modulus in each of the investigated designs. Results are shown as mean values with the corresponding standard deviations. For the linear regressions, the coefficient of determination is shown.

**Figure 8.**Correlation between the ratio a/d (width (a) and strut diameter (d)) and elastic modulus in each of the investigated designs. Results are shown as mean values with the corresponding standard deviations. For the linear regressions, the coefficient of determination is shown.

**Figure 9.**Correlation between compressive strength and elastic modulus in each of the investigated designs. Results are shown as mean values with the corresponding standard deviations. For the linear regressions, the coefficient of determination is shown.

**Figure 10.**Relationship between elastic modulus for a design and the ratio between porosity and modulus ϑ correlation.

**Figure 11.**Relationship between E-modulus and the E-modulus-porosity correlation in comparison with examples from literature with regard to Ti6Al4V and in comparison with cancellous and cortical bone.

**Table 1.**Overview of the geometric parameters for the unit cell variants of the three investigated designs (cubic, pyramidal and twisted, n = 5 for each configuration) as well as the dimensions for the large-size scaffold for mechanical testing. All values are derived from CAD data and are given in mm.

Basic Design | Configu-Ration | Unit Structure | Scaffold Structure | ||||||
---|---|---|---|---|---|---|---|---|---|

Height (c) | Width (a) | Width (a_{1}) | Depth (b) | Depth (b_{1}) | Strut Diameter (d) | Height (H) | Diameter (D) | ||

Cube | |||||||||

C1 | 4.0 | 4.0 | --- | 4.0 | --- | 2.2 | 30.1 | 26.1 | |

C2 | 4.0 | 4.0 | --- | 4.0 | --- | 1.8 | 29.1 | 25.6 | |

C3 | 3.0 | 3.0 | --- | 3.0 | --- | 1.8 | 20.2 | 16.8 | |

C4 | 3.0 | 3.0 | --- | 3.0 | --- | 1.1 | 19.1 | 16.1 | |

C5 | 2.0 | 2.0 | --- | 2.0 | --- | 0.8 | 10.6 | 8.1 | |

Truncated Pyramid | |||||||||

P1 | 6.0 | 6.0 | 3.0 | 6.0 | 3.0 | 2.2 | 37.2 | 35.2 | |

P2 | 4.0 | 4.0 | 2.0 | 4.0 | 2.0 | 1.5 | 25.4 | 23.2 | |

P3 | 3.0 | 3.0 | 1.5 | 3.0 | 1.5 | 1.1 | 19.6 | 17.6 | |

P4 | 3.0 | 3.0 | 1.5 | 3.0 | 1.5 | 1.0 | 19.7 | 17.6 | |

Twisted | |||||||||

T1 | 4.0 | 2.83 | --- | 2.83 | --- | 1.1 | 18.0 | 17.0 | |

T2 | 4.0 | 2.83 | --- | 2.83 | --- | 1.0 | 18.0 | 17.0 | |

T3 | 3.0 | 2.12 | --- | 2.12 | --- | 0.9 | 18.0 | 17.0 |

Design | Elastic Modulus (GPa) | Compressive Strength (MPa) | Compressive Strain (%) |
---|---|---|---|

C1 | 18.9 ± 1.2 | 510.7 ± 67.8 | 4.2 ± 2.6 |

C2 | 10.9 ± 1.0 | 369.7 ± 40.3 | 4.0 ± 0.7 |

C3 | 21.6 ± 1.9 | 761.1 ± 108.6 | 4.6 ± 1.3 |

C4 | 7.2 ± 0.5 | 203.1 ± 50.4 | 2.7 ± 0.6 |

C5 | 9.3 ± 0.3 | 184.4 ± 2.1 | 3.2 ± 0.3 |

P1 | 5.7 ± 0.3 | 153.9 ± 15.5 | 3.0 ± 0.3 |

P2 | 6.9 ± 0.8 | 164.1 ± 44.8 | 2.6 ± 0.7 |

P3 | 5.2 ± 0.6 | 142.8 ± 13.3 | 2.8 ± 0.1 |

P4 | 3.4 ± 0.2 | 121.1 ± 10.9 | 3.6 ± 0.2 |

T1 | 21.4 ± 0.7 | 316.9 ± 2.3 | 2.8 ± 0.2 |

T2 | 16.7 ± 0.6 | 215.0 ± 3.0 | 3.2 ± 0.2 |

T3 | 26.3± 1.9 | 402.9 ± 7.9 | 4.1 ± 0.1 |

**Table 3.**Overview of energy-relevant process parameters which were used in the building of all models.

Parameter | Description | Unit | Process Parameter |
---|---|---|---|

P | Laser Power | W | 275 |

v | Scan speed | mm/s | 805 |

d | Hatch spacing | µm | 120 |

t | Layer thickness | µm | 30 |

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**MDPI and ACS Style**

Weißmann, V.; Wieding, J.; Hansmann, H.; Laufer, N.; Wolf, A.; Bader, R. Specific Yielding of Selective Laser-Melted Ti6Al4V Open-Porous Scaffolds as a Function of Unit Cell Design and Dimensions. *Metals* **2016**, *6*, 166.
https://doi.org/10.3390/met6070166

**AMA Style**

Weißmann V, Wieding J, Hansmann H, Laufer N, Wolf A, Bader R. Specific Yielding of Selective Laser-Melted Ti6Al4V Open-Porous Scaffolds as a Function of Unit Cell Design and Dimensions. *Metals*. 2016; 6(7):166.
https://doi.org/10.3390/met6070166

**Chicago/Turabian Style**

Weißmann, Volker, Jan Wieding, Harald Hansmann, Nico Laufer, Andreas Wolf, and Rainer Bader. 2016. "Specific Yielding of Selective Laser-Melted Ti6Al4V Open-Porous Scaffolds as a Function of Unit Cell Design and Dimensions" *Metals* 6, no. 7: 166.
https://doi.org/10.3390/met6070166