# Computational Study of the Influence of α/β-Phase Ratio and Porosity on the Elastic Modulus of Ti-Based Alloy Foams

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{4})

_{2}CO

_{3}as a space-holder. Chen et al. [41] obtained a lower elastic modulus for Ti-6Al-4V foams via the electron beam melting method, in which the elastic modulus values were between 2.6 and 2.0 GPa for a porosity between 80.1 and 81.5%. Furthermore, Ti-6Al-4V foams synthetized by the selective laser melting method [41] exhibited a larger elastic modulus when compared to all Ti-based alloys. The latter is due to the pore shape being different from other foams, since they exhibited a diamond unit cell shape in the top and bottom views and an ellipsoid shape in the mantle view. Pore sizes were considered higher when compared to the rest of the ternary Ti-based alloys, being between 2500 and 4000 μm. The pore structure was designed using a Computer Aid Design (CAD) which allowed smooth surfaces to be obtained for the pore surfaces. The Gibson–Ashby model was applied, with the parameters being determined as α and n being equal to 1.5 and 2, respectively, in order to be acceptably fitted. On the other hand, Chen et al. [41,42] reported higher elastic values for foams having a lower porosity (43 to 71%), 55.9 to 9.7 GPa, respectively.

## 2. Computational Modeling and Simulation

#### 2.1. RVE-FEM Method

- (a)
- Microstructure Model Generation

- (i)
- First step, using the parameters listed in Table 1, the microstructures were generated. The β-phase amount was variated from 0 to 90.5%. Differences between designed and obtained β-phase were observed, with the highest difference being seen in the designed β-phase, showing 40%. The RVE model was designed to be at least five times larger than grain sizes. The computational procedure consisted of periodically and randomly inserting volumes of grains into a matrix cube until the desired volumetric fraction was achieved. Grains are considered as one type of phase, and the space between grains (grain boundary) is considered as a second type of phase (Figure 1a).
- (ii)
- Second step, the models were exported and loaded in the SpaceClaim CAD software (SpaceClaim Corporation, Version 2019 R3, Manufacturer, Ansys INC-SPACECLAIM CORP, Ciudad Concord, MA, USA).
- (iii)
- Third step, using the draw options, the cubic model was transformed into the cylinder model, where colors represented each phase, α or β (Figure 1b).

- (b)
- Porosity model generation

- (i)
- (ii)
- Second step, the models were exported and loaded in the SpaceClaim CAD software.
- (iii)
- Third step, using the draw options, the cubic model was transformed into cylinder models.
- (iv)
- For each of the seven porosity models, the α/β ratio phase was added (Table 3). There are 7 porosity models and 11 microstructure models, which gives 77 configurations to be simulated. Porosity values were taken from porosity reported for human bones [48,49,50,51]. The simulation parameters used in this study are listed in Table 4. The elastic modulus values for the α-phase and β-phase were obtained from first-principles calculations made for Ti-xTM (TM = V, Cr, Nb, Mo) and ternary Ti-15TM-yAl alloys [52]. The calculated alloy compositions are similar to that of the Ti-6Al-4V alloy; thus, the elastic modulus values for the α-phase and β-phase were considered.

#### 2.2. Image Analysis

#### 2.3. Simulations

## 3. Results and Discussion

#### 3.1. Effect of the Microstructure on E Values

#### 3.2. Effect That Microstructure and Porosity Have on E Values

_{f}), Equation (11); and (iii) aspect ratio (AR), Equation (12), where x

_{min}and x

_{max}are the smallest and largest pore size dimensions. The circularity and shape factor decreased when porosity increased from 0.712 (C), 0.700 (Ff) for a porosity of 29% to 0.583 (C), 0.600 (Ff) for a porosity of 53.4%, respectively (Figure 8b). The aspect ratio values changed from 1.752 to 2.319, and this is explained by the pore coalescence producing elongated pores (Figure 6).

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Analytical Models to Determine E Values

_{0}was considered the foam and bulk material density, respectively, and α and n are parameters that depend on pore structure. Those parameters vary as α between 0.1 and 4 and n between 1.8 and 2.2.

_{c}is the critical porosity at which E = 0, and m is a constant that depends on pore distribution geometry.

_{c}is the critical porosity, and a is a constant that depends on the packing geometry factor (a is equal to 1 for spherical-shaped pores).

_{f}= 4pA/PE

^{2}, A is the pore area, and PE is the experimental perimeter of the pore.

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**Figure 1.**(

**a**) Grain shape morphology and (

**b**) step sequence used to obtain cylinder models with microstructures composed of α- and β-phases. Phase one can be α or β, and this is similar for phase two.

**Figure 3.**Variation of E as a function of β-phase amount for two conditions: (i) β-phase as matrix (microstructure B) and (ii) α-phase as matrix (microstructure A). Moreover, elastic modulus values were calculated using the mixing rules.

**Figure 4.**Directional deformation in height (in mm) under compression for the bulk samples with β-phase amounts of (

**a**) 10%, (

**b**) 20%, (

**c**) 30%, (

**d**) 40%, (

**e**) 50%, (

**f**) 60%, (

**g**) 70%, (

**h**) 80%, and (

**i**) 90% for a microstructure composed as α-phase grains + intergranular β-phase (microstructure A).

**Figure 5.**Directional stress in height (in mm) under compression for the bulk samples with β-phase amounts of (

**a**) 10%, (

**b**) 20%, (

**c**) 30%, (

**d**) 40%, (

**e**) 50%, (

**f**) 60%, (

**g**) 70%, (

**h**) 80%, and (

**i**) 90% for a microstructure composed as α-phase grains + intergranular β-phase (microstructure A).

**Figure 6.**Transversal images of different models showing the pore coalescence. (

**a**) 29.4%, (

**b**) 34.9%, (

**c**) 39.6%, (

**d**) 44.2%, (

**e**) 49.3%, (

**f**) 52.5%, (

**g**) 53.4%.

**Figure 8.**(

**a**) Equivalent mean pore size and maximum pore size variation and (

**b**) pore shape characterization, circularity, shape factor, and aspect ratio as a function of porosity.

**Figure 9.**Directional deformation in height (in μm) under compression for the foam samples with 0% β-phase amount. (

**a**) 29.4%, (

**b**) 34.9%, (

**c**) 39.6%, (

**d**) 44.2%, (

**e**) 49.3%, (

**f**) 52.5%, (

**g**) 53.4%.

**Figure 10.**Directional stress in height (in mm) under compression for the foam samples with 0% β-phase amount. (

**a**) 29.4%, (

**b**) 34.9%, (

**c**) 39.6%, (

**d**) 44.2%, (

**e**) 49.3%, (

**f**) 52.5%, (

**g**) 53.4%.

Designed β-Phase Amount (v/v%) | Obtained β-Phase Amount (v/v%) | Difference between Designed and Obtained β-Phase |
---|---|---|

10 | 12.0 | 2.03 |

20 | 20.0 | 0. |

30 | 33.2 | 3.2 |

40 | 34.7 | −5.3 |

50 | 45.5 | −4.5 |

60 | 61.6 | 1.6 |

70 | 71.3 | 1.3 |

80 | 81.4 | 1.4 |

90 | 90.5 | 0.5 |

Morphological Parameters | Characteristic of Porosity |
---|---|

Distribution of porosity | Homogenous |

Shape pore | Sphere-cylinder |

Aspect ratio | 1.1 |

Distribution pore function | Normal distribution |

Average pore size | 250 μm |

Standard deviation | 50 μm |

Designed Porosity (v/v%) | Obtained (Or Real) Porosity (v/v%) | Difference between Designed and Obtained Porosity |
---|---|---|

29 | 29.4 | 0.4 |

34 | 34.8 | 0.8 |

38 | 37.0 | −1.0 |

43 | 41.8 | −1.2 |

47 | 49.2 | 2.2 |

52 | 52.5 | 0.5 |

56 | 53.4 | −2.6 |

**Table 4.**Mechanical parameters used in the simulations [52].

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## Share and Cite

**MDPI and ACS Style**

Aguilar, C.; Henriquez, J.; Salvo, C.; Alfonso, I.; Araya, N.; Muñoz, L.
Computational Study of the Influence of α/β-Phase Ratio and Porosity on the Elastic Modulus of Ti-Based Alloy Foams. *Materials* **2023**, *16*, 4064.
https://doi.org/10.3390/ma16114064

**AMA Style**

Aguilar C, Henriquez J, Salvo C, Alfonso I, Araya N, Muñoz L.
Computational Study of the Influence of α/β-Phase Ratio and Porosity on the Elastic Modulus of Ti-Based Alloy Foams. *Materials*. 2023; 16(11):4064.
https://doi.org/10.3390/ma16114064

**Chicago/Turabian Style**

Aguilar, Claudio, Javier Henriquez, Christopher Salvo, Ismelí Alfonso, Nicolas Araya, and Lisa Muñoz.
2023. "Computational Study of the Influence of α/β-Phase Ratio and Porosity on the Elastic Modulus of Ti-Based Alloy Foams" *Materials* 16, no. 11: 4064.
https://doi.org/10.3390/ma16114064