# Synthesis and Mechanical Characterization of Binary and Ternary Intermetallic Alloys Based on Fe-Ti-Al by Resonant Ultrasound Vibrational Methods

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}) and elasticity modulus of 205 GPa [5,6], make them potential candidates for the manufacture of structural parts, including in aggressive environments for applications like heat exchangers, engine or turbine components.

_{0}, L2

_{1}, B2) and their prototypes (CsCl, ${\mathrm{MgZn}}_{2}$, AuCu, ${\mathrm{HfGa}}_{2}$) are also given. The CsCl (B2) Structure is diatomic with equal numbers of atoms of each type. The L1

_{0}structure is a tetragonal distortion of the fcc structure. In the Heusler (L2

_{1}) Structure, all of the atoms are located on the sites of a body centered cubic lattice. The Mg atoms of the ${\mathrm{MgZn}}_{2}$ hexagonal Laves Structure (C14) are located on the sites of the hexagonal diamond structure.

## 2. Experimental Techniques and Configurations

#### 2.1. Fabrication of Binary and Ternary Alloy Samples

_{x}Fe

_{y}, Ti

_{x}Al

_{y}, Fe

_{x}Al

_{y}, Ti

_{x}Al

_{y}Fe

_{z}, where x, y, z are the percentage weight concentrations in the chemical formula) presented in this work were synthesized using an arc furnace (Compact Arc melter MAM-1, Edmund Büler GMBH, Hechingen, Germany), operating under vacuum and flush of argon to avoid oxidation. To ensure uniformity and complete fusion of the alloys, a vacuum was applied, then argon was flushed and this procedure was repeated five times for each alloy. The alloys were prepared from pellets of pure elements, with purity levels for Fe → 99.98%, Ti → 99.99%, Al → 99.98% (the concentrations are in % weight). All the samples were obtained in disc form (with diameter of 2.0 cm) using the same single copper mold. It should be noted that two types of geometries can be obtained from the furnace molds, in disc or baguette form.

#### 2.2. Preparation of the Surfaces and Size of the Alloy Disc Samples

## 3. Experimental Methods for Vibration Spectrum Data Acquisition

#### 3.1. Configuration Using Shear Wave Transducers

## 4. Configuration with Piezoelectric Discs

## 5. The Ingredients for Estimating Alloy Disc Sample Elastic Moduli

#### 5.1. First Model for Solving the Direct Problem

#### 5.2. Second Model for Solving the Direct Problem

#### 5.3. Solving the Inverse Problem of Vibrational Spectroscopy

#### 5.4. How to Identify and Classify the Modes of Vibration in the Response Spectrum of the Specimen

## 6. Results

#### 6.1. Identification and Classification of Vibration Modes in the Spectrum and Validation of the Interaction Model Used for Inversion

#### 6.2. Validation of the Interaction Model for the Inverse Problem Using Synthetic Data

**k**-point mesh of $36\times 36\times 36$ [41] to achieve reasonable precision therefore necessitating long hours of computation.

#### 6.3. Comparison between the Two Experimental Resonance Ultrasound Vibrational Methods

## 7. Discussion

#### 7.1. Comparison with Some Published Results in the Literature

#### 7.2. The Influence of the Formed Phases and Their Crystallographic Structures on the Mechanical Properties

^{3}). The parameters that fitted the exponential equation were as follows; $a=2999$, $b=2.1e2$ (with 95% confidence bounds). Therefore, with the knowledge of iron concentration in the alloy, the density of the FeAl alloy can be determined from the curve or predicted from the equation. This works perfectly for the FeAl alloys in Table 3 and Table 4.

^{3}) in order to fit the data to an exponential equation of the type $E(\rho )=a-bexp(-c\rho )$ (constants obtained $a=203.6$, $b=161.8$, $c=5.8e-4$, with 95% confidence bounds). In summary, if the Fe concentration in the FeAl alloy is known, then the density of the alloy can be retrieved from the projection on the curve in Figure 7 or its corresponding equation. Then, its Young’s modulus can be recovered from the projection of the density on the curve in Figure 8 or its exponential equation.

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The first vibration spectroscopy experiment using shear wave transducers. The diagram depicts the setup employed.

**Figure 2.**The second vibration spectroscopy experimental setup. (

**a**) diagram of the setup employed; (

**b**) upper panel, side view photograph of the piezoelectric exciter and sensor, lower panel, front view of the thick disc alloy specimen placed between the piezoelectric exciter and the sensor discs.

**Figure 3.**The natural vibration modal deformations of the thick discs, (

**a**) the first flexural mode; (

**b**) the first compressional mode.

**Figure 4.**Comparison between the spectra obtained using the shear wave and the piezoelectric disc transducers for the same disc ternary sample ${\mathrm{Fe}}_{63}{\mathrm{Ti}}_{29}{\mathrm{Al}}_{8}$ (composition given in percentage by weight).

**Figure 5.**The vibration spectrum obtained using the piezoelectric disc transducers for the ternary disc sample ${\mathrm{Fe}}_{52}{\mathrm{Ti}}_{22}{\mathrm{Al}}_{26}$ (composition given in percentage by weight). (

**a**) point concentrated force excitation setup for the longitudinal modes only; (

**b**) edge excitation for all modes; (

**c**) dotted lines are the 3D finite element method computation of the disc response picked at a node on the opposite face where excitation is applied. The first shear (SH) and Longitudinal (Long.) modes are also shown; (

**d**) edge excitation responses showing all the modes with two different values of the pair (E, $\nu $).

**Figure 6.**The variation of Young’s modulus, retrieved from data generated by the setup employing the thin piezoelectric disc transducers, versus the ternary Fe-Ti-Al alloy density.

**Figure 7.**The variation of the density of the binary FeAl alloy as a function of the iron content. The data point shown by the pentagram is from reference [6].

**Figure 8.**The variation of the Young’s modulus of the binary alloy FeAl as a function of its density. The data were generated by the setup employing the thin PZT disc transducers. The data point shown by the pentagram is from reference [6].

Base Alloy Studied | Phases | Prototype | Space Group | Crystal Structure [12] | References |
---|---|---|---|---|---|

FeTi | FeTi | CsCl | Pm-3m | B2 | [10,13,14] |

Fe_{2}Ti | MgZn_{2} | P6_{3}6/mmc | C14 | ||

TiAl | TiAl | AuCu | P4/mmm | L1_{0} | [13,14,15] |

TiAl_{2} | HfGa_{2} | I4_{1}/amd | - | ||

FeAl | FeAl | CsCl | Pm-3m | B2 | [10,13,14] |

FeAl_{2} | FeAl_{2} | P1 | - | ||

FeTiAl | Fe_{2}AlTi | - | Pm-3m | L2_{1} | [10,13,14] |

**Table 2.**Mechanical properties calculated by ab initio (density functional theory), and resonance frequencies of the single crystal metals computed using 3D finite element method (${F}_{1}$, ${F}_{2}$). ${E}_{1}^{r}$ and ${E}_{2}^{r}$ are the recovered Young’s moduli (in GPa) using the synthetic resonance frequencies (${F}_{1}$, ${F}_{2}$) and the second interaction model (Equation (2)).

Element | Density (Kg/m^{3}) | Young’s Modulus | Poisson Ratio (ν) | F_{1} (Hz) | F_{2} (Hz) | ${\mathit{E}}_{\mathbf{1}}^{\mathit{r}}$ | ${\mathit{E}}_{\mathbf{2}}^{\mathit{r}}$ |
---|---|---|---|---|---|---|---|

Fe | 7874 Ref. [45] | 212 Ref. [46] | 0.27 | 70480 | 110577 | 213.27 | 210.05 |

Ti | 4500 Ref. [47] | 114.6 Ref. [44,48] | 0.3 | 67932 | 109053 | 111.14 | 114.60 |

Al | 2707 Ref. [45] | 69.3 Ref. [45,49] | 0.3 | 68230 | 109530 | 67.27 | 69.36 |

**Table 3.**The resonance frequencies (${f}_{1}$, ${f}_{2}$) recovered from the vibration spectra obtained using piezoelectric disc transducers and the corresponding Young’s moduli (${E}_{1}$, ${E}_{2}$) retrieved using Equation (2). The final 3D FEM Young’s moduli (E) after adjustment to fit experimental resonance frequencies. The difference between ${E}_{2}$ and E is given in percentage. The † indicates ab initio calculated results found in the literature. The values of ${K}_{1}$ = 4.6356, ${K}_{2}$ = 7.3284, and the initial $\nu $ = 0.27.

Composition | Composition | ${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ | Density | ${\mathit{E}}_{1}$ | ${\mathit{E}}_{2}$ | E (Gpa) | ${\mathit{F}}_{1}$ (Hz) | ${\mathit{F}}_{2}$ (Hz) | Difference Percentage | Reference |
---|---|---|---|---|---|---|---|---|---|---|---|

(wt. %) | (at. %) | (Hz) | (Hz) | (Kg/m^{3}) | (GPa) | (GPa) | 3D FEM | 3D FEM | 3D FEM | ${\mathit{E}}_{1}$ − E (%) | E (GPa), $\mathit{\nu}$ |

${\mathrm{Fe}}_{47}{\mathrm{Ti}}_{23}{\mathrm{Al}}_{30}$ | ${\mathrm{Fe}}_{35}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{45}$ | 72,600 | 114,200 | 6469 | 185.92 | 184.04 | 184.04 | 72,682 | 114000 | 1.02 | |

${\mathrm{Fe}}_{50}{\mathrm{Ti}}_{23}{\mathrm{Al}}_{27}$ | ${\mathrm{Fe}}_{38}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{42}$ | 73,000 | 114,600 | 6580 | 191.20 | 188.54 | 188.54 | 72,942 | 114,440 | 1.41 | |

${\mathrm{Fe}}_{52}{\mathrm{Ti}}_{22}{\mathrm{Al}}_{26}$ | ${\mathrm{Fe}}_{40}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{40}$ | 73,800 | 116,000 | 6682 | 198.44 | 196.16 | 196.16 | 73,883 | 115,836 | 1.16 | |

${\mathrm{Fe}}_{57}{\mathrm{Ti}}_{22}{\mathrm{Al}}_{21}$ | ${\mathrm{Fe}}_{45}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{35}$ | 73,800 | 115,800 | 6748 | 199.31 | 197.42 | 197.42 | 73,665 | 115,637 | 0.96 | |

${\mathrm{Fe}}_{63}{\mathrm{Ti}}_{29}{\mathrm{Al}}_{8}$ | ${\mathrm{Fe}}_{55}{\mathrm{Ti}}_{30}{\mathrm{Al}}_{15}$ | 73,000 | 115,600 | 6888 | 200.14 | 200.82 | 200.82 | 73,665 | 115,575 | 0.34 | |

${\mathrm{Fe}}_{65}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{15}$ | ${\mathrm{Fe}}_{56}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{24}$ | 73,000 | 115,400 | 6925 | 201.22 | 201.20 | 201.20 | 73,450 | 115,238 | 0.01 | |

${\mathrm{Fe}}_{68}{\mathrm{Ti}}_{24}{\mathrm{Al}}_{8}$ | ${\mathrm{Fe}}_{60}{\mathrm{Ti}}_{25}{\mathrm{Al}}_{15}$ | 73,000 | 115,800 | 6972 | 202.59 | 203.97 | 203.97 | 73,704 | 115,638 | 0.68 | |

${\mathrm{Fe}}_{78}{\mathrm{Ti}}_{14}{\mathrm{Al}}_{8}$ | ${\mathrm{Fe}}_{70}{\mathrm{Ti}}_{15}{\mathrm{Al}}_{15}$ | 72,800 | 115,200 | 7263 | 209.89 | 209.56 | 209.56 | 73,196 | 114,038 | 0.16 | |

${\mathrm{Fe}}_{80}{\mathrm{Al}}_{20}$ | ${\mathrm{Fe}}_{66}{\mathrm{Al}}_{34}$ | 70,600 | 111,200 | 7412 | 201.44 | 199.56 | 199.56 | 70,706 | 110,933 | 0.94 | 200 Ref. [6] |

${\mathrm{Fe}}_{60}{\mathrm{Al}}_{40}$ | ${\mathrm{Fe}}_{42}{\mathrm{Al}}_{58}$ | 82,200 | 130,400 | 5330 | 196.37 | 197.73 | 197.73 | 82,997 | 130,216 | 0.69 | “ |

${\mathrm{Fe}}_{60}{\mathrm{Ti}}_{40}$ | ${\mathrm{Fe}}_{56}{\mathrm{Ti}}_{44}$ | 70,800 | 111,600 | 7050 | 192.69 | 191.57 | 191.57 | 71,033 | 111,445 | 0.58 | 191.66, $\nu $ = 0.287 Ref. [47] † |

${\mathrm{Fe}}_{50}{\mathrm{Ti}}_{50}$ | ${\mathrm{Fe}}_{46}{\mathrm{Ti}}_{54}$ | 70,000 | 110,000 | 6950 | 185.69 | 183.47 | 183.47 | 70,013 | 109,845 | 1.21 | 182.38, $\nu $ = 0.28 Ref. [50] † |

${\mathrm{Ti}}_{55}{\mathrm{Al}}_{45}$ | ${\mathrm{Ti}}_{40}{\mathrm{Al}}_{60}$ | 88,200 | 140,000 | 3880 | 164.58 | 165.92 | 165.92 | 89,109 | 139,805 | 0.81 | 160–176 Ref. [54] |

${\mathrm{Ti}}_{45}{\mathrm{Al}}_{55}$ | ${\mathrm{Ti}}_{32}{\mathrm{Al}}_{68}$ | 91,200 | 144,200 | 3609 | 163.37 | 163.70 | 163.70 | 91,774 | 143,986 | 0.02 | 161.99, $\nu $ = 0.265 Ref. [52] † |

**Table 4.**The resonance frequencies (${f}_{1}$, ${f}_{2}$) recovered from the vibration spectra obtained using the shear wave transducers and the corresponding Young’s moduli (${E}_{1}$, ${E}_{2}$) retrieved using Equation (2). The final 3D FEM Young’s moduli (E) after adjustment to fit experimental resonance frequencies. The difference between ${E}_{2}$ and E is given in percentage. The † indicate ab initio calculated results found in the literature. The values of ${K}_{1}$ = 4.6356, ${K}_{2}$ = 7.3284, and the initial $\nu $ = 0.27.

Composition | Composition | ${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ | Density | ${\mathit{E}}_{1}$ | ${\mathit{E}}_{2}$ | E (Gpa) | ${\mathit{F}}_{1}$ (Hz) | ${\mathit{F}}_{2}$ (Hz) | Difference Percentage | Reference |
---|---|---|---|---|---|---|---|---|---|---|---|

(wt. %) | (at. %) | (Hz) | (Hz) | (Kg/m^{3}) | (GPa) | (Gpa) | 3D FEM | 3D FEM | 3D FEM | ${\mathit{E}}_{2}$ − E (%) | E (GPa), $\mathit{\nu}$ |

${\mathrm{Fe}}_{47}{\mathrm{Ti}}_{23}{\mathrm{Al}}_{30}$ | ${\mathrm{Fe}}_{35}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{45}$ | 72,400 | 113,900 | 6469 | 184.80 | 183.10 | 184.80 | 72,512 | 137,700 | 0.92 | |

${\mathrm{Fe}}_{50}{\mathrm{Ti}}_{23}{\mathrm{Al}}_{27}$ | ${\mathrm{Fe}}_{38}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{42}$ | 73,600 | 114,800 | 6580 | 192.20 | 189.10 | 192.20 | 73,000 | 114,770 | 1.61 | |

${\mathrm{Fe}}_{52}{\mathrm{Ti}}_{22}{\mathrm{Al}}_{26}$ | ${\mathrm{Fe}}_{40}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{40}$ | 72,200 | 116,000 | 6682 | 189.90 | 196.10 | 189.90 | 73,900 | 115,936 | 3.26 | |

${\mathrm{Fe}}_{57}{\mathrm{Ti}}_{22}{\mathrm{Al}}_{21}$ | ${\mathrm{Fe}}_{45}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{35}$ | 73,200 | 115700 | 6748 | 197.10 | 197.08 | 197.10 | 73,600 | 115,500 | 0.01 | |

${\mathrm{Fe}}_{63}{\mathrm{Ti}}_{29}{\mathrm{Al}}_{8}$ | ${\mathrm{Fe}}_{55}{\mathrm{Ti}}_{30}{\mathrm{Al}}_{15}$ | 72,700 | 114,800 | 6888 | 198.50 | 198.05 | 198.50 | 73,150 | 114,700 | 0.23 | |

${\mathrm{Fe}}_{65}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{15}$ | ${\mathrm{Fe}}_{56}{\mathrm{Ti}}_{20}{\mathrm{Al}}_{24}$ | 72,600 | 115,000 | 6925 | 199.02 | 199.80 | 199.02 | 73,050 | 114,600 | 0.39 | |

${\mathrm{Fe}}_{68}{\mathrm{Ti}}_{24}{\mathrm{Al}}_{8}$ | ${\mathrm{Fe}}_{60}{\mathrm{Ti}}_{25}{\mathrm{Al}}_{15}$ | 72,800 | 116,800 | 6972 | 201.40 | 207.50 | 201.40 | 72,931 | 115,300 | 3.03 | |

${\mathrm{Fe}}_{78}{\mathrm{Ti}}_{14}{\mathrm{Al}}_{8}$ | ${\mathrm{Fe}}_{70}{\mathrm{Ti}}_{15}{\mathrm{Al}}_{15}$ | 72,600 | 114,700 | 7263 | 208.73 | 208.47 | 208.73 | 72,900 | 114,400 | 0.12 | |

${\mathrm{Fe}}_{80}{\mathrm{Al}}_{20}$ | ${\mathrm{Fe}}_{66}{\mathrm{Al}}_{34}$ | 70400 | 110,200 | 7412 | 200.03 | 196.30 | 200.03 | 70,650 | 110,800 | 1.86 | 200 Ref. [6] |

${\mathrm{Fe}}_{60}{\mathrm{Al}}_{40}$ | ${\mathrm{Fe}}_{42}{\mathrm{Al}}_{58}$ | 82,400 | 130,200 | 5330 | 197.30 | 197.10 | 197.30 | 82,840 | 130,000 | 0.10 | “ |

${\mathrm{Fe}}_{60}{\mathrm{Ti}}_{40}$ | ${\mathrm{Fe}}_{56}{\mathrm{Ti}}_{44}$ | 70,400 | 111400 | 7050 | 190.50 | 190.80 | 190.50 | 70740 | 111,000 | 0.16 | 191.66, $\nu $ = 0.287 Ref. [47] † |

${\mathrm{Fe}}_{50}{\mathrm{Ti}}_{50}$ | ${\mathrm{Fe}}_{46}{\mathrm{Ti}}_{54}$ | 69,200 | 108,700 | 6950 | 181.47 | 179.16 | 181.47 | 69,174 | 108,700 | 1.27 | 182.38, $\nu $ = 0.28 Ref. [50] † |

${\mathrm{Ti}}_{55}{\mathrm{Al}}_{45}$ | ${\mathrm{Ti}}_{40}{\mathrm{Al}}_{60}$ | 88,700 | 139,800 | 3880 | 166.45 | 165.40 | 166.45 | 88,521 | 139,270 | 0.63 | 160–176 Ref. [54] |

${\mathrm{Ti}}_{45}{\mathrm{Al}}_{55}$ | ${\mathrm{Ti}}_{32}{\mathrm{Al}}_{68}$ | 90,800 | 145,700 | 3609 | 162.60 | 166.90 | 162.60 | 91,460 | 143,000 | 2.64 | 161.99, $\nu $ = 0.265 Ref. [52] † |

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**MDPI and ACS Style**

Chanbi, D.; Ogam, E.; Amara, S.E.; Fellah, Z.E.A.
Synthesis and Mechanical Characterization of Binary and Ternary Intermetallic Alloys Based on Fe-Ti-Al by Resonant Ultrasound Vibrational Methods. *Materials* **2018**, *11*, 746.
https://doi.org/10.3390/ma11050746

**AMA Style**

Chanbi D, Ogam E, Amara SE, Fellah ZEA.
Synthesis and Mechanical Characterization of Binary and Ternary Intermetallic Alloys Based on Fe-Ti-Al by Resonant Ultrasound Vibrational Methods. *Materials*. 2018; 11(5):746.
https://doi.org/10.3390/ma11050746

**Chicago/Turabian Style**

Chanbi, Daoud, Erick Ogam, Sif Eddine Amara, and Z. E. A. Fellah.
2018. "Synthesis and Mechanical Characterization of Binary and Ternary Intermetallic Alloys Based on Fe-Ti-Al by Resonant Ultrasound Vibrational Methods" *Materials* 11, no. 5: 746.
https://doi.org/10.3390/ma11050746