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Development of Non-Destructive Dynamic Characterization Technique for MMCs: Predictions of Mechanical Properties for Al@Al_{2}O_{3} Composites

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## Abstract

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## 1. Introduction

## 2. Materials

#### 2.1. Experimentation

#### 2.2. Characterization

## 3. Methods

#### 3.1. Non-Invasive PSD-Based Experimental Setup

#### 3.2. Response Measurement

#### 3.3. Evaluation of Young’s Modulus

#### 3.4. Numerical Modeling

^{®}v. 6.1., Stockholm, Sweden). Material properties of modeled geometry have been selected from the previous literature and assigned to the individual components of the composite model [27]. The listed geometrical parameters of the composite for homogenization are shown in Table 1.

#### 3.4.1. Create a Unit Cell Model

#### 3.4.2. Homogenization Analysis

- A macroscopic constitutive material model has been made after integrating the Equation (2).
- Conducting NMTs on a unit cell model using Finite Element mesh
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- Provide the macroscopic displacement gradient H followed by the relative displacement vector ${q}^{\left[J\right]}$ of the external points,$${q}^{\left[J\right]}=H\xb7{L}^{\left[J\right]}$$
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- The BVP equations are illustrated as per the following equations.$$\left.\begin{array}{c}\nabla y\xb7P=0\\ F=\nabla y\xb7\omega \end{array}\right\}in\phantom{\rule{4.pt}{0ex}}{y}_{0}$$$$P=\tau \left(F\right),$$$$\left.\begin{array}{c}{\omega}^{\left[J\right]}-{\omega}^{[-J]}=\tilde{H}\xb7{L}^{\left[J\right]}\\ {\tilde{T}}^{\left[J\right]}=\frac{1}{\partial {{y}_{0}}^{\left[J\right]}}{\int}_{\partial {{y}_{0}}^{\left[J\right]}}^{.}P.{E}^{\left[J\right]}ds\end{array}\right\}on\phantom{\rule{4.pt}{0ex}}\partial {{y}_{0}}^{\left[J\right]}$$On imposing,$${\omega}^{\left[J\right]}-{\omega}^{[-J]}={q}^{\left[J\right]},$$
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- Obtain 1st-Piola–Kirchhoff (PK) or nominal stress ${P}_{iJ}$ at each incremental step n for all loading patterns $\alpha $ by directly solving the extended micro-scale BVP with the response force vector,$${P}_{iJ}=\frac{{R}_{i}\left[J\right]}{\left|\partial {{Y}_{0}}^{\left[J\right]}\right|},$$

- Identifying macroscopic parameters for materials
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- Calculate the macro-scale 2nd PK stress using NMT data.$${S}^{[n,\alpha ]}=\left({\left({F}^{[n,\alpha ]}\right)}^{-1}{P}^{[n,\alpha ]}\right),$$
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- Create a function with the material parameters p
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- Identify the macroscopic material parameters p by solving the obtained algebraic Equation (8), where G and b are the coefficient matrix.$$Gp=b.$$
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- Macroscopic FE-analysis

Solve the macro-scale BVP using the assumed constitutive model with identified material parameters

#### 3.4.3. Numerical Procedure for Macro Model

## 4. Results

#### 4.1. SEM, EDS and XRD Analysis of Al@Al${}_{2}$O${}_{3}$ Composite Powder

#### 4.2. Experimental Studies

#### 4.3. Comparison of Young’s Modulus for Validation

#### 4.4. Numerical Studies

#### 4.4.1. Modal Analysis

#### 4.4.2. Frequency Response

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**(

**a**) Low magnification and (

**b**) high magnification SEM micrographs of Al@Al${}_{2}$O${}_{3}$ composite powder.

**Figure 8.**Transfer function of the different composition of the Al${}_{2}$O${}_{3}$ in Aluminum matrix.

**Figure 10.**Mode shapes of Al-15 vol.% Al${}_{2}$O${}_{3}$ composite at model frequencies: (

**a**) Mode 1—456 kHz, (

**b**) Mode 2—457 kHz, (

**c**) Mode 3—522.6 kHz, (

**d**) Mode 4—711.9 kHz, (

**e**) Mode 5—712 kHz, and (

**f**) Mode 6—839.6 kHz. Color bar shows the variation in displacement in the sample.

**Figure 11.**Frequency response of the sample with the Al${}_{2}$O${}_{3}$ NPs content in the composite material.

Parameters | Sample | ||
---|---|---|---|

Al-5%Al${}_{2}$O${}_{3}$ | Al-10%Al${}_{2}$O${}_{3}$ | Al-15%Al${}_{2}$O${}_{3}$ | |

Radius of Al${}_{2}$O${}_{3}$ NP, R (nm) | 25 | 25 | 25 |

Volume of Al${}_{2}$O${}_{3}$ NP, Vn (nm${}^{3}$) | 65,416.67 | 65,416.67 | 65,416.67 |

Volume ratio, (Vn/[Vn+Vm]) | 0.05 | 0.1 | 0.15 |

Volume of Al matrix, Vm (nm${}^{3}$) | 621,458.3 | 588,750 | 55,6041.7 |

Side length of unit cell, a (nm) | 85.33699 | 83.81279 | 82.23104 |

Parameters | Sample | ||
---|---|---|---|

Al-5%Al${}_{2}$O${}_{3}$ | Al-10%Al${}_{2}$O${}_{3}$ | Al-15%Al${}_{2}$O${}_{3}$ | |

Young’s Modulus (GPa) | 131.97 | 151.68 | 170.76 |

Poison Ratio | 0.3416 | 0.332 | 0.3288 |

Density (g/cm${}^{3}$) | 2.76 | 2.83 | 2.89 |

Radius (mm) | 12.5 | 12.5 | 12.5 |

Thickness (mm) | 10 | 10 | 10 |

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**MDPI and ACS Style**

Pingale, A.D.; Gautam, D.; Owhal, A.; Deshwal, D.; Belgamwar, S.U.; Rao, V.K.P.
Development of Non-Destructive Dynamic Characterization Technique for MMCs: Predictions of Mechanical Properties for Al@Al_{2}O_{3} Composites. *NDT* **2023**, *1*, 22-34.
https://doi.org/10.3390/ndt1010003

**AMA Style**

Pingale AD, Gautam D, Owhal A, Deshwal D, Belgamwar SU, Rao VKP.
Development of Non-Destructive Dynamic Characterization Technique for MMCs: Predictions of Mechanical Properties for Al@Al_{2}O_{3} Composites. *NDT*. 2023; 1(1):22-34.
https://doi.org/10.3390/ndt1010003

**Chicago/Turabian Style**

Pingale, Ajay D., Diplesh Gautam, Ayush Owhal, Dhruv Deshwal, Sachin U. Belgamwar, and Venkatesh K. P. Rao.
2023. "Development of Non-Destructive Dynamic Characterization Technique for MMCs: Predictions of Mechanical Properties for Al@Al_{2}O_{3} Composites" *NDT* 1, no. 1: 22-34.
https://doi.org/10.3390/ndt1010003