# Components of the Shear Modulus and Their Dependence on Temperature and Plastic Deformation of a Metallic Glass

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## Abstract

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## 1. Introduction

## 2. Experimental

## 3. Results and Discussion

#### 3.1. Shear Modulus Components of Predeformed Samples in the Initial and Relaxed States

#### 3.2. Separation of Non-Relaxation Components of the Shear Modulus

#### 3.3. Separation of the Relaxation Component and Its Dependence on Plastic Deformation

#### 3.4. Dependence of the Shear Modulus Components on the Defect Concentration and Plastic Deformation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Temperature dependences of the normalized shear modulus g in the initial (run 1) and relaxed (run 2) states of a sample plastically predeformed by ${\epsilon}_{pl}=20\%$. The solid and dashed lines give temperature dependences of the sum of harmonic, anharmonic and electronic components denoted as ${g}_{norel}$ for the initial state (run 1) and after relaxation (run 2). The calorimetric glass transition temperature ${T}_{g}$ is indicated by the arrow. The error is less than symbols’ size.

**Figure 2.**Temperature dependences of the first derivative of the normalized shear modulus over temperature $dg/dT$ in the initial (run 1) and relaxed (run 2) states of a sample predeformed by ${\epsilon}_{pl}=20\%$. The derivative of the non-relaxation component is given by the straight line. The inset gives temperature dependence of the second derivative ${d}^{2}g/d{T}^{2}$ in the relaxed (run 2) state. The glass transition temperature ${T}_{g}$ and the temperature of structural relaxation onset ${T}_{sr}$ are indicated by the arrows.

**Figure 3.**Temperature dependences of the relaxation component of the normalized shear modulus of bulk glassy Zr${}_{46}$Cu${}_{45}$Al${}_{7}$Ti${}_{2}$ in the undeformed state (${\epsilon}_{pl}=0$) and after deformation by different ${\epsilon}_{pl}$ as indicated. The error is about the symbols’ size.

**Figure 4.**The integral magnitude of the relaxation contribution to the shear modulus ${S}_{rel}$ calculated with Equation (9) and the change of the defect concentration $\Delta c$ determined according to Equation (8) as a function of plastic deformation ${\epsilon}_{pl}$. The lines give square least-fit approximations. It is seen that $\Delta Srel$ is proportional to $\Delta c$.

**Figure 5.**Dependence of the anharmonic coefficient ${\alpha}_{anh}$ and electronic coefficient ${\alpha}_{el}$ defined by Equation (1) on plastic deformation ${\epsilon}_{pl}$. The solid lines give least-square-fits of the data points. The errors are about the symbols’ size.

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**MDPI and ACS Style**

Makarov, A.; Kretova, M.; Afonin, G.; Kobelev, N.; Khonik, V. Components of the Shear Modulus and Their Dependence on Temperature and Plastic Deformation of a Metallic Glass. *Metals* **2022**, *12*, 1964.
https://doi.org/10.3390/met12111964

**AMA Style**

Makarov A, Kretova M, Afonin G, Kobelev N, Khonik V. Components of the Shear Modulus and Their Dependence on Temperature and Plastic Deformation of a Metallic Glass. *Metals*. 2022; 12(11):1964.
https://doi.org/10.3390/met12111964

**Chicago/Turabian Style**

Makarov, Andrei, Marina Kretova, Gennadii Afonin, Nikolai Kobelev, and Vitaly Khonik. 2022. "Components of the Shear Modulus and Their Dependence on Temperature and Plastic Deformation of a Metallic Glass" *Metals* 12, no. 11: 1964.
https://doi.org/10.3390/met12111964