# A fluctuation method to calculate the third order elastic constants in crystalline solids

## Abstract

This paper derives exact expressions of the isothermal third order elastic constants (TOE) in crystalline solids in terms of the kinetic and potential energies of the system. These expressions reveal that the TOE constants consist of a Born component and a relaxation component. The Born component is simply the third derivative of the system's potential energy with respect to the deformation, while the relaxation component is related to the non-uniform rearrangements of the atoms when the system is subject to a macroscopic deformation. Further, based on the general expressions derived here, a direct (fluctuation) method of computing the isothermal TOE constants is developed. Numerical examples of using this fluctuation method are given to compute the TOE constants of single crystal iron.

- Authors:

- School of Materials Science and Engineering, Beihang University, Beijing 100191 (China)
- Department of Civil and Environmental Engineering, Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208 (United States)

- Publication Date:

- OSTI Identifier:
- 22410254

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Applied Physics

- Additional Journal Information:
- Journal Volume: 117; Journal Issue: 20; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-8979

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; CALCULATION METHODS; DEFORMATION; ELASTICITY; FLUCTUATIONS; IRON; MONOCRYSTALS; POTENTIAL ENERGY; RELAXATION; SOLIDS

### Citation Formats

```
Chen, Zimu, and Qu, Jianmin, E-mail: j-qu@northwestern.edu.
```*A fluctuation method to calculate the third order elastic constants in crystalline solids*. United States: N. p., 2015.
Web. doi:10.1063/1.4921354.

```
Chen, Zimu, & Qu, Jianmin, E-mail: j-qu@northwestern.edu.
```*A fluctuation method to calculate the third order elastic constants in crystalline solids*. United States. doi:10.1063/1.4921354.

```
Chen, Zimu, and Qu, Jianmin, E-mail: j-qu@northwestern.edu. Thu .
"A fluctuation method to calculate the third order elastic constants in crystalline solids". United States. doi:10.1063/1.4921354.
```

```
@article{osti_22410254,
```

title = {A fluctuation method to calculate the third order elastic constants in crystalline solids},

author = {Chen, Zimu and Qu, Jianmin, E-mail: j-qu@northwestern.edu},

abstractNote = {This paper derives exact expressions of the isothermal third order elastic constants (TOE) in crystalline solids in terms of the kinetic and potential energies of the system. These expressions reveal that the TOE constants consist of a Born component and a relaxation component. The Born component is simply the third derivative of the system's potential energy with respect to the deformation, while the relaxation component is related to the non-uniform rearrangements of the atoms when the system is subject to a macroscopic deformation. Further, based on the general expressions derived here, a direct (fluctuation) method of computing the isothermal TOE constants is developed. Numerical examples of using this fluctuation method are given to compute the TOE constants of single crystal iron.},

doi = {10.1063/1.4921354},

journal = {Journal of Applied Physics},

issn = {0021-8979},

number = 20,

volume = 117,

place = {United States},

year = {2015},

month = {5}

}