# American Institute of Mathematical Sciences

September  2021, 26(9): 4999-5021. doi: 10.3934/dcdsb.2020322

## A comparative study of atomistic-based stress evaluation

 1 School of Mathematics and Statistics, Wuhan University, Hubei Key Laboratory in Computational Mathematics (Wuhan University), Wuhan, Hubei, China 430072 2 School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, China 430072

* Corresponding author: J. Z. Yang

Received  May 2020 Revised  August 2020 Published  September 2021 Early access  November 2020

This paper presents a comparative study on several issues of the microscopic stress definitions. Firstly, we derived an Irving-Kirkwood formulation for Cauchy stress evaluation in Eulerian coordinates. We showed that quantities, such as density and momentum, should to be defined properly on microscopic level in order to guarantee the conservation relations on macroscopic level. Secondly, the relation between Cauchy and first Piola-Kirchhoff stress was investigated both theoretically and numerically. At zero temperature, classical pointwise relation between these two stress is satisfied both in Virial and Hardy formulation. While at finite temperature, temporal averaging is required to guarantee this relation for Virial formulation. For Hardy formulation, an additional term need to be included in the classical relation between the Cauchy stress and the first Piola-Kirchhoff stress. Meanwhle, the linear relation between the Cauchy stress and the first Piola-Kirchhoff stress with respect to the temperature are obtained in both Virial and Hardy formulations. The thermal expansion coefficients are also studied by using quasi-harmonic approximation. Thirdly, different from that in the Lagrangian coordinates case, where the time averaging procedure can be performed in a post-processing manner when the kernel function is separable, the stress evaluation in Eulerian system must be evaluated spatially and temporally at the same time, even in separable kernel case. This can be seen from the comparison of the two procedures. Numerical examples were provided to illustrate our investigations.

Citation: Shuyang Dai, Fengru Wang, Jerry Zhijian Yang, Cheng Yuan. A comparative study of atomistic-based stress evaluation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4999-5021. doi: 10.3934/dcdsb.2020322
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The averaged Virial stress of the Al systems under different temperature and deformation. From top to bottom panel, the Virial-PK stress $\langle\boldsymbol{\sigma}_{PK}^{Virial}\rangle$, the Virial-Cauchy stress $\langle\boldsymbol{\sigma}_{Cauchy}^{Virial}\rangle$ and the term $\langle\boldsymbol{\sigma}_{Cauchy}^{Virial}\rangle-\frac{1}{|\textbf{A}|}\langle\boldsymbol{\sigma}_{PK}^{Virial}\rangle\textbf{A}^{T}$(line Ⅲ) are shown respectively. For comparison we also plot the $\langle|V|^{-1}\sum_{i = 1}^{N}f_{i}\otimes \textbf{u}_{i}\rangle$(line Ⅰ) and $\langle|V|^{-1}\sum_{i = 1}^{N}m_{i}\textbf{v}_{i}\otimes \textbf{v}_{i}\rangle$(line Ⅱ) in the bottom panel pictures. The left and right panel show the results obtained based on different deformation matrix $\textbf{A}_{1}$ and $\textbf{A}_{2}$, respectively
The averaged Hardy stress of the $\alpha$-Fe systems under different temperature and deformation. From top to bottom panel, the Hardy-PK stress $\langle\boldsymbol{\sigma}_{PK}^{Hardy}\rangle$, the Hardy-Cauchy stress $\langle\boldsymbol{\sigma}_{Cauchy}^{Hardy}\rangle$ and the difference term $\langle\boldsymbol{\sigma}_{Cauchy}^{Hardy}\rangle-\frac{1}{|\textbf{A}|}\langle\boldsymbol{\sigma}_{PK}^{Hardy}\rangle\textbf{A}^{T}$ are shown respectively. The left and right panel show the results obtained based on different deformation matrix $\textbf{A}_{1}$ and $\textbf{A}_{2}$, respectively
Two additional terms in (3.24). In the right picture: Line Ⅰ is $\sum_{i}\langle m_i\textbf{v}_i\otimes\textbf{v}_i\rangle\varphi(\textbf{AX}-\textbf{AX}_i)$; Line Ⅱ is $\sum_{i}\langle\textbf{f}_i\otimes\textbf{u}_i\rangle\varphi(\textbf{AX}-\textbf{AX}_i)$; Line Ⅲ is $\sum_{i}\langle\textbf{f}_i\otimes\textbf{u}_i+m_i\textbf{v}_i\otimes\textbf{v}_i\rangle\varphi(\textbf{AX}-\textbf{AX}_i)$
$\widehat{\boldsymbol{\sigma}}_{Cauchy,11}(\textbf{x},t)-\boldsymbol{\sigma}_{Cauchy,11}^{GIK}(\textbf{x},t)$ under different temperature and deformation. From left to right, we use the kernel $\tau_{1}$ and $\tau_{2}$ respectively; From top to bottom, we apply the deformation $\textbf{A}_{0} = \textbf{I}, \textbf{A}_{1}$ to the system respectively
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