# 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 and Continuous Dynamical Systems - B, 2021, 26 (9) : 4999-5021. doi: 10.3934/dcdsb.2020322
##### References:
 [1] N. C. Admal and E. B. Tadmor, A unified interpretation of stress in molecular systems, Journal of elasticity, 100 (2010), 63-143.  doi: 10.1007/s10659-010-9249-6. [2] N. C. Admal, J. Marian and G. Po, The atomistic representation of first strain-gradient elastic tensors, J. Mech. Phys. Solids, 99 (2017), 93-115.  doi: 10.1016/j.jmps.2016.11.005. [3] N. C. Admal and E. B. Tadmor, Material fields in atomistics as pull-backs of spatial distributions, J. Mech. Phys. Solids, 89 (2016), 59-76.  doi: 10.1016/j.jmps.2016.01.006. [4] I. Bitsanis, J. J. Magda, M. Tirrell and H. T. Davis, Molecular dynamics of flow in micropores, The Journal of chemical physics, 87 (1987), 1733-1750.  doi: 10.1063/1.453240. [5] Y. Chen and A. Diaz, Physical foundation and consistent formulation of atomic-level fluxes in transport processes, Phys. Rev. E, 98 (2018), 052113. doi: 10.1103/PhysRevE.98.052113. [6] Y. Chen, The origin of the distinction between microscopic formulas for stress and Cauchy stress, Europhysics Letters, 116 (2016), 34003. doi: 10.1209/0295-5075/116/34003. [7] R. Clausius, Xvi. on a mechanical theorem applicable to heat, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 40 (1870), 122-127.  doi: 10.1080/14786447008640370. [8] T. J. Delph, Local stresses and elastic constants at the atomic scale, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 461 2005, 1869-1888. doi: 10.1098/rspa.2004.1421. [9] R. M. Elder, W. D. Mattson and T. W. Sirk, Origins of error in the localized virial stress, Chemical Physics Letters, 731 (2019), 136580. doi: 10.1016/j.cplett.2019.07.008. [10] T. Hao and Z. M. Hossain, Atomistic mechanisms of crack nucleation and propagation in amorphous silica, Phys. Rev. B, 100 (2019), 014204. doi: 10.1103/PhysRevB.100.014204. [11] R. J. Hardy, Atomistic formulas for local properties in systems with many-body interactions, The Journal of Chemical Physics, 145 (2016), 204103. doi: 10.1063/1.4967872. [12] R. J. Hardy, Formulas for determining local properties in molecular-dynamics simulations: Shock waves, The Journal of Chemical Physics, 76 (1982), 622-628.  doi: 10.1063/1.442714. [13] J. H. Irving and J. G. Kirkwood, The statistical mechanical theory of transport processes. iv. the equations of hydrodynamics, The Journal of chemical physics, 18 (1950), 817-829.  doi: 10.1063/1.1747782. [14] N. Kalyanasundaram, M. Wood, J. B. Freund and H. T. Johnson, Stress evolution to steady state in ion bombardment of silicon, Mechanics Research Communications, 35 (2008), 50-56.  doi: 10.1016/j.mechrescom.2007.08.009. [15] L. T. Kong, Phonon dispersion measured directly from molecular dynamics simulations, Computer Physics Communications, 182 (2011), 2201-2207.  doi: 10.1016/j.cpc.2011.04.019. [16] L. T. Kong, G. Bartels, C. Campañá, C. Denniston and M. H. Müser, Implementation of green's function molecular dynamics: An extension to lammps, Computer Physics Communications, 180 (2009), 1004-1010.  doi: 10.1016/j.cpc.2008.12.035. [17] J. C. Maxwell, On reciprocal figures, frames, and diagrams of forces, Cambridge University Press, 2011, 161-207. doi: 10.1017/CBO9780511710377.014. [18] A. I. Murdoch and D. Bedeaux, On the physical interpretation of fields in continuum mechanics, International Journal of Engineering Science, 31 (1993), 1345-1373.  doi: 10.1016/0020-7225(93)90002-C. [19] A. I. Murdoch and D. Bedeaux, Continuum equations of balance via weighted averages of microscopic quantities, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 445 (1994), 157-179. doi: 10.1098/rspa.1994.0054. [20] W. Noll, Die herleitung der grundgleichungen der thermomechanik der kontinua aus der statistischen mechanik, J. Rational Mech. Anal., 4 (1955), 627-646. [21] R. Parthasarathy, A. Misra and L. Ouyang, Finite-temperature stress calculations in atomic models using moments of position, Journal of Physics: Condensed Matter, 30 (2018), 265901. doi: 10.1088/1361-648X/aac52f. [22] E. R. Smith, D. M. Heyes and D. Dini, Towards the Irving-Kirkwood limit of the mechanical stress tensor, The Journal of Chemical Physics, 146 (2017), 224109. doi: 10.1063/1.4984834. [23] E. R. Smith, P. E. Theodorakis, R. V. Craster and O. K. Matar, Moving contact lines: Linking molecular dynamics and continuum-scale modeling, Langmuir, 34 (2018), 12501-12518.  doi: 10.1021/acs.langmuir.8b00466. [24] E. B. Tadmor and R. E. Miller, Modeling Materials: Continuum, Atomistic and Multiscale Techniques, Cambridge University Press, 2011.  doi: 10.1017/CBO9781139003582. [25] D. H. Tsai, The virial theorem and stress calculation in molecular dynamics, The Journal of Chemical Physics, 70 (1979), 1375-1382.  doi: 10.1063/1.437577. [26] J. Z. Yang, X. Wu and X. Li, A generalized irving-kirkwood formula for the calculation of stress in molecular dynamics models, The Journal of Chemical Physics, 137 (2012), 134104. doi: 10.1063/1.4755946. [27] J. Z. Yang, C. Mao, X. Li and C. Liu, On the cauchy-born approximation at finite temperature, Computational Materials Science, 99 (2015), 21-28.  doi: 10.1016/j.commatsci.2014.11.030. [28] X. W. Zhou, R. B. Sills, D. K. Ward and R. A. Karnesky, Atomistic calculations of dislocation core energy in aluminium, Phys. Rev. B, 95 (2017), 054112. doi: 10.1103/PhysRevB.95.054112. [29] J. A. Zimmerman, R. E. Jones and J. A. Templeton, A material frame approach for evaluating continuum variables in atomistic simulations, Journal of Computational Physics, 229 (2010), 2364-2389.  doi: 10.1016/j.jcp.2009.11.039. [30] J. A. Zimmerman, E. B. WebbⅢ, J. J. Hoyt, R. E. Jones, P. Klein and D. J. Bammann, Calculation of stress in atomistic simulation, Modelling and Simulation in Materials Science and Engineering, 12 (2004), S319. doi: 10.1088/0965-0393/12/4/S03.

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##### References:
 [1] N. C. Admal and E. B. Tadmor, A unified interpretation of stress in molecular systems, Journal of elasticity, 100 (2010), 63-143.  doi: 10.1007/s10659-010-9249-6. [2] N. C. Admal, J. Marian and G. Po, The atomistic representation of first strain-gradient elastic tensors, J. Mech. Phys. Solids, 99 (2017), 93-115.  doi: 10.1016/j.jmps.2016.11.005. [3] N. C. Admal and E. B. Tadmor, Material fields in atomistics as pull-backs of spatial distributions, J. Mech. Phys. Solids, 89 (2016), 59-76.  doi: 10.1016/j.jmps.2016.01.006. [4] I. Bitsanis, J. J. Magda, M. Tirrell and H. T. Davis, Molecular dynamics of flow in micropores, The Journal of chemical physics, 87 (1987), 1733-1750.  doi: 10.1063/1.453240. [5] Y. Chen and A. Diaz, Physical foundation and consistent formulation of atomic-level fluxes in transport processes, Phys. Rev. E, 98 (2018), 052113. doi: 10.1103/PhysRevE.98.052113. [6] Y. Chen, The origin of the distinction between microscopic formulas for stress and Cauchy stress, Europhysics Letters, 116 (2016), 34003. doi: 10.1209/0295-5075/116/34003. [7] R. Clausius, Xvi. on a mechanical theorem applicable to heat, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 40 (1870), 122-127.  doi: 10.1080/14786447008640370. [8] T. J. Delph, Local stresses and elastic constants at the atomic scale, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 461 2005, 1869-1888. doi: 10.1098/rspa.2004.1421. [9] R. M. Elder, W. D. Mattson and T. W. Sirk, Origins of error in the localized virial stress, Chemical Physics Letters, 731 (2019), 136580. doi: 10.1016/j.cplett.2019.07.008. [10] T. Hao and Z. M. Hossain, Atomistic mechanisms of crack nucleation and propagation in amorphous silica, Phys. Rev. B, 100 (2019), 014204. doi: 10.1103/PhysRevB.100.014204. [11] R. J. Hardy, Atomistic formulas for local properties in systems with many-body interactions, The Journal of Chemical Physics, 145 (2016), 204103. doi: 10.1063/1.4967872. [12] R. J. Hardy, Formulas for determining local properties in molecular-dynamics simulations: Shock waves, The Journal of Chemical Physics, 76 (1982), 622-628.  doi: 10.1063/1.442714. [13] J. H. Irving and J. G. Kirkwood, The statistical mechanical theory of transport processes. iv. the equations of hydrodynamics, The Journal of chemical physics, 18 (1950), 817-829.  doi: 10.1063/1.1747782. [14] N. Kalyanasundaram, M. Wood, J. B. Freund and H. T. Johnson, Stress evolution to steady state in ion bombardment of silicon, Mechanics Research Communications, 35 (2008), 50-56.  doi: 10.1016/j.mechrescom.2007.08.009. [15] L. T. Kong, Phonon dispersion measured directly from molecular dynamics simulations, Computer Physics Communications, 182 (2011), 2201-2207.  doi: 10.1016/j.cpc.2011.04.019. [16] L. T. Kong, G. Bartels, C. Campañá, C. Denniston and M. H. Müser, Implementation of green's function molecular dynamics: An extension to lammps, Computer Physics Communications, 180 (2009), 1004-1010.  doi: 10.1016/j.cpc.2008.12.035. [17] J. C. Maxwell, On reciprocal figures, frames, and diagrams of forces, Cambridge University Press, 2011, 161-207. doi: 10.1017/CBO9780511710377.014. [18] A. I. Murdoch and D. Bedeaux, On the physical interpretation of fields in continuum mechanics, International Journal of Engineering Science, 31 (1993), 1345-1373.  doi: 10.1016/0020-7225(93)90002-C. [19] A. I. Murdoch and D. Bedeaux, Continuum equations of balance via weighted averages of microscopic quantities, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 445 (1994), 157-179. doi: 10.1098/rspa.1994.0054. [20] W. Noll, Die herleitung der grundgleichungen der thermomechanik der kontinua aus der statistischen mechanik, J. Rational Mech. Anal., 4 (1955), 627-646. [21] R. Parthasarathy, A. Misra and L. Ouyang, Finite-temperature stress calculations in atomic models using moments of position, Journal of Physics: Condensed Matter, 30 (2018), 265901. doi: 10.1088/1361-648X/aac52f. [22] E. R. Smith, D. M. Heyes and D. Dini, Towards the Irving-Kirkwood limit of the mechanical stress tensor, The Journal of Chemical Physics, 146 (2017), 224109. doi: 10.1063/1.4984834. [23] E. R. Smith, P. E. Theodorakis, R. V. Craster and O. K. Matar, Moving contact lines: Linking molecular dynamics and continuum-scale modeling, Langmuir, 34 (2018), 12501-12518.  doi: 10.1021/acs.langmuir.8b00466. [24] E. B. Tadmor and R. E. Miller, Modeling Materials: Continuum, Atomistic and Multiscale Techniques, Cambridge University Press, 2011.  doi: 10.1017/CBO9781139003582. [25] D. H. Tsai, The virial theorem and stress calculation in molecular dynamics, The Journal of Chemical Physics, 70 (1979), 1375-1382.  doi: 10.1063/1.437577. [26] J. Z. Yang, X. Wu and X. Li, A generalized irving-kirkwood formula for the calculation of stress in molecular dynamics models, The Journal of Chemical Physics, 137 (2012), 134104. doi: 10.1063/1.4755946. [27] J. Z. Yang, C. Mao, X. Li and C. Liu, On the cauchy-born approximation at finite temperature, Computational Materials Science, 99 (2015), 21-28.  doi: 10.1016/j.commatsci.2014.11.030. [28] X. W. Zhou, R. B. Sills, D. K. Ward and R. A. Karnesky, Atomistic calculations of dislocation core energy in aluminium, Phys. Rev. B, 95 (2017), 054112. doi: 10.1103/PhysRevB.95.054112. [29] J. A. Zimmerman, R. E. Jones and J. A. Templeton, A material frame approach for evaluating continuum variables in atomistic simulations, Journal of Computational Physics, 229 (2010), 2364-2389.  doi: 10.1016/j.jcp.2009.11.039. [30] J. A. Zimmerman, E. B. WebbⅢ, J. J. Hoyt, R. E. Jones, P. Klein and D. J. Bammann, Calculation of stress in atomistic simulation, Modelling and Simulation in Materials Science and Engineering, 12 (2004), S319. doi: 10.1088/0965-0393/12/4/S03.
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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