# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021176
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## On the Cauchy-Born approximation at finite temperature for alloys

 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  December 2020 Revised  May 2021 Early access July 2021

In this paper, we present the procedure of generalization and implementation of the Cauchy-Born approximation to the calculation of stress at finite temperature for alloy system in which the effects of inner displacement should be incorporated. With the help of quasi-harmonic approximation, a closed form of the first Piola-Kirchhoff stress is derived as a summation of pure deformation contribution and linear term due to thermal effects. For alloy system with periodic boundary condition, a further simplified formulation of stress based on some invariance constraints is derived in reciprocal space by using Fourier transformation, in which the temperature effect can be efficiently taking account. Several numerical examples are performed for various crystalline systems to validate our generalization procedure of finite temperature Cauchy-Born (FTCB) method for alloy.

Citation: Shuyang Dai, Fengru Wang, Jerry Zhijian Yang, Cheng Yuan. On the Cauchy-Born approximation at finite temperature for alloys. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021176
##### References:
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Neyertz, A general pressure tensor calculation for molecular dynamics simulations, Molecular Physics, 84 (1995), 577-595.  doi: 10.1080/00268979500100371.  Google Scholar [13] 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.  Google Scholar [14] G. Cicotti, D. Frenkel and I. McDonald, Simulation of Liquids and Solids. Molecular Dynamics and Monte Carlo Methods in Statistical Mechanics, North Holland, 1987. Google Scholar [15] 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.  Google Scholar [16] W. A. Curtin and R. E. Miller, Atomistic/continuum coupling in computational materials science, Model. Simul. Mater. Sci. Engrg., 11 (2003), R33–R68. doi: 10.1088/0965-0393/11/3/201.  Google Scholar [17] S. Dai, Y. Xiang and D. J. Srolovitz, Structure and energy of (111) low-angle twist boundaries in Al, Cu and Ni, Acta Materialia, 61 (2013), 1327-1337.  doi: 10.1016/j.actamat.2012.11.010.  Google Scholar [18] M. S. Daw and M. I. Baskes, Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals, Physical Review B, 29 (1984), 6443. doi: 10.1103/PhysRevB.29.6443.  Google Scholar [19] J. Du, C. Wang and T. Yu, Construction and application of multi-element EAM potential (Ni-Al-Re) in $\gamma$/$\gamma'$ Ni-based single crystal superalloys, Modelling and Simulation in Materials Science and Engineering, 21 (2013), 015007. Google Scholar [20] L. M. Dupuy, E. B. Tadmor, R. E. Miller and R. Phillips, Finite-temperature quasicontinuum: Molecular dynamics without all the atoms, Physical Review Letters, 95 (2005), 060202. doi: 10.1103/PhysRevLett.95.060202.  Google Scholar [21] W. E and P. Ming, Cauchy–born rule and the stability of crystalline solids: Static problems, Archive for Rational Mechanics and Analysis, 183 (2007), 241-297.  doi: 10.1007/s00205-006-0031-7.  Google Scholar [22] W. E and P. Ming, Cauchy-born rule and the stability of crystalline solids: Dynamic problems, Acta Mathematicae Applicatae Sinica, English Series, 23 (2007), 529-550.  doi: 10.1007/s10255-007-0393.  Google Scholar [23] S. Foiles, M. Baskes and M. Daw, Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys, Physical Review B, 33 (1986), 7983. Google Scholar [24] D. Frenkel and B. Smit, Understanding molecular simulation: From algorithms to applications, Academic Press, 50 (2002). doi: 10.1063/1.881812.  Google Scholar [25] G. Friesecke and F. Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice, Journal of Nonlinear Science, 12 (2002), 445-478.  doi: 10.1007/s00332-002-0495-z.  Google Scholar [26] 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.  Google Scholar [27] S. Katsura, T. Morita, S. Inawashiro, T. Horiguchi and Y. Abe, Lattice green's function. Introduction, Journal of Mathematical Physics, 12 (1971), 892-895.  doi: 10.1063/1.1665662.  Google Scholar [28] C. Kittel and D. F. Holcomb, Introduction to solid state physics, American Journal of Physics, 35 (1967), 547-548.   Google Scholar [29] J. Knap and M. Ortiz, An analysis of the quasicontinuum method, Journal of the Mechanics and Physics of Solids, 49 (2001), 1899-1923.  doi: 10.1016/S0022-5096(01)00034-5.  Google Scholar [30] R. Lesar, R. Najafabadi and D. J. Srolovitz, Finite-temperature defect properties from free-energy minimization, Physical Review Letters, 63 (1989), 624-627.  doi: 10.1103/PhysRevLett.63.624.  Google Scholar [31] X. Li, J. Z. Yang and W. E, A multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks, Journal of Computational Physics, 229 (2010), 3970-3987.  doi: 10.1016/j.jcp.2010.01.039.  Google Scholar [32] J. Lutsko, D. Wolf and S. Yip, Free energy calculation via MD: Methodology and application to bicrystals, Le Journal de Physique Colloques, 49 (1988), 375-379.  doi: 10.1051/jphyscol:1988543.  Google Scholar [33] J. C. Maxwell, I. On reciprocal figures, frames, and diagrams of forces, Transactions of the Royal Society of Edinburgh, 26 (1890), 161-207.  doi: 10.1017/CBO9780511710377.014.  Google Scholar [34] M. Mendelev, D. Srolovitz, G. Ackland and S. Han, Effect of fe segregation on the migration of a non-symmetric $\Sigma$5 tilt grain boundary in Al, Journal of Materials Research, 20 (2005), 208-218.   Google Scholar [35] R. Miller and E. Tadmor, The quasicontinuum method: Overview, applications and current direction, J. Comput. Aid. Mater. Des., 9 (2002), 203-239.   Google Scholar [36] H. J. Monkhorst and J. D. Pack, Special points for brillouin-zone integrations, Physical Review B, 13 (1976), 5188-5192.  doi: 10.1103/PhysRevB.13.5188.  Google Scholar [37] R. Najafabadi and D. J. Srolovitz, Order-disorder transitions at and segregation to (001) Ni-Pt surfaces, Surface Science, 286 (1993), 104-115.   Google Scholar [38] S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (1984), 255-268.   Google Scholar [39] S. Okamura, A. Miyazaki, S. Sugimoto, N. Tezuka and K. Inomata, Large tunnel magnetoresistance at room temperature with a Co$_2$FeAl full-heusler alloy electrode, Applied Physics Letters, 86 (2005), 232503. Google Scholar [40] R. Pathria, Statistical Mechanics, International Series in Natural Philosophy, Pergamon, 2017. Google Scholar [41] V. Sorkin, R. S. Elliott and E. B. Tadmor, A local quasicontinuum method for 3d multilattice crystalline materials: Application to shape-memory alloys, Modelling and Simulation in Materials Science and Engineering, 22 (2014), 055001. Google Scholar [42] E. B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids, Philosophical Magazine A, 73 (1996), 1529-1563.  doi: 10.1080/01418619608243000.  Google Scholar [43] E. B. Tadmor, G. S. Smith, N. Bernstein and E. Kaxiras, Mixed finite element and atomistic formulation for complex crystals, Phys. Rev. B, 59 (1999), 235-245.  doi: 10.1103/PhysRevB.59.235.  Google Scholar [44] E. B. Tadmor and R. E. Miller, Modeling Materials: Continuum, Atomistic and Multiscale Techniques, Cambridge University Press, 2011.  doi: 10.1017/CBO9781139003582.  Google Scholar [45] J. Tersoff, New empirical approach for the structure and energy of covalent systems, Physical Review B Condensed Matter, 37 (1988), 6991. doi: 10.1103/PhysRevB.37.6991.  Google Scholar [46] V. K. Tewary, Green-function method for lattice statics, Advances in Physics, 22 (1973), 757-810.  doi: 10.1080/00018737300101389.  Google Scholar [47] D. R. Trinkle, Lattice green function for extended defect calculations: Computation and error estimation with long-range forces, Physical Review B, 78 (2008), 014110. doi: 10.1103/PhysRevB.78.014110.  Google Scholar [48] D. Tsai, The virial theorem and stress calculation in molecular dynamics, The Journal of Chemical Physics, 70 (1979), 1375-1382.  doi: 10.1063/1.437577.  Google Scholar [49] G. J. Wagner and W. K. Liu, Coupling of atomic and continuum simulations using a bridging scale decomposition, J. Comput. Phys., 190 (2003), 249-274.   Google Scholar [50] Y. Xiang, H. Wei, P. Ming and W. E, A generalized Peierls/Nabarro model for curved dislocations and core structures of dislocation loops in al and cu, Acta Materialia, 56 (2008), 1447-1460.  doi: 10.1016/j.actamat.2007.11.033.  Google Scholar [51] S. Xiao and W. Yang, Temperature-related Cauchy–Born rule for multiscale modeling of crystalline solids, Computational Materials Science, 37 (2006), 374-379.  doi: 10.1016/j.commatsci.2005.09.007.  Google Scholar [52] 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.  Google Scholar [53] J. Z. Yang, X. Wu and X. Li, A generalized irving-kirkwood formula for the calculation of stress in molecular dynamics models, Journal of Chemical Physics, 137 (2012), 134104. doi: 10.1063/1.4755946.  Google Scholar [54] M. Zhou, A new look at the atomic level Virial stress: On continuum-molecular system equivalence, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 459 (2003), 2347-2392.  doi: 10.1098/rspa.2003.1127.  Google Scholar [55] 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.  Google Scholar [56] X. Zhou, R. Johnson and H. Wadley, Misfit-energy-increasing dislocations in vapor-deposited CoFe/NiFe multilayers, Physical Review B, 69 (2004), 144113. doi: 10.1103/PhysRevB.69.144113.  Google Scholar [57] J. A. Zimmerman, E. B. WebbIII, 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. Google Scholar

show all references

##### 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.  Google Scholar [2] 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.  Google Scholar [3] 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.  Google Scholar [4] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Cengage Learning, 1976. Google Scholar [5] C. A. Becker, F. Tavazza, Z. T. Trautt and R. A. B. de Macedo, Considerations for choosing and using force fields and interatomic potentials in materials science and engineering, Current Opinion in Solid State and Materials Science, 17 (2013), 277-283.  doi: 10.1016/j.cossms.2013.10.001.  Google Scholar [6] T. Belytschko and S. Xiao, Coupling methods for continuum modelwith molecular model, Int. J. Multi. Comput. Engrg., 1 (2003), 115-126.   Google Scholar [7] I. Bitsanis, J. J. Magda, M. Tirrell and H. Davis, Molecular dynamics of flow in micropores, The Journal of Chemical Physics, 87 (1987), 1733-1750.  doi: 10.1063/1.453240.  Google Scholar [8] X. Blanc, C. L. Bris and P. L. Lions, From molecular models to continuum mechanics, Archive for Rational Mechanics and Analysis, 164 (2002), 341-381.  doi: 10.1007/s00205-002-0218-5.  Google Scholar [9] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Courier Corporation, 1986.  Google Scholar [10] P. E. Blöchl, O. Jepsen and O. K. Andersen, Improved tetrahedron method for brillouin-zone integrations, Physical Review B, 49 (1994), 16223. Google Scholar [11] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, 1954.   Google Scholar [12] D. Brown and S. Neyertz, A general pressure tensor calculation for molecular dynamics simulations, Molecular Physics, 84 (1995), 577-595.  doi: 10.1080/00268979500100371.  Google Scholar [13] 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.  Google Scholar [14] G. Cicotti, D. Frenkel and I. McDonald, Simulation of Liquids and Solids. Molecular Dynamics and Monte Carlo Methods in Statistical Mechanics, North Holland, 1987. Google Scholar [15] 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.  Google Scholar [16] W. A. Curtin and R. E. Miller, Atomistic/continuum coupling in computational materials science, Model. Simul. Mater. Sci. Engrg., 11 (2003), R33–R68. doi: 10.1088/0965-0393/11/3/201.  Google Scholar [17] S. Dai, Y. Xiang and D. J. Srolovitz, Structure and energy of (111) low-angle twist boundaries in Al, Cu and Ni, Acta Materialia, 61 (2013), 1327-1337.  doi: 10.1016/j.actamat.2012.11.010.  Google Scholar [18] M. S. Daw and M. I. Baskes, Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals, Physical Review B, 29 (1984), 6443. doi: 10.1103/PhysRevB.29.6443.  Google Scholar [19] J. Du, C. Wang and T. Yu, Construction and application of multi-element EAM potential (Ni-Al-Re) in $\gamma$/$\gamma'$ Ni-based single crystal superalloys, Modelling and Simulation in Materials Science and Engineering, 21 (2013), 015007. Google Scholar [20] L. M. Dupuy, E. B. Tadmor, R. E. Miller and R. Phillips, Finite-temperature quasicontinuum: Molecular dynamics without all the atoms, Physical Review Letters, 95 (2005), 060202. doi: 10.1103/PhysRevLett.95.060202.  Google Scholar [21] W. E and P. Ming, Cauchy–born rule and the stability of crystalline solids: Static problems, Archive for Rational Mechanics and Analysis, 183 (2007), 241-297.  doi: 10.1007/s00205-006-0031-7.  Google Scholar [22] W. E and P. Ming, Cauchy-born rule and the stability of crystalline solids: Dynamic problems, Acta Mathematicae Applicatae Sinica, English Series, 23 (2007), 529-550.  doi: 10.1007/s10255-007-0393.  Google Scholar [23] S. Foiles, M. Baskes and M. Daw, Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys, Physical Review B, 33 (1986), 7983. Google Scholar [24] D. Frenkel and B. Smit, Understanding molecular simulation: From algorithms to applications, Academic Press, 50 (2002). doi: 10.1063/1.881812.  Google Scholar [25] G. Friesecke and F. Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice, Journal of Nonlinear Science, 12 (2002), 445-478.  doi: 10.1007/s00332-002-0495-z.  Google Scholar [26] 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.  Google Scholar [27] S. Katsura, T. Morita, S. Inawashiro, T. Horiguchi and Y. Abe, Lattice green's function. Introduction, Journal of Mathematical Physics, 12 (1971), 892-895.  doi: 10.1063/1.1665662.  Google Scholar [28] C. Kittel and D. F. Holcomb, Introduction to solid state physics, American Journal of Physics, 35 (1967), 547-548.   Google Scholar [29] J. Knap and M. Ortiz, An analysis of the quasicontinuum method, Journal of the Mechanics and Physics of Solids, 49 (2001), 1899-1923.  doi: 10.1016/S0022-5096(01)00034-5.  Google Scholar [30] R. Lesar, R. Najafabadi and D. J. Srolovitz, Finite-temperature defect properties from free-energy minimization, Physical Review Letters, 63 (1989), 624-627.  doi: 10.1103/PhysRevLett.63.624.  Google Scholar [31] X. Li, J. Z. Yang and W. E, A multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks, Journal of Computational Physics, 229 (2010), 3970-3987.  doi: 10.1016/j.jcp.2010.01.039.  Google Scholar [32] J. Lutsko, D. Wolf and S. Yip, Free energy calculation via MD: Methodology and application to bicrystals, Le Journal de Physique Colloques, 49 (1988), 375-379.  doi: 10.1051/jphyscol:1988543.  Google Scholar [33] J. C. Maxwell, I. On reciprocal figures, frames, and diagrams of forces, Transactions of the Royal Society of Edinburgh, 26 (1890), 161-207.  doi: 10.1017/CBO9780511710377.014.  Google Scholar [34] M. Mendelev, D. Srolovitz, G. Ackland and S. Han, Effect of fe segregation on the migration of a non-symmetric $\Sigma$5 tilt grain boundary in Al, Journal of Materials Research, 20 (2005), 208-218.   Google Scholar [35] R. Miller and E. Tadmor, The quasicontinuum method: Overview, applications and current direction, J. Comput. Aid. Mater. Des., 9 (2002), 203-239.   Google Scholar [36] H. J. Monkhorst and J. D. Pack, Special points for brillouin-zone integrations, Physical Review B, 13 (1976), 5188-5192.  doi: 10.1103/PhysRevB.13.5188.  Google Scholar [37] R. Najafabadi and D. J. Srolovitz, Order-disorder transitions at and segregation to (001) Ni-Pt surfaces, Surface Science, 286 (1993), 104-115.   Google Scholar [38] S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (1984), 255-268.   Google Scholar [39] S. Okamura, A. Miyazaki, S. Sugimoto, N. Tezuka and K. Inomata, Large tunnel magnetoresistance at room temperature with a Co$_2$FeAl full-heusler alloy electrode, Applied Physics Letters, 86 (2005), 232503. Google Scholar [40] R. Pathria, Statistical Mechanics, International Series in Natural Philosophy, Pergamon, 2017. Google Scholar [41] V. Sorkin, R. S. Elliott and E. B. Tadmor, A local quasicontinuum method for 3d multilattice crystalline materials: Application to shape-memory alloys, Modelling and Simulation in Materials Science and Engineering, 22 (2014), 055001. Google Scholar [42] E. B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids, Philosophical Magazine A, 73 (1996), 1529-1563.  doi: 10.1080/01418619608243000.  Google Scholar [43] E. B. Tadmor, G. S. Smith, N. Bernstein and E. Kaxiras, Mixed finite element and atomistic formulation for complex crystals, Phys. Rev. B, 59 (1999), 235-245.  doi: 10.1103/PhysRevB.59.235.  Google Scholar [44] E. B. Tadmor and R. E. Miller, Modeling Materials: Continuum, Atomistic and Multiscale Techniques, Cambridge University Press, 2011.  doi: 10.1017/CBO9781139003582.  Google Scholar [45] J. Tersoff, New empirical approach for the structure and energy of covalent systems, Physical Review B Condensed Matter, 37 (1988), 6991. doi: 10.1103/PhysRevB.37.6991.  Google Scholar [46] V. K. Tewary, Green-function method for lattice statics, Advances in Physics, 22 (1973), 757-810.  doi: 10.1080/00018737300101389.  Google Scholar [47] D. R. Trinkle, Lattice green function for extended defect calculations: Computation and error estimation with long-range forces, Physical Review B, 78 (2008), 014110. doi: 10.1103/PhysRevB.78.014110.  Google Scholar [48] D. Tsai, The virial theorem and stress calculation in molecular dynamics, The Journal of Chemical Physics, 70 (1979), 1375-1382.  doi: 10.1063/1.437577.  Google Scholar [49] G. J. Wagner and W. K. Liu, Coupling of atomic and continuum simulations using a bridging scale decomposition, J. Comput. Phys., 190 (2003), 249-274.   Google Scholar [50] Y. Xiang, H. Wei, P. Ming and W. E, A generalized Peierls/Nabarro model for curved dislocations and core structures of dislocation loops in al and cu, Acta Materialia, 56 (2008), 1447-1460.  doi: 10.1016/j.actamat.2007.11.033.  Google Scholar [51] S. Xiao and W. Yang, Temperature-related Cauchy–Born rule for multiscale modeling of crystalline solids, Computational Materials Science, 37 (2006), 374-379.  doi: 10.1016/j.commatsci.2005.09.007.  Google Scholar [52] 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.  Google Scholar [53] J. Z. Yang, X. Wu and X. Li, A generalized irving-kirkwood formula for the calculation of stress in molecular dynamics models, Journal of Chemical Physics, 137 (2012), 134104. doi: 10.1063/1.4755946.  Google Scholar [54] M. Zhou, A new look at the atomic level Virial stress: On continuum-molecular system equivalence, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 459 (2003), 2347-2392.  doi: 10.1098/rspa.2003.1127.  Google Scholar [55] 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.  Google Scholar [56] X. Zhou, R. Johnson and H. Wadley, Misfit-energy-increasing dislocations in vapor-deposited CoFe/NiFe multilayers, Physical Review B, 69 (2004), 144113. doi: 10.1103/PhysRevB.69.144113.  Google Scholar [57] J. A. Zimmerman, E. B. WebbIII, 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. Google Scholar
Demonstration of B2 alloys structure. Left: Undeformed lattice. Right: Deformed lattice
Results for $\alpha$-Fe system. The left panel shows the comparisons of the PK stress in Virial formulation ($\sigma_{11}$) calculated by (2.11) from MD simulations (red dashed lines) and (2.52) from our FTCB method (blue solid lines) under different temperatures and deformations. The right panel shows the relative error between two methods. From top to bottom the deformation gradient is $\boldsymbol F_0, \boldsymbol F_1, \boldsymbol F_2$
Results for NiAl alloy system. The left panel shows the comparisons of the PK stress in Virial formulation ($\sigma_{11}$) calculated by (2.11) from MD simulations (red dashed lines) and (2.52) from our FTCB method (blue solid lines) under different temperatures and deformations. The right panel shows the relative error between two methods. From top to bottom the deformation gradient is $\boldsymbol F_0, \boldsymbol F_1, \boldsymbol F_2$
Results for FeAl alloy system. The left panel shows the comparisons of the PK stress in Virial formulation ($\sigma_{11}$) calculated by (2.11) from MD simulations (red dashed lines) and (2.52) from our FTCB method (blue solid lines) under different temperatures and deformations. The right panel shows the relative error between two methods. From top to bottom the deformation gradient is $\boldsymbol F_0, \boldsymbol F_1, \boldsymbol F_2$
Results for Si system. The left panel shows the comparisons of the PK stress in Virial formulation ($\sigma_{11}$) calculated by (2.11) from MD simulations (red dashed lines) and (2.52) from our FTCB method (blue solid lines) under different temperatures and deformations. The right panel shows the relative error between two methods. From top to bottom the deformation gradient is $\boldsymbol F_0, \boldsymbol F_1, \boldsymbol F_2$
Results for Co$_2$FeAl system. The left panel shows the comparisons of the PK stress in Virial formulation ($\sigma_{11}$) calculated by (2.11) from MD simulations (red dashed lines) and (2.52) from our FTCB method (blue solid lines) under different temperatures and deformations. The right panel shows the relative error between two methods. From top to bottom the deformation gradient is $\boldsymbol F_0, \boldsymbol F_1, \boldsymbol F_2$. Notice that the result obtained by FTCB method without $\boldsymbol \xi = \boldsymbol 0$ are shown by brown solid lines
Optimized values of fitting parameters of the EAM potential for NiAl
 Parameter Ni Al Cross parameter NiAl $f_e$ $2.81\times 10^{-3}$ $2.23\times10^{-3}$ $\alpha(eV)$ $3.5442\times 10^{-1}$ $r_e(\rm{Å})$ 2.50 2.85 $\beta$ 7.2547 $\chi(\rm{Å}^{-1})$ $2.8411$ $2.5268$ $\gamma(eV)$ $1.0466\times10^{-3}$ $n$ $3.0447\times10^{-1}$ $4.4658\times10^{-1}$ $\kappa$ $-3.9796$ $s$ $1.0000$ $2.9236$ $r_0(\rm{Å})$ $2.5222$ $\rho_e$ $3.5985\times10^{-2}$ $7.8227\times10^{-2}$ $h(\rm{Å})$ $5.1505\times10^{-1}$ $d_{t_it_i}^{t_i}(eV)$ $-2.3014\times10^1$ $1.4317\times10^{-1}$ $r_c(\rm{Å})$ $5.1786$ $\alpha(eV)$ $1.2510\times10^{-2}$ $1.0034\times10^{-1}$ $d_{NiAl}^{Ni}(eV)$ $-2.1818$ $\beta$ $1.0000\times10^{-3}$ $8.1857$ $d_{NiAl}^{Al}(eV)$ $1.0676$ $\gamma(eV)$ $-3.5163$ $4.0514\times10^{-3}$ $\kappa$ $7.5831$ $-5.2299\times10^{-1}$ $r_0(\rm{Å})$ $2.4890$ $2.8638$ $h(\rm{Å})$ $4.8984\times10^{-1}$ $6.4596\times10^{-1}$ $r_c(\rm{Å})$ $5.0338$ $7.2958$
 Parameter Ni Al Cross parameter NiAl $f_e$ $2.81\times 10^{-3}$ $2.23\times10^{-3}$ $\alpha(eV)$ $3.5442\times 10^{-1}$ $r_e(\rm{Å})$ 2.50 2.85 $\beta$ 7.2547 $\chi(\rm{Å}^{-1})$ $2.8411$ $2.5268$ $\gamma(eV)$ $1.0466\times10^{-3}$ $n$ $3.0447\times10^{-1}$ $4.4658\times10^{-1}$ $\kappa$ $-3.9796$ $s$ $1.0000$ $2.9236$ $r_0(\rm{Å})$ $2.5222$ $\rho_e$ $3.5985\times10^{-2}$ $7.8227\times10^{-2}$ $h(\rm{Å})$ $5.1505\times10^{-1}$ $d_{t_it_i}^{t_i}(eV)$ $-2.3014\times10^1$ $1.4317\times10^{-1}$ $r_c(\rm{Å})$ $5.1786$ $\alpha(eV)$ $1.2510\times10^{-2}$ $1.0034\times10^{-1}$ $d_{NiAl}^{Ni}(eV)$ $-2.1818$ $\beta$ $1.0000\times10^{-3}$ $8.1857$ $d_{NiAl}^{Al}(eV)$ $1.0676$ $\gamma(eV)$ $-3.5163$ $4.0514\times10^{-3}$ $\kappa$ $7.5831$ $-5.2299\times10^{-1}$ $r_0(\rm{Å})$ $2.4890$ $2.8638$ $h(\rm{Å})$ $4.8984\times10^{-1}$ $6.4596\times10^{-1}$ $r_c(\rm{Å})$ $5.0338$ $7.2958$
Suggested parameters of the Tersoff potential for silicon
 $A(eV)$ $1.8308\times 10^3$ $\alpha$ 0 $\lambda_1(\rm{Å}^{-1})$ 2.4799 $B(eV)$ $4.7118\times 10^2$ $\beta$ $1.0999\times 10^{-6}$ $\lambda_2(\rm{Å}^{-1})$ 1.7322 $R(\rm{Å})$ 2.85 $n$ $7.8734\times 10^{-1}$ $D(\rm{Å})$ 0.15 $c$ $1.0039\times 10^5$ $d$ $1.6218\times 10^1$ $h$ $-5.9826\times 10^{-1}$
 $A(eV)$ $1.8308\times 10^3$ $\alpha$ 0 $\lambda_1(\rm{Å}^{-1})$ 2.4799 $B(eV)$ $4.7118\times 10^2$ $\beta$ $1.0999\times 10^{-6}$ $\lambda_2(\rm{Å}^{-1})$ 1.7322 $R(\rm{Å})$ 2.85 $n$ $7.8734\times 10^{-1}$ $D(\rm{Å})$ 0.15 $c$ $1.0039\times 10^5$ $d$ $1.6218\times 10^1$ $h$ $-5.9826\times 10^{-1}$
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