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June  2022, 27(6): 3131-3153. doi: 10.3934/dcdsb.2021176

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 Published  June 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (6) : 3131-3153. doi: 10.3934/dcdsb.2021176
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 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.

[3]

N. C. AdmalJ. 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.

[4]

N. W. Ashcroft and N. D. Mermin, Solid State Physics, Cengage Learning, 1976.

[5]

C. A. BeckerF. TavazzaZ. 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.

[6]

T. Belytschko and S. Xiao, Coupling methods for continuum modelwith molecular model, Int. J. Multi. Comput. Engrg., 1 (2003), 115-126. 

[7]

I. BitsanisJ. J. MagdaM. Tirrell and H. Davis, Molecular dynamics of flow in micropores, The Journal of Chemical Physics, 87 (1987), 1733-1750.  doi: 10.1063/1.453240.

[8]

X. BlancC. 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.

[9]

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Courier Corporation, 1986.

[10]

P. E. Blöchl, O. Jepsen and O. K. Andersen, Improved tetrahedron method for brillouin-zone integrations, Physical Review B, 49 (1994), 16223.

[11] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, 1954. 
[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.

[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.

[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.

[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.

[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.

[17]

S. DaiY. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

D. Frenkel and B. Smit, Understanding molecular simulation: From algorithms to applications, Academic Press, 50 (2002). doi: 10.1063/1.881812.

[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.

[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.

[27]

S. KatsuraT. MoritaS. InawashiroT. Horiguchi and Y. Abe, Lattice green's function. Introduction, Journal of Mathematical Physics, 12 (1971), 892-895.  doi: 10.1063/1.1665662.

[28]

C. Kittel and D. F. Holcomb, Introduction to solid state physics, American Journal of Physics, 35 (1967), 547-548. 

[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.

[30]

R. LesarR. 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.

[31]

X. LiJ. 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.

[32]

J. LutskoD. 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.

[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.

[34]

M. MendelevD. SrolovitzG. 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. 

[35]

R. Miller and E. Tadmor, The quasicontinuum method: Overview, applications and current direction, J. Comput. Aid. Mater. Des., 9 (2002), 203-239. 

[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.

[37]

R. Najafabadi and D. J. Srolovitz, Order-disorder transitions at and segregation to (001) Ni-Pt surfaces, Surface Science, 286 (1993), 104-115. 

[38]

S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (1984), 255-268. 

[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.

[40]

R. Pathria, Statistical Mechanics, International Series in Natural Philosophy, Pergamon, 2017.

[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.

[42]

E. B. TadmorM. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids, Philosophical Magazine A, 73 (1996), 1529-1563.  doi: 10.1080/01418619608243000.

[43]

E. B. TadmorG. S. SmithN. 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.

[44] E. B. Tadmor and R. E. Miller, Modeling Materials: Continuum, Atomistic and Multiscale Techniques, Cambridge University Press, 2011.  doi: 10.1017/CBO9781139003582.
[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.

[46]

V. K. Tewary, Green-function method for lattice statics, Advances in Physics, 22 (1973), 757-810.  doi: 10.1080/00018737300101389.

[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.

[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.

[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. 

[50]

Y. XiangH. WeiP. 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.

[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.

[52]

J. Z. YangC. MaoX. 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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

N. C. AdmalJ. 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.

[4]

N. W. Ashcroft and N. D. Mermin, Solid State Physics, Cengage Learning, 1976.

[5]

C. A. BeckerF. TavazzaZ. 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.

[6]

T. Belytschko and S. Xiao, Coupling methods for continuum modelwith molecular model, Int. J. Multi. Comput. Engrg., 1 (2003), 115-126. 

[7]

I. BitsanisJ. J. MagdaM. Tirrell and H. Davis, Molecular dynamics of flow in micropores, The Journal of Chemical Physics, 87 (1987), 1733-1750.  doi: 10.1063/1.453240.

[8]

X. BlancC. 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.

[9]

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Courier Corporation, 1986.

[10]

P. E. Blöchl, O. Jepsen and O. K. Andersen, Improved tetrahedron method for brillouin-zone integrations, Physical Review B, 49 (1994), 16223.

[11] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, 1954. 
[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.

[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.

[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.

[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.

[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.

[17]

S. DaiY. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

D. Frenkel and B. Smit, Understanding molecular simulation: From algorithms to applications, Academic Press, 50 (2002). doi: 10.1063/1.881812.

[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.

[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.

[27]

S. KatsuraT. MoritaS. InawashiroT. Horiguchi and Y. Abe, Lattice green's function. Introduction, Journal of Mathematical Physics, 12 (1971), 892-895.  doi: 10.1063/1.1665662.

[28]

C. Kittel and D. F. Holcomb, Introduction to solid state physics, American Journal of Physics, 35 (1967), 547-548. 

[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.

[30]

R. LesarR. 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.

[31]

X. LiJ. 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.

[32]

J. LutskoD. 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.

[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.

[34]

M. MendelevD. SrolovitzG. 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. 

[35]

R. Miller and E. Tadmor, The quasicontinuum method: Overview, applications and current direction, J. Comput. Aid. Mater. Des., 9 (2002), 203-239. 

[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.

[37]

R. Najafabadi and D. J. Srolovitz, Order-disorder transitions at and segregation to (001) Ni-Pt surfaces, Surface Science, 286 (1993), 104-115. 

[38]

S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Molecular Physics, 52 (1984), 255-268. 

[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.

[40]

R. Pathria, Statistical Mechanics, International Series in Natural Philosophy, Pergamon, 2017.

[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.

[42]

E. B. TadmorM. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids, Philosophical Magazine A, 73 (1996), 1529-1563.  doi: 10.1080/01418619608243000.

[43]

E. B. TadmorG. S. SmithN. 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.

[44] E. B. Tadmor and R. E. Miller, Modeling Materials: Continuum, Atomistic and Multiscale Techniques, Cambridge University Press, 2011.  doi: 10.1017/CBO9781139003582.
[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.

[46]

V. K. Tewary, Green-function method for lattice statics, Advances in Physics, 22 (1973), 757-810.  doi: 10.1080/00018737300101389.

[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.

[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.

[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. 

[50]

Y. XiangH. WeiP. 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.

[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.

[52]

J. Z. YangC. MaoX. 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.

[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.

[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.

[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.

[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.

[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.

Figure 1.  Demonstration of B2 alloys structure. Left: Undeformed lattice. Right: Deformed lattice
Figure 2.  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 $
Figure 3.  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 $
Figure 4.  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 $
Figure 5.  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 $
Figure 6.  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
Table 3.1.  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 $
Table 3.2.  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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