Advertisement

Generalized Continua Concepts in Coarse-Graining Atomistic Simulations

  • Shuozhi Xu
  • Ji Rigelesaiyin
  • Liming Xiong
  • Youping Chen
  • David L. McDowellEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 90)

Abstract

Generalized continuum mechanics (GCM) has attracted increased attention in the context of multiscale materials modeling, an example of which is a bottom-up GCM model, called the atomistic field theory (AFT). Unlike most other GCM models, AFT views a crystalline material as a continuous collection of lattice points; embedded within each point is a unit cell with a group of discrete atoms. As such, AFT concurrently bridges the discrete and continuous descriptions of materials, two fundamentally different viewpoints. In this chapter, we first review the basics of AFT and illustrate how it is realized through coarse-graining atomistic simulations via a concurrent atomistic-continuum (CAC) method. Important aspects of CAC, including its advantages relative to other multiscale methods, code development, and numerical implementations, are discussed. Then, we present recent applications of CAC to a number of metal plasticity problems, including static dislocation properties, fast moving dislocations and phonons, as well as dislocation/grain boundary interactions. We show that, adequately replicating essential aspects of dislocation fields at a fraction of the computational cost of full atomistics, CAC is established as an effective tool for coarse-grained modeling of various nano/micro-scale thermal and mechanical problems in a wide range of monatomic and polyatomic crystalline materials.

Notes

Acknowledgements

These results are in part based upon work supported by the National Science Foundation as a collaborative effort between Georgia Tech (CMMI-1232878) and University of Florida (CMMI-1233113). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors thank Dr. Jinghong Fan, Dr. Qian Deng, Dr. Shengfeng Yang, Dr. Xiang Chen, Mr. Rui Che, and Mr. Weixuan Li for helpful discussions, Mr. Kevin Chu for building the Python scripting interface in PyCAC, and Dr. Aleksandr Blekh for arranging execution of PyCAC via MATIN. The work of SX was supported in part by Georgia Tech Institute for Materials and in part by the Elings Prize Fellowship in Science offered by the California NanoSystems Institute (CNSI) on the UC Santa Barbara campus. SX also acknowledges support from the Center for Scientific Computing from the CNSI, MRL: an NSF MRSEC (DMR-1121053). LX acknowledges the support from the Department of Energy, Office of Basic Energy Sciences under Award Number DE-SC0006539. The work of LX was also supported in part by the National Science Foundation under Award Number CMMI-1536925. DLM is grateful for the additional support of the Carter N. Paden, Jr. Distinguished Chair in Metals Processing. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575.

References

  1. 1.
    Maugin, G.A.: Non-Classical Continuum Mechanics: A Dictionary. Springer, Singapore (2016)zbMATHGoogle Scholar
  2. 2.
    Maugin, G.A.: Some remarks on generalized continuum mechanics. Math. Mech. Solids 20(3), 280–291 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Maugin, G.A.: Generalized continuum mechanics: various paths. In: Continuum Mechanics Through the Twentieth Century, pp. 223–241. Springer (2013)Google Scholar
  4. 4.
    Maugin, G.A.: Continuum Mechanics Through the Twentieth Century, Solid Mechanics and Its Applications, vol. 196, pp. 978–994. Springer, Berlin (2013)Google Scholar
  5. 5.
    Maugin, G.A.: Generalized continuum mechanics: what do we mean by that? In: Maugin, G., Metrikine, A. (eds.) Mechanics of Generalized Continua. Advances in Mechanics and Mathematics, pp. 3–13. Springer, New York, NY (2010)Google Scholar
  6. 6.
    Maugin, G.A.: A historical perspective of generalized continuum mechanics. In: Altenbach, H., Maugin, G., Erofeev, V. (eds.) Mechanics of Generalized Continua. Advanced Structured Materials, vol. 7. pp. 3–19 (2011)Google Scholar
  7. 7.
    Maugin, G.A., Metrikine, A.V.: Mechanics of Generalized Continua: One Hundred Years After the Cosserats. Springer, New York (2010)zbMATHGoogle Scholar
  8. 8.
    Chen, Y., Lee, J.: Atomistic formulation of a multiscale field theory for nano/micro solids. Philos. Mag. 85(33–35), 4095–4126 (2005)Google Scholar
  9. 9.
    Chen, Y.: Reformulation of microscopic balance equations for multiscale materials modeling. J. Chem. Phys. 130(13), 134706 (2009)Google Scholar
  10. 10.
    Chen, Y., Lee, J., Xiong, L.: A generalized continuum theory and its relation to micromorphic theory. J. Eng. Mech. 135(3), 149–155 (2009)Google Scholar
  11. 11.
    Chen, Y., Zimmerman, J., Krivtsov, A., McDowell, D.L: Assessment of atomistic coarse-graining methods. Int. J. Eng. Sci. 49(12), 1337–1349 (2011)zbMATHGoogle Scholar
  12. 12.
    Cosserat, E., Cosserat, F.: Théorie des corps déformables, vol. 3, pp. 17–29, Paris (1909)Google Scholar
  13. 13.
    Chen, Y., Lee, J.D., Eskandarian, A.: Micropolar theory and its applications to mesoscopic and microscopic problems. Comput. Model. Eng. Sci. 5(1), 35–43 (2004)zbMATHGoogle Scholar
  14. 14.
    Eringen, A.C.: Theory of micropolar elasticity. In: Microcontinuum Field Theories, pp. 101–248. Springer (1999)Google Scholar
  15. 15.
    Eringen, A.C.: Microcontinuum Field Theories: I. Foundations and Solids. Springer, New York (1999)zbMATHGoogle Scholar
  16. 16.
    Eringen, A.C.: Mechanics of Micromorphic Continua. Springer (1968)Google Scholar
  17. 17.
    Chen, Y., Lee, J.D.: Connecting molecular dynamics to micromorphic theory. (I). Instantaneous and averaged mechanical variables. Phys. A 322, 359–376 (2003)zbMATHGoogle Scholar
  18. 18.
    Chen, Y., Lee, J.D.: Connecting molecular dynamics to micromorphic theory. (II). Balance laws. Phys. A 322, 377–392 (2003)zbMATHGoogle Scholar
  19. 19.
    Chen, Y., Lee, J., Eskandarian, A.: Atomistic counterpart of micromorphic theory. Acta Mech. 161(1–2), 81–102 (2003)zbMATHGoogle Scholar
  20. 20.
    Chen, Y., Lee, J.D.: Determining material constants in micromorphic theory through phonon dispersion relations. Int. J. Eng. Sci. 41(8), 871–886 (2003)Google Scholar
  21. 21.
    Chen, Y., Lee, J.D., Eskandarian, A.: Atomistic viewpoint of the applicability of microcontinuum theories. Int. J. Solids Struct. 41(8), 2085–2097 (2004)zbMATHGoogle Scholar
  22. 22.
    Chen, Y., Lee, J.D., Eskandarian, A.: Examining the physical foundation of continuum theories from the viewpoint of phonon dispersion relation. Int. J. Eng. Sci. 41, 61–83 (2003)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Hoover, W.G.: Computational Statistical Mechanics. Elsevier (1991)Google Scholar
  24. 24.
    Chen, Y.: Local stress and heat flux in atomistic systems involving three-body forces. J. Chem. Phys. 124(5), 054113 (2006)Google Scholar
  25. 25.
    Chen, Y., Diaz, A.: Local momentum and heat fluxes in transient transport processes and inhomogeneous systems. Phys. Rev. E 94(5), 053309 (2016)Google Scholar
  26. 26.
    Chen, Y.: The origin of the distinction between microscopic formulas for stress and Cauchy stress. EPL 116(3), 34003 (2016)Google Scholar
  27. 27.
    Espanol, P.: Statistical mechanics of coarse-graining. In: Novel Methods in Soft Matter Simulations, pp. 69–115. Springer (2004)Google Scholar
  28. 28.
    Izvekov, S., Voth, G.A.: Multiscale coarse-graining of liquid-state systems. J. Chem. Phys. 123(13), 134105 (2005)Google Scholar
  29. 29.
    Izvekov, S., Voth, G.A.: A multiscale coarse-graining method for biomolecular systems. J. Phys. Chem. B 109(7), 2469–2473 (2005)Google Scholar
  30. 30.
    Noid, W., Chu, J.W., Ayton, G.S., Krishna, V., Izvekov, S., Voth, G.A., Das, A., Andersen, H.C.: The multiscale coarse-graining method. I. A rigorous bridge between atomistic and coarse-grained models. J. Chem. Phys. 128(24), 244114 (2008)Google Scholar
  31. 31.
    Noid, W., Liu, P., Wang, Y., Chu, J.W., Ayton, G.S., Izvekov, S., Andersen, H.C., Voth, G.A.: The multiscale coarse-graining method. II. Numerical implementation for coarse-grained molecular models. J. Chem. Phys. 128(24), 244115 (2008)Google Scholar
  32. 32.
    Tadmor, E.B., Ortiz, M., Phillips, R.: Quasicontinuum analysis of defects in solids. Philos. Mag. A 73, 1529–1563 (1996)Google Scholar
  33. 33.
    Dupuy, L.M., Tadmor, E.B., Miller, R.E., Phillips, R.: Finite-temperature quasicontinuum: molecular dynamics without all the atoms. Phys. Rev. Lett. 95, 060202 (2005)Google Scholar
  34. 34.
    Kulkarni, Y., Knap, J., Ortiz, M.: A variational approach to coarse-graining of equilibrium and non-equilibrium atomistic description at finite temperature. J. Mech. Phys. Solids 56, 1417–1449 (2008)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Shenoy, V.B., Miller, R., Tadmor, E.B., Phillips, R., Ortiz, M.: Quasicontinuum models of interfacial structure and deformation. Phys. Rev. Lett. 80, 742–745 (1998)Google Scholar
  36. 36.
    Rudd, R.E., Broughton, J.Q.: Coarse-grained molecular dynamics and the atomic limit of finite elements. Phys. Rev. B 58(10), R5893 (1998)Google Scholar
  37. 37.
    Irving, J., Kirkwood, J.G.: The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18(6), 817–829 (1950)MathSciNetGoogle Scholar
  38. 38.
    Kittel, C.: Introduction to Solid State Physics. Wiley, Inc (1956)Google Scholar
  39. 39.
    Deng, Q, Xiong, L., Chen, Y.: Coarse-graining atomistic dynamics of fracture by finite element method. Int. J. Plast. 26(9), 1402–1414Google Scholar
  40. 40.
    Xiong, L., Chen, Y.: Coarse-grained simulations of single-crystal silicon. Modell. Simul. Mater. Sci. Eng. 17, 035002 (2009)Google Scholar
  41. 41.
    Xiong, L., Chen, Y., Lee, J.D.: Atomistic simulation of mechanical properties of diamond and silicon carbide by a field theory. Model. Simul. Mater. Sci. Eng. 15(5), 535 (2007)Google Scholar
  42. 42.
    Xiong, L., Tucker, G., McDowell, D.L., Chen, Y.: Coarse-grained atomistic simulation of dislocations. J. Mech. Phys. Solids 59(2), 160–177 (2011)zbMATHGoogle Scholar
  43. 43.
    Xu, S., Che, R., Xiong, L., Chen, Y., McDowell, D.L.: A quasistatic implementation of the concurrent atomistic-continuum method for FCC crystals. Int. J. Plast. 72, 91–126 (2015)Google Scholar
  44. 44.
    Xu, S., Payne, T.G., Chen, H., Liu, Y., Xiong, L., Chen, Y., McDowell, D.L.: PyCAC: The concurrent atomistic-continuum simulation environment. J. Mater. Res. (2018) in press, https://doi.org/10.1557/jmr.2018.8Google Scholar
  45. 45.
    Shilkrot, L.E., Curtin, W.A., Miller, R.E.: A coupled atomistic/continuum model of defects in solids. J. Mech. Phys. Solids 50, 2085–2106 (2002)zbMATHGoogle Scholar
  46. 46.
    Shilkrot, L.E., Miller, R.E., Curtin, W.A.: Coupled atomistic and discrete dislocation plasticity. Phys. Rev. Lett. 89, 025501 (2002)Google Scholar
  47. 47.
    Xu, S., Xiong, L., Deng, Q., McDowell, D.L.: Mesh refinement schemes for the concurrent atomistic-continuum method. Int. J. Solids Struct. 90, 144–152 (2016)Google Scholar
  48. 48.
    Zbib, H.M., de la Rubia, T.D., Bulatov, V.: A multiscale model of plasticity based on discrete dislocation dynamics. ASME J. Eng. Mater. Technol. 124(1), 78–87 (2002)Google Scholar
  49. 49.
    Hochrainer, T., Zaiser, M., Gumbsch, P.: A three-dimensional continuum theory of dislocation systems: kinematics and mean-field formulation. Philos. Mag. 87, 1261–1282 (2007)Google Scholar
  50. 50.
    Arsenlis, A., Cai, W., Tang, M., Rhee, M., Oppelstrup, T., Hommes, G., Pierce, T.G., Bulatov, V.V.: Enabling strain hardening simulations with dislocation dynamics. Model. Simul. Mater. Sci. Eng. 15, 553–595 (2007)Google Scholar
  51. 51.
    El-Azab, A., Deng, J., Tang, M.: Statistical characterization of dislocation ensembles. Philos. Mag. 87(8–9), 1201–1223 (2007)Google Scholar
  52. 52.
    Devincre, B., Hoc, T., Kubin, L.: Dislocation mean free paths and strain hardening of crystals. Science 320(5884), 1745–1748 (2008)Google Scholar
  53. 53.
    Motz, C., Weygan, D., Senger, J., Gumbsch, P.: Initial dislocation structures in 3-D discrete dislocation dynamics and their influence on microscale plasticity. Acta Mater. 57(6), 1744–1754 (2009)Google Scholar
  54. 54.
    Zaiser, M., Sandfeld, S.: Scaling properties of dislocation simulations in the similitude regime. Model. Simul. Mater. Sci. Eng. 22:065012, (2014)Google Scholar
  55. 55.
    Groma, I., Zaiser, M., Ispanovity, P.D.: Dislocation patterning in a two-dimensional continuum theory of dislocations. Phys. Rev. B 93, 214110 (2016)Google Scholar
  56. 56.
    Xia, S., El-Azab, A.: Computational modelling of mesoscale dislocation patterning and plastic deformation of single crystals. Model. Simul. Mater. Sci. Eng. 23(5), 55009 (2015)Google Scholar
  57. 57.
    Xiong, L., Chen, Y.: Effects of dopants on the mechanical properties of nanocrystalline silicon carbide thin film. Comput. Model. Eng. Sci. 24, 203–214 (2008)Google Scholar
  58. 58.
    Xiong, L., Chen, Y.: Coarse-grained simulations of single-crystal silicon. Model. Simul. Mater. Sci. Eng. 17, 035002 (2009)Google Scholar
  59. 59.
    Deng, Q., Chen, Y.: A coarse-grained atomistic method for 3D dynamic fracture simulation. Int. J. Multiscale Comput. Eng. 11, 227–237 (2013)Google Scholar
  60. 60.
    Xiong, L., Deng, Q., Tucker, G., McDowell, D.L., Chen, Y.: A concurrent scheme for passing dislocations from atomistic to continuum domains. Acta Mater. 60, 899–913 (2012)Google Scholar
  61. 61.
    Xiong, L., Deng, Q., Tucker, G., McDowell, D.L., Chen, Y.: Coarse-grained atomistic simulations of dislocations in Al, Ni and Cu crystals. Int. J. Plast. 38, 86–101 (2012)Google Scholar
  62. 62.
    Xiong, L., McDowell, D.L., Chen, Y.: Nucleation and growth of dislocation loops in Cu, Al and Si by a concurrent atomistic-continuum method. Scr. Mater. 67, 633–636 (2012)Google Scholar
  63. 63.
    Xiong, L., Chen, Y.: Coarse-grained atomistic modeling and simulation of inelastic material behavior. Acta Mech. Solida Sin. 25, 244–261 (2012)Google Scholar
  64. 64.
    Xiong, L., McDowell, D.L., Chen, Y.: Sub-THz Phonon drag on dislocations by coarse-grained atomistic simulations. Int. J. Plast. 55, 268–278 (2014)Google Scholar
  65. 65.
    Xiong, L., Xu, S., McDowell, D.L., Chen, Y.: Concurrent atomistic-continuum simulations of dislocation-void interactions in fcc crystals. Int. J. Plast. 65, 33–42 (2015)Google Scholar
  66. 66.
    Xiong, L., Rigelesaiyin, J., Chen, X., Xu, S., McDowell, D.L., Chen, Y.: Coarse-grained elastodynamics of fast moving dislocations. Acta Mater. 104, 143–155 (2016)Google Scholar
  67. 67.
    Yang, S., Xiong, L., Deng, Q., Chen, Y.: Concurrent atomistic and continuum simulation of strontium titanate. Acta Mater. 61, 89–102 (2013)Google Scholar
  68. 68.
    Yang, S., Chen, Y.: Concurrent atomistic and continuum simulation of bi-crystal strontium titanate with tilt grain boundary. Proc. Roy. Soc. A 471, 20140758 (2015)Google Scholar
  69. 69.
    Yang, S., Zhang, N., Chen, Y.: Concurrent atomistic-continuum simulation of polycrystalline strontium titanate. Philos. Mag. 95, 2697–2716 (2015)Google Scholar
  70. 70.
    Yang, S., Chen, Y.: Concurrent atomistic-continuum simulation of defects in polyatomic ionic materials. In: Weinberger, C., Tucker, G. (eds.) Multiscale Materials Modeling for Nanomechanics. Springer International Publishing, Switzerland (2016)Google Scholar
  71. 71.
    Chen, X., Xiong, L., McDowell, D.L., Chen, Y.: Effects of phonons on mobility of dislocations and dislocation arrays. Scr. Mater. 137, 22–26 (2017)Google Scholar
  72. 72.
    Chen, X., Li, W., Xiong, L., Li, Y., Yang, S., Zheng, Z., McDowell, D.L., Chen, Y.: Ballistic-diffusive phonon heat transport across grain boundaries. Acta Mater. 136, 355–365 (2017)Google Scholar
  73. 73.
    Chen, X., Diaz, A., Xiong, L., Chen, Y.: Passing waves from atomistic to continuum. J. Comput. Phys. 354, 393–402 (2018)MathSciNetGoogle Scholar
  74. 74.
    Chen, X., Li, W., Diaz, A., Li, Y., McDowell, D.L., Chen, Y.: Recent progress in the concurrent atomistic-continuum method and its application in phonon transport. MRS Commun. 7(4), 785–797 (2017)Google Scholar
  75. 75.
    Li, J: AtomEye: an efficient atomistic configuration viewer. Model. Simul. Mater. Sci. Eng. 11(2), 173 (2003)Google Scholar
  76. 76.
    Stukowski, A: Visualization and analysis of atomistic simulation data with OVITO—the Open Visualization Tool. Model. Simul. Mater. Sci. Eng. 18(1), 015012 (2010)Google Scholar
  77. 77.
    Jones, J.E.: On the determination of molecular fields. II. From the equation of state of a gas. Proc. R. Soc. Lond. A 106(738), 463–477 (1924)Google Scholar
  78. 78.
    Daw, M.S., Baskes, M.I.: Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B 29(12), 6443–6453 (1984)Google Scholar
  79. 79.
    Xu, S.: The concurrent atomistic-continuum method: Advancements and applications in plasticity of face-centered cubic metals. Ph.D. Dissertation, Georgia Institute of Technology (2016)Google Scholar
  80. 80.
    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Oxford University Press, USA (1989)zbMATHGoogle Scholar
  81. 81.
    Verlet, L.: Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 159, 98–103 (1967)Google Scholar
  82. 82.
    Swope, W.C., Andersen, H.C., Berens, P.H., Wilson, K.R.: A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: Application to small water clusters. J. Chem. Phys. 76(1), 637–649 (1982)Google Scholar
  83. 83.
    Xu, S., Xiong, L., Chen, Y., McDowell, D.L.: Sequential slip transfer of mixed-character dislocations across Σ3 coherent twin boundary in FCC metals: A concurrent atomistic-continuum study. npj Comput. Mater. 2, 15016 (2016)Google Scholar
  84. 84.
    Xu, S., Xiong, L., Chen, Y., McDowell, D.L.: A concurrent atomistic-continuum study of slip transfer of sequential mixed character dislocations across symmetric tilt grain boundaries in Ni. JOM 69, 814–821 (2017)Google Scholar
  85. 85.
    McDowell, D.L.: A perspective on trends in multiscale plasticity. Int. J. Plast. 26, 1280–1309 (2010)zbMATHGoogle Scholar
  86. 86.
    Xu, S., Xiong, L., Chen, Y., McDowell, D.L.: An analysis of key characteristics of the Frank-Read source process in FCC metals. J. Mech. Phys. Solids 96, 460–476 (2016)Google Scholar
  87. 87.
    Xu, S., Xiong, L., Chen, Y., McDowell, D.L.: Shear stress- and line length-dependent screw dislocation cross-slip in FCC Ni. Acta Mater. 122, 412–419 (2017)Google Scholar
  88. 88.
    Xu, S., Xiong, L., Chen, Y., McDowell, D.L.: Edge dislocations bowing out from a row of collinear obstacles in Al. Scr. Mater. 123, 135–139 (2016)Google Scholar
  89. 89.
    Xu, S., Xiong, L., Chen, Y., McDowell, D.L.: Validation of the concurrent atomistic-continuum method on screw dislocation/stacking fault interactions. Crystals 7, 120 (2017)Google Scholar
  90. 90.
    Xiong, L., Chen, X., Zhang, N., McDowell, D.L., Chen, Y.: Prediction of phonon properties of 1D polyatomic systems using concurrent atomistic-continuum simulation. Arch. Appl. Mech. 84, 1665–1675 (2014)Google Scholar
  91. 91.
    Rice, J.R.: Inelastic constitutive relations for solids: An internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455 (1971)zbMATHGoogle Scholar
  92. 92.
    Muschik, W.: Non-Equilibrium Thermodynamics with Application to Solids. Springer, New York (1993)zbMATHGoogle Scholar
  93. 93.
    Hull, D., Bacon, D.J.: Introduction to Dislocations, 5th edn. Butterworth-Heinemann, Oxford, UK (2011)Google Scholar
  94. 94.
    Anderson, P.M., Hirth, J.P., Lothe, J.: Theory of Dislocations, 3rd edn. Cambridge University Press (2017)Google Scholar
  95. 95.
    Nye, J.F.: Some geometrical relations in dislocated crystals. Acta Mater. 1(2), 153–162 (1953)Google Scholar
  96. 96.
    Hill, R., Sneddon, I.N. (eds.): Progress in Solid Mechanics, vol. 1, p. 330. North-Holland Publishing Company (1960)Google Scholar
  97. 97.
    Mishin, Y., Mehl, M.J., Papaconstantopoulos, D.A., Voter, A.F., Kress, J.D.: Structural stability and lattice defects in copper: Ab initio, tight-binding, and embedded-atom calculations. Phys. Rev. B 63(22), 224106 (2001)Google Scholar
  98. 98.
    Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995)zbMATHGoogle Scholar
  99. 99.
    Hirel, P.: Atomsk: A tool for manipulating and converting atomic data files. Comput. Phys. Commun. 197, 212–219 (2015)Google Scholar
  100. 100.
    Hartley, C.S., Mishin, Y.: Representation of dislocation cores using Nye tensor distributions. Mater. Sci. Eng. A 400, 18–21 (2005)Google Scholar
  101. 101.
    Gurrutxaga-Lerma, B., Balint, D.S., Dini, D., Eakins, D.E., Sutton, A.P.: A dynamic discrete dislocation plasticity method for the simulation of plastic relaxation under shock loading. Proc. R. Soc. A 469, 20130141 (2013)MathSciNetzbMATHGoogle Scholar
  102. 102.
    Chen, X., Chernatynskiy, A., Xiong, L., Chen, Y.: A coherent phonon pulse model for transient phonon thermal transport. Comput. Phys. Commun. 195, 112–116 (2015)Google Scholar
  103. 103.
    Ramesh, K.T.: Nanomaterials: Mechanics and Mechanisms. Springer (2009)Google Scholar
  104. 104.
    Kacher, J., Eftink, B.P., Cui, B., Robertson, I.M.: Dislocation interactions with grain boundaries. Curr. Opin. Solid State Mater. Sci. 18, 227–243 (2014)Google Scholar
  105. 105.
    Counts, W.A., Braginsky, M.V., Battaile, C.C., Holm, E.A.: Predicting the Hall-Petch effect in fcc metals using non-local crystal plasticity. Int. J. Plast. 24, 1243–1263 (2008)zbMATHGoogle Scholar
  106. 106.
    Spearot, D.E., Sangid, M.D.: Insights on slip transmission at grain boundaries from atomistic simulations. Curr. Opin. Solid State Mater. Sci. 18, 188–195 (2014)Google Scholar
  107. 107.
    Stukowski, A.: Structure identification methods for atomistic simulations of crystalline materials. Model. Simul. Mater. Sci. Eng. 20, 045021 (2012)Google Scholar
  108. 108.
    Mishin, Y., Farkas, D., Mehl, M.J., Papaconstantopoulos, D.A.: Interatomic potentials for monoatomic metals from experimental data and ab initio calculations. Phys. Rev. B 59, 3393 (1999)Google Scholar
  109. 109.
    Voter, A.F., Chen, S.P.: Accurate interatomic potentials for Ni, Al, and Ni3Al. Mater. Res. Soc. Symp. Proc. 82, 175 (1987)Google Scholar
  110. 110.
    Angelo, J.E., Moody, N.R., Baskes, M.I.: Trapping of hydrogen to lattice-defects in nickel. Model. Simul. Mater. Sci. Eng. 3, 289 (1995)Google Scholar
  111. 111.
    Foiles, S.M., Hoyt, J.J.: Computation of grain boundary stiffness and mobility from boundary fluctuations. Acta Mater. 54, 3351 (2006)Google Scholar
  112. 112.
    Zhou, X.W., Johnson, R.A., Wadley, H.N.G.: Misfit-energy-increasing dislocations in vapor-deposited CoFe/NiFe multilayers. Phys. Rev. B 69, 144113 (2004)Google Scholar
  113. 113.
    Lipkin, D.M., Clarke, D.R., Beltz, G.E.: A strain-gradient model of cleavage fracture in plastically deforming materials. Acta Mater. 44, 4051–4058 (1996)Google Scholar
  114. 114.
    Hussein, A.M., El-Awady, J.A.: Quantifying dislocation microstructure evolution and cyclic hardening in fatigued face-centered cubic single crystals. J. Mech. Phys. Solids 91, 126–144 (2016)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Shuozhi Xu
    • 1
  • Ji Rigelesaiyin
    • 2
  • Liming Xiong
    • 2
  • Youping Chen
    • 3
  • David L. McDowell
    • 4
    • 5
    Email author
  1. 1.California NanoSystems Institute, University of California, Santa BarbaraSanta BarbaraUSA
  2. 2.Department of Aerospace EngineeringIowa State UniversityAmesUSA
  3. 3.Department of Mechanical and Aerospace EngineeringUniversity of FloridaGainesvilleUSA
  4. 4.Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA
  5. 5.School of Materials Science and EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations