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The Quasicontinuum Method: Theory and Applications

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Multiscale Materials Modeling for Nanomechanics

Part of the book series: Springer Series in Materials Science ((SSMATERIALS,volume 245))

Abstract

The quasicontinuum (QC) method has become a popular technique to bridge the gap between atomistic and continuum length scales in crystalline solids. In contrast to many other concurrent scale-coupling methods, the QC method only relies upon constitutive information on the lowest scale (viz., on interatomic potentials) and thus avoids empirical constitutive laws at the larger scales. This is achieved by the application of a seamless coarse-graining scheme to the discrete atomistic ensemble and the careful selection of a small set of representative atoms. Since its inception almost two decades ago, many different variants and flavors of the QC method have been developed, not only to study the mechanics of solids but also to describe such physical phenomena as mass and heat transfer, or to efficiently describe fiber networks and truss structures. Here, we review the theoretical fundamentals and give a (non-exhaustive) overview of the state of the art in QC theory, computational methods, and applications. We particularly emphasize the fully nonlocal QC formulation which adaptively ties atomistic resolution to moving defects, and we illustrate simulation results based on this framework. Finally, we point out challenges and open questions associated with the QC methodology.

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References

  1. A. Abdulle, P. Lin, A.V. Shapeev, Numerical methods for multilattices. Multiscale Model. Simul. 10 (3), 696–726 (2012)

    Article  Google Scholar 

  2. F.F. Abraham, J.Q. Broughton, N. Bernstein, E. Kaxiras, Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture. Europhys. Lett. 44 (6), 783 (1998)

    Google Scholar 

  3. J.S. Amelang, D.M. Kochmann, Surface effects in nanoscale structures investigated by a fully-nonlocal energy-based quasicontinuum method, Mechanics of Materials 90, 166–184 (2015)

    Article  Google Scholar 

  4. J.S. Amelang, G.N. Venturini, D.M. Kochmann, Summation rules for a fully-nonlocal energy-based quasicontinuum method, J. Mech. Phys. Solids 82, 378–413 (2015)

    Article  Google Scholar 

  5. J.S. Amelang, A fully-nonlocal energy-based formulation and high-performance realization of 842 the quasicontinuum method. PhD thesis, California Institute of Technology (2015)

    Google Scholar 

  6. M. Ariza, I. Romero, M. Ponga, M. Ortiz, Hotqc simulation of nanovoid growth under tension in copper. Int. J. Fract. 174, 75–85 (2012)

    Article  Google Scholar 

  7. L.A.A. Beex, R.H.J. Peerlings, M.G.D. Geers, A quasicontinuum methodology for multiscale analyses of discrete microstructural models. Int. J. Numer. Methods Eng. 87 (7), 701–718 (2011)

    Article  Google Scholar 

  8. L.A.A. Beex, C.W. Verberne, R.H.J. Peerlings, Experimental identification of a lattice model for woven fabrics: application to electronic textile. Compos. Part A Appl Sci. Manuf. 48, 82–92 (2013)

    Article  Google Scholar 

  9. L.A.A. Beex, R.H.J. Peerlings, M.G.D. Geers, A multiscale quasicontinuum method for dissipative lattice models and discrete networks. J. Mech. Phys. Solids 64, 154–169 (2014)

    Article  Google Scholar 

  10. L.A.A. Beex, R.H.J. Peerlings, M.G.D. Geers, A multiscale quasicontinuum method for lattice models with bond failure and fiber sliding. Comput. Methods Appl. Mech. Eng. 269, 108–122 (2014)

    Article  Google Scholar 

  11. L.A.A. Beex, R.H.J. Peerlings, M.G.D. Geers, Central summation in the quasicontinuum method. J. Mech. Phys. Solids 70, 242–261 (2014)

    Article  Google Scholar 

  12. T. Belytschko, S.P. Xiao, Coupling methods for continuum model with molecular model. Int. J. Multiscale Comput. Eng. 1 (1), 115–126 (2003)

    Article  Google Scholar 

  13. S. Brinckmann, D.K. Mahajan, A. Hartmaier, A scheme to combine molecular dynamics and dislocation dynamics. Model. Simul. Mater. Sci. Eng. 20 (4), 045001 (2012)

    Google Scholar 

  14. J.Q. Broughton, F.F. Abraham, N. Bernstein, E. Kaxiras, Concurrent coupling of length scales: methodology and application. Phys. Rev. B 60, 2391–2403 (1999)

    Article  Google Scholar 

  15. P.W. Chung, Computational method for atomistic homogenization of nanopatterned point defect structures. Int. J. Numer. Methods Eng. 60 (4), 833–859 (2004)

    Article  Google Scholar 

  16. J.D. Clayton, P.W. Chung, An atomistic-to-continuum framework for nonlinear crystal mechanics based on asymptotic homogenization. J. Mech. Phys. Solids 54 (8), 1604–1639 (2006)

    Article  Google Scholar 

  17. W.A. Curtin, R.E. Miller, Atomistic/continuum coupling in computational materials science. Model. Simul. Mater. Sci. Eng. 11 (3), R33 (2003)

    Google Scholar 

  18. X.D. Dai, Y. Kong, J.H. Li, B.X. Liu, Extended finnin-sinclair potential for bcc and fcc metals and alloys. J. Phys. Condens. Matter 18, 4527–4542 (2006)

    Article  Google Scholar 

  19. M.S. Daw, M.I. Baskes, Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B 29, 6443–6453 (1984)

    Article  Google Scholar 

  20. K. Dayal, J. Marshall, A multiscale atomistic-to-continuum method for charged defects in electronic materials, in Presented as the Society of Engineering Science 2011 Annual Technical Conference, Evanston (2011)

    Google Scholar 

  21. D.J. Diestler, Z.B. Wu, X.C. Zeng, An extension of the quasicontinuum treatment of multiscale solid systems to nonzero temperature. J. Chem. Phys. 121, 9279–9282 (2004)

    Article  Google Scholar 

  22. M. Dobson, R.S. Elliott, M. Luskin, E.B. Tadmor, A multilattice quasicontinuum for phase transforming materials: cascading cauchy born kinematics. J. Comput. Aided Mater. Des. 14, 219–237 (2007)

    Article  Google Scholar 

  23. M. Dobson, R.S. Elliott, M. Luskin, E.B. Tadmor, A multilattice quasicontinuum for phase transforming materials: cascading cauchy born kinematics. J. Comput. Aided Mater. Des. 14 (1), 219–237 (2007)

    Article  Google Scholar 

  24. M. Dobson, M. Luskin, C. Ortner, Accuracy of quasicontinuum approximations near instabilities. J. Mech. Phys. Solids 58, 1741–1757 (2010)

    Article  Google Scholar 

  25. M. Dobson, M. Luskin, C. Ortner, Sharp stability estimates for the force-based quasicontinuum approximation of homogeneous tensile deformation. Multiscale Model. Simul. 8, 782–802 (2010)

    Article  Google Scholar 

  26. M. Dobson, M. Luskin, C. Ortner, Stability, instability, and error of the force-based quasicontinuum approximation. Arch. Ration. Mech. Anal. 197, 179–202 (2010)

    Article  Google Scholar 

  27. L.M. Dupuy, E.B. Tadmor, R.E. Miller, R. Phillips, Finite-temperature quasicontinuum: molecular dynamics without all the atoms. Phys. Rev. Lett. 95, 060202 (2005)

    Article  Google Scholar 

  28. B. Eidel, A. Stukowski, A variational formulation of the quasicontinuum method based on energy sampling in clusters. J. Mech. Phys. Solids 57, 87–108 (2009)

    Article  Google Scholar 

  29. M.I. Espanol, D.M. Kochmann, S. Conti, M. Ortiz, A gamma-convergence analysis of the quasicontinuum method. SIAM Multiscale Model. Simul. 11, 766–794 (2013)

    Article  Google Scholar 

  30. F. Feyel, J.-L. Chaboche, Fe2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre sic/ti composite materials. Comput. Methods Appl. Mech. Eng. 183 (3–4), 309–330 (2000)

    Article  Google Scholar 

  31. S.M. Foiles, M.I. Baskes, M.S. Daw, Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys. Phys. Rev. B 33, 7983–7991 (1986)

    Article  Google Scholar 

  32. V. Gavini, K. Bhattacharya, M. Ortiz, Quasi-continuum orbital-free density-functional theory: a route to multi-million atom non-periodic DFT calculation. J. Mech. Phys. Solids 55 (4), 697–718 (2007)

    Article  Google Scholar 

  33. M. Gunzburger, Y. Zhang, A quadrature-rule type approximation to the quasi-continuum method. Multiscale Model. Simul. 8 (2), 571–590 (2010)

    Article  Google Scholar 

  34. S. Hai, E.B. Tadmor, Deformation twinning at aluminum crack tips. Acta Mater. 51, 117–131 (2003)

    Article  Google Scholar 

  35. K. Hardikar, V. Shenoy, R. Phillips, Reconciliation of atomic-level and continuum notions concerning the interaction of dislocations and obstacles. J. Mech. Phys. Solids 49 (9), 1951–1967 (2001)

    Article  Google Scholar 

  36. L. Huai-Bao, L. Jun-Wan, N. Yu-Shan, M. Ji-Fa, W. Hong-Sheng, Multiscale analysis of defect initiation on the atomistic crack tip in body-centered-cubic metal Ta. Acta Phys. Sin. 60 (10), 106101 (2011)

    Google Scholar 

  37. M. Iyer, V. Gavini, A field theoretical approach to the quasi-continuum method. J. Mech. Phys. Sol. 59 (8), 1506–1535 (2011)

    Article  Google Scholar 

  38. S. Izvekov, G.A. Voth, A multiscale coarse-graining method for biomolecular systems. J. Phys. Chem. B 109 (7), 2469–2473 (2005)

    Article  Google Scholar 

  39. R.A. Johnson, Alloy models with the embedded-atom method. Phys. Rev. B 39, 12554–12559 (1989)

    Article  Google Scholar 

  40. C.L. Kelchner, S.J. Plimpton, J.C. Hamilton, Dislocation nucleation and defect structure during surface indentation. Phys. Rev. B 58, 11085–11088 (1998)

    Article  Google Scholar 

  41. W.K. Kim, M. Luskin, D. Perez, A.F. Voter, E.B. Tadmor, Hyper-qc: an accelerated finite-temperature quasicontinuum method using hyperdynamics. J. Mech. Phys. Solids 63, 94–112 (2014)

    Article  Google Scholar 

  42. J. Knap, M. Ortiz, An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49 (9), 1899–1923 (2001)

    Article  Google Scholar 

  43. D.M. Kochmann, G.N. Venturini, A meshless quasicontinuum method based on local maximum-entropy interpolation. Model. Simul. Mater. Sci. Eng. 22, 034007 (2014)

    Article  Google Scholar 

  44. Y. Kulkarni, Coarse-graining of atomistic description at finite temperature. PhD thesis, California Institute of Technology (2007)

    Google Scholar 

  45. Y. Kulkarni, J. Knap, M. Ortiz, A variational approach to coarse graining of equilibrium and non-equilibrium atomistic description at finite temperature. J. Mech. Phys. Solids 56, 1417–1449 (2008)

    Article  Google Scholar 

  46. S. Kwon, Y. Lee, J.Y. Park, D. Sohn, J.H. Lim, S. Im, An efficient three-dimensional adaptive quasicontinuum method using variable-node elements. J. Comput. Phys. 228 (13), 4789–4810 (2009)

    Article  Google Scholar 

  47. J. Li, J. Mei, Y. Ni, H. Lu, W. Jiang, Two-dimensional quasicontinuum analysis of the strengthening and weakening effect of cu/ag interface on nanoindentation. J. Appl. Phys. 108 (5), 054309 (2010)

    Google Scholar 

  48. J. Li, H. Lu, Y. Ni, J. Mei, Quasicontinuum study the influence of misfit dislocation interactions on nanoindentation. Comput. Mater. Sci. 50 (11), 3162–3170 (2011)

    Article  Google Scholar 

  49. X.H. Li, M. Luskin, C. Ortner, A.V. Shapeev, Theory-based benchmarking of the blended force-based quasicontinuum method. Comput. Methods Appl. Mech. Eng. 268, 763–781 (2014)

    Article  Google Scholar 

  50. W.K. Liu, H.S. Park, D. Qian, E.G. Karpov, H. Kadowaki, G.J. Wagner, Bridging scale methods for nanomechanics and materials. Comput. Methods Appl. Mech. Eng. 195 (13–16), 1407–1421 (2006)

    Article  Google Scholar 

  51. H. Lu, Y. Ni, J. Mei, J. Li, H. Wang, Anisotropic plastic deformation beneath surface step during nanoindentation of fcc al by multiscale analysis. Comput. Mater. Sci. 58, 192–200 (2012)

    Article  Google Scholar 

  52. M. Luskin, C. Ortner, An analysis of node-based cluster summation rules in the quasicontinuum method. SIAM J. Numer. Anal. 47 (4), 3070–3086 (2009)

    Article  Google Scholar 

  53. M. Luskin, C. Ortner, B. Van Koten, Formulation and optimization of the energy-based blended quasicontinuum method. Comput. Methods Appl. Mech. Eng. 253, 160–168 (2013)

    Article  Google Scholar 

  54. J. Marian, G. Venturini, B.L. Hansen, J. Knap, M. Ortiz, G.H. Campbell, Finite-temperature extension of the quasicontinuum method using langevin dynamics: entropy losses and analysis of errors. Model. Simul. Mater. Sci. Eng. 18 (1), 015003 (2010)

    Google Scholar 

  55. J. Marshall, K. Dayal, Atomistic-to-continuum multiscale modeling with long-range electrostatic interactions in ionic solids. J. Mech. Phys. Solids 62, 137–162 (2014). Sixtieth anniversary issue in honor of Professor Rodney Hill

    Google Scholar 

  56. C. Miehe, J. Schröder, J. Schotte, Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials. Comput. Methods Appl. Mech. Eng. 171, 387–418 (1999)

    Article  Google Scholar 

  57. R.E. Miller, E.B. Tadmor, A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model. Simul. Mater. Sci. Eng. 17, 053001 (2009)

    Article  Google Scholar 

  58. R.E Miller, M. Ortiz, R. Phillips, V. Shenoy, E.B. Tadmor, Quasicontinuum models of fracture and plasticity. Eng. Fract. Mech. 61, 427–444 (1998)

    Google Scholar 

  59. A.K. Nair, D.H. Warner, R.G. Hennig, W.A. Curtin, Coupling quantum and continuum scales to predict crack tip dislocation nucleation. Scr. Mater. 63 (12), 1212–1215 (2010)

    Article  Google Scholar 

  60. M. Ortiz, L. Stainier, The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Eng. 171 (3–4), 419–444 (1999)

    Article  Google Scholar 

  61. C. Ortner, A priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1d. Math. Comput. 80 (275), 1265–1285 (2011)

    Article  Google Scholar 

  62. H.S. Park, E.G. Karpov, W.K. Liu, P.A. Klein, The bridging scale for two-dimensional atomistic/continuum coupling. Philos. Mag. 85 (1), 79–113 (2005)

    Article  Google Scholar 

  63. J.Y. Park, S. Im, Adaptive nonlocal quasicontinuum for deformations of curved crystalline structures. Phys. Rev. B 77, 184109 (2008)

    Article  Google Scholar 

  64. J.Y. Park, C.-H. Park, J.-S. Park, K.-J. Kong, H. Chang, S. Im, Multiscale computations for carbon nanotubes based on a hybrid qm/qc (quantum mechanical and quasicontinuum) approach. J. Mech. Phys. Solids 58 (2), 86–102 (2010)

    Article  Google Scholar 

  65. R. Phillips, D. Rodney, V. Shenoy, E. Tadmor, M. Ortiz, Hierarchical models of plasticity: dislocation nucleation and interaction. Model. Simul. Mater. Sci. Eng. 7, 769–780 (1999)

    Article  Google Scholar 

  66. M. Ponga, M. Ortiz, M.P. Ariza, Finite-temperature non-equilibrium quasi-continuum analysis of nanovoid growth in copper at low and high strain rates. Mech. Mater. 90, 253–267 (2015)

    Article  Google Scholar 

  67. A. Ramasubramaniam, M. Itakura, E.A. Carter, Interatomic potentials for hydrogen in α-iron based on density functional theory. Phys. Rev. B 79, 174101 (2009)

    Article  Google Scholar 

  68. I. Ringdalen Vatne, E. Ostby, C. Thaulow, Multiscale simulations of mixed-mode fracture in bcc-fe. Model. Simul. Mater. Sci. Eng. 19 (8), 085006 (2011)

    Google Scholar 

  69. R.E. Rudd, J.Q. Broughton, Coarse-grained molecular dynamics: nonlinear finite elements and finite temperature. Phys. Rev. B 72, 144104 (2005)

    Article  Google Scholar 

  70. J. Schröder, Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Stabilitätsproblemen. Habilitation thesis, Universität Stuttgart (2000)

    Google Scholar 

  71. V.B. Shenoy, R. Miller, E.B. Tadmor, R. Phillips, M. Ortiz, Quasicontinuum models of interfacial structure and deformation. Phys. Rev. Lett. 80, 742–745 (1998)

    Article  Google Scholar 

  72. V.B. Shenoy, V. Shenoy, R. Phillips, Finite temperature quasicontinuum methods. Mater. Res. Soc. Symp. Proc. 538, 465–471 (1999)

    Article  Google Scholar 

  73. M.S. Shephard, C. Picu, D.K. Datta, Composite grid atomistic continuum method: an adaptive approach to bridge continuum with atomistic analysis. Int. J. Multiscale Comput. Eng. 2 (3) (2004)

    Google Scholar 

  74. L.E. Shilkrot, R.E. Miller, W.A. Curtin, Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics. J. Mech. Phys. Solids 52, 755–787 (2004)

    Article  Google Scholar 

  75. T. Shimokawa, J.J. Mortensen, J. Schiøtz, K.W. Jacobsen, Matching conditions in the quasicontinuum method: removal of the error introduced at the interface between the coarse-grained and fully atomistic region. Phys. Rev. B 69, 214104 (2004)

    Article  Google Scholar 

  76. T. Shimokawa, T. Kinari, S. Shintaku, Interaction mechanism between edge dislocations and asymmetrical tilt grain boundaries investigated via quasicontinuum simulations. Phys. Rev. B 75, 144108 (2007)

    Article  Google Scholar 

  77. V. Sorkin, R.S. Elliott, E.B. Tadmor, A local quasicontinuum method for 3d multilattice crystalline materials: application to shape-memory alloys. Model. Simul. Mater. Sci. Eng. 22 (5), 055001 (2014)

    Google Scholar 

  78. P. Suryanarayana, Coarse-graining Kohn-Sham density functional theory. PhD thesis, California Institute of Technology (2011)

    Google Scholar 

  79. P. Suryanarayana, K. Bhattacharya, M. Ortiz, Coarse-graining Kohn-Sham Density Functional Theory. J. Mech. Phys. Solids 61 (1), 38–60 (2013)

    Article  Google Scholar 

  80. E.B. Tadmor, S. Hai, A Peierls criterion for the onset of deformation twinning at a crack tip. J. Mech. Phys. Solids 51 (5), 765–793 (2003)

    Article  Google Scholar 

  81. E.B. Tadmor, R.E. Miller, http://qcmethod.org/ (2016)

  82. E.B. Tadmor, M. Ortiz, R. Phillips, Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (6), 1529–1563 (1996)

    Article  Google Scholar 

  83. E.B. Tadmor, R. Miller, R. Philipps, M. Ortiz, Nanoindentation and incipient plasticity. J. Mater. Res. 14, 2233–2250 (1999)

    Article  Google Scholar 

  84. E.B. Tadmor, U.V. Waghmare, G.S. Smith, E. Kaxiras, Polarization switching in PbTiO3: an ab initio finite element simulation. Acta Mater. 50 (11), 2989–3002 (2002)

    Article  Google Scholar 

  85. E.B. Tadmor, F. Legoll, W. Kim, L. Dupuy, R. Miller, Finite-temperature quasi-continuum. Appl. Mech. Rev. 65 (1), 010803 (2013)

    Google Scholar 

  86. Z. Tang, H. Zhao, G. Li, N.R. Aluru, Finite-temperature quasicontinuum method for multiscale analysis of silicon nanostructures. Phys. Rev. B 74, 064110 (2006)

    Article  Google Scholar 

  87. G.N. Venturini, Topics in multiscale modeling of metals and metallic alloys. PhD thesis, California Institute of Technology (2011)

    Google Scholar 

  88. G.N. Venturini, K. Wang, I. Romero, M.P. Ariza, M. Ortiz, Atomistic long-term simulation of heat and mass transport. J. Mech. Phys. Solids 73, 242–268 (2014)

    Article  Google Scholar 

  89. A.F. Voter, A method for accelerating the molecular dynamics simulation of infrequent events. J. Chem. Phys. 106 (11), 4665–4677 (1997)

    Article  Google Scholar 

  90. A.F. Voter, F. Montalenti, T.C. Germann, Extending the time scale in atomistic simulation of materials. Annu. Rev. Mater. Res. 32 (1), 321–346 (2002)

    Article  Google Scholar 

  91. K.G. Wang, M. Ortiz, M.P. Ariza, Long-term atomistic simulation of hydrogen diffusion in metals. Int. J. Hydrog. Energy 40 (15), 5353–5358 (2015)

    Article  Google Scholar 

  92. X. Wang, X. Guo, Quasi-continuum model for the finite deformation of single-layer graphene sheets based on the temperature-related higher order cauchy-born rule. J. Comput. Theor. Nanosci. 10 (1), 154–164 (2013)

    Article  Google Scholar 

  93. L. Ward, A. Agrawal, K.M. Flores, W. Windl, Rapid production of accurate embedded-atom method potentials for metal alloys. ArXiv e-prints (2012)

    Google Scholar 

  94. S. Xiao, W. Yang, A temperature-related homogenization technique and its implementation in the meshfree particle method for nanoscale simulations. Int. J. Numer. Methods Eng. 69 (10), 2099–2125 (2007)

    Article  Google Scholar 

  95. Q. Yang, E. Biyikli, A.C. To, Multiresolution molecular mechanics: statics. Comput. Methods Appl. Mech. Eng. 258, 26–38 (2013)

    Article  Google Scholar 

  96. W. Yang, S. Xiao, The applications of meshfree particle methods at the nanoscale, in Computational Science – ICCS 2005, ed. by V.S. Sunderam, G.D. Albada, P.M.A. Sloot, J. Dongarra. Lecture Notes in Computer Science, vol. 3516 (Springer, Berlin/Heidelberg, 2005), pp. 284–291

    Google Scholar 

  97. H. Yoshihiko, Y. Nobuhiro, S.V. Dmitriev, K. Masanori, T. Shingo, Large scale atomistic simulation of cu/al2o3 interface via quasicontinuum analysis. J. Jpn. Inst. Metals 69 (1), 90–95 (2005)

    Article  Google Scholar 

  98. W. Yu, S. Shen, Initial dislocation topologies of nanoindentation into copper film with a nanocavity. Eng. Fract. Mech. 77 (16), 3329–3340 (2010)

    Article  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge support from the National Science Foundation (NSF) under grant number CMMI-123436.

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Authors and Affiliations

  1. California Institute of Technology, Graduate Aerospace Laboratories, 1200 E California Blvd MC 105-50, Pasadena, CA, 91125, USA

    Dennis M. Kochmann

  2. Harvey Mudd College, 301 Platt Blvd, Claremont, CA, 91711, USA

    Jeffrey S. Amelang

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Correspondence to Dennis M. Kochmann .

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  1. Drexel University, Philadelphia, Pennsylvania, USA

    Christopher R. Weinberger

  2. Drexel University, Philadelphia, Pennsylvania, USA

    Garritt J. Tucker

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Kochmann, D.M., Amelang, J.S. (2016). The Quasicontinuum Method: Theory and Applications. In: Weinberger, C., Tucker, G. (eds) Multiscale Materials Modeling for Nanomechanics. Springer Series in Materials Science, vol 245. Springer, Cham. https://doi.org/10.1007/978-3-319-33480-6_5

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