Energy and force transition between atoms and continuum in quasicontinuum method

  • Chang, Shu-Wei (Department of Civil Engineering, National Taiwan University) ;
  • Liao, Ying-Pao (Department of Civil Engineering, National Taiwan University) ;
  • Huang, Chang-Wei (Department of Civil Engineering, Chung-Yuan Christian University) ;
  • Chen, Chuin-Shan (Department of Civil Engineering, National Taiwan University)
  • Received : 2013.10.23
  • Accepted : 2013.11.11
  • Published : 2014.03.25


We present a full energy and force formulation of the quasicontinuum method with non-local and local transition elements. Non-local transition elements are developed to transmit inhomogeneity from the atomistic to the continuum regions. Local transition elements are developed to resolve the mathematical mismatch between non-local atoms and the local continuum. The rationale behind these transition elements is provided by analyzing the energy and force transitions between atoms and continuum under the Cauchy-Born rule. We show that breakdown of the Cauchy-Born rule occurs for slaved atoms of local elements within the cutoff of non-local atoms. The inadequacy of the Cauchy-Born rule at the transition region naturally leads to the need of atomistic treatment of transition slaved and transition representative atoms. Such an atomistic treatment together with a full or cutoff sampling allows non-local transition elements containing these transition entities to transmit inhomogeneity. Different force formulations for transition representative atoms and pure local representative atoms allow the local transition elements to resolve non-local and local mismatches. The method presented herein is validated by force calculations in an unstressed perfect crystal as well as an unrelaxed grain boundary model. A nanoindentation simulation in 3D is conducted to demonstrate the accuracy and efficiency of the proposed method.


  1. Chan, C.Y., Chen, Y.Y., Chang, S.W. and Chen, C.S. (2011), "Atomistic studies of nanohardness size effects", Int. J. Theoretical Appl. Multiscale Mech., 2(1), 62-71.
  2. Chen, C.S., Wang, C.K. and Chang, S.W. (2008), "Atomistic simulation and investigation of nanoindentation, contact pressure and nanohardness", Interact. Multiscale Mech., 1(4), 411-422.
  3. Chen, J. and Lee, J.D. (2010), "Atomistic analysis of nano/micro biosensors", Interact. Multiscale Mech., 3(2), 111-121.
  4. Curtin, W.A., Miller, R.E. (2003), Atomistic/continuum coupling in computational materials science. Modeling and Simulation in Materials Sciences and Engineering, 11, R33-R68.
  5. Jeong, J., Cho, M. and Choi, J. (2011), "Effective mechanical properties of micro/nano-scale porous materials considering surface effects", Interact. Multiscale Mech., 4(2), 107-122.
  6. Knap, J. and Ortiz, M. (2001), "An analysis of the quasicontinuum method", J. Mech. Physics Solids, 49, 1899-1923.
  7. Kulkarni, Y. (2007), "Coarse-graining of atomistic description at finite temperature", Ph.D. Thesis, California Institute of Technology.
  8. Lai, C.W. and Chen, C.S. (2013), "Influence of indenter shape on nanoindentation: an atomistic study", Interact. Multiscale Mech., 6(3), 301-316.
  9. Miller, R., Ortiz, M., Phillips, R., Shenoy, V. and Tadmor, E.B. (1998a), "Quasicontinuum models of fracture and plasticity", Eng. Fract. Mech., 61, 427-444.
  10. Miller, R., Tadmor, E.B., Phillips, R. and Ortiz, M. (1998b), "Quasicontinuum simulation of fracture at the atomic scale", Model. Simul. Mater. Sci. Eng., 6, 607-638.
  11. Miller, R. and Tadmor, E.B. (2003), "The quasicontinuum method: overview, applications and current directions", J. Comput.-Aided Mater. Des., 9, 203-39.
  12. Press, W.H., Vetterling, W.T., Teukolsky, S.A. and Flannery, B.P. (2000), Numerical Recipes in C++: The Art of Scientific Computing, Second Edition, Cambridge University Press.
  13. Rodney, D. and Phillips, R. (1999), "Structure and strength of dislocation junctions: an atomic level analysis", Phys. Rev. Lett., 82, 1704-1707.
  14. Shen, L. (2013), "Molecular dynamics study of Al solute-dislocation interactions in Mg alloys", Interact. Multiscale Mech., 6(2), 127-136.
  15. Shenoy, V.B., Miller, R., Tadmor, E.B., Phillips, R. and Ortiz, M. (1998), "Quasicontinuum models of 560 interfacial structure and deformation", Phys. Rev. Lett., 80, 742-745.
  16. Shenoy, V.B., Miller, R., Tadmor, E.B., Rodney, D., Phillips, R. and Ortiz, M. (1999a), "An adaptive finite element approach to atomic scale mechanics - the quasicontinuum method", J. Mech. Physics Solids, 47, 611-641.
  17. Shenoy, V., Shenoy, V. and Phillips, R. (1999), "Finite temperature quasicontinuum methods", Mater. Res. Soc. Symp. Proc., 538, 465-471.
  18. Shenoy, V.B., Phillips, R. and Tadmor, E.B. (2000), "Nucleation of dislocations beneath a plane strain indenter", J. Mech. Phys. Solids, 48, 649-673.
  19. Tadmor, E.B., Ortiz, M. and Phillips, R. (1996), "Quasicontinuum analysis of defects in solids", Philosophical Magazine A, 73, 1529-1563.
  20. Tadmor, E.B., Miller, R. and Phillips, R. (1999), "Nanoindentation and incipient plasticity", J. Mater. Res., 14, 2233-2250.
  21. Tadmor, E.B., Waghmare, U.V., Smith, G.S. and Kaxiras, E. (2002), "Polarization switching in $PbTiO_{3}$: An ab initio finite element simulation", Acta Mat., 50, 2989-3002.
  22. Teng, H., Lee, C.H. and Chen, J.S. (2011), "On the continuum formulation for modeling DNA loop formation, Interact. Multiscale Mech., 4(3), 219-237.
  23. Wang, Y.C., Wu, C.Y., Chen, C. and Yang, D.S. (2013), "Molecular dynamics studies of interaction between hydrogen and carbon nano-carriers", Interact. Multiscale Mech., 6(3), 271-286.
  24. Zhao, H. and Aluru, N.R. (2008), "Molecular dynamics simulation of bulk silicon under strain", Interact. Multiscale Mech., 1(2), 303-315.