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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.452 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2013005838
pages 527-541


Hossein Talebi
Institute of Structural Mechanics, Bauhaus University-Weimar
M. Silani
Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, 84156-83111, Iran
S. P. A. Bordas
Institute of Mechanics and Advanced Materials, Theoretical and Computational Mechanics, Cardiff University, Cardiff, CF24 3AA, United Kingdom
Pierre Kerfriden
Institute of Mechanics and Advanced Materials, Theoretical and Computational Mechanics, Cardiff University, Cardiff, CF24 3AA, United Kingdom
Timon Rabczuk
Institute of Structural Mechanics, Bauhaus-Universitat Weimar, Marienstr. 15, D-99423 Weimar, Germany


We propose a method to couple a three-dimensional continuum domain to a molecular dynamics domain to simulate propagating cracks in dynamics. The continuum domain is treated by an extended finite element method to handle the discontinuities. The coupling is based on the bridging domain method, which blends the continuum and atomistic energies. The Lennard-Jones potential is used to model the interactions in the atomistic domain, and the Cauchy-Born rule is used to compute the material behavior in the continuum domain. To our knowledge, it is the first time that a three dimensional extended bridging domain method is reported. To show the suitability of the proposed method, a three-dimensional crack problem with an atomistic region around the crack front is solved. The results show that the method is capable of handling crack propagation and dislocation nucleation.


  1. ABAQUS 6.11, Standard User‘s Manual.

  2. Aghaei, A., Abdolhosseini Qomi, M., Kazemi, M., and Khoei, A., Stability and size-dependency of cauchy-born hypothesis in three-dimensional applications. DOI: 10.1016/j.ijsolstr.2009.01.013

  3. Anciaux, G., Ramisetti, S., and Molinari, J., A finite temperature bridging domain method for MD-FE coupling and application to a contact problem. DOI: 10.1016/j.cma.2011.01.012

  4. Babuška, I., Caloz, G., and Osborn, J., Special finite element methods for a class of second order elliptic problems with rough coefficients. DOI: 10.1137/0731051

  5. Babu&#353;ka, I. and Melenk, J. M., The partition of unity method. DOI: 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N

  6. Barbieri, E., Petrinic, N., Meo, M., and Tagarielli, V. L., A new weight-function enrichment in meshless methods for multiple cracks in linear elasticity. DOI: 10.1002/nme.3313

  7. Belytschko, T. and Black, T., Elastic crack growth in finite elements with minimal remeshing. DOI: 10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S

  8. Belytschko, T. and Guidault, P. A., Bridging domain methods for coupled atomistic&mdash;continuum models with <i>L</i><sup>2</sup> or <i>H</i><sup>1</sup> couplings. DOI: 10.1002/nme.2461

  9. Belytschko, T. and Xiao, S. P., A bridging domain method for coupling continua with molecular dynamics. DOI: 10.1016/j.cma.2003.12.053

  10. Belytschko, T. and Xu, M., Conservation properties of the bridging domain method for coupled molecular/continuum dynamics. DOI: 10.1002/nme.2323

  11. Ben Dhia, H. R. G., The arlequin method as a flexible engineering design tool. DOI: 10.1002/nme.1229

  12. Borst, R. D., Aubertin, P., and R&#233;thor&#233;, J., A coupled molecular dynamics and extended finite element method for dynamic crack propagation. DOI: 10.1002/nme.2675

  13. Borst, R. D., Aubertin, P., and R&#233;thor&#233;, J., Energy conservation of atomistic/continuum coupling. DOI: 10.1002/nme.2542

  14. B&#252;hler, M., Atomistic Modeling of Materials Failure.

  15. Cavalcante Neto, J. B., Wawrzynek, P. A., Carvalho, M. T. M., Martha, L. F., and Ingraffea, A. R., An algorithm for threedimensional mesh generation for arbitrary regions with cracks. DOI: 10.1007/PL00007196

  16. Cleri, F., Wolf, D., Yip, S., and Phillpot, S. R., Atomistic simulation of dislocation nucleation and motion from a crack tip. DOI: 10.1016/S1359-6454(97)00214-0

  17. Duarte, C. and Oden, J., An HP adaptive method using clouds. DOI: 10.1016/S0045-7825(96)01085-7

  18. Duflot, M. and Nguyen-Dang, H., A meshless method with enriched weight functions for fatigue crack growth. DOI: 10.1002/nme.948

  19. Fries, T., A corrected xfem approximation without problems in blending elements. DOI: 10.1002/nme.2259

  20. Galland, F., Gravouil, A., Malvesin, E., and Rochette, M., A global model reduction approach for 3D fatigue crack growth with confined plasticity. DOI: 10.1016/j.cma.2010.08.018

  21. Gracie, R. and Belytschko, T., Concurrently coupled atomistic and XFEM models for dislocations and cracks. DOI: 10.1002/nme.2488

  22. Gracie, R. and Belytschko, T., An adaptive concurrent multiscale method for the dynamic simulation of dislocations. DOI: 10.1002/nme.3112

  23. Gracie, R., Oswald, J., and Belytschko, T., On a new extended finite element method for dislocations: Core enrichment and nonlinear formulation. DOI: 10.1016/j.jmps.2007.07.010

  24. Haile, J., Molecular Dynamics Simulation: Elementary Methods.

  25. Kelchner, C., Plimpton, S., and Hamilton, J., Dislocation nucleation and defect structure during surface indentation. DOI: 10.1103/PhysRevB.58.11085

  26. Kerfriden, P., Gosselet, P., Adhikari, S., Bordas, S., and Passieux, J., POD-based model order reduction for the simulation of strong nonlinear evolutions in structures: Application to damage propagation. DOI: 10.1088/1757-899X/10/1/012165

  27. Kerfriden, P., Gosselet, P., Adhikari, S., and Bordas, S. P. A., Bridging proper orthogonal decomposition methods and augmented Newton&#8211;Krylov algorithms: An adaptive model order reduction for highly nonlinear mechanical problems. DOI: 10.1016/j.cma.2010.10.009

  28. Kerfriden, P., Passieux, J., and Bordas, S., Local/global model order reduction strategy for the simulation of quasi-brittle fracture. DOI: 10.1002/nme.3234

  29. Lange, F., The interaction of a crack front with a second-phase dispersion. DOI: 10.1080/14786437008221068

  30. Menouillard, T., R&#233;thor&#233;, J., Combescure, A., and Bung, H., Efficient explicit time stepping for the extended finite element method (X-FEM). DOI: 10.1002/nme.1718

  31. Mi, C., Buttry, D., Sharma, P., and Kouris, D., Atomistic insights into dislocation-based mechanisms of void growth and coalescence. DOI: 10.1016/j.jmps.2011.05.008

  32. Miller, R. and Tadmor, E., The quasicontinuum method: Overview, applications and current directions. DOI: 10.1023/A:1026098010127

  33. Nguyen, V., Rabczuk, T., Bordas, S., and Duflot, M., Meshless methods: A review and computer implementation aspects. DOI: 10.1016/j.matcom.2008.01.003

  34. Nose, S., Constant-temperature molecular dynamics. DOI: 10.1088/0953-8984/2/S/013

  35. Oden, J. and Prudhomme, S., Estimation of modeling error in computational mechanics. DOI: 10.1006/jcph.2002.7183

  36. Oden, J. and Vemaganti, K., Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. Part I. Error estimates and adaptive algorithms. DOI: 10.1006/jcph.2000.6585

  37. Plimpton, S., Fast parallel algorithms for short-range molecular dynamics. DOI: 10.1006/jcph.1995.1039

  38. Plimpton, S., Atomistic Stress Simulator (WARP).

  39. Pozrikidis, C., On the applicability of the Cauchy-Born rule. DOI: 10.1016/j.commatsci.2009.03.031

  40. Press, W., Numerical Recipes in FORTRAN: the art of scientific computing.

  41. Rapaport, D., The Art of Molecular Dynamics Simulation.

  42. Ravi-Chandar, K. and Knauss,W. G., An experimental investigation into dynamic fracture: II Microstructural aspects. DOI: 10.1007/BF01152313

  43. Robert, J., Comments on virial theorems for bounded systems. DOI: 10.1119/1.13390

  44. Shenoy, V., Miller, R., Tadmor, E., Rodney, D., Phillips, R., and Ortiz, M., An adaptive finite element approach to atomic-scale mechanics&#8211;the quasicontinuum method.

  45. Simpson, R. and Trevelyan, J., Evaluation of <i>J</i><sub>1</sub> and <i>J</i><sub>2</sub> integrals for curved cracks using an enriched boundary element method. DOI: 10.1016/j.engfracmech.2010.12.006

  46. Simpson, R. and Trevelyan, J., A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics. DOI: 10.1016/j.cma.2010.06.015

  47. Strouboulis, T., Babu&#353;ka, I., and Copps, K., The design and analysis of the generalized finite element method. DOI: 10.1016/S0045-7825(99)00072-9

  48. Subramaniyan, A. and Sun, C., Continuum interpretation of virial stress in molecular simulations. DOI: 10.1016/j.ijsolstr.2008.03.016

  49. Sun, Y., Izumi, S., Hara, S., and Sakai, S., Anisotropy behavior of dislocation nucleation from a sharp corner in copper. DOI: 10.1299/jcst.5.54

  50. Tadmor, E., Ortiz, M., and Phillips, R., Quasicontinuum analysis of defects in solids. DOI: 10.1080/01418619608243000

  51. Talebi, H., Samaniego, C., Samaniego, E., and Rabczuk, T., On the numerical stability and mass-lumping schemes for explicit enriched meshfree methods. DOI: 10.1002/nme.3275

  52. Talebi, H., Zi, G., Silani, M., Samaniego, E., and Rabczuk, T., A simple circular cell method for multilevel finite element analysis. DOI: 10.1155/2012/526846

  53. Ventura, G., Xu, J., and Belytschko, T., A vector level set method and new discontinuity approximations for crack growth by EFG. DOI: 10.1002/nme.471

  54. Xiao, S. P. and Belytschko, T., Coupling methods for continuum model with molecular model.

  55. Xiong, L., Deng, Q., Tucker, G., McDowell, D., and Chen, Y., A concurrent scheme for passing dislocations from atomistic to continuum domains. DOI: 10.1016/j.actamat.2011.11.002

  56. Zhang, Y., Hughes, T., and Bajaj, C., An automatic 3D mesh generation method for domains with multiple materials. DOI: 10.1016/j.cma.2009.06.007

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