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MD/FE Multiscale Modeling of Contact

  • Srinivasa Babu Ramisetti
  • Guillaume Anciaux
  • Jean-Francois Molinari
Chapter
Part of the NanoScience and Technology book series (NANO)

Abstract

Limitations of single scale approaches to study the complex physics involved in friction have motivated the development of multiscale models. We review the state-of-the-art multiscale models that have been developed up to date. These have been successfully applied to a variety of physical problems, but that were limited, in most cases, to zero Kelvin studies. We illustrate some of the technical challenges involved with simulating a frictional sliding problem, which by nature generates a large amount of heat. These challenges can be overcome by a proper usage of spatial filters, which we combine to a direct finite-temperature multiscale approach coupling molecular dynamics with finite elements. The basic building block relies on the proper definition of a scale transfer operator using the least square minimization and spatial filtering. Then, the restitution force from the generalized Langevin equation is modified to perform a two-way thermal coupling between the two models. Numerical examples are shown to illustrate the proposed coupling formulation.

Keywords

Molecular Dynamic Coarse Scale Random Force Multiscale Method High Frequency Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This material is based on the work supported by the Swiss National Foundation under Grant no 200021_122046/1 and the European Research Council Starting Grant no 240332.

References

  1. 1.
    H. Czichos, Tribology (Elsevier, Amsterdam 1978)Google Scholar
  2. 2.
    A. Majumdar, B. Bhushan, Role of fractal geometry in roughness characterization and contact mechanics of surfaces. J. Tribol. 112(2), 205–216 (1990)CrossRefGoogle Scholar
  3. 3.
    B.N.J. Persson, Contact mechanics for randomly rough surfaces. Surf. Sci. Rep. 61(4), 201–227 (2006)ADSCrossRefGoogle Scholar
  4. 4.
    S.B. Ramisetti, C. Campa, Anciaux, J.F. Molinari, M.H. Mser, M.O. Robbins, The autocorrelation function for island areas on self-affine surfaces. J. Phys. Condens. Matter 23(21), 215004 (2011)Google Scholar
  5. 5.
    B. Luan, M.O. Robbins, The breakdown of continuum models for mechanical contacts. Nature 435(7044), 929–932 (2005)ADSCrossRefGoogle Scholar
  6. 6.
    G.V. Dedkov, Experimental and theoretical aspects of the modern nanotribology. Phys. Status Solidi A 179(1), 375 (2000)CrossRefGoogle Scholar
  7. 7.
    J. Gao, W.D. Luedtke, D. Gourdon, M. Ruths, J.N. Israelachvili, U. Landman, Frictional forces and Amontons’ law: From the molecular to the macroscopic scale. J. Phys. Chem. B 108(11), 3410–3425 (2004)CrossRefGoogle Scholar
  8. 8.
    J.O. Koskilinna, M. Linnolahti, T.A. Pakkanen, Friction coefficient for hexagonal boron nitride surfaces from ab initio calculations. Tribol. Lett. 24(1), 37–41 (2006)CrossRefGoogle Scholar
  9. 9.
    M. Renouf, F. Massi, N. Fillot, A. Saulot, Numerical tribology of a dry contact. Tribol. Int. 44(78), 834–844 (2011)CrossRefGoogle Scholar
  10. 10.
    J.F. Jerier, J.F. Molinari, Normal contact between rough surfaces by the discrete element method. Tribol. Int. 47, 1–8 (2012)CrossRefGoogle Scholar
  11. 11.
    V.S. Deshpande, A. Needleman, E. Van der Giessen, Discrete dislocation plasticity analysis of static friction. Acta Mater. 52(10), 3135–3149 (2004)CrossRefGoogle Scholar
  12. 12.
    S. Hyun, L. Pei, J.F. Molinari, M.O. Robbins, Finite-element analysis of contact between elastic self-affine surfaces. Phys. Rev. E 70(2), 026117 (2004)ADSCrossRefGoogle Scholar
  13. 13.
    P. Wriggers, T.A. Laursen, Computational Contact Mechanics (Springer, Dordrecht, 2008)Google Scholar
  14. 14.
    B. Luan, M.O. Robbins, Contact of single asperities with varying adhesion: Comparing continuum mechanics to atomistic simulations. Phys. Rev. E 74(2), 026111 (2006)ADSCrossRefGoogle Scholar
  15. 15.
    Y. Mo, I. Szlufarska, Roughness picture of friction in dry nanoscale contacts. Phys. Rev. B 81(3), 035405 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    P. Spijker, G. Anciaux, J.F. Molinari, The effect of loading on surface roughness at the atomistic level. Comput. Mech. 50(3), 273–283 (2011)CrossRefGoogle Scholar
  17. 17.
    T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (Dover Publications, New York, 2000)Google Scholar
  18. 18.
    O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis & Fundamentals (Elsevier Butterworth-Heinemann, Amsterdam, 2005)Google Scholar
  19. 19.
    D.C. Rapaport, The Art of Molecular Dynamics Simulation (Cambridge University Press, 2004)Google Scholar
  20. 20.
    K. Komvopoulos, J. Yang, Dynamic analysis of single and cyclic indentation of an elasticplastic multi-layered medium by a rigid fractal surface. J. Mech. Phys. Solids 54(5), 927–950 (2006)ADSzbMATHCrossRefGoogle Scholar
  21. 21.
    K. Komvopoulos, Z.Q. Gong, Stress analysis of a layered elastic solid in contact with a rough surface exhibiting fractal behavior. Int. J. Solids Struct. 44(78), 2109–2129 (2007)zbMATHCrossRefGoogle Scholar
  22. 22.
    K. Komvopoulos, Effects of multi-scale roughness and frictional heating on solid body contact deformation. C. R. Mnique 336(12), 149–162 (2008)zbMATHGoogle Scholar
  23. 23.
    S. Hyun, M.O. Robbins, Elastic contact between rough surfaces: Effect of roughness at large and small wavelengths. Tribol. Int. 40(10–12), 1413–1422 (2007)CrossRefGoogle Scholar
  24. 24.
    H.J.C. Berendsen, Simulating the Physical World: Hierarchical Modeling from Quantum Mechanics to Fluid Dynamics (Cambridge University Press, 2007)Google Scholar
  25. 25.
    M. Griebel, S. Knapek, G. Zumbusch, Numerical Simulation in Molecular Dynamics: Numerics, Algorithms, Parallelization, Applications (Springer, November 2010)Google Scholar
  26. 26.
    J. Rottler, M.O. Robbins, Macroscopic friction laws and shear yielding of glassy solids. Comput. Phys. Commun. 169(13), 177–182 (2005)ADSCrossRefGoogle Scholar
  27. 27.
    O.M. Braun, A.G. Naumovets, Nanotribology: Microscopic mechanisms of friction. Surf. Sci. Rep. 60(67), 79–158 (2006)ADSCrossRefGoogle Scholar
  28. 28.
    H.H. Yu, P. Shrotriya, Y.F. Gao, K.S. Kim, Micro-plasticity of surface steps under adhesive contact: Part I surface yielding controlled by single-dislocation nucleation. J. Mech. Phys. Solids 55(3), 489–516 (2007)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    C. Campa\(\tilde{\rm n}\)á, M.H. Müser, Contact mechanics of real vs. randomly rough surfaces: a Green’s function molecular dynamics study. EPL (Europhysics Letters) 77(3), 38005 (2007)Google Scholar
  30. 30.
    H.J. Kim, W.K. Kim, M.L. Falk, D.A. Rigney, MD simulations of microstructure evolution during high-velocity sliding between crystalline materials. Tribol. Lett. 31(1), 67–67 (2008)CrossRefGoogle Scholar
  31. 31.
    C. Yang, B.N.J. Persson, Contact mechanics: contact area and interfacial separation from small contact to full contact. J. Phys.: Condens. Matter 20(21), 215214 (2008)ADSGoogle Scholar
  32. 32.
    T. Liu, G. Liu, P. Wriggers, S. Zhu, Study on contact characteristic of nanoscale asperities by using molecular dynamics simulations. J. Tribol. 131(2), 022001–022001 (2009)CrossRefGoogle Scholar
  33. 33.
    P. Spijker, G. Anciaux, J.F. Molinari, Dry sliding contact between rough surfaces at the atomistic scale. Tribol. Lett. 44(2), 279–285 (2011)CrossRefGoogle Scholar
  34. 34.
    P. Spijker, G. Anciaux, J.F. Molinari, Relations between roughness, temperature and dry sliding friction at the atomic scale. Tribol. Int. 59, 222–229 (2013)CrossRefGoogle Scholar
  35. 35.
    F.F. Abraham, R. Walkup, H. Gao, M. Duchaineau, T.D.D.L. Rubia, M. Seager, Simulating materials failure by using up to one billion atoms and the world’s fastest computer: work-hardening. Proc. Nat. Acad. Sci. 99(9), 5783–5787 (2002)ADSCrossRefGoogle Scholar
  36. 36.
    J. Broughton, F. Abraham, N. Bernstein, E. Kaxiras, Concurrent coupling of length scales: Methodology and application. Phys. Rev. B 60(4), 2391–2403 (1999)ADSCrossRefGoogle Scholar
  37. 37.
    R. Miller, E.B. Tadmor, R. Phillips, M. Ortiz, Quasicontinuum simulation of fracture at the atomic scale. Modell. Simul. Mater. Sci. Eng. 6(5), 607–638 (1998)ADSCrossRefGoogle Scholar
  38. 38.
    V.B. Shenoy, R. Miller, E.B. Tadmor, R. Phillips, M. Ortiz, Quasicontinuum models of interfacial structure and deformation. Phys. Rev. Lett. 80(4), 742–745 (1998)ADSCrossRefGoogle Scholar
  39. 39.
    W.A. Curtin, R.E. Miller, Atomistic/continuum coupling in computational materials science. Modell. Simul. Mater. Sci. Eng. 11(3), R33–R68 (2003)ADSCrossRefGoogle Scholar
  40. 40.
    W.K. Liu, E.G. Karpov, S. Zhang, H.S. Park, An introduction to computational nanomechanics and materials. Comput. Methods Appl. Mech. Eng. 193(17–20), 1529–1578 (2004)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    H.S. Park, W.K. Liu, An introduction and tutorial on multiple-scale analysis in solids. Comput. Methods Appl. Mech. Eng. 193(17–20), 1733–1772 (2004)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    G. Lu, E. Kaxiras, An overview of multiscale simulations of materials. Handbook of Theoretical and Computational Nanotechnology (American Scientific Publishers, Stevenson Ranch, 2005), p. 10Google Scholar
  43. 43.
    R.E. Miller, E.B. Tadmor, A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Modell. Simul. Mater. Sci. Eng. 17(5), 053001 (2009)ADSCrossRefGoogle Scholar
  44. 44.
    J.M. Wernik, S.A. Meguid, Coupling atomistics and continuum in solids: status, prospects, and challenges. Int. J. Mech. Mater. Des. 5(1), 79–110 (2009)CrossRefGoogle Scholar
  45. 45.
    E. Weinan, B. Engquist, X. Li, W. Ren, E. Vanden-Eijnden, Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2(3), 367–450 (2007)zbMATHMathSciNetGoogle Scholar
  46. 46.
    E.B. Tadmor, M. Ortiz, R. Phillips, Quasicontinuum analysis of defects in solids. Philos. Mag. A 73(6), 1529–1563 (1996)ADSCrossRefGoogle Scholar
  47. 47.
    R. Miller, E.B. Tadmor, The quasicontinuum method: overview, applications and current directions. J. Comput. Aided Mater. Des. 9(3), 203–239 (2002)ADSCrossRefGoogle Scholar
  48. 48.
    L.E. Shilkrot, R.E. Miller, W.A. Curtin, Coupled atomistic and discrete dislocation plasticity. Phys. Rev. Lett. 89(2), 025501 (2002)ADSCrossRefGoogle Scholar
  49. 49.
    S.P. Xiao, T. Belytschko, A bridging domain method for coupling continua with molecular dynamics. Comput. Methods Appl. Mech. Eng. 193(17–20), 1645–1669 (2004)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    S. Kohlhoff, P. Gumbsch, H.F. Fischmeister, Crack propagation in b.c.c. crystals studied with a combined finite-element and atomistic model. Philos. Mag. A 64(4), 851–878 (1991)ADSCrossRefGoogle Scholar
  51. 51.
    G.J. Wagner, W.K. Liu, Coupling of atomistic and continuum simulations using a bridging scale decomposition. J. Comput. Phys. 190(1), 249–274 (2003)ADSzbMATHCrossRefGoogle Scholar
  52. 52.
    R. Miller, M. Ortiz, R. Phillips, V. Shenoy, E.B. Tadmor, Quasicontinuum models of fracture and plasticity. Eng. Fract. Mech. 61(3–4), 427–444 (1998)CrossRefGoogle Scholar
  53. 53.
    V.B. Shenoy, R. Miller, E.B. Tadmor, D. Rodney, R. Phillips, M. Ortiz, An adaptive finite element approach to atomic-scale mechanicsthe quasicontinuum method. J. Mech. Phys. Solids 47(3), 611–642 (1999)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    J. Knap, M. Ortiz, An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49(9), 1899–1923 (2001)ADSzbMATHCrossRefGoogle Scholar
  55. 55.
    V. Shenoy, V. Shenoy, R. Phillips, Finite temperature quasicontinuum methods. MRS Online Proc. Libr. 538 (1998)Google Scholar
  56. 56.
    L.M. Dupuy, E.B. Tadmor, R.E. Miller, R. Phillips, Finite-temperature quasicontinuum: molecular dynamics without all the atoms. Phys. Rev. Lett. 95(6), 060202 (2005)ADSCrossRefGoogle Scholar
  57. 57.
    Z. Tang, H. Zhao, G. Li, N.R. Aluru, Finite-temperature quasicontinuum method for multiscale analysis of silicon nanostructures. Phys. Rev. B 74(6), 064110 (2006)ADSCrossRefGoogle Scholar
  58. 58.
    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(4), 1417–1449 (2008)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  59. 59.
    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. Modell. Simul. Mater. Sci. Eng. 18(1), 015003 (2010)ADSCrossRefGoogle Scholar
  60. 60.
    E.B. Tadmor, F. Legoll, W.K. Kim, L.M. Dupuy, R.E. Miller, Finite-temperature quasi-continuum. Appl. Mech. Rev. 65(1), 010803–010803 (2013)ADSCrossRefGoogle Scholar
  61. 61.
    L.E. Shilkrot, W.A. Curtin, R.E. Miller, A coupled atomistic/continuum model of defects in solids. J. Mech. Phys. Solids 50(10), 2085–2106 (2002)ADSzbMATHCrossRefGoogle Scholar
  62. 62.
    E. Van der Giessen, A. Needleman, Discrete dislocation plasticity: a simple planar model. Modell. Simul. Mater. Sci. Eng. 3(5), 689 (1995)ADSCrossRefGoogle Scholar
  63. 63.
    B. Shiari, R.E. Miller, W.A. Curtin, Coupled atomistic/discrete dislocation simulations of nanoindentation at finite temperature. J. Eng. Mater. Technol. 127(4), 358–368 (2005)CrossRefGoogle Scholar
  64. 64.
    S. Qu, V. Shastry, W.A. Curtin, R.E. Miller, A finite-temperature dynamic coupled atomistic/discrete dislocation method. Modell. Simul. Mater. Sci. Eng. 13(7), 1101 (2005)ADSCrossRefGoogle Scholar
  65. 65.
    H.B. Dhia, Problémes mécaniques multi-échelles: la méthode arlequin. Comptes Rendus de l’Académie des Sciences - Series IIB - Mechanics-Physics-Astronomy, 326(12):899–904 (1998)Google Scholar
  66. 66.
    H.B. Dhia, G. Rateau, The arlequin method as a flexible engineering design tool. Int. J. Numer. Meth. Eng. 62(11), 14421462 (2005)CrossRefGoogle Scholar
  67. 67.
    P.T. Bauman, H.B. Dhia, N. Elkhodja, J.T. Oden, S. Prudhomme, On the application of the arlequin method to the coupling of particle and continuum models. Comput. Mech. 42(4), 511–530 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  68. 68.
    S. Prudhomme, H.B. Dhia, P.T. Bauman, N. Elkhodja, J.T. Oden, Computational analysis of modeling error for the coupling of particle and continuum models by the arlequin method. Comput. Methods Appl. Mech. Eng. 197(4142), 3399–3409 (2008)ADSzbMATHCrossRefGoogle Scholar
  69. 69.
    T. Belytschko, S.P. Xiao, Coupling methods for continuum model with molecular model. Int. J. Multiscale Comput. Eng. 1(1), 115–126 (2003)CrossRefGoogle Scholar
  70. 70.
    G. Anciaux, O. Coulaud, J. Roman, G. Zerah, Ghost force reduction and spectral analysis of the 1D bridging method, Technical report (INRIA, HAL, 2008)Google Scholar
  71. 71.
    G. Anciaux, S.B. Ramisetti, J.F. Molinari, A finite temperature bridging domain method for MD-FE coupling and application to a contact problem. Comput. Methods Appl. Mech. Eng. 205208, 204212 (2011)Google Scholar
  72. 72.
    H.S. Park, E.G. Karpov, P.A. Klein, W.K. Liu, Three-dimensional bridging scale analysis of dynamic fracture. J. Comput. Phys. 207(2), 588–609 (2005)ADSzbMATHCrossRefGoogle Scholar
  73. 73.
    S.A. Adelman, Generalized langevin equation approach for atom/solid-surface scattering: collinear atom/harmonic chain model. J. Chem. Phys. 61(10), 4242–4246 (1974)ADSCrossRefGoogle Scholar
  74. 74.
    S.A. Adelman, Generalized langevin theory for gas/solid processes: dynamical solid models. J. Chem. Phys. 65(9), 3751–3762 (1976)ADSMathSciNetCrossRefGoogle Scholar
  75. 75.
    S.A. Adelman, Generalized langevin equation approach for atom/solid-surface scattering: general formulation for classical scattering off harmonic solids. J. Chem. Phys. 64(6), 2375–2389 (1976)ADSMathSciNetCrossRefGoogle Scholar
  76. 76.
    W. Cai, M. de Koning, V.V. Bulatov, S. Yip, Minimizing boundary reflections in coupled-domain simulations. Phys. Rev. Lett. 85(15), 3213–3216 (2000)ADSCrossRefGoogle Scholar
  77. 77.
    E. Weinan, Z. Huang, Matching conditions in atomistic-continuum modeling of materials. Phys. Rev. Lett. 87(13), 135501 (2001)ADSCrossRefGoogle Scholar
  78. 78.
    E. Weinan, Z. Huang, A dynamic atomisticcontinuum method for the simulation of crystalline materials. J. Comput. Phys. 182(1), 234–261 (2002)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  79. 79.
    G.J. Wagner, E.G. Karpov, W.K. Liu, Molecular dynamics boundary conditions for regular crystal lattices. Comput. Methods Appl. Mech. Eng. 193(1720), 1579–1601 (2004)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  80. 80.
    E.G. Karpov, H.S. Park, W.K. Liu, A phonon heat bath approach for the atomistic and multiscale simulation of solids. Int. J. Numer. Meth. Eng. 70(3), 351–378 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  81. 81.
    N. Mathew, R.C. Picu, M. Bloomfield, Concurrent coupling of atomistic and continuum models at finite temperature. Comput. Methods Appl. Mech. Eng. 200(5–8), 765–773 (2011)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  82. 82.
    S.B. Ramisetti, G. Anciaux, J.F. Molinari, Spatial filters for bridging molecular dynamics with finite elements at finite temperatures. Comput. Methods Appl. Mech. Eng. 253, 28–38 (2013)ADSzbMATHCrossRefGoogle Scholar
  83. 83.
    G. Anciaux, J.F. Molinari, Sliding of rough surfaces and energy dissipation with a 3D multiscale approach. Int. J. Numer. Meth. Eng. 83(8–9), 1255–1271 (2010)zbMATHCrossRefGoogle Scholar
  84. 84.
    B.N.J. Persson, O. Albohr, U. Tartaglino, A.I. Volokitin, E. Tosatti, On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion. J. Phys.: Condens. Matter 17(1), R1–R62 (2005)ADSGoogle Scholar
  85. 85.
    H.O. Peitgen, D. Saupe, Y. Fisher, M. McGuire, R.F. Voss, M.F. Barnsley, R.L. Devaney, B.B. Mandelbrot, The Science of Fractal Images, 1st edn. (Springer, New York, 1988)Google Scholar
  86. 86.
    R.F. Voss, Random fractal forgeries, in Fundamental Algorithms for Computer Graphics, ed. by R.A. Earnshaw (Springer, Heidelberg, 1985), pp. 805–835CrossRefGoogle Scholar
  87. 87.
    G. Anciaux, Simulation multi-échelles des solides par une approche couplée dynamique moléculaire/éléments finis. De la modélisation á la simulation haute performance. Ph.D. thesis, University of Bordeaux (INRIA, CEA), France, July 2007Google Scholar
  88. 88.
    J. Fish, M.A. Nuggehally, M.S. Shephard, C.R. Picu, S. Badia, M.L. Parks, M. Gunzburger, Concurrent AtC coupling based on a blend of the continuum stress and the atomistic force. Comput. Methods Appl. Mech. Eng. 196(4548), 4548–4560 (2007)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  89. 89.
    K. Fackeldey, R. Krause, Multiscale coupling in function spaceweak coupling between molecular dynamics and continuum mechanics. Int. J. Numer. Meth. Eng. 79(12), 15171535 (2009)CrossRefGoogle Scholar
  90. 90.
    K. Fackeldey, The Weak Coupling Method for Coupling Continuum Mechanics with Molecular Dynamics. Ph.D. thesis, Bonn, February 2009Google Scholar
  91. 91.
    S.B. Ramisetti, G. Anciaux, J.F. Molinari, A concurrent atomistic and continuum coupling method with applications to thermo-mechanical problems. Submitted, 2013Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Srinivasa Babu Ramisetti
    • 1
  • Guillaume Anciaux
    • 2
  • Jean-Francois Molinari
    • 2
  1. 1.University of EdinburghEdinburghUK
  2. 2.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland

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