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MD/FE Multiscale Modeling of Contact

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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.

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References

  1. H. Czichos, Tribology (Elsevier, Amsterdam 1978)

    Google Scholar 

  2. A. Majumdar, B. Bhushan, Role of fractal geometry in roughness characterization and contact mechanics of surfaces. J. Tribol. 112(2), 205–216 (1990)

    Article  Google Scholar 

  3. B.N.J. Persson, Contact mechanics for randomly rough surfaces. Surf. Sci. Rep. 61(4), 201–227 (2006)

    Article  ADS  Google Scholar 

  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. B. Luan, M.O. Robbins, The breakdown of continuum models for mechanical contacts. Nature 435(7044), 929–932 (2005)

    Article  ADS  Google Scholar 

  6. G.V. Dedkov, Experimental and theoretical aspects of the modern nanotribology. Phys. Status Solidi A 179(1), 375 (2000)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  9. M. Renouf, F. Massi, N. Fillot, A. Saulot, Numerical tribology of a dry contact. Tribol. Int. 44(78), 834–844 (2011)

    Article  Google Scholar 

  10. J.F. Jerier, J.F. Molinari, Normal contact between rough surfaces by the discrete element method. Tribol. Int. 47, 1–8 (2012)

    Article  Google Scholar 

  11. V.S. Deshpande, A. Needleman, E. Van der Giessen, Discrete dislocation plasticity analysis of static friction. Acta Mater. 52(10), 3135–3149 (2004)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  13. P. Wriggers, T.A. Laursen, Computational Contact Mechanics (Springer, Dordrecht, 2008)

    Google Scholar 

  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)

    Article  ADS  Google Scholar 

  15. Y. Mo, I. Szlufarska, Roughness picture of friction in dry nanoscale contacts. Phys. Rev. B 81(3), 035405 (2010)

    Article  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  17. T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (Dover Publications, New York, 2000)

    Google Scholar 

  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. D.C. Rapaport, The Art of Molecular Dynamics Simulation (Cambridge University Press, 2004)

    Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  22. K. Komvopoulos, Effects of multi-scale roughness and frictional heating on solid body contact deformation. C. R. Mnique 336(12), 149–162 (2008)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  24. H.J.C. Berendsen, Simulating the Physical World: Hierarchical Modeling from Quantum Mechanics to Fluid Dynamics (Cambridge University Press, 2007)

    Google Scholar 

  25. M. Griebel, S. Knapek, G. Zumbusch, Numerical Simulation in Molecular Dynamics: Numerics, Algorithms, Parallelization, Applications (Springer, November 2010)

    Google Scholar 

  26. J. Rottler, M.O. Robbins, Macroscopic friction laws and shear yielding of glassy solids. Comput. Phys. Commun. 169(13), 177–182 (2005)

    Article  ADS  Google Scholar 

  27. O.M. Braun, A.G. Naumovets, Nanotribology: Microscopic mechanisms of friction. Surf. Sci. Rep. 60(67), 79–158 (2006)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    ADS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  39. W.A. Curtin, R.E. Miller, Atomistic/continuum coupling in computational materials science. Modell. Simul. Mater. Sci. Eng. 11(3), R33–R68 (2003)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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. 10

    Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  47. R. Miller, E.B. Tadmor, The quasicontinuum method: overview, applications and current directions. J. Comput. Aided Mater. Des. 9(3), 203–239 (2002)

    Article  ADS  Google Scholar 

  48. L.E. Shilkrot, R.E. Miller, W.A. Curtin, Coupled atomistic and discrete dislocation plasticity. Phys. Rev. Lett. 89(2), 025501 (2002)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  55. V. Shenoy, V. Shenoy, R. Phillips, Finite temperature quasicontinuum methods. MRS Online Proc. Libr. 538 (1998)

    Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  62. E. Van der Giessen, A. Needleman, Discrete dislocation plasticity: a simple planar model. Modell. Simul. Mater. Sci. Eng. 3(5), 689 (1995)

    Article  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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. H.B. Dhia, G. Rateau, The arlequin method as a flexible engineering design tool. Int. J. Numer. Meth. Eng. 62(11), 14421462 (2005)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  69. 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 

  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. 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. 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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  74. S.A. Adelman, Generalized langevin theory for gas/solid processes: dynamical solid models. J. Chem. Phys. 65(9), 3751–3762 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  77. E. Weinan, Z. Huang, Matching conditions in atomistic-continuum modeling of materials. Phys. Rev. Lett. 87(13), 135501 (2001)

    Article  ADS  Google Scholar 

  78. E. Weinan, Z. Huang, A dynamic atomisticcontinuum method for the simulation of crystalline materials. J. Comput. Phys. 182(1), 234–261 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    ADS  Google Scholar 

  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. R.F. Voss, Random fractal forgeries, in Fundamental Algorithms for Computer Graphics, ed. by R.A. Earnshaw (Springer, Heidelberg, 1985), pp. 805–835

    Chapter  Google Scholar 

  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 2007

    Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  90. K. Fackeldey, The Weak Coupling Method for Coupling Continuum Mechanics with Molecular Dynamics. Ph.D. thesis, Bonn, February 2009

    Google Scholar 

  91. S.B. Ramisetti, G. Anciaux, J.F. Molinari, A concurrent atomistic and continuum coupling method with applications to thermo-mechanical problems. Submitted, 2013

    Google Scholar 

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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.

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Ramisetti, S.B., Anciaux, G., Molinari, JF. (2015). MD/FE Multiscale Modeling of Contact. In: Gnecco, E., Meyer, E. (eds) Fundamentals of Friction and Wear on the Nanoscale. NanoScience and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-10560-4_14

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