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Predictive multiscale theory for design of heterogeneous materials

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Abstract

A general multiscale theory for modeling heterogeneous materials is derived via a nested domain based virtual power decomposition. Three variations on the theory are proposed; a concurrent approach, a simplified hierarchical approach and a statistical power equivalence approach. Deformation at each scale of analysis is solved either (a) by direct numerical simulation (DNS) of the microstructure or (b) by higher order homogenization of the microstructure. If the latter approach is chosen, a set of multiscale homogenized constitutive relations must be derived. This is demonstrated using a computational cell modeling technique for a four scale metal alloy. In each variation of the theory, a transfer of information occurs between the scales giving a coupled formulation. The concurrent approach achieves a more comprehensive coupling than the hierarchical approach, making it more accurate for dynamic fast time scale simulations. The power equivalence approach is strongly coupled and is useful for performing larger scale simulations as the expensive multiple scale DNS boundary value problems are replaced with statistical higher order continua. Furthermore, these continua may be solved on a single spatial discretisation using an extended finite element framework, making the theory applicable within existing high performance computing codes.

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References

  1. Lee JH, Shishidou T, Zhao YJ, Freeman AJ and Olson GB (2005). Strong interface adhesion in Fe/TiC. Philos Magaz 85(31): 3683–3697

    Article  Google Scholar 

  2. Beredsen HJC, Postma JPM, DiNola A and Haak JR (1984). Molecular dynamics with coupling to an external bath. J Chem Phys 81: 3684–3690

    Article  Google Scholar 

  3. Haile JM (1992). Molecular omulation. Wiley, New York

    Google Scholar 

  4. Liu WK, Karpov EG and Park HS (2005). Nano-mechanics and materials: theory, multiscale methods and applications. Wiley, New York

    Google Scholar 

  5. Allen MP and Tildesley DJ (1987). Computer simulation of liquids. Oxford University Press, New York

    MATH  Google Scholar 

  6. Hoover WG (1986). Molecular dynamics. Springer, Berlin

    Google Scholar 

  7. Rapaport DC (1995). The art of molecular dynamics simulation. Cambridge University Press, New York

    Google Scholar 

  8. Abraham FF and Gao H (2000). How fast can cracks propagate?. Phys Rev Lett 84(14): 3113–3116

    Article  Google Scholar 

  9. Hao S, Liu WK and Chang CT (2000). Computer implementation of damage models by finite element and meshfree methods. Computat Methods Appl Mech Eng 187: 401–440

    Article  MATH  Google Scholar 

  10. Biner SB and Morris JR (2003). The effects of grain size and dislocation source density on the strengthening behaviour of polycrystals: a two-dimensional discrete dislocation simulation. Philos Magaz 83: 3677–3690

    Article  Google Scholar 

  11. Liu WK, Jun S, Qian D (2005) Computational nanomechanics of materials. In: Rieth M, Schommers W (eds) Handbook of theoretical and computational nanotechnology, vol X. American Scientific, Stevension Ranch. (in press)

  12. Christopher D, Smith R and Richter A (2001). Atomistic modelling of nanoindentation in iron and silver. Nanotechnology 12: 372–383

    Article  Google Scholar 

  13. Lilleodden ET, Zimmerman JA, Foiles SM and Nix WD (2003). Atomistic simulations of elastic deformation and dislocation nucleation during nanoindentation. J Mech Phys Solids 51: 901–920

    Article  MATH  Google Scholar 

  14. Zhou SJ, Beazley DM, Lomdahl PS and Holian BL (1997). Large scale molecular dynamics simulations of three-dimensional ductile failure. Phys Rev Lett 78: 479–482

    Article  Google Scholar 

  15. Zhang S, Mielke SL, Khare R, Troya D, Ruoff RS, Schatz GC and Belytschko T (2005). Mechanics of defects in carbon nanotubes: atomistic and multiscale simulations. Phys Rev B 71: 115403

    Article  Google Scholar 

  16. Arroyo M and Belytschko T (2002). An atomistic-based finite deformation membrane for single layer crystalline films. J Mech Phys Solids 50: 1941–1977

    Article  MATH  MathSciNet  Google Scholar 

  17. Belytschko T, Xiao SP, Schatz GC and Ruoff RS (2002). Atomistic simulations of nanotube fracture. Phys Rev B 65: 235430

    Article  Google Scholar 

  18. Mura T (1991). Micromechanics of defects in solids, 2nd edn. Kluwer, Dordrecht

    Google Scholar 

  19. Nemat-Nasser S and Hori M (1999). Micromechanics: overall properties of heterogeneous materials. North-Holland, Amsterdam

    Google Scholar 

  20. Alber I, Bassani JL, Khantha M, Vitek V and Wang GJ (1992). Grain boundaries as heterogeneous systems: Atomic and continuum elastic properties. Philos Trans Phys Sci Eng 339: 555–586

    Article  Google Scholar 

  21. Hao S, Liu WK and Chang CT (2000). Computer implementation of damage models by finite element and meshfree methods. Computat Methods Appl Mech Eng 187: 401–440

    Article  MATH  Google Scholar 

  22. Hao S, Liu WK, Moran B, Vernerey F and Olson GB (2004). Multiscale constitutive model and computational framework for the design of ultrahigh strength, high toughness steels. Computat Methods Appl Mech Eng 193: 1865–1908

    Article  MATH  Google Scholar 

  23. Hao S, Moran B, Liu WK and Olson GB (2003). A hierarchical multiphysics constitutive model for steels design. J Comput-Aided Mater Des 10: 99–142

    Article  Google Scholar 

  24. McVeigh C and Liu WK (2006). Prediction of central bursting during axisymmetric cold extrusion of a metal alloy containing particles. Int J Solids Struct 43(10): 3087–3105

    Article  MATH  Google Scholar 

  25. Belytschko T, Fish J and Engelman BE (1988). A finite element with embedded localization zones. Comput Methods Appl Mech Eng 70: 59–89

    Article  MATH  Google Scholar 

  26. Belytschko T and Tabbara M (1993). h-adaptive finite element methods for dynamic problems, with emphasis on localization. Int J Numer Methods Eng 36: 4245–4265

    Article  MATH  Google Scholar 

  27. Nemat-Nasser S (2004). Plasticity: a treatise on the finite deformation of heterogeneous inelastic materials. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  28. Abraham FF, Broughton JQ, Bernstein N and Kaxiras E (1998). Spanning the length scales in dynamic simulation. Comput Phys 12: 538–546

    Article  Google Scholar 

  29. Abraham FF, Broughton JQ, Bernstein N and Kaxiras E (1998). Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture. Europhys Lett 44: 783–787

    Article  Google Scholar 

  30. Abraham FF, Broughton JQ, Bernstein N and Kaxiras E (1999). Concurrent coupling of length scales: methodology and application. Phys Rev B 60: 2391–2402

    Article  Google Scholar 

  31. Adelman SA and Doll JD (1974). Generalized Langevin equation approach for atom/solid-surface scattering—collinear atom/harmonic chain model. J Chem Phys 61: 4242–4245

    Article  Google Scholar 

  32. Broughton JQ, Abraham FF, Bernstein N and Kaxiras E (1999). Concurrent coupling of length scales: methodology and application. Phys Rev B 60: 2391–2402

    Article  Google Scholar 

  33. Cai W, de Koning M, Bulatov VV and Yip S (2000). Minimizing boundary reflections in coupled domain simulations. Phys Rev Lett 85: 3213–3216

    Article  Google Scholar 

  34. Curtin WA and Miller RE (2003). Atomistic/continuum coupling in computational materials science. Model Simulation Mater Sci Eng 11: R33–R68

    Article  Google Scholar 

  35. Weinan E and Huang Z (2001). Matching conditions in atomistic-continuum modeling materials. Phys Rev Lett 87: 135501

    Article  Google Scholar 

  36. Weinan E and Huang Z (2002). A dynamic atomistic-continuum method for the simulation of crystalline materials. J Comput Phys 182: 234–260

    Article  MATH  MathSciNet  Google Scholar 

  37. W E, Engquist B, Huang Z (2003) Heterogeneous multiscale method: A general methodology for multiscale modelling.Physical Review B, 67:092101

    Google Scholar 

  38. Fish J and Wen J (2004). Discrete-to-continuum bridging based on multigrid principles. Comput Methods Appl Mech Eng 193: 1693–1711

    Article  MATH  Google Scholar 

  39. Kadowaki H and Liu WK (2004). Bridging multi-scale method for localization problems. Comput Methods Appl Mech Eng 193: 3267–3302

    Article  MATH  Google Scholar 

  40. Kadowaki H and Liu WK (2005). A multiscale approach for the micropolar continuum model. Comput Model Eng Sci 7: 269–282

    MATH  MathSciNet  Google Scholar 

  41. Karpov EG, Wagner GJ and Liu WK (2005). A Green’s function approach to deriving non-reflecting boundary conditions in molecular dynamics simulations. Int J Numer Methods Eng 62: 1250–1262

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  43. Lu G, Kaxiras E (2005) Overview of multiscale simulations of materials. In: Rieth M, Schommers W (eds) Handbook of Theoretical and Computational Nanotechnology, vol X. American Scientific, Stevension Ranch, pp 1–33

  44. Park HS, Karpov EG, Klein PA and Liu WK (2005). The bridging scale for two-dimensional atomistic/continuum coupling. Philos Magaz 85: 79–113

    Article  Google Scholar 

  45. Park HS, Karpov EG, Klein PA and Liu WK (2005). Three-dimensional bridging scale analysis of dynamic fracture. J comput Phys 207: 588–609

    Article  MATH  Google Scholar 

  46. Park HS and Liu WK (2004). An introduction and tutorial on multiplescale analysis in solids. Comput Methods Appl Mech Eng 193: 1733–1772

    Article  MATH  MathSciNet  Google Scholar 

  47. Picu RC (2000). Atomistic-continuum simulation of nano-indentation in molybdenum. J Comput Aided Mater Des 7: 77–87

    Article  Google Scholar 

  48. Tang S, Hou TY and Liu WK (2006). Mathematical framework of bridging scale method. Int J Numer Methods Eng 65: 1688–1713

    Article  MATH  MathSciNet  Google Scholar 

  49. Wagner GJ and Liu WK (2003). Coupling of atomistic and continuum simulations using a bridging scale decomposition. J Comput Phys 190: 249–274

    Article  MATH  Google Scholar 

  50. Wagner GJ and Liu WK (2001). Hierarchical enrichment for bridging scales and mesh-free boundary conditions. Int J Numer Methods Eng 50: 507–524

    Article  MATH  MathSciNet  Google Scholar 

  51. Fleck NA, Muler GM, Ashby MF and Hutchinson JW (1994). Strain gradient plasticity: theory and experiment. Acta Metallurgy Mater 42: 475–487

    Article  Google Scholar 

  52. Gao H, Huang Y, Nix WD and Hutchinson JW (1999). Mechanism-based strain gradient Plasticity-I. Theory J Mech Phys Solids 47: 1239–1263

    Article  MATH  MathSciNet  Google Scholar 

  53. Huang Y, Gao H, Nix WD and Hutchinson JW (2000). Mechanism-based strain gradient plasticity-II. Anal J Mech Phys Solids 48: 99–128

    Article  MATH  MathSciNet  Google Scholar 

  54. Mindlin RD (1964). Micro-structure in linear elasticity. Arch Rational Mech Anal 16: 15–78

    Article  MathSciNet  Google Scholar 

  55. Vernerey F (2006) PhD Thesis, Northwestern University

  56. McVeigh C, Vernerey F, Liu WK, Brinson LC (2006) Multiresolution analysis for material design. Comput Methods Appl Mech Eng

  57. Geers MGD, Ubachs RLJM and Engelen RAB (2003). Strongly nonlocal gradient-enhanced finite strain elastoplasticity. Int J Numer Meth Eng 56: 2039–2068

    Article  MATH  Google Scholar 

  58. Engelen RAB, Geers MGD and Baaijens FTP (2003). Nonlocal implicit gradient-enhanced elasto-plasticity for the modeling of softening behaviour. Int J Plast 19: 403–433

    Article  MATH  Google Scholar 

  59. Lions JL (1981). Some methods in the mathematical analysis of systems and their control. Science Press, Beijing, 1

    MATH  Google Scholar 

  60. Sanchez-Palenia E (1980). Non-homogeneous media and vibration theory. Springer, Berlin, 45

    Google Scholar 

  61. Francfort GA (1983). Homogenization and linear thermoelasticity. SIAM J Math Anal 14: 696–708

    Article  MATH  MathSciNet  Google Scholar 

  62. Guedes JM and Kikuchi N (1990). Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput Methods Appl Mech Eng 83: 143–198

    Article  MATH  MathSciNet  Google Scholar 

  63. Xiao SP and Belytschko T (2004). A bridging domain method for coupling continua with molecular dynamics. Comput Methods Appl Mech Eng 193: 1645–1669

    Article  MATH  MathSciNet  Google Scholar 

  64. Berenger J (1994). A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114: 185–200

    Article  MATH  MathSciNet  Google Scholar 

  65. To AC and Li S (2005). Perfectly Matched Multiscale Simulations. Phys Rev B 72: 035414

    Article  Google Scholar 

  66. Zohdi T and Wriggers P (2005). Introduction to computational micromechanics. Springer, Heidelberg

    MATH  Google Scholar 

  67. Zohdi TI, Oden JT and Rodin GJ (1996). Hierarchical modeling of heterogeneous bodies. Comput Methods Appl Mech Eng 138: 273–298

    Article  MATH  MathSciNet  Google Scholar 

  68. Oden JT and Zohdi TI (1997). Analysis and adaptive modeling of highly heterogeneous elastic structures. Comput Methods Appl Mech Eng 148: 367–391

    Article  MATH  MathSciNet  Google Scholar 

  69. Oden JT, Prudhomme S, Romkes A, Bauman P (2007) Multi-scale modeling of physical phenomena: adaptive control of models. Comput Methods Appl Mech Eng (to appear)

  70. Takano N, Uetsuji Y, Kashiwagi Y and Zako M (1999). Hierarchical modelling of textile composite materials and structures by the homogenization method. Model Simul Mater Sci Eng 7: 207–231

    Article  Google Scholar 

  71. Fish J and Chen W (2001). Higher-order homogenization of initial/boundary-value problem. J Eng Mech 127(12): 1223–1230

    Article  Google Scholar 

  72. Cosserat E and Cosserat F (1909). Theorie des corps deformables. A Hermann et Fils, Paris

    Google Scholar 

  73. Germain P (1973). The method of virtual power in continuum mechanics. part 2: microstrucure. SIAM J Appl Math 25: 556–575

    Article  MathSciNet  Google Scholar 

  74. Mindlin RD (1964). Micro-structure in linear elasticity. Arch Ration Mech Anal 16: 15–78

    Article  MathSciNet  Google Scholar 

  75. Gao H, Huang Y, Nix WD and Hutchinson JW (1999). Mechanism-based strain gradient plasticity-theory. J Mech Phys Solids 47: 1239–1263

    Article  MATH  MathSciNet  Google Scholar 

  76. Goods SH, Brown LM (1979) The effects of second phases on the mechanical properties of alloys. Acta Metallurgica

  77. Hao S, Liu WK, Moran B, Vernerey F and Olson GB (2004). Multiscale constitutive model and computational framework for the design of ultrahigh strength, high toughness steels. Comput Methods Appl Mech Eng 193: 1865–1908

    Article  MATH  Google Scholar 

  78. Hao S, Moran B, Liu WK and Olson GB (2003). A hierarchical multiphysics constitutive model for steels design. J Comput-Aided Mater Des 10: 99–142

    Article  Google Scholar 

  79. McVeigh C, Vernerey F, Liu WK, Moran B, Olson GB (2006) An interactive microvoid shear localization mechanism in high strength steels. J Mech Phys Solids (accepted)

  80. Kouznetsova V, Geers MGD and Brekelmans WAM (2002). Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int J Numer Methods Eng 54(8): 1235–1260

    Article  MATH  Google Scholar 

  81. Bammann DJ, Chiesa ML, Johnson GC (1996) Modeling large deformation and failure in manufacturing processes. In: Tatsumi, Wannabe, Kambe (eds) Theor. App. Mech., Elsevier Science, p 259

  82. Fish J and Chen W (2004). Space-time multiscale model for wave propagation in heterogeneous media. Comp Meth Appl Mech Eng 193: 4837–4856

    Article  MATH  MathSciNet  Google Scholar 

  83. Nagai G, Fish J, Watanabe K (2004) Stabilized nonlocal model for wave propagation in heterogeneous media. Comput Mech 33(2)

  84. Fish J, Chen W and Li R (2007). Generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions. Comp Methods Appl Mech Eng 196: 908–922

    Article  MATH  MathSciNet  Google Scholar 

  85. Mesarovicy S and Padbidri J (2005). Minimal kinematic boundary conditions for simulations of disordered microstructures. Philos Magaz 85(1): 65–78

    Article  Google Scholar 

  86. Liu WK, Belytschko T and Mani A (1986). Random field finite elements. Int J Numer Methods Eng 23(10): 1831–1845

    Article  MATH  MathSciNet  Google Scholar 

  87. Liu WK, Mani A and Belytschko T (1987). Finite element methods in probabilistic mechanics. Prob Eng Mech 2(4): 201–213

    Article  Google Scholar 

  88. Qian D, Wagner GJ and Liu WK (2004). A multiscale projection method for the analysis of carbon nanotubes. Comput Methods Appl Mech Eng 193: 1603–1632

    Article  MATH  Google Scholar 

  89. Girifalco LA and Weizer VG (1958). Application of the morse potential function to cubic metals. Phys Rev 114: 687–690

    Article  Google Scholar 

  90. Daw MS, Foiles SM and Baskes MI (1993). The embedded-atom method: a review of theory and applications. Mater Sci Reports 9: 251–310

    Article  Google Scholar 

  91. Brenner DW (1990). Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys Rev B 42: 9458–9471

    Article  Google Scholar 

  92. Biswas R and Hamann DR (1985). Interatomic potential for silicon structural energies. Phys Rev Lett 55: 2001–2004

    Article  Google Scholar 

  93. Biswas R and Hamann DR (1987). New classical models for silicon structural energies. Phys Rev B 36: 6434–6445

    Article  Google Scholar 

  94. Tersoff J (1988). New empirical approach for the structure and energy of covalent systems. Phys Rev B 37: 6991–7000

    Article  Google Scholar 

  95. Tersoff J (1986). New empirical model for the structural properties of silicon. Phys Rev Lett 56: 632–635

    Article  Google Scholar 

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  1. Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208-3111, USA

    Wing Kam Liu & Cahal McVeigh

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  1. Wing Kam Liu
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  2. Cahal McVeigh
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Correspondence to Wing Kam Liu or Cahal McVeigh.

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Submitted to a Special Issue of Computational Mechanics in Honor of Professor Ladeveze’s 60th Birthday.

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Liu, W.K., McVeigh, C. Predictive multiscale theory for design of heterogeneous materials. Comput Mech 42, 147–170 (2008). https://doi.org/10.1007/s00466-007-0176-8

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