Computational Mechanics

, Volume 53, Issue 4, pp 687–737 | Cite as

The archetype-genome exemplar in molecular dynamics and continuum mechanics

  • M. Steven Greene
  • Ying Li
  • Wei Chen
  • Wing Kam LiuEmail author
Original Paper


We argue that mechanics and physics of solids rely on a fundamental exemplar: the apparent properties of a system depend on the building blocks that comprise it. Building blocks are referred to as archetypes and apparent system properties as the system genome. Three entities are of importance: the archetype properties, the conformation of archetypes, and the properties of interactions activated by that conformation. The combination of these entities into the system genome is called assembly. To show the utility of the archetype-genome exemplar, this work presents the mathematical ingredients and computational implementation of theories in solid mechanics that are (1) molecular and (2) continuum manifestations of the assembly process. Both coarse-grained molecular dynamics (CGMD) and the archetype-blending continuum (ABC) theories are formulated then applied to polymer nanocomposites (PNCs) to demonstrate the impact the components of the assembly triplet have on a material genome. CGMD simulations demonstrate the sensitivity of nanocomposite viscosities and diffusion coefficients to polymer chain types (archetype), polymer–nanoparticle interaction potentials (interaction), and the structural configuration (conformation) of dispersed nanoparticles. ABC simulations show the contributions of bulk polymer (archetype) properties, occluded region of bound rubber (interaction) properties, and microstructural binary images (conformation) to predictions of linear damping properties, the Payne effect, and localization/size effects in the same class of PNC material. The paper is light on mathematics. Instead, the focus is on the usefulness of the archetype-genome exemplar to predict system behavior inaccessible to classical theories by transitioning mechanics away from heuristic laws to mechanism-based ones. There are two core contributions of this research: (1) presentation of a fundamental axiom—the archetype-genome exemplar—to guide theory development in computational mechanics, and (2) demonstrations of its utility in modern theoretical realms: CGMD, and generalized continuum mechanics.


Materials genome Archetype  Coarse-graining Continuum mechanics  Molecular dynamics Polymer 



This work is supported by NSF CMMI Grants 0823327 and 0928320, as well as by NSF IDR CMM Grant I 1130948. M. Steven Greene warmly thanks the National Science Foundation GRFP for its support. Y. Li acknowledges financial support from Ryan Fellowship and Royal E. Cabell Terminal Year Fellowship at Northwestern University. W.K. Liu was also supported by the World Class University Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology. The authors would also like to thank Assistant Professor Khalil Elkhodary at the American University in Cairo, Egypt and Professor Shan Tang at Chongqing University in China for their challenging discussions and assistance with clarifying many of the concepts presented herein.

Supplementary material


  1. 1.
    ABAQUS theory manual (2011) Version 6.11Google Scholar
  2. 2.
    Abberton BC, Liu WK, Keten S (2013) Coarse-grained simulation of molecular mechanisms of recovery in thermally activated shape-memory polymers. J Mech Phys Solids. doi: 10.1016/j.jmps.2013.08.003
  3. 3.
    Accelrys (NASDAQ:ACCL) (2012) Materials studio. Available online at Accessed 18 Oct 2013
  4. 4.
    Akutagawa K, Yamaguchi K, Yamamoto A, Heguri H (2008) Mesoscopic mechanical analysis of filled elastomer with 3d-finite element analysis and transmission electron microtomography. Rubber Chem Technol 81:182–189Google Scholar
  5. 5.
    Askes H, Metrikine AV, Pichugin AV, Bennett T (2008) Four simplified gradient elasticity models for the simulation of dispersive wave propagation. Philos Mag 88(28–29):3415–3443Google Scholar
  6. 6.
    Bažant ZP, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech 128(11):1119Google Scholar
  7. 7.
    Bažant ZP, Planas J (1998) Fracture and size effect in concrete and other quasibrittle materials. CRC Press, Boca RatonGoogle Scholar
  8. 8.
    Belytschko T, Mullen R (1978) On dispersive properties of finite element solutions. In: Achenbach J, Miklowitz J (eds) Modern problems in wave propagation. Wiley, New York, pp 67–82Google Scholar
  9. 9.
    Brini E, Algaer EA, Ganguly P, Li C, Rodríguez-Ropero F, van der Vegt NF (2013) Systematic coarse-graining methods for soft matter simulations: a review. Soft Matter 9(7):2108–2119Google Scholar
  10. 10.
    Brinson HF, Brinson LC (2008) Polymer engineering science and viscoelasticity: an introduction. Springer, New YorkGoogle Scholar
  11. 11.
    Brinson LC, Schmidt I, Lammering R (2004) Stress-induced transformation behavior of a polycrystalline niti shape memory alloy: micro and macromechanical investigations via in situ optical microscopy. J Mech Phys Solids 52(7):1549–1571zbMATHGoogle Scholar
  12. 12.
    Clifton T, Ferreira P (2013) Does dark energy really exist? Sci Am 58–65 (special edition: Extreme physics, probing the mysteries of the cosmos)Google Scholar
  13. 13.
    Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, West SussexGoogle Scholar
  14. 14.
    Deng H, Liu Y, Gai D, Dikin DA, Putz KW, Chen W, Brinson LC, Burkhart C, Poldneff M, Jiang B, Papakonstantopoulos GJ (2012) Utilizing real and statistically reconstructed microstructures for the viscoelastic modeling of polymer nanocomposites. Compos Sci Technol 72(14):1725–1732Google Scholar
  15. 15.
    Dill K (2002) Molecular driving forces: statistical thermodynamics in chemistry & biology. Garland Science, New YorkGoogle Scholar
  16. 16.
    Doi M, Edwards S (1988) The theory of polymer dynamics, vol 73. Oxford University Press, New YorkGoogle Scholar
  17. 17.
    Dupres S, Long DR, Albouy PA, Sotta P (2009) Local deformation in carbon black-filled polyisoprene rubbers studied by nmr and X-ray diffraction. Macromolecules 42(7):2634–2644Google Scholar
  18. 18.
    Elkhodary KI, Greene MS, Tang S, Belytschko T, Liu WK (2013a) Archetype blending continuum theory. Comput Methods Appl Mech Eng 254:309–333MathSciNetGoogle Scholar
  19. 19.
    Elkhodary KI, Tang S, Liu WK (2013b) Inclusion clusters in the archetype-blending continuum theory. In: Handbook of micromechanics and nanomechanics. Pan Stanford Publishing, SingaporeGoogle Scholar
  20. 20.
    Eringen AC (1999) Microcontinuum field theories I: foundation and solids. Springer, New YorkGoogle Scholar
  21. 21.
    Eringen AC, Suhubi ES (1964) Nonlinear theory of simple microelastic solids. Int J Eng Sci 2(189–203):389–404MathSciNetGoogle Scholar
  22. 22.
    Faller R (2004) Automatic coarse graining of polymers. Polymer 45(11):3869–3876Google Scholar
  23. 23.
    Fish J, Kuznetsov S (2010) Computational continua. Int J Numer Methods Eng 84(7):774–802zbMATHMathSciNetGoogle Scholar
  24. 24.
    Fleck NA, Hutchinson JW (1997) Strain gradient plasticity. Adv Appl Mech 33:295–361Google Scholar
  25. 25.
    Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42(2):475–487Google Scholar
  26. 26.
    Forester T, Smith W (1998) Shake, rattle, and roll: efficient constraint algorithms for linked rigid bodies. J Comput Chem 19(1):102–111Google Scholar
  27. 27.
    Forrest JA, Dalnoki-Veress K, Stevens JR, Dutcher JR (1996) Effect of free surfaces on the glass transition temperature of thin polymer films. Phys Rev Lett 77(10):2002–2005Google Scholar
  28. 28.
    Frenkel D, Smit B (2001) Understanding molecular simulation: from algorithms to applications, vol 1. Academic press, New YorkGoogle Scholar
  29. 29.
    Fröhlich J, Niedermeier W, Luginsland HD (2005) The effect of filler–filler and filler–elastomer interaction on rubber reinforcement. Compos Part A Appl S 36(4):449–460Google Scholar
  30. 30.
    Gao H, Huang Y, Nix WD, Hutchinson JW (1999) Mechanism-based strain gradient plasticity: i. theory. J Mech Phys Solids 47(6):1239–1263zbMATHMathSciNetGoogle Scholar
  31. 31.
    Geers MGD, Kouznetsova VG, Brekelmans WAM (2010) Multi-scale computational homogenization: trends and challenges. J Comput Appl Math 234(7):2175–2182zbMATHGoogle Scholar
  32. 32.
    Genomic Science Program (2011) About the human genome project. Available online at Accessed 8 Oct 2013
  33. 33.
    Germain P (1973) The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J Appl Math 25(3):556–575zbMATHMathSciNetGoogle Scholar
  34. 34.
    Ghanem R, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New YorkzbMATHGoogle Scholar
  35. 35.
    Gibson LJ, Ashby MF (1997) Cellular solids: structure and properties, 2nd edn. Cambridge University Press, New YorkGoogle Scholar
  36. 36.
    Gonella S, Greene MS, Liu WK (2011) Characterization of heterogeneous solids via wave methods in computational microelasticity. J Mech Phys Solids 59(5):959–974zbMATHGoogle Scholar
  37. 37.
    Gonzalez J, Knauss WG (1998) Strain inhomogeneity and discontinuous crack growth in a particulate composite. J Mech Phys Solids 46(10):1981–1995zbMATHGoogle Scholar
  38. 38.
    Greene MS, Liu Y, Chen W, Liu WK (2011) Computational uncertainty analysis in multiresolution materials via stochastic constitutive theory. Comput Methods Appl Mech Eng 200(1–4):309–325MathSciNetGoogle Scholar
  39. 39.
    Greene MS, Gonella S, Liu WK (2012) Microelastic wave field signatures and their implications for microstructure identification. Int J Solids Struct 49(22):3148–3157Google Scholar
  40. 40.
    Greene MS, Xu H, Tang S, Chen W, Liu WK (2013) A generalized uncertainty propagation criterion from benchmark studies of microstructured material systems. Comput Methods Appl Mech Eng 254:271–291MathSciNetGoogle Scholar
  41. 41.
    Greer JR, Oliver WC, Nix WD (2005) Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta Mater 53(6):1821–1830Google Scholar
  42. 42.
    Gurtin ME, Anand L (2005) A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part i: small deformations. J Mech Phys Solids 53(7):1624–1649zbMATHMathSciNetGoogle Scholar
  43. 43.
    Hao S, Liu WK, Qian D (2000) Localization-induced band and cohesive model. J Appl Mech 67(4):803–812zbMATHGoogle Scholar
  44. 44.
    Harmandaris V, Kremer K (2009) Dynamics of polystyrene melts through hierarchical multiscale simulations. Macromolecules 42(3):791–802Google Scholar
  45. 45.
    Hoogerbrugge P, Koelman J (1992) Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhys Lett 19(3):155Google Scholar
  46. 46.
    Hoover W (1985) Canonical dynamics: equilibrium phase-space distributions. Phys Rev A 31(3):1695Google Scholar
  47. 47.
    Hua CC, Schieber JD (1998) Segment connectivity, chain-length breathing, segmental stretch, and constraint release in reptation models. i. Theory and single-step strain predictions. J Chem Phys 109:10018–10027Google Scholar
  48. 48.
    Hughes T, Cottrell J, Bazilevs Y (2005) Isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39–41):4135–4195zbMATHMathSciNetGoogle Scholar
  49. 49.
    Hwang KC, Jiang H, Huang Y, Gao H, Hu N (2002) A finite deformation theory of strain gradient plasticity. J Mech Phys Solids 50(1):81–99zbMATHMathSciNetGoogle Scholar
  50. 50.
    Jensen MK, Khaliullin R, Schieber JD (2012) Self-consistent modeling of entangled network strands and linear dangling structures in a single-strand mean-field slip-link model. Rheol Acta 51(1):21–35Google Scholar
  51. 51.
    Jordan J, Jacob KI, Tannenbaum R, Sharaf MA, Jasiuk I (2005) Experimental trends in polymer nanocomposites: a review. Mater Sci Eng A Struct 393:1–11 (review article)Google Scholar
  52. 52.
    Kamberaj H, Low R, Neal M (2005) Time reversible and symplectic integrators for molecular dynamics simulations of rigid molecules. J Chem Phys 122(224):114Google Scholar
  53. 53.
    Karásek L, Sumita M (1996) Characterization of dispersion state of filler and polymer–filler interactions in rubber–carbon black composites. J Mater Sci 31(2):281–289Google Scholar
  54. 54.
    Koiter WT (1964) Couple stresses in the theory of elasticity, i and ii. Proc K Ned Akad van Wet Ser B 67(1):17–44zbMATHGoogle Scholar
  55. 55.
    Kopacz A, Patankar N, Liu W (2012) The immersed molecular finite element method. Comput Methods Appl Mech Eng 233–236:28–39MathSciNetGoogle Scholar
  56. 56.
    Kremer K, Grest G (1990) Dynamics of entangled linear polymer melts: a molecular-dynamics simulation. J Chem Phys 92:5057Google Scholar
  57. 57.
    Kremer K, Müller-Plathe F (2002) Multiscale simulation in polymer science. Mol Simul 28(8–9):729–750Google Scholar
  58. 58.
    Kröger M (1999) Efficient hybrid algorithm for the dynamic creation of wormlike chains in solutions, brushes, melts and glasses. Comput Phys Commun 118(2):278–298Google Scholar
  59. 59.
    Kröger M (2004) Simple models for complex nonequilibrium fluids. Phys Rep 390(6):453–551MathSciNetGoogle Scholar
  60. 60.
    Kröger M, Hess S (2000) Rheological evidence for a dynamical crossover in polymer melts via nonequilibrium molecular dynamics. Phys Rev Lett 85(5):1128–1131Google Scholar
  61. 61.
    Lakes R (1993) Materials with structural hierarhcy. Nature 361(6412):511–514Google Scholar
  62. 62.
    Lakes RS (1999) Viscoelastic solids. CRC Press, Boca RatonGoogle Scholar
  63. 63.
    Leblanc JL (2000) Elastomer–filler interactions and the rheology of filled rubber compounds. J Appl Polym Sci 78(8):1541– 1550Google Scholar
  64. 64.
    Li Y, Kröger M, Liu WK (2011) Primitive chain network study on uncrosslinked and crosslinked \(cis\)-polyisoprene polymers. Polymer 52(25):5867–5878Google Scholar
  65. 65.
    Li Y, Kröger M, Liu WK (2012a) Nanoparticle effect on the dynamics of polymer chains and their entanglement network. Phys Rev Lett 109(11):118,001Google Scholar
  66. 66.
    Li Y, Kröger M, Liu WK (2012b) Nanoparticle geometrical effect on structure, dynamics and anisotropic viscosity of polyethylene nanocomposites. Macromolecules 45(4):2099–2112Google Scholar
  67. 67.
    Li Y, Tang S, Abberton B, Kröger M, Burkhart C, Jiang B, Papakonstantopoulos G, Poldneff M, Liu WK (2012c) A predictive multiscale computational framework for viscoelastic properties of linear polymers. Polymer 53(25):5935–5952Google Scholar
  68. 68.
    Li Y, Abberton B, Kröger M, Liu WK (2013) Challenges in multiscale modeling of polymer dynamics. Polymers 5(2):751– 832Google Scholar
  69. 69.
    Litvinov VM, Orza RA, Klüppel M, van Duin M, Magusin PCMM (2011) Rubber–filler interactions and network structure in relation to stress–strain behavior of vulcanized, carbon black filled epdm. Macromolecules 44(12):4887–4900Google Scholar
  70. 70.
    Liu WK, Belytschko T, Mani A (1986a) Probabilistic finite elements for nonlinear structural dynamics. Comput Methods Appl Mech Eng 56(1):61–81zbMATHGoogle Scholar
  71. 71.
    Liu WK, Belytschko T, Mani A (1986b) Random field finite elements. Int J Numer Methods Eng 23(10):1831–1845zbMATHMathSciNetGoogle Scholar
  72. 72.
    Liu WK, Karpov EG, Zhang S, Park H (2004) An introduction to computational nanomechanics and materials. Comput Methods Appl Mech Eng 193(17):1529–1578zbMATHMathSciNetGoogle Scholar
  73. 73.
    Liu WK, Karpov E, Park H, Wiley J (2006) Nano mechanics and materials: theory, multiscale methods and applications. Wiley, New YorkGoogle Scholar
  74. 74.
    Liu Y, Greene MS, Chen W, Dikin DA, Liu WK (2012) Computational microstructure characterization and reconstruction for stochastic multiscale material design. Comput Aided Des 45(1):65–76Google Scholar
  75. 75.
    Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice Hall, Englewood CliffsGoogle Scholar
  76. 76.
    Martyna G, Klein M, Tuckerman M (1992) Nosé-hoover chains: the canonical ensemble via continuous dynamics. J Chem Phys 97:2635Google Scholar
  77. 77.
    Matsen M (2001) The standard gaussian model for block copolymer melts. J Phys Condens Matter 14(2):R21Google Scholar
  78. 78.
    McVeigh C, Liu WK (2008) Linking microstructure and properties through a predictive multiresolution continuum. Comput Methods Appl Mech Eng 197(41–42):3268–3290zbMATHMathSciNetGoogle Scholar
  79. 79.
    McVeigh C, Liu WK (2010) Multiresolution continuum modeling of micro-void assisted dynamic adiabatic shear band propagation. J Mech Phys Solids 58(2):187–205zbMATHMathSciNetGoogle Scholar
  80. 80.
    McVeigh C, Vernerey F, Liu WK, Moran B, Olson G (2007) An interactive micro-void shear localization mechanism in high strength steels. J Mech Phys Solids 55(2):225–244Google Scholar
  81. 81.
    Milano G, Goudeau S, Müller-Plathe F (2005) Multicentered gaussian-based potentials for coarse-grained polymer simulations: linking atomistic and mesoscopic scales. J Polym Sci Part B 43(8):871–885Google Scholar
  82. 82.
    Miller RE, Tadmor EB (2009) A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model Simul Mater Sci 17(5):053,001Google Scholar
  83. 83.
    Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16(1):51–78zbMATHMathSciNetGoogle Scholar
  84. 84.
    Mindlin RD (1965) Second gradient of strain and surface-tension in linear elasticity. Int J Solids Struct 1(4):417–438Google Scholar
  85. 85.
    Müller-Plathe F (2002) Scale-hopping in computer simulations of polymers. Soft Mater 1(1):1–31Google Scholar
  86. 86.
    Müller-Plathe F (2012) Ibisco:it is boltzmann inversion software for coarse graining simulations. Available online at Accessed 18 Oct 2013
  87. 87.
    Mura T (1987) Micromechanics of defects in solids mechanics of elastic and inelastic solids, 2nd edn. Kluwer Academic, DordrechtGoogle Scholar
  88. 88.
    Naraghi M, Arshad SN, Chasiotis I (2011) Molecular orientation and mechanical property size effects in electrospun polyacrylonitrile nanofibers. Polymer 52(7):1612–1618Google Scholar
  89. 89.
    National Science and Technology Council (2011) Materials genome initiative for global competitiveness. Tech. rep, Office of Science and Technology PolicyGoogle Scholar
  90. 90.
    Nemat-Nasser S, Hori M (1999) Micromechanics: ovrall properties of heterogeneous materials. Elsevier, New YorkGoogle Scholar
  91. 91.
    Nix WD, Gao H (1998) Indentation size effects in crystalline materials: a law for strain gradient plasticity. J Mech Phys Solids 46(3):411–425zbMATHGoogle Scholar
  92. 92.
    Noid W, Chu J, Ayton G, Krishna V, Izvekov S, Voth G, Das A, Andersen H (2008a) The multiscale coarse-graining method. i. A rigorous bridge between atomistic and coarse-grained models. J Chem Phys 128(24):244,114Google Scholar
  93. 93.
    Noid W, Liu P, Wang Y, Chu J, Ayton G, Izvekov S, Andersen H, Voth G (2008b) The multiscale coarse-graining method. ii. Numerical implementation for coarse-grained molecular models. J Chem Phys 128(24):244,115Google Scholar
  94. 94.
    Oden JT, Prudhomme S (2011) Control of modeling error in calibration and validation processes for predictive stochastic models. Int J Numer Methods Eng 87(1–5):262–272zbMATHGoogle Scholar
  95. 95.
    Olson GB (2000) Designing a new material world. Science 288:993–998Google Scholar
  96. 96.
    Ostoja-Starzewski M (1998) Random field models of heterogeneous materials. Int J Solids Struct 35(19):2429–2455zbMATHGoogle Scholar
  97. 97.
    Ostoja-Starzewski M (2006) Material spatial randomness: from statistical to representative volume element. Probab Eng Mech 21(2):112–132Google Scholar
  98. 98.
    Padding J, Briels W (2011) Systematic coarse-graining of the dynamics of entangled polymer melts: the road from chemistry to rheology. J Phys Condens Matter 23(23):233,101Google Scholar
  99. 99.
    Papakonstantopoulos G, Doxastakis M, Nealey P, Barrat J, de Pablo J (2007) Calculation of local mechanical properties of filled polymers. Phys Rev E 75(3):031,803Google Scholar
  100. 100.
    Papargyri-Beskou S, Polyzos D, Beskos DE (2009) Wave dispersion in gradient elastic solids and structures: a unified treatment. Int J Solids Struct 46(21):3751–3759zbMATHGoogle Scholar
  101. 101.
    Park H, Karpov E, Liu WK (2004) A temperature equation for coupled atomistic/continuum simulations. Comput Methods Appl Mech Eng 193(17):1713–1732zbMATHMathSciNetGoogle Scholar
  102. 102.
    Parrinello M, Rahman A (1981) Polymorphic transitions in single crystals: a new molecular dynamics method. J Appl Phys 52(12):7182–7190Google Scholar
  103. 103.
    Paul W, Smith GD (2004) Structure and dynamics of amorphous polymers: computer simulations compared to experiment and theory. Rep Prog Phys 67(7):1117Google Scholar
  104. 104.
    Plimpton S et al (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117(1):1–19zbMATHGoogle Scholar
  105. 105.
    Qian D, Wagner GJ, Liu WK (2004) A multiscale projection method for the analysis of carbon nanotubes. Comput Methods Appl Mech Eng 193(17):1603–1632zbMATHGoogle Scholar
  106. 106.
    Qiao R, Deng H, Putz KW, Brinson LC (2011) Effect of particle agglomeration and interphase on the glass transition temperature of polymer nanocomposites. J Polym Sci Polym Phys 49(10):740–748Google Scholar
  107. 107.
    Reith D, Pütz M, Müller-Plathe F (2003) Deriving effective mesoscale potentials from atomistic simulations. J Comput Chem 24(13):1624–1636Google Scholar
  108. 108.
    Rho JY, Kuhn-Spearing L, Zioupos P (1998) Mechanical properties and the hierarchical structure of bone. Med Eng Phys 20:92–102Google Scholar
  109. 109.
    Rouse P Jr (1953) A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J Chem Phys 21:1272Google Scholar
  110. 110.
    Schieber JD, Neergaard J, Gupta S (2003) A full-chain, temporary network model with sliplinks, chain-length fluctuations, chain connectivity and chain stretching. J Rheol 47:213Google Scholar
  111. 111.
    Stankovich S, Dikin DA, Dommett GHB, Kohlhaas KM, Zimney EJ, Stach EA, Piner RD, Nguyen ST, Ruoff RS (2006) Graphene-based composite materials. Nature 442(7100):282–286Google Scholar
  112. 112.
    Starr F, Douglas J (2011) Modifying fragility and collective motion in polymer melts with nanoparticles. Phys Rev Lett 106(11):115,702Google Scholar
  113. 113.
    Stölken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46(14):5109– 5115Google Scholar
  114. 114.
    Sun H (1998) Compass: An ab initio force-field optimized for condensed-phase applications overview with details on alkane and benzene compounds. J Phys Chem B 102(38):7338–7364Google Scholar
  115. 115.
    Szleifer I, Carignano M (2009) Tethered polymer layers. Adv Chem Phys 94:165–260Google Scholar
  116. 116.
    Tang S, Greene MS, Liu WK (2011) A variable constraint tube model for size effects in polymer nano-structures. Appl Phys Lett 99(191):910Google Scholar
  117. 117.
    Tang S, Greene MS, Liu WK (2012a) A renormalization approach to model interaction in microstructured solids: application to porous elastomer. Comput Methods Appl Mech Eng 217–220:213–225MathSciNetGoogle Scholar
  118. 118.
    Tang S, Greene MS, Liu WK (2012b) Two-scale mechanism-based theory of nonlinear viscoelasticity. J Mech Phys Solids 60(2):199–226zbMATHMathSciNetGoogle Scholar
  119. 119.
    Tang S, Kopacz AM, Chan S, Olson GB, Liu WK (2013a) Three-dimensional ductile fracture analysis with a hybrid multiresolution approach and microtomography. J Mech Phys Solids 61(11):2108–2124Google Scholar
  120. 120.
    Tang S, Kopacz AM, OKeeffe SC, Olson GB, Liu WK (2013b) Concurrent multiresolution finite element: formulation and algorithmic aspects. Comput Mech. doi: 10.1007/s00466-013-0874-3
  121. 121.
    Thurner PJ, Erickson B, Jungmann R, Schriock Z, Weaver JC, Fantner GE, Schitter G, Morse DE, Hansma PK (2007) High-speed photography of compressed human trabecular bone correlates whitening to microscopic damage. Eng Frac Mech 74(12):1928–1941Google Scholar
  122. 122.
    Tian R, Chan S, Tang S, Kopacz AM, Wang JS, Jou HJ, Siad L, Lindgren LE, Olson GB, Liu WK (2010) A multiresolution continuum simulation of the ductile fracture process. J Mech Phys Solids 58(10):1681–1700zbMATHGoogle Scholar
  123. 123.
    Ting CS, Sachse W (1978) Measurement of ultrasonic dispersion by phase comparison of continuous harmonic waves. J Acoust Soc Am 64(3):852–857Google Scholar
  124. 124.
    Torquato S (2002) Statistical description of microstructures. Annu Rev Mater Res 32:77–111Google Scholar
  125. 125.
    Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11(1):385–414zbMATHMathSciNetGoogle Scholar
  126. 126.
    Tschöp W, Kremer K, Batoulis J, Bürger T, Hahn O (1998) Simulation of polymer melts. i. Coarse-graining procedure for polycarbonates. Acta Polym 49(2–3):61–74Google Scholar
  127. 127.
    Tuckerman M, Berne B, Martyna G (1992) Reversible multiple time scale molecular dynamics. J Chem Phys 97:1990Google Scholar
  128. 128.
    Uchic MD, Groeber MA, Dimiduk DM, Simmons JP (2006) 3d microstructural characterization of nickel superalloys via serial-sectioning using a dual beam fib-sem. Scripta Materialia 55(1):23–28 Google Scholar
  129. 129.
    Vacatello M (2001) Monte carlo simulations of polymer melts filled with solid nanoparticles. Macromolecules 34(6):1946–1952Google Scholar
  130. 130.
    Vernerey F, Liu WK, Moran B (2007) Multi-scale micromorphic theory for hierarchical materials. J Mech Phys Solids 55(12):2603–2651zbMATHMathSciNetGoogle Scholar
  131. 131.
    Vernerey F, Liu WK, Moran B, Olson G (2008) A micromorphic model for the multiple scale failure of heterogeneous materials. J Mech Phys Solids 56(4):1320–1347zbMATHMathSciNetGoogle Scholar
  132. 132.
    Vernerey FJ, Liu WK, Moran B, Olson G (2009) Multi-length scale micromorphic process zone model. Comput Mech 44(3):433–445zbMATHGoogle Scholar
  133. 133.
    Wagner G, Karpov E, Liu W (2004) Molecular dynamics boundary conditions for regular crystal lattices. Comput Methods Appl Mech Eng 193(17):1579–1601zbMATHMathSciNetGoogle Scholar
  134. 134.
    Wagner GJ, Liu WK (2003) Coupling of atomistic and continuum simulations using a bridging scale decomposition. J Comput Phys 190(1):249–274zbMATHGoogle Scholar
  135. 135.
    Wang L, Hu H (2005) Flexural wave propagation in single-walled carbon nanotubes. Phys Rev B 71(19):195,412Google Scholar
  136. 136.
    Xiao SP, Belytschko T (2004) A bridging domain method for coupling continua with molecular dynamics. Comput Methods Appl Mech Eng 193(17–20):1645–1669zbMATHMathSciNetGoogle Scholar
  137. 137.
    Xu H, Greene MS, Deng H, Dikin D, Brinson LC, Liu WK, Burkhart C, Papakonstantopoulos GJ, Poldneff M, Chen W (2013) Stochastic reassembly strategy for managing information complexity in heterogeneous materials analysis and design. J Mech Des (In press)Google Scholar
  138. 138.
    Yin D, Zhang Y, Peng Z, Zhang Y (2003) Effect of fillers and additives on the properties of sbr vulcanizates. J Appl Polym Sci 88(3):775–782Google Scholar
  139. 139.
    Yin X, Chen W, To A, McVeigh C, Liu WK (2008) Statistical volume element method for predicting microstructure-constitutive property relations. Comput Methods Appl Mech Eng 197(43–44):3516–3529zbMATHMathSciNetGoogle Scholar
  140. 140.
    Yurekli K, Krishnamoorti R, Tse MF, McElrath KO, Tsou AH, Wang HC (2001) Structure and dynamics of carbon black-filled elastomers. J Polym Sci Polym Phys 39(2):256–275Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • M. Steven Greene
    • 1
  • Ying Li
    • 2
  • Wei Chen
    • 2
    • 3
  • Wing Kam Liu
    • 4
    • 5
    • 6
    Email author
  1. 1.Theoretical and Applied MechanicsNorthwestern UniversityEvanstonUSA
  2. 2.Mechanical EngineeringNorthwestern UniversityEvanstonUSA
  3. 3.Wilson-Cook Professor in Engineering Design, Mechanical EngineeringNorthwestern UniversityEvanstonUSA
  4. 4.Walter P. Murphy Professor of Mechanical EngineeringNorthwestern UniversityEvanstonUSA
  5. 5.World Class University Program in Sungkyunkwan UniversitySeoulKorea
  6. 6.King Abdulaziz University (KAU)JeddahSaudi Arabia

Personalised recommendations