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Coupling atomistics and continuum in solids: status, prospects, and challenges

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Abstract

The recent focus of the scientific community on multiscale computer modeling techniques of nano-engineered materials stems from the desire to develop more realistic methodologies that are capable of accurately describing the varied time and length scales associated with this class of materials. Of importance is the ability to model the atomistic region using the appropriate techniques such as quantum mechanics/molecular dynamics, and the continuum region using homogenized properties. The continuity of atomistic and continuum regions in a solid necessitates a seamless coupling between these two regions. This is carried out using a transition region. In view of the large discrepancy between length and time scales in atomistic and continuum regions, the development of the transition region has been the main concern of the research community. It is the purpose of this review to critically discuss the issues concerning the transition region and the efforts made by the scientific community in treating them. In particular, this review addresses issues concerning the coupling of molecular dynamics to finite element modeling techniques. Three aspects of this review are accordingly considered. The first is concerned with the current state of atomistic–continuum coupling techniques in computational mechanics. The second is concerned with present the research conducted in the Engineering Mechanics and Design Laboratory at the University of Toronto in the field of nano-reinforced interfaces. Finally, we present the limitations of the current techniques and suggestions for improvements.

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Abbreviations

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {a}_{i} \) :

Acceleration of ith atom

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r}_{i} \) :

Position of ith atom

[B]:

Matrix containing derivatives of shape functions

r ij :

Distance between atom i and atom j

C i :

Set of atoms around representative atom i

R f(t):

Bridging scale random force term

{d}:

FE nodal displacement vector

t :

Time

[D]:

Elasticity matrix

{T}:

FE surface traction vector

{ε}:

FE strain vector

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u} \) :

Interpolated displacement field

E a :

Atomistic energy

U :

FE internal energy

E α :

Representative atom energy

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {U}_{i} \) :

Displacement on node i

E c :

Continuum energy

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {v}_{i} \) :

Velocity of ith atom

E e :

Elemental energy

\( V( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} } ) \) :

Interatomic potential energy

E i :

Energy of atom i

V i :

Individual atom potential

E QC :

Quasicontinuum energy

V ij :

Two-body potential

f α :

Force on representative atom α

V ijk :

Three-body potential

f i :

EAM electron density dependent embedding energy

W :

FE external work

f(t):

Bridging scale interatomic force

u, v, w :

x, y, z displacements in FE

f imp(t):

Bridging scale impedance force

ρ :

Mass density

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {F}_{i} \) :

Force acting on ith atom

θ(t−τ):

Bridging scale time history kernel

\( \left\{ \Im \right\} \) :

FE combined body and applied force vector

Π:

FE potential energy

g c :

Atomic level force experienced by cluster atom c

{σ}:

FE stress vector

m i :

Mass of ith atom

μ :

Lennard-Jones potential well depth parameter

M A :

Atomic mass matrix

ψ :

Lennard-Jones hard sphere radius parameter

M :

Finite element mass matrix

n α :

Representative atom weight function

[N]:

Shape function matrix

N e :

Number of atoms in element e

N n :

Number of nodes

N rep :

Number of representative atoms

AFEM:

Atomic scale finite element method

CADD:

Combined atomistic discrete dislocation method

CB:

Cauchy-Born rule

CFRP:

Carbon fibre reinforced plastic

CGMD:

Coarse grained molecular dynamics

CNT:

Carbon nanotube

DFT:

Density functional theory

EA:

Pure epoxy adhesive

EAM:

Embedded atom method potential

EANP:

Epoxy adhesive reinforced with nanopowder

EANT:

Epoxy adhesive reinforced with carbon nanotubes

FE:

Finite element

FEAt:

Finite element atomistic method

GFRP:

Glass fibre reinforced plastic

GLARE:

Glass reinforced aluminum composites

LJ:

Lennard-Jones potential

MD:

Molecular dynamics

MEMS:

Micro-electrical-mechanical devices

MM:

Micromechanics

MPM:

Material point method

NRPC:

Nano-reinforced polymer composites

QC:

Quasicontinuum method

QM:

Quantum mechanics

PDE:

Partial differential equation

RVE:

Representative volume element

SWCNT:

Single-walled carbon nanotube

SW:

Stillinger-Weber potential

MWCNT:

Multi-walled carbon nanotube

TB:

Tight binding method

References

  • Abraham, F.F.: Madly spanning the length scales in dynamic fracture. Comp. Model. Eng. Sci. 1(4), 63–69 (2000)

    MathSciNet  Google Scholar 

  • Abraham, F.F., Broughton, J.Q.: Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture. Europhys. Lett. 44(6), 783–787 (1998). doi:10.1209/epl/i1998-00536-9

    Article  Google Scholar 

  • Abraham, F.F., Walkup, R., Gao, H., Duchaineau, M., De la Rubia, T.D., Seager, M.: Simulating materials failure by using up to one billion atoms and the world’s fastest computer: brittle fracture. Proc. Natl. Acad. Sci. USA 99(9), 5777–5782 (2002). doi:10.1073/pnas.062012699

    Article  Google Scholar 

  • Adelzadeh, M., Shodja, H.M., Rafii-Tabar, H.: Computational modeling of the interaction of two edge cracks, and two edge cracks interacting with a nanovoid, via an atomistic finite element method. Comput. Mater. Sci. 42, 186–193 (2008). doi:10.1016/j.commatsci.2007.07.012

    Article  Google Scholar 

  • Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Claredon Press, Oxford (1987)

    MATH  Google Scholar 

  • Almeida, T.S., Coutinho, K., Cabral, B.J.C., Canuto, S.: Electronic properties of liquid ammonia: a sequential molecular dynamics/quantum mechanics approach. J. Chem. Phys. 128, 014506 (2008)

    Google Scholar 

  • An, W., Wu, X., Yang, J.L., Zeng, X.C.: Adsorption and surface reactivity on single-walled boron nitride nanotubes containing stone-wales defects. J. Phys. Chem. C 111, 14105–14112 (2007). doi:10.1021/jp072443w

    Google Scholar 

  • Aoyagi, T., et al.: A general-purpose coarse-grained molecular dynamics program. Comput. Phys. Commun. 145, 267–279 (2002). doi:10.1016/S0010-4655(02)00271-0

    Article  MATH  Google Scholar 

  • Baker, A., Dutton, S., Kelly, D.: Composite Materials for Aircraft Structures, 2nd edn. American Institute of Aeronautics and Astronautics, USA (2004)

    Google Scholar 

  • Bockenheimer, C., Valeske, B., Possart, W.: Network structure in epoxy aluminium bonds after mechanical treatment. Int. J. Adhes. Adhes. 22, 349–356 (2002). doi:10.1016/S0143-7496(02)00014-3

    Article  Google Scholar 

  • Broughton, J., Abraham, F.: Concurrent coupling of length scales methodology and application. Phys. Rev. B 60, 4 (1999). doi:10.1103/PhysRevB.60.2391

    Google Scholar 

  • Buehler, M., Kong, Y., Gao, H.: Deformation mechanisms of very long single-wall carbon nanotubes subject to compressive loading. J. Eng. Mater. Technol. 126, 245–249 (2004)

    Google Scholar 

  • Cao, G., Chen, X.: Buckling behavior of single-walled carbon nanotubes and a targeted molecular mechanics approach. Phys. Rev. B, 74, 10pp (2006a)

  • Cao, G., Chen, X.: Buckling of single-walled carbon nanotubes upon bending: molecular dynamics simulations and the finite element method. Phys. Rev. B. 73, 11pp (2006b)

  • Cao, G., Chen, X.: The effects of chirality and boundary conditions on the mechanical properties of single-walled carbon nanotubes. Int. J. Solids Struct. 44, 5447–5465 (2007). doi:10.1016/j.ijsolstr.2007.01.005

    Article  MATH  Google Scholar 

  • Carlsson, A.: Beyond pair potentials in elemental transition metals and semiconductors. Solid. State. Phys. 43, 1–91 (1990). doi:10.1016/S0081-1947(08)60323-9

    Article  Google Scholar 

  • Cavallotti, C., Di Stanislao, M., Moscatelli, D., Veneroni, A.: Materials computation towards technological impact: the multiscale approach to thin films deposition. Electrochim. Acta 50, 4566–4575 (2005). doi:10.1016/j.electacta.2004.10.092

    Article  Google Scholar 

  • Chandraseker, K.I., Mukherjee, S.: Atomistic-continuum and ab initio estimation of the elastic moduli of single-walled carbon nanotubes. Comput. Mater. Sci. 40, 147–158 (2007). doi:10.1016/j.commatsci.2006.11.014

    Article  Google Scholar 

  • Clementi, E.: Global scientific and engineering simulations on scalar, vector and parallel LCAP-type supercomputers. Philos. Trans. R. Soc. Lond. A 326, 445–470 (1988). doi:10.1098/rsta.1988.0097

    Article  MATH  Google Scholar 

  • Curtin, W., Miller, R.: Atomistic/continuum coupling in computational materials science. Model Simul. Mater. Sci. Eng. 11, R33–R68 (2003). doi:10.1088/0965-0393/11/3/201

    Article  Google Scholar 

  • Daw, M., Baskes, M.: Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B. 29, 12 (1984). doi:10.1103/PhysRevB.29.6443

    Article  Google Scholar 

  • Ding, F.: Theoretical study of the stability of defects in single-walled carbon nanotubes as a function of their distance from the nanotube end. Phys. Rev. B. 72, 245–409 (2005)

    Google Scholar 

  • Dupuy L., Tadmor, E., Miller, R., Phillips, R.: Finite-temperature quasicontinuum: molecular dynamics without all the atoms. PRL 95 (2005)

  • Endo, M., Hayashi, T., Kim, Y.A., Terrones, M., Dresselhaus, M.S.: Applications of carbon nanotubes in the twenty-first century. Philos. Trans. R. Soc. Lond. A 362, 2223–2238 (2004). doi:10.1098/rsta.2004.1437

    Article  Google Scholar 

  • Ercolessi, F.: A Molecular Dynamics Primer. Spring College in Computational Physics, Trieste (1997)

  • Esfarjani, K., Gorjizadeh, N., Nasrollahi, Z.: Molecular dynamics of single wall carbon nanotube growth on nickel surface. Comput. Mater. Sci. 36, 117–120 (2006). doi:10.1016/j.commatsci.2005.02.022

    Article  Google Scholar 

  • Fiedler, B., Gojny, F.H., Wichmann, M.H.G., Nolte, M.C.M., Schulte, K.: Fundamental aspects of nano-reinforced composites. Compos. Sci. Technol. 66, 3115–3125 (2006). doi:10.1016/j.compscitech.2005.01.014

    Article  Google Scholar 

  • Fish, J., Nuggehally, M.A., Shepard, M.S., Picu, C.R., Badia, S., Parks, M.L., et al.: Concurrent AtC coupling based on a blend of the continuum stress and the atomistic force. Comput. Methods Appl. Mech. Eng. 196, 4548–4560 (2007). doi:10.1016/j.cma.2007.05.020

    Article  Google Scholar 

  • Frankland, S.J.V., Caglar, A., Brenner, D.W., Griebel, M.: Molecular simulation of the influence of chemical cross-links on the shear strength of carbon nanotube-polymer interfaces. J. Phys. Chem. B. 106, 12 (2002). doi:10.1021/jp015591+

    Article  Google Scholar 

  • Galano, A., Francisco-Marquez, M.: Reactivity of silicon and germanium doped CNTs toward aromatic sulfur compounds: a theoretical approach. Chem. Phys. 345, 87–94 (2008). doi:10.1016/j.chemphys.2008.01.040

    Article  Google Scholar 

  • Gao, X.L., Li, K.: A shear-lag model for carbon nanotube-reinforced polymer composites. Int. J. Solids Struct. 42, 1649–1667 (2005). doi:10.1016/j.ijsolstr.2004.08.020

    Article  MATH  Google Scholar 

  • Gates, T.S., Odegard, G.M., Frankland, S.J.V., Clancy, T.C.: Computational materials: multi-scale modeling and simulation of nanostructured materials. Compos. Sci. Technol. 65, 2416–2434 (2005). doi:10.1016/j.compscitech.2005.06.009

    Article  Google Scholar 

  • Ghoniem, N., Busso, E., Kioussis, N., Huang, H.: Multiscale modeling of nanomechanics and micromechanics an overview. Philos. Mag. V83(31–34), 3475–3528 (2003). doi:10.1080/14786430310001607388

    Article  Google Scholar 

  • Gong, S.X., Meguid, S.A.: On the elastic fields of an elliptical inhomogeneity under plane deformation. Proc. R. Soc. Lond. A. 443(1919), 457–471 (1993)

    Google Scholar 

  • Gong, N., Liang, Y., Yao, Y., Liu, B.: Static and dynamic analysis of carbon nanotube cantilever based on molecular dynamics simulation. Key Eng. Mater. 375–376, 631–635 (2008)

    Article  Google Scholar 

  • Gumbsch, P.: An atomistic study of brittle fracture: toward explicit failure criteria from atomistic modeling. J. Mater. Res. 10(11), 2897–2907 (1995). doi:10.1557/JMR.1995.2897

    Article  Google Scholar 

  • Gumbsch, P., Beltz, G.: On the continuum versus atomistic descriptions of dislocation nucleation and cleavage in nickel. Model. Simul. Mater. Sci. Eng. 3, 597–613 (1995). doi:10.1088/0965-0393/3/5/002

    Article  Google Scholar 

  • Guo, Z., Yang, W.: MPM/MD handshaking method for multiscale simulation and its application to high energy cluster impacts. Int. J. Mech. Sci. 48, 145–159 (2006). doi:10.1016/j.ijmecsci.2005.08.007

    Article  MATH  Google Scholar 

  • Guo, X., Leung, A.Y.T., Jiang, H., He, X.Q., Huang, Y.: Critical strain of carbon nanotubes: an atomic-scale finite element study. J. Appl. Mech. 74, 347–351 (2007)

    Google Scholar 

  • Hao, S., Moran, B., Liu, W., Olson, G.: A hierarchical multi-physics model for design of high toughness steels. J. Comput. Aided Mater. Des. 10, 99–142 (2003). doi:10.1023/B:JCAD.0000036813.66891.41

    Article  Google Scholar 

  • Haslam, A.J., Moldovan, D., Phillpot, S.R., Wolf, D., Gleiter, H.: Combined atomistic and mesoscale simulation of grain growth in nanocrystalline thin films. Comput. Mater. Sci. 23, 15–32 (2002). doi:10.1016/S0927-0256(01)00218-X

    Article  Google Scholar 

  • Hockney, R.W.: The potential calculation and some applications. Methods Comput. Phys. 9, 135–211 (1970)

    Google Scholar 

  • Hu, N., Fukunaga, H., Lu, C., Kameyama, M., Yan, B.: Prediction of elastic properties of carbon nanotube reinforced composites. Proc. R. Soc. 461, 1685–1710 (2005). doi:10.1098/rspa.2004.1422

    Article  Google Scholar 

  • Hughes, T.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Englewood Cliffs, NJ Prentice-Hall (1987)

    MATH  Google Scholar 

  • Izvekov, S., Voth, G.: A multiscale coarse-graining method for biomolecular systems. Phys. Chem. B. 109, 2469–2473 (2005)

    Google Scholar 

  • Jones, J.: On the determination of molecular fields: II from the equation of state of a gas. Proc. R. Soc. A106, 463–477 (1924). doi:10.1098/rspa.1924.0082

    Article  Google Scholar 

  • Karpov, E., Wagner, G., Liu, W.: A green’s function approach to deriving non-reflecting boundary conditions in molecular dynamics simulations. Int. J. Numer. Methods Eng. 62, 1250–1262 (2005). doi:10.1002/nme.1234

    Article  MATH  Google Scholar 

  • Knap, J., Ortiz, M.: An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49, 1899–1923 (2001). doi:10.1016/S0022-5096(01)00034-5

    Article  MATH  Google Scholar 

  • Kohlhoff, S., Gumbsch, P., Fischmeister, H.: Crack propagation in b.c.c crystals studied with a combined finite-element and atomistic model. Philos. Mag. A 64(4), 851–878 (1991). doi:10.1080/01418619108213953

    Article  Google Scholar 

  • Kojic, M., Filipovic, N., Tsuda, A.: A mesoscopic bridging scale method for fluids and coupling dissipative particle dynamics with continuum finite element method. Comput. Methods Appl. Eng. 197, 821–833 (2008). doi:10.1016/j.cma.2007.09.011

    Article  MathSciNet  MATH  Google Scholar 

  • Leung, A.Y.T., Guo, X., He, X.Q., Kitipornchai, S.: A continuum model for zigzag single-walled carbon nanotubes. Appl. Phys. Lett. 86, 083110 (2005)

    Google Scholar 

  • Leung, A.Y.T., Guo, X., He, X.Q., Jiang, H., Hunag, Y.: Postbuckling of carbon nanotubes by atomic-scale finite element. J. Appl. Phys. 99, 124308 (2006)

    Google Scholar 

  • Li, C., Chou, T.: Elastic moduli of multi-walled carbon nanotubes and the effect of van der waals forces. Compos. Sci. Technol. 63, 1517–1524 (2003a). doi:10.1016/S0266-3538(03)00072-1

  • Li, C., Chou, T.: A structural mechanics approach for the analysis of carbon nanotubes. Int. J. Solids Struct. 40, 2487–2499 (2003b). doi:10.1016/S0020-7683(03)00056-8

    Article  MATH  Google Scholar 

  • Li, C., Chou, T.: Multiscale modeling of compressive behavior of carbon nanotube/polymer composites. Compos. Sci. Technol. 66, 2409–2414 (2006). doi:10.1016/j.compscitech.2006.01.013

    Article  Google Scholar 

  • Liew, K.M., Wong, C.H., He, X.Q., Tan, M.J., Meguid, S.A.: Nanomechanics of single and multiwalled carbon nanotubes. Phys. Rev. B. 69, 11 (2004). doi:10.1103/PhysRevB.69.115429

    Article  Google Scholar 

  • Liu, W., Karpov, E., Zhang, S., Park, H.: An introduction to computational nanomechanics and materials. Comput. Methods Appl. Mech. Eng. 193, 1529–1578 (2004a). doi:10.1016/j.cma.2003.12.008

    Article  MATH  MathSciNet  Google Scholar 

  • Liu, B., Huang, Y., Jiang, H., Qu, S., Hwang, K.C.: The atomic-scale finite element method. Comput. Methods Appl. Mech. Eng. 193, 1849–1864 (2004b). doi:10.1016/j.cma.2003.12.037

    Article  MATH  Google Scholar 

  • Liu, B., Jiang, H., Huang, Y., Qu, S., Yu, M.-F., Hwang, K.C.: Atomic-scale finite element method in multiscale computation with applications to carbon nanotubes. Phys. Rev. B. 72, 035435 (2005)

    Google Scholar 

  • Liu, W.K., Park, H.S., Qian, D., Karpov, E.G., Kadowaki, H., Wagner, G.J.: Bridging scale methods for nanomechanics and materials. Comput. Methods Appl. Mech. Eng. 195, 1407–1421 (2006). doi:10.1016/j.cma.2005.05.042

    Article  MATH  MathSciNet  Google Scholar 

  • Lordi, V., Yao, N.: Molecular mechanics of binding in carbon-nanotube-polymer composites. J. Mater. Res. 15, 12 (2000)

    Google Scholar 

  • Lu, G., Tadmor, E.B., Kaxiras, E.: From electrons to finite elements: a concurrent multiscale approach for metals. Phys. Rev. B. 73, 124108 (2006a)

  • Lu, H., Daphalapurkar, N., Wang, B., Roy, S., Komanduri, R.: Multiscale simulation from atomistic to continuum–coupling molecular dynamics (MD) with the material point method (MPM). Philos. Mag 86(20), 2971–2994 (2006b). doi:10.1080/14786430600625578

    Article  Google Scholar 

  • Meguid, S.A., Chen, B.J.: Modelling temperature-dependent fracture nucleation of SWCNTs using atomistic-based continuum theory. Int. J. Sol. Struct. 44(11–12), 3828–3839 (2007)

    MATH  Google Scholar 

  • Meguid, S.A., Sun, Y.: On the tensile and shear strength of nano-reinforced composite interfaces. Mater. Des. 325, 289–296 (2004)

    Google Scholar 

  • Meguid, S.A., Sun, Y.: Intelligent condition monitoring of aerospace composites: part I—nano reinforced surfaces & interfaces. Int. J. Mech. Mater. Des. 37–52 (2005)

  • Meguid, S.A., Wang, X.D.: On the dynamic interaction between a microdefect and a main crack. Proc. R. Soc. Lond. A. 448(1934), 449–464 (1995)

    Google Scholar 

  • Meguid, S.A., Wang, X.D.: The dynamic interaction of a microcrack with a main crack under antiplane loading. Int. J. Solids Struct. 31(8), 1085–1097 (1994). doi:10.1016/0020-7683(94)90165-1

    Article  MATH  Google Scholar 

  • Micheal Lai, W., Rubin, D., Krempl, E.: Introduction to Continuum Mechanics, Revised Edition. Pergamon Press, NY (1978)

    Google Scholar 

  • Miller, R., Tadmor, E.: The quasicontinuum method: overview, applications and current directions. J. Comput. Aided. Mater. Des. 9, 203–239 (2002). doi:10.1023/A:1026098010127

    Article  Google Scholar 

  • Miller, R., Tadmor, E., Phillips, R., Ortiz, M.: Quasicontinuum simulation of fracture at the atomic scale. Model. Simul. Mater. Sci. Eng. 6, 607–638 (1998a). doi:10.1088/0965-0393/6/5/008

    Article  Google Scholar 

  • Miller, R., Ortiz, M., Phillips, R., Shenoy, V., Tadmor, E.: Quasicontinuum models of fracture and plasticity. Eng. Fract. Mech. 61, 427–444 (1998b). doi:10.1016/S0013-7944(98)00047-2

    Article  Google Scholar 

  • Miller, R.E., Shilkrot, L.E., Curtin, W.A.: A coupled atomistics and discrete dislocation plasticity simulation of nanoindentation into single crystal films. Acta Mater. 52, 271–284 (2004). doi:10.1016/j.actamat.2003.09.011

    Article  MathSciNet  Google Scholar 

  • Mylvaganam, K., Vodenitcharova, T., Zhang, L.C.: The bending-kinking analysis of a single-walled carbon nanotube–a combined molecular dynamics and continuum mechanics technique. J. Mater. Sci. 41, 3341–3347 (2006). doi:10.1007/s10853-005-5389-7

    Article  Google Scholar 

  • Namilae, S., Chandra, N.: Multiscale model to study the effect of interfaces in carbon nanotube-based composites. J. Eng. Mater. Technol. 127 222–232 (2005)

    Google Scholar 

  • Odegard, G.M., Frankland, S.J.V.: Effect of nanotube functionalisation on the elastic properties of polyethylene nanotube composites. AIAA J. 43(8), 1828–1835 (2005). doi:10.2514/1.9468

    Article  Google Scholar 

  • Odegard, G.M., Gates, T.S., Nicholson, L.M., Wise, K.E.: Equivalent-continuum modeling of nano-structured materials. Compos. Sci. Technol. 62, 1869–1880 (2002). doi:10.1016/S0266-3538(02)00113-6

    Article  Google Scholar 

  • Odegard, M., Gates, T.S., Wise, K.E., Park, C., Siochi, E.J.: Constitutive modeling of nanotube-reinforced polymer composites. Compos. Sci. Technol. 63, 1671–1687 (2003). doi:10.1016/S0266-3538(03)00063-0

    Article  Google Scholar 

  • Pantano, A., Parks, D.M., Boyce, M.C.: Mechanics of deformation of single- and multi-wall carbon nanotubes. J. Mech. Phys Solids 52, 789–821 (2004). doi:10.1016/j.jmps.2003.08.004

    Article  MATH  Google Scholar 

  • Park, H., Liu, W.: An introduction and tutorial on multiple-scale analysis in solids. Comput. Methods Appl. Mech. Eng. 193, 1733–1772 (2004). doi:10.1016/j.cma.2003.12.054

    Article  MATH  MathSciNet  Google Scholar 

  • Park, H.S., Karpov, E.G., Liu, W.K., Klein, P.A.: The bridging scale for two-dimensional atomistic/continuum coupling. Philos. Mag. 85(1), 79–113 (2005). doi:10.1080/14786430412331300163

    Article  Google Scholar 

  • Qian, D., Wagner, G., Liu, W.: A multiscale projection method for the analysis of carbon nanotubes. Comput. Methods Appl. Mech. Eng. 193(17–20), 1603–1632 (2004). doi:10.1016/j.cma.2003.12.016

    Article  MATH  Google Scholar 

  • Qu, S., Shastry, V., Curtin, W., Miller, R.: A finite-temperature dynamic coupled atomistic/discrete dislocation method. Model. Simul. Mater. Sci. Eng. 13, 1101–1118 (2005). doi:10.1088/0965-0393/13/7/007

    Article  Google Scholar 

  • Rafii-Tabar, H.: Modelling the nano-scale phenomena in condensed matter physics via computer-based numerical simulations. Phys. Rep. 325, 239–310 (2000). doi:10.1016/S0370-1573(99)00087-3

    Article  Google Scholar 

  • Rodney, D., Phillips, R.: Structure and strength of dislocation junctions: an atomic level analysis. PRL 82, 1704–1707 (1999). doi:10.1103/PhysRevLett.82.1704

    Article  Google Scholar 

  • Rudd, R.E.: Coupling of length scales in MEMS modelling: the atomic limit of finite elements. Int. Soc. Opt. Eng. 4019, 16–25 (2000)

    Google Scholar 

  • Rudd, R.E., Broughton, J.Q.: Coarse-grained molecular dynamics and the atomic limit of finite elements. Phys. Rev. B. 58, 10 (1998). doi:10.1103/PhysRevB.58.R5893

    Article  Google Scholar 

  • Rudd, R.E., Broughton, J.Q.: Coupling of length scales and atomistic simulation of MEMS resonators. Int. Soc. Opt. Eng. 3680, 104–113 (1999)

    Google Scholar 

  • Rudd, R., Broughton, J.: Concurrent coupling of length scales in solid state systems. Phys. Status Solidif. 217, 251 (2000). doi :10.1002/(SICI)1521-3951(200001)217:1<251::AID-PSSB251>3.0.CO;2-A

    Article  Google Scholar 

  • Rudd, R.E., Broughton, J.Q.: Coarse-grained molecular dynamics: nonlinear finite elements and finite temperature. Phys. Rev. B. 72, 144104 (2005)

    Google Scholar 

  • Shenoy, V., Miller, R., Tadmor, E., Phillips, R., Ortiz, M.: Quasicontinuum models of interfacial structure and deformation. Phys. Rev. Lett. 80, 742–745 (1998). doi:10.1103/PhysRevLett.80.742

    Article  Google Scholar 

  • Shenoy, V., Miller, R., Tadmor, E., Rodney, D., Phillips, R., Ortiz, M.: An adaptive finite element approach to atomic-scale mechanics–the quasicontinuum method. J. Mech. Phys Solids 47, 611–642 (1999a). doi:10.1016/S0022-5096(98)00051-9

    Article  MATH  MathSciNet  Google Scholar 

  • Shenoy, V., Miller, R., Tadmor, E., Rodney, D., Phillips, R., Ortiz, M.: An adaptive finite element approach to atomic-scale mechanics—the quasicontinuum method. J. Mech. Phys. Solids 47, 611–642 (1999b). doi:10.1016/S0022-5096(98)00051-9

    Article  MATH  MathSciNet  Google Scholar 

  • Shi, D., Feng, X., Hang, H., Huang, Y., Hwang, K.: Multiscale analysis of fracture of carbon nanotubes embedded in composites. Int. J. Fract. 134, 369–386 (2005). doi:10.1007/s10704-005-3073-1

    Article  Google Scholar 

  • Shiari, B., Miller, R., Curtin, W.: Coupled atomistic/discrete dislocation simulations of nanoindentation at finite temperatures, J. Eng. Mater. Technol. 127, 358–368 (2005)

    Google Scholar 

  • Shilkrot, L.E., Miller, R.E., Curtin, W.A.: Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics. J. Mech. Phys. Solids. 52, 755–787 (2004). doi:10.1016/j.jmps.2003.09.023

    Article  MATH  MathSciNet  Google Scholar 

  • Shlarl, B., Miller, R.E., Klug, D.D.: Multiscale modeling of solids at the nanoscale: dynamic approach. Can. J. Phys. 86, 391–400 (2008). doi:10.1139/P07-145

    Article  Google Scholar 

  • Spencer, A.J.M.: Continuum Mechanics. Dover Publications, Dover edition, NY (2004)

    MATH  Google Scholar 

  • Srivastava, D., Wei, C.: Nanomechanics of carbon nanotubes and composites. Appl. Mech. Rev 56, 2 (2003). doi:10.1115/1.1538625

    Article  Google Scholar 

  • Stillinger, F., Weber, T.: Computer simulation of local order in condensed phases of silicon. Phys. Rev. B 31, 8 (1985). doi:10.1103/PhysRevB.31.5262

    Article  Google Scholar 

  • Sulsky, D., Zhou, S., Schreyer, H.: Application of a particle-in-cell-method to solid mechanics. Comput. Phys. Commun. 87, 236–252 (1995). doi:10.1016/0010-4655(94)00170-7

    Article  MATH  Google Scholar 

  • Sun, Y.: Influence of nanofillers on bond strength and toughness, University of Toronto, Ph.D. Thesis, 2007

  • Tadmor, E., Phillips, R., Ortiz, M.: Mixed atomistic and continuum models of deformation in solids. Langmuir 12, 4529–4534 (1996a). doi:10.1021/la9508912

    Article  Google Scholar 

  • Tadmor, E., Ortiz, M., Phillips, R.: Quasicontinuum analysis of defects in solids. Philos. Mag. A 73(6), 1529–1563 (1996b). doi:10.1080/01418619608243000

    Article  Google Scholar 

  • Tadmor, E., Miller, R., Phillips, R., Ortiz, M.: Nanoindentation and incipient plasticity. J. Mater. Res. 14, 2233–2250 (1999). doi:10.1557/JMR.1999.0300

    Article  Google Scholar 

  • Troya, D., Mielke, S.L., Schatz, G.C.: Carbon nanotube fracture-differences between quantum mechanical mechanisms and those of empirical potentials. Chem. Phys. Lett. 382, 133–141 (2003). doi:10.1016/j.cplett.2003.10.068

    Article  Google Scholar 

  • Tserpes, K.I., Papanikos, P., Labeas, G.: Sp.G. Pantelakis, Multi-scale modeling of tensile behavior of carbon nanotube-reinforced composites. Theor. Appl. Fract. Mech. 49, 51–60 (2008). doi:10.1016/j.tafmec.2007.10.004

    Article  Google Scholar 

  • Verlet, L.: Computer experiments on classical fluids. I. Thermodynamical properties of lennard-jones molecules. Phys. Rev. 159, 98–103 (1967). doi:10.1103/PhysRev.159.98

    Article  Google Scholar 

  • Vvedensky, D.: Multiscale modeling of nanostructures. J. Phys. Condens. Matter 16, R1537–R1576 (2004). doi:10.1088/0953-8984/16/50/R01

    Article  Google Scholar 

  • Wagner, G.J., Liu, W.K.: Coupling of atomistic and continuum simulations using a bridging scale decomposition. J. Comput. Phys. 190, 249–274 (2003). doi:10.1016/S0021-9991(03)00273-0

    Article  MATH  Google Scholar 

  • Wagner, G., Karpov, E., Liu, W.: Molecular dynamics boundary conditions for regular crystal lattices. Comput. Methods Appl. Mech. Eng. 193, 1579–1601 (2004). doi:10.1016/j.cma.2003.12.012

    Article  MATH  MathSciNet  Google Scholar 

  • Wang, X.D., Meguid, S.A.: Dynamic interaction between a matrix crack and a circular inhomogeneity with a distinct interphase. Int. J. Solids Struct. 36, 517–531 (1999). doi:10.1016/S0020-7683(98)00039-0

    Article  MATH  Google Scholar 

  • Wang, C., Zhou, G., Liu, H., Wu, J., Qiu, Y., Gu, B., et al.: Chemical functionalisation of carbon nanotubes by carboxyl groups on stone-wales defects: a density functional theory study. J. Phys. Chem. B 110, 10266–10271 (2006). doi:10.1021/jp060412f

    Article  Google Scholar 

  • Wong, C.H., Liew, K.M., He, X.Q., Tan, M.J., Meguid, S.A.: Modeling and simulation of multi-walled carbon nanotubes using molecular dynamics simulation. NSTI. Nanotech. 3, 2004 (2004)

    Google Scholar 

  • Yakobson, B.I., Brabec, C.J., Bernhole, J.: Nanomechanics of carbon tubes: instabilities beyond linear response. Phys. Rev. Lett. 76, 14 (1996). doi:10.1103/PhysRevLett.76.2511

    Article  Google Scholar 

  • Zeng, Q.H., Yu, A.B., Lu, G.Q.: Multiscale modeling and simulation of polymer nanocomposites. Prog. Polym. Sci. 33, 191–269 (2008). doi:10.1016/j.progpolymsci.2007.09.002

    Article  Google Scholar 

  • Zhao, X., Cummings, P.T.: Molecular dynamics study of carbon nanotube oscillators revisited. J. Chem. Phys. 124, 134–705 (2006)

    Google Scholar 

  • Zhao, J., Ding, Y.: Silicon carbide nanotubes functionalized by transition metal atoms: a density-functional study. J. Phys. Chem. C 112, 2558–2564 (2008). doi:10.1021/jp073722m

    Article  Google Scholar 

  • Zienkiewicz, O.C.: The Finite Element Method, vol 1–2, 4th edn. McGraw-Hill, London (1991)

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Acknowledgements

The authors wish to acknowledge the financial support provided by Natural Sciences and Engineering Research Council (NSERC) of Canada.

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  1. Engineering Mechanics and Aerospace Design Lab, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, ON, Canada, M5S 3G8

    J. M. Wernik & S. A. Meguid

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Wernik, J.M., Meguid, S.A. Coupling atomistics and continuum in solids: status, prospects, and challenges. Int J Mech Mater Des 5, 79–110 (2009). https://doi.org/10.1007/s10999-008-9087-x

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