Abstract
This paper presents a multiscale continuum field theory and its application in modeling and simulation of nano/micro systems. The theoretical construction of the continuum field theory will be briefly introduced. In the simulation model, a single crystal can be discretized into finite element mesh as in a continuous medium. However, each node is a representative unit cell, which contains a specified number of discrete and distinctive atoms. Governing differential equations for each atom in all nodes are obtained. Material behaviors of a given system subject to the combination of mechanical loadings and temperature field can be obtained through numerical simulations. In this work, the nanoscale size effect in single crystal bcc iron is studied, the phenomenon of wave propagation is simulated and wave speed is obtained. Also, dynamic crack propagation in a multiscale model is simulated to demonstrate the advantage and applicability of this multiscale continuum field theory.
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References
Stan M, Yip S (2009) Design and evaluation of nuclear fuels and structural materials: predictive modeling and high-performance simulations, white paper for the Joint Office of Science and Office of Nuclear Energy Workshop on Advanced Modeling and Simulation for Nuclear Fission Energy Systems, Washington DC
Tadmor EB, Ortiz M, Phillips R (1996) Quasicontinuum analysis of defects in solids. Phil Mag A 73: 1529–1563
Ericksen JL (1984) The Cauchy and born hypothesis for crystals. In: Gurtin M (eds) Phase transformations and material instabilities in solids. Academic Press, New York, pp 61–77
Rudd RE, Broughton JQ (1998) Coarse-grained molecular dynamics and atomic limit of finite elements. Phys Rev B 58: 5893–5896
Rudd RE, Broughton JQ (2000) Concurrent coupling of length scales in solid state systems. Phys Stat Sol 217: 251–291
Abraham F, Broughton J, Bernstein N, Kaxiras E (1998) Spanning the length scales in dynamic simulation. Comput Phys 12: 538–546
Broughton J, Bernstein N, Kaxiras E, Abraham F (1999) Concurrent coupling of length scales: methodology and application. Phys Rev B 60: 2391–2403
Rudd RE, Broughton JQ (1999) Atomistic simulation of MEMS resonators trough the coupling of length scales. J Model Simul Microsyst 1(1): 29–38
Li XT, E W (2005) Multiscale modeling of dynamics of solids at finite temperature. J Mech Phys Solids 53: 1650–1685
E W, Engquist B (2003) The heterogeneous multi-scale methods. Comm Math Sci 1(1): 87–132
Wagner GJ, Karpov EG, Liu WK (2004) Molecular dynamics boundary conditions for regular crystal lattices. Comput Methods Appl Mech Eng 193(17–20): 1579–1601
Qian D, Wagner GJ, Liu WK (2004) A multiscale projection method for the analysis of carbon nanotubes. Comput Methods Appl Mech Eng 193: 1603–1632
Wagner GJ, Liu WK (2003) Coupling of atomic and continuum simulations using a bridging scale decomposition. J Comput Phys 190: 249–274
Karpov EG, Yu H, Park H, Liu WK, Wang J, Qian D (2006) Multiscale boundary conditions in crystalline solids: theory and application to nanoindentation. Int J Solids Struct 43(21): 6359–6379
Park HS, Karpov EG, Liu WK, Klein PA (2005) The bridging scale for three-dimensional atomistic/continuum coupling. Phil Mag 85(1): 79–113
Xiao SP, Belytschko T (2004) A bridging domain method for coupling continua with molecular dynamics. Comput Methods Appl Mech Eng 193: 1645–1669
Vernerey FJ, Liu WK, Moran B (2007) Multi-scale micromorphic theory for hierarchical materials. J Mech Phys Solid 55(12): 2603–2651
Vernerey FJ, Liu WK, Moran B, Olson GB (2008) A micromorphic model for the multiple scale failure of heterogeneous materials. J Mech Phys Solids 56(4): 1320–1347
McVeigh C, Liu WK (2008) Linking microstructure and properties through a predictive multiresolution continuum. Comput Methods Appl Mech Eng 197: 3268–3290
McVeigh C, Liu WK (2009) Multiresolution modeling of ductile reinforced brittle composites. J Mech Phys Solids 57: 244–267
To AC, Li S (2005) Perfectly matched multiscale simulations. Phys Rev B 72: 035414
Li S, Liu X, Agrawal A, To AC (2006) Perfectly matched multiscale simulations for discrete systems: extension to multiple dimensions. Phys Rev B 74: 045418
Chen Y, Lee JD (2005) Atomistic formulation of a multiscale theory for nano/micro physics. Phil Mag 85: 4095–4126
Chen Y (2006) Local stress and heat flux in atomistic systems involving three-body forces. J Chem Phys 124: 054113
Chen Y, Lee JD (2006) Conservation laws at nano/micro scales. J Mech Mater Struct 1: 681–704
Chen Y, Lee JD, Xiong L (2006) Stresses and strains at nano/micro scales. J Mech Mater Struct 1: 705–723
Xiong L, Chen Y, Lee JD (2007) Atomistic simulation of mechanical properties of diamond and silicon by a field theory. Model Simul Mater Sci Eng 15: 535–551
Lei Y, Lee JD, Zeng X (2008) Response of a rocksalt crystal to electromagnetic wave modeled by a multiscale field theory. Interact Multiscale Mech 1(4): 467–476
Chen Y (2009) Reformulation of microscopic balance equations for multiscale materials modeling. J Chem Phys 130(13): 134706
Lee JD, Wang XQ, Chen Y (2009) Multiscale material modeling and its application to a dynamic crack propagation problem. Theor Appl Fracture Mech 51: 33–40
Lee JD, Wang XQ, Chen Y (2009) Multiscale computational for nano/micro material. J Eng Mech 135: 192–202
Irvine JH, Kirkwood JG (1950) The statistical theory of transport processes. IV. The equations of hydrodynamics. J Chem Phys 18: 817
Hardy RJ (1982) Formulas for determining local properties in molecular-dynamics simulations: shock waves. J Chem Phys 76(1): 622–628
Cheung KS, Yip S (1991) Atomic-level stress in an inhomogeneous system. J Appl Phys 70(10): 5688–5690
Haile JM (1992) Molecular dynamics simulation. Wiley, New York
Eringen AC, Suhubi ES (1964) Nonlinear theory of simple micro-elastic solids-I. Int J Eng Sci 2: 189–203
Zeng XW, Chen YP, Lee JD (2006) Determining material constants in nonlocal micromorphic theory through phonon dispersion relations. Int J Eng Sci 44: 1334–1345
Finnis MW, Sinclair JE (1984) A simple empirical N-body potential for transition metals. Phil Mag A 50(1): 45–55
Finnis MW, Sinclair JE (1986) Erratum: a simple empirical N-body potential for transition metals. Phil Mag A 53(1): 161
Klotz S, Braden M (2000) Phonon dispersion of bcc iron to 10 GPa. Phys Rev Lett 85: 3209–3212
Hai S, Tadmor EB (2003) Deformation twinning at aluminum crack tips. Acta Materialia 51: 117–131
Guo Y-F, Zhao D-L (2007) Atomistic simulation of structure evolution at crack tip in bcc-iron. Mater Sci Eng A 448: 281–286
Dove M (1993) Introduction to lattice dynamics. Cambridge University Press, Cambridge
Acknowledgments
The authors acknowledge the support by National Science Foundation under Award Number CMMI-0646674.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Zeng, X., Wang, X., Lee, J.D. et al. Multiscale modeling of nano/micro systems by a multiscale continuum field theory. Comput Mech 47, 205–216 (2011). https://doi.org/10.1007/s00466-010-0538-5
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DOI: https://doi.org/10.1007/s00466-010-0538-5