A mathematical homogenization perspective of virial stress

International Journal for Numerical Methods in Engineering

Volumn 67, Issue 2, 2006, Pages 189-207

  Chen, Wen ^a   Fish, Jacob ^a  

^a RENSSELAER POLYTECHNIC INSTITUTE   (United States)

Author keywords

Homogenization; Multiscale; Virial

Indexed keywords

ATOMS; COMPUTATIONAL METHODS; CONTINUUM MECHANICS; PROBLEM SOLVING; STRESSES;

CAUCHY STRESS; CONTINUUM EQUATION; CONTINUUM STRESS; GENERALIZED MATHEMATICAL HOMOGENIZATION (GMH); VIRIAL STRESS;

MOLECULAR DYNAMICS;

ATOMS; COMPUTATIONAL METHODS; CONTINUUM MECHANICS; MOLECULAR DYNAMICS; PROBLEM SOLVING; STRESSES;

EID: 33745933710     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.1622     Document Type: Article

References (22)

1. Clausius R. On a mechanical theory applicable to heat. Philosophical Magazine 1870; 40:122-127.
2. Maxwell JC. Van der Waals on the continuity of the gaseous and liquid states. Nature 1874; 10:477-480.
3. Cheung KS, Yip S. Atomic-level stress in an inhomogeneous system. Journal of Applied Physics 1991; 10:5688-5690.
4. Cormier J, Rickman JM, Delph TJ. Stress calculation in atomistic simulations of perfect and imperfect solids. Journal of Applied Physics 2001; 89:99-104.
5. Lutsko JF. Stress and elastic constants in anisotropic solids: molecular dynamics techniques. Journal of Applied Physics 1988; 64:1152-1154.
6. Irving JH, Kirkwood JG. The statistical mechanical theory of transport processes, IV The equations of hydrodynamics. Journal of Chemical Physics 1950; 18:817-829.
7. Hardy RJ. Formulas for determining local properties in molecular dynamics simulations: shock waves. Journal of Chemical Physics 1982; 76:622-628.
8. Zhou M. A new look at the atomic level virial stress - on continuum-molecular system equivalence. Proceedings of the Royal Society of London, Series A 2003; 459:2347-2392.
9. Zhou M, McDowell DL. Equivalent continuum for dynamically deforming atomistic particle systems. Philosophical Magazine A 2002; 82:2547-2574.
10. Srolovitz D, Maeda K, Vitek V, Egami T. Structural defects in amorphous solids: statistical analysis of a computer model. Philosophical Magazine A 1981; 44:847-866.
11. Horstemeyer MF, Baskes MI. Atomic finite deformation simulations: a discussion on length scale effects in relation to mechanical stress. Journal of Engineering Materials and Technology 1999; 121:114-119.
12. Alber I, Bassani JL, Khantha M, Vitek V, Wang GJ. Grain boundaries as heterogeneous systems: atomic and continuum elastic properties. Philosophical Transactions of the Royal Society of London, Series A 1992; 339:555-586.
13. E W, Engquist B. The heterogeneous multi-scale methods. Communications in Mathematical Sciences 2002; 1:87-132.
14. Stillinger FH, Weber TA. Computer simulation of local order in condensed phases of silicon. Physical Review B 1985; 31:5262-5271.
15. Rapaport DC. The Art of Molecular Dynamics Simulation. Cambridge University Press: Cambridge, 1995.
16. Tersoff J. Empirical Interatomic potential for carbon, with applications to amorphous carbon. Physical Review Letters 1988; 61:2879-2882.
17. Zohdi TI. Computational design of swarms. International Journal for Numerical Methods in Engineering 2004; 57:2205-2219.
18. Fish J, Chen W, Tang Y Space-time generalized mathematical homogenization of atomistic media at finite temperatures. International Journal for Multiscale Computational Engineering 2005, in press.
19. Chen W, Fish J. A generalized space-time mathematical homogenization theory for bridging atomistic and continuum scales. International Journal for Numerical Methods in Engineering 2005, in press.
20. Born M, Huang K. Dynamical Theory of Crystal Lattices. Clarendon Press: Oxford, 1954.
21. Belytschko T, Liu WK, Moran B. Nonlinear Finite Elements for Continua and Structures. Wiley: New York, 2001.
22. Hansen JP. An introduction to molecular dynamics, with application to the glass transition. In Computer Simulation in Materials Science, Interatomic Potentials, Simulation Techniques and Applications, Meyer M, Pontikis V (eds). Kluwer: Dordrecht, 1991.