A Lagrangian-based continuum homogenization approach applicable to molecular dynamics simulations

Abstract The continuum notions of effective mechanical quantities as well as the conditions that give meaningful deformation processes for homogenization problems with large deformations are reviewed. A continuum homogenization model is presented and recast as a Lagrangian-based approach for heterogeneous media that allows for an extension to discrete systems simulated via molecular dynamics (MD). A novel constitutive relation for the effective stress is derived so that the proposed Lagrangian-based approach can be used for the determination of the “stress–deformation” behavior of particle systems. The paper is concluded with a careful comparison between the proposed method and the Parrinello–Rahman approach to the determination of the “stress–deformation” behavior for MD systems. When compared with the Parrinello–Rahman method, the proposed approach clearly delineates under what conditions the Parrinello–Rahman scheme is valid.

[1]  P. Chadwick.  Continuum Mechanics: Concise Theory and Problems , 1976 .

[2]  J. Ericksen.  The Cauchy and Born Hypotheses for Crystals. , 1983 .

[3]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[4]  H. C. Andersen.  Molecular dynamics simulations at constant pressure and/or temperature , 1980 .

[5]  R. Hill,et al.  On macroscopic effects of heterogeneity in elastoplastic media at finite strain , 1984, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  Hamilton's Principle , 1968, Nature.

[7]  P. Germain.  Sur certaines définitions liées à l'énergie en mécanique des solides , 1982 .

[8]  M. Parrinello,et al.  Polymorphic transitions in single crystals: A new molecular dynamics method , 1981 .

[9]  Francesco Costanzo,et al.  On the definitions of effective stress and deformation gradient for use in MD: Hill’s macro-homogeneity and the virial theorem , 2005 .

[10]  Min Zhou,et al.  A new look at the atomic level virial stress: on continuum-molecular system equivalence , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[11]  Francesco Costanzo,et al.  On the notion of average mechanical properties in MD simulation via homogenization , 2004 .

[12]  Berend Smit,et al.  Understanding molecular simulation: from algorithms to applications , 1996 .

[13]  C. Stolz.  General Relationships between Micro and Macro Scales for the Non-linear Behaviour of Heterogeneous Media , 1986 .

[14]  M. Gurtin,et al.  Phase Transformation and Material Instabilities in Solids , 1984 .

[15]  P. M. Squet.  Local and Global Aspects in the Mathematical Theory of Plasticity , 1985 .

[16]  G. Zanzotto.  On the material symmetry group of elastic crystals and the Born Rule , 1992 .

[17]  C. Brooks.  Computer simulation of liquids , 1989 .

[18]  Leonard Meirovitch,et al.  Methods of analytical dynamics , 1970 .

[19]  S. Passman,et al.  Hamilton's principle in continuum mechanics , 1985 .

[20]  A. Spencer.  Continuum Mechanics , 1967, Nature.

[21]  Joseph Zarka,et al.  Modelling small deformations of polycrystals , 1986 .

[22]  M. Parrinello,et al.  Crystal structure and pair potentials: A molecular-dynamics study , 1980 .

[23]  A. Sawczuk,et al.  Plasticity today : modelling, methods, and applications , 1985 .

[24]  G. Maugin.  The Thermomechanics of Plasticity and Fracture , 1992 .

[25]  R. Hill.  On constitutive macro-variables for heterogeneous solids at finite strain , 1972, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.