From Molecular Models¶to Continuum Mechanics

By Xavier Blanc, C. L. Bris, Pierre-Louis Lions. 2002

246 citations recorded by the source corpus.

Abstract We present here a limiting process allowing us to write some continuum mechanics models as a natural asymptotic of molecular models. The approach is based on the hypothesis that the macroscopic displacement is equal to the microscopic one. We carry out the corresponding calculations in the case of two-body energies, including higher order terms, and also in the case of Thomas-Fermi type models.

Subfields: Geophysics, Physical and Theoretical Chemistry, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c503974e69944806157bfb062f4ced5c6a91687d

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In Geophysics

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In Statistical and Nonlinear Physics

Citations in the corpus

Principles of Multiscale Modeling (2012, methodology): “Adaptive finite element methods pioneered by Babuska and Rheinboldt [3, 4], which are based on rigorous a posteriori error estimates.”
Asymptotic expansions by Γ-convergence (2008, background): “It is interesting that another approximation at order ε with the same structure as in (35) but different W (u) and C can be formally obtained by a pointwise limit [10,66].”
Cauchy–Born Rule and the Stability of Crystalline Solids: Static Problems (2007, result): “B, L B and L assumed that the microscopic displacement of the atoms follows a smooth macroscopic disp lacement field, and derived, in the continuum limit, both bulk and surface energ y expressions from atomistic models [5].”
From atoms to crystals: a mathematical journey (2005, background): “In [29] and [26], we have investigated a strategy consisting of passing to the macroscopic limit for a given macroscopic deformation of the sample.”
On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity (2007, background): “[6, 7, 8] and Le Bris & Lions [26] for a discussion of the relation between molecular models and models on a continuum level.”
Atomistic to continuum limits for computational materials science (2007, methodology): “The work [30], announced and summarized in [29], however presents a rigorous and systematic study of such a question, in a variety of settings.”