A derivation of continuum nonlinear plate theory from atomistic models

Multiscale Modeling and Simulation

Volumn 5, Issue 2, 2006, Pages 664-694

  Schmidt, Bernd ^a  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

convergence; Discrete to continuum limits; Plate theory

Indexed keywords

CONTINUUM MECHANICS; CONVERGENCE OF NUMERICAL METHODS; ELASTICITY; MATHEMATICAL MODELS; THREE DIMENSIONAL;

ATOMISTIC MODELS; CONTINUUM ELASTICITY; DISCRETE-TO-CONTINUUM LIMITS; PLATE THEORY; THREE-DIMENSIONAL ELASTICITY;

ATOMS;

EID: 34250164600     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/050646251     Document Type: Article

References (23)

1. X. BLANC, C. LEBRIS, AND P.-L. LIONS, Convergence de modèles moléculaires vers des modèles de mécanique des milieux continus C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), pp. 949-956.
2. X. BLANC, C. LEBRIS, AND P.-L. LIONS, From molecular models to continuum mechanics, Arch. Ration. Mech. Anal., 164 (2002), pp. 341-381.
3. A. BRAIDES, Nonlocal variational limits of discrete systems, Commun. Contemp. Math., 2 (2000), pp. 285-297.
4. A. BRAIDES AND M. S. GELLI, Limits of discrete systems with long-range interactions, J. Convex Anal., 9 (2002), pp. 363-399.
5. A. BRAIDES AND M. S. GELLI, Continuum limits of discrete systems without convexity hypotheses, Math. Mech. Solids, 7 (2002), pp. 41-66.
6. S. CONTI, G. DOLZMANN, B. KIRCHHEIM, AND S. MÜLLER, Sufficient conditions for the validity of the Cauchy-Born rule close to SO(n), J. Eur. Math. Soc. (JEMS), 8 (2006), pp. 515-539.
7. G. DAL MASO, An Introduction to Γ-Convergence, Birkhäuser, Boston, Basel, Berlin, 1993.
8. L. C. EVANS AND R. F. GARIEPY, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, London, New York, Washington, DC, 1992.
9. G. FRIESECKE AND R. D. JAMES. A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods, J. Mech. Phys. Solids, 48 (2000), pp. 1519-1540.
10. G. FRIESECKE, R. D. JAMES, AND S. MÜLLER, Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Acad. Sci. Paris, 334 (2002), pp. 173-178.
11. G. FRIESECKE, R. D. JAMES, AND S. MÜLLER, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), pp. 1461-1506.
12. G. FRIESECKE, R. D. JAMES, AND S. MÜLLER, A hierarchy of plate models derived from non-linear elasticity by Γ-convergence, Arch. Ration. Mech. Anal., 180 (2006), pp. 183-236.
13. G. FRIESECKE, R. D. JAMES, M. G. MORA, AND S. MÜLLER, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence, C. R. Acad. Sci. Paris, 336 (2003), pp. 697-702.
14. G. FRIESECKE AND F. THEIL, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice, J. Nonlinear Sci., 12 (2002), pp. 445-478.
15. H. LE DRET AND A. RAOULT, La modèle membrane non linéaire comme limite variationnelle de l'élasticité non linéaire tridimensionnelle, C. R. Acad. Sci. Paris, 317 (1993), pp. 221-226.
16. H. LE DRET AND A. RAOULT, The nonlinear membrane model as a variational limit of three-dimensional elasticity, J. Math. Pures Appl. (9), 74 (1995), pp. 549-578.
17. LE DRET AND A. RAOULT, The membrane shell model in nonlinear elasticity: A variational asymptotic derivation, J. Nonlinear Sci., 6 (1996), pp. 59-84.
18. A. E. H. LOVE, A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, Cambridge, UK, 1927.
19. B. SCHMIDT, On the passage from atomic to continuum theory for thin films, Arch. Ration. Mech. Anal., submitted.
20. B. SCHMIDT, Qualitative properties of a continuum theory for thin films, Ann. Inst. H. Poincaré Anal. Non Linéaire, submitted.
21. B. SCHMIDT, Effective Theories for Thin Elastic Films, Ph.D. thesis, Universität Leipzig, Leipzig, Germany, 2006.
22. F. THEIL, Surface energies in a two-dimensional mass-spring model for crystals, submitted; also available online from http:// www.maths.warwick.ac.uk/~theil/pub.html (2004).
23. J. WLOKA, Partielle Differentialgleichungen, Teubner, Stuttgart, 1982.