×

zbMATH — the first resource for mathematics

On the passage from atomic to continuum theory for thin films. (English) Zbl 1156.74028
Summary: We give a rigorous derivation of a continuum theory from atomic models for thin films. This scheme has been proposed by G. Friesecke and R. D. James in [J. Mech. Phys. Solids 48, No. 6–7, 1519–1540 (2000; Zbl 0984.74009)]. The resulting continuum energy expression is obtained by integrating a stored energy density which not only depends on the deformation gradient, but also on \(\nu-1\) director fields when \(\nu\) is the (fixed) number of atomic film layers.

MSC:
74K35 Thin films
74A25 Molecular, statistical, and kinetic theories in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alicandro R., Cicalese M.: A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36, 1–37 (2004) · Zbl 1070.49009 · doi:10.1137/S0036141003426471
[2] Alicandro, R., Braides, A., Cicalese, M.: Continuum limits of discrete thin films with superlinear growth densities. Preprint 2005. http://cvgmt.sns.it/papers/alibracic05/ · Zbl 1148.49010
[3] Antman S.S.: Nonlinear Problems of Elasticity. Springer, Berlin (1995) · Zbl 0820.73002
[4] Anzellotti G., Baldo S., Percivale D.: Dimension reduction in variational problems, asymptotic development in \(\Gamma\)-convergence and thin structures in elasticity. Asymptotic Anal 9, 61–100 (1994) · Zbl 0811.49020
[5] Blanc X., LeBris C.: Periodicity of the infinite-volume ground state of a one-dimensional quantum model. Nonlinear Anal. Theory Methods Appl. 48A, 791–803 (2002) · Zbl 0992.82043 · doi:10.1016/S0362-546X(00)00215-7
[6] Blanc X., LeBris C., Lions P.-L.: Convergence de modèles moléculaires vers des modèles de mécanique des milieux continus. C. R. Acad. Sci. Paris 332, 949–956 (2001) · Zbl 0986.74006
[7] Blanc X., LeBris C., Lions P.-L.: From molecular models to continuum mechanics. Arch. Rational Mech. Anal. 164, 341–381 (2002) · Zbl 1028.74005 · doi:10.1007/s00205-002-0218-5
[8] Braides A.: Nonlocal variational limits of discrete systems. Commun. Contemp. Math. 2, 285–297 (2000) · Zbl 0957.49011
[9] Braides A., Gelli M.S.: Limits of discrete systems with long-range interactions. J. Convex Anal. 9, 363–399 (2002) · Zbl 1031.49022
[10] Braides A., Gelli M.S.: Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7, 41–66 (2002) · Zbl 1024.74004 · doi:10.1177/1081286502007001229
[11] Ciarlet P.G.: Mathematical Elasticity vol. I: Three-dimensional Elasticity. North-Holland, Amsterdam (1988) · Zbl 0648.73014
[12] Ciarlet P.G.: Mathematical Elasticity vol. II: Theory of Plates. North-Holland, Amsterdam (1997) · Zbl 0888.73001
[13] Dacorogna B.: Direct Methods in the Calculus of Variations. Springer, Berlin (1989) · Zbl 0703.49001
[14] Dal Maso G.: An Introduction to \(\Gamma\)-convergence. Birkhäuser, Boston (1993) · Zbl 0816.49001
[15] Euler, L.: Methodus Inveniendi Lineas Curvas, Additamentum I: De Curvis Elasticis (1744). In: Opera Omnia Ser. Prima vol. XXIV, pp. 231–297. Orell Füssli, Bern 1952
[16] Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992) · Zbl 0804.28001
[17] Friesecke G., James R.D.: A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods. J. Mech. Phys. Solids 48, 1519–1540 (2000) · Zbl 0984.74009 · doi:10.1016/S0022-5096(99)00091-5
[18] Friesecke G., James R.D., Müller S.: Rigorous derivation of nonlinear plate theory and geometric rigidity. C. R. Acad. Sci. Paris 334, 173–178 (2002) · Zbl 1012.74043
[19] Friesecke G., James R.D., Müller S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55, 1461–1506 (2002) · Zbl 1021.74024 · doi:10.1002/cpa.10048
[20] Friesecke G., James R.D., Müller S.: A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Rational Mech. Anal. 180, 183–236 (2006) · Zbl 1100.74039 · doi:10.1007/s00205-005-0400-7
[21] Friesecke G., James R.D., Mora M.G., Müller S.: Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence. C. R. Acad. Sci. Paris 336, 697–702 (2003) · Zbl 1140.74481
[22] Friesecke G., James R.D., Mora M.G., Müller S.: Derivation of the nonlinear bending-torsion theory for inextensible rods by \(\Gamma\)-convergence. Calc. Var. Partial Differ. Equ. 18, 287–305 (2003) · Zbl 1053.74027 · doi:10.1007/s00526-003-0204-2
[23] von Kármán, T.: Festigkeitsprobleme im Maschinenbau. In: Encyclopädie der Mathematischen Wissenschaften, vol. IV/4, pp. 311–385, Leipzig, 1910 · JFM 41.0907.02
[24] Kirchhoff G.: Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math. 40, 51–88 (1850) · ERAM 040.1086cj · doi:10.1515/crll.1850.40.51
[25] Le Dret H., Raoult A.: La modèle membrane non linéaire comme limite variationnelle de l’élasticité non linéaire tridimensionnelle. C. R. Acad. Sci. Paris 317, 221–226 (1993) · Zbl 0781.73037
[26] Le Dret H., Raoult A.: The nonlinear membrane model as a variational limit of three-dimensional elaticity. J. Math. Pures Appl. 74, 549–578 (1995) · Zbl 0847.73025
[27] Le Dret H., Raoult A.: The membrane shell model in nonlinear elaticity: a variational asymptotic derivation. J. Nonlinear Sci. 6, 59–84 (1996) · Zbl 0844.73045 · doi:10.1007/BF02433810
[28] Love A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (1927) · JFM 53.0752.01
[29] Phillips R.: Crystals, Defects and Microstructures. Cambridge University Press, Cambridge (2001)
[30] Schmidt B.: Qualitative properties of a continuum theory for thin films. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, 43–75 (2008) · Zbl 1142.74026 · doi:10.1016/j.anihpc.2006.09.001
[31] Schmidt B.: A derivation of continuum nonlinear plate theory from atomistic models. SIAM Multiscale Model. Simul. 5, 664–694 (2006) · Zbl 1117.49018 · doi:10.1137/050646251
[32] Schmidt, B.: Effective Theories for Thin Elastic Films. Ph.D. thesis, University of Leipzig, 2006 · Zbl 1126.74002
[33] Sutton A.P.: Electronic Structure of Materials. Oxford University Press, Oxford (1994) · Zbl 0812.11007
[34] Theil F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262, 209–236 (2005) · Zbl 1113.82016 · doi:10.1007/s00220-005-1458-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.