Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Thin elastic films: The impact of higher order perturbations


Authors: Irene Fonseca, Gilles Francfort and Giovanni Leoni
Journal: Quart. Appl. Math. 65 (2007), 69-98
MSC (2000): Primary 49J45, 74K15
DOI: https://doi.org/10.1090/S0033-569X-06-01035-7
Published electronically: November 15, 2006
MathSciNet review: 2313149
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The asymptotic behavior of an elastic thin film penalized by a van der Wals type interfacial energy is investigated when both its thickness and the magnitude of the additional energy vanish in the limit. Keeping track of both mid-plane and out-of-plane deformations (through the introduction of the Cosserat vector), the resulting behavior strongly depends upon the ratio between thickness and interfacial energy.


References [Enhancements On Off] (What's this?)

  • 1. E. ACERBI, N. FUSCO.
    Semicontinuity problems in the calculus of variations.
    Arch. Rat. Mech. Anal., 86, 1984, 125-145. MR 0751305 (85m:49021)
  • 2. L. AMBROSIO, S. MORTOLA, V.M. TORTORELLI.
    Functionals with linear growth defined on vector valued BV functions.
    J. Math. Pures Appl. 9, 1991, 269-323. MR 1113814 (92j:49004)
  • 3. J.M. BALL, F. MURAT.
    $ W^{1,p}$ quasiconvexity and variational problems for multiple integrals.
    J. Funct. Anal. 58, 1984, 225-253. MR 0759098 (87g:49011a)
  • 4. K. BHATTACHARYA, R.D. JAMES.
    A theory of thin films of martensitic materials with applications to microactuators.
    J. Mech. Phys. Solids 47, 1999, 531-576. MR 1675215 (2000h:74063)
  • 5. G. BOUCHITTÉ, I.  FONSECA, M.L.  MASCARENHAS.
    Bending moment in membrane theory.
    J. Elasticity 73, 2003, 75-99. MR 2057737 (2005c:74051)
  • 6. A. BRAIDES, A. DEFRANCESCHI.
    Homogenization of multiple integrals.
    Oxford Lecture Series in Mathematics and its Applications, 12, Oxford, 1998. MR 1684713 (2000g:49014)
  • 7. A. BRAIDES, I. FONSECA, G.A. FRANCFORT.
    3D-2D asymptotic analysis for inhomogeneous thin films.
    Indiana Univ. Math. J. 49, 2000, 1367-1404. MR 1836533 (2002j:35025)
  • 8. B. DACOROGNA.
    Direct methods in the calculus of variations.
    Applied Mathematical Sciences, 78. Springer-Verlag, Berlin, 1989. MR 0990890 (90e:49001)
  • 9. E. DE GIORGI, G. LETTA.
    Une notion générale de convergence faible pour des fonctions croissantes d'ensemble.
    Ann. Sc. Norm. Sup. Pisa Cl. Sci. 4, 1977, 61-99. MR 0466479 (57:6357)
  • 10. L.C. EVANS.
    Weak convergence methods for nonlinear partial differential equations.
    CBMS Regional Conference Series in Mathematics, 74. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1034481 (91a:35009)
  • 11. L.C. EVANS, R.F. GARIEPY.
    Measure theory and fine properties of functions.
    Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660 (93f:28001)
  • 12. I. FONSECA, G.A. FRANCFORT.
    3D-2D asymptotic analysis of an optimal design problem for thin films.
    J. Reine Angew. Math. 505, 1998, 173-202. MR 1662252 (99k:73091)
  • 13. I. FONSECA, D. KINDERLEHRER, P. PEDREGAL.
    Energy functionals depending on elastic strain and chemical composition.
    Calc. Var. Partial Differential Equations 2, no. 3, 1994, 283-313. MR 1385072 (97f:73011)
  • 14. I. FONSECA, G. LEONI, R. PARONI. On lower semicontinuity in $ BH^{p}$ and 2-quasiconvexification. Calc. Var. Partial Differential Equations 17 (2003), no. 3, 283-309. MR 1989834 (2004d:49030)
  • 15. I. FONSECA, S. MÜLLER, P. PEDREGAL. Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998), no. 3, 736-756. MR 1617712 (99e:49013)
  • 16. H. LE DRET, A. RAOULT.
    The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity.
    J. Math. Pures. Appl. 74, 1995, 549-578. MR 1365259 (97d:73009)
  • 17. P.A. LOEB, E. TALVILA.
    Lusin's theorem and Bochner integration.
    Sci. Math. Jpn. 60-1, 2004, 113-120. MR 2072104 (2005j:28018)
  • 18. Y.C. SHU.
    Heterogeneous thin films of martensitic materials.
    Arch. Rat. Mech. Anal 153, 2000, 39-90. MR 1772534 (2002a:74099)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 49J45, 74K15

Retrieve articles in all journals with MSC (2000): 49J45, 74K15


Additional Information

Irene Fonseca
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
Email: fonseca@andrew.cmu.edu

Gilles Francfort
Affiliation: L.P.M.T.M., Université Paris-Nord, 93430 Villetaneuse, France
Email: francfor@lpmtm.univ-paris13.fr

Giovanni Leoni
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Email: giovanni@andrew.cmu.edu

DOI: https://doi.org/10.1090/S0033-569X-06-01035-7
Received by editor(s): March 26, 2006
Published electronically: November 15, 2006
Additional Notes: The research of I. Fonseca was partially funded by the U.S. National Science Foundation DMS Grants 0103799 & 0401763, and by the Center for Nonlinear Analysis under the U.S. National Science Foundation DMS Grants 9803791 & 0405343.
This work progressed through various visits of G. Francfort to Carnegie Mellon and that author is grateful to the Center for Nonlinear Analysis for its hospitality.
The research of G. Leoni was partially funded by the U.S. National Science Foundation DMS Grant 0405423
Article copyright: © Copyright 2006 Brown University

American Mathematical Society