Thin elastic films: The impact of higher order perturbations

Fonseca, Irene; Francfort, Gilles; Leoni, Giovanni

doi:10.1090/S0033-569X-06-01035-7

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

Emilio Acerbi and Nicola Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), no. 2, 125–145. MR 751305, DOI https://doi.org/10.1007/BF00275731
L. Ambrosio, S. Mortola, and V. M. Tortorelli, Functionals with linear growth defined on vector valued BV functions, J. Math. Pures Appl. (9) 70 (1991), no. 3, 269–323. MR 1113814
J. M. Ball and F. Murat, $W^{1,p}$-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal. 58 (1984), no. 3, 225–253. MR 759098, DOI https://doi.org/10.1016/0022-1236%2884%2990041-7
K. Bhattacharya and R. D. James, A theory of thin films of martensitic materials with applications to microactuators, J. Mech. Phys. Solids 47 (1999), no. 3, 531–576. MR 1675215, DOI https://doi.org/10.1016/S0022-5096%2898%2900043-X
Guy Bouchitté, Irene Fonseca, and M. Luísa Mascarenhas, Bending moment in membrane theory, J. Elasticity 73 (2003), no. 1-3, 75–99 (2004). MR 2057737, DOI https://doi.org/10.1023/B%3AELAS.0000029996.20973.92
Andrea Braides and Anneliese Defranceschi, Homogenization of multiple integrals, Oxford Lecture Series in Mathematics and its Applications, vol. 12, The Clarendon Press, Oxford University Press, New York, 1998. MR 1684713
A. Braides, I. Fonseca, and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J. 49 (2000), no. 4, 1367–1404. MR 1836533, DOI https://doi.org/10.1512/iumj.2000.49.1822
Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890
E. De Giorgi and G. Letta, Une notion générale de convergence faible pour des fonctions croissantes d’ensemble, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 1, 61–99 (French). MR 466479
Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1034481
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
Irene Fonseca and Gilles Francfort, 3D-2D asymptotic analysis of an optimal design problem for thin films, J. Reine Angew. Math. 505 (1998), 173–202. MR 1662252, DOI https://doi.org/10.1515/crll.1998.505.173
Irene Fonseca, David Kinderlehrer, and Pablo Pedregal, Energy functionals depending on elastic strain and chemical composition, Calc. Var. Partial Differential Equations 2 (1994), no. 3, 283–313. MR 1385072, DOI https://doi.org/10.1007/BF01235532
Irene Fonseca, Giovanni Leoni, and Roberto Paroni, On lower semicontinuity in $BH^p$ and 2-quasiconvexification, Calc. Var. Partial Differential Equations 17 (2003), no. 3, 283–309. MR 1989834, DOI https://doi.org/10.1007/BF01235532
Irene Fonseca, Stefan Müller, and Pablo Pedregal, Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal. 29 (1998), no. 3, 736–756. MR 1617712, DOI https://doi.org/10.1137/S0036141096306534
Hervé Le Dret and Annie Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9) 74 (1995), no. 6, 549–578 (English, with English and French summaries). MR 1365259
Peter A. Loeb and Erik Talvila, Lusin’s theorem and Bochner integration, Sci. Math. Jpn. 60 (2004), no. 1, 113–120. MR 2072104
Y. C. Shu, Heterogeneous thin films of martensitic materials, Arch. Ration. Mech. Anal. 153 (2000), no. 1, 39–90. MR 1772534, DOI https://doi.org/10.1007/s002050000088

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
MR Author ID: 67965
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
MR Author ID: 321623
Email: giovanni@andrew.cmu.edu

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