Second-order energies on thin structures: Variational theory and non-local effects

Journal of Functional Analysis

Volumn 204, Issue 1, 2003, Pages 228-267

  Bouchitté, Guy ^a   Fragalà, Ilaria ^b  

^a UNIVERSITÉ DE TOULON   (France)

^b POLITECNICO DI MILANO   (Italy)

Author keywords

Cosserat field; Hessian matrix; Positive measures; Relaxation

Indexed keywords

EID: 0142024049     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-1236(02)00181-7     Document Type: Article

References (35)

1. E. Acerbi, G. Buttazzo, D. Percivale, A variational definition for strain energy of an elastic string, J. Elasticity 25 (1991) 137-148.
2. G. Bellettini, G. Bouchitté, I. Fragalà, BV functions with respect to a measure and relaxation of metric integral functionals, J. Convex Anal. 6 (2) (1999) 349-366.
3. K. Bhattacharya, R.D. James, A theory of thin films of martensitic materials with applications to microactuators, J. Mech. Phys. Solids 47 (3) (1999) 531-576.
4. G. Bouchitté, G. Buttazzo, Characterization of optimal shapes and masses through Monge-Kantorovich equation, J. Eur. Math. Soc. 3 (2001) 139-168.
5. G. Bouchitté, G. Buttazzo, I. Fragalà, Mean curvature of a measure and related variational problems, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. IV XXV (1997) 179-196.
6. G. Bouchitté, G. Buttazzo, I. Fragalà, Convergence of Sobolev spaces on varying manifolds, J. Geom. Anal. 11 (3) (2001) 399-422.
7. G. Bouchitté, G. Buttazzo, P. Seppecher, Energies with respect to a measure and applications to low dimensional structures, Calc. Var. 5 (1997) 37-54.
8. G. Bouchitté, I. Fragalà, Homogenization of thin structures by two-scale method with respect to measures, SIAM J. Math. Anal. 32 (6) (2001) 1198-1226.
9. G. Bouchitté, I. Fragalà, Homogenization of elastic problems on thin structures: a measure-fattening approach, J. Convex Anal., to appear.
10. G. Bouchitté, I. Fragalà, M. Rajesh, Homogenization of second-order energies on periodic thin structures, submitted.
11. F. Bourquin, P. Ciarlet, G. Geymonat, A. Raoult, Γ-convergence et analyse asymptotique des plaques minces, C. R. Acad. Sci. Paris. Sér. I Math. 315 (9) (1992) 1017-1024.
12. G. Buttazzo, Semicontinuity, Relaxation, and Integral Representation in the Calculus of Variations, in: Pitman Research Notes Mathematical Series, Vol. 207, Longman, Harlow, 1989.
13. C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, in: Lecture Notes in Mathematics, Vol. 580, Springer, Berlin, 1977.
14. P.G. Ciarlet, P. Destuynder, Une justification du modèle biharmonique en théorie linéaire des plaques, C. R. Acad. Sci. Paris. Sér. A-B 285 (13) (1977) A851-A854.
15. P.G. Ciarlet, P. Destuynder, Une justification du modèle non linéaire en théorie des plaques, C. R. Acad. Sci. Paris. Sér. A-B 287 (1) (1978) A33-A36.
16. P.G. Ciarlet, H. Le Dret, Justification de la condition aux limites d'encastrement d'une plaque par une méthode asymptotique, C. R. Acad. Sci. Paris. Sér. I Math. 307 (20) (1988) 1015-1018.
17. D. Cioranescu, J. Saint Jean Paulin, Homogenization of Reticulated Structures, in: Applied Mathematical Sciences, Vol. 136, Springer, New York, 1999.
18. G. Dal Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993.
19. M.C. Delfour, J.P. Zolésio, Intrinsic differential geometry and theory of thin shells. Quaderni della Scuola Normale Superiore di Pisa, 2000.
20. I. Fonseca, G. Francfort, 3D-2D asymptotic analysis of an optimal design problem for thin films, J. Reine Angew. Math. 505 (1998) 173-202.
21. I. Fonseca, E. Zappale, Multiscale relaxation of convex functionals, Preprint of the Center for Nonlinear Analysis, Carnegie Mellon University, 2002.
22. I. Fragalà, Lower semicontinuity of μ-quasiconvex integrals, on ESAIM: COCV, to appear.
23. I. Fragalà, C. Mantegazza, On some notions of tangent space to a measure, Proc. Roy. Soc. Edinburgh A 129 (1999) 331-342.
24. P. Germain, Mécanique, Tomes I and II, Ellipses éditeur, Paris, 1986.
25. P. Hajlasz, P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (688) (2000) 1-101.
26. L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer, Berlin, 1983.
27. A.D. Ioffe, On lower semicontinuity of integral functionals. Parts I and II, SIAM J. Control Optim. 15 (4,6) (1977) 521-538, 991-1000.
28. L. Landau, E. Lifchitz, Théorie de l'Élasticité, MIR, Moscow, 1967.
29. 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.
30. H. Le Dret, A. Raoult, The membrane shell model in nonlinear elasticity: a variational asymptotic derivation, J. Nonlinear Sci. 6 (1996) 59-84.
31. J.M. Lee, Riemannian Manifolds: an Introduction to Curvature, in: Graduate Texts in Mathematics, Vol. 176, Springer, Berlin, 1997.
32. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, Cambridge, 1927.
33. C. Mantegazza, Curvature varifolds with boundary, J. Differential Geom. 43 (1996) 807-843.
34. L. Simon, Lectures on geometric measure theory, Vol. 3, Proceedings, Centre for Mathematical Analysis, Australian National University, 1983.
35. M. Valadier, Multi-applications measurables à valeurs convexes compactes, J. Math. Pures Appl. 50 (1971) 265-297.