The energy density of martensitic thin films via dimension reduction

Interfaces and Free Boundaries

Volumn 6, Issue 4, 2004, Pages 439-459

  Freddi, Lorenzo ^a   Paroni, Roberto ^b  

^a UNIVERSITY OF UDINE   (Italy)

^b UNIVERSITY OF SASSARI   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 11144305090     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/109     Document Type: Article

References (31)

1. ACERBI, E., BUTTAZZO, G., & PERCIVALE, D. A variational definition of the strain energy for an elastic string. J. Elasticity 25 (1991), 137-148. Zbl 0734.73094 MR 1111364
2. ACERBI, E. & Fusco, N. Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal. 86 (1984), 125-145. Zbl 0565.49010 MR 0751305
3. ANZELLOTTI, G., BALDO, S., & PERCIVALE, D. Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity. Asymptot. Anal. 9 (1994), 61-100. Zbl 0811.49020 MR 1285017
4. BALL, J. M. A version of the fundamental theorem for Young measures. PDEs and Continuum Models of Phase Transitions, Lecture Notes in Phys. 344, Springer (1989), 207-215. Zbl 0991.49500 MR 1036070
5. BALL, J. M. & JAMES, R. D. Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987), 13-52. Zbl 0629.49020 MR 0906132
6. BALL, J. M. & JAMES, R. D. Proposed experimental tests of a theory of fine microstructure and the two well problem. R. Soc. London Philos. Trans. Ser A 338 (1992), 389-450. Zbl 0758.73009
7. BĚLÍK, P. & LUSKIN, M. A computational model for the indentation and phase transformation of a martensitic thin film. J. Mech. Phys. Solids 50 (2002), 1789-1815. Zbl 1004.74059 MR 1915331
8. BĚLÍK, P. & LUSKIN, M. A total-variation surface energy model for thin films of martensitic cristals. Interfaces Free Bound. 4 (2002), 71-88. Zbl 1014.49013 MR 1877536
9. BHATTACHARYA, K. & BRAIDES, A. Thin films with many small cracks. R. Soc. London Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), 823-840. Zbl 1011.74042 MR 1898090
10. BHATTACHARYA, K. & JAMES, R. D. A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids 47 (1999), 531-576. Zbl 0960.74046 MR 1675215
11. BOCEA, M. Young measures minimizers in the asymptotic analysis of thin films. Preprint (2003).
12. BOCEA, M. & FONSECA, I. Equi-integrability results for 3D-2D dimension reduction problems. ESAIM Control Optim. Calc. Var. 7 (2002), 443-470. Zbl pre01966585 MR 1925037
13. BRAIDES, A., FONSECA, I., & FRANCFORT, G. 3d-2d asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J. 49 (2000), 1367-1404. Zbl 0987.35020 MR 1836533
14. BUTTAZZO, G. Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Res. Notes Math. Ser. 207, Longman (1989). Zbl 0669.49005 MR 1020296
15. CIARLET, P. G. & DESTUYNDER, P. A justification of the two-dimensional linear plate model. J. Mécanique 18 (1979), 315-344. Zbl 0415.73072 MR 0533827
16. DACOROGNA, B. Direct Methods in the Calculus of Variations. Appl. Math. Sci. 78, Springer (1989). Zbl 0703.49001 MR 0990890
17. DAL MASO, G. An Introduction to Γ-convergence. Birkhäuser(1993). Zbl 0816.49001 MR 1201152
18. FREDDI, L. & PARONI, R. A 3D-1D Young measure theory of an elastic string. Asymptot. Anal 39 (2004), 61-89. MR 2083576
19. JAMES, R. D. & KINDERLEHRER, D. Theory of diffusionless phase transitions. PDEs and Continuum Models of Phase Transitions, Lecture Notes in Phys. 344, Springer (1989), 51-84. Zbl 0991.74504 MR 1036063
20. KINDERLEHRER, D. & PEDREGAL, P. Characterizations of Young measures generated by gradients. Arch. Rat. Mech. Anal. 115 (1991), 329-365. Zbl 0754.49020 MR 1120852
21. KINDERLEHRER, D. & PEDREGAL, P. Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994), 59-90. Zbl 0808.46046 MR 1274138
22. LE DRET, H. & RAOULT, A. The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995), 549-578. Zbl 0847.73025 MR 1365259
23. LUSKIN, M. On the computation of crystalline microstructure. Acta Numer. 5 (1996), 191-257. Zbl 0867.65033 MR 1624603
24. MÜLLER, S. Variational models for microstructure and phase transitions. Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996), Lecture Notes in Math. 1713, Springer (1999), 85-210. Zbl 0968.74050 MR 1731640
25. MURAT, F. & TARTAR, L. Calculus of variations and homogenization. Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl. 31, Birkhäuser (1997), 139173. Zbl 0939.35019 MR 1493042
26. PARONI, R. From discrete to continuum problems: a Young measure approach. Z. Angew. Math. Phys. 54 (2003), 328-348. Zbl pre01968509 MR 1967332
27. PEDREGAL, P. Parametrized Measures and Variational Principles. Progr. Nonlinear Differential Equations Appl. 130, Birkhäuser (1997). Zbl 0879.49017 MR 1452107
28. SANTOS, P. M. & ZAPPALE, E. Second-order analysis for thin structures. Nonlinear Anal. 56 (2004), 679-713. Zbl pre02056770 MR 2036786
29. SHU, Y. C. Heterogeneous thin films of martensitic materials. Arch. Rat. Mech. Anal. 153 (2000), 39-90. Zbl 0959.74043 MR 1772534
30. VALADIER, M. Young measures. Methods of Nonconvex Analysis, Lecture Notes in Math. 1446, Springer (1990), 152-188. Zbl 0738.28004 MR 1079763
31. YOUNG, L. C. Lectures on the Calculus of Variations and Optimal Control Theory. W. B. Saunders (1969). Zbl 0177.37801 MR 0259704