A total-variation surface energy model for thin films of martensitic Crystals

Interfaces and Free Boundaries

Volumn 4, Issue 1, 2002, Pages 71-88

  Bělík, Pavel ^a   Luskin, Mitchell ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Bounded variation; Martensite; Surface energy; Thin film

Indexed keywords

EID: 0002572859     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/53     Document Type: Article

References (28)

1. Acerbi, E., Buttazzo, G., & Percivale, D. A variational definition for the strain energy of an elastic string. J. Elast. 25. (1991) 137-148.
2. Adams, R. Sobolev Spaces. Academic, New York (1975).
3. Anzelloti, G., Baldo, S., & Percivale, D. Dimension reduction in variation problems, asymptotic development in/-convergence and thin structures in elasticity. Asymptotic Anal. 9, (1994) 61-100.
4. Ball, J. M. & James, R. D. Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100, (1987) 13-52.
5. 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.
6. Bhattacharya, K., Li, B., & Luskin, M. The simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation. Arch. Ration. Mech. Anal. 149, (1999) 123- 154.
7. Brenner, S. C. & Scott, L. R. The Mathematical Theory of Finite Element Methods. Springer, New York (1994).
8. Beli'k, P., Brule, T. & Luskin, M. On the numerical modeling of deformations of pressurized martensitic thin films. Math. Modelling Numer. Anal. 35, (2001) 525-548.
9. Beli'k, P. & Luskin, M. Stability of microstructure for tetragonal to monoclinic martensitic transformations. Math. Modelling Numer. Anal. 34. (2000) 663-685.
10. Beli'k, P. & Luskin, M. A computational model for the indentation and phase transformation of a martensitic thin film. Preprint, (2001).
11. Clarlet, P. G. The Finite Element Method for Elliptic Problems. North-Holland. Amsterdam (1978).
12. Dong, J. W., Chen, L. C., Palmstrom, C. J., James, R. D., & McKernan, S. Molecular beam epitaxy growth of ferromagnetic single crystal (001) Ni2MnGa on (001) GaAs. Appl. Phys. Lett. 75, (1999)1443-1445.
13. Le Dret, H. & Raoult, A. The nonlinear membrane model as variational limit of nonlinear three- dimensional elasticity. J. Math. Pure Appl. 73, (1995) 549-578.
14. Efendiev, Y. & Luskin, M. Stability of microstructures for some martensitic transformations. Math. Comput. Modelling, to appear.
15. Evans, L. C. & Gariepy, R. F. Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton. FL (1992).
16. Fonseca, I. & Francfort, G. 3d-2d asymptotic analysis of an optimal design problem for thin films. J. Reine Angew. Math. 505.(1998) 173-202.
17. Fonseca, I. Phase transitions of elastic solid materials. Arch. Ration. Mech. Anal. 107, (1989) 195-223.
18. Glustl, E. Minimal Surfaces and Functions of Bounded Variation. Birkhauser, Basel (1984).
19. Kohn, R. & Muller, S. Branching of twins near an austenite-Twinned-martensite interface. Phil. Mag. A 66, (1992a) 697-715.
20. Kohn, R. & Muller, S. Relaxation and regularization of nonconvex variational problems. In Proceedings of the Second International Conference on Partial Differential Equations, (Rendiconti del Seminario Matematico e Fisico di Milano 62). pp. 89-113. (1992b).
21. Kohn, R. & Muller, S. Surface energy and microstructure in coherent phase transitions. Comm. Pure Appl. Math. 47, (1994) 405-435.
22. Krulevitch, P., Lee, A. P., Ramsey, P. B., Trevino, J. C., Hamilton, J. & Northrup, M. A. Thin film shape memory microactuators. J. MEMS 5. (1996) 270-282.
23. Li, B. & Luskin, M. Finite element analysis of microstructure for the cubic to tetragonal transformation. SIAM J. Numer. Anal. 35, (1998) 376-392.
24. Luskin, M. Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math. 75, (1996a) 205-221.
25. Luskin, M. On the computation of cry stalline microstructure. Acta Numer. 5. (1996b) 191-257.
26. Modica, L. & Mortola, S. II limite nella T-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5) 14, (1977a) 526-529.
27. Modica, L. & Mortola, S. Un esempio di T-convergenza. Boll. Un. Mat. Ital. B (5) 14. (1977b) 285-299.
28. Muller, S. Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Part. Diff. Equ. 1.(1993) 169-204.