Mathematical Problems in Mechanics
Liquid-like behavior of shape memory alloys
[Comportement liquide d'alliages à mémoire de forme]

Présenté par : John Ball

Dolzmann, Georg1 ; Kirchheim, Bernd2
1 Mathematics Department, University of Maryland, College Park, MD 20742, USA
2 Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany
Comptes Rendus. Mathématique, Tome 336 (2003) no. 5, pp. 441-446.

Dans cette Note, nous démontrons que la matrice identité est un point intérieur de l'envelope quasiconvexe Kqc d'un ensmble compact K⊂{X∈𝕄 3,3 : det X=1} lorsque Kqc contient une configuration de type triple puits. Ceci est le cas en particulier pour les transformations de phase de cubique à tétragonal et de cubique à orthorhombique, et répond à une question discutée dans S. Müller, Microstructures, phase transitions and geometry, in: A. Balog et al. (Eds.), Proceedings European Congress of Mathematics, Progr. Math., Birkhäuser, 1998.

In this Note, we prove that the identity matrix is an inner point of the quasiconvex hull Kqc of a compact set K⊂{X∈𝕄 3,3 : det X=1} whenever Kqc contains a three-well configuration. This is in particular the case for the cubic to tetragonal and the cubic to orthorhombic phase transformations, and answers a question discussed in S. Müller, Microstructures, phase transitions and geometry, in: A. Balog et al. (Eds.), Proceedings European Congress of Mathematics, Progr. Math., Birkhäuser, 1998.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00082-7
Dolzmann, Georg 1 ; Kirchheim, Bernd 2

1 Mathematics Department, University of Maryland, College Park, MD 20742, USA
2 Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany 
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Dolzmann, Georg; Kirchheim, Bernd. Liquid-like behavior of shape memory alloys. Comptes Rendus. Mathématique, Tome 336 (2003) no. 5, pp. 441-446. doi : 10.1016/S1631-073X(03)00082-7. http://www.numdam.org/articles/10.1016/S1631-073X(03)00082-7/

[1] Ball, J.M.; James, R.D. Proposed experimental tests of a theory of fine microstructure and the two-well problem, Philos. Trans. Roy. Soc. London Ser. A, Volume 338 (1992), pp. 389-450

[2] Bhattacharya, K. Self-accommodation in martensite, Arch. Rational Mech. Anal., Volume 120 (1992), pp. 201-244

[3] Bhattacharya, K.; Dolzmann, G. Relaxation of some multi-well problems, Proc. Roy. Soc. Edinburgh Sect. A, Volume 131 (2001), pp. 279-320

[4] Friesecke, G. Quasiconvex hulls and recoverable strains in shape-memory alloys, TMR Meeting ‘Phase Transitions in Crystalline Solids’, Max Planck Institute for Mathematics in the Sciences, Leipzig, 2000

[5] Müller, S. Variational methods for microstructure and phase transitions (Bethuel, F.; Huisken, G.; Müller, S.; Steffen, K.; Hildebrandt, S.; Struwe, M., eds.), Calculus of Variations and Geometric Evolution Problems, Proc. C.I.M.E. Summer School, Cetraro, 1996, Lecture Notes in Math., 1713, Springer, 1999

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