Liquid-like behavior of shape memory alloys

Georg Dolzmann; Bernd Kirchheim

doi:10.1016/S1631-073X(03)00082-7

Mathematical Problems in Mechanics

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

Présenté par : John Ball

Georg Dolzmann ¹ ; Bernd Kirchheim 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

Comptes Rendus. Mathématique, Volume 336 (2003) no. 5, pp. 441-446.

Résumé
Abstract

Dans cette Note, nous démontrons que la matrice identité est un point intérieur de l'envelope quasiconvexe K^qc d'un ensmble compact $K \subset {X \in 𝕄^{3, 3} : \det X = 1}$ lorsque K^qc 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 K^qc of a compact set $K \subset {X \in 𝕄^{3, 3} : \det X = 1}$ whenever K^qc 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 : 2002-08-28
Accepté le : 2003-02-04
Publié le : 2003-03-01

DOI : 10.1016/S1631-073X(03)00082-7

Affiliations des auteurs :

Georg Dolzmann ¹ ; Bernd Kirchheim 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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     author = {Georg Dolzmann and Bernd Kirchheim},
     title = {Liquid-like behavior of shape memory alloys},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {441--446},
     publisher = {Elsevier},
     volume = {336},
     number = {5},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00082-7},
     language = {en},
}

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%A Bernd Kirchheim
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Georg Dolzmann; Bernd Kirchheim. Liquid-like behavior of shape memory alloys. Comptes Rendus. Mathématique, Volume 336 (2003) no. 5, pp. 441-446. doi : 10.1016/S1631-073X(03)00082-7. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00082-7/

Bibliographie
Cité par

[1] J.M. Ball; R.D. James 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] K. Bhattacharya Self-accommodation in martensite, Arch. Rational Mech. Anal., Volume 120 (1992), pp. 201-244

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

[4] G. Friesecke 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] S. Müller Variational methods for microstructure and phase transitions (F. Bethuel; G. Huisken; S. Müller; K. Steffen; S. Hildebrandt; M. Struwe, 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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