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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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

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Allan Chan; Sergio Conti Energy scaling and branched microstructures in a model for shape-memory alloys with SO(2) invariance, M $^{3}$ AS. Mathematical Models Methods in Applied Sciences, Volume 25 (2015) no. 6, pp. 1091-1124 | DOI:10.1142/s0218202515500281 | Zbl:1311.49030
Sergio Conti; Georg Dolzmann Relaxation of a model energy for the cubic to tetragonal phase transformation in two dimensions, M $^{3}$ AS. Mathematical Models Methods in Applied Sciences, Volume 24 (2014) no. 14, pp. 2929-2942 | DOI:10.1142/s0218202514500419 | Zbl:1304.49027
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