Geometry of martensite needles in shape memory alloys

Conti, Sergio; Lenz, Martin; Lüthen, Nora; Rumpf, Martin; Zwicknagl, Barbara

Comptes Rendus. Mathématique, Tome 358 (2020) no. 9-10, pp. 1047-1057.

Nous étudions la géométrie des domaines en forme d’aiguille dans les alliages à mémoire de forme. Les domaines en forme d’aiguille sont omniprésents dans les martensites près des interfaces macroscopiques entre régions laminées dans des directions différentes, ou près d’interfaces macroscopiques entre austénite et martensites jumelées. Leur géométrie résulte de l’influence relative de la non-convexité locale de la densité d’énergie effective et des interactions à longue portée (linéaires) engendrées par le champ de déformation élastique, et est pour le moment assez mal comprise. Nous présentons un modèle d’optimisation de forme bi-dimensionnel basé sur l’élasticité non-linéaire et étudions son approximation numérique. Nos résultats montrent que le profil effilé des aiguilles peut être expliqué dans le cadre de l’élasticité non-linéaire, mais pas dans le cadre linéarisé. L’amincissement et la flexion qui en résultent reproduisent les caractéristiques principales observées expérimentalement sur le Ni${}_{65}$Al${}_{35}$.

We study the geometry of needle-shaped domains in shape-memory alloys. Needle-shaped domains are ubiquitously found in martensites around macroscopic interfaces between regions which are laminated in different directions, or close to macroscopic austenite/twinned-martensite interfaces. Their geometry results from the interplay of the local nonconvexity of the effective energy density with long-range (linear) interactions mediated by the elastic strain field, and is up to now poorly understood. We present a two-dimensional shape optimization model based on finite elasticity and discuss its numerical solution. Our results indicate that the tapering profile of the needles can be understood within finite elasticity, but not with linearized elasticity. The resulting tapering and bending reproduce the main features of experimental observations on Ni${}_{65}$Al${}_{35}$.

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