Quasiconvexity at the Boundary and the Nucleation of Austenite

Abstract

ABSTRACT. Motivated by experimental observations of H. Seiner et al., we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy stabilized as a single variant of martensite. In the experiments the nucleation process was induced by localized heating and it was observed that, regardless of where the localized heating was applied, the nucleation points were always located at one of the corners of the sample -a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure, involving austenite and a simple laminate of martensite, can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant in the interior, faces and edges, respectively, provided the specimen is suitably cut.MSC (2010): 74N15, 49K30 quasiconvexity at the boundary, microstructure, phase transitions, nucleation

Necessary and sufficient conditions for the strong local minimality of C1 extremals on a class of non-smooth domains

“…In particular, it was shown that, in this simplified model, the quasiconvexity conditions hold in the interior and at edges, thus preventing the localised nucleation of the high temperature phase; in contrast, and consistent with observations, quasiconvexity was lost at certain corners allowing for nucleation. The sufficiency result presented in the current work could be used to strengthen the result of [5] in a different modelling regime where variations that are not localised are also considered. This may indeed be an interesting direction to pursue.…”

“…For example, models based on energy minimisation, and consequently the techniques of the vectorial Calculus of Variations, have been very successful in materials science where typical specimens are polyhedral. Indeed, the work presented here has been largely motivated by [5] where, in a simplified model, a set of quasiconvexity conditions at edges and corners of a (convex) polyhedral domain was employed to explain remarkable experimental observations in a shape-memory alloy (see [6]). In particular, it was shown that, in this simplified model, the quasiconvexity conditions hold in the interior and at edges, thus preventing the localised nucleation of the high temperature phase; in contrast, and consistent with observations, quasiconvexity was lost at certain corners allowing for nucleation.…”

Nucleation barriers for the cubic‐to‐tetragonal phase transformation in the absence of self‐accommodation

“…In this sense, Theorem (and its counterpart [, Theorem 1]) is a quantitative version of the result by Battacharya on self‐accommodation of martensite for the cubic‐to‐tetragonal phase transformation in the geometrically linear theory [, Chapter 9]. We note that the energetics for the nucleation at corners for the cubic‐to‐tetragonal transformation have been considered by Bella and Goldman in [] and by Ball and Koumatos in []. Furthermore, related results for the cubic‐to‐orthorhombic phase transformation in two space dimensions have been studied by Rüland in [].…”

“…With the assumptions on 3 and , this can be rewritten as By Young's inequality the volume term to the right can be absorbed and we get…”

Incompatible Sets of Gradients and Metastability

“…The constrained theory has been justified as a limiting theory for Young measures of low energy sequences by Forclaz [38] using Γ -convergence, but under assumptions not allowing W (A) → ∞ as det A → 0+; the proof is based on replacing ϕ by kϕ in (6.1) and letting k → ∞ (a similar procedure to letting |T |/κ → 0 but which does not require additional smoothness assumptions on ϕ). A more general Γ -convergence analysis including the austenite energy well and allowing W (A) → ∞ as det A → 0+ is given by [16,Proposition 1]. The design of orientations was based on the minimization of (6.4), which can be done in the following way by minimizing its integrand (see Chu [26], Chu & James [27]).…”