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

Abstract

The austenite‐to‐martensite phase transformation is characterized by the creation and growth of small nuclei of the new martensitic phase. Within a geometrically linear description and including an interfacial energy, we show that the minimal energy scales like V1113 in the volume V of the nucleus for V≫1 when primarily only two martensite variants are present. This complements the findings in [Knüpfer, Kohn, Otto, CPAM 2012] where it has been shown that the minimal energy of the transient nuclei has the lower scaling V911 if all three martensite variants are present. The higher scaling of the minimal energy is related to a failure of self‐accommodation. The proof requires new analytical ideas and tools.

On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

“…Our result should be viewed in the context of scaling laws in the calculus of variations in general and more specifically in the modelling of shape-memory alloys and related phase transformation problems (see [37] and [45] for surveys on this). In the context of the modelling of shape-memory alloys, scaling laws, providing some insights on the possible behaviour of energy minimizers, have been deduced in various settings [3,5,6,8,9,14,15,20,[34][35][36][38][39][40][41][42]55]. For certain models, in subsequent steps, even finer properties (such as for instance almost periodicity results) have been derived [13].…”

“…For (36) to hold for some m ∈ N a necessary condition is that α satisfies α| log( )| > log(Md). Under this assumption on α (which will be satisfied by our choice of α in terms of , see below), ( 36) is satisfied for m > 1 α − 2.…”

Asymptotic shape of isolated magnetic domains

“…Nucleation and related problems in shape memory alloys have been e.g. considered in [43][44][45]. There are various other related models from materials science where dilute/large volume regimes are relevant, such as epitaxial growth (e.g.…”

Asymptotic Self-Similarity of Minimizers and Local Bounds in a Model of Shape-Memory Alloys

“…In their model one partial derivative is constrained to take only two values, leading to the characteristic non-convexity. Their work has been meanwhile generalized to different volume fractions [19,59], to vectorial settings in the context of linearized elasticity [3,10,11,23,36,38,39,49,52] and to geometrically nonlinear formulations [12,50]. These vectorial generalizations have confirmed that the Kohn-Müller scalar model indeed captures the correct scaling of the energy.…”