Piecewise affine stress-free martensitic inclusions in planar nonlinear elasticity

Abstract

We consider a partial differential inclusion problem which models stress-free martensitic inclusions in an austenitic matrix, based on the standard geometrically nonlinear elasticity theory. We show that for specific parameter choices there exist piecewise affine continuous solutions for the square-to-oblique and the hexagonal-to-oblique phase transitions. This suggests that for specific crystallographic parameters the hysteresis of the phase transformation will be particularly small.

Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys

“…Specifically, transition-state theory explains that the transformation from austenite to martensite is strongly influenced by the energetics of the critical nucleus, which is a small inclusion of martensite in an austenitic matrix. It is known that stress-free inclusions with interfaces of finite total area (or length, in two dimensions) are possible only for special material parameters [33,37,38,39,49,48,27,12].…”

Exact Constructions in the (Non-linear) Planar Theory of Elasticity: From Elastic Crystals to Nematic Elastomers

“…(i) Provide necessary and sufficient conditions for a geometrically non-linear, "single layer" Conti construction associated with a phase transformation for general n ∈ N (see Sections 2.2 and 2.3). This builds on and generalises the argument from [CKZ17]. (ii) Discuss the iterability of the single layer constructions from (i).…”

“…A class of particularly symmetric, exactly stress-free deformations had been studied by Conti [Con08] in specific set-ups (we will also refer to these as "Conti constructions"), see also the precursors in [MŠ99,CT05]. It is the purpose of this article to study these structures systematically in the sequel, following and extending ideas from [CKZ17] and treating elastic and nematic liquid crystal elastomers in a unified framework.…”

“…Let us comment on some of these aspects in more detail: On the one hand, these specific solutions are of particular interest as not only their bulk energy vanishes, but also their surface energy, measured for instance in terms of the BV norm of ∇u is finite (see Section 2.4.1 for some remarks on energetics). As a consequence, materials which exhibit such structures are candidates for materials with low hysteresis as nucleation has low energy barriers (both in purely bulk but also in bulk and surface energy models) [CKZ17], see also [ZRM09] for more information on hysteresis in shape-memory alloys.…”

“…To this end, we rely on the geometrically non-linear constructions from [CKZ17] which we investigate for a general n-well problem before passing to the limit n → ∞. As in [CKZ17] we obtain necessary and sufficient conditions on the wells in order for the corresponding Conti constructions to exist. We remark that in the context of the two-dimensional, geometrically linearised hexagonal-to-rhombic phase transformation by completely different methods (relying on the characterisation of homogeneous deformations involving four variants of martensite) necessary conditions had been derived in an OxPDE summer project by Stuart Patching [Pat14].…”