Non-Laminate Microstructures in Monoclinic-I Martensite

Abstract

We study the symmetrised rank-one convex hull of monoclinic-I martensite (a twelve-variant material) in the context of geometrically-linear elasticity. We construct sets of T 3 s, which are (non-trivial) symmetrised rankone convex hulls of three-tuples of pairwise incompatible strains. Moreover we construct a five-dimensional continuum of T 3 s and show that its intersection with the boundary of the symmetrised rank-one convex hull is fourdimensional. We also show that there is another kind of monoclinic-I martensite with qualitatively different semi-convex hulls which, so far as we know, has not been experimentally observed.Our strategy is to combine understanding of the algebraic structure of symmetrised rank-one convex cones with knowledge of the faceting structure of the convex polytope formed by the strains.

Two Kinds of Monoclinic‐I Martensite and some Insight into the Microstructures they Form

“…not compatible) strains of E more closely, again following [4]. It turns out that there are eight 3-tuples of pairwise incompatible strains: {1, 6, 12}, {1, 8, 10}, {2, 5, 11}, {2, 7, 9}, {3, 6, 9}, {3, 8, 11}, {4, 5, 10}, {4, 7, 12}.…”

“…Surprisingly to us, the facet structure of monoclinic-I martensite depends on whether < δ, = δ or > δ, see [4]. This means that the symmetrised lamination, rank-one, quasiconvex and convex hulls/envelopes all qualitatively depend on which of these cases occur.…”

“…Hence, L(E) ⊆ R(E) ⊆ Q(E) ⊆ C(E). For a definition of the mentioned semi-convex hulls and their relation see the references given in [4].…”

“…For L(E) = C(E) it is necessary and sufficient that all edges that are extremal in C(E) be pairwise compatible, see [6] and [4]. This result motivated us to investigate the boundary of C(E) and to understand its facet structure better.…”

Bounds on the recoverable deformations of polycrystalline SMAs at finite strain

“…assuming small deformations with respect to a reference configuration [2,[5][6][7][8][9][10][11][12][13]. In this paper, we focus on upper bounds of the recoverable strains of martensitic polycrystals, in the geometrically non-linear setting.…”