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

Abstract

In this article we deduce necessary and sufficient conditions for the presence of "Conti-type", highly symmetric, exactly-stress free constructions in the geometrically non-linear, planar n-well problem, generalising results of [CKZ17]. Passing to the limit n → ∞, this allows us to treat solid crystals and nematic elastomer differential inclusions simultaneously. In particular, we recover and generalise (non-linear) planar tripole star type deformations which were experimentally observed in [MA80a, MA80b, KK91]. Further we discuss the corresponding geometrically linearised problem.

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].…”

Minimal Energy for Geometrically Nonlinear Elastic Inclusions in Two Dimensions

“…is attained on the line [c, d] and is solved by t * := t − r c−d |c−d| for some r with 0 < r < C 1 2 . Here, the error bound for r is a consequence of (15). Using v(t) as a competitor and inserting the bound for r c,t implies…”

“…We also refer to [4,5,50] for the study of quasiconvexity at the boundary. Further, highly symmetric, low energy nucleation mechanisms have been explored in [25] and [15] both in the geometrically linear and nonlinear theories in two dimensions. In the geometrically nonlinear settings substantially less is known in terms of nucleation properties due to the presence of the nonlinear gauge group.…”

Semi-discrete modeling of systems of wedge disclinations and edge dislocations via the Airy stress function method

“…We refer to [69] for the classification of over forty types of disclinations that can be constructed in MgCd alloys undergoing the hexagonal-to-orthorhombic transformation. Among them, we recall examples of beautiful, self-similar, martensitic microstructures containing a dipole of wedge disclinations (see in particular [74] and [75]), for which models and computations are produced in [85,23] and mathematical theories are derived in [21]. Examples of complex self-similar microstructures incorporating disclinations emerging from the nucleation and evolution of needle-shaped regions occupied by martensitic phase are described in [65,66] (see also [12,22] for computations and stochastic models).…”