Landau theory of structures in tetragonal-orthorhombic ferroelastics

Abstract

A Landau expansion of the elastic energy in the strains is used to study two-dimensional structures in tetragonal-orthorhombic ferroelastics with constraints. Local energy minima are found with respect to the components of the displacement and so the strains satisfy the compatibility relation; this interdependence of the strains, combined with the constraints, can give rise to a subtle frustration. Extraordinarily, a complex energy surface with many bulk metastable states results purely from boundary conditions, without bulk inhomogeneities ͑such as impurities͒ of any sort. Some settings require twin walls in only one set of tetragonal 110-type planes; only two variants appear, and the dilatational and shear strains are localized near the surface. Tip splitting can occur when twin walls collide with fixed boundaries. Other settings require both 110 and 11 0 walls and so all four variants appear. The structures resulting from collisions of the two twin families are so complex that the ground state of a large system cannot be found with confidence. Strange walls appear between variants with identical deviatoric strain. The dilatational and shear strains are large also in the bulk. Walls wobble, bow, and bend counterintuitively, and pairs sometimes pinch in. Study of the rotation is shown to be essential for understanding some aspects of the structures, particularly collisions of orthogonal twin bands. The ferroelastic-ferromagnet analogy is found to be misleading in important respects. Tip splitting, pinching-in, wall wobbling, and other phenomena are seen in electron microscopy of YBa 2 Cu 3 O 7Ϫ␦ and other materials.

Microstructure and mechanical properties of constrained shape-memory alloy nanograins and nanowires

“…The effect of clamping 13,14 and non-transforming layers 15,16 has been studied theoretically. However, the influence of geometry and size has not been systematically studied.…”

Computations of geometrically linear and nonlinear Ginzburg-Landau mo dels for martensitic pattern formation

“…There are several other computations of geometrically nonlinear models for martensitic phase transformations in the literature [5,6,16,22,27,28], but none of these has compared geometrically linear and geometrically nonlinear models. Only Dondl et al [16], Jacobs [22] and Kloucek and Luskin [24] simulate geometrically nonlinear models which include a capillarity term.…”

“…Only Dondl et al [16], Jacobs [22] and Kloucek and Luskin [24] simulate geometrically nonlinear models which include a capillarity term. Neither Dondl et al [16] nor Kloucek and Luskin [24] simulate macrotwin interfaces.…”

“…Neither Dondl et al [16] nor Kloucek and Luskin [24] simulate macrotwin interfaces. Jacobs [22] uses conjugate gradient energy minimisation to find low energy states. Jacobs [22] shows that tip splitting is either a low energy or long lived metastable energy state, but this study does not use very high spatial resolution or show that the persistence of the split tips is due to geometric nonlinearity which introduces many metastable states.…”

Finite volume analysis of nonlinear thermo-mechanical dynamics of shape memory alloys

“…In the 2D case, we focus our attention on the square-to-rectangular transformations that can be regarded as a 2D analog of the realistic cubicto-tetragonal and tetragonal-to-orthorhombic transformations [11,12]. It is known that for this kind of transformations, the free energy function φ can be constructed by taking advantage of a Landau free energy function F L .…”