Zig-zag twins and helical phase transformations

Abstract

We demonstrate the large bending deformation induced by an array of permanent magnets (applied field ∼0.02 T) designed to minimize poles in the bent state of the crystal. Planar cantilevers of NiMnGa (5M modulated martensite) ferromagnetic shape memory alloy deform into an arched shape according to theory, with a zig-zag microstructure that complies with the kinematic and magnetic compatibility between adjacent twin variants. A general theory of bent and twisted states is given, applicable to both twinning and austenite/martensite transformations. Some of these configurations achieve order-of-magnitude amplification of rotation and axial strain. We investigate also atomistic analogues of these bent and twisted configurations with perfect interfaces between phases. These mechanisms of large deformation, induced by small magnetic fields or temperature changes, have potential application to the development of new actuation technologies for micro-robotic systems.

Keywords: Ni2MnGa; bending; continuum mechanics; ferromagnetic shape memory; martensitic phase transformation.

Figure 1.

Predicted two-variant microstructure viewed down the [001] axis (a) and a three-dimensional view (b). Figures drawn with η₁=1.013, η₂=0.952. Note the precise absence of poles on the internal interfaces in the deformed configuration. (a) Reference (ii), deformed (i) and (b) three-dimensional view with magnetization. (Online version in colour.)

Figure 2.

(a) Alternating variants in the reference configuration. (b) Schematic picture of the energy wells (excluding magnetism). The circles represent SO(3)U₁ and SO(3)U₂, and lines connecting deformation gradients indicate that they differ by a matrix of rank one.

Figure 3.

Experimental realization of a two-variant microstructure in NiMnGa. (a) The permanent magnet array. Sliding the oriented NiMnGa rod past this array produced the structure seen in (b). The magnetization in (b) is added by hand. (Online version in colour.)

Figure 4.

Predicted austenite/single-variant martensite microstructure using measured lattice parameters and the energy-well structure of thealloy Ti₅₀Ni_40.75Pd_9.25. The long direction of the reference configuration is parallel to [1,−1,0] while its lateral face normals are (1,1,0) and (0,0,1). (a) Reference configuration (austenite) and (b) austenite (red), martensite (blue). (Online version in colour.)

Figure 5.

Schematic of the general method of producing compatible bent and twisted microstructures, shown with r=1 and illustrating the consistency condition (4.1).

Figure 6.

A zero-energy helical microstructure drawn accurately with the lattice parameters of Ti₅₀Ni_40.75Pd_9.25. The reference configuration (a) is an unstressed single crystal of austenite that has been cut out in a helical shape. The deformedconfiguration (b) consists of austenite regions (red) and two distinct variants (blue, green) of martensite, all of which are zero free-energy, unstressed states. (a) Reference configuration (austenite) and (b) austenite (red), martensite (blue, green). (Online version in colour.)

Figure 7.

Nearest-neighbour generators: the isometry g₁ maps the red atom to the yellow atom, while the isometry g₂ maps the red atom to thegreen one. (Online version in colour.)

Figure 8.

Perfect interface between helical states. See text. (Online version in colour.)