The energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a -limit of three-dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v : U → R n , U ⊂ R n . We show that the L 2 -distance of ∇v from a single rotation matrix is bounded by a multiple of the L 2 -distance from the group SO(n) of all rotations.

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Reversibility of structural phase transformations has profound technological implications in a wide range of applications from fatigue life in shape-memory alloys (SMAs) to magnetism in multiferroic oxides. The geometric nonlinear theory of martensite universally applicable to all structural transitions has been developed. It predicts the reversibility of the transitions as manifested in the hysteresis behaviour based solely on crystal symmetry and geometric compatibilities between phases. In this article, we report on the verification of the theory using the high-throughput approach. The thin-film composition-spread technique was devised to rapidly map the lattice parameters and the thermal hysteresis of ternary alloy systems. A clear relationship between the hysteresis and the middle eigenvalue of the transformation stretch tensor as predicted by the theory was observed for the first time. We have also identified a new composition region of titanium-rich SMAs with potential for improved control of SMA properties.

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Materials undergoing reversible solid-to-solid martensitic phase transformations are desirable for applications in medical sensors and actuators, eco-friendly refrigerators and energy conversion devices. The ability to pass back and forth through the phase transformation many times without degradation of properties (termed 'reversibility') is critical for these applications. Materials tuned to satisfy a certain geometric compatibility condition have been shown to exhibit high reversibility, measured by low hysteresis and small migration of transformation temperature under cycling. Recently, stronger compatibility conditions called the 'cofactor conditions' have been proposed theoretically to achieve even better reversibility. Here we report the enhanced reversibility and unusual microstructure of the first martensitic material, Zn45Au30Cu25, that closely satisfies the cofactor conditions. We observe four striking properties of this material. (1) Despite a transformation strain of 8%, the transformation temperature shifts less than 0.5 °C after more than 16,000 thermal cycles. For comparison, the transformation temperature of the ubiquitous NiTi alloy shifts up to 20 °C in the first 20 cycles. (2) The hysteresis remains approximately 2 °C during this cycling. For comparison, the hysteresis of the NiTi alloy is up to 70 °C (refs 9, 12). (3) The alloy exhibits an unusual riverine microstructure of martensite not seen in other martensites. (4) Unlike that of typical polycrystal martensites, its microstructure changes drastically in consecutive transformation cycles, whereas macroscopic properties such as transformation temperature and latent heat are nearly reproducible. These results promise a concrete strategy for seeking ultra-reliable martensitic materials.

Abstract. We report results from a systematic program of changing composition of alloys in the system TiNiX, X= Cu, Pt, Pd, Au, to pursue certain special lattice parameters that have been identified previously with low hysteresis. We achieve λ 2 = 1, where λ 2 is the middle eigenvalue of the transformation strain matrix, for alloys with X = Pt, Pd, Au. In all cases there is a sharp drop of the graph of hysteresis vs. composition at the composition where λ 2 = 1. When the size of the hysteresis is replotted vs. λ 2 we obtain an universal graph for these alloys, which also agrees with trends in an earlier combinatorial study of alloys in the system TiNiCu. Motivated by these experimental results, we present a new theory for the size of the hysteresis based on the growth from a small scale of fully developed austenite martensite needles. In this theory the energy of the transition layer plays a critical role. New methods for calculation the optimal layer are developed that rely on Γ-convergence arguments, the small parameter being |λ 2 − 1|. The limiting energy of the transition layer is found to be governed by a nonstandard linear elasticity problem. Overall, the results point to a simple systematic method of achieving low hysteresis and a high degree of reversibility in transforming materials.