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.

An axisymmetric particle sedimenting in an otherwise quiescent Newtonian fluid, in the Stokes regime, retains its initial orientation. For the special case of a spheroidal geometry, we examine analytically the effects of weak inertia and viscoelasticity in driving the particle towards an eventual steady orientation independent of initial conditions. The generalized reciprocal theorem, together with a novel vector spheroidal harmonics formalism, is used to find closed-form analytical expressions for the O(Re) inertial torque and the O(De) viscoelastic torque acting on a sedimenting spheroid of an arbitrary aspect ratio. Here, Re = UL/ν is the Reynolds number, with U being the sedimentation velocity, L the semi-major axis and ν the fluid kinematic viscosity, and is a measure of the inertial forces acting at the particle scale. The Deborah number, De = (λU)/L, is a dimensionless measure of the fluid viscoelasticity, with λ being the intrinsic relaxation time of the underlying microstructure. The analysis is valid in the limit Re, De 1, and the effects of viscoelasticity are therefore modelled using the constitutive equation of a second-order fluid. The inertial torque always acts to turn the spheroid broadside-on, while the final orientation due to the viscoelastic torque depends on the ratio of the magnitude of the first (N 1 ) to the second normal stress difference (N 2 ), and the sign (tensile or compressive) of N 1 . For the usual case of near-equilibrium complex fluids -a positive and dominant N 1 (N 1 > 0, N 2 < 0 and |N 1 /N 2 | > 1) -both prolate and oblate spheroids adopt a longside-on orientation. The viscoelastic torque is found to be remarkably sensitive to variations in κ in the slender-fibre limit (κ 1), where κ = L/b is the aspect ratio, b being the radius of the spheroid (semi-minor axis). The angular dependence of the inertial and viscoelastic torques turn out to be identical, and one may then characterize the long-time orientation of the sedimenting spheroid based solely on a critical value (El c ) of the elasticity number, El = De/Re. For El < El c (> El c ), inertia (viscoelasticity) prevails with the spheroid settling broadside-on (longside-on). The analysis shows that El c ∼ O[(1/ln κ)] for κ 1, and the viscoelastic torque thus dominates for a slender rigid fibre. For a slender fibre alone, we also briefly analyse the effects of elasticity on fibre orientation outside the second-order fluid regime.

The cofactor conditions, introduced in James and Zhang (2005), are conditions of compatibility between phases in martensitic materials. They consist of three subconditions: i) the condition that the middle principal stretch of the transformation stretch tensor U is unity (λ 2 = 1), ii) the condition a · U cof(U 2 − I)n = 0, where the vectors a and n are certain vectors arising in the specification of the twin system, and iii) the inequality trU 2 + det U 2 − (1/4)|a| 2 |n| 2 ≥ 2. Together, these conditions are necessary and sufficient for the equations of the crystallographic theory of martensite to be satisfied for the given twin system but for any volume fraction f of the twins, 0 ≤ f ≤ 1. This contrasts sharply with the generic solutions of the crystallographic theory which have at most two such volume fractions for a given twin system of the form f * and 1 − f * . In this paper we simplify the form of the cofactor conditions, we give their specific forms for various symmetries and twin types, we clarify the extent to which the satisfaction of the cofactor conditions for one twin system implies its satisfaction for other twin systems. In particular, we prove that the satisfaction of the cofactor conditions for either Type I or Type II twins implies that there are solutions of the crystallographic theory using these twins that have no elastic transition layer. We show that the latter further implies macroscopically curved, transition-layer-free austenite/martensite interfaces for Type I twins, and planar transitionlayer-free interfaces for Type II twins which nevertheless permit significant flexibility (many deformations) of the martensite. We identify some real material systems nearly satisfying the cofactor conditions. Overall, the cofactor conditions are shown to dramatically increase the number of deformations possible in austenite/martensite mixtures without the presence of elastic energy needed for coexistence. In the context of earlier work that links the special case λ 2 = 1 to reversibility (Cui et al., 2006;Zhang et al., 2009;Zarnetta et al., 2010), it is expected that satisfaction of the cofactor conditions for Type I or Type II twins will lead to further lowered hysteresis and improved resistance to transformational fatigue in alloys whose composition has been tuned to satisfy these conditions.

It is well known that, under inertialess conditions, the orientation vector of a torque-free neutrally buoyant spheroid in an ambient simple shear flow rotates along so-called Jeffery orbits, a one-parameter family of closed orbits on the unit sphere centred around the direction of the ambient vorticity (Jeffery, Proc. R. Soc. Lond. A, vol. 102, 1922, pp. 161-179). We characterize analytically the irreversible drift in the orientation of such torque-free spheroidal particles of an arbitrary aspect ratio, across Jeffery orbits, that arises due to weak inertial effects. The analysis is valid in the limit Re, St 1, where Re = (γ L 2 ρ f )/µ and St = (γ L 2 ρ p )/µ are the Reynolds and Stokes numbers, which, respectively, measure the importance of fluid inertial forces and particle inertia in relation to viscous forces at the particle scale. Here, L is the semimajor axis of the spheroid, ρ p and ρ f are the particle and fluid densities, γ is the ambient shear rate, and µ is the suspending fluid viscosity. A reciprocal theorem formulation is used to obtain the contributions to the drift due to particle and fluid inertia, the latter in terms of a volume integral over the entire fluid domain. The resulting drifts in orientation at O(Re) and O(St) are evaluated, as a function of the particle aspect ratio, for both prolate and oblate spheroids using a vector spheroidal harmonics formalism. It is found that particle inertia, at O(St), causes a prolate spheroid to drift towards an eventual tumbling motion in the flow-gradient plane. Oblate spheroids, on account of the O(St) drift, move in the opposite direction, approaching a steady spinning motion about the ambient vorticity axis. The period of rotation in the spinning mode must remain unaltered to all orders in St. For the tumbling mode, the period remains unaltered at O(St). At O(St 2 ), however, particle inertia speeds up the rotation of prolate spheroids. The O(Re) drift due to fluid inertia drives a prolate spheroid towards a tumbling motion in the flow-gradient plane for all initial orientations and for all aspect ratios. Interestingly, for oblate spheroids, there is a bifurcation in the orientation dynamics at a critical aspect ratio of approximately 0.14. Oblate spheroids with aspect ratios greater than this critical value drift in a direction opposite to that for prolate spheroids, and eventually approach a spinning motion about the ambient vorticity axis starting from any initial † Email address for correspondence: sganesh@jncasr.ac.in ‡ Present address: Aerospace Engineering and Mechanics, University of Minnesota, MN, USA. § These authors contributed equally to this work. 632 V. Dabade, N. K. Marath and G. Subramanian orientation. For smaller aspect ratios, a pair of non-trivial repelling orbits emerge from the flow-gradient plane, and divide the unit sphere into distinct basins of orientations that asymptote to the tumbling and spinning modes. With further decrease in the aspect ratio, these repellers move away from the flow-gradient plane, eventually coalesci...