Original Article Journal of Intelligent Material Systems and Structures 0(0) 1–10 Ó The Author(s) 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1045389X13493354 jim.sagepub.com Analytical and numerical modeling of shape memory alloy Negator springs for constant-force, long-stroke actuators Andrea Spaggiari and Eugenio Dragoni Abstract This article explores the merits of shape memory alloy Negator springs as powering elements for solid-state actuators. A Negator spring is a spiral spring made of strip of metal wound on the flat with an inherent curvature such that, in repose, each coil wraps tightly on its inner neighbor. The unique characteristic of Negator springs is the nearly constant force needed to unwind the strip for very large, theoretically infinite deflections. Moreover, the flat shape, having a high area-over-volume ratio, grants improved bandwidth compared to any solution with solid wires or helical springs. The shape memory alloy material is modeled as elastic in austenitic range while an exponential continuum law is used to describe the martensitic behavior. The mathematical model of the mechanical behavior of shape memory alloy Negator springs is provided, and their performances as active elements in constant-force, long-stroke actuators are assessed. The shape memory alloy Negator spring is also simulated in a commercial finite element software, ABAQUS, and its mechan- ical behavior is estimated through finite element analyses. The analytical and the numerical predictions are in good agree- ment, both in martensitic and in austenitic ranges. Keywords actuator, shape memory, optimization Introduction Shape memory alloys (SMAs) are smart materials successfully used in the field of compact solid-state actuation and microactuation because of their high power-over-mass and force-over-mass ratios as shown by Funakubo (1986). Usually, shape memory actuators are based on SMA wires or helical springs. The use of SMA wires ensures a high actuating force, a minimal amount of active material and good mechanical band- width (i.e. operating frequency) of the actuator. In con- trast, Nespoli et al. (2010) showed that SMA wires are undermined by poor strokes and considerable length of the overall construction, when compared to helical SMA springs, which are more compact but affected by lower output forces and lower actuation frequencies. Moreover, the SMA materials, especially compared to electromagnetic linear actuators, cannot match their strokes, but they are extremely lighter and, being a solid-state material, their application is straightforward and requires less moving parts. SMA helical springs have complementary perfor- mances with respect to SMA wires: the stroke per unit of length of the actuator is greatly enhanced but only at the cost of much more active material. Moreover, the electric current needed to supply the SMA springs is much higher than for the wires, and the bandwidth is drastically reduced. In addition, an SMA spring used as actuator keeps the linear behavior of a traditional spring, and its stroke is reduced by the backup element necessary to restore the initial position of the actuator. This linear characteristic does not fit well with the fact that the external load is often constant, which results in reduced stroke and output force of the actuator. The ideal force–stroke characteristic of an actuator is con- stant, in order to increase the actuator efficiency and to avoid oversizing. In the technical literature, there are Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Reggio Emilia, Italy Corresponding author: Andrea Spaggiari, University of Modena and Reggio Emilia, via Amendola, 2, Campus S. Lazzaro, 42122 Reggio Emilia, Italy. Email: andrea.spaggiari@unimore.it