Finite Element Modelling of a NiTi SMA Wire

The development of a Finite Element Model of a NiTi shape memory alloy (SMA) wire in a commercial explicit finite element software, Ls-Dyna, is presented. A user-defined material (UMAT) model has been developed by implementing one of the earliest one-dimensional SMA constitutive models, the Tanaka model [1], into Ls-Dyna through a FORTRAN code. The aim is to apply this model to develop a SMA-actuated morphing wing. Morphing has attracted considerable attention among researchers for the past few decades because of the potential of providing optimum flight conditions at various flight missions. A combination of the morphing wing with smart materials such as SMAs offers further advantages such as a significant reduction in weight and system complexity, compared to an actuation achieved by mechanical motors or hydraulic systems. The simplest example of an actuated structure is presented, that is a SMA wire connected to a linear spring in series. The SMA wire was modelled as a beam element with one integration point, which is equivalent to a truss element, whereas the linear spring was modelled as a discrete element. One complete heating-cooling cycle was applied on the SMA wire. Upon heating, a reverse transformation (martensite-to-austenite) occurred, caused the wire to shorten and consequently extended the spring. Hence the stress in the wire increased while the SMA strain decreased until the end of the transformation. Upon cooling, a forward martensitic transformation (austenite-to-martensite) took place and reduced the stiffness of the wire. As a result, the spring contracted and the wire extended, and so the stress decreased while the SMA strain increased until the end of the transformation. This finite element prediction of the thermomechanical behaviour was compared to an analytical solution for small displacement, and a close agreement was achieved. A parameter analysis was then carried out to analyse the dependence of the thermomechanical behaviour on several parameters such as the length and cross-sectional area of the SMA wire, as well as the spring stiffness (stiffness of the actuated structure). As expected, the FE model showed that the recovery stress (maximum stress at the end of heating cycle) increased whereas the recovery strain (maximum strain at the end of heating cycle) decreased with increase in the SMA length and spring stiffness, and with decrease in the SMA cross-sectional area. The user-defined SMA model was further tested as a design tool to morph a pre-curved corrugated plate. The results showed that for large diameter SMA wires, increasing the number of SMA wires in each cell resulted in a small increment in the tip deflection. A parallel configuration is preferable than a ‘V’ configuration for cells consisted of two SMA wires, because slightly higher tip deflection can be achieved. This is mainly due to the influence of the SMA length on the thermomechanical behaviour. Finally, the SMA model was applied on a morphing wing consists of six corrugated plates between its leading and trailing edges. The resulted trailing edge vertical deflection is 36.6 mm, about 10% of the chord length. In conclusion, this work provides a foundation for future exploration of SMA-actuated morphing wings using Ls-Dyna, such as an optimization of the internal structure using LS-OPT, and fluid-structure interaction (FSI) simulations to include the effect of incoming air flow on the movement of the wings