5.1. SMA Modelling Approach
The structural morphing of the tabs is obtained through the SMA technology. The compactness, jointly to the relatively large energy density, makes this technology a good candidate for this specific steady (<1 Hz) application. This frequency upper limit is strictly related to thermal inertia of the material. In principle, the higher is the power supply, the shorter is the time of activation, and the more efficient is the cooling of the shorter in the de-activation phase. Many works can be found in the literature dealing with the speed of actuation of the SMA material and with strategies aimed at increasing it. A key role is played by the surface of the actuator, being strictly related to the heat transfer. To cite some examples, one recalls the work of Song et al. [
31], focusing on an SMA-based actuator arriving at a frequency of 35 Hz; in this work, the strategy to overcome the above-mentioned upper limitation of 1 Hz is presented. In another work [
32], S Sunjai Nakshatharan et al. investigated the effect of pre-stress on the actuation speed of an antagonistic SMA system and demonstrated the capability of achieving cyclic actuation in a period of 0.5 s.
Another critical aspect is the environmental temperature variation that impacts on needed power, time of activation and de-activation, and is, potentially, responsible for undesired activations. The starting activation temperature is kept generally over the upper limit of the operational scenario to avoid undesired activation. Designers can play on two parameters to control the activation temperature: the composition of the alloy and the preload level, the former assuring a wide excursion window for the transformation temperatures (more than one hundred of Celsius degrees), the latter causing an increase in the original activation temperatures regulated by the alloy transformation temperature/stress ratio.
To guarantee both inwards and outwards deflections, the antagonistic configuration, shown in
Figure 10, was investigated. The original tab, illustrated in the sketch on the left, was milled preserving the root hinge region and the tip while shaping a flat middle plate on most of the chord. Root and tip were then linked to two SMA rods, as illustrated in the scheme on the right.
Two SMA rods were connected to the tip of the inner bending beams in an antagonistic way to provide both upward and downward deflections of the flaps. The contraction of one SMA actuator upon heating resulted in the extension of the opposing SMA actuator mechanically. Then, the contraction by heating of the extended actuator will reverse the actuation.
The SMA rods were mounted with a certain pre-stress to have enough martensite phase exploitable for the actuation, that is to say, potentially transformable by heating into austenite to produce strain recovery at a macroscopic level. When outwards deflection is required, the upper SMA is heated; its contraction thus causes the upwards movement of the tip. In the same way, when inwards deflection is required, the lower SMA is activated. Although it is possible to imagine the activation of an SMA rod when the other one is still warm and in austenite phase, in this work, only the case of the activation of an SMA when the other one is cold and in martensite phase was considered.
The main tab structure is prone only to deflections occurring in the plane containing the SMA elements; no constraint enforces this type of motion other than the geometry of the cross section of the main tab whose lowest inertia moment drives the planar deflection. To model the system, the non-linear MSC/Nastran SOL400 solver was used in combination with an SMA dedicated card, collecting all the information for the alloy constitutive law (austenite, martensite elastic modulus, Poisson ratio, transformation stresses at reference temperature, maximum recoverable strain, material density, transformation stress/temperature gradients). The data collected in
Table 3 were used for the simulation and hereafter presented. A non-linear static analysis was implemented to simulate the different working steps of the SMA (pre-stretching and connection to the structure, upward activation, reverse to neutral position, downward activation, reverse to neutral position). A uniform incremental temperature load was assigned to the SMA material, passing from the initial value to the final one foreseen, per each step of the simulation.
The finite element model shown in
Figure 11 was realized. The root region was not modelled; in fact, being remarkably more rigid than the other parts, it was replaced by constraints. The flat plate was simulated through beam elements. For the two SMA rods, hexahedral parabolic elements were considered; solid non-linear elements are, in fact, mandatory for the implementation of the SMA constitutive law. Finally, the tip was simulated through rigid connections among the edges of the SMA rods and the flat plate.
Three moments of the life of the system were simulated:
Integration of the SMA rods.
Activation/heating of the upper SMA and then restoring/cooling into the neutral configuration.
Activation/heating of the lower SMA and then restoring/cooling into the neutral configuration.
The integration of the SMA rods, originally in the austenite phase, consists of clamping the SMAs at the root region and stretching them up to bring the edges at the tip connection. During this operation, the applied load causes martensite production. Displacements along the rod axes were imposed during the simulation; as the edges coincided with the tip rigid connection, multiple constraints were imposed to pin the overlapping nodes. Then, the edges of the SMA were released to allow their elastic recovery and to achieve equilibrium with the tab structure. In
Figure 12, the initial configuration (SMA before stretching) and the fully integrated configuration (SMA stretched, connected, and in equilibrium with the flat plate) are illustrated.
The stress level within the components was assumed as design constraint. 1/3 of the ultimate stress was assumed as allowable for the structure material; the allowable arose to the 80% of the ultimate stress for the SMA material; this choice is justified by the necessity to exploit as much as possible the SMA actuation capability, strictly related to the amount of martensite generated during the pre-load. Another design constraint considered was represented by structural instability. The elastic recovery of the SMA rods in fact produces an axial force on the flat plate potentially causing its collapse. To prevent this event, the pre-load level of each SMA rod was assumed to be lower than 1/3 of the instability load of the flat plate. In this way, a margin of 33% was assumed also in view of the compressive action due to the stretched skin, currently not considered.
5.2. Parametrization and Results
On the basis of the model illustrated above, a parametric study was organized. The target was to maximize the deflection of the tip of the tab, meeting the requirements in terms of structural safety discussed in the previous section.
In
Table 4, the parameters are considered, and their variation range are reported. The width of the plate was kept constant at 40 mm, while a maximum temperature of 500 °C was considered to guarantee full activation, even crossing the effective limit of complete conversion of the martensite. Despite the generally critical role played by the weight for aerospace applications, this parameter was not considered at this stage of the work. From one side, 16 SMA rods (each with a length of about 90 mm and a diameter no greater than 4 mm) scarcely contributed to the over-all weight (0.11 over 24 kg), and, from the other side, this additional parameter could have led astray the parameterization process with losses in terms of actuation performance.
The morphing tab system was conceived to withstand the operational loads without the contribution of the SMA rods. In this sense, attention was paid to the action during the parameterization process due to the pre-stretching of the SMAs on the tab’s main structures to prevent from any instability, even in fully deflected conditions.
Some considerations drove the selection of the mentioned parameters:
The thickness of the plate increases the robustness of the structure but, at the same time, reduces the actuation performance.
The SMA diameter increases its authority and reduces the stress level; however, large diameters may cause the collapse of the structure.
Finally, the shorter the SMA the higher the stretching needed for the connection to the structure, with a, consequently, higher amount of martensite production and, thus, strain recoverability; however, large stretching causes higher stress levels within the SMA and greater transmitted forces, potentially causing the collapse of the flat plate.
A total of 125 configurations were simulated by considering 5 different values for each parameter. Only nine configurations met the safety requirements. They are compared in
Figure 13 in terms of tip displacement, normalized with respect to the maximum one and its safety margins that are referred to the beam and the SMA material. The case/configuration 18 (red thick line), clearly enveloping all the others, was selected for this preliminary design. Its main features and performance are summarized in
Table 5 and
Table 6, respectively.
Then, the evolution of the displacement of the tip and the stress produced in the beam and in the SMA rods were computed and plotted in
Figure 14 against the iteration step. The different phases were highlighted using different colors. The simulation started with the stretching of the SMA rods (yellow region); the beam was not affected by this operation, and thus both tip displacement and beam stress were zero. Then, the edges of the SMA rods were linked to the tip and released to enable elastic recovery (green region); no displacement of the tip occurs, owing to the symmetry of the loading, but the stress level rises in the beam and slightly decreases within the SMA rods. At the end of this phase, the upper SMA is heated (red region on the left); after having achieved the activation temperature threshold, the transformation starts, and the tip moves upwards until all the martensite is transformed into austenite (225 °C). At this point, even if the temperature further rises up to the imposed limit (500 °C), a plateau region is observed for the displacement and for the stress level of the beam and the SMA rods. It is also worth noting that, due to the presence of austenite, the stress level of the activated SMA is higher than the antagonistic. The cooling then starts (blue region on the left); the temperature of the SMA slows down to room temperature, and, as the transformation threshold is achieved, the plateau ends, and the displacement and the stress level quickly return to the neutral configuration. At this point, the activation and de-activation of the lower SMA starts; the same behavior was observed, as shown by the last two red and blue regions.
Finally, a rough estimate of the power consumption can be addressed. Assuming, per each SMA rod, a length of 90 cm, and considering a helicoid heater with a 2 mm square cable covering 50% of the SMA surface with 20 turns, a heating surface of 416 mm
2 can be obtained. Multiplying this latter value by a power per unit surface of 0.069 W/mm
2 needed by this type of heater [
33], a power consumption of 28.7 W can be obtained. Considering an antagonistic actuation (only one SMA rod per each couple is heated), a total power of 8 tabs × 28.7 W = 230 W is required. However, it is worth noting that this estimate is very conservative. In fact, the declared power per unit surface refers to a heater temperature of 400 °C (almost two times the needed one).