2.1. Layout of the actuated corrugated laminate

A functional sketch of the concept proposed in this research activity is presented in Fig. 1, where the SMA actuator, namely a wire or a system of wires, acts in a parallel configuration with an elastic spring. The spring stiffness represents the elastic properties in the actuation direction of a corrugated laminate element. Thanks to the properties of the SMA material, a contraction can be induced by increasing the wire temperature, and the SMA actuator compresses the laminate, acting against its stiffness and thus producing a varying stress within the wire itself. Then, when the temperature is reduced, a phase transformation occurs in the SMA and the laminate stiffness should be sufficient to re-elongate the wire, thus recovering the initial configuration and rearming the wire for the subsequent cycle.

Figure 1. Simplified model of the actuated corrugated laminate.

Considering a corrugated laminate of a given width w in the non-morphing direction, the number of wires per unit width defines the density of the SMA actuators, $\rho_{wire}$. This parameter influences the maximum achievable contraction, the forces required for the re-elongation of the actuators (rearming), the weight and the energy costs of the actuation system.

The physical implementation of the concept is based on the embedment of the SMA wires between two corrugated laminates, as shown in Fig. 2. The two corrugated laminates, with a square-rounded profile, are joined to form a twin corrugated configuration, as it was defined in Ref. (Reference Airoldi, Fournier, Borlandelli, Bettini and Sala6). The same configuration was also adopted in Ref. (Reference Thill, Etches, Bond, Potter and Weaver3) to increase the bending stiffness in a morphing skin application. The sensitivity studies carried out in Ref. (Reference Airoldi, Fournier, Borlandelli, Bettini and Sala6), by comparing different types of simple and combined configurations of corrugated profiles, proved that this type of corrugated element provides excellent bending and shear stiffness in the non-morphing directions, if compared with a flat laminate of the same weight. Accordingly, this configuration can offer significant stiffness contributions if used in the structure of a morphing surface, while retaining the capability to be elongated or contracted in one direction. Moreover, the twin configuration involves some advantages for the integration of the SMA wires. Indeed, wires can be integrated along the bonding line of the two square-rounded profiles. Therefore, actuators can work inside protected closed cells, whichever solution is adopted for the physical integration. Since temperature controls the activity of SMA actuators, such a protected environment could be exploited to implement appropriate strategies to optimise temperature management.

Figure 2. Corrugated configuration: (a) geometry of the base cell and (b) twin corrugated configuration with a SMA wire in the mid-plane.

The base-cell geometry that was chosen for the concept implementation is illustrated in Fig. 2(a), and geometrical dimensions are given in Table 1. Demonstrators were designed according to the three-cell arrangement shown in Fig. 2(b). The SMA wires were integrated between the two square-rounded profiles by bonding them directly to the laminates or by making them pass through channels obtained between the two adjoined profiles. In both cases, they were let free to contract for a length $L_{f}$, exemplified in Fig. 2(b), whose value actually depends on the details of the method adopted for their physical integration.

Table 1 Corrugated base-cell geometrical parameters

2.2. Numerical evaluation of the corrugated laminate stiffness

The composite material used for the production of the corrugated laminate is a SEAL CC90/ET443 plain-weave carbon-reinforced fabric, pre-impregnated with epoxy matrix. Its properties were characterised in Ref. (Reference Bettini, Airoldi, Sala, Di Landro, Ruzzene and Spadoni37) and are reported in Table 2. Following the approach presented in Ref. (Reference Airoldi, Fournier, Borlandelli, Bettini and Sala6), a finite element model was developed within the Simulia/Abaqus Standard code, to represent a one-element-wide slice of the corrugated element. In the model, continuum shell elements (SC8R elements⁽³⁸⁾) were used to represent as closely as possible the geometry of the twin corrugated configuration. In this work, lay-ups of fabric plies with a 0/90 orientation of fibres, with the 0° oriented axis along the corrugation profile, were chosen to study the development of the proposed actuated laminate concept. Although such a choice could be suboptimal for an aircraft skin, it minimises the risk of activating non-linear matrix-dominated responses during the deformation and the effects of anti-clastic curvatures due to Poisson’s ratio, thus simplifying the prediction and the analysis of the response for the demonstrator to be designed and tested. However, the actuated corrugated laminate could be designed and produced by adopting different lay-ups, optimised for specific structural requirements, as confirmed in the numerical analysis presented in Section 3.

Table 2 Mechanical properties of the material

The contour shown in Fig. 3 represents the normal stress along the corrugated profile of the external and internal plies of the continuum shell elements, for a displacement of 1mm. The force per unit width divided by the elongation $\Delta p/p$ provides the stiffness of the corrugated element cell, $k_{corr}$, which is reported in Fig. 3 for different values of the number of plies, $n_{ply}$. It can be observed that the cell stiffness increases with the cube of the thickness, thus indicating that the deformation mechanism is dominated by the bending behaviour of the laminate.

Figure 3. Finite element model of a single cell of the corrugated laminate.

Once the lay-up has been selected, the axial stiffness $K_{corr}$, referred to a complete twin corrugated plate, having a width w and a number of cells $n_{cell}$, can be computed as

(1) \begin{equation}{K_{corr}} = {k_{corr}} \cdot {w \over {{n_{cell}} \cdot p}}\end{equation}

2.3. Experimental characterisation of the SMA

The design of the actuated corrugated laminate requires the definition of the functional parameters of the active SMA element. The actuator chosen for the manufacture of the demonstrators is a Smartflex® wire with a diameter, $d_{wire}$, of 0.2mm, produced by SAES Getters S.p.A., made of a near-equiatomic NiTi (49–51%) SMA, stabilised for actuation purposes. The NiTi wire was characterised through differential scanning calorimetry by using a TA Instruments DSC Q100, to evaluate the transition temperatures in stress-free conditions and then subjected to mechanical tests, by using a TA Instruments DMA Q800.

The DSC analysis identified a temperature for austenitic transformation (austenitic finish temperature, $A_{f}$^{(Reference Otsuka and Wayman39)}) of 73.8°C. From the engineering point of view, $A_{f}$ is the temperature that is required to complete the wire contraction and the consequent full actuation, but its value increases with the stress applied to the wire, according to the Clausius–Clapeyron equation of the SMA in Equation (2), which defines the relationship between the applied stress and the transformation temperature by means of a constant coefficient, $CC_{Af}$^{(Reference Otsuka and Wayman39)}.

(2) \begin{equation}{{d{\sigma _{wire}}} \over {dT}} = C{C_{Af}}\end{equation}

Therefore, the complete characterisation of the actuator required strain-recovery tests (i.e. actuation cycles at various fixed levels of applied stress with cycling temperature) to obtain the stroke recovered at different loads, the residual strain after thermal cycling and the temperatures for phase transformation under stress, in conditions close to the operational ones.

In these tests, performed by using the DMA apparatus, the fixed stress levels applied were increased progressively in steps of 50MPa from an initial value of 50MPa to a final level of 300MPa, for a total number of six cycles. Temperature was varied cyclically from −50 to +200°C, with a rate of 5°C/min. The strain versus temperature curves recorded are presented in Fig. 4, whilst Fig. 5(a) indicates the procedure used to identify the transition temperatures from each one of the curves obtained at different stress levels. The cycles started with the wires in austenite phase, subjected to the prescribed stress. Then, the wires were cooled down below the martensite start temperature, $M_{s}$, where the stress applied led to the detwinning of the martensite phase. Detwinning continued with cooling, until a maximum elongation was reached at a martensite finish temperature, $M_{f}$. At the end of cooling, the cycle was reversed, and heating led to the transformation of the wire back into the austenitic phase. In the heating branch, the austenite start temperature, $A_{s}$, and austenite finish temperature, $A_{f}$, were identified. Once the transformation temperatures from the curves at different stress levels were known, the stress – temperature phase diagram shown in Fig. 5(b) was drawn. A linear regression of the data provided a rate of 8.62MPa/K for $A_{f}$ temperature (Clausius–Clapeyron coefficient $CC_{Af}$). Analogous rates were defined for the other transition temperatures: $M_{s}$ $(CC_{Ms}=7.35 \mathrm{MPa/K})$, $M_f(CC_{Mf}=5.65 \mathrm{MPa/K})$ and $A_{s}(CC_{As}=8.17 \mathrm{MPa/K})$.

Figure 4. Strain recovery tests at fixed stress levels (from 50MPa to 300MPa in steps of 50MPa) with cycling temperature.

Figure 5. SMA characterisation: (a) example of transformation temperatures with applied stress of 100MPa and (b) stress–temperature phase diagram.

It can be seen from Fig. 4 that, the larger the applied stress, the greater the strain recoverable during the martensite transformation. This effect is influenced by the lower stiffness of martensite with respect to austenite phase, which led to increasing differences in mechanical strain for increasing stress.

At the end of the cycles, the curves exhibit a residual strain, which tended to be higher as the applied stress was increased. This effect indicates the occurrence of irreversible phenomena during the temperature cycle. Actually, it is known that the functional performance of SMA actuators can change under repeated cycling, particularly if the stress applied and the stroke recovered are high^{(Reference Fumagalli, Butera and Coda26)}. Therefore, in engineering applications, operational stress and recoverable strain should be limited to guarantee durability. Specifically, according to the manufacturer, the wire adopted in this research is optimised to work at 150MPa with a 3.5% stroke, and in such conditions a minimum of 200,000 cycles is guaranteed.

From the engineering point of view, the results obtained in the tests performed on the SMA wire highlight that the corrugated laminate could be actuated by heating the wire at a temperature slightly higher than $100^{\circ}\mathrm{C}$. The transformation in martensite phase will start at temperature lower than $50^{\circ}\mathrm{C}$. The stress required to detwin the wire in the martensite phase is relatively low, thus indicating that rearming could be achieved through the forces exerted by the corrugated laminate. However, in the proposed application, the actuation and the cooling take place at neither constant temperature nor stress. In particular, the application of a decreasing stress during cooling does not guarantee the complete rearming of the wire under the action of the elastic forces exerted by the laminate. This feature must be taken into consideration in the analysis and interpretation of experimental results.

To define the elastic moduli of the material in the different phases, a further DMA mechanical characterisation was performed, loading and unloading the wire under displacement control at a fixed temperature. Such tests provided the elastic modulus for the wire at different temperatures, revealing a significant increment on passing from the martensite phase, where a value $E_{m}$ of about 20GPa was recorded, to the austenite phase, characterised by a modulus $E_{a}$ of 57GPa. Such information is fundamental to evaluate the stiffness of the wire system embedded in the corrugated laminate beyond the $A_{f}$ temperature, which will be required to predict the capability of the actuator to deform the composite element and the response of the actuated structure. Considering a wire element in austenite phase with a length $L_{f}$, the stiffness $k_{wire}$ can be estimated through the following expression:

(3) \begin{equation}{k_{wire}} = {{{E_a}\left( {\pi d_{wire}^2/4} \right)} \over {{L_f}}} = {{{E_a}{A_{wire}}} \over {{L_f}}}\end{equation}

where $A_{wire}=0.0314 \mathrm{mm}^{2}$ is the cross section of each wire.

The total stiffness K _wire of the actuation system made of a number of wires n _wire across the width w of a corrugated laminate consisting of n _cell actuated cells is expressed as in Eq. 4:

(4) \begin{equation}{K_{wire}} = {k_{wire}}{{{n_{wire}}} \over {{n_{cell}}}}\end{equation}

3.1. Evaluation of the system performance and design choices

Assuming that the simplified Turner’s model is applicable, the deformation produced by the temperature in a SMA wire is considered to be a combination of the strain related to the phase transformation, $\varepsilon^{m}$, and the conventional thermal expansion, $\varepsilon^{t}$. According to Turner’s model^{(Reference Turner32,Reference Turner33)} , the total response can be represented by introducing an equivalent coefficient of thermal expansion, $\alpha^{eq}$, which depends on temperature and provides the strain $\varepsilon^{T}=\varepsilon^{m}+\varepsilon^{t}$, produced by a temperature variation with respect to a reference temperature $T_{0}$:

(5) \begin{equation}{\varepsilon ^T} = {\alpha ^{eq}}\left( T \right)\left( {T - {T_0}} \right) = {\alpha ^{eq}}\left( T \right)\Delta T\end{equation}

Considering the whole actuation system of n _wire wires embedded in a corrugated laminate of n _cell, the total free length of the wires will be n _cellL _f, according to the scheme presented in Fig. 2, so that the contraction u ^T of the free actuation system would be:

(6) \begin{equation}{u^T} = {n_{cell}}{L_f}{\varepsilon ^T} = {n_{cell}}{L_f}{\alpha ^{eq}}\left( T \right)\Delta T\end{equation}

In the presence of a force F _wire applied to the actuator, a total displacement u is obtained. If the stiffness K _wire is evaluated through Eqs. 3 and 4, the constitutive response of the actuator can be written as

(7) \begin{equation}{F_{wire}} = {K_{wire}}\left( {u - {u^T}} \right) = {K_{wire}}u - {F^T}\end{equation}

If the system sketched in Fig. 1 is not subjected to other external forces, F _wire is in equilibrium with the force exerted by the mechanical response of the laminate, F _corr. The force $-F^{T}=-K_{wire}{{u}^{T}}$ can be defined as an equivalent thermal force exerted by the wire system if it was completely constrained, that is for $u=0$. Since the stiffness of the corrugated laminated defined in Eq. 1, K _corr, is known, the equilibrium of the actuated corrugated laminate for a generic displacement u, without external forces, reads

(8) \begin{equation}{K_{wire}}\!\left( {u - {u^T}} \right) + {K_{corr}}u=0.\end{equation}

This equation can be rewritten and solved for u as shown in Eq. 9:

(9) \begin{equation}u = {{{K_{wire}}{u^T}} \over {{K_{corr}} + {K_{wire}}}} = {{{u^T}} \over {{K_{corr}}/{K_{wire}}+1}}\end{equation}

According to Eq. 9, the ratio of the displacement of the actuated corrugated laminate to the displacement $u^{T}$ depends on the ratio $K_{corr}/K_{wire}$, which, following Eqs. 1,3 and 4, can be expressed as

(10) \begin{equation}{{{K_{corr}}} \over {{K_{wire}}}} = {{{k_{corr}}} \over {{E_a}{A_{wire}}}}{w \over {{n_{wire}}}}{{{L_f}} \over p} = {{{k_{corr}}} \over {{E_a}{A_{wire}}}}{{{L_f}} \over p}{1 \over {{\rho _{wire}}}},\end{equation}

where $\rho_{wire}=w/n_{wire}$ is the wire density along the width of the corrugated laminate. Considering a fixed profile for the corrugated laminate with a lay-up made of 0/90 oriented fabric plies and given the SMA wire type, the ratio between the contraction induced on the corrugated laminate, u, and u ^T, which is the contraction that would be obtained by the free SMA wire with the same L _f, depends on the number of ply, n _ply, the wire density, $\rho_{wire}$, and the L _f/p ratio. The ratio u/u ^T is plotted versus the wire density for different values of n _ply and L _f/p in Fig. 6.

Figure 6. Contraction of the actuated corrugated laminate versus wire density.

It can be seen that a wire density of 100 wires per metre is adequate to achieve 90% of the maximum potential contraction for n _ply = 3, whereas 300 wires per metre and 1000 wires per metre are required to obtain the same performance for n _ply = 5 and n _ply = 7, respectively. A reduction of L _f/p improves the u/u ^T ratio, but it should be observed that the absolute value of u ^T is actually proportional to L _f, as indicated in Eq. (6).

Considering such results, a lay-up with three fabric plies having 0/90 orientation was selected for the demonstrators. The integration of the wires was accomplished in two different ways, thus developing two kinds of demonstrators. In a first version, the wires were bonded to the laminate, so that they were partially free to contract within the demonstrator ( $L_{f}/p\approx0.5$), whereas a second version was designed by allowing the wires to contract completely along the whole demonstrator length. Therefore, this second solution is closer to the case with L _f/p = 1.0. In both cases, wires were integrated with a fixed pre-elongation $\varepsilon^{I}=4\%$, in order to achieve a free maximum contraction $\varepsilon^{T}=\varepsilon^{I}$ when actuated at a temperature beyond A _f.

The choice of these parameters enabled an evaluation of the stress acting in the SMA wires as a function of the wire density. Indeed, once the displacement u has been evaluated through Eq. (9), a stress recovery procedure can be applied by considering that the force acting in the wire system is $-F_{wire}=-K_{wire}(u-u^{T})$. The stress acting in the single wire, $\sigma_{wire}$, is evaluated by dividing this force by the total cross section of the wire system and considering the solution for u obtained from Eq. (9):

(11) \begin{equation}{\sigma _{wire}} = {{ - {K_{wire}}\left( {u - {u^T}} \right)} \over {{n_{wire}}{A_{wire}}}} = {{{u^T}{K_{wire}}} \over {{n_{wire}}{A_{wire}}}}{{{K_{corr}}} \over {{K_{corr}} + {K_{wire}}}}\end{equation}

Substitution of the expressions for u ^T, K _wire and K _corr given in Eqs. (1), (3), (4) and (6) leads to Eq. (12):

(12) \begin{equation}{\sigma _{wire}} = {\varepsilon ^T}{E_a}\left( {{1 \over {1 + {{{K_{wire}}} \over {{K_{corr}}}}}}} \right) = {\varepsilon ^T}{E_a}\left( {{1 \over {1 + {{{E_a}{A_{wire}}} \over {{k_{corr}}}}{p \over {{L_f}}}{\rho _{wire}}}}} \right)\!,\end{equation}

where K _wire/K _corr is the inverse of the ratio defined in Eq. (10).

For the chosen design parameters, the contraction u and the values of $\sigma_{wire}$ estimated from Eq. (9) and Eq. (12) are plotted versus $\rho_{wire}$ in Fig. 7, for $\varepsilon^{I}=4\%$. The influence of L _f on the recovered displacement is apparent, as well as the importance of a correct selection of $\rho_{wire}$ to maintain the stress levels in the wires below acceptable limits. For the conditions at hand, a density higher than 80 wires/m is adequate, for the solution with L _f/p = 0.5 to satisfy the condition $\sigma_{wire}<150 \mathrm{MPa}$, introduced in Section 2.3, whereas a higher density is required for L _f/p = 1.0. Accordingly, the number of wires in the SMA actuation system of the demonstrators was set to obtain two levels of $\rho_{wire}$, namely 80 wires/m and 120 wires/m.

Figure 7. Contraction and stress in SMA wires for different L _f/p ratio versus wire density.

Although the configuration with L _f/p = 0.5 achieves lower contractions than the one with L _f/p = 1.0, it reduces the stress levels on the wires and increases the u/u ^T ratio. It is also worth noting that the adoption of different corrugated profiles could lead to configurations with $0.5<L_{f}/p<1.0$, so that the preferred solution may depend on specific requirements. Moreover, a configuration with $L_{f}/p=1.0$ involves wires that pass through channels obtained between the two adjoined profiles, which can apply forces only at the ends of the panel. This may not be adequate for large or curved panels and still requires, eventually, a method for the connection between wires and the ends of the corrugated laminate. Such reasons motivated the manufacture of a demonstrator with bonded wires (80 wires/m) and of two demonstrators with tubed passing-through wires, with densities of 80 wires/m and 120 wires/m.

3.2. Calibration of Turner’s approach for the finite element models

Numerical models of the demonstrators were developed to analyse in more detail their response and to validate an approach, based on the application of Turner’s method^{(Reference Turner32,Reference Turner33)} , for predicting the performance of the actuated systems based on the concept presented in this paper.

The first step in the development of the numerical approach was the identification of the equivalent coefficient of thermal expansion, $\alpha^{eq}$, introduced in Eq. (5). Since a detailed representation of the actuation transient is beyond the scope of this work, the calibration is required mainly to represent the capability of strain recovery of the wire under the action of the actuation stimulus. Accordingly, the characterisation of the wires presented in Section 2.3 provides sufficient data for the definition of the law $\alpha^{eq}(T)$. However, to refine the calibration, an additional experiment was conducted in a thermal chamber, with a single wire connected to a spring with a stiffness of 0.7N/mm and then actuated to record the behaviour in variable stress conditions. The stiffness value of the spring was chosen to represent approximately the effect of the corrugated laminate on a single wire, considering a wire density of 80 wires/m. In this experiment, the total strain recorded, $\varepsilon$, was considered to be the sum of the elastic strain, $\varepsilon^{e}$, and the overall strain due to temperature variation, $\varepsilon^{T}=\varepsilon^{m}+\varepsilon^{t}$. Since the coefficient $\alpha^{eq}$ takes into account all the strains due to temperature, the following expression holds:

(13) \begin{equation}{\alpha ^{eq}}\left( T \right) = {{\varepsilon - {\varepsilon ^e}} \over {T - {T_0}}} = {{\varepsilon - {\sigma \over {E\left( T \right)}}} \over {T - {T_0}}} = {{{\varepsilon ^T}} \over {T - {T_0}}},\end{equation}

where E(T) is the Young’s modulus of the wire measured in the DMA tests, varying from E _m to E _a, according to the mechanical tests discussed in Section 2, and T ₀ is an arbitrarily chosen reference temperature (in this case $21^{\circ}\mathrm{C}$).

The calibration of $\alpha^{eq}$ was verified in a simple numerical model of the spring/SMA wire system, which was developed by using the Simulia/Abaqus Standard code. The model, shown in Fig. 8(a), is referred to an 80-mm-long wire, represented by using four beam elements (type B31⁽³⁸⁾), connected with a spring element (type SpringA⁽³⁸⁾). The Young’s modulus and the thermal expansion coefficient of the material attributed to beam elements varied with temperature according to the laws determined for E(T) and $\alpha^{eq}(T)$. The nodes at the ends of the wire and of the spring model were constrained, and the temperature was increased from $21^{\circ}\mathrm{C}$ to $147^{\circ}\mathrm{C}$. The numerical curve of contraction versus temperature, reported in Fig. 8(b) indicates an appreciable correlation with the actuation branch of the experimental temperature-controlled cycle.

Figure 8. Model of the SMA wire and spring system: (a) contour of displacement, (b) numerical–experimental correlation.

3.3. Numerical models of the actuated corrugated laminates

The finite element mesh of the demonstrators was developed including a representation of the adhesive layers between the two adjoining profiles. Such a choice led to a highly refined mesh, with typical element length lower than 0.5mm. A detail of the finite element model is shown in Fig. 9(a). The corrugated composite profiles were modelled by using laminated bi-linear shell elements (type S4⁽³⁸⁾), which were characterised with the lay-up of three 0/90 oriented fabric plies having the properties reported in Table 1. Solid elements (type C3D8R⁽³⁸⁾) were used to represent the adhesive layers, which were extended to fill the cusps between the adjoining profiles. An isotropic elastic material, with Young’s modulus E _AD = 3.5GPa and a Poisson’s ratio of 0.33, was considered for the adhesive. The wires were modelled by means of the same beam element (type B31⁽³⁸⁾) and the same temperature-varying stiffness and CTEs, calibrated as discussed in the previous sub-section. A length of 0.3mm was adopted for the wire mesh.

Figure 9. Finite element model of the actuated laminate: (a) detail of the mesh, (b) longitudinal normal stress for the 120 wires/m version with tubed wires, (c) shear stress in material axis for the 120 wires/m version with tubed wires and [+45/−45/+45] lay-up and (d) stress state in the adhesive for the specimen with 80 wires/m and bonded SMA actuators.

The meshes of the wires were produced separately and connected to the rest of the model. To model the specimen versions with wires passing through channels between the corrugated profiles, the nodes at the wire ends were included in two rigid bodies created with all the nodes of the shells and the bricks at the ends of the corrugated laminate. No additional interactions were introduced between the wires and the adhesive layers. For the version with bonded SMA actuators, only the rigid body at the fixed end was defined, without including the wire nodes. Then, surfaces were created for each wire, including all the wire elements. A surface-based constraint (tie constraint⁽³⁸⁾) was enforced between such surfaces and the nodes of the adhesive elements, with a radius of influence of 0.8mm.

Analyses were conducted by raising the temperature from an initial condition of $21^{\circ}\mathrm{C}$ to a final value of $147^{\circ}\mathrm{C}$. Non-linear geometrical effects were taken into account.

3.4. Numerical results and discussion

The actuation pulses on the demonstrators with 0/90 lay-ups were analysed using the finite element models. The equivalent CTE attributed to the wires led to a contraction of the whole model, as shown in Fig. 9(b), referred to the contour of the normal stress in longitudinal direction, at the maximum contraction, for the analyses of the 120 wires/m element with tubed wires. Moreover, the same wire configuration was also adopted to perform an analysis with a [+45/−45/+45] lay-up, which yielded the contour shown in Fig. 9(c), referred to the shear stress in material axis. Such analysis was carried out to assess the performance of a different type of lay-up, although it was not included in the experimental assessment.

The quantitative results obtained with 0/90 lay-ups were analysed and compared with the analytical predictions in Table 3, where L _f/p = 0.5 and L _f/p = 1.05 refer to the cases with bonded and tubed wires, respectively. The value of 1.05 was adopted since, in the final design of the demonstrator, wires resulted longer than the three times the cell pitch, due to integration reasons. Two additional analyses were conducted for the cases with tubed SMA wires, by reducing the Young’s modulus attributed to the adhesive, E _AD, to 0.1GPa, in order to investigate the influence of the adhesive stiffness on the results.

Table 3 Numerical results

The displacements obtained from the numerical models are in good agreement with the analytical model predictions, although the stress in the SMA wires obtained from the FE analyses were higher than those predicted by the analytical models. However, the results obtained by reducing the adhesive Young’s modulus to 0.1GPa show that most of the discrepancy can be attributed to the stiffening effect of the adhesive on the twin corrugated laminates. The influence of such an effect on the displacement was very limited, since the K _wire/K _corr was very high for the configurations considered, and so the displacement u was very close to the asymptotic limit u ^T (Figs. 6 and 7). Nevertheless, it can be assumed that the presence of the adhesive in the cusps may oppose the bending of the composite laminates in those regions, and this may increase the overall stiffness of the laminate. Indeed, if the Young’s modulus of the adhesive is reduced, the discrepancies between the numerical and analytical stress levels decreases from about 34% to 11%. Overall, the comparison with the FEM analyses confirms the validity of the analytical predictions.

According to the numerical results, the solution with bonded wires was characterised by stress levels in the composite and in the adhesive higher than those in the configuration with tubed wires. Indeed, for the bonded wire case, a significant state of stress was detected in the wire egress region, as shown in Fig. 9(d). The stress values, with peaks higher than 50MPa, indicate that a physical integration based on bonding in a real application requires an adhesive adapted to high-temperature applications to guarantee long-term performance. Moreover, the analysis shows that the wires in the egress region are subjected to a contraction, which may provide an additional contribution to the overall displacement.

Finally, the results obtained with a [+45/−45/+45] lay-up confirm that the adoption of different types of lay-up is possible. In the analysis performed, the shear stress in material axis remained limited to 20MPa, as shown in Fig. 9(c), a value that does not indicate the risk of activation of an inelastic matrix-dominated response in the fabric plies. The deformed shape and the maximum contraction (2.98mm) are analogous to those obtained with the 0/90 oriented plies.

4.1. Configuration with bonded wires

A simply corrugated laminate panel with the selected 0/90 lay-up with three plies and the profile shown in Fig. 2 was produced following the technique presented in Ref. (Reference Airoldi, Fournier, Borlandelli, Bettini and Sala6), based on the use of a metallic mould and an elastomeric counter-mould (Airpad^TM). Curing was carried out in a heated plate press, at a temperature of 125°C and an applied pressure of 3 bar. A series of elements were cut from the laminate, with a width w = 75mm and a length of 76mm, including three cells. Such elements were bonded together to manufacture the demonstrators of the actuated corrugated panel. The first configuration was produced with $\rho_{wire}=80 \mathrm{wires/m}$, corresponding to six SMA wires, of the same type as characterised in Section 2. SMA actuators with an initial elongation $\varepsilon^{I}=4\%$ were bonded between the two simply corrugated elements.

The technological process is described in Fig. 10(a). A structural epoxy-based toughened adhesive (Hysol^® 9497) was used, mixed with 0.1mm-diameter microspheres, to ensure thickness and a more viscous consistency of the mixture. Actually, the type of adhesive selected may not be adapted for bonding the wires in real applications, due to the relevant stress observed in the FE analysis, which could lead to a significant reduction of fatigue performance at high temperature^{(Reference Beber, Schneider and Brede40)}. Adhesives more suited for medium/high-temperature applications, such as those indicated in Ref. (Reference Da Silva41), could be used. However, the adoption of an epoxy-based toughened adhesive was considered adequate for these technological trials, aimed to a preliminary assessment of bonding as a method to integrate the SMA wires in the proposed concept. After the application of the adhesive paste, the SMA wires were positioned on one of the composite elements, by applying a slight tension. Then, the other element was positioned and pressed by using a plate. The curing process was carried out at $22^{\circ}\mathrm{C}$ for 7days, to avoid any activation of the SMA wires and to achieve optimal mechanical properties of the adhesive. The result is presented in Fig. 10(b). It can be seen that the adhesive filled the cusps formed by the adjoined profiles along the bonding line, thus confirming the adoption of the nominal ratio L _f/p = 0.5 used in the analytical and numerical models.

Figure 10. Actuated corrugated laminate with bonded SMA wires (80 wires/m): (a) manufacturing process, (b) specimen produced.

The specimen was tested on a purpose-built tool, designed to measure the contraction achieved in the condition of minimum friction, shown in Fig. 11. One end of the twin corrugated laminate was gripped between two bars fixed on a metallic plate. Screw terminals were used to connect the SMA wires to electrical cables. The displacement was monitored using a laser transducer, which measured the variation of the distance between a thin fixed vertical plate and the moveable end of the specimen.

Figure 11. Experimental set-up for the measurement of contraction.

The actuator system was controlled by a power switch, driven by a Direct-Current (DC) power supply. This allowed tuning of the heating and cooling time of SMA wires by using a square wave. The input waveform and the displacement time history were managed by running LabVIEW DAQ software. In this first test, actuation was performed with three actuation pulses at 0.9A per wire with a duration of 5s, followed by an interval of 15s during which no current was provided, to allow the cooling of the SMA wires. The displacement time history recorded is provided in Fig. 12. During the first cycle, a displacement of 1.43mm was measured. However, the initial length was not completely recovered after the cooling, with a residual contraction of about 0.3mm. Actually, after the first peak, the following cycles obtained a stabilised contraction with an average value of about 1.11mm.

Figure 12. Displacement history for the twin corrugated with bonded SMA wires (80 wires/m).

The predictions of the analytical and the numerical models for this configuration, obtained by applying Eq. 9 with L _f/p = 0.5, were 1.38mm and 1.48mm, respectively. The results of the first peak was in acceptable agreement with such values, especially if the uncertainties related to the behaviour of the wire part embedded into the adhesive layer are considered. Since the maximum stress predicted by Eq. 12 was 92MPa, which is a relatively low value, it is probable that the stress exerted during the elastic recovery of the corrugated laminate decreased too early, thus opposing complete recovery of the initial length.

4.2. Configurations with tubed SMA wires and different wire densities

A second configuration of the actuated corrugated laminate was developed with SMA wires passing through PTFE tubes bonded between the profiles. Such a solution presents some appealing aspects with respect to the solution with wires bonded, which has several disadvantages. First, the bonding reduces the free length of the actuation wires and consequently the contraction that can be achieved. Besides, the positioning of the wires requires care to avoid detwinning, and a slow room-temperature bonding process is required to avoid activation. Moreover, in a real application, wires could not be replaced in case of breakage. Finally, the adhesive would be directly subjected to heating cycles in operational conditions, which could degrade the mechanical properties^{(Reference Beber, Schneider and Brede40)}. On the contrary, an integration solution that allows the SMA wires to slide through the bonded zones eliminates most of the aforementioned drawbacks, although it is characterised by higher stress levels in the wires and can exert forces only at the ends of the panels.

Two versions based on this integration method were produced, with 80 wires/m and 120 wires/m. The technological process (see Fig. 13(a)) was basically the same as performed for the production of the version with bonded wires. In this case, the deposition of the SMA wires was replaced by the deposition of PTFE tubes of about 0.8mm external diameter. To avoid the collapse of such tubes during the curing process, steel wires with a diameter of 0.3mm were introduced into these polymeric tubes. A hot-temperature curing process was carried out at $80^{\circ}\mathrm{C}$ for 1.5h followed by 1day at $22^{\circ}\mathrm{C}$. Thereafter, the steel wires were removed and the SMA wires, elongated with a pre-strain $\varepsilon^{I}=4\%$, were introduced into the channels.

Figure 13. Actuated corrugated composite with SMA wires passing through tubes: (a) manufacturing process and (b) specimen produced with 120 wires/m.

The SMA wires were connected to electrical cables by means of screw terminals, serving also as a mechanical constraint for the wires inserted into the tubes. A thin aluminium plate (0.5mm thick) was interposed between the specimen and the terminals to provide a plane restraint surface for the terminals. In this second configuration, the working length of the wire practically coincides with the whole length of the corrugated specimen, and was slightly higher than three periods, so that L _f/p = 1.05. The double corrugated laminate tested is presented in Fig. 13(b) in the version with 120 wires/m, corresponding to nine wires.

Application of Eq. 9 provided an estimation of the contraction u = 2.79mm and u = 2.86mm for the configuration with 80 wires/m and 120 wires/m, respectively. The predictions of the FEM models were very close to such values, as shown in Table 3.

Tests were carried out following a procedure analogous to that presented for the bonded version, though an actuation pulse of 7s was adopted, followed by an interval of 150s. In the first tests with the 80 wires/m configuration, conducted with a current of 0.9A/wire, a contraction of about 2.0mm was achieved, which was significantly lower than the expected value. Actually, the configuration with L _f/p = 1.05 and 80 wires/m is characterised by a very high expected wire stress of 186MPa according to Eq. 12 or 249MPa according to the finite element model. Therefore, the limited value of contraction could be explained by considering the increment of A _s and A _f with the stress, as presented in Fig. 5(b). Such increment, and the loss of heat at the wire–PTFE tube interfaces, opposes the complete transformation in austenite, which would have required a greater amount of heat to reach the necessary temperature. To verify this hypothesis, single-cycle tests were conducted on the specimen by increasing the current up to 1.4A/wire, with the results provided in Fig. 14. It can be seen that the peak displacement reached the expected value for a current higher than 1.2A/wire. The fully developed plateau during the 7s of actuation phase indicates that full actuation was achieved. Actuation tests with three cycles of actuation at 1.3A/wire confirmed an average contraction of 3.1mm. It should be observed that, above a current of 1.3A/wire, the specimen underwent evident overheating. Considering the rearming phase, the initial length of the wire was almost completely recovered in all the tests. This result is certainly influenced by the initial stress value, which was significantly higher than in the specimen with bonded wires. However, it is worth noting that the complete recovery required a time longer than the 15s made available in the experiment presented in Fig. 12.

Figure 14. Displacement at different current intensities for the specimen with tubed wires and 80 wires/m.

For the specimen with 120 wires/m, the best results were achieved with a square wave of current with an amplitude of 1.0A/wire and a heating time of 7s, followed by an interval of 150s (Fig. 15). Tests confirmed an average displacement peak value of 3.18mm, which was slightly higher than the predictions of the analytical and numerical models. In general, it can be observed that, in the experiments with tubed wires, the contraction at full actuation was slightly higher than the analytical and numerical predictions, and this could be related to the presence of an additional elongation induced in the wire during handling or at the moment of insertion into the PTFE tubes, which increased $\varepsilon^{I}$ beyond the nominal 4% level. For the case with bonded wires, presented in Fig. 12, such an effect could have been compensated by the contraction of the wire in the adhesive layer, as evidenced in FEM analyses. It can be observed that the configuration with 120 wires/m passing through the PTFE tubes achieved full actuation with 1.0A/wire, without the need of further increasing the current intensity, as in the case presented in Fig. 14. This is probably related to the relatively low wire stress, which is 128MPa according to the analytical model and 172MPa in the finite element model, which was found to be influenced by the stiffness attributed to the adhesive in the cusps. These levels are lower than the ones for the 80 wires/m configuration and probably did not lead to an excessive increment of transformation temperature. However, the stress levels resulted adequate to recover almost completely the initial length during the cooling phase, as shown in Fig. 15, though the recovery was complete only after 100s after the termination of the actuation pulse. The initial and contracted configuration of the specimen are shown in the pictures presented in Fig. 16.

Figure 15. Displacement history for the double corrugated laminate with 120 wires/m (actuation pulse = 1A/wire)

Figure 16. Actuation experiments performed on the specimens with tubed SMA wires and 120 wire/m: (a) initial state, (b) fully actuated state.

Overall, the tests performed confirmed the expected behaviour of the actuated element. It can be observed that actuation was almost completely obtained in a relatively short time (about 1s), although the amperage required can be significant. Such performance is highly influenced by the wire diameter, since small wires are quickly heated but are characterised by a high convective heat transfer coefficient^{(Reference Balakrishnan, Dinh, Phan, Dao and Nguyen42)}. Since SMA elements are certainly suited for low-frequency actuation, a reduction of the heat transfer coefficient and an increment of the actuation time could be acceptable for applications. Such aspects, and the possibility of controlling the temperature inside the corrugated cells through appropriate strategies, could be exploited to reduce the amount of current required by the proposed actuated system.