Effect of pseudoelastic hysteresis of shape memory alloy springs on the aeroelastic behavior of a typical airfoil section

Abstract

The literature on aeroelasticity includes studies on the use of smart materials as sensors and actuators in vibration control problems. Although different smart materials are available, shape memory alloys have received growing attention in aerospace applications. The hysteretic response of shape memory alloys exhibiting pseudoelasticity provides energy dissipating and damping capabilities for these materials, and therefore, the effectiveness of the pseudoelastic behavior of shape memory alloys has been investigated for passive structural vibration control. However, its effect on the aeroelastic behavior of lifting surfaces has not been covered in the literature. Hence, this article addresses the modeling and analysis of a 2-degree-of-freedom typical aeroelastic section with shape memory alloy springs introduced through the pitch degree of freedom. A state-space model is employed for the simulations of the coupled system, and a two-state approximation to Theodorsen aerodynamics is used for the determination of the aerodynamic loads. The effects of the hysteretic behavior of the shape memory alloy springs on the aeroelastic behavior of the typical section are investigated at the flutter boundary and at post-flutter regime.

Keywords

shape memory alloy pseudoelastic hysteresis aeroelasticity

Introduction

Several research groups have investigated the use of smart materials in vibration control problems. In general, the main control techniques using smart materials are the passive and active ones. Although different smart materials are available, the shape memory alloys (SMAs) have received growing attention in passive control since their hysteretic behavior provides energy dissipating and damping capabilities. The stress-induced hysteretic behavior of SMAs is due to austenite–martensite phase transformations observed when the alloy is subjected to cyclic stress–strain loading above the austenite finish temperature (Lagoudas, 2008).

The effectiveness of the pseudoelastic hysteresis of SMAs for passive structural damping augmentation has been investigated in the literature. The energy dissipation associated with the superelastic plateau of the martensitic phase has been employed to enhance the vibration performance and safety of civil structures (Alam et al., 2007; Janke et al., 2005; Saadat et al., 2002). Bachmann et al. (2012) investigated the use of piezoelectric damping and also SMA damping for noise and vibration control in novel open rotor aircraft engines with composite fan blades. The results show that embedded SMA wires (in the low-temperature martensitic phase) can passively improve the mechanical damping of the host structure. Moreover, piezoelectric shunt damping increases the damping of the blade further than the SMA with the drawback of more additional weight for the structure. Gandhi and Chapuis (2002) presented an experimentally validated finite element model of a clamped-free beam with tip mass and symmetrically mounted pseudoelastic SMA wires. The influence and effectiveness of the SMA damper configuration with varying excitation force amplitudes were investigated. Rezaei et al. (2012) investigated the nonlinear free vibration and damping of a clamped–clamped Euler–Bernoulli beam with embedded pre-strained SMA wires employed in the pseudoelastic state.

SMA-based vibration isolation devices with adaptive capabilities have also been reported in the literature (Aguiar et al., 2013; Ibrahim, 2008; Khan et al., 2004; Mishra et al., 2013; Savi et al., 2011). These adaptive absorbers simultaneously use the variation of mechanical properties due to temperature-induced phase transformations and energy dissipation related to the hysteretic behavior related to stress-induced transformations (Aguiar et al., 2013; Savi et al., 2011).

Several constitutive models have described the behavior of SMAs. The phenomenological models assume previously known mathematical functions to estimate the martensite fraction of an SMA. The functions are based on experimental stress–strain tests and can be adapted for different SMA behaviors. The classical one dimensional (1D) model of Tanaka et al. (1986) uses an exponential function, and the hardening effect is ignored during phase transformation in the stress–strain relation. Later, a cosine function was proposed by Liang and Rogers (1990), which was also adapted by Brinson (1993). Brinson (1993) models the martensite fraction as stress-induced and temperature-induced fractions enabling the modeling of the shape memory effect (SME) at low temperatures.

Researchers have also proposed various models to represent the behavior of helical springs made from SMAs, which range from simple equivalent 1D models (Aguiar et al., 2010, 2013; An et al., 2012; Baghani et al., 2012; Liang and Rogers, 1997) to accurate solutions considering the complex stress and strain distributions in the springs (Lagoudas et al., 2012; Mirzaeifar et al., 2011; Saleeb et al., 2013). The literature shows that simple equivalent models have good correspondence to experimental results for springs with large spring index and small helix angle (Aguiar et al., 2010; An et al., 2012). Recently, Enemark et al. (2014) presented an experimentally verified modeling of an SMA spring. Brinson’s model was employed to describe the behavior of the SMA spring with two modifications: a new smooth hardening function and a new sub-loop function.

The literature has also reported on the use of SMAs in aerospace problems. Barbarino et al. (2014) present a comprehensive review on SMA actuators to morphing aircraft. Hartl and Lagoudas (2007) review the aerospace applications of SMA actuators in fixed-wing and rotorcraft cases, including the use of SMAs in propulsion systems (Calkins et al., 2006; Hartl et al., 2010). In the particular case of aeroelastic investigations, research includes the control of the dynamic properties of structural panels of aircraft with SMA elements in order to change the flutter behavior (Tawfik et al., 2002). Barzegari et al. (2012) present the modeling and analysis of a cantilever wing with embedded SMA wires. The authors report improvements in the aeroelastic behavior (increasing stability margin and decreasing limit cycle amplitudes) due to the induced in-plane load of the SMA wires.

This article investigates the effect of hysteretic response of SMAs on the dynamic aeroelastic behavior of a 2-degree-of-freedom (2DOF) typical section. SMA springs are introduced through the pitch DOF. The well-known lumped parameter aeroelastic equations governing a 2DOF typical section (Hodges and Pierce, 2011) are modified in order to include the energy dissipation capability of the SMA springs. Brinson’s (1993) model is employed to describe the behavior of the SMA, and the helical spring model follows Liang and Rogers (1997). Thermal coupling is not included in the modeling of this article. Isothermal conditions are assumed since the SMA springs mostly operate under elastic regime. Although inelastic regimes are achieved, phase transformations are incomplete, and therefore, negligible temperature variations are assumed. A two-state approximation to Theodorsen aerodynamics is used in order to determine the unsteady aerodynamic loads. The effects of the hysteretic behavior of SMA springs under different preload levels on the aeroelastic behavior of the typical section are investigated at the linear flutter speed of the typical aeroelastic section and later at the post-flutter regime.

Mathematical models

This section presents the modeling of the typical aeroelastic section with SMA springs introduced to the pitch DOF. First, the model of SMA springs is presented. Although the recent literature includes models representing the complex stress and strain distributions in SMA springs (Lagoudas et al., 2012; Mirzaeifar et al., 2011), a simple model is assumed in this work since the main focus is to discuss the general behavior of a dynamic system (in this case an aeroelastic system) with SMA springs, in agreement with the simplified analyses followed by others (An et al., 2012; Barzegari et al., 2012; Bil et al., 2013; Hadi et al., 2010; Rezaei et al., 2012). Later, the governing equations of a typical aeroelastic section with steel springs are presented along with the required modifications to include SMA springs.

Model of an SMA coil spring

As stated in Mirzaeifar et al. (2011), simple spring models are acceptably accurate for some practical elastic springs and for describing the global force–displacement behavior of SMA springs. Curvature effects should be considered for precise representation of stress and martensite fraction distributions (Lagoudas, 2008; Mirzaeifar et al., 2010, 2011). Since the main motivation in this article is to investigate the behavior of an aeroelastic system in the presence of pseudoelastic hysteresis, the hypotheses of linear shear stress distribution and constant phase transformation in the cross-sectional area of the SMA spring are assumed, in agreement with the assumptions followed by others (Aguiar et al., 2010, 2013; An et al., 2012; Barzegari et al., 2012; Bil et al., 2013; Hadi et al., 2010; Rezaei et al., 2012) where the global model dynamics is the main focus rather than the precise SMA internal state.

In this article, the martensite fraction for an SMA undergoing phase transformation is given by Brinson’s (1993) equations and is denoted by $ξ$ . The austenite to martensite transformation ( $A \to M$ or forward) due to mechanical loading is given by

ξ^{A \to M} = \frac{1 - ξ_{0}}{2} \cos {\frac{π}{σ_{s}^{min} - σ_{f}^{\min}} [σ - σ_{f}^{M}]} + \frac{1 + ξ_{0}}{2}

(1)

when the mechanical stress ( $σ$ ) is between the martensite start stress ( $σ_{s}^{M}$ ) and the martensite finish stress ( $σ_{f}^{M}$ ). Such critical stresses are given by

σ_{s}^{M} = σ_{s}^{\min} + C_{M} (T - M_{s})

(2a)

σ_{f}^{M} = σ_{f}^{\min} + C_{M} (T - M_{s})

(2b)

which include minimum stresses for the onset ( $σ_{s}^{\min}$ ) and completion ( $σ_{f}^{\min}$ ) of forward transformation. Equations (2a) and (2b) show that the critical stresses increase with increasing temperature ( $T$ ) above the martensite start temperature ( $M_{s}$ ) proportionally to the material constant $C_{M}$ .

The martensite to austenite transformation ( $M \to A$ or reverse) during mechanical unloading is given by

ξ^{M \to A} = \frac{ξ_{0}}{2} {\cos [\frac{π}{A_{f} - A_{s}} (T - A_{s} - \frac{σ}{C_{A}})] + 1}

(3)

for stress values between the austenite start ( $σ_{s}^{A}$ ) and austenite finish ( $σ_{f}^{A}$ ) stresses. Such critical stresses are given by

σ_{s}^{A} = C_{A} (T - A_{s})

(4a)

σ_{f}^{A} = C_{A} (T - A_{f})

(4b)

where $A_{s}$ is the austenite start temperature, $A_{f}$ is the austenite finish temperature, and $C_{A}$ is the material constant which relates the critical stresses for reverse transformation to the temperature of the material.

The initial value of the martensitic fraction ( $ξ_{0}$ in equations (1) and (3)) is defined at the beginning of each transformation (forward or reverse). Since this value assumes the current value of the martensitic fraction (prior to the onset of transformation), the presence of some initial martensite (e.g. due to incomplete transformations) can be accounted for.

Based on the mechanical constitutive relation of Brinson (1993) and on the pure shear criterion (Liang and Rogers, 1992, 1997), a 1D shear stress and strain relation for SMAs can be given as

τ - τ_{0} = G (ξ) γ - G (ξ_{0}) γ_{0} + \bar{Ω} (ξ) ξ - \bar{Ω} (ξ_{0}) ξ_{0} + \bar{Θ} (T - T_{0})

(5)

where $τ$ is the shear stress, $G (ξ)$ is the shear modulus, $γ$ is the shear strain, $\bar{Ω} (ξ)$ is the phase transformation strain factor, $\bar{Θ}$ is related to the thermal expansion coefficient, and subscript “0” denotes the initial state. Equation (5) is modified to

τ = G (ξ) γ + \bar{Ω} (ξ) ξ

(6)

when assuming an austenitic phase, stress-free and strain-free initial conditions ( $ξ_{0} = τ_{0} = γ_{0} = 0$ ), constant temperature throughout the loading and unloading processes ( $T = T_{0}$ ), as well as considering only the pseudoelastic effect (which requires $T \geq A_{f}$ and thus allows the temperature-induced martensitic fraction to be neglected).

The shear stress ( $τ$ ) is related to the axial stress ( $σ$ ) by von Mises criterion for pure shear given as $σ = \sqrt{3} τ$ (Liang and Rogers, 1992, 1997), which is used for the critical stresses as well (i.e. $τ_{s}^{M} = σ_{s}^{M} / \sqrt{3}$ and so on). The martensitic fraction–dependent shear modulus is given by

G (ξ) = \frac{D (ξ)}{2 (1 + μ_{P})}

(7)

where $μ_{P}$ is Poisson’s ratio and $D (ξ)$ is Young’s modulus for an arbitrary martensitic fraction. The modulus is given by

D (ξ) = D_{A} + ξ (D_{M} - D_{A})

(8)

where $D_{M}$ is the fully martensitic modulus and $D_{A}$ is the fully austenitic modulus (Tanaka et al., 1986).

The transformation strain is given by

\bar{Ω} (ξ) = - G (ξ) ε_{res}

(9)

where $ε_{res}$ is the SMA maximum residual strain.

Based on conventional spring design, Liang and Rogers (1997) expressed the deflection of an SMA spring as

y = \frac{2 π R^{2} N}{r} γ

(10)

where $R$ is the mean spring radius, $r$ is the spring wire radius, $N$ is the number of active coils, and $γ$ is obtained from equation (6) as

γ = \frac{τ}{G (ξ)} - \frac{\bar{Ω} (ξ)}{G (ξ)} ξ

(11)

which accounts for the elastic and inelastic strains.

For a known stress input, the martensitic fraction can be calculated from equation (1) or (3) (depending on the transformation), the shear strain can be obtained from equation (11), and the spring deflection from equation (10). However, in the aeroelastic problem considered in this study, the stress is unknown, whereas the spring deflection is known (to be calculated at each time step). Since the stress depends on the martensitic fraction and vice versa, equation (6) is solved iteratively in this work. First, the shear strain is calculated by solving equation (10) for $γ$ . The shear strain is used a priori to estimate the shear stress with an elastic relation ( $τ = G γ$ ). The stress estimation is used only to identify whether the spring is operating in the elastic regime or in the inelastic regime (undergoing transformation). The inelastic regime is identified for a forward transformation if $τ_{s}^{M} \leq τ \leq τ_{f}^{M}$ and the mechanical loading is increasing. The inelastic regime is also identified for a reverse transformation if $τ_{s}^{A} \geq τ \geq τ_{f}^{A}$ and the mechanical loading is decreasing. If a transformation is identified, stress values (denoted by $τ_{test}$ ) in the corresponding transformation range are assumed (i.e. $τ_{test} = [τ_{s}^{M}, τ_{f}^{M}]$ or $τ_{test} = [τ_{s}^{A}, τ_{f}^{A}]$ and a discretization on the order of $10^{- 5} τ_{s}^{M}$ is suggested). Values for the martensitic fraction (denoted by $ξ_{test}$ ) are calculated for the stress range being considered (using equation (1) or (3)). Each pair of stress and corresponding martensitic fraction is verified in the following equation

τ_{test} - [G (ξ) γ + \bar{Ω} (ξ) ξ_{test}] = 0

(12)

and the pair ( $τ_{test}, ξ_{test}$ ) which satisfies equation (12) is assumed as the previously unknown shear stress and martensitic fraction. Having obtained these quantities, the martensitic fraction–dependent parameters can be updated.

Liang and Rogers (1997) report the stiffness of an SMA spring as

k_{c} (ξ) = \frac{r^{4}}{4 R^{3} N} G (ξ)

(13)

which will be used in this article to calculate the equivalent stiffness of the pitch DOF and the elastic component of the SMA springs. The inelastic component can also be obtained since the shear stress at the SMA wire’s perimeter was assumed in the model derivation. The shear stress at the outer diameter can be expressed as

τ = \frac{2 R}{π r^{3}} F

(14)

where $F$ is an external force. The force is related to shear stress by the expression $π r^{3} / 2 R$ in equation (14). Using the same expression, the inelastic term in equation (6) ( $\bar{Ω} (ξ) ξ$ ) becomes a force-related term given by

Y (ξ) = \frac{π r^{3}}{2 R} \bar{Ω} (ξ) ξ

(15a)

Y (ξ) = - \frac{π r^{3}}{2 R} G (ξ) ξ ε_{res}

(15b)

when equation (9) is combined with equation (15a).

By substituting equation (14) into equation (11) and then equation (11) into equation (10) and rearranging the resulting equation (which can be found in Liang and Rogers (1997))

f^{sma} = k_{c} (ξ) y + Y (ξ)

(16)

gives the force of an SMA spring. In the next item, the spring deflection will be expressed as a function of the pitch angle.

Aeroelastic model associated with SMA springs

A typical aeroelastic section can provide a useful insight into the behavior of an aerodynamic surface under airflow excitation. Lumped parameter-based derivation of the governing equations for a typical section can be found on textbooks on aeroelasticity such as Bisplinghoff et al. (1996) and Hodges and Pierce (2011). Figure 1(a) shows the schematic of a 2DOF typical section. The plunge and pitch displacement variables are denoted by $h$ and $α$ , respectively. The plunge displacement is measured at the elastic axis (positive in the downward direction), and the pitch angle is measured about the elastic axis (positive in the clockwise direction). In addition, $b$ is the semichord length of the airfoil section, $x_{α}$ is the dimensionless chordwise offset of the elastic axis from the centroid (CG), $k_{h}$ is the stiffness per length (in the span direction) in the plunge DOF, $k_{α}$ is the stiffness per length in the pitch DOF, $d_{α}$ is the damping coefficient per unit length in the pitch DOF, $d_{h}$ is the damping coefficient per unit length in the plunge DOF (both omitted in Figure 1 for clarity), $U_{\infty}$ is the airflow speed, $M_{α}$ is the aerodynamic moment per unit length, and $L$ is the aerodynamic lift force per unit length.

Figure 1(b) shows the modified typical section investigated in this article. The torsional pitch spring is replaced by a pair of SMA coil springs (denoted by $S_{1}$ and $S_{2}$ in Figure 1(b)), each one attached to the end of a rigid bar (assumed without mass) passing through the elastic axis. Please note that the plunge spring was omitted in Figure 1(b) for clarity. The distance from each end of the bar to the elastic axis is $w$ (and the length of the bar is $2 w$ ).

This article presents an extension of the work by Sousa et al. (2011), who considered a typical section model for wind energy harvesting purposes. In the referred article, only steel springs were considered, and piezoelectric coupling was added to the plunge DOF. The piezoaeroelastic model was cast into state-space representation and successfully verified against experimental results. In this work, the electromechanical coupling is not considered, and a pair of SMA coil springs (denoted by $S_{1}$ and $S_{2}$ in Figure 1(b)) is modeled in the pitch DOF (replacing the torsional steel spring).

The equations of motion for the typical section (with steel springs) of Figure 1(a) can be given as

I_{α} \overset{\cdot\cdot}{α} + {mx}_{α} b \overset{\cdot\cdot}{h} + d_{α} \overset{\cdot}{α} + k_{α} α = M_{α}

(17a)

{mx}_{α} b \overset{\cdot\cdot}{α} + (m + m_{f}) \overset{\cdot\cdot}{h} + d_{h} \overset{\cdot}{h} + k_{h} h = - L

(17b)

where $m$ is the airfoil mass per length (in the span direction), $m_{f}$ is the effective fixture mass (connecting the airfoil to the plunge springs—it is zero in the ideal representation of Figure 1) per length, and $I_{α}$ is the moment of inertia per length.

Equations (17a) and (17b) are expressed in a dimensionless form as

r_{α}^{2} α " + x_{α} \bar{h} " + ζ_{α} α' + η_{α}^{2} r_{α}^{2} α = {\bar{M}}_{α}

(18a)

x_{α} α " + μ \bar{h} " + ζ_{h} \bar{h}' + \bar{h} = - \bar{L}

(18b)

where $\bar{h} = {hb}^{- 1}$ is the plunge displacement in semichords, $μ = (m + m_{f}) m^{- 1}$ is the ratio of the translating mass to the rotating mass, $r_{α} = {(I_{α} m^{- 1} b^{- 2})}^{1 / 2}$ is the dimensionless radius of gyration, $ζ_{α} = d_{α} {({mb}^{2} ω_{h})}^{- 1}$ is the damping ratio of the pitch DOF, and $ζ_{h} = d_{h} {(m ω_{h})}^{- 1}$ is the damping ratio of the plunge DOF. The uncoupled and undamped pitch and plunge natural frequencies are $ω_{α} = {(k_{α} I_{α}^{- 1})}^{1 / 2}$ and $ω_{h} = {(k_{h} m^{- 1})}^{1 / 2}$ , respectively. The pitch-to-plunge frequency ratio is $η_{α} = ω_{α} ω_{h}^{- 1}$ . The dimensionless aerodynamic moment is ${\bar{M}}_{α} = M_{α} {({mb}^{2} ω_{h}^{2})}^{- 1}$ , and the dimensionless aerodynamic lift is $\bar{L} = L {(mb ω_{h}^{2})}^{- 1}$ . A dimensionless time is defined as $\bar{t} = ω_{h} t$ , where $t$ is the dimensional time. The time derivatives are $\overset{\cdot}{x} = ω_{h} x'$ and $\overset{\cdot\cdot}{x} = ω_{h}^{2} x "$ , where the overdot denotes differentiation with respect to $t$ , the prime denotes differentiation with respect to $\bar{t}$ , and $x$ represents an arbitrary time-dependent variable.

Considering the SMA springs configuration of Figure 1(b), the dimensionless elastic moment term in equation (18a) ( $η_{α}^{2} r_{α}^{2} α$ ) may be replaced by

ϑ (α, ξ_{1}, ξ_{2}) = \frac{w}{{mlb}^{2} ω_{h}^{2}} [- f_{1}^{sma} (α, ξ_{1}) + f_{2}^{sma} (α, ξ_{2})]

(19)

which is the equivalent dimensionless moment due to the SMA springs; $ξ_{i}$ is the martensitic fraction of each spring, and the subscript $i$ assumes 1 or 2 for each SMA spring ( $S_{1}$ or $S_{2}$ ) of Figure 1(b). The forces $f_{i}^{sma}$ of each SMA spring are calculated from equation (16) as

f_{i}^{sma} = k_{c} (ξ_{i}) y_{i} (α) + Y (ξ_{i}), i = 1 or 2

(20)

and the axial deflections of SMA springs $S_{1}$ and $S_{2}$ , respectively, are

y_{1} (α) = y_{0} - w α

(21a)

y_{2} (α) = y_{0} + w α

(21b)

where $α$ is the pitch displacement and $y_{0}$ is the pre-deflection of the springs that results in mechanical preload (i.e. bias stress). The opposite signs (negative or positive) refer to the out-of-phase deformation of the springs $S_{1}$ and $S_{2}$ .

Although in this work the steel pitch spring (Figure 1(a)) is replaced by the SMA springs of Figure 1(b), one should keep the same elastic moment (for the steel spring or for the SMA springs) for comparison purposes. In order to obtain the same elastic moment, the distance ( $w$ ) from each coil spring to the elastic axis can be given by

w = \sqrt{\frac{k_{α} l}{2 k_{c} (ξ)}}

(22)

where $k_{α}$ is the torsional stiffness per unit length of the typical section with steel torsional spring and $k_{c}$ is given by equation (13) for a fully austenitic phase. Therefore, the modified model exhibits the same aeroelastic behavior (the same flutter speed and flutter frequency as well as the same amplitudes and frequencies at any airflow speed) of the typical section with steel springs (assuming that all other parameters are not changed) when no stress-induced phase transformation is achieved in the SMA springs.

In this article, the unsteady aerodynamic loads (lift and aerodynamic moment) due to arbitrary motions of the airfoil section are obtained from Jones’ exponential approximation (Jones, 1938) of Wagner’s indicial function. Wagner’s function is an approximation to the generalized Theodorsen function able to represent arbitrary (e.g. non-harmonic) airfoil motions (Bisplinghoff et al., 1996; Theodorsen, 1935). The aeroelastic equations governing the typical section with SMA springs can be written in state-space form along with the state-space representation of the aerodynamic model proposed by Edwards et al. (1979). The model of Edwards et al. (1979) results in two augmented states in the aeroelastic equations. The state-space representation is convenient for numerical solution with traditional integration schemes (e.g. Runge–Kutta methods).

The state-space form is presented as

x' = Ax + {ax}_{c}

(23)

where $Ax$ accounts for the linear portion of the model (everything but the SMA springs) and ${ax}_{c}$ accounts for the nonlinear portion (the effects of the SMA springs). The aeroelastic state vector, $x$ , is defined as

x = {\begin{matrix} α & \bar{h} & α' & \bar{h}' & a_{aug} & a'_{aug} \end{matrix}}^{T}

(24)

where $a_{aug}$ is the augmented aerodynamic state. The state matrix $A$ is given by

A = [\begin{matrix} 0 & I & 0 \\ - M^{- 1} K & - M^{- 1} B & M^{- 1} \tilde{D} \\ {\tilde{I}}^{- 1} {\tilde{E}}_{1} & {\tilde{I}}^{- 1} {\tilde{E}}_{2} & {\tilde{I}}^{- 1} {\tilde{F}}_{p} \end{matrix}]

(25)

where $M$ , $K$ , and $B$ are related to mass, stiffness, and damping, respectively, and are given by

M = M_{s} - κ M_{nc}

(26a)

K = K_{s} - κ u^{2} (K_{nc} + 0.5 {RS}_{1})

(26b)

B = B_{s} - κ u (B_{nc} + 0.5 {RS}_{2})

(26c)

where subscript $s$ denotes the structural contribution, and the matrices $M_{s}$ , $K_{s}$ , and $B_{s}$ are given by

M_{s} = [\begin{matrix} r_{α}^{2} & x_{α} \\ x_{α} & μ \end{matrix}]

(27a)

K_{s} = [\begin{matrix} f_{K} & 0 \\ 0 & 1 \end{matrix}]

(27b)

B_{s} = [\begin{matrix} ζ_{α} & 0 \\ 0 & ζ_{h} \end{matrix}]

(27c)

and the term “ $f_{K}$ ” in equation (27b) can be changed according to the typical section model being considered. For a model with a torsional steel spring (as in Figure 1(a)) or with a pair of equivalent steel coil springs (similar to Figure 1(b))

f_{K} = r_{α}^{2} η_{α}^{2} = \frac{2 w^{2} k_{c}}{{mlb}^{2} ω_{h}^{2}}

(28)

and the term “ ${ax}_{c}$ ” is not considered in equation (23). This form is convenient for stability (eigenvalue) analysis. However, for a pair of SMA springs (as in Figure 1(b))

f_{K} = 0

(29)

and the term “ ${ax}_{c}$ ” is kept as is in equation (23).

The matrix $a$ is given by

a = [\begin{matrix} 0 & 0 & 0 \\ - M^{- 1} K_{1} & M^{- 1} K_{2} & 0 \\ 0 & 0 & 0 \end{matrix}]

(30)

and

x_{c} = {\begin{matrix} y_{1} (α) & 1 & y_{2} (α) & 1 & 0 & 0 \end{matrix}}^{T}

(31)

The matrices $K_{1}$ and $K_{2}$ (in equation (30)) are given by

K_{i} = \frac{w}{{mlb}^{2} ω_{h}^{2}} [\begin{matrix} k_{c} (ξ_{i}) & Y (ξ_{i}) \\ 0 & 0 \end{matrix}], i = 1 or 2

(32)

since SMA springs are modeled.

In equation (25), $I$ is the identity matrix, $\tilde{I} = ω_{h}^{- 1} I$ , $\tilde{D} = ω_{h}^{- 2} D$ , ${\tilde{E}}_{1} = ω_{h}^{- 2} E_{1}$ , ${\tilde{E}}_{2} = ω_{h}^{- 1} E_{2}$ , and ${\tilde{F}}_{p} = ω_{h}^{- 2} F_{p}$ . In equations (26a) to (26c), $κ = ρ b^{2} m^{- 1}$ , where $ρ$ is the local air density, and $u = ω_{h}^{- 1} b^{- 1} U_{\infty}$ is the dimensionless airflow speed, where $U_{\infty}$ is the (dimensional) airflow speed. Matrices $M_{nc}$ , $K_{nc}$ , $B_{nc}$ , $D$ , $E_{1}$ , $E_{2}$ , $F$ , $R$ , $S_{1}$ , and $S_{2}$ (in equations (25) to (26c)) represent the aerodynamic model and can be found in Edwards et al. (1979). Such matrices depend on the airflow speed, semichord length, and position of the elastic axis.

In this work, the equations are solved by a classical fourth-order Runge–Kutta method. At each time step, the pitch displacement is calculated and the spring deflection is obtained. The shear strain is calculated and the shear stress is updated through one of two possibilities. If the current stress is in the critical range (for forward or reverse phase transformations), the iterative procedure described in section “Model of an SMA coil spring” (equation (12)) is performed, and the SMA properties are updated (to be employed in the next time step of the aeroelastic solution). Otherwise, the martensite fraction and the martensite-dependent parameters remain unchanged. In such a case, the stress is obtained from the linear stress–strain relation of equation (6). This procedure is repeated for each SMA spring used in the aeroelastic model.

Case studies

This section reports three case studies using the previously described models. In the first case study, the constitutive behavior of an SMA is discussed. In the second case, the effect of preloaded SMA springs on the aeroelastic behavior of the typical section at the linear flutter speed is investigated. In the third case study, the effect of preloaded SMA springs on the aeroelastic behavior of the typical section in the post-flutter regime is investigated. The SMA parameters shown in Table 1 are used in the simulations of all cases. The parameters are based on Aguiar et al. (2013) and are estimated for a commercially available SMA spring that is suitable for the physical dimensions of an existing experimental typical section described in Sousa et al. (2011).

Table 1.

Thermomechanical properties of the SMA.

$M_{f}$	$302 K$	Martensite finish temperature
$M_{s}$	$315 K$	Martensite start temperature
$A_{s}$	$316 K$	Austenite start temperature
$A_{f}$	$331 K$	Austenite finish temperature
$C_{M}$	$4 MPa K^{- 1}$	Stress–temperature constant for martensite
$C_{A}$	$6 MPa K^{- 1}$	Stress–temperature constant for austenite
$σ_{s}^{\min}$	$100 MPa$	Stress for onset of detwinning at $T \leq M_{s}$
$σ_{f}^{\min}$	$170 MPa$	Stress for completion of detwinning at $T \leq M_{s}$
$ε_{res}$	$0.067$	Maximum residual strain
$G_{M}$	$11.5 GPa$	Martensitic shear modulus
$G_{A}$	$14.5 GPa$	Austenitic shear modulus
$μ_{P}$	$0.3$	Poisson’s ratio

SMA: shape memory alloy.

Thermomechanical behavior of the SMA spring

In this case study, the SMA behavior is described for different conditions of the material temperature and shear stress. The effect of temperature on transformation stresses is shown in Figure 2. The stress-induced phase transformation starts when $τ \geq τ_{s}^{\min}$ (where $τ_{s}^{\min} = σ_{s}^{\min} / \sqrt{3}$ ) and completes only for $τ \geq τ_{f}^{\min}$ (where $τ_{f}^{\min} = σ_{f}^{\min} / \sqrt{3}$ ). The critical stresses are constant when $T \leq M_{s}$ . For $T > M_{s}$ , the critical stresses increase with increasing temperature, and a linear relation is assumed.

Figure 3 shows the shear stress–strain behavior for different temperatures. In all cases, 100% initial austenitic phase is assumed. The curves for $T < A_{f}$ exhibit a residual strain after a complete unloading, which is related to the SME. The hysteretic behavior for $T \geq A_{f}$ is related to pseudoelasticity. As expected, the hysteresis loop moves upward in the stress–strain plane as the temperature increases. Additionally, the area inside the hysteresis loop decreases with increasing temperature due to higher values of $C_{A}$ with respect to $C_{M}$ (see Table 1).

The particular case of $T = A_{f}$ (in Figure 3) is detailed in Figure 4. The stress represents a mechanical loading–unloading cycle. As the stress increases from $τ_{s}^{M}$ up to $τ_{f}^{M}$ , the martensite fraction grows from 0% to 100% (Figure 4(a)). After the complete phase transformation, the SMA mechanical properties remain unchanged. During mechanical unloading, the austenite phase is recovered when the stress decreases from $τ_{s}^{A}$ to $τ_{f}^{A}$ . The SMA is fully austenite and exhibits no residual deformation when completely unloaded. The corresponding change in shear modulus is shown in Figure 4(b). The change in the material properties and increase of the inelastic strain are related to the hysteretic behavior shown in Figure 3, which is related to the energy dissipation.

Effect of pseudoelastic hysteresis on the aeroelastic behavior of a typical section

The effect of the pseudoelastic hysteresis of SMA springs on the aeroelastic behavior of a typical section is investigated and discussed. The material properties provided in Table 1 and the properties of the typical section given in Table 2 are used. The dimensionless parameters of Table 2 are based on dimensional (physical) parameters from Sousa et al. (2011), who considered an airfoil whose semichord length is $0.125 m$ . The SMA coil springs considered have $0.95 mm$ wire diameter, $8 mm$ coil diameter, and $16.5$ active coils. The SMA springs are $36.7 mm$ long with the minimum allowed length of $20 mm$ . The SMA temperature is assumed constant at $331 K$ in all simulations. At such a temperature, the SMA is fully austenite for a stress-free state. The corresponding stiffness of each coil spring is $k_{c} = 175 N m^{- 1}$ (using $w = 0.085 m$ , please check Figure 1(b)). The critical stresses for the phase transformation are $τ_{s}^{M} = 95 MPa$ , $τ_{f}^{M} = 135 MPa$ , $τ_{s}^{A} = 52 MPa$ , and $τ_{f}^{A} = 0 MPa$ (see Figure 2). In all simulations, the initial condition is assumed a plunge displacement ( $h_{0} = 10 mm$ or ${\bar{h}}_{0} = 0.08$ ).

Table 2.

Dimensionless properties of the typical aeroelastic section (based on Sousa et al., 2011).

$a$	$- 0.5$	Elastic axis position
$μ$	$2.597$	Plunge-to-pitch mass ratio
$x_{α}$	$0.26$	Distance from elastic axis to CG
$r_{α}$	$0.504$	Radius of gyration
$η_{α}$	$0.51$	Frequency ratio
$ζ_{α}$	$0.088$	Pitch damping ratio
$ζ_{h}$	$0.0035$	Plunge damping ratio

Effect of pseudoelastic hysteresis on the aeroelastic behavior of the typical section at the linear flutter speed

In this case study, the effect of pseudoelastic hysteresis on the aeroelastic behavior of the typical section at the linear flutter speed is investigated. It is important to remember that the linear steel spring usually employed in the pitch DOF of a typical section was replaced by a pair of SMA springs. When linear steel springs are considered in the typical section, the usual linear aeroelastic behavior is verified: the typical section is stable for airflow speeds smaller than the linear flutter speed and unstable for airflow speeds larger than the linear flutter speed. The same linear aeroelastic behavior is verified when no stress-induced phase transformations are achieved in the SMA springs replacing a linear steel spring. This way, the term “linear flutter speed” refers to the flutter boundary of a typical section with steel springs or the flutter boundary of the modified typical section (with SMA springs) when no stress-induced phase transformations are achieved.

Figure 5 displays the flutter boundary, pitch and plunge displacements, as well as shear stress of each SMA spring introduced in the pitch DOF. Figure 5(a) shows the real part of the eigenvalue for the plunge DOF ( $λ_{h}$ ) for increasing airflow speed ( $U_{\infty}$ ). The linear flutter speed ( $U_{LF}$ ) is predicted as $11.6 m s^{- 1}$ , in agreement with Sousa et al. (2011). The dimensionless plunge and pitch time histories at the linear flutter speed are shown in Figure 5(b) and (c). The pitch SMA springs are subjected to deformations due to the pitch displacement, reaching the maximum length of $47.7 mm$ and minimum length of $25.7 mm$ (larger than the minimum allowed length of $20 mm$ ). The maximum achieved shear stress of each coil spring in the pitch DOF is $47 MPa$ (Figure 5(d)), which is lower than the critical stress for the onset of the phase transformation ( $95 MPa$ ). Therefore, the typical section with SMA springs behaves exactly as a linear aeroelastic system (since stress-induced phase transformations are not achieved). The amplitudes of oscillation (the pitch displacement in particular) would be excessively large in order to reach the critical stress value. Therefore, a set of preload values is considered (f ₀ = 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, and 4.5 N symmetrically for both springs) in this article to fully use the damping capability of SMAs and investigate its effect on the aeroelastic behavior of the typical section.

The aeroelastic behavior of the typical section at the linear flutter speed with increasing preload is shown in Figure 6. Figure 6(a) to (d) shows that both plunge and pitch displacements decrease with increasing preload on the SMA springs (for preload values larger than $1.7 N$ ). For the preload value of $4.5 N$ , the plunge amplitude is $54 %$ smaller than the plunge amplitude of the typical section with no preload on the SMA springs (Figure 6(b)), and the pitch amplitude is $56 %$ smaller than the pitch amplitude of Figure 6(d). However, the pitch and plunge DOFs become less sensitive to the variations in the preload values for $f_{0} > 4 N$ .

For comparison purposes, the plunge and pitch amplitudes of a typical section with a pair of steel springs in pitch DOF (that gives the same pitch elastic moment of the SMA springs with no preload) are also presented in Figure 6(b) and (d). Pitch and plunge amplitudes are insensitive to applied preload on the steel springs. Moreover, the same mechanical outputs are obtained for preload values smaller than $1.5 N$ of both cases. Effects due to preload other than stress increase (which would be expected also for steel springs, e.g. friction) are neglected because of the low forces involved.

The variation of the resulting shear stress (due to preload and pitch displacement contributions) with increasing preload level is shown in Figure 7. The pitch angle required for stress-induced phase transformation (leading to pseudoelastic effect) decreases with increasing preload value. For preload values larger than $2 N$ , the resulting shear stress is enough to induce the onset of forward phase transformations ( $τ > τ_{s}^{M}$ ). The transformations lead to a hysteretic behavior of the SMA springs and, consequently, to the aeroelastic behavior presented in Figure 6. Very large pitch angles would be required for stress-induced phase transformation in the range $0 \leq f_{0} \leq 1.5 N$ . Consequently, the typical section with SMA springs behaves as a linear aeroelastic system (as well as the one with steel springs) for this preload range. For $f_{0} \geq 2 N$ , the resulting shear stress is almost constant (and slightly larger than $τ_{s}^{M}$ , the dotted line in Figure 7) since the shear stress is quite constant when the martensitic plateau (transformation region) is reached. In addition, one should note that the amount of shear stress due to pitch displacement is the same for $0 \leq f_{0} \leq 2 N$ and decreases with increasing preload value since the aeroelastic displacements decrease for preload values larger than $2 N$ (Figure 6). For $f_{0} \geq 4 N$ , the resulting shear stress is almost fully due to preload on the SMA spring.

The particular case of $f_{0} = 3 N$ is shown in Figure 8 (the case with no preload, $f_{0} = 0 N$ , is also shown for reference). Figures 8(a) and (b) show the dimensionless plunge and pitch time histories obtained slightly above the linear flutter speed of Figure 5(a). The unstable behavior for $f_{0} = 0 N$ (growing oscillations) is converted to limit cycle oscillations (LCOs) when the preload value is increased to $3 N$ . Figure 8(c) to (f) shows the corresponding change in the length of SMA spring $S_{1}$ , the shear stress, the martensite fraction, and the stiffness (due to the change in the shear modulus), respectively. Figure 8(g) shows the shear stress against the length of the SMA spring.

Figures 9(a) and (b) show a detail of the shear stress and a detail of the corresponding martensite fraction, respectively, for the case of Figure 8 ( $f_{0} = 3 N$ ). Critical stresses are displayed, and an incomplete phase transformation can be observed.

Figure 6(c) shows that the equilibrium position of the steady-state pitch response changes with increasing preload. By increasing the preload value, the critical stress for a complete reverse phase transformation ( $τ_{f}^{A}$ ) (unloading) is never reached. After the first forward phase transformation, the martensite fraction is always larger than zero. Additionally, the SMA springs undergo different stress states due to the initial condition ( ${\bar{h}}_{0} = 0.08$ or $h_{0} = 10 mm$ ). Although symmetric springs with respect to the chord line are assumed, an imbalance in the martensite fraction occurs. Since the elastic modulus of SMAs depends on the martensite fraction, the springs exhibit a small difference in stiffness, which leads to different equilibrium positions. However, it is important to remember that the previously discussed aeroelastic behavior (Figures 6 and 7) is only related to the damping capabilities of the preloaded SMA springs (not due to the small stiffness variation). Figure 10 confirms such a behavior by showing the pitch response for $f_{0} = 4.5 N$ and two different initial conditions, namely, ${\bar{h}}_{0} = 0.08$ and ${\bar{h}}_{0} = - 0.08$ .

Effect of pseudoelastic hysteresis on the aeroelastic behavior of the typical section at the post-flutter regime

The effect of pseudoelastic hysteresis of SMA springs on the aeroelastic behavior of the typical section at the linear flutter boundary was discussed in the previous case study. Pitch and plunge amplitudes decrease with increasing preload on the SMA springs. Also, unstable aeroelastic motions at an airflow speed slightly above the linear flutter speed were converted to LCOs due to the nonlinear (hysteretic) response of the preloaded SMA springs (above certain preload value). The presence of nonlinearities in aeroelastic systems often results in bifurcations with LCOs above or below the linear flutter speed (Dowell et al., 2003). Sometimes, nonlinearities are welcome in aeroelastic systems since the catastrophic linear flutter behavior above the linear flutter speed can be replaced by LCOs of acceptable amplitudes. Therefore, this case study investigates the effect of pseudoelastic hysteresis of preloaded SMA springs on the aeroelastic behavior of the typical section at the post-flutter regime. The same set of preload values of section “Effect of pseudoelastic hysteresis on the aeroelastic behavior of the typical section at the linear flutter speed” is considered. Due to the limitations of the aerodynamic model employed to calculate the aerodynamic loads (linear aerodynamics that is valid for small mechanical motions), the aeroelastic behavior is investigated in this case study for airflow speeds ranging from the linear flutter speed up to $15 m s^{- 1}$ (about $30 %$ above the linear flutter speed). In this range, the maximum plunge displacement is always smaller than twice the initial condition in the plunge ( $\bar{h} < 2 {\bar{h}}_{0}$ ; ${\bar{h}}_{0} = 0.08$ ), and the pitch displacement is smaller than 0.2 rad (for specific preload value).

Figure 11(a) shows the variation of the maximum airflow speed that satisfies the limiting solution (in terms of pitch and plunge amplitudes) with increasing preload on the SMA springs. Figure 11(b) shows the resulting shear stress (due to preload-induced stress and pitch-induced stress) for each maximum airflow speed that satisfies the limiting solution of Figure 11(a). Linear aeroelastic behavior is verified for preload values smaller than $1.5 N$ since stress-induced transformations are not verified for this range of preloads and pitch displacement. Therefore, the typical section is unstable for any airflow speed higher than $11.6 m s^{- 1}$ (the linear flutter speed) and stable for airflow speeds below the linear flutter speed. Persistent and bounded oscillations of acceptable amplitude (LCOs) are obtained for the gray area of Figure 11(a). For airflow speeds larger than the limiting solution, the response amplitude becomes very large (LCOs of large amplitudes), and the aerodynamic model is not valid. Since the aeroelastic displacements at the post-flutter regime are larger than the displacements at the flutter boundary (since the aerodynamic loads increase with airflow speed), the shear stress contribution due to pitch displacement is slightly larger in Figure 11(b) than in Figure 7 ( $τ$ exceeds the $τ_{s}^{M}$ line) for preload values larger than $1.5 N$ (nonlinear aeroelastic behavior). The resultant shear stress is also constant for larger preload values due to the same reasons discussed in the previous case study.

Figure 12(a) to (f) shows the time histories of dimensionless plunge displacement, dimensionless pitch displacement, elongation of SMA spring ( $S_{1}$ ), shear stress, martensite fraction, and stiffness of pitch DOF for $f_{0} = 3 N$ and $U_{\infty} = 14 m s^{- 1}$ . The elongation of the spring (Figure 12(c)) and the corresponding shear stress (Figure 12(d)) show that stress-induced phase transformation is achieved due to the preload condition at the airflow speed of $14 m s^{- 1}$ . Although the martensite fraction (Figure 12(e)) is approximately $3 %$ , LCOs of acceptable amplitude are verified in the post-flutter regime due to the nonlinear pseudoelastic behavior of the SMA springs (Figure 12(g)).

Figure 13 shows the variation of the shear stress of one of the SMA pitch springs for increasing preload levels at the linear flutter speed and also at the highest airflow speed for LCOs of acceptable amplitude (limiting solution of Figure 11(a)) under a post-flutter condition. The springs ( $S_{1}$ and $S_{2}$ in Figure 1(b)) are under loading–unloading conditions due to the harmonic motion of the typical section at flutter speed or harmonic motion of LCOs in post-flutter condition. The upper boundaries in Figure 13 (preload-induced stress plus pitch displacement-induced stress) are related to forward phase transformations (loading stages). In such case, the elongation of spring $S_{1}$ is due to pitch displacements in the opposite direction to the preload. The lower boundaries are related to reverse phase transformations (unloading stages), and the resultant stress is the preload-induced stress minus the stress from the negative extreme of the pitch displacement (maximum spring shortening). During unloading stages, the pitch motion has the same direction of the preload. Depending on the preload level, the spring can be under tension for the entire pitch range. Therefore, positive stress values may also be observed in the lower boundaries of Figure 13 as the preload increases. The full recovery of the austenitic phase is not observed in the unloading stages of Figure 13 due to the positive stresses related to larger preload values.

Figure 14(a) shows the martensite fractions of springs $S_{1}$ and $S_{2}$ , and Figure 14(b) shows the stiffness of each spring obtained at both linear flutter speed and limiting solution of each preload in the post-flutter regime. The stiffness of each spring is identical in the post-flutter regime for the range of preloads considered. At the linear flutter speed, the martensite fraction and the stiffness of $S_{1}$ and $S_{2}$ springs are different for higher preload values. Since the pitch and plunge displacements increase with increasing airflow speed, the austenitic phase recovery is more evident for $U_{\infty} > U_{LF}$ (the lower boundary in Figure 13). Therefore, the effect of preload and initial condition on the equilibrium position of the pitch steady-state response at the linear flutter speed (Figures 6(c) and 10) is not observed in the post-flutter condition (Figures 12(b) and 14(a) and (b)). The aeroelastic behavior of the typical section at the flutter boundary or at post-flutter condition is insensitive to a variation of stiffness shown in Figure 14(b) (assuming that no hysteretic behavior is present). Therefore, the hysteretic response of the SMA springs (and consequently the damping capabilities of the SMA springs) is the specific factor modifying the aeroelastic behavior of the typical section.

Conclusion

In this article, the modeling and analysis of a 2DOF typical aeroelastic section with SMA springs introduced through the pitch DOF are presented. Brinson’s (1993) model is employed to describe the behavior of the SMA material undergoing phase transformation (due to mechanical loading–unloading processes). A model for SMA coil springs based on the pure shear assumption is derived following Liang and Rogers (1992, 1997). Isothermal conditions are assumed since the predicted phase transformations are incomplete, and a small region in the inelastic plateau is achieved ( $ξ < 0.1$ in all the cases presented). The equations of motion of a 2DOF typical aeroelastic section with steel springs are modified to include SMA springs to the pitch DOF. The aerodynamic loads are obtained following Edwards et al. (1979), who propose a state-space representation to Jones’ approximation to Wagner’s indicial function.

The effect of the pseudoelastic response of SMA springs on the aeroelastic behavior of the typical section is investigated in two case studies. A set of preload values ( $0 \leq f_{0} \leq 4.5 N$ ) is considered in order to bias the stress of the SMA springs toward the inelastic plateau. Initially, the aeroelastic behavior of the typical section at the linear flutter speed is investigated. For preload values smaller than $1.7 N$ , the typical section with SMA springs behaves like a linear aeroelastic system (with steel springs in pitch DOF). For higher preload levels, stress-induced phase transformations are verified at relatively small pitch angles. Therefore, the linear aeroelastic behavior is modified due to the nonlinear hysteretic behavior of the SMA springs. The amplitude of aeroelastic oscillations decreases with increasing preload level due to the damping capabilities related to the pseudoelastic hysteresis. For the preload value of $4.5 N$ , the amplitudes of pitch and plunge DOFs are more than $50 %$ smaller than the amplitudes of the cases with no preload.

The effect of the pseudoelastic behavior of the SMA springs on the aeroelastic behavior of the typical section at the post-flutter regime is investigated in another case study. For a specific range of preload values on the SMA springs ( $1.5 \leq f_{0} \leq 4.5 N$ ), the catastrophic linear flutter behavior observed in the post-flutter condition (above the linear flutter speed) is replaced by LCOs of acceptable amplitudes due to the energy dissipating and damping capabilities of the SMA springs. It is shown that the hysteretic response of the SMA springs helps in bringing the response amplitude to acceptable levels and increases the upper limit of the post-flutter speed of acceptable response amplitude by $30 %$ .

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by CAPES (DS-00011/07-0) and partially funded by CNPq (401777/2012-0). This work was also partially funded by CNPq (574001/2008-5) and FAPEMIG (APQ-0076-09) through the INCT-EIE.

References

Aguiar

RAA

Savi

Pacheco

PMCL

(2010) Experimental and numerical investigations of shape memory alloy helical springs. Smart Materials and Structures 19(2): 025008 (9 pp.).

Aguiar

RAA

Savi

Pacheco

PMCL

(2013) Experimental investigation of vibration reduction using shape memory alloys. Journal of Intelligent Material Systems and Structures 24(2): 247–261.

Alam

Youssef

Nehdi

(2007) Utilizing shape memory alloys to enhance the performance and safety of civil infrastructure. Canadian Journal of Civil Engineering 34(9): 1075–1086.

S-M

Ryu

Cho

. (2012) Engineering design framework for a shape memory alloy coil spring actuator using a static two-state model. Smart Materials and Structures 21(5): 055009 (16 pp.).

Bachmann

Oliveira

Sigg

. (2012) Passive damping of composite blades using embedded piezoelectric modules or shape memory alloy wires. Smart Materials and Structures 21(7): 075027 (13 pp.).

Baghani

Naghdabadi

Arghavani

(2012) A semi-analytical study on helical springs made of shape memory polymer. Smart Materials and Structures 21(4): 045014 (11 pp.).

Barbarino

Saavedra Flores

Ajaj

. (2014) A review on shape memory alloys with applications to morphing aircraft. Smart Materials and Structures 23(6): 063001 (19 pp.).

Barzegari

Dardel

Fathi

. (2012) Aeroelastic characteristics of cantilever wing with embedded shape memory alloys. Acta Astronautica 79: 189–202.

Bil

Massey

Abdullah

(2013) Wing morphing control with shape memory alloy actuators. Journal of Intelligent Material Systems and Structures 24(7): 879–898.

10.

Bisplinghoff

Ashley

Halfman

(1996) Aeroelasticity. Mineola, NY: Dover Publications, Inc. (Corrected Republication of Addison-Wesley, 1955).

11.

Brinson

(1993) One-dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. Journal of Intelligent Material Systems and Structures 4(2): 229–242.

12.

Calkins

Butler

Mabe

(2006) Variable geometry chevrons for jet noise reduction. In: Proceedings of the 27th AIAA aeroacoustics conference, Cambridge, MA, 8–10 May, 11 pp.

13.

Dowell

Edwards

Strganac

(2003) Nonlinear aeroelasticity. Journal of Aircraft 40(5): 857–874.

14.

Edwards

Ashley

Breakwell

(1979) Unsteady aerodynamic modeling for arbitrary motions. AIAA Journal 17(4): 365–374.

15.

Enemark

Savi

Santos

(2014) Nonlinear dynamics of a pseudoelastic shape memory alloy system—theory and experiment. Smart Materials and Structures 23(8): 085018 (17 pp.).

16.

Gandhi

Chapuis

(2002) Passive damping augmentation of a vibrating beam using pseudoelastic shape memory alloy wires. Journal of Sound and Vibration 250(3): 519–539.

17.

Hadi

Yousefi-Koma

Moghaddam

. (2010) Developing a novel SMA-actuated robotic module. Sensors and Actuators A: Physical 162: 72–81.

18.

Hartl

Lagoudas

(2007) Aerospace applications of shape memory alloys. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 221: 535–552.

19.

Hartl

Lagoudas

Calkins

. (2010) Use of a Ni60Ti shape memory alloy for active jet engine chevron application: I. Thermomechanical characterization. Smart Materials and Structures 19(1): 015020 (14 pp.).

20.

Hodges

Pierce

(2011) Introduction to Structural Dynamics and Aeroelasticity (Cambridge Aerospace Series, Vol. 15). 2nd ed. New York, NY: Cambridge University Press.

21.

Ibrahim

(2008) Recent advances in nonlinear passive vibration isolators. Journal of Sound and Vibration 314(3–5): 371–452.

22.

Janke

Czaderski

Motavalli

. (2005) Applications of shape memory alloys in civil engineering structures—overview, limits and new ideas. Materials and Structures 38(5): 578–592.

23.

Jones

(1938) Operational Treatment of the Nonuniform-Lift Theory in Airplane Dynamics (NACA Technical Note 667), 11 pp. National Advisory Commitee for Aeronautics, Washington, DC.

24.

Khan

Lagoudas

Mayes

. (2004) Pseudoelastic SMA spring elements for passive vibration isolation: part I—modeling. Journal of Intelligent Material Systems and Structures 15(6): 415–441.

25.

Lagoudas

(ed.) (2008) Shape Memory Alloys: Modeling and Engineering Applications. New York: Springer.

26.

Lagoudas

Hartl

Chemisky

. (2012) Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys. International Journal of Plasticity 32–33: 155–183.

27.

Liang

Rogers

(1990) One-dimensional thermomechanical constitutive relations for shape memory materials. Journal of Intelligent Material Systems and Structures 1(2): 207–234.

28.

Liang

Rogers

(1992) A multi-dimensional constitutive model for shape memory alloys. Journal of Engineering Mathematics 26: 429–443.

29.

Liang

Rogers

(1997) Design of shape memory alloy springs with applications in vibration control. Journal of Intelligent Material Systems and Structures 8(4): 314–322.

30.

Mirzaeifar

DesRoches

Yavari

(2010) Exact solutions for pure torsion of shape memory alloy circular bars. Mechanics of Materials 42(8): 797–806.

31.

Mirzaeifar

DesRoches

Yavari

(2011) A combined analytical, numerical, and experimental study of shape-memory-alloy helical springs. International Journal of Solids and Structures 48(3–4): 611–624.

32.

Mishra

Gur

Chakraborty

(2013) An improved tuned mass damper (SMA-TMD) assisted by a shape memory alloy spring. Smart Materials and Structures 22(9): 095016 (16 pp.).

33.

Rezaei

DAH

Kadkhodaei

Nahvi

(2012) Analysis of nonlinear free vibration and damping of a clamped-clamped beam with embedded prestrained shape memory alloy wires. Journal of Intelligent Material Systems and Structures 23(10): 1107–1117.

34.

Saadat

Salichs

Noori

. (2002) An overview of vibration and seismic applications of NiTi shape memory alloy. Smart Materials and Structures 11: 218–229.

35.

Saleeb

Dhakal

Hosseini

. (2013) Large scale simulation of NiTi helical spring actuators under repeated thermomechanical cycles. Smart Materials and Structures 22(9): 094006 (20 pp.).

36.

Savi

Paula

Lagoudas

(2011) Numerical investigation of an adaptive vibration absorber using shape memory alloys. Journal of Intelligent Material Systems and Structures 22(1): 67–80.

37.

Sousa

Anicézio

De Marqui

Jr . (2011) Enhanced aeroelastic energy harvesting by exploiting combined nonlinearities: theory and experiment. Smart Materials and Structures 20(9): 094007 (8 pp.).

38.

Tanaka

Kobayashi

Sato

(1986) Thermomechanics of transformation pseudoelasticity and shape memory effect in alloys. International Journal of Plasticity 2: 59–72.

39.

Tawfik

J-J

Mei

(2002) Thermal post-buckling and aeroelastic behaviour of shape memory alloy reinforced plates. Smart Materials and Structures 11(2): 297–307.

40.

Theodorsen

(1935) General theory of aerodynamic instability and the mechanism of flutter. NACA Report 496, pp. 413–433. National Advisory Commitee for Aeronautics. Langley Field, VA. May 2, 1934.