Cosserat modeling for deformation configuration of shape memory alloy unimorph actuators

Abstract

Shape memory alloys (SMAs) can contract their length via a crystalline phase transition that is dependent upon their temperature and stress state. SMAs have been used as linear micro-actuators due to their high strength to weight ratio and compact structure. However, the relatively low linear contraction ( $~$ 4%–5% in length) limits their use. To remedy this, the SMA can be offset from a passive structure, which acts to magnify the deformation. The resulting amount of deformation depends upon the material properties and geometry of both the SMA and the passive structure. In this work, geometrically exact beam theory (also known as Cosserat theory) is coupled with SMA constitutive relations to model the maximum deformation configuration of these actuators. Four of these actuators of various lengths were fabricated and tested to verify the model. For the four actuators tested, the mean squared error between the experimental results and the Cosserat model were between 0.0702 mm (0.1% error) for the shortest actuator (66 mm in length) and 3.59 mm (2.7% error) for the longest actuator (135 mm in length). These results show that the closed form solution derived for this Cosserat beam model can accurately model the deformation of these active structures.

Keywords

Cosserat model geometrically exact beam theory shape memory alloy actuators

1. Introduction

Shape memory alloys (SMAs) are metallic alloys that can produce and recover appreciable strain. The strain production is the result of the material undergoing a phase transformation. The strain can be recovered through two different effects: the shape memory effect and the pseudoelastic effect (Tanaka et al., 1986). The shape memory effect is strain recovery through varying the temperature of the material, whereas the pseudoelastic effect is produced by cycling the stress to initiate the phase transition (Li et al., 2011; Zotov et al., 2014). The most commonly used and best performing SMA material is nitinol, which is a nickel titanium alloy that can achieve approximately 4% linear strain while maintaining a high number of cycles (Brinson and Huang, 1996; Ivshin and Pence, 1994; Liang and Rogers, 1997; Prahlad and Chopra, 2001). Since the strain production is a material property, these metals are viable options as micro-actuators when compared to traditional forms of actuation, such as electric motors and pneumatic actuators. However, the small strain has limited their use compared to other forms of soft actuators that can achieve higher levels of strain, such as dielectric elastomers (Brochu and Pei, 2010; Hunter and Lafontaine, 1992; Mirvakili and Hunter, 2018).

In order to achieve higher levels of strain, the SMA can be paired with a passive material in a unimorph or bimorph configuration (Kennedy et al., 2020; Song et al., 2016). These configurations have been previously used in piezoelectric composite actuators to increase their deformation. For SMA unimorph actuators, a SMA wire is routed along the length of the passive material. When the SMA is heated and transitions to its high temperature austenite phase, the wire shortens in length, but the passive layer does not; this results in a bending motion of the entire structure. The deformation of these SMA unimorph actuators is a function of material properties of the SMA wire, material properties of the passive layer, and the geometry of the actuator. Varying any of these properties results in a different deformation profile of the actuator (Brinson et al., 1997; Villanueva et al., 2010). These bending actuators have been used to create biomimetic aquatic robots (Jin et al., 2016; Kim et al., 2012; Wang et al., 2008) and grippers (Rodrigue et al., 2017).

Since classical beam modeling methods (such as linear beam theory) assume small deflections (on the order of the thickness of the beam), these methods cannot be used to accurately model the large deformation of these beam actuators. The transfer matrix approach is one numerical technique used to model large deformations, wherein the method splits the beams into smaller segments, then classic beam theory is applied to each smaller segment before recombining the segments to determine the final beam deformation (Villanueva et al., 2010). The shear-lag model has also been applied to SMA embedded systems (Lagoudas and Tadjbakhsh, 1992). Bending mechanics coupled with heat transfer and phase kinetic models have been used to model the quasi-static transient response of a SMA-actuated catheter (Veeramani et al., 2008). Finite element techniques and numerical approaches have also been used to model and optimize the deformation profile of SMA bending actuators (Crews and Buckner, 2012; Machairas et al., 2019; Simone et al., 2019; Wang and Shahinpoor, 1997). Additionally, the Galerkin method has been applied to novel piezoelectric composite actuators to characterize their performance (Ma et al., 2019, 2022; Zhou et al., 2020a, 2020b).

Geometrically exact beam theory, also known as Cosserat theory, has been previously used to model the deformation of highly flexible systems (Antman, 2005; Lacarbonara, 2013) such as drill strings (Vlajic et al., 2014), MEMS components (Wang et al., 2004), and elastic beams (Lacarbonara and Yabuno, 2006; Son et al., 2008). Recently, it has been applied to active structures, such as soft robots (Zhou et al., 2015). In this paper, geometrically exact beam theory will be outlined and applied to a shape memory alloy unimorph actuator. Then, the closed form solution will be compared to experimental deformation profiles of SMA unimorph actuators. While the geometrically exact beam theory has been used to model other actuators, the authors believe this is the first time that it has been used for a shape memory alloy morphing actuator. It should be noted that the design parameters (the offset distance $d$ , the Young’s modulus of the two structures, etc.) are included in this geometrically exact formulation. Moreover, this technique, namely, setting the SMA’s geometry approximately equal to the beam geometry while estimating the maximum force from the SMA to predict the deformation, could also be applied to other designs of actuators. Since this method admits a closed form solution, this Cosserat model could be used to inform the design process of SMA morphing actuators.

2. Geometrically exact beam theory for SMA unimorph actuator

In geometrically exact beam theory, the three-dimensional structure is characterized by a line deforming in space provided that the aspect ratio (i.e. length to diameter) is sufficiently large. The material line, often called the Cosserat curve, can be any smooth line that connects all the cross sections of the beam. However, the line is often chosen as the line that connects the centers of mass of each cross section, which leads to a simplification of the equations of motion. It is assumed that the cross section at any given material point along the curve remains rigid in the system, which means that the position of individual material points in the cross section remain unchanged relative to each other during deformation (Lacarbonara, 2013). In this work, due to the aspect ratio of the SMA and the passive structure, it is assumed that the structure is neither shearable nor extensible. Therefore, if the position of the material line is known, the configuration of the three dimensional structure is also known in its entirety.

Every point along the material line is described by a position vector, $\vec{r} (s, t)$ , which is parameterized by the initial arc length parameter, $s$ , which spans from $0$ to $L$ (the length of the rod), and time, $t$ . Since the actuator to be tested in this study moves in a single plane, the theory is simplified to its two dimensional form. The configuration of the rod in Euclidean space with mutually orthogonal unit vectors, $\hat{i}$ and $\hat{j}$ , can be described as:

\vec{r} (s, t) = x (s, t) \hat{i} + y (s, t) \hat{j}

(1)

Both $x$ and $y$ are functions of both $s$ and $t$ , where $x$ and $y$ describe the position of the material line at $s$ and time, $t$ . In order to determine the deformation of the structure, it is convenient to attach a coordinate frame to the material line of the beam. This orthogonal coordinate frame consists of two unit vectors for the two dimensional representation. Because the beam is assumed to be unshearable, the unit vector $\hat{a}$ is normal to the cross section at $s$ , $\hat{a}$ is equal to the unit vector of the tangent vector, $d \vec{r} / ds$ , and $\hat{b}$ is orthogonal to $\hat{a}$ , as shown in Figure 1.

The local coordinate frame is related to the global coordinate system through the following rotational transformation using equations (2a) and (2b).

\hat{a} (s, t) = \cos (θ) \hat{i} + \sin (θ) \hat{j}

(2a)

\hat{b} (s, t) = - \sin (θ) \hat{i} + \cos (θ) \hat{j}

(2b)

Here, $θ (s, t)$ is the angle between $\hat{a}$ and $\hat{i}$ , which is also a function of $s$ and $t$ .

2.1. Equations of motion

The general equations of motion per unit length for a Cosserat rod are constructed by performing a force and moment balance on the system (Antman, 2005):

{\vec{n}}_{s} + \vec{f} = ρ A {\vec{r}}_{tt} + ρ I {\hat{b}}_{tt}

(3a)

{\vec{m}}_{s} + {\vec{r}}_{s} \times \vec{n} + l \hat{k} = ρ I \hat{b} \times {\vec{r}}_{tt} + ρ J \hat{b} \times {\hat{b}}_{tt}

(3b)

The subscripts of the variables denotes partial differentiation with respect to the variable in the subscript. The vectors $\vec{f}$ and $l \hat{k}$ represent the external force and moment vectors per unit length applied to the system, respectively. The internal force is represented by the vector, $\vec{n}$ , and the moment is represented by the vector, $\vec{m}$ . The density of the material is $ρ$ , while $A$ , $I$ , and $J$ represent the cross sectional area, mass moment of inertia, and area moment of inertia of the beam, respectively.

For a beam undergoing planar motion in two dimensions, the internal force and moment vectors can be decomposed into components in the local or global reference frame of the beam, $\hat{a}$ and $\hat{b}$ . These vectors then have the form shown below:

\vec{n} (s, t) = N_{1} (s, t) \hat{i} + N_{2} (s, t) \hat{j}

(4a)

\vec{n} (s, t) = N (s, t) \hat{a} + H (s, t) \hat{b}

(4b)

\vec{m} (s, t) = M (s, t) \hat{k}

(4c)

Here, $N$ is the force in the $\hat{a}$ direction, and $H$ is the force in the $\hat{b}$ direction. Since the beam is assumed to be inextensible and unshearable, $N$ is the axial force, while $H$ is the shear force. The quantity $M$ is the moment about the $\hat{k}$ direction.

Time is included as a parameter in the initial setup of the equations of motion, as the general form of this model is a function of both time and space. These equations will then be simplified to only consider a static analysis. Only the austenite modulus was used to preserve the simplicity of the model, since the maximum deformation state would be primarily austenite, resulting in very little difference between the modulus and austenite modulus. Time-domain solutions from this geometrically exact model would need to include hysteretic material constitutive relations of the SMA to properly account for phase transitions between the martensite and austenite.

2.2. Full model with continuous SMA force

The goal of this study is to model the maximum deformation configuration of a SMA embedded unimorph actuator. Therefore, the time response of the actuator will not be calculated, and the time derivatives in equations (3a) and (3b) will be set to zero as the system will be assumed to be static in its maximum deformed configuration. The equations of motion are then modified to include the spatial derivative of the force produced by the SMA actuator and the cross product of the position vector spatial derivative and the SMA force vector.

{\vec{n}}_{s} + {\vec{F}}_{sma, s} = 0

(5a)

{\vec{m}}_{s} + {\vec{r}}_{s} \times \vec{n} + {\vec{r}}_{sma, s} \times {\vec{F}}_{sma} = 0

(5b)

Since it is assumed that $\hat{a}$ is the unit vector in the direction of $d \vec{r} / ds$ and that the SMA wire is held parallel at a constant offset to the material line of the beam for this study, ${\vec{F}}_{sma} = - F_{sma} \hat{a}$ . The magnitude of the SMA force is constant, but the direction changes spatially as the beam curves (i.e. as $θ$ changes). It is also assumed that no external forces, such as gravity, are acting on the system.

In order to model the force generated by the SMA wire as a continuous force throughout the length of the beam, the location of the SMA wire at every point, $s$ , needs to be known. The SMA position vector can be related to the configuration of the beam by the angle, $θ$ , and the SMA offset, $d$ :

{\vec{r}}_{sma} (x, y, θ) = \vec{r} + d \hat{b}

(6a)

{\vec{r}}_{sma} (x, y, θ) = (x - d \sin (θ)) \hat{i} + (y + d \cos (θ)) \hat{j}

(6b)

In order to solve the static equilibrium equations, boundary conditions need to be specified. The boundary conditions for this model are as follows:

x (0) = 0

(7a)

y (0) = 0

(7b)

θ (0) = 0

(7c)

N_{1} (L) = q F_{sma} \cos (θ (L))

(7d)

N_{2} (L) = q F_{sma} \sin (θ (L))

(7e)

M (L) = qd F_{sma}

(7f)

The number of wires, $q$ , the offset distance of the wire from the material line, $d$ , and the force from the SMA, $F_{sma}$ , all affect the moment, $M$ . This set of equations can be solved for the kinematic unknown, $θ$ .

2.3. Pure moment case

For the following discussion, the assumption is made that the force of the SMA is negligible as compared to the moment that it applies at the end of the beam. Using this assumption, the equations (5a) and (5b) can be further simplified:

{\vec{n}}_{s} = 0

(8a)

{\vec{m}}_{s} + {\vec{r}}_{s} \times \vec{n} = 0

(8b)

Integration of equation (8a) shows that the internal force is constant throughout the rod:

\vec{n} = N_{1} \hat{i} + N_{2} \hat{j}

(9)

The Euler-Bernoulli constitutive relation is also assumed throughout this paper, which is shown below in equation (10):

M = EJ θ_{s}

(10)

Here, $E$ is the elastic modulus of the passive material. Using this relation and substituting equation (9) into equations (8a) and (8b), the following differential equations are found:

M_{s} - N_{1} y_{s} + N_{2} x_{s} = 0

(11a)

M = EJ θ_{s}

(11b)

x_{s} = \cos (θ)

(11c)

y_{s} = \sin (θ)

(11d)

This is a fourth order system with unknown parameters $N_{1}$ and $N_{2}$ . In order to obtain a solvable system of equations, equation (11a) can be integrated with respect to $s$ , resulting in a set of third order differential equations:

M - N_{1} y + N_{2} x + M_{o} = 0

(12a)

M = EJ θ_{s}

(12b)

x_{s} = \cos (θ)

(12c)

y_{s} = \sin (θ)

(12d)

Again using the assumption that the force of the SMA is negligible as compared to the moment at the end of the beam, the boundary conditions (7a)–(7f) can be written as:

x (0) = 0

(13a)

y (0) = 0

(13b)

θ (0) = 0

(13c)

N_{1} (L) = 0

(13d)

N_{2} (L) = 0

(13e)

M (L) = qd F_{sma}

(13f)

For this simplified system, an analytical closed form solution can be obtained. From equation (8a), it is known that the internal reaction force in the structure does not change with respect to $s$ . Along with the boundary conditions in equations (13d) and (13e), the internal reaction force can be written as follows:

N_{1} (s) = 0

(14a)

N_{2} (s) = 0

(14b)

Using this relationship with equation (11a), it can be determined that the internal reaction moment also does not vary with respect to $s$ and is constant throughout the beam. Combining this with equation (12a), it can be shown that the internal reaction moment is equal and opposite to the moment boundary condition. Equation (12b) can be integrated and solved for $θ$ using the boundary condition, $θ (s = 0) = 0$ :

θ = \frac{Ms}{EJ}

(15)

Using the relationship shown in equation (15), equations (12c) and (12d) can be integrated in order to solve for $x$ and $y$ as functions of $s$ :

x (s) = \frac{EJ}{M} \sin (\frac{Ms}{EJ})

(16a)

y (s) = - \frac{EJ}{M} \cos (\frac{Ms}{EJ}) + \frac{EJ}{M}

(16b)

The two models, which are presented in the Full model with continuous SMA force section (equations (5a)–(7f)) and the Pure moment case section (equations (16a) and (16b)), were solved using parameters from the experimental actuators tested in this work. These parameters are listed in Table 1. The equations (5a)–(7f) were solved numerically using Mathematica, and the analytical solution (equations (16a) and (16b)) were both plotted in Figure 2. In this figure, the overlapping green and black curves represent the solutions from these two sets of equations, and they are found to be indistinguishable from each other. Since the solution of the unsimplified set of equations and the solution of the simplified model that uses the assumption are indistinguishable, this provides a justification that the assumption was reasonable. For this reason, equations (16a) and (16b) will be used to model the system in conjunction with the force from the SMA as calculated in the next section.

Table 1.

Material properties and parameters.

Property	Value	Units
Elastic modulus	200	GPa
Austenite modulus	70	GPa
Width	12.7	mm
Thickness	0.1016	mm
SMA diameter	0.1	mm
Offset	1.2192	mm
SMA Prestrain	4.25	%
SMA wires	2	-

2.4. Shape memory alloy force

In order to solve for the deformed configuration of the system, the force being produced by the SMA wire needs to be calculated. The state of an SMA material is dependent upon the temperature, stress, current phase fraction, and direction of the phase transition of the material. The SMA’s one dimensional constitutive relation developed by Boyd and Lagoudas (Boyd and Lagoudas, 1994; Brinson and Huang, 1996; Shu et al., 1997) will be used to relate the strain of the material to the stress. Therefore, the force of the actuator can be calculated with the cross sectional area of the SMA wire. The constitutive relation takes the form:

σ = E_{sma} ε^{e}

(17a)

= E_{sma} [ε - ε^{transf} - α (T - T^{0})]

(17b)

The stress is equal to the elastic modulus of the SMA material multiplied by the elastic strain, $ε^{e}$ , of the SMA material. The elastic strain can be decomposed into three different strains: (1) the total strain of the SMA material in the current configuration of the actuator, $ε$ , (2) the transformation strain, $ε^{transf}$ , and (3) the strain produced by thermal expansion or contraction, where $α$ is the thermal expansion coefficient, $T$ is the current temperature, and $T^{0}$ is the reference temperature for the material. As will be described later, it is assumed that the configuration of the wire is the same as the host structure with an offset of a distance, $d$ . This assumptions enables the calculation of the strain from the current configuration. Typically, the strain resulting from thermal effects is neglected, since it is significantly smaller than the other two. Thus, the equation simplifies to the following:

σ = E_{sma} [ε - ε^{transf}]

(18)

The total strain, $ε$ , is a function of the length of the SMA wire in the current deformed configuration of the actuator:

ε = \frac{L_{sma} - L}{L} + ε^{prestrain}

(19)

Here, $L_{sma}$ is the current length of the SMA wire in the deformed configuration, $L$ is the length of the undeformed beam, and $ε^{prestrain}$ is the prestrain imposed on the wire before being routed into the actuator. Using the closed form solution used previously (equations (6b), (16a), and (16b)) and the assumption that the SMA wire is held at a constant offset, $d$ , from the beam, the functions for the $x$ and $y$ positions of the SMA wire at every point, $s$ , can be formulated as follows:

x_{sma} (s) = \sin (\frac{Ms}{EJ}) (\frac{EJ}{M} - d)

(20a)

y_{sma} (s) = \cos (\frac{Ms}{EJ}) (d - \frac{EJ}{M}) + \frac{EJ}{M}

(20b)

The material properties, $E$ and $J$ , are material properties of the beam, while $M$ is an unknown parameter. The length of the SMA wires based on the deformed configuration of the beam can be calculated as:

L_{sma} = \int_{0}^{L} \sqrt{x_{sma, s}^{2} + y_{sma, s}^{2}} ds

(21a)

x_{sma, s} (s) = (\frac{M}{EJ}) \cos (\frac{Ms}{EJ}) (\frac{EJ}{M} - d)

(21b)

y_{sma, s} (s) = - (\frac{M}{EJ}) \sin (\frac{Ms}{EJ}) (d - \frac{EJ}{M})

(21c)

Here, $x_{sma, s} (s)$ and $y_{sma, s} (s)$ are the spatial derivatives of the $x$ and $y$ functions of the SMA wire with respect to $s$ . Therefore, the length of the SMA wire can be calculated as:

L_{sma} = L - \frac{Md}{EJ} L

(22)

Since the stress on the SMA is changing as the material transitions, the wire cannot recover its entire prestrain as the material follows a loading path on the material stress-strain curve (Veeramani et al., 2008). The $ε^{transf}$ term in equation (18) accounts for the part of the prestrain that remains unrecovered during the transformation process. The amount of this unrecovered strain is related to the stress that the passive material imposes on the SMA material as:

ε^{transf} = \frac{F_{sma} d^{2}}{EJ}

(23)

This yields the following equations to determine the maximum deformed configuration of the SMA embedded beam:

x (s) = \frac{EJ}{F_{sma} d} \sin (\frac{F_{sma} d}{EJ} s)

(24a)

y (s) = - \frac{EJ}{F_{sma} d} \cos (\frac{F_{sma} d}{EJ} s) + \frac{EJ}{F_{sma} d}

(24b)

The force in the SMA wires, $F_{sma}$ , is calculated by substituting equations (19), (22), and (23) into (18) and multiplying by the area to get units of force:

F_{sma} = \frac{1}{4} q E_{sma} π D_{sma}^{2} (\frac{- 2 F_{sma} d^{2}}{EJ} + ε^{prestrain})

(25)

Here, $D_{sma}$ is the diameter of the SMA wire, and $E_{sma}$ is the elastic modulus of the austenite phase of the SMA. In order to account for the two wires running the length of the beam, the cross sectional area in $σ = F / A$ will be $A = 0.25 q π D_{sma}^{2}$ as in the leading term of equation (25).

3. SMA unimorph actuator design, fabrication, and experiment

3.1. Design and fabrication

Four SMA actuators were designed and fabricated using a unimorph structure to achieve high deformation motion. The active layer of the actuator consists of a 100 $μ$ m diameter SMA wire (Flexinol, Dynalloy, Irvine, CA). The passive layer is composed of 1095 spring steel (McMaster-Carr, Elmhurst, IL), with dimensions shown in Table 1. A 3D printed structure was adhered to the spring steel with cyanoacrylate glue to fix the SMA wire at an offset from the resting configuration of the beam as shown in Figure 3. These structures were printed using a Formlabs Form 3 (Formlabs, Durham, NC) using the high temperature resin to ensure that the structure did not break during heating cycles of the SMA. The structure was printed with the smallest possible print layer thickness (25 µm) to ensure the highest possible resolution and print quality. The 3D printed structure was printed as a singular structure. Each offset holder was connected to the adjacent offset holders by a connecting rod on each side. These connecting rods ensured that the offset holders were 10 mm apart and in a straight line when the offset holders were adhered to the beam. Once the structure was adhered, the connecting rods were removed to ensure that they did not affect the bending of the beam. The final structure is shown in Figure 3. The open structure limits friction induced onto the SMA wire and parasitic heating or cooling effects from the passive layer.

The SMA wire is routed from the fixed end to the free end, then it is looped back to the fixed end; this allows all the electrical connections to be placed at the fixed end, as shown in Figure 4. This ensures that no electrical wires or connections affect the deformation of the structure. Actuator parameters and material properties of the passive metal and SMA wires used in the model are shown in Table 1.

3.2. Experimental setup and processing

An Edgertronic SC2+ camera (Sanstreak Corp., San Jose, CA) was used to collect kinematic data at 200 frames per second to track circular markers at increments approximately 10 mm apart to approximate the movement of the actuator as seen in Figure 4(b). The SMA actuator is realized as a cantilever, with one end fixed and the other end free. The actuator was positioned so that it hung vertically with the free end pointing downwards; it should be noted that the effects of gravity were neglected. The SMA wire was heated by running a constant 0.2 mA current through it. The 0.2 mA current was selected based on the SMA specifications to actuate the wire in approximately 1 second. Since the purpose of this study is to investigate the maximum deformation of the actuator and not the time response, the current was left “on” to hold the SMA actuator in its maximum deformation configuration until the camera capture was completed; a representative time history of the free end of an actuator is shown in Figure 5 along with the initial and deformed configuration of a 135 mm actuator in Figure 6.

MATLAB was used for image processing to calculate the maximum deformation configuration of the actuator with respect to its initial position. The first (initial) and last (maximum deformation) frame were used for processing. In the initial configuration, equation (16b) is modified to include an unknown parameter, $M_{fab}$ :

y (s) = - \frac{EJ}{M_{fab}} \cos (\frac{M_{fab} s}{EJ}) + \frac{EJ}{M_{fab}}

(26)

This $M_{fab}$ term is the small moment applied to the passive layer from the fabrication process of connecting the SMA wire, which is used to account for any initial bend in the experimental actuator. The resulting curve fit of $M_{fab}$ was added to the moment created by the SMA wires to determine the final deformed configuration. This was done to mitigate any errors introduced from placing the markers relative to the midline of the beam as well as taking into account the initial configuration of the beam. The experimental data was also compared to an ANSYS 2021 Static Structural simulation. The calculated moment, $M + M_{fab}$ , was applied to the free end of the beam with a fixed support on the fixed end. The ANSYS simulation used beam elements with an element size of 0.1 mm. An iterative solver was used with the large deflection option enabled.

4. Results

Four actuators of various lengths (66, 107, 125, and 135 mm) were fabricated and tested to compare with the model equations (24a)–(25). The experimental results, Cosserat model, and ANSYS simulation results are shown in Figures 7 to 10. The mean squared error (MSE) between the markers of the experimental actuators and the Cosserat model, for both the initial and deformed configurations, are shown in Table 2; the MSE between the markers of the experimental actuators and ANSYS simulations are compared in this table as well. The percentage error was calculated as $\frac{MSE}{Length} \times 100 %$ .

Table 2.

Mean squared error (MSE).

Length	Initial MSE	Cosserat deformed MSE	FEA deformed MSE
66 mm	2.76 $\times 10^{- 5}$ mm	0.0702 mm (0.1% error)	0.0707 mm (0.1% error)
107 mm	6.95 $\times 10^{- 4}$ mm	2.15 mm (2.0% error)	2.12 mm (2.0% error)
125 mm	1.78 $\times 10^{- 3}$ mm	2.93 mm (2.3% error)	2.95 mm (2.4% error)
135 mm	2.89 $\times 10^{- 4}$ mm	3.59 mm (2.7% error)	3.63 mm (2.7% error)

The FEA results agree with the model results using equations (24a)–(25). Differences between the assumptions made in the modeling and the experimental actuators likely result in the error shown in Figures 7 to 10. The model slightly overpredicts the deflection in all four cases. This overprediction is likely because the model presented in this work assumes that the SMA wire is at a constant offset from the midline of the passive beam. However, the wire is only held at this offset at discrete increments along the length of the beam, resulting in a piecewise linear representation of the assumed curvature. Therefore, the model overpredicts the moment on the end of the beam, as well as the strain and deflection. This is also supported by the FEA results, wherein the same moment was used as the boundary condition and the results are also overpredictions. If the number of wire support structures were increased, the curvature of the SMA would more closely match the model, resulting in a better approximation of the moment on the end. Along with this assumption, dimensional tolerances of both the manufactured beam and 3D printed offset holders could also cause error between the model and experiment due to the millimeter scale of the parts used. The results show the importance of the initial (resting) configuration and taking the small moment created by the fabrication process into account for modeling the maximum deformation. Without the addition of the $M_{fab}$ term to the moment created by the SMA wire, the errors for each length would have been larger.

5. Summary

Shape memory alloys exhibit the ability to contract and change their shape based upon their temperature and stress states. This ability comes from a transition in the phases of the material based upon those states. Since this property is a product of the material, it has mainly been used as a micro-actuator in the form of small wires that contract linearly. However, one of the limiting factors in the use of these materials as actuators is their low recoverable strain, which is approximately 4%–5%, when used as a linear actuator. In order to increase their performance as an actuator, SMAs can be paired with a passive, axially stiff material. With the SMA offset at a known distance from the midline of the passive material, the structure will bend when the SMA changes phase and contracts. Bending actuators like this can be used to design biomimetic robots, such as grippers. Modeling the deformation of these actuators is crucial to expanding their use.

In this work, geometrically exact beam theory was applied to model the deformation of a constant cross section beam actuated by a shape memory alloy wire. The deformation was coupled to the SMA constitutive relation to simultaneously solve for the stress in the SMA and the deformation of the beam in order to predict the maximum deformation of the bending actuator. Four actuators over various lengths were fabricated and tested, and the SMA was activated by electrically heating the wire while capturing the motion with a camera. The mean squared error between the markers in the experiment and the Cosserat model prediction were 0.0702 mm (0.1% error), 2.15 mm (2.0% error), 2.93 mm (2.3% error), and 3.59 mm (2.7% error) for actuators of length 66, 107, 125, and 135 mm, respectively.

Overall, this closed form solution predicted the experimental results with high accuracy for the actuators tested and holds promise for the modeling of more complex active structures in both static and dynamic usage. Accurately modeling these structures and their deformed configuration combined with SMA’s self sensing capability could allow for shape control of these actuators.

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 authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This material is based upon research supported by the Office of Naval Research Young Investigator Award Number N00014-19-1-2413.

ORCID iD

Scott Kennedy

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