ANALYTICAL TREATMENT OF CONTROLLED BEAM DEFLECTIONS USING SMA WIRES

Transcription

1 ANALYTICAL TREATMENT OF CONTROLLED BEAM DEFLECTIONS USING SMA WIRES L. C. Brinson, M. S. Huang Department of Mechanical Engineering Northwestern University Evanston, IL C. Boller, W. Brand Daimler Benz AG, Forschung und Technik Stuttgart, Germany Abstract In this paper the active control of beam deflection through heating and cooling of Shape Memory Alloy (SMA) wires is examined. A phenomenological constitutive law for SMA wires is coupled with beam theory to provide predictions of beam shape upon temperature change in the SMA actuator. Both the linear and nonlinear beam theory are presented, enabling calculation of large deflections. Examples for a single wire attached at the tip of a uniform beam are given, but the procedure can easily be generalized for other configurations and utilized in control algorithms. Issues of design constraints for shape control with shape memory wires are addressed and the model is qualitatively verified by experiments. Introduction The importance of advanced material systems is rapidly increasing as ever more stringent demands are placed by our society and environment on the development of new technological systems. Smart material systems play an important role in innovative technology, providing materials that can act as both control elements and structural members (such as piezoelectrics, shape memory alloys, or magnetostrictive materials). These materials consequently offer great possibilities for self-controlling structures, enabling these structures to adapt themselves to various loading conditions in the sense of structural optimization. The technological advantages of each class of these materials over traditional materials arise from special capabilities due to unique microstructure or molecular properties. However, these unique properties necessarily add complexity to the experimental -1-

2 analysis, the constitutive descriptions and the structural implementation of these materials, issues that must be addressed and understood before the full potential of smart structures can be realized. This paper focuses on Shape Memory Alloy materials and provides a technique to describe shape control of a structural member through a basic analytical and numerical formulation. The analysis necessarily includes direct coupling to a constitutive law based on the micromechanics of SMAs and illustrates how the advances made in constitutive modeling enable prediction of the material performance as components in active engineering structures as well as of the engineering structures themselves. The shape memory effect has been documented as early as 1951 (Chang and Read) and the materials aspect of the phase transformations responsible for the macroscale behavior has received much attention through the intervening years (Krishnan and Brown, 1973; Warlimont et al., 1974; Delaey, 1991). In the past decade, the intense interest in smart structures applications has led to the formulation of a variety of constitutive models, mechanical experiments, and commercial applications (Tanaka and Nagaki, 1982; Falk, 1983; Patoor, Eberhardt and Berveiller, 1988; Abeyaratne and Knowles, 1991; Pence, 1992; Sun and Hwang, 1993; Boyd and Lagoudas, 1994). Among the applications of SMAs, shape control via wires attached externally or embedded in a structure have been considered; couplings and connectors are in use; frequency control via induced stresses have been explored; medical devices relying on the pseudoelastic effect have been proposed (Harrison and Hodgson, 1976; Owen, 1976; Wayman, 198; Funakubo, 1987; Rogers, Liang and Jia, 1989; Wayman and Duerig, 199; Humbeeck, Chandrasekaran and Delaey, 1991; Beauchamp et al., 1992; Falcioni, 1992). This paper brings together basic work on constitutive modeling with a simplified application example: control of tip deflection of a beam with an externally attached SMA wire. Even in this simplified case, the analysis is nontrivial due to the coupling of the beam equation and constitutive law, but the problem is tractable and similar approaches could be the basis for analysis of structures with higher complexity. The first section of the paper reviews general SMA behavior and describes the constitutive model used subsequently. The next sections formulate first the linear and then the nonlinear coupled -2-

3 beam and constitutive law equations and describes the solution techniques. Then results are presented and discussed and implication of the method for active structures is given. Background Shape memory behavior is due to a reversible thermoelastic crystalline phase transformation between a high symmetry parent phase (austenite) and a low symmetry product phase (martensite); the phase changes occur as a function of both stress and temperature. Formation of the martensitic phase under uniaxial or shear stress results in the formation of preferred crystalline variant orientations (via detwinning ) which lead to a large, recoverable strain (order of 1%) (Funakubo, 1987; Delaey, 1991). This capability for reversible, controllable large-strains is the basis of much interest in SMAs as control materials. Large shape changes can be induced easily and reproducibly with these materials, or alternatively, in a constrained situation, large stresses can be imparted to the connected structural components. The stress-temperature behavior of a given SMA alloy can be described by a phase diagram as in Figure 1. Figure 1 is obtained for a given material by performing tensile tests at different temperatures to delineate the transformation strips [M], [A] and [d]. On loading paths with components in the direction of the transformation strip vectors, transformation from the parent to product phase (or vice versa) occurs in the respective strips. At high temperatures, the unloaded material is austenitic and upon reaching a critical stress during loading a transformation to detwinned (or oriented) martensite occurs, which is reversed upon unloading as the material returns to the austenite phase. (See loading path 1 in Figure 1.) Here, the material undergoes a large strain during the [M] transformation strip, which is recovered in a hysteresis loop as the material passes through the [A] strip. At lower temperatures, upon loading an austenitic or twinned martensite material will also undergo transformation to detwinned martensite upon passing through the [M] or [d] strips (path 2 in Figure 1). Here no strain is recovered upon unloading, as the material remains in its detwinned state. Strain recovery can be achieved by subsequently heating the material through the [A] strip. The loading paths considered in this paper are more complex, since the SMA wire is constrained by the elastic beam during loading/heating. -3-

4 σ crit 1 P σ cr f σ cr s [d] Mt,d M d [t] 2 C M Mt,dA [M] C A [A] A Figure 1: M f M s A s A f T Phase diagram: Critical stresses for conversion of twins and austenite-martensite crystal transformations as functions of temperature. Transformation regions in the white strips and transformation strip vectors indicated by small arrows. Region of irreversible plastic deformation denoted by P, pure austenite by A, pure detwinned martensite by M d ; coexistance regions in hatched domains (e.g., in M t,d A twinned and detwinned martensite and austenite can all coexist). Due to the nature of the transformations (formation of preferred variants related to axis of loading), shape memory materials are predominantly characterized and utilized in wire (1-D) form and several 1-D constitutive models are able to replicate experimental results. The model chosen for use in this study is phenomenological, consistent with underlying thermodynamics, with a final engineering form utilizing only clearly defined material constants. This constitutive law, with origins in work by Tanaka (1982) and recently modified by Brinson (1993), relates the stress (σ) to the strain (ε), temperature (T) and detwinned martensite fraction (ξ S ) in the material: σ σ = D( ξ) ε D( ξ ) ε + Ω( ξ) ξ Ω( ξ ) ξ + Θ ( T T ). (1.1) S S A notable feature of this constitutive model is that it distinguishes between the twinned, ξ T, and detwinned, ξ S, martensite in the material by means of internal variables, with the total martensite fraction of the material given by ξ = ξs + ξt. (1.2) Here ξ T corresponds to temperature-induced (or twinned ) martensite and is not associated with additional transformation strain, while ξ S corresponds to stress-induced (or detwinned ) martensite and is associated with the recoverable transformation strain. This distinction is consistent with the micromechanics of SMAs and is the fundamental -4-

5 reason that this constitutive law is successful in capturing basic SMA behavior at all temperatures and initial states, where other similar approaches have difficulty. The constitutive relation (1.1) must also be coupled with transformation kinetics equations, which relate the martensite fraction to the stress and temperature ξ = ξ( σ, T ) (1.3) Expressions for (1.3) can be found elsewhere (Tanaka, 1986; Liang and Rogers, 199; Brinson, 1993). These expressions mathematically describe the evolution of the martensite fraction with stress based on a critical stress-temperature phase diagram (Figure 1). The exact form of equation (1.3) simply describes the rate and progress of that transformation. More detail on the phase diagram can be found in other references (Brinson, 1993; Bekker and Brinson, 1994). The problem examined in this paper will utilize this constitutive law and associated kinetic relations (Brinson, 1993) to study controlled deflections of a beam by heating and cooling an attached SMA wire. Note that most characterization experiments on SMAs to establish the phase diagram are performed by isothermal tests (as loading paths 1 and 2). However, a few constrained tests (iso-strain) where temperature is varied have provided results for location of transformation strips that coincide with the isothermal tests (Dye, 199; Liang, 199; Ditman, 1993). Hence, although the example here will involve a general loading path on the phase diagram, we assume that the kinetic relations as previously defined hold. Beam Model Shape Memory Alloys have been proposed for use in a wide array of engineering structures. However, the complexity of most applications combined with the intricate nonlinear behavior of the SMA material itself makes analysis and design difficult. For this reason, we have chosen here to examine a simple beam under control of an externally attached SMA wire to elucidate the essentials of the interaction. Such a model is representative of many shape control applications. Although beam models have been examined (Jia and Rogers, 1989; Chaudhry and Rogers, 1991), the coupling between the beam equations and the SMA constitutive models has only been partially considered. It is the latter that is the focus of the work here. -5-

6 Elastic beam x SMA wire Offset device z Figure 2: SMA wire and elastic beam configuration before and after deformation The simple elastic beam and SMA wire arrangement is shown in Figure 2. Before the SMA wire is attached to the beam, it is subjected to a tensile loading and unloading procedure in order to provide some fraction of detwinned martensite (residual deformation) in the wire (as in loading path 2 in Figure 1); the wire is then attached to the beam with zero prestress, but with residual strain. (Thus the beginning temperature of the system must be below A f so that martensite can exist in the absence of stress; see Fig. 1.) Note that the SMA wire is attached to the right end of the beam with an offset device of variable length. A larger offset will increase the beam deflection for a given load in the wire. The left end of the wire is attached to the wall at the base of the beam. The wire is to be heated or cooled uniformly and the subsequent deformation of the beam (as the SMA undergoes inverse transformation to austenite upon heating) will be described as a function of temperature. In general shape control applications, either the wire temperature will change due to overall temperature change in the system or a current will be run through the wire to achieve resistive heating. In either case, the temperature of the SMA is the control parameter. In the subsequent two sections we will discuss both small and large beam deflections (linear and nonlinear analyses). In each case, the first step is to formulate the appropriate beam equations and the link those to the SMA equations. Since the shape memory wire actuator is under simple tension in this arrangement, the SMA constitutive law (1.1) and kinetic laws (1.3) describe its response based on the temperature and the load in the -6-

7 wire. The load on the SMA wire is provided by the beam s resistance to bending as the wire attempts to recover the original residual strain upon heating. Thus it is the load on the wire, which is also the load on the beam, that is the direct coupling mechanism between the two bodies. x ζ l w(x) N M Q F d η δ Figure 3: Free body diagram of deflected beam The initial beam equations can be formulated by examining a free body diagram of the beam (Figure 3). The force applied on the beam by the SMA wire, F, can be decomposed into horizontal, N, and vertical, Q, components N = F Q = F ζ dsin( w'( ζ)) ( ζ dsin( w'( ζ)) ) + ( η+ dcos( w'( ζ)) t/ 2) 2 2 η+ dcos( w'( ζ)) t/ 2 ( ζ dsin( w'( ζ)) ) + ( η+ dcos( w'( ζ)) t/ 2) 2 2 (2.1) (2.2) where coordinates (ζ,η) define the location of the tip of the beam as shown in Figure 3 and t is the thickness of the beam. Equilibrium of the beam then requires: M = N( η+ dcos( w ( ζ)) w( x) ) Q( ζ x dsin( w ( ζ)) ) (2.3) This equation will be combined with the usual beam equation M = κ ( x) (2.4) EI where κ is the curvature of the neutral axis, with cantilever boundary conditions at x=, w()=w'()=. Linear Analysis If the deflections are small compared to the length (w'(x) small), the curvature κ can be expressed by -7-

8 M = κ ( x) = w ( x) (2.5) EI Hence the governing equations (2.3) and (2.4) simplify to N w x = + EI wx Q EI x Qζ Nη Nd ( ) ( ) EI (2.6) This second order, nonhomogeneous differential equation can be solved by standard methods, providing η w( x) = C1cos( kx) + C2sin( kx) + x+ d ζ (2.7) N where k =. The constants C 1 and C 2 are determined from boundary conditions EI yielding 1 h η w( x) = dcos( kx) sin( kx) + x + d (2.8) k ζ ζ Equation (2.8) depends on the location of the tip of the beam ζ and η which are subject to two further constraints ζ 2 w( ζ) = η, 1 + w ( x) dx = l (2.9) Equations (2.8) and (2.9) could be solved iteratively to find the deflection for a known load, F. In the case of the shape control application, however, the load F is not known, rather a given temperature change in the SMA wire causes phase transformation in a prestrained wire which then induces a load F as the beam resists bending. Since the amount of transformation strain recovered at a given temperature is dependent on the load in the wire and the load in the wire is dependent on the resistance of the beam to deflection, the SMA constitutive equation must be coupled with the final beam equations. To perform the coupled analysis, first note that the length of the SMA wire (originally l s ) when the beam is deflected is given by 2 2 l s = ζ + η (2.1) and thus the strain in the SMA wire is given by l ε = s ζ + η l s 2 2 (2.11) -8-

9 Since the SMA wire is in uniaxial tension, the stress is directly related to the force, F, (σ=f/a) and this together with equation (2.11) can be used with Equations (1.1) and (1.3) to complete the system of equations. Note that the initial conditions, σ, ε, ξ must be specified for the SMA wire. Nonlinear Analysis For the nonlinear analysis, the assumption of w'(x) small is no longer valid and the exact expression for curvature must be used w ( x) κ = 2 ( 1 + w ( x) ) 3 2 (2.12) Hence the governing differential equation for the beam deformation becomes nonlinear and can be written as w ( x) N = + { } + w x EI wx Q EI x 1 ( ) Q N Qdsin w ( ) Ndcos w ( ) 2 1 ( ) 32 ζ η ζ ζ (2.13) EI ( ) Due to the difficulty of solving this nonlinear governing equation coupled with the SMA constitutive law for the wire, we have chosen to present a direct numerical approach here. The steps leading towards an analytical solution (producing several equations which can then be solved numerically via Newton s method) can be found elsewhere (Brand et al., 1994). To solve the problem directly with a numerical approach, first separate the problem into two functions: 1) The wire strain as function of the stress using the SMA constitutive law (equation (1.1) and transformation kinetics equations): ε1 = f( 1 σ) (2.14) 2) The wire strain as a function of the load using beam theory. In this case, a certain load applied to the beam results in deformation of the beam; from that deformation, the tip deflection and hence length of SMA wire can be calculated (see below for details of this calculation). Define this wire strain from beam theory as ε = f( σ) (2.15) 2 2 Subtracting ε 2 from ε 1 : ε ε = f ( σ) f ( σ) = f( σ) (2.16) -9-

10 At a particular temperature, there is only one stress for which f(σ)= and a bisection search method is used here to find the desired stress. The σ max and σ min for the first iteration are set to be the critical buckling load at 7 C (maximum operating temperature for this study) and zero, respectively. (This same critical buckling load is used as the maximum for the first step in the search method in all cases since any load larger than the maximum will serve.) Since the real stress must be bounded by σ min and σ max f(σ min )*f(σ max ) (2.17) The bisection method is then applied by defining σ i = (σ min +σ max )/2. If f(σ max )*f(σ i ) then the desired stress is between σ max and σ i and we redefine σ min =σ i. Otherwise, the desired stress is between σ min and σ i, in which case we redefine σ max =σ i. Iteration based on equation (2.17) and successive bisections then continues until the desired convergence on the stress, σ. The second strain (or f 2 (σ) ) in equation (2.15) bears further discussion. To find f 2 (σ) (via equation 2.11) the bending moment, which depends on the shape of the beam, must be known. A gradual approach method is used to obtain the appropriate relationship. Initially it is assumed that the beam has no deformation (w 1 (x)=, ζ 1 =l, η 1 =), and the load (determined from the stress, σ min or σ max, being used during iteration) is applied at the tip of the offset device. The moment distribution, M 1 (x), on the beam is calculated based on the equilibrium relationship from equation (2.3). Then a new shape of the beam, w 2 (x) is calculated from the differential equation (2.13) and boundary conditions w ( x) Mwx ( ( ), x) = BC s: w()=, w'()= (2.18) 2 3/ 2 ( 1 + ( w ( x)) ) EI via a 4th order Runge-Kutta method. In this process, the known function value and slope at one point are used to estimate the function value of next point. (At each step, n, wx ( n ) is calculated from point x n 1.) Equation (2.9) is used at each step to ensure that at the final step, nf: nf l = Δ x ( w( xn) w( xn 1)) (2.19) n= 1 Subsequently, the new tip deflection and x-displacement (ζ 2, η 2 ) are calculated and used with the w 2 (x) again in equation (2.3) to determine a new moment distribution, M 2 (x), which is used in equation (2.13) to obtain w 3 (x), etc. The process continues until w m (x) is -1-

11 within the desired tolerance of w m-1 (x). (One should note that in this process, the moment is ever increasing and hence the beam deflection likewise increases.) It should be noted that no attempt was made here to optimize the solution method and improvements on the procedure can certainly be made. We want only to point out that the solution can be obtained, that it is robust and can be used in design purposes. The current numerical algorithm could conceivably be used in control algorithms for a given application since once the geometry and materials are fixed, look up tables for the roots can be made which would allow the near instantaneous solution necessary for a feedback loop. Alternatively the analytical solution (Brand et al., 1994) could be extended and used. Results and Discussion In order to first verify the beam theory without reference to the SMA actuator and constitutive law, a short series of experiments were run with the beam arrangement illustrated in Figure 1, where the beam was arranged vertically and SMA wire was replaced by a steel wire with weights fixed to the lower end. Buckling was avoided by the eccentric loading of the beam via the offset device as shown, allowing controlled bending of the beam. Several beam geometries were considered with parameters given in Table 1. The results of the experiments are shown in Figure 4 where the deflection of the beam tip with increasing applied load is plotted. Note that both the linear and nonlinear theory are accurate at small deflections, while with the larger deflections, the nonlinear theory tracks the experimental data more accurately. The results are acceptable to within experimental error and thus with this and other evidence we feel confident to proceed with the coupled SMA beam response. Table 1: Dimension of the beams of 5 beam cases. (Unit: meter) Case 1 Case 2 Case 3 Case 4 Case 5 beam length beam width beam thickness wire diameter offset device

12 .18 Deflection at end points (m) Case#: experiments linear nonlinear Load in the wire (N) 3 4 Figure 4: Comparison of experimental beam deflections with linear and nonlinear theory. SMA wire not considered here. Figures 5 through 1 show results of the coupled beam and SMA constitutive law theory described earlier and represent the potential shape control capabilities with SMAs while also illustrating a few points of concern in terms of design and prediction. The material properties for the SMA actuator (Table 3) are taken from data in the literature (Dye, 199; Liang, 199) and used in the constitutive and kinetic laws (1.1) - (1.3). Two particular cases are chosen for representation here and the beam data is given in Table 2: case A represents a stiffer beam and case B a softer beam. Both cases use the same SMA wire which is prestrained by 3%. This pre-strain results in 45% detwinned martensite in the material when the wire is initially fixed to the beam. The temperature range of study is such that the wire can exist in the martensitic state at the low end of the range and the austenitic state at the higher end of the range. After the wire is fixed to the beam with no prestress, the temperature is cycled and the results predicted by both the linear and nonlinear methods above. Ideally with this method one can provide continuous shape control to the beam dependent on temperature. -12-

13 Table 2: Beam dimensions for good control capability (A) and poor control capability (B, which is the same as case 2 in Table 1) Initial strain for both cases is.3. Case A Case B length.3.4 width.1.5 thickness.2.1 offset device.3.3 SMA diameter.13.6 In Figure 5, the stress in the SMA wire is plotted as a function of temperature. The light dashed lines on the plot indicate the transformation strips described earlier in Figure 1. As the temperature is increased initially, there is very little response in the wire until the critical transformation temperature is reached. At this stage, the detwinned martensite in the SMA wire begins the inverse transformation to the parent phase, which would result in stress-free strain recovery if the wire were unconstrained. Since the wire is attached to the beam, however, the beam provides a resistance to that deformation and hence the stress in the wire continues to increase as the temperature increases. Figure 6 shows that as the temperature increases, the martensite fraction decreases, indicating transformation. As this occurs, the tip of the beam is deflected as the SMA material recovers the initial strain, which can be seen in Figure 7. The distinction between case A (stiffer beam) and case B (softer beam) can be clearly seen in all figures. First examining Figure 5, one notes that the softer beam provides less resistance to the austenitic transformation and hence the transformation is completed at a much lower temperature and at a much lower stress than for the stiffer beam. However, as case B was chosen here to illustrate, with a very weak beam the stress induced during Table 3: Material properties for the Nitinol alloy used in the examples (Dye (199), Liang (199)) Moduli, Density Transformation Temperatures Transformation constants maximum residual strain Da = MPa Mf = 9 C CM = 8 MPa/ C ε L =. 67 Dm = MPa M s = C CA = 13. 8MPa/ C Θ= 55. MPa/ C As = C cr σ s = 1MPa 3 ρ= kg m A f = 49 C cr σ f = 17MPa -13-

14 4 Stress in SMA wire (x1 6 Pa) Case A linear nonlinear Case B linear nonlinear cool heat Figure 5: Temperature ( C) Stress in the SMA actuator as a function of temperature. Case A shows good control capability, while case B shows poor control capability (see Table 2) the semi-constrained transformation is quite small and upon cooling no inverse transformation to martensite occurs. This response is the typical one-way shape memory effect: a load induced strain can be recovered upon heating, but the strain does not reappear upon cooling. * In contrast, with a stiffer beam the temperature and stress at the end of transformation are much greater and upon cooling, the material undergoes an inverse transformation to recover the detwinned martensite fraction. This response has been demonstrated by using bias springs in series with a SMA actuator; here the beam itself provides the same resistance to deformation as the spring in those cases (Funakubo, 1987; Liang and Rogers, 1992). Note that the critical transformation stresses are material parameters and thus the choice of the particular SMA alloy for the actuator then becomes an important one, as is demonstrated. Also, the size of the actuator must be appropriate for the device: in case B here the SMA wire was chosen smaller than in case A, however the wire was still improperly sized. Case B would require an even smaller diameter SMA actuator to enable * excluding the two-way shape memory effect which can be induced upon proper material training. -14-

15 .5.4 linear, case A nonlinear, case A nonlinear, case B Martensite Fraction, ξ S.3.2 cool heat heat Temperature ( C) Figure 6: Stress-dependent martensite fraction in the wire with temperature. Cases as defined above. the same continuous, repeatable shape control shown by case A. One other point of interest is the maximum stresses obtained. In each case here, the material transformed fully to austenite upon heating as seen by the exit from the transformation strip in Figure 5 and the zero value of the martensite fraction in Figure 6. The size of the wire and beam here were carefully chosen to demonstrate reasonable behavior. The maximum stress in the wire is on the order of that seen in NiTi wire in reversible transformation regions and the critical stress for irreversible plastic deformation is avoided. These points illustrate the usefulness of models such as the one proposed here in the design stages to enable proper selection of actuator composition and size for given structural dimensions. The deflection of the beam tip in Figure 7 ranges from zero to 8 mm (for case A), quite significant shape response for a 4 mm length beam. The resulting hot and cold shapes are illustrated in Figure 8. The w(x) deformation curve of the beam is plotted to scale here so that the magnitude of the deformation and actual shape are represented. Note that the cold shape does not quite fully recover the original shape of the beam, as can also be seen by the small residual deflection in Figure 7. However, all subsequent temperature cycles return to this same cold shape (provided the SMA material has stable cyclic behavior; see -15-

16 later note). In Figures 5-7, case A shows only the first cycle response for a full temperature cycle between extremes. Figure 9 shows the distinction between the first cycle (beginning from an unstressed wire) and subsequent cycles. The origin of this behavior is the residual stress in the wire upon cooling: as the material passes through the martensite transformation under stress, it returns to the point where that martensite fraction was created under load initially. Only by unloading the specimen or device will the original shape be recovered. In cases where this is undesirable, the actuator could be fixed to the structure with an initial pre-stress, where the structure is designed so that the initial stress level would be the desired cold shape. In this event, the first cycle would be identical to all subsequent cycles. This phenomena of residual stress upon cooling in restrained recovery cases (exhibited as a difference between the first cycle and subsequent cycles) has been documented experimentally (Liang, 199; Mooi, 1992). The necessity of the nonlinear theory is also clearly demonstrated in this example. Since beam deflections are quite large, the linear response dramatically overpredicts the stresses in the SMA wire as well as the maximum beam deflection. For applications where precision is required, the nonlinear analysis would be essential Deflection at the end point (m) linear, case A nonlinear, case A nonlinear, case B heat cool heat. 2 4 Temperature ( C) 6 8 Figure 7: Tip deflection of the beam as controlled by temperature in the SMA actuator. Cases as described in Table

17 Undeformed shape Cool shape Hot shape Figure 8: Shape of undeformed beam, after heated up to 8 C, and after cooled down to 2 C. The results shown so far only illustrate full transformation, with temperature changes between the extremes of austenitic and martensitic behavior. Figures 9a and 9b also show the closed hysteresis loop and cyclic behavior between these temperature extremes, with the distinction between the first and subsequent cycles, as mentioned earlier, illustrated. In real applications, shape control would be a continuous and immediate response to any temperature change and would not necessarily involve full transformations in each cycle. Figure 1a illustrates an example of this nature with a particular temperature profile versus time and the resulting tip deflection. It is seen here that the shape of the beam can be well controlled by temperature change in the SMA wire. Figure 1b shows the stress history in the SMA wire during temperature cycling for the temperature history from figure 1a. Figure 1 illustrates several important issues. First, note that the tip deflections at the two 3 minima are different. This response captured by the SMA constitutive model is a real phenomena and occurs in this case because the stress history in the SMA wire prior to those two 3 states are different. In Figure 1b, by tracking the arrow numbers it is seen that the first 3 state is reached after arrow 2 with a previous stress maximum of 27 MPa, while the second 3 state is reached after arrow 5 with a previous stress maximum of 325 MPa. Shape memory response under partial transformation cycles is dependent upon the most recent transformation return point (Cory and McNichols, 1985; Ortín, 1991) and work is still underway to fully implement correct representation of this behavior into the kinetic law used here. The fact that the structural shape response in -17-

18 complicated temperature cycling is path dependent will be of critical interest in any applications where partial cycling occurs and precision is required. A given temperature will not correspond to a given deflection in general, rather the previous history must be known and accounted for in order to predict material response. A second issue that should be mentioned here is that most SMA materials undergo a change in behavior upon cyclic loading, resulting in appreciable differences in response during the first 1-15 cycles, after which the response stabilizes. Features of the initial changes that occur are decrease of the width of the hysteresis loop and a decrease in the maximum recoverable strain. It is assumed here that the material properties would be characterized after this initial stabilization. The consistency of SMA response under arbitrary load cycling have not been studied in depth to date and further work in this area would be required to assure accuracy of modeling and desirable performance of SMA response in applications. The shape control response can be actively implemented through resistive heating of the SMA wires, where a control program would regulate the current through the wires as desired for any given time to impart the desired deformation. Alternatively, the SMA actuator could function passively in response to the structure s environmental temperature as designed from the outset. Note that temperature variations throughout the SMA wire are not considered in the analysis outlined here. However, given appropriate thermal boundary conditions on the SMA wire, the appropriate heat transfer problem can be solved for the resistive heating case (Brinson, Bekker and Hwang, 1995) to obtain the temperature profile in the wire. This temperature profile would then simply be coupled into the SMA constitutive behavior and would only affect the determination of ε 1 in equation (2.14). The remainder of the solution would remain unchanged. Thus, using current as a control variable instead of temperature in the wire would be simple to implement. Insulated end conditions for the SMA wire would give rise to a nearly constant temperature distribution in the wire and would correspond to the results shown here. -18-

19 Deflection (x1-3 m) Temperature ( o C) Time Figure 9a: Beam tip deflection for more cyclic temperature profile, full transformation only Stress in SMA wire (x1 6 Pa) Temperature ( C) Figure 9b: Stress response in SMA wire with cyclic heating for temperature profile of Figure 9a. Difference between first cycle (starting with wire unstressed) and subsequent cycles illustrated

20 Deflection (x1-3 m) Temperature ( C) Time 6 8 Figure 1a: Beam tip deflection for more complex temperature profile. Stress in SMA wire (x1 6 Pa) Temperature ( C) Figure 1b: Stress response in SMA wire with cyclic heating for temperature profile of Figure 1a. -2-

21 8 Tip deflection (mm) beam size: l=4mm w=25mm t=1mm Figure 11: Temperature ( C) Experimental data for beam tip deflection using k-alloy (NiTiCu; 6mm diameter) as actuator on beam configuration 1 12 The final result to be included here is a preliminary experimental result for control of beam deflection with a SMA wire (as in Figure 2) through resistive heating. Unfortunately, the actuator material that was available for this study at Daimler Benz is a copper based alloy (NiTiCu) that undergoes a double phase transition and is not well characterized. Consequently, the results cannot be compared directly to the theory at this stage. However, the experimental results in Figure 11 clearly demonstrate all the features of previous predicted results for a beam controlled with a NiTi wire: the beam begins deflection as the wire is heated; after the transformation in the SMA is complete, the deflection remains constant; the shape is recovered in a hysteresis loop as the SMA cools; and a residual tip deflection upon cooling is seen. These experimental results then confirm the power of the predictive scheme developed here, and also point out the limitation: in order for the theory to function properly as would be required in application, all material properties must be well characterized and the constitutive law must be able to adapt to the particular material response idiosyncrasies. In the case shown here, the SMA actuator material undergoes an complex double-staged transformation which could be captured by a revision of the constitutive law. (This would -21-

22 require extensive testing for verification, the scope of which is beyond the current study.) However, the fact that the beam shape was controllable by temperature change in the SMA actuator and that the response is qualitatively comparable to the previous predictions for the NiTi alloy indicate that active shape control with SMAs by use of control schemes such as that described here is feasible. Conclusions This paper has considered active shape control of beams with shape memory alloy actuators and in particular has examined the potential of analytical predictions using temperature of the SMA as the control parameter. A constitutive law for shape memory materials was coupled with a nonlinear beam deflection analysis and a numerical approach was taken for final solution of the governing equations of the problem. The results agreed qualitatively well with preliminary experimental data and further experiments are planned. The results indicate that the shape of the beam can be controlled quite well with this method, provided material parameters are initially chosen correctly. In this particular example, if the effective beam stiffness is too low compared to the recovery forces generated in the SMA wire, the capability of multi-cycle temperature shape control is lost. The analysis provided here for a deflection of a simple beam could be extended readily to more sophisticated structures with attached or embedded SMA wire actuators. The key is to appropriately couple the structural response with the SMA actuator response via the SMA constitutive law. In order to accomplish this task, the SMA actuator material must be well characterized and the cyclic response known to be stable. Predictive models such as the one demonstrated here would facilitate design of applications using shape memory materials. The optimal geometry and arrangement of components could be determined; the best material properties for the SMA actuators and other structural parts could also be chosen based on predictive analysis. Given design plans could also be simulated and performance evaluated with the aid of similar models. Acknowledgments The authors gratefully acknowledge the support of the National Science Foundation (LCB, MSH). -22-

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