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Design and Characterization of SMA Helical Springs
This chapter presents the design, modeling and implementation of a NiTi SMA helical spring intented to be the active element of a miniature actuator for tactile stimulation. Existing constitutive models are used to predict its thermomechanical behavior. Following an experimental re-identification of the parameters involved in modeling, actuation speed and force-displacement experimental results are presented to validate simulation. The cyclic performance is then evaluated through a fatigue life analysis. Finally, some micro-structural observations on the effects of heat treatment and fatigue are presented and discussed.
Modeling of shape memory alloys
Extensive work has been devoted to model the SMA thermomechanical behavior. Ap-proaches range from atomic interactions ([Patoor87], [Falk89], [Kafka94], [Lexcellent96], [Nae03]) and thermodynamic formulations ([Ortin89a], [Ortin91b], [Berveiller91], [Boyd96a], [Boyd96b]) to phenomenological models based on experimental data ([Tanaka86], [Liang90], [Brinson93], [Graesser94], [Ivshin94]). A comprehensive review of work done in SMA mo-deling can be found in [Birman97] (195 references in 1997).
Models agree, regardless of their approach, that an SMA element can be considered as a three-element system in which thermal energy is converted into mechanical work. Figure 2.1 illustrates the SMA block diagram model.
In spite of their high quality, most of these models are difficult to use in practice: they lead to quite complex mathematical equations that require burdensome numerical solu-tion. Moreover, they involve parameters not clearly identified and depend on previously determined experimental data.
This chapter discusses the main features associated with SMA modeling through existing constitutive models suitable for easy analytical prediction of the SMA behavior: they use only the geometry of the SMA element and few material properties extracted from the manufacturer’s data sheets or from the literature. The models here presented can be applied to design SMA helical springs that are optimal in time response and force performance.
Modeling in this chapter starts developing the thermal dynamics of SMAs through a classic heat transfer analysis, then it integrates Liang-Rogers kinetic law to take into account the phase transformation phenomenon. Finally, the mechanical behavior is predicted using a Tanaka based model.
The thermal dynamics of SMAs
Heat transfer analysis
Let us start analyzing the SMA thermal behavior considering a system in thermal equi-librium (Figure 2.2) where Qs is the heat generated by an external source and Qconduction, Qconvection and Qradiation are the different dissipation forms of heat.
Concerning the characteristic length l, it is commonly defined as the volume of the material divided by its surface area (l=V/S ). However, in the case of thin metallic cylinders or wires (a particularly popular shape in SMAs), it is more pertinent to consider the characteristic length equal to the cylinder’s diameter (l=d ) when the element is placed perpendicularly to the flow and equal to its length (l=L) when placed parallel to the flow.
Parameter determination through an example
Now that equation (2.5) relates temperature to an input heat source, let us analyze the thermal behavior of an SMA at this point to highlight the basic concepts of its actuation.
Consider a NiTi SMA helical spring of wire diameter d, mean diameter D and N active coils to be electrically actuated at standard conditions (air at room temperature).
For electrically driven SMAs, heat is generated in accordance with the Lenz-Joule law: Qs = i2R, with: R = ρe φ (2.8)
where i is the electrical current applied and R is the element’s electrical resistance, which can be calculated from the material’s resistivity ρe, the element’s length L and its cross sectional area φ.
Thus, if the Biot number is valid, equation (2.8) together with classical geometry of a helical spring can further reduce equation (2.5) to:
Te + 4ρei2 (1 − e−at) (heating) 4h
T (t) = π2d3h with a = (2.9)
Te + (To (cooling) ρCd
− Te)e−at
Note that the only geometrical parameter that influences the thermal response of the SMA spring is the wire diameter d, which is evident if we consider that a spring is essentially a wire wound into a helix.
Consider a set of NiTi wires of diameter: 100, 150, 200 and 250 µm. The thermal properties of NiTi, directly extracted from the manufacturer’s data sheets ([Dynalloy]), as well as the air properties of interest for this study are given in table 2.1.
Figure 2.3(a) shows the evolution of heat-exchange coefficient h for this set of wires. Note that natural convection becomes more efficient as the diameter decreases.
Being the Biot number valid for all cases, equation (2.9) can be used. Figure 2.3(b) shows the time response of this set of wires to a step current of 500 mA. The duration of the pulse varies according to the time in which the SMA achieves its final temperature of transformation (Af ). Note that the actuation time increases with the diameter.
Figure 2.3(c) shows the time response to a set of step currents applied to the 200 µm diameter wire. The amplitude is varied from 400 to 800 mA. Note that the greater the current, the faster the SMA heats.
Figure 2.3(d) shows the effect of coefficient h during the heating process. Note that the greater the current, the less the influence of convection. Figure 2.3(e) shows the corresponding cooling response under conditions of free and forced convection with the environment. Note the acceleration effect of forced air convection on cooling.
A simple method to induce forced convection is by flowing air from an external fan. Compared to free convection, forced convection decreases the cooling time, for example, by a factor of 2 when using a 1 m/s air flow rate.
Thus, the total time response is composed of the time required for heating up and cooling down the SMA. Figure 2.3(f) shows the importance of using forced convection on cooling, mostly for continuous cyclic operation. Note that the SMA can be actuated 2 times during a free convection period by using forced convection.
Phase transformation
Unfortunately, actual SMA thermal behavior is more complex. As seen in chapter 1, SMAs, unlike traditional materials, show hysteresis during both M→A and A→M trans-formations. Physically, these hystereses are a dissipation and assimilation of latent heat due to phase transformation that tend to slow down both heating and cooling processes. Consequently, it is necessary to include this effect in the thermal analysis.
A successful empirical relation has been proposed by Liang and Rogers [Liang90], which describes the amount of martensite fraction ξ transformed on a temperature T. For cons-tant load conditions, ξ can be written for cooling as:
SMA mechanical behavior
Considering the simple geometry of a helical spring, Tobushi and Tanaka [Tobushi91b] have proposed and experimentally validated a simple elastoplastic constitutive relation to predict its SMA load-deflection behavior. This relation suggests a stress-strain-temperature dependence that is summarized in this section.
Strain-deflection behavior
The mechanical problem consists of a simple uniaxial tension exerted along the helical spring. The helical spring is mainly defined by its mean diameter D, the wire’s diameter d and the number of active coils N. Assuming that d is small in comparison to D, it can be considered that the cross section of the wire is only submitted to pure torsion.
Suppose then that only pure torsional stress τ acts in the cross section φ of the wire.
Therefore, the deformation in φ is only torsional strain γ.
Recall from classical theory of torsional elements, that the distribution of τ along φ is not uniform: a higher stress is observed in the outer diameter where the fibers are the most loaded (Figure 2.6(a)). Analogously for an SMA helical spring, during the loading process, the elastic to plastic transformation proceeds from the outer diameter to the center of the circular section. Due to the geometry of the section, the transformed zones are concentric rings (Figure 2.6(b)).
Behavior of the plastic region: evolution of rb
When an SMA helical spring is loaded and unloaded at a constant temperature above As, the torsional stress τ induced in the cross section of the wire varies as shown schematically in figure 2.8 [Tobushi91b].
In the loading process, an elastic behavior is first observed. In this region τ is proportional to the radius r (Figure 2.8(a)). When τ reaches the critical stress τ m, the SIM transfor-mation starts. Upon an increase of load, the plastic region expands from the surface into the center and reaches the boundary of the elastic region at radius rb (Figure 2.8(b)).
In the unloading process, τ first decreases proportionally to r following an elastic be-havior (Figure 2.8(c)). When τ reaches the critical stress τ a, the reverse transformation starts and proceeds into the center upon a decrease of load (Figure 2.8(d)). When the reverse transformation is over, the original configuration is recovered following once again an elastic behavior (Figure 2.8(e)).
Material constants
Concerning the material constants G, cm and ca, it can be considered:
(i) The shear modulus of elasticity G is defined by: G = E (2.29) 2(1+ν)
where E is the Young’s modulus of elasticity and ν is the Poisson’s ratio. SMAs are characterized by having a temperature dependent Young’s modulus: low in martensitic state that completely changes to a high one in austenitic state (Figure 2.9(a)). Both E and ν can be directly extracted from the manufacturer’s data sheets.
(ii) In the stress-temperature phase diagram, the material constant’s cm and ca describe the influence of stress on the phase transformation (Figure 2.9(b)). For helical springs, the values of cm and ca are estimated by the relationship between the torsional stress, torsional strain and temperature at the surface of the wire in torsion. Both values are commonly found in the literature for NiTi alloys.
Deformation behavior of an SMA helical spring
Let us analyze the mechanical behavior of an SMA helical spring by using equation (2.28). Consider an SMA helical spring of dimensions: wire diameter d=200 µm, mean spring diameter D=1.3 mm and the number of active coils N=12.
Figure 2.10 shows the simulated relations between load P and deflection δ in the loading-unloading processes at different constant temperatures for a maximum deflection of δmax=10 mm.
The simulated loading curves indicate that, in order to induce a 10 mm deflection to the SMA spring, the necessary load to be applied is approximately 180 mN at 20◦C, 410 mN at 78◦C and 720 mN at 98◦C. On the other hand, the unloading curves indicate the expected mechanical hysteresis of the SMA when the original configuration is being recovered. Note that for temperatures at or above the austenite state, the memory shape is perfectly recovered but for martensite ones, it does not.
Figure 2.11 shows the relation between load P and the boundary radius rb for the three constant temperatures considered. As mentioned in section 2.3.4, during the loading process, the SIM transformation starts at a point Sm on the surface of the wire and expands into the center. Upon an increase of P, the SIM transformation will reach a point Bm at δmax. During the unloading process, a point Ba is first reached following an elastic behavior. A further decrease of P triggers the reverse transformation, which contracts toward the surface until it reaches point Fa. Note that no reverse transformation starts for martensitic temperatures.
As previously mentioned, equation (2.28) permits to establish a relationship between load, deflection and temperature. Thus, an operational zone can be defined for an SMA element [Bellouard00]. Figure 2.12(a) shows the phenomenological representation of the spring’s SMA behavior in this 3D space. Note that, as introduced in figure 2.10, the tensile force developed by the spring against a load increases with temperature.
From figure 2.12(a) it is also possible to obtain the spring’s behavior at constant stress. Figure 2.12(b) shows the analytical behavior in the deflection-temperature space at dif-ferent stress constant values.
Table of contents :
Introduction
1 Toward a New Electronic Travel Aid for the Blind
1.1 Blind mobility aids
1.1.1 Classic aids
1.1.2 Electronic travel aids
1.1.3 Synthesis and discussion
1.2 Intelligent Glasses: a visuo-tactile ETA for the blind
1.2.1 Vision system
1.2.2 Tactile display
1.3 Criteria for the design of the IG tactile display
1.3.1 Guidelines for tactile displays: a psychophysiology approach
1.3.2 Choice of actuation technology
1.4 Essential features of shapememory alloys
1.5 Conclusion
2 Design and Characterization of SMA Helical Springs
2.1 Modeling of shapememory alloys
2.2 The thermal dynamics of SMAs
2.2.1 Heat transfer analysis
2.2.2 Heat-exchange coefficient h
2.2.3 Parameter determination through an example
2.2.4 Phase transformation
2.3 SMA mechanical behavior
2.3.1 Strain-deflection behavior
2.3.2 Temperature dependence
2.3.3 Load-deflection-temperature relation
2.3.4 Behavior of the plastic region: evolution of rb
2.3.5 Material constants
2.3.6 Deformation behavior of an SMA helical spring
2.4 Experimental procedure
2.4.1 Material
2.4.2 Parameter identification
2.4.3 Phase transformation parameters
2.4.4 Resistivity/Electrical resistance
2.5 Experimental evaluation of SMA thermomechanical properties
2.5.1 Thermal behavior of the NiTi spring
2.5.2 Mechanical behavior of the NiTi spring
2.5.3 Micro-structural observations on the NiTi specimens
2.6 Conclusion
3 Miniature SMA Actuator for Tactile Binary Information Display
3.1 SMA based actuators
3.1.1 Two-way actuator using a mass
3.1.2 Biased actuator
3.1.3 SMA antagonist actuator
3.1.4 Bistablemechanisms
3.2 An SMA based actuator for touch stimulation
3.2.1 Proposed solutions
3.3 Prototype and performance
3.4 Conclusion
4 A Low-Cost Highly-Portable Tactile Display with SMAs
4.1 Tactile display
4.1.1 Design and implementation
4.1.2 Prototype
4.2 Drive system
4.2.1 Electronic drive
4.2.2 Crossbar circuit analysis and control method
4.2.3 Software
4.3 A thermal array structure
4.4 Conclusion
5 Preliminary Psychophysical Evaluation of the Tactile Display
5.1 First evaluation of the tactile display
5.1.1 Experimental procedure
5.2 Experiment 1: tactile acuity and Braille
5.2.1 Session 1 – Location of a tactile stimulus
5.2.2 Session 2 – Location of tactile stimuli
5.2.3 Session 3 – Bimodal perception
5.3 Experiment 2: shape recognition
5.3.1 Session 1 – Primitive shapes
5.3.2 Session 2 – Various shapes
5.4 Experiment 3: space interaction and navigation
5.4.1 Session 1 – Direction recognition
5.4.2 Session 2 – Environment structure
5.4.3 Session 3 – Navigation and tactilemaps
5.5 Conclusion
Conclusions
Bibliography
Publications
Appendixes
