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Hindawi Publishing CorporationSmart Materials ResearchVolume 2011, Article ID 167195, 5 pagesdoi:10.1155/2011/167195

Research Article

A New Approach to the Design of Helical Shape MemoryAlloy Spring Actuators

Esuff Khan and Sivakumar M. Srinivasan

Department of Applied Mechanics, IIT Madras, Chennai 600036, India

Correspondence should be addressed to Sivakumar M. Srinivasan, [email protected]

Received 21 July 2011; Revised 24 October 2011; Accepted 25 October 2011

Academic Editor: Xianglong Meng

Copyright © 2011 E. Khan and S. M. Srinivasan. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Shape memory alloys (SMAs) are smart materials that have the ability to recover their original shape by eliminating residualdeformations, when subjected to adequate temperature rise (Shape memory effect). This special behavior attracts the use of SMAsas efficient stroke/force actuators. Most of the engineering applications require helical springs as actuators and proper design ofSMA helical spring actuators is very important. In the traditional design approach of SMA spring (Waram, 1993), only strainbetween linear zones was considered in order to simplify the design and to improve the fatigue life. Only modulus differencebetween high-temperature and low-temperature phase was utilized, and the transformation strain was not considered as thetotal transformation strain will be more and will degrade the performance of actuator. In the present design, we have shownthat transformation strain can be restricted by using hard stops and the partial transformation strain can be used to improvingthe capacity of SMA spring actuator. A comparison of the traditional design approach of SMA spring and the proposed designprocedure has been made to give an idea of its effect on the design and the related parameters.

1. Introduction

Shape memory alloys (SMAs) are smart materials whichundergo solid-to-solid phase transformations under thermo-mechanical loading exhibiting special intrinsic propertiessuch as the pseudoelasticity and the shape memory effect(SME). The shape memory effect of SMAs provides possibili-ties of using it as actuators [1]. Unlike other known actuators,the SMA actuators are nonlinear in behavior because of thenature of SMA as a material. This paper presents a simpleprocedure for the design of helical SMA spring actuators,taking into account their nonlinear behavior and the intro-duction of hard stops.

SMAs generally exist in two phases: austenite, a high-temperature phase, and martensite, a low-temperaturephase. Martensite is soft in nature, can undergo finite defor-mations during loading, and leaves residual strains whenunloaded. During loading, the stress and strain response isessentially linear until it reaches a critical stress value. Atthis critical stress value, it could enter into transformationzone where twinned martensite gets converted to detwinned

martensite, undergoing large strains for small stress incre-ments. This strain is called the transformation strain. Atany point, if the temperature is increased well beyond theaustenite temperature, the SMA recovers the residual strain(phase transformation from martensite to austenite), pro-ducing an actuation stroke. During this phase transforma-tion process of an SMA, large loads and/or displacements canbe generated in a relatively short period of time making thiscomponent an interesting mechanical actuator.

Because of such remarkable properties, SMAs have founda number of applications in different areas [2]. Han et al. [3]showed how SMA spring actuator can be used to enhancethe buckling capacity of columns. Lee and Lee [4] exploredthe application of SMA spring actuator in active cathetermedical device. Spinella and Dragoni [5] showed that theactuator performance can be improved by using the hollowSMA springs. They proved that by emptying the inefficientmaterial from the wire center, the hollow section features alower mass, lower cooling time, and lower heating energythan its solid counterpart for given strength, stiffness, anddeflection. Thus, it becomes necessary to design the SMA

2 Smart Materials Research

0

50

100

150

200

250

0 0.5 1 1.5 2 2.5

Strain (%)

Stress versus strain

MartensiteAustenite

300

Stre

ss (

MP

a)

Figure 1: Stress versus strain response in linear zone.

actuator appropriate to an application to allow for efficientperformance.

In literature, the design approaches proposed so far haveassumed a linear mechanical response for both martensiteand austenite [2–6]. Essentially transformation inducedchange in modulus is assumed in these approaches to be re-sponsible for the stroke in the actuator. Waram’s designapproach [6] for SMA spring is one such popular designapproach that considers mainly the modulus differencebetween the martensite and austenite phases in the design asshown in Figure 1. The reason for such a design approachbeing popular could be that in the transformation region,SMA undergoes large strains for small increment of stressvalue. In addition, allowing high deformations that occurin this transformation regime may render degradation inits functional as well as structural performance. In Waram’sdesign, the martensite phase strain is restricted in design toallow for a very good fatigue life. The stroke obtainable fromsuch designs is low compared to that which can be achievedby allowing for transformation strains.

Also, it should be stressed that unlike in most otheractuators, the material of SMA actuators is nonlinear inmechanical response beyond a critical stress. The responsebeyond this stress is involved in the effective actuator actionand that could make the design complex because of thenonlinearity involved. In this paper, we propose an approachwhich can utilize the partial transformation strain into thespring design. Transformation strain is considered to berestricted by using the external hard stop, so that onlypartial detwinning takes place during loading. By includingthe partial transformation strain into the design, the actu-ation capacity of SMA helical spring may improve withoutcompromising significantly on its fatigue life.

In this paper, the design parameters are analyzed withthe consideration of transformation strains. The nonlinearbehavior of martensite phase is idealized as an elastoplastic

response to reduce the complexity in the formulation andis compared with the traditional linear approach. Linearapproach is first presented in the next section (Section 2).Then, the nonlinear analysis and design approach is dis-cussed in Section 3. The comparison between the twoapproaches based on different design parameters is made inSection 4 before making concluding remarks on the need fornonlinear approach to the design of helical SMA springs.

2. Linear Approach (Waram’s Design [6])

In the standard design procedure, the aim is to arrive at thewire diameter, d, the spring diameter, D, and the numberof turns, n, for a spring that will deliver the required force,P, and a stroke, S, in a full actuation cycle. The appropriatevalues of shear moduli, Gh, in the hot and, Gl, in thecold states, the maximum allowable shear stress, τc, in theaustenitic state, and the limit on shear strain, γ, in the coldstate are the input parameters in this design procedure. Theprocedure is described below [6], in brief.

The maximum shear stress allowable in the austenitestate, τc, provided from fatigue considerations puts a con-straint on maximum allowable force on the spring, Pmax. Theshear stress in the wire and the force on the spring can berelated using

τc = W8CPmax

πd2, (1)

where the Spring Index, C = d/D, generally taken to be be-tween 6 and 10 and W , is the Wahl correction factor.

From (1), it is possible to find out the required wirediameter of the spring given the maximum allowable designload. Equation (2) is an expression that allows one todecide on the number of coils using “d,” “D,” S and theallowable shear strain difference Δγ = (γl − γh) where γlis the maximum low-temperature martensite shear strainallowable and γh is the high temperature shear strain. Thelimit on the low temperature martensitic shear strain is toensure adequate cycle life without functional degradation.

The number of coils (n) can be calculated by

n = Sd

πΔγD2. (2)

However, a closer look at the design equations reveals thatonce the value of P and S are chosen, the required numberof coils (n) gets automatically fixed without requiring anexternally imposed limit on the shear strain recommendedin the standard design.

The deflection of the spring (δ) in hot or cold state,assuming the material to be elastic is given by

δ = 8PD3n

Gd4, (3)

where P is the force exerted by the spring and G, the rigidityor shear modulus in the appropriate state.

The stiffness (K) can be calculated as

K = Gd4

8D3n, (4)

Smart Materials Research 3

w

w

Austenticdeflection

δh

δlMartensiticdeflection

Stroke

Hot ColdO A B

Hot

Figure 2: Schematic diagram spring actuator.

0

100

200

300

400

500

600

700

800

0 2 4 6 8 10

Stre

ss (

MP

a)

Strain (%)

MartensiteAustenite

Stroke

Stress versus strain

Figure 3: Idealized stress versus strain response of SMA.

where G pertains to the appropriate shear modulus cold, Gl

or hot, Gh.Therefore, the total load to stroke relationship can be

obtained for the spring in the cold and the hot state as shownin Figure 2. Kl and Kh stand for low- and high-temperaturestiffness.

The maximum possible stroke (Smax) can now be ex-pressed as

Smax = P[

1Kl− 1

Kh

]= 8PD3n

d4

[1Gl− 1

Gh

]. (5)

In the above approach, the stroke depends upon the differ-ence in the reciprocal of the shear moduli of the martensiteand the austenite phases.

3. Proposed Nonlinear Approach

In this approach, partial transformation strain is utilized.Considering the total transformation strain will degrade thecyclic life of actuator. The transformation strain can be

Hard stops

Austenticdeflection

Martensiticdeflection

Stroke

Hot ColdO A B

Hot

Dd

w

w

Figure 4: SMA spring actuator with hard stop.

γ

τ

τγ

Figure 5: Simplified martensite phase response.

restricted using hard stops, which are highly stiff materials.By using these hard stops as shown in Figures 3 and 4, nowin the martensite phase, the material undergoes large strain,and appropriate actuation will give more stroke compared tolinear approach. Figure 5 is the simplified martensite phasebehavior of SMA which will consist of both twinned anddetwinned martensite. In Figure 6 the stress distribution overthe wire cross-section is shown. Till τ reaches τy , the materialbehaves linearly, and then transformation starts from theouter region and move toward the inner core.

At the Maximum Elastic Torque:

Ty = J

cτy = 1

2πc3τy ,

φy =Lγyc

,

(6)

where Ty , τy , γy are the maximum torque, stress, and strain,respectively, at which transformation starts, J , the polar ofmoment of inertia, and φy the angle of twist at maximumstress.

4 Smart Materials Research

τ

= τγ

CO ρ

τmax

(a) Linear

τ

CO ρ

τγ

ρy

(b) Non-Linear

Figure 6: Stress distribution over the wire cross section.

Practically, a helical spring is subjected to both torsionalshear stress and direct shear stress but a closer look at thestress values gives an understanding that the torsional shearstress is much higher (2 × spring index) compared to thedirect shear stress, and therefore, the direct shear stress effectis neglected in this study.

As the torque is increased, a transformed region developsaround an elastic core:

ρyγy= ρ

γ,

T = 23πc3τy

(1− 1

4

ρ3y

c3

)= 4

3Ty

(1− 1

4

φ3y

φ3

).

(7)

Also the above equation can be simplified as:

P = 23π

Cc2τy

(1− 1

4

γ3y

γ3

). (8)

From the above equation, it is possible to calculate thewire diameter (d = 2 ∗ c) once the total strain value isprovided. The value of the spring index (C) is assumed tobe 6.As ρy → 0, the torque approaches a limiting value, (seeFigure 7):

Tp = 43Ty = critical torque. (9)

Since there is no warping in circular prismatic members,(2) can also be used to find out the number of turns. Oncethe diameter (d) and number of turns (n) are calculated, theremaining design parameters are derived.

3.1. Comparison of Linear and Nonlinear Approaches. Using(1) and (8), a comparison can be made between linear andnonlinear analyses based on design parameters using samplecalculations. From (1), by providing the values of requiredmaximum actuation force (Pmax), spring index (C), andtransformation stress (τc), we can calculate the required wirediameter. Similarly, from (8), we can calculate the requiredwire diameter for the nonlinear zone by providing the above

φY 2φY 3φY φ

YTY

Tp = 4/3TY

0

T

Figure 7: Torque versus angle of twist.

parameters along with the total strain (γ) value. In Figure 8,the variation of wire diameter (d) with respect to strain (γ)at a constant deformation (δ5 = 5 mm deformation) and fordifferent actuation force values (P = 5 N, 10 N, 15 N, 20 N)is shown. The transformation shear stress was consideredto be 250 MPa and the strain at the end of linear zonewas considered as 2%. The spring index was assumed tobe 6. Using (2), the number of turns (n) can be calculatedby providing the values of wire diameter (d), deformation(δ) and the total strain (γ). Using the same equation,relationship between strain and the number of turns (n)can be obtained which is plotted for different actuationforce values. Number of turns is directly proportional to thedeformation (δ) with δ = 5 mm.

From Figure 8, it is noted that as the strain enters intotransformation region (>2%), there is an abrupt drop in thedesign diameter till strain reaches a value of around 3%.Then, it stabilizes to a particular value for higher strains.

Smart Materials Research 5

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1 0.12

Wir

e di

amet

er (

mm

)

Strain

L P5 δ5NL P5 δ5L P10 δ5NL P10 δ5


Figure 8: Variation of diameter (d) with respect to total strain inmartensite phase.

0.5

1.5

2.5

3.5

4.5

Nu

mbe

r of

tu

rns

0 0.02 0.04 0.06 0.08 0.1 0.12

Strain

0

1

2

3

4



Figure 9: Variation of “n” with respect to total strain in martensitephase.

It can be noted that as for higher strain values, (8) asymp-totically reduces to

P = 23π

Cc2τy. (10)

So, for large values of shear strain, the wire radius can befound out by simplifying the above equation as

c =√

3F2πτy

. (11)

Figure 9 shows that the required number of turns var-iation follows the same trend as required wire diameter as

the observed in Figure 8. While the difference between thetwo models is considerable for lower values of strain, it comesdown and stabilizes for higher strain values. However, valuesobtained from the nonlinear model are more than the linearmodel values due to the fact that number of turns (n) isinversely proportional to the wire diameter (d)—(2). Also,with increase in the actuation force, the difference betweentwo curves is coming down. Thus, when transformationstrain is taken into consideration, it is suggested to use (11)and (2) for the calculating design parameters.

4. Conclusions

The design parameters are analyzed in the paper with theconsideration of transformation strains in the design of SMAactuator springs. An idealized nonlinear behavior of SMAis used to reduce the complexity in the formulation and iscompared with the traditional linear approach. Additionally,external hard stops are assumed in the analysis to constrainfor allowable total transformation strain that is closelyassociated with the fatigue life of the spring. While therequired wire diameter for the same actuation force is lowerwhen nonlinear behavior is considered, the number of turnsneeds is higher for the same stroke. It is shown that theeffect of transformation strain can be taken into account inthe design procedure modified for the nonlinear behavior ofthe SMA spring. New simplified relations are also obtainedfor higher strain values. Further studies have to be carriedout to quantitatively define the fatigue-sensitive problemsassociated with the transformation strain.

References

[1] D. J. Leo, Engineering Analysis of Smart Material Systems, JohnWiley & Sons, New York, NY, USA, 2007.

[2] R. A. A. de Aguiar et al., “Shape memory alloy helical springs:modeling, simulation and experimental analysis,” in ABCMSymposium Series in Solid Mechanics in Brazil, vol. 2, pp. 169–182, 2009.

[3] H. P. Han, K. K. Ang, Q. Wang, and F. Taheri, “Bucklingenhancement of epoxy columns using embedded shape mem-ory alloy spring actuators,” Composite Structures, vol. 72, no. 2,pp. 200–211, 2006.

[4] H. J. Lee and J. J. Lee, “Evaluation of the characteristics ofa shape memory alloy spring actuator,” Smart Materials andStructures, vol. 9, no. 6, pp. 817–823, 2000.

[5] I. Spinella and E. Dragoni, “Analysis and design of hollow heli-cal springs for shape memory actuators,” Journal of IntelligentMaterial Systems and Structures, vol. 21, no. 2, pp. 185–199,2010.

[6] T. C. Waram, Actuator Design Using Shape Memory Alloys, TomWaram, Ontario, Canada, 2nd edition, 1993.

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