research article nonlinear analysis of actuation ... · the maximum de ection of the sma composite...

Research ArticleNonlinear Analysis of Actuation Performance of Shape MemoryAlloy Composite Film Based on Silicon Substrate

Shuangshuang Sun1 and Xiance Jiang2

1 College of Electromechanical Engineering Qingdao University of Science amp Technology Qingdao 266061 China2Department of Mechanical and Aeronautical Engineering Naval Aeronautical Engineering Academy Qingdao BranchQingdao 266041 China

Correspondence should be addressed to Shuangshuang Sun sunkirasohucom

Received 25 September 2013 Revised 24 January 2014 Accepted 12 February 2014 Published 20 March 2014

Academic Editor Mahmoud Kadkhodaei

Copyright copy 2014 S Sun and X Jiang This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The mechanical model of the shape memory alloy (SMA) composite film with silicon (Si) substrate was established by the methodof mechanics of composite materials The coupled action between the SMA film and Si substrate under thermal loads was analyzedby combining static equilibrium equations geometric equations and physical equations The material nonlinearity of SMA andthe geometric nonlinearity of bending deformation were both considered By simulating and analyzing the actuation performanceof the SMA composite film during one cooling-heating thermal cycle it is found that the final cooling temperature boundarycondition and the thickness of SMA film have significant effects on the actuation performance of the SMA composite film Besidesthe maximum deflection of the SMA composite film is affected obviously by the geometric nonlinearity of bending deformationwhen the thickness of SMA film is very large

1 Introduction

Microactuators are key parts for microactuating in micro-electro-mechanical systems (MEMS) Compared with othermicroactuators bimorph thin film SMA microactuatorscomposed of thin film SMAs and Si substrates have been paidmuch more attention recently [1ndash3] due to the simple struc-ture large work density and rapid response Some prototypesof bimorph thin film SMA microactuators have been fabri-cated for instancemicrovalves micropumpsmicrogrippersand so forth The review of the SMASi composite film usedas microactuators in MEMS can be made to the references ofNing et al [4] and Fu et al [5 6]

To make SMASi composite film be used as good-performance and high-efficient microactuators in MEMSand also to reduce the cost of research and developmentpredicting and analyzing the actuation performance ofSMASi composite film theoretically are very necessary Afew researchers have done some work in this aspect Forexample Wang [7] studied the actuating performance ofSMASi composite film by the bimetal theory [8] for linear-elastic materials But the theoretical result is about four times

of that of tests Cheng [9] made some optimization researchon the dynamic actuation characteristics of TiNi(Cu)Sicomposite diaphragm by the commercial software ANSYSand gave some qualitative comparison with experimentsWang et al [10] simulated the actuation displacements of aSiTiNi bimorph structure by ANSYS and compared themwith test results yet larger differences exist between them Itis noted that these simulations by ANSYS were all based ongeometric linear analysis and the nonlinear physical relationof SMAwas input by discrete data Authors in [11] have madesome preliminary studies on the modeling and actuationcharacteristics of NiTiSi composite film by the method ofmechanics of materials but only the material nonlinearity ofSMA was considered Generally the actual thickness of theSMASi composite film used as microactuators is very smallwith sizes from a few micrometers to dozens of micrometersBut the largest bending displacement of SMA composite filmalong thickness is usually approach to or much larger thanthe size of thickness during working which produces thegeometric nonlinearity of bending deformation So the effectsof geometric nonlinearity of deformation on the actuationperformance of SMASi composite film should be considered

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 728950 7 pageshttpdxdoiorg1011552014728950

2 Mathematical Problems in Engineering

when modeling and predicting Yet little work was reportedin this aspect

We have done some work [12] on the actuation perfor-mance of NiTiSi composite film by considering both thematerial nonlinearity of SMA and the geometric nonlinearityof bending deformation yet only the maximum actuationdisplacement of simply-supported NiTiSi composite filmvarying with three final cooling temperatures and the maxi-mum actuation displacement of cantilever NiTiSi compositefilm varying with SMA thickness were studied As for therelations between the maximum actuation displacementand the phase transformation fraction and stress of SMAduring thermal cycles they were not of concern Additionallyhow boundary conditions affect the actuation performanceof NiTiSi composite film was not discussed theoreticallyand numerically Based on our previous work [11 12] theSMASi composite film will be further studied in this paperto comprehensively predict its actuation performance byconsidering both the material nonlinearity of SMA and thegeometric nonlinearity of bending deformationThe SMASicomposite film subjected to thermal loads will be modeledaccording to the method of mechanics of composite mate-rials The effects of the final cooling temperature boundarycondition thickness of SMAfilm and geometric nonlinearityof deformation on the actuation performance of SMASicomposite film will be discussed in detail The influence ofthe phase transformation fraction and stress of SMA on themaximum actuation displacement of SMASi composite filmwill also be explored

2 Modeling of SMASi Composite Film

The SMASi composite film is considered as a compositebeam in this paper Figure 1(a) shows the schematic diagramof an SMASi composite film element before working where119905119886and 119905

119898denote the thickness of the SMA film and Si

substrate respectively Figure 1(b) shows the working state ofa microsegment of the SMA composite film element underthermal loads where119872 is the bendingmoment in each cross-section of the SMA composite film and d120579 is the relativerotation angle of the left and the right cross-sections of themicrosegment with length d119909

To analyze the actuation performance of the SMA com-posite film under thermal loads the coordinate system isestablished as shown in Figure 1(a) The coupled actionbetween the SMAfilm and the Si substrate will be analyzed bycombining static equilibriumequations geometric equationsand physical equations in the following sections

21 Static Equilibrium Equations Because no mechanicalloads are applied on the SMASi composite film except forthermal loads the following static equilibrium equations [13]must be satisfied for the microsegment in Figure 1(b)

119865119909= int

119905119898

0

120590119898sdot 119887 sdot 119889119910 + 120590

119886sdot 119887 sdot 119905119886= 0

119872119909119911

= int

119905119898

0

120590119898sdot 119910 sdot 119887 sdot 119889119910 + 120590

119886sdot 119910119886sdot 119887 sdot 119905119886cong 119872

(1)

where 119865119909and 119872

119909119911denote resultant internal forces and

internal moments in each cross-section of the microsegmentin Figure 1(b) 120590

119898and 120590

119886denote the stress of Si substrate

and the SMA film respectively 119887 is the width of theSMA composite film 119910

119886is the transverse coordinate

of the midheight of SMA film with 119910119886

= minus1199051198862 The

bending moment 119872 has the approximate expression [11]of 119872 = 120590

119886119905119886119887(119905119898

+ 119905119886)2 Note that the stress of SMA is

considered to have no change of gradient along its thicknessas the thickness of SMA is usually smaller than that of Sisubstrate

22 Geometric Equations Assume the curvature radius andstrain of the interface in the SMA composite beam tobe 1205880and 1205760 From the geometric relation shown in Figure 1

we have the following expression

1205760= 1205880(d120579d119909

) minus 1 = 12058801198960minus 1 (2)

where 1198960= d120579d119909

Then the strain 120576119898

of an arbitrary fiber 119887119887 with thedistance 119910 to the interface in Si substrate can be expressedas

120576119898= 1205760+ 1198960119910 (3)

For the SMA film the strain 120576119886of any points is assumed

to be equal with the following approximate expression

120576119886= 1205760+ 1198960(minus

119905119886

2) (4)

And the curvature 119896 of themidplane of the SMA compos-ite film can be written as

119896 =1

[1205880+ ((119905119886+ 119905119898) 2 minus 119905

119886)] (5)

23 Physical Equations Assume that the constitutive relationof Si substrate is linear elastic though geometrically nonlineardeformation will occur in the SMASi composite film duringworking Then the stress-strain relationship of Si substrateunder thermal loads can be written as

120590119898= (120576119898minus 120572119898sdot Δ119879) sdot 119864

119898 (6)

where 120590119898 120572119898 and 119864

119898denote the stress thermal expansion

coefficient and elastic modulus of Si substrate respectivelyΔ119879 is the change of temperature

For the SMA film the Liang-Rogersrsquo constitutive model[14] modified by Wang [7] is employed in the following

120590119886= 119863 (120585) 120576

119886+ Ω (120585) 120585 + 120579 (120585) 119879 + 119878

0 (7)

where 1198780= 1205901198860

minus 119863(1205850)1205761198860

minus Ω(1205850)1205850minus 120579(1205850)1198790 119863Ω and

120579 denote the elastic modulus phase transformation modulusand thermal elastic modulus of SMA respectively Theyare all the functions of martensite fraction 120585 119879 representstemperatureThe quantities with the subscript 0 denote initialconditions In addition we have three expressions 119863(120585) =

119863119860+ 120585(119863

119872minus 119863119860) Ω(120585) = minus120576

119871119863(120585) and 120579(120585) = minus120572119863(120585)

Here 119863119860and 119863

119872are the austenite and martensite elastic

modulus 120576119871is the maximum residual strain and 120572 is the

thermal expansion coefficient of SMA

Mathematical Problems in Engineering 3

Si substrate film

O

ta

tm

SMA film

xy

dx

y

bb

(a)

M M

y

1205880

d120579

b998400 b998400

(b)

Figure 1 Schematic diagram of the SMASi composite film element (a) before thermal loads are applied and (b) microsegment after thermalloads are applied

24 Phase Transformation Equations of SMA When SMAundergoes phase transformations the following phase trans-formation kinetics proposed by Liang-Rogers [14] are em-ployed

Conversion frommartensite to austenite for119862119860(119879minus119860

119891) lt

120590119886lt 119862119860(119879 minus 119860

119904)

120585 =1205850

2cos [119886

119860(119879 minus 119860

119904) + 119887119860120590119886] + 1 (8)

Conversion from austenite to martensite for 119862119872(119879 minus

119872119904) lt 120590119886lt 119862119872(119879 minus119872

119891)

120585 =(1 minus 120585

0)

2cos [119886

119872(119879 minus119872

119891) + 119887119872120590119886] +

(1 + 1205850)

2 (9)

In (8) and (9) 119886119860 119886119872 119887119860 and 119887

119872arematerial constants of

SMA with the relationship 119886119860= 120587(119860

119891minus119860119904) 119886119872

= 120587(119872119878minus

119872119891) 119887119860

= minus119886119860119862119860 and 119887

119872= minus119886119872119862119872 Additionally 119872

119878

119872119891 119860119904 and 119860

119891denote the martensite start martensite fin-

ish austenite start and austenite finish temperatures of SMAin the stress-free state respectively 119862

119860and 119862

119872are material

parameters that describe the relationship of temperature andthe critical stress to induce transformations

Combining (1)ndash(7) we get the following three expres-sions

120590119886=

120572119898Δ119879 minus (120576

119871120585 + 120572119879) + 119878

0119863 (120585)

1119863 (120585) + [119905119886(71199052

119898+ 12119905119886119905119898+ 61199052

119886)] (119864

1198981199053

119898) (10)

1205760= 120572119898sdot Δ119879 minus

120590119886sdot 119905119886(7119905119898+ 6119905119886)

119864119898sdot 1199052

119898

(11)

1198960=

12120590119886sdot 119905119886(119905119898+ 119905119886)

119864119898sdot 1199053

119898

(12)

Note that (10) is a nonlinear equation when phasetransformations occurs in SMA as 120590

119886 120585 and 119879 are coupled

together If the phase transformation is from martensite toaustenite then 120590

119886can be obtained by solving (10) and (8)

by using the subroutine ZREAL given in Visual Fortran Ifthe phase transformation is from austenite to martensite

then 120590119886can be obtained by solving (10) and (9) by using the

same subroutine But if there are no phase transformationsin SMA 120585 is always a constant Then (10) is a linear equationabout 120590

119886and 119879 The current stress of SMA corresponding to

the current temperature can be directly obtained by solvingthis equation

Once 120590119886is obtained then 120576

0and 1198960can be calculated out

from (11) and (12) respectively And the curvature 119896 of themidplane can be obtained from (5)

3 Bending Deformation of SMASiComposite Film

The maximum displacement produced in working is an im-portant index of weighting the actuation capability of SMASicomposite film In this section the maximum deflection(ie displacement of centroid in each cross-section along thethickness direction) expressions of SMASi composite filmwill be derived for two common boundary conditions bothends are simply supported (S-S) and one end is fixed and theother is free (F-F)

The general expression of the deflection equation of SMASi composite film can be written as

d2119908d1199092

[1 + (d119908d119909)2]32

= 119896 (13)

where 119908 denotes the deflection of SMASi composite film Itis defined to be positive along the positive 119910-axis directionshown in Figure 1

In the case of small deformations (13) can be simplifiedas the following linear differential equation

d2119908d1199092

= 119896 (14)

Solving the differential equations of (13) and (14) by com-bining boundary conditions we get the following expressionsof the maximum deflection of SMA composite film


Table 1 Material properties of NiTi-SMA and Si [7]

NiTi SMA

119872119891= 23∘C 119862

119872= 113MPa∘C

119872119904= 50∘C 119862

119860= 45MPa∘C

119860119904= 62∘C 119863

119872= 13GPa

119860119891= 100

∘C 119863119860= 30GPa

120572 = 88 times 10minus6∘C 120576

119871= 1

Si 119864119898= 1805GPa 120572

119898= 233 times 10

minus6∘C

Under S-S boundary condition

119908119873

max =1

119896minus1

119896

radic1 minus (119871 sdot 119896

2)

2

119908119871

max =1

81198961198712

(15)

Under F-F boundary condition

119908119873

max = minus1

119896+1

119896

radic1 minus (119896119871)2

119908119871

max = minus1

21198961198712

(16)

where 119908119873

max and 119908119871

max represent the maximum deflectionsobtained from the nonlinear differential equation of (13) andthe linear differential equation of (14) respectively 119871 denotesthe total length of the SMASi composite film

4 Numerical Simulation and Discussions

In this section the actuation performance of the NiTi-SMA composite film with Si substrate in a cooling-heatingthermal cycle will be simulated and discussed The NiTi-SMA composite film is cooled first from the crystallizationtemperature 119879cry (550

∘C) to the final cooling temperature 119879119891

and then it is heated from 119879119891to 119879cry again

Material parameters of NiTi-SMA and Si used in sim-ulation are taken from Wang [7] and shown in Table 1Unless otherwise specified the geometric sizes of the SMAcomposite film are 119905

119886= 5 120583m 119905

119898= 20 120583m and 119871 = 3mm

The boundary condition is S-SThe initial stress and strain ofNiTi SMA at 119879cry are zero 119879119891 = 30

∘C The simulation resultsare given in Figures 2ndash4

To observe the influence of the final cooling temperature119879119891on actuation performance of SMA composite film the

SMA composite film at 119879cry are firstly cooled to 30∘C

40∘C and 50

∘C respectively and then heated to 119879cry againIn the three cooling-heating thermal cycles the maxi-mum deflection 119908

119873

max of SMA composite film martensitefraction 120585 and stress 120590

119886of SMA varyingwith temperature are

given in Figures 2(a) 2(b) and 2(c) respectivelyFrom Figure 2(a) we can see that 119908119873max increases first

with decreasing temperature during cooling due to the actionof thermal stresses produced between SMA and Si sub-strate then it decreases rapidly with decreasing temperaturewhen SMA undergoes phase transformation from austeniteto martensite for each case of 119879

119891 In addition the lower

the value of 119879119891is the more obviously the value of 119908119873max

decreases This is because that more austenite transforms tomartensite at lower119879

119891(see Figure 2(b)) For119879

119891= 30∘C 119908119873max

decreases to zero at about 40∘C during cooling and then

it continues to decrease to some negative value with theongoing decreasing of temperature During heating 119908119873max foreach case of 119879

119891has a slight change first and then its value

increases rapidly with increasing temperature once reversephase transformation from martensite to austenite occurs inSMA When SMA ends reverse phase transformation thevalue of 119908

119873

max decreases again with increasing temperaturedue to thermal stress between SMA and Si substrate Itrecovers to zero again when temperature is raised to 119879cry

Figure 2(b) shows martensite fraction of SMA varyingwith temperature during the three cooling-heating thermalcycles It can be seen from this figure that martensitefraction increases rapidly from zero with the decreasing oftemperature during cooling when SMA undergoes phasetransformation from austenite to martensite And the lowerthe value of 119879

119891is the larger the martensite fraction is at

the end of cooling For instance the martensite fractionis about 081 for 119879

119891= 30

∘C And it is about 007 for119879119891

= 50∘C While martensite fraction decreases rapidly

with increasing temperature during heating when reversephase transformation occurs in SMA When temperature israised to 119879cry all martensite is completely transformed intoaustenite

Figure 2(c) illustrates the variation of SMA stress withtemperature during the three cooling-heating thermal cyclesWe can see that the stress of SMA reaches very high valuebefore phase transformation occurs in SMA during thecooling stage for each case of 119879

119891 This is induced by the

bimetal effect between SMA and Si substrate But the stressof SMA decreases dramatically with temperature once SMAundergoes phase transformation from austenite tomartensiteduring the ongoing cooling stage For 119879

119891= 30

∘C themaximum tensile stress of SMA is decreased with the largestamplitude which decreases to zero first at about 40∘C andthen continues to decrease to negative values The stress ofSMA increases dramatically when reverse phase transforma-tion occurs in SMA during the heating stage and it becomeszero again when temperature reaches 119879cry

In summary Figures 2(a) 2(b) and 2(c) tell us thatthermal stresses produced between SMA and Si duringcooling can be released completely by the shape memoryeffect of SMA And the maximum deflection of the NiTi-SMA composite film induced by thermal stresses can alsobe recovered completely if 119879

119891is properly set during cooling

This is very important for the design of bimorph thin filmSMAmicroactuators

Themaximumdeflections119908119873max varyingwith temperaturefor SMA composite films under S-S and F-F boundaryconditions are given in Figure 3 to observe effects of differentboundary conditions during the cooling-heating thermalcycle

It can be seen from Figure 3 that boundary conditionshave significant effects on the maximum deflection of SMAcomposite film Under S-S boundary condition the max-imum deflection keeps approximately all positive valuesduring the cooling-heating thermal cycle While most partof the maximum deflection of SMA composite film keepsnegative values under F-F boundary condition Additionally


0 100 200 300 400 500 600

000

Max

imum

defl

ectio

n of

SM

A co

mpo

site fi

lm (m

)

Temperature (∘C)

Cooling

Heating

Tf = 30∘CTf = 40∘C

Tf = 50∘C

900times 10minus5

750

600

450

300

150

minus150

(a)

0 100 200 300 400 500 600

Temperature (∘C)

Tf = 30∘CTf = 40∘C

Tf = 50∘C

Cooling

Heating

09

08

07

06

05

04

03

02

01

00

minus02

minus01

Mar

tens

ite fr

actio

n of

SM

A

(b)

0 100 200 300 400 500 600

Temperature (∘C)

Tf = 30∘CTf = 40∘C

Tf = 50∘C

Cooling

Heating

800E + 007

600E + 007

400E + 007

200E + 007

000E + 000

minus200E + 007

Stre

ss o

f SM

A (M

Pa)

(c)

Figure 2 Effects of the final cooling temperature 119879119891on SMA composite film (a) the maximum deflection of SMA composite film varying

with temperature (b) the martensite fraction of SMA varying with temperature and (c) the stress of SMA varying with temperature

the largest absolute value of the maximum deflection underF-F boundary condition is more than four times of thatunder S-S boundary condition during the whole thermalcycle This results from higher thermal stresses producedin SMA composite film under F-F boundary condition Wealso can see from Figure 3 that the largest absolute valueof the maximum deflection under each boundary conditionrecovers fully to zero first and then increases slightly to thereverse direction when SMA undergoes phase transforma-tion from austenite to martensite And the absolute valueof the maximum deflection under each boundary conditionincreases dramatically once the reverse phase transformationoccurs in SMA It begins to decrease slowly after the reverse

phase transformation is finished during the ongoing heatingstage

Figures 4(a) and 4(b) show effects of SMA thicknesson the maximum deflection of SMA composite film underF-F and S-S boundary conditions respectively Both 119908

119873

maxand 119908

119871

max are given in the two figures to observe influence ofthe geometric nonlinearity of bending deformation

From Figure 4 we can see that the maximum deflectionsof 119908119873max and 119908

119871

max increase with the increasing of the SMAthickness This is probably due to the fact that the SMASicomposite filmwill produce higher thermal stresses when theSMA film is thicker But 119908119871max is always smaller than 119908

119873

maxfor each SMA thickness under each boundary condition


0 100 200 300 400 500 600

Temperature (∘C)

05

15times 10minus4

10

00

minus05

minus35

minus30

minus20

minus15

minus10

S-S boundary conditionF-F boundary condition

Max

imum

defl

ectio

n (m

)

Cooling

Cooling

Heating

Heating

minus25

Figure 3 Effects of boundary conditions on the maximum deflec-tion of SMA composite film

In addition the difference between 119908119873

max and 119908119871

max becomesmore obviouswhen the thickness of SMAfilmbecomes largerespecially under F-F boundary condition This demonstratesthat the geometric nonlinearity of bending deformationshould be considered when the thickness of SMA film is verylarge

5 Conclusions

The mechanical model of the SMA composite film with Sisubstrate was established by the method of mechanics ofcompositematerials in this paperThe actuation performanceof the SMASi composite film subjected to thermal loads wassimulated and analyzed during one cooling-heating thermalcycle The effects of the final cooling temperature boundarycondition SMA thickness and geometric nonlinearity on theactuation performance of the NiTi-SMA composite filmwerediscussed The following conclusions can be drawn from thepresent study

(1) The maximum deflection produced by thermalstresses recoversmore obviously whenmore austenitetransforms to martensite during cooling for SMASicomposite film with lower 119879

119891

(2) The maximum actuation displacement along thick-ness under F-F boundary condition is more thanfour times of that under S-S boundary condition forSMASi composite film during one cooling-heatingthermal cycle

(3) The maximum deflection of SMASi composite filmincreases with increasing the thickness of SMA filmwhether under F-F boundary condition or under S-Sboundary condition

2 4 6 8 10 12 14

60times 10minus4

55

50

45

40

35

30

25

20

15

Max

imum

defl

ectio

n (m

)

Thickness of SMA film (120583m)

wNmax

wLmax

(a)

2 4 6 8 10 12 14


08

06

04

02

16times 10minus4

14

12

10

Max

imum

defl

ectio

n (m

)

wNmax

wLmax

(b)

Figure 4 Effects of SMA thickness on the maximum deflectionof SMA composite film (a) F-F boundary condition and (b) S-Sboundary condition

(4) Effects of geometric nonlinearity of bending deforma-tion on the actuation displacement of SMASi com-posite film becomesmore obvious when the thicknessof SMA film becomes larger especially under F-Fboundary condition

This study can provide some helpful guidance for thedesign and application of bimorph thin film SMA microac-tuators in MEMS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

The authors acknowledge the financial support of the Pro-motive Research Fund for Excellent Young and Middle-aged Scientists of Shandong Province (no BS2010CL016) theProject of Shandong Province Higher Educational Scienceand Technology Program (no J13LB08) and the Project ofScience and Technology Development Plan of Qingdao (no13-1-4-150-jch)

References

[1] D Xu L Wang G Ding Y Zhou A Yu and B Cai ldquoChar-acteristics and fabrication of NiTiSi diaphragm micropumprdquoSensors and Actuators A Physical vol 93 no 1 pp 87ndash92 2001

[2] J Teng and P D Prewett ldquoFocused ion beam fabrication ofthermally actuated bimorph cantileversrdquo Sensors and ActuatorsA Physical vol 123-124 pp 608ndash613 2005

[3] J H Cao B L Wang C Ye and H Wang ldquoMicropumpdesign based onNiTi shapememory alloyrdquo ScienceampTechnologyInformation vol 29 pp 548ndash549 2011

[4] Z H Ning Z G Wang Y Q Fu and X T Zu ldquoAdvancesof investigation on TiNi-based shape memory alloy filmsrdquoMaterials Review vol 20 no 2 pp 118ndash125 2006

[5] Y Q Fu H J Du W M Huang S Zhang and M Hu ldquoTiNi-based thin films in MEMS applications a reviewrdquo Sensors andActuators A Physical vol 112 no 2-3 pp 395ndash408 2004

[6] Y Q Fu J K Luo A J Flewitt W M Huang and W IMilne ldquoThin film shape memory alloys and microactuatorsrdquoInternational Journal of Computational Materials Science andSurface Engineering vol 2 no 3-4 pp 209ndash226 2009

[7] L WangTheory and techniques of thin film shape memory alloymicroactuators based on silicon substrate [PhD thesis] ShanghaiJiaotong University 2000

[8] S Timoshenko and J Gere Mechanics of Materials Van Nos-trand Reinhold New York NY USA 1972

[9] X L Cheng Optimization research of dynamic actuating char-acteristics of TiNi(Cu)Si composite diaphragm based on MEMStechnology [PhD thesis] Shanghai Jiaotong University 2002

[10] Y P Wang Y B Wu X B Zhang C C Zhang and C SYang ldquoSimulation and fabrication of an SMA electro-thermalactuator with Si TiNi bimorph structurerdquoMicronanoelectronicTechnology vol 47 no 2 pp 93ndash98 2010

[11] S S Sun and J Dong ldquoModeling and simulation on theactuation performance of shape memory alloySi compositediaphragmrdquo Journal of Shanghai Jiaotong University vol 44 no8 pp 1145ndash1149 2010

[12] S S Sun and X C Jiang ldquoNonlinear analysis of actuationperformance of NiTiSi composite filmrdquoAppliedMechanics andMaterials vol 437 pp 572ndash576 2013

[13] G L Shen and G K Hu Mechanics of Composite MaterialsPublishing House of Tsinghua University Beijing China 2006

[14] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 8 no 4 pp285ndash302 1997

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Stochastic AnalysisInternational Journal of


when modeling and predicting Yet little work was reportedin this aspect

We have done some work [12] on the actuation perfor-mance of NiTiSi composite film by considering both thematerial nonlinearity of SMA and the geometric nonlinearityof bending deformation yet only the maximum actuationdisplacement of simply-supported NiTiSi composite filmvarying with three final cooling temperatures and the maxi-mum actuation displacement of cantilever NiTiSi compositefilm varying with SMA thickness were studied As for therelations between the maximum actuation displacementand the phase transformation fraction and stress of SMAduring thermal cycles they were not of concern Additionallyhow boundary conditions affect the actuation performanceof NiTiSi composite film was not discussed theoreticallyand numerically Based on our previous work [11 12] theSMASi composite film will be further studied in this paperto comprehensively predict its actuation performance byconsidering both the material nonlinearity of SMA and thegeometric nonlinearity of bending deformationThe SMASicomposite film subjected to thermal loads will be modeledaccording to the method of mechanics of composite mate-rials The effects of the final cooling temperature boundarycondition thickness of SMAfilm and geometric nonlinearityof deformation on the actuation performance of SMASicomposite film will be discussed in detail The influence ofthe phase transformation fraction and stress of SMA on themaximum actuation displacement of SMASi composite filmwill also be explored

2 Modeling of SMASi Composite Film

The SMASi composite film is considered as a compositebeam in this paper Figure 1(a) shows the schematic diagramof an SMASi composite film element before working where119905119886and 119905

119898denote the thickness of the SMA film and Si

substrate respectively Figure 1(b) shows the working state ofa microsegment of the SMA composite film element underthermal loads where119872 is the bendingmoment in each cross-section of the SMA composite film and d120579 is the relativerotation angle of the left and the right cross-sections of themicrosegment with length d119909

To analyze the actuation performance of the SMA com-posite film under thermal loads the coordinate system isestablished as shown in Figure 1(a) The coupled actionbetween the SMAfilm and the Si substrate will be analyzed bycombining static equilibriumequations geometric equationsand physical equations in the following sections

21 Static Equilibrium Equations Because no mechanicalloads are applied on the SMASi composite film except forthermal loads the following static equilibrium equations [13]must be satisfied for the microsegment in Figure 1(b)

119865119909= int

119905119898

0

120590119898sdot 119887 sdot 119889119910 + 120590

119886sdot 119887 sdot 119905119886= 0

119872119909119911

= int

119905119898

0

120590119898sdot 119910 sdot 119887 sdot 119889119910 + 120590

119886sdot 119910119886sdot 119887 sdot 119905119886cong 119872

(1)

where 119865119909and 119872

119909119911denote resultant internal forces and

internal moments in each cross-section of the microsegmentin Figure 1(b) 120590

119898and 120590

119886denote the stress of Si substrate

and the SMA film respectively 119887 is the width of theSMA composite film 119910

119886is the transverse coordinate

of the midheight of SMA film with 119910119886

= minus1199051198862 The

bending moment 119872 has the approximate expression [11]of 119872 = 120590

119886119905119886119887(119905119898

+ 119905119886)2 Note that the stress of SMA is

considered to have no change of gradient along its thicknessas the thickness of SMA is usually smaller than that of Sisubstrate

22 Geometric Equations Assume the curvature radius andstrain of the interface in the SMA composite beam tobe 1205880and 1205760 From the geometric relation shown in Figure 1

we have the following expression

1205760= 1205880(d120579d119909

) minus 1 = 12058801198960minus 1 (2)

where 1198960= d120579d119909

Then the strain 120576119898

of an arbitrary fiber 119887119887 with thedistance 119910 to the interface in Si substrate can be expressedas

120576119898= 1205760+ 1198960119910 (3)

For the SMA film the strain 120576119886of any points is assumed

to be equal with the following approximate expression

120576119886= 1205760+ 1198960(minus

119905119886

2) (4)

And the curvature 119896 of themidplane of the SMA compos-ite film can be written as

119896 =1

[1205880+ ((119905119886+ 119905119898) 2 minus 119905

119886)] (5)

23 Physical Equations Assume that the constitutive relationof Si substrate is linear elastic though geometrically nonlineardeformation will occur in the SMASi composite film duringworking Then the stress-strain relationship of Si substrateunder thermal loads can be written as

120590119898= (120576119898minus 120572119898sdot Δ119879) sdot 119864

119898 (6)

where 120590119898 120572119898 and 119864

119898denote the stress thermal expansion

coefficient and elastic modulus of Si substrate respectivelyΔ119879 is the change of temperature

For the SMA film the Liang-Rogersrsquo constitutive model[14] modified by Wang [7] is employed in the following

120590119886= 119863 (120585) 120576

119886+ Ω (120585) 120585 + 120579 (120585) 119879 + 119878

0 (7)

where 1198780= 1205901198860

minus 119863(1205850)1205761198860

minus Ω(1205850)1205850minus 120579(1205850)1198790 119863Ω and

120579 denote the elastic modulus phase transformation modulusand thermal elastic modulus of SMA respectively Theyare all the functions of martensite fraction 120585 119879 representstemperatureThe quantities with the subscript 0 denote initialconditions In addition we have three expressions 119863(120585) =

119863119860+ 120585(119863

119872minus 119863119860) Ω(120585) = minus120576

119871119863(120585) and 120579(120585) = minus120572119863(120585)

Here 119863119860and 119863

119872are the austenite and martensite elastic

modulus 120576119871is the maximum residual strain and 120572 is the

thermal expansion coefficient of SMA


Si substrate film

O

ta

tm

SMA film

xy

dx

y

bb

(a)

M M

y

1205880

d120579

b998400 b998400

(b)




119891) lt

120590119886lt 119862119860(119879 minus 119860

119904)

120585 =1205850

2cos [119886

119860(119879 minus 119860

119904) + 119887119860120590119886] + 1 (8)


119872119904) lt 120590119886lt 119862119872(119879 minus119872

119891)

120585 =(1 minus 120585

0)

2cos [119886

119872(119879 minus119872

119891) + 119887119872120590119886] +

(1 + 1205850)

2 (9)

In (8) and (9) 119886119860 119886119872 119887119860 and 119887



119891minus119860119904) 119886119872

= 120587(119872119878minus

119872119891) 119887119860

= minus119886119860119862119860 and 119887


119878

119872119891 119860119904 and 119860



119860and 119862

119872are material



120590119886=

120572119898Δ119879 minus (120576

119871120585 + 120572119879) + 119878

0119863 (120585)

1119863 (120585) + [119905119886(71199052

119898+ 12119905119886119905119898+ 61199052

119886)] (119864

1198981199053

119898) (10)

1205760= 120572119898sdot Δ119879 minus

120590119886sdot 119905119886(7119905119898+ 6119905119886)

119864119898sdot 1199052

119898

(11)

1198960=

12120590119886sdot 119905119886(119905119898+ 119905119886)

119864119898sdot 1199053

119898

(12)


119886 120585 and 119879 are coupled














d2119908d1199092

[1 + (d119908d119909)2]32

= 119896 (13)



d2119908d1199092

= 119896 (14)




NiTi SMA

119872119891= 23∘C 119862

119872= 113MPa∘C

119872119904= 50∘C 119862

119860= 45MPa∘C

119860119904= 62∘C 119863

119872= 13GPa

119860119891= 100

∘C 119863119860= 30GPa

120572 = 88 times 10minus6∘C 120576

119871= 1

Si 119864119898= 1805GPa 120572

119898= 233 times 10

minus6∘C


119908119873

max =1

119896minus1

119896


2)

2

119908119871

max =1

81198961198712

(15)


119908119873

max = minus1

119896+1

119896

radic1 minus (119896119871)2

119908119871

max = minus1

21198961198712

(16)

where 119908119873

max and 119908119871







119886= 5 120583m 119905

119898= 20 120583m and 119871 = 3mm





40∘C and 50


119873








119891(see Figure 2(b)) For119879

119891= 30∘C 119908119873max





119873





119891= 30







119891= 30








0 100 200 300 400 500 600

000

Max

imum

defl

ectio

n of

SM

A co

mpo

site fi

lm (m

)

Temperature (∘C)

Cooling

Heating

Tf = 30∘CTf = 40∘C

Tf = 50∘C

900times 10minus5

750

600

450

300

150

minus150

(a)

0 100 200 300 400 500 600

Temperature (∘C)

Tf = 30∘CTf = 40∘C

Tf = 50∘C

Cooling

Heating

09

08

07

06

05

04

03

02

01

00

minus02

minus01

Mar

tens

ite fr

actio

n of

SM

A

(b)

0 100 200 300 400 500 600

Temperature (∘C)

Tf = 30∘CTf = 40∘C

Tf = 50∘C

Cooling

Heating

800E + 007

600E + 007

400E + 007

200E + 007

000E + 000

minus200E + 007

Stre

ss o

f SM

A (M

Pa)

(c)






119873

maxand 119908

119871



119871


119873



0 100 200 300 400 500 600

Temperature (∘C)

05

15times 10minus4

10

00

minus05

minus35

minus30

minus20

minus15

minus10


Max

imum

defl

ectio

n (m

)

Cooling

Cooling

Heating

Heating

minus25



max and 119908119871


5 Conclusions



119891



2 4 6 8 10 12 14

60times 10minus4

55

50

45

40

35

30

25

20

15

Max

imum

defl

ectio

n (m

)


wNmax

wLmax

(a)

2 4 6 8 10 12 14


08

06

04

02

16times 10minus4

14

12

10

Max

imum

defl

ectio

n (m

)

wNmax

wLmax

(b)







Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Si substrate film

O

ta

tm

SMA film

xy

dx

y

bb

(a)

M M

y

1205880

d120579

b998400 b998400

(b)




119891) lt

120590119886lt 119862119860(119879 minus 119860

119904)

120585 =1205850

2cos [119886

119860(119879 minus 119860

119904) + 119887119860120590119886] + 1 (8)


119872119904) lt 120590119886lt 119862119872(119879 minus119872

119891)

120585 =(1 minus 120585

0)

2cos [119886

119872(119879 minus119872

119891) + 119887119872120590119886] +

(1 + 1205850)

2 (9)

In (8) and (9) 119886119860 119886119872 119887119860 and 119887



119891minus119860119904) 119886119872

= 120587(119872119878minus

119872119891) 119887119860

= minus119886119860119862119860 and 119887


119878

119872119891 119860119904 and 119860



119860and 119862

119872are material



120590119886=

120572119898Δ119879 minus (120576

119871120585 + 120572119879) + 119878

0119863 (120585)

1119863 (120585) + [119905119886(71199052

119898+ 12119905119886119905119898+ 61199052

119886)] (119864

1198981199053

119898) (10)

1205760= 120572119898sdot Δ119879 minus

120590119886sdot 119905119886(7119905119898+ 6119905119886)

119864119898sdot 1199052

119898

(11)

1198960=

12120590119886sdot 119905119886(119905119898+ 119905119886)

119864119898sdot 1199053

119898

(12)


119886 120585 and 119879 are coupled














d2119908d1199092

[1 + (d119908d119909)2]32

= 119896 (13)



d2119908d1199092

= 119896 (14)




NiTi SMA

119872119891= 23∘C 119862

119872= 113MPa∘C

119872119904= 50∘C 119862

119860= 45MPa∘C

119860119904= 62∘C 119863

119872= 13GPa

119860119891= 100

∘C 119863119860= 30GPa

120572 = 88 times 10minus6∘C 120576

119871= 1

Si 119864119898= 1805GPa 120572

119898= 233 times 10

minus6∘C


119908119873

max =1

119896minus1

119896


2)

2

119908119871

max =1

81198961198712

(15)


119908119873

max = minus1

119896+1

119896

radic1 minus (119896119871)2

119908119871

max = minus1

21198961198712

(16)

where 119908119873

max and 119908119871







119886= 5 120583m 119905

119898= 20 120583m and 119871 = 3mm





40∘C and 50


119873








119891(see Figure 2(b)) For119879

119891= 30∘C 119908119873max





119873





119891= 30







119891= 30








0 100 200 300 400 500 600

000

Max

imum

defl

ectio

n of

SM

A co

mpo

site fi

lm (m

)

Temperature (∘C)

Cooling

Heating

Tf = 30∘CTf = 40∘C

Tf = 50∘C

900times 10minus5

750

600

450

300

150

minus150

(a)

0 100 200 300 400 500 600

Temperature (∘C)

Tf = 30∘CTf = 40∘C

Tf = 50∘C

Cooling

Heating

09

08

07

06

05

04

03

02

01

00

minus02

minus01

Mar

tens

ite fr

actio

n of

SM

A

(b)

0 100 200 300 400 500 600

Temperature (∘C)

Tf = 30∘CTf = 40∘C

Tf = 50∘C

Cooling

Heating

800E + 007

600E + 007

400E + 007

200E + 007

000E + 000

minus200E + 007

Stre

ss o

f SM

A (M

Pa)

(c)






119873

maxand 119908

119871



119871


119873



0 100 200 300 400 500 600

Temperature (∘C)

05

15times 10minus4

10

00

minus05

minus35

minus30

minus20

minus15

minus10


Max

imum

defl

ectio

n (m

)

Cooling

Cooling

Heating

Heating

minus25



max and 119908119871


5 Conclusions



119891



2 4 6 8 10 12 14

60times 10minus4

55

50

45

40

35

30

25

20

15

Max

imum

defl

ectio

n (m

)


wNmax

wLmax

(a)

2 4 6 8 10 12 14


08

06

04

02

16times 10minus4

14

12

10

Max

imum

defl

ectio

n (m

)

wNmax

wLmax

(b)







Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











NiTi SMA

119872119891= 23∘C 119862

119872= 113MPa∘C

119872119904= 50∘C 119862

119860= 45MPa∘C

119860119904= 62∘C 119863

119872= 13GPa

119860119891= 100

∘C 119863119860= 30GPa

120572 = 88 times 10minus6∘C 120576

119871= 1

Si 119864119898= 1805GPa 120572

119898= 233 times 10

minus6∘C


119908119873

max =1

119896minus1

119896


2)

2

119908119871

max =1

81198961198712

(15)


119908119873

max = minus1

119896+1

119896

radic1 minus (119896119871)2

119908119871

max = minus1

21198961198712

(16)

where 119908119873

max and 119908119871







119886= 5 120583m 119905

119898= 20 120583m and 119871 = 3mm





40∘C and 50


119873








119891(see Figure 2(b)) For119879

119891= 30∘C 119908119873max





119873





119891= 30







119891= 30








0 100 200 300 400 500 600

000

Max

imum

defl

ectio

n of

SM

A co

mpo

site fi

lm (m

)

Temperature (∘C)

Cooling

Heating

Tf = 30∘CTf = 40∘C

Tf = 50∘C

900times 10minus5

750

600

450

300

150

minus150

(a)

0 100 200 300 400 500 600

Temperature (∘C)

Tf = 30∘CTf = 40∘C

Tf = 50∘C

Cooling

Heating

09

08

07

06

05

04

03

02

01

00

minus02

minus01

Mar

tens

ite fr

actio

n of

SM

A

(b)

0 100 200 300 400 500 600

Temperature (∘C)

Tf = 30∘CTf = 40∘C

Tf = 50∘C

Cooling

Heating

800E + 007

600E + 007

400E + 007

200E + 007

000E + 000

minus200E + 007

Stre

ss o

f SM

A (M

Pa)

(c)






119873

maxand 119908

119871



119871


119873



0 100 200 300 400 500 600

Temperature (∘C)

05

15times 10minus4

10

00

minus05

minus35

minus30

minus20

minus15

minus10


Max

imum

defl

ectio

n (m

)

Cooling

Cooling

Heating

Heating

minus25



max and 119908119871


5 Conclusions



119891



2 4 6 8 10 12 14

60times 10minus4

55

50

45

40

35

30

25

20

15

Max

imum

defl

ectio

n (m

)


wNmax

wLmax

(a)

2 4 6 8 10 12 14


08

06

04

02

16times 10minus4

14

12

10

Max

imum

defl

ectio

n (m

)

wNmax

wLmax

(b)







Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600

000

Max

imum

defl

ectio

n of

SM

A co

mpo

site fi

lm (m

)

Temperature (∘C)

Cooling

Heating

Tf = 30∘CTf = 40∘C

Tf = 50∘C

900times 10minus5

750

600

450

300

150

minus150

(a)

0 100 200 300 400 500 600

Temperature (∘C)

Tf = 30∘CTf = 40∘C

Tf = 50∘C

Cooling

Heating

09

08

07

06

05

04

03

02

01

00

minus02

minus01

Mar

tens

ite fr

actio

n of

SM

A

(b)

0 100 200 300 400 500 600

Temperature (∘C)

Tf = 30∘CTf = 40∘C

Tf = 50∘C

Cooling

Heating

800E + 007

600E + 007

400E + 007

200E + 007

000E + 000

minus200E + 007

Stre

ss o

f SM

A (M

Pa)

(c)






119873

maxand 119908

119871



119871


119873



0 100 200 300 400 500 600

Temperature (∘C)

05

15times 10minus4

10

00

minus05

minus35

minus30

minus20

minus15

minus10


Max

imum

defl

ectio

n (m

)

Cooling

Cooling

Heating

Heating

minus25



max and 119908119871


5 Conclusions



119891



2 4 6 8 10 12 14

60times 10minus4

55

50

45

40

35

30

25

20

15

Max

imum

defl

ectio

n (m

)


wNmax

wLmax

(a)

2 4 6 8 10 12 14


08

06

04

02

16times 10minus4

14

12

10

Max

imum

defl

ectio

n (m

)

wNmax

wLmax

(b)







Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600

Temperature (∘C)

05

15times 10minus4

10

00

minus05

minus35

minus30

minus20

minus15

minus10


Max

imum

defl

ectio

n (m

)

Cooling

Cooling

Heating

Heating

minus25



max and 119908119871


5 Conclusions



119891



2 4 6 8 10 12 14

60times 10minus4

55

50

45

40

35

30

25

20

15

Max

imum

defl

ectio

n (m

)


wNmax

wLmax

(a)

2 4 6 8 10 12 14


08

06

04

02

16times 10minus4

14

12

10

Max

imum

defl

ectio

n (m

)

wNmax

wLmax

(b)







Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article nonlinear analysis of actuation ... · the maximum de ection of the sma composite...

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