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published online 17 February 2014Journal of Intelligent Material Systems and StructuresReza Mehrabi, Mahmoud Kadkhodaei, Masood Taheri Andani and Mohammad Elahinia

torsion loading−Microplane modeling of shape memory alloy tubes under tension, torsion, and proportional tension

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Original Article

Journal of Intelligent Material Systemsand Structures1–12� The Author(s) 2014Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X14522532jim.sagepub.com

Microplane modeling of shape memoryalloy tubes under tension, torsion, andproportional tension–torsion loading

Reza Mehrabi1,2,3, Mahmoud Kadkhodaei1, Masood Taheri Andani2 andMohammad Elahinia2

AbstractIn this study, a three-dimensional thermomechanical constitutive model based on the microplane theory is proposed tosimulate the behavior of shape memory alloy tubes. The three-dimensional model is implemented in ABAQUS byemploying a user material subroutine. In order to validate the model, the numerical results of this approach are com-pared with new experimental findings for a NiTi superelastic torque tube under tension, pure torsion, and proportionaltension–torsion performed in stress- and strain-controlled manners. The numerical and experimental results are inagreement indicating the capability of the proposed microplane model in capturing the behavior of shape memory alloytubes. This model is capable of predicting both superelasticity and shape memory effect by providing closed-form rela-tionships for calculating the strain components in terms of the stress components.

KeywordsShape memory alloy, microplane, tube, tension–torsion, constitutive model

Introduction

Shape memory alloys (SMAs) are attractive candidatesfor different applications in mechanical, civil, medical,and aerospace systems (Hartl and Lagoudas, 2007;Saadat et al., 2002). SMAs have significant advantagesover conventional actuation methods. Their signifi-cantly reduced weight, size, complexity, and large defor-mation make them suitable as actuators (Nespoli et al.,2010).

Mathematical modeling of SMAs has been the topicof many works conducted by researchers in the lastdecade. So far, several modeling platforms have beendeveloped for these materials. Investigation of the tor-sional behavior in SMAs has received extensive atten-tion due to the growing number of applications beingdeveloped and proposed for SMA tubes and rods. Forexample, Shishkin (1994) proposed a torsional modelfor solid SMA rods. He provided an analytical frame-work to correlate thermomechanical diagrams in ten-sion, compression, and torsion. Keefe (1994) as well asKeefe and Carman (1997) investigated NiTiCu torquetubes with different wall thicknesses and proposed anexponential relationship between shear strain and shearstress for SMAs under torsion. This relationship couldmodel the recovery torque behavior with respect totemperature at fully austenite temperatures. Prahlad

and Chopra (2007) and Mehrabi et al. (2012) worked onthe material modeling and experimental characterizationof SMA rod and tube actuators undergoing pure torsiondeformations. They obtained the model parameters fromthe experimental results. Thamburaja (2005) and Panet al. (2007) developed and implemented a constitutivemodel for detwinning and martensite reorientation ofSMAs in ABAQUS. Phenomenological modeling is acommon approach in which the global mechanical beha-viors of SMAs are investigated by macroscopic energyfunctions that are dependent on internal variables.Among the available macroscopic models, some (Boydand Lagoudas, 1996; Arghavani et al., 2010; Lagoudaset al., 2012; Mehrabi et al., 2012; Oliveira et al., 2010;Panico and Brinson, 2007; Saleeb et al., 2011; Zaki,

1Department of Mechanical Engineering, Isfahan University of Technology,

Isfahan, Iran.2Dynamic and Smart Systems Laboratory, Mechanical, Industrial, and

Manufacturing Engineering Department, University of Toledo, Toledo,

OH, USA.3Department of Mechanical Engineering, School of Engineering, Vali-e-Asr

University, Rafsanjan, Iran

Corresponding author:

Mahmoud Kadkhodaei, Department of Mechanical Engineering, Isfahan

University of Technology, Isfahan 84156-83111, Iran.

Email: [email protected]

by Reza Mehrabi on February 18, 2014jim.sagepub.comDownloaded from

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2012) are developed for three-dimensional (3D) multiax-ial loadings.

Another phenomenological approach for modelingthe SMAs is ‘‘microplane theory.’’ Microplane methodwas first introduced by Bazant (1984), Bazant and Prat(1988a, 1988b), and Carol and Bazant (1997) to studythe behavior of quasi-brittle materials, such as concrete,soil, fiber composite, and stiff foams. In this approach,one-dimensional (1D) constitutive law is considered forassociated normal and tangential stress/strain compo-nents on any arbitrary plane, called microplane, at eachmaterial point. Then a homogenization process isemployed to generalize the 1D equation to obtain a 3Dmacroscopic model. Brocca et al. (2002) proposed thefirst SMA model based on microplane theory. In theirmodel, shear stress on each microplane was divided intotwo perpendicular components within the plane. In themicro-level constitutive relations, shear and normalmoduli were equal, and the evolution equations forshear directions on all microplanes at a point wereobtained from the same phase diagram. Kadkhodaeiet al. (2007, 2008) proposed the idea of utilizing oneresultant shear component within each plane and apply-ing the volumetric–deviatoric split for constitutive equa-tions. They showed that microplane formulations withtwo shear components have a directional bias natureand may result in prediction of unfeasible behaviors,such as producing shear strain during pure axial loadingor axial strain during pure shear loading. They alsoshowed that the use of identical evolution equations forall microplanes at a point does not coincide with thephysical principles relating the martensite volume frac-tion. Mehrabi and Kadkhodaei (2013) proposed a 3Dphenomenological model based on microplane theory toshow the ability of this approach in predicting marten-site reorientation in nonproportional loadings. Almostall the previous works published on the microplane the-ory lack a robust evaluation of experimental data forloading conditions more complicated than simple ten-sion. This issue is thoroughly addressed in this work.

Some experimental studies under tension, torsion,proportional, and nonproportional loadings have beenperformed on SMA tubes to assess the mathematicalmodels. Lim and McDowell (1995) carried out experi-ments on superelastic NiTi tubes to investigate theirresponse under tension/torsion loading modes. Theyreported a rate-dependent behavior due to heat genera-tion during the stress-induced phase transformation.Sittner et al. (1995) performed combined tension–torsion experiments on superelastic CuAlZnMnpolycrystalline thin-wall tubes. They showed that theinelastic strains in proportional and nonproportionalcombined loadings are fully reversible if the axial strainis reversible. Sun and Li (2002) reported an experimen-tal study on the behavior of polycrystalline NiTi micro-tubes under tension, torsion, and tension–torsioncombined loadings. They found that during torsion, the

stress–strain curve exhibits monotonic hardening, thestress-induced transformation is axially homogeneousthroughout the whole tube, and the transformationstrain in torsion is smaller than that under tension.Wang et al. (2010) and Grabe and Bruhns (2008, 2009)experimentally investigated the superelastic stress–strainresponse of NiTi under combined tension–torsion load-ing conditions and showed that the loading path signifi-cantly affects the mechanical responses of the material.

Investigation of the NiTi thin-walled tubes’ behaviorunder tension, torsion, and proportional tension–torsionis the focus of this study. To this end, microplane theoryis utilized to predict superelasticity (SE) and shape mem-ory effect (SME). 1D constitutive relations are consid-ered for normal and tangential components on anygeneric plane passing through a material point. The inte-gral form of the principle of complementary virtualwork is then used to generalize 1D model and obtain themacroscopic 3D constitutive equations. Since all thematerial parameters needed for the proposed model arenot measured and reported in the existing experimentalworks, conducting new tests on tension and torsion formaterial parameter derivation is inevitable. The pro-posed model is verified against the new experimentaldata obtained for a NiTi thin-walled tube under uniaxialtension, pure torsion, and proportional tension–torsionin stress- and strain-controlled loading/unloading. Thenumerical results show that the developed microplaneformulation successfully reproduces the experimentalresults. This model is further used to predict the torque–angle and shear stress–strain responses of thin-walledtubes with different thicknesses below the austenite starttemperature, when shape memory phenomenon takesplace. The numerical results indicate the ability of theclosed-form microplane model in capturing the SE andSME behavior of SMAs.

Microplane model

The SMA microplane model considers the possibility ofmartensitic transformation on several planes with dif-ferent orientations, and it generally obtains the trans-formation strain as a superposition of shear-inducedtransformation strains to reproduce the actual physicalbehavior of SMAs. Three main steps of this theory canbe summarized as follows: (1) projection of the stress,(2) definition of 1D constitutive laws on the micro-level,and (3) homogenization process on the material pointto generalize the 1D relations to 3D ones. These threemain steps are schematically shown in Figure 1.

According to Figure 2, the normal and shear stressvectors on any microplane passing through a materialpoint are considered as the projection of macroscopicstress tensor in the form of

sN =N : s, sT =T : s ð1Þ

2 Journal of Intelligent Material Systems and Structures



where sN is the normal stress, sT is the shear stress,and the tensors N and T have the Cartesian compo-nents as

Nij = ninj, Tij =(nitj + njti)

2, ti =

siknk � sN niffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisjrsjsnrns � s2N

pð2Þ

in which ni represents the components of the unit nor-mal vector (n) on the plane. The normal stress is dividedinto the volumetric stress, sV , and the deviatoric stress,sD, in the form of

sV =V : s=smm

3, sD =sN � sV =D : s ð3Þ

where the tensors V and D have the Cartesian compo-nents as

Vij =dij3, Dij = ninj �

dij3

ð4Þ

In the homogenization process, the weak form ofmicro–macro equilibrium equations can be constructedusing the principle of complementary virtual work as(Carol et al., 2001)

ðV

e : dsdV=

ðV

(eN dsN + eT dsT )dV ð5Þ

where V is the surface of a unit sphere representing allpossible microplane orientations passing through amaterial point. The left term in equation (5) is integra-tion on a unit sphere and can be simplified as4p=3e : ds

By substituting equations (1) and (3) into equation(5) and considering the independence of individual

Figure 1. Microplane theory based on the volumetric–deviatoric–tangential split.

Figure 2. Stress components on a microplane.

Mehrabi et al. 3



components of the virtual stress tensor, the macro-scopic strain is stated in the following close form

e= eV1+3

4p

ðV

(eDN+ eTT)dV ð6Þ

in which the second-order unit tensor is denoted by 1(Kronecker delta).

To define the constitutive laws on the micro-level,1D constitutive relations between the projected stressesand the corresponding strains are considered.Deviatoric part of the normal stress on a microplane isprojected from macroscopic stress by equation (3) whiletangential stress is projected by equation (1). Sincemartensitic transformation induces shear deformations,shear is assumed to be the only source of inelastic beha-vior. Consequently, a 1D SMA constitutive law is usedfor the shear component while a linear elastic stress–strain relation is used for the normal components, thatis, the deviatoric part has no inelastic response. Thelocal constitutive equations for the volumetric and thedeviatoric parts of the normal strain as well as the elas-tic part of the tangential strain are defined as

eV =sV

E0V, eD =

sD

E0D, eeT =

sT

E0Tð7Þ

where E0V , E0D, and E

0T are local components of the linear

elastic stiffness tensor. By substituting equation (7) intoequation (6) and evaluating this integral followed bycomparing it with constitutive equations of linear elasti-city, the relations between local and global componentsof the modulus are derived as (Kadkhodaei et al., 2007)

E0V =E(j)

(1� 2y) , E0D =

E(j)

(1+ y), E0T =

E(j)

(1+ y)ð8Þ

where E and y are Young’s modulus and Poisson’sratio, respectively. So the final local constitutive equa-tions are obtained by substituting equation (8) intoequation (7)

eV =(1� 2y)sV

E(j), eD =

(1+ y)sDE(j)

, eeT =(1+ y)sT

E(j)

ð9Þ

In fact, decomposition of normal microplane strainis defined as eN = eV + eD, and tangential microplanestrain is decomposed as eT = eeT + e

trT . Moreover, the

inelastic tangential strain is considered to be in the formof

etrT = e�j(�s, T ) ð10Þ

where e� is the axial maximum recoverable strain of anSMA and j(�s, T ) is the martensite volume fraction as afunction of the effective stress, �s, and temperature. In

this article, for a thin-walled tube, the effective stress ismacroscopically stated as

�s =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 + 3t2p

ð11Þ

where s is the macroscopic tensile stress and t is themacroscopic shear stress.

Here, the relationships suggested by Brinson (1993)are utilized for j(�s, T ).Referring to Figure 3, the evolu-tion equations for j(�s, T ) at different regions in thephase diagram are as follows:

Conversion to detwinned martensite

For T.TMs and scrs +CM (T � TMs )\�s\scrf +

CM (T � TMs )

j(�s, T )=1� j0

2

cosp

scrs � scrf3 �s � scrf � CM (T � TMs )h i( )

+1+ j0

2

ð12Þ

For T\TMs and scrs \�s\s

crf

j(�s, T )=1� j0

2cos

p

scrs � scrf3 ½�s � scrf �

( )+

1+ j02

ð13Þ

Conversion to austenite

For T.TAs and CA(T � TAf )\�s\CA(T � TAs )

j(�s, T )=j02

1+ cosp

TAf � T AsT � TAs �

�s

CA

� �" #( )ð14Þ

Figure 3. Critical stress–temperature phase diagram fortransformation.




in which j0, TMf , T

Ms , T

As , and T

Af are the initial marten-

site fraction, martensite finish, martensite start, austenitestart, and austenite finish temperatures, respectively.Also, CM and CA represent the slope of martensite andaustenite strips in the stress–temperature phase diagram,and scrs and s

crf are the start critical stress and final criti-

cal stress in the stress–temperature phase diagram inwhich the forward and reverse transformations takeplace, as shown in Figure 3.

The effective elastic modulus in transformation stepis calculated in terms of the martensite volume fractionusing the Reuss model for SMAs (Brinson and Huang,1996)

1

E(j)=

(1� j)EA

+j

EMð15Þ

where EA and EM are the elastic moduli of pure auste-nite and pure martensite phases, respectively.

Substituting equations (3), (4), (9), and (10) in equa-tion (6), the total strain is

eij = eeij + e

trij ð16aÞ

eeij = �y

E(j)sssdij +

1+ y

E(j)srs �

3

4p

ðV

(NrsNij + TrsTij)dV

ð16bÞ

etrij = e� � j(�s, T ) � 3

4p

ðV

TijdV ð16cÞ

(16)

In calculation of the strain, the above integrals areevaluated numerically by using a 42-point Gaussianintegration formula for a sphere surface (Bazant andOh, 1986).

Algorithm and formulation used in user materialsubroutine

The computational algorithm in user material subrou-tine (UMAT) is outlined in Table 1. Microplane modelprovides closed-form relationships to calculate strainsin terms of stresses. Calculation of equation (16) isstraightforward because the magnitude of martensitevolume fraction can be determined in closed form andthe derivation of the material Jacobian is straightfor-ward. The implicit solver of ABAQUS (UMAT) imple-ments these formulations. So the material Jacobian(DDSDDE) is calculated, and then stress is updated.Moreover, since the proposed model is a quasi-staticmodel, there is no time-discrete equation and no trialvalue for the parameters.

Experiment and material details

All mechanical tests were performed using a BoseElectroForce machine (Figure 4). All experiments werecontrolled by axial force and torsional torque. Thedeformation was recorded by axial displacement androtation. Johnson Matthey supplied Nitinol thin-walltubes with an outer diameter of 4.5 mm, a thickness of0.3 mm, and a gage length of 14 mm. The transforma-tion temperatures of the samples were characterized by

Table 1. Algorithm for constitutive modeling of SMA.

1. Read De and sn (strain increment and stress evaluated from the previous step) from ABAQUS.2. Check for transformation according to the phase diagram

If transformation happened, calculate(a) Martensite volume fraction: j(�s, T), equations (12) to (14)(b) The transformation strain: etr , equation (16c)

Otherwise go to stage 33. Calculate the elastic strain: ee, equation (16b)4. Calculate strain

e= ee + etr , equation (16a)5. Calculate Jacobian matrix

Cet�1

=∂e

∂s= � y

Edijdpq +

1+ y

E

3

4p

ðV

(NijNpq + TijTpq)dV

+1+ y

Esrs

3

4p

ðV

Tij∂Trs∂Tpq

+ Trs∂Tij∂Tpq

� �dV+

9

8pe�

Spq�s

dj(�s, T)

d�s

ðV

TijdV+ e�j(�s, T)

3

4p

ðV

∂Tij∂Tpq

dV

6. Calculate stress incremental tensorDs=Cet : De

7. Update stresssn+ 1 =sn +Ds

8. End the program

Mehrabi et al. 5



the differential scanning calorimetry (DSC) and areshown in Table 2.

Samples were trained prior to testing at the constanttemperature of 323 K to obtain a stable response andremove residual strains possibly generated through themanufacturing processes. Stable response was observedin specimens after about 30 loading/unloading cycles.The loading rate was under 10�3 s�1 to be near the iso-thermal boundary conditions that are used in the pres-ent simulations. Three uniaxial characterization testswere performed at T = 296 K, T = 313 K, and T =323 K to calibrate the material parameters. The mate-rial parameters calibrated from these uniaxial experi-ments are listed in Table 2. The uniaxial tension, pure

torsion, and proportional tests were carried out at theconstant temperature of 296 K.

Results and discussion

This section is divided into three main parts. At first,the SE of torque tubes with different thicknesses in ten-sion and torsion is investigated. In the following, withthe same material parameters, microplane predictionsare compared with experimental data obtained in

Figure 4. Experimental setup.

Table 2. Calibrated material properties of the NiTi torque tubein uniaxial loading to be used in microplane formulation.

Symbols Values Units

EA 20,000 MPa

EM 13,300 MPa

yA = yM 0.33

TMf 241 K

TMs 258 K

TAs 268 K

TAf 288 K

scrs 20 MPa

scrf 100 MPa

CM 6 MPa/K

CA 8.2 MPa/K

e� 0.038Figure 5. Schematic representation of the finite elementmodel for a thin-walled tube under pure torsion: (a)undeformed shape, (b) deformed shape, and (c) shear stress–strain at three different points on the cross section.




proportional tension–torsion loadings. In the last part,stress-induced martensite at TMs \T\T

As is investigated.

Figure 5(a) shows an undeformed thin-walled SMAtube with the outer diameter of 4.5 mm, a thickness of0.3 mm, and a length of 14 mm subjected to torsion atone end, while the other end is fixed. Figure 5(b) showsshear stress contour in deformed tube under a 30� rota-tion. It is clear that shear stress on the outer diameter isnearly constant. Shear stress–strain curves in threepoints at different places are shown in Figure 5(c). Asshown, results of these three different places on thecross section are in a good agreement.

To model an SMA thin-walled tube in ABAQUS(Abaqus 6.9, 2009), 3D eight-node continuous solid brick(C3D8R)-type elements are used with 1240 elements inthe whole finite element (FE) model. To make sure thatthe quantity of the elements is adequate, some sets of ele-ments are assigned to the model in which similar resultsare obtained by increasing the number of elements.

In numerical simulations produced by the developedUMAT, the time step for each loading and unloadingstep is fixed at 0.01, and each step is divided into 100increments. In each increment with a specific strainincrement, strain is calculated by equation (16), andthen the Jacobian matrix and stress incremental are cal-culated. All the reported results belong to the outer dia-meter of the thin-walled tube.

Investigation of SMA superelastic behavior in tensionand torsion

The ability of the proposed microplane model in cap-turing SE is investigated here. The uniaxial stress–strainresponse obtained from the microplane approach iscompared with experimental result at the temperatureof 296 K and is shown in Figure 6. The agreementbetween the simulation results and experiment indicates

the reasonable accuracy of the obtained materialparameters.

It has been reported that the mechanical characteris-tics are significantly different in tension and torsion(Wang et al., 2010). In other words, an SMA tubebehaves differently under torsion in contrast with ten-sion (Sun and Li, 2002). According to these results, thesame material parameters are used in the correspondingmodel for the study of pure torsion of NiTi torque tubeexcept axial maximum recoverable strain, which isdefined as e�=

ffiffiffi3p

= 0:0219. Although this assumptionsolved the discrepancy issues in torsional modeling, itneeds to be investigated further in a more fundamentalapproach. Another possible assumption is the vonMises behavior of the material. The other possibleassumption is dominance of shear mode in transforma-tion. Figure 7 shows the comparison of microplanemodel with experimental data at the temperature of296 K. The shear strain transformation in pure torsionshould be

ffiffiffi3p

e�j (Kadkhodaei et al., 2008), whichaccording to equation (16c) is

gtr = 2etr12 = 2e�j

3

4p

ðV

T12dV= 2 3 0:0219

3 1 3 0:895= 0:039 ð17Þ

There is a 3.3% error in microplane prediction thatis due to the numerical evolution of the integrals. Thediscrepancy between the experimental data and micro-plane results in forward and reverse transformationregions, which may be due to the kinetics formulas thatare utilized to estimate j. Also, some discrepancies dur-ing the elastic loading and unloading are mainly due tothe assumption of constant modulus of elasticity forpure austenite (EA) and pure martensite (EM ), which isnot the exact case in reality. Moreover, it is seen thatshear strain in forward transformation begins from

Figure 6. Comparison of the microplane model withexperimental data for a thin-walled tube with 0.3 mm thicknessat T = 296K.

Figure 7. Comparison of the microplane model withexperimental data for a thin-walled tube with 0.3 mm thicknessat T = 296K.

Mehrabi et al. 7



about 0.02 and finishes at 0.074, which approximatelycorrespond to 8� and 28� of rotation, respectively.

The proposed formulation is based on small straintheory, and hence, such large angles of rotations mayimpose artificial strains and is considered as anothersource of discrepancy observed in Figure 7. Therefore,small rotation is applied in pure torsion so that the pro-posed model would be applicable without error.However, extension of the proposed model to largestrain domain needs to be more thoroughly investi-gated in future studies.

The effect of wall thickness of the tube on thetorque–angle response is depicted in Figure 8. Threetubes are simulated while subjected to a same amountof rotation. As shown, by increasing the thickness, thetorque required for full transformation is increased,and it shows that thicker tubes need more torque tocomplete the transformation.

Superelastic proportional tension–torsion

To demonstrate aspects of the proposed model, pro-portional loading in stress- and strain-controlled man-ners is experimentally performed, and the findings arecompared with the obtained numerical results using thesame material parameters extracted in uniaxial loading.The studied proportional loading paths in stress- andstrain-controlled cases are shown in Figures 9 and 11,respectively. In the stress-controlled case, maximumaxial stress and shear stress are 230 and 101 MPa,respectively. At least two repetitions are carried out foreach loading path.

The proposed model predictions are qualitativelysimilar to the experimental results, as shown in Figure10. Uniaxial stress–strain and shear stress–strainresponses are shown in Figure 10(a) and (b). Figure10(c) shows effective stress versus effective strain using

equation (11) for the effective stress and the followingequation for the effective strain

�e=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2 +

g2

3

rð18Þ

where e is the macroscopic tensile strain and g is themacroscopic shear strain. There is a reasonable agree-ment between microplane predictions and experimentalresults.

One more proportional loading case in a strain-controlled manner in the form of Figure 11 is consid-ered, where maximum displacement is 0.5 mm andmaximum rotation is reached to 0.26 rad. Again, tworepetitions are carried out for each step and experimen-tal results are shown in Figure 12. Referring to the dis-cussion provided for Figure 7, some differencesbetween the numerical and experimental results may beattributed to the utilized small strain formulation inthis work.

Stress-induced martensite at TMs \T\TAs

In this part, torsion of thin-walled NiTi tubes with dif-ferent thicknesses is studied by considering a constantloading/unloading cycle at T = 265K, which is underTAs . These tubes are initially in the austenite phase, andthe material parameters are the same as those shown inTable 2. In the loading step, a portion of austenite istransformed to martensite. Then, in the unloading part,martensite fraction remains constant and a residualstrain appears in the tube. This residual strain can berecovered by increasing the temperature to the austenitefinish temperature. The corresponding torque–twistangle curves are shown in Figure 13(a) using micro-plane model. It is shown that by increasing the wallthickness, the torque required for the full

Figure 8. Torque–angle of rotation diagram for differentthicknesses at T = 296K using microplane model.

Figure 9. Proportional loading path (stress-controlled).




transformation increases so that the martensitic trans-formation is not fully completed for the thickness of0.4 mm.

Shear stress–strain diagrams corresponding toFigure 13(a) are shown in Figure 13(b). A tube with 0.2mm thickness shows the complete transformation whilethe thickness of 0.4 mm does not. The results shown inFigure 13(a) and (b) indicate that at a relatively smallthickness, the martensitic transformation is completed.This, however, is not the case at a greater thickness.

In Figure 14, the ability of the microplane model inthe simulation of SME at T = 265K for a tube with thethickness of 0.2 mm is depicted. This figure shows theability of the proposed microplane model in predictingstrain recovery after the removal of stress in the heatingprocess.

Figure 10. Comparison of the microplane model with the experimental data in proportional loading, stress-controlled: (a) axialstress–strain, (b) shear stress–strain, and (c) effective stress–effective strain.

Figure 11. Proportional loading path (strain-controlled).

Mehrabi et al. 9



Figure 12. Comparison of the microplane model with the experimental data in proportional loading, strain-controlled: (a) axialstress–strain, (b) shear stress–strain, and (c) effective stress–effective strain.

Figure 13. (a) Torque–angle of rotation and (b) shear stress–strain diagram for thin-walled tubes with different wall thicknesses atT = 265 K obtained from microplane model.




Conclusion

In this article, behavior of NiTi tubes under tension,torsion, and proportional loading is studied. For thispurpose, a 3D constitutive model is developed by utiliz-ing the microplane theory, in which 1D constitutiverelations for normal and shear directions of all micro-planes at a material point are generalized to 3D macro-scopic constitutive equations by employing ahomogenization technique. The results of the proposedmicroplane approach are compared with the experi-mental results for a superelastic NiTi tube. Resultsshow that an increase in the wall thickness increasesthe necessary torque to induce a complete transforma-tion in SMA torque tubes with constant outer dia-meters. Microplane formulation is also able todemonstrate the characteristics of proportional loadingpaths. The proposed model predicts an SME at tem-peratures below the austenite start temperature. Thisindicates the capability of the proposed model in pre-dicting the SMA behavior as a simple approach. Thisproposed model provides closed-form relations for theinelastic strain in terms of the applied stresses. Anadvantage of this approach is the simplicity of derivingthe material properties in uniaxial tests and using themfor torsion, as well as other loading conditions. One ofthe future steps of this modeling approach is an investi-gation of the maximum recoverable strain in torsion inorder to find a fundamental solution.

Acknowledgements

The authors would like to acknowledge collaboration ofProfessor Arbab Chirani at Laboratoire Brestois deMécanique et des Systèmes, ENSTA Bretagne/UBO/ENIB,Technopôle Brest-Iroise, Plouzané, France, for the DSC tests.

Author’s note

R.M. is currently a visiting PhD student at the Dynamic andSmart Systems Laboratory, Mechanical, Industrial, andManufacturing Engineering Department, University ofToledo, Toledo, OH, USA.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This study was financially supported by the Iranian Ministryof Science, Research and Technology (to R.M.).

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