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Journal of Intelligent Material Systems and Structures http://jim.sagepub.com/ Microplane modeling of shape memory alloy tubes under tension, torsion, and proportional tension− torsion loading Reza Mehrabi, Mahmoud Kadkhodaei, Masood Taheri Andani and Mohammad Elahinia Journal of Intelligent Material Systems and Structures published online 17 February 2014 DOI: 10.1177/1045389X14522532 The online version of this article can be found at: http://jim.sagepub.com/content/early/2014/02/10/1045389X14522532

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Original Article

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

Journal of Intelligent Material Systems and Structures 1–12 Ó The Author(s) 2014 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1045389X14522532 jim.sagepub.com

Reza Mehrabi1,2,3, Mahmoud Kadkhodaei1, Masood Taheri Andani2 and Mohammad Elahinia2

Abstract In this study, a three-dimensional thermomechanical constitutive model based on the microplane theory is proposed to simulate the behavior of shape memory alloy tubes. The three-dimensional model is implemented in ABAQUS by employing a user material subroutine. In order to validate the model, the numerical results of this approach are compared with new experimental findings for a NiTi superelastic torque tube under tension, pure torsion, and proportional tension–torsion performed in stress- and strain-controlled manners. The numerical and experimental results are in agreement indicating the capability of the proposed microplane model in capturing the behavior of shape memory alloy tubes. This model is capable of predicting both superelasticity and shape memory effect by providing closed-form relationships for calculating the strain components in terms of the stress components. Keywords Shape memory alloy, microplane, tube, tension–torsion, constitutive model

Introduction Shape memory alloys (SMAs) are attractive candidates for different applications in mechanical, civil, medical, and aerospace systems (Hartl and Lagoudas, 2007; Saadat et al., 2002). SMAs have significant advantages over conventional actuation methods. Their significantly reduced weight, size, complexity, and large deformation make them suitable as actuators (Nespoli et al., 2010). Mathematical modeling of SMAs has been the topic of many works conducted by researchers in the last decade. So far, several modeling platforms have been developed for these materials. Investigation of the torsional behavior in SMAs has received extensive attention due to the growing number of applications being developed and proposed for SMA tubes and rods. For example, Shishkin (1994) proposed a torsional model for solid SMA rods. He provided an analytical framework to correlate thermomechanical diagrams in tension, compression, and torsion. Keefe (1994) as well as Keefe and Carman (1997) investigated NiTiCu torque tubes with different wall thicknesses and proposed an exponential relationship between shear strain and shear stress for SMAs under torsion. This relationship could model the recovery torque behavior with respect to temperature at fully austenite temperatures. Prahlad

and Chopra (2007) and Mehrabi et al. (2012) worked on the material modeling and experimental characterization of SMA rod and tube actuators undergoing pure torsion deformations. They obtained the model parameters from the experimental results. Thamburaja (2005) and Pan et al. (2007) developed and implemented a constitutive model for detwinning and martensite reorientation of SMAs in ABAQUS. Phenomenological modeling is a common approach in which the global mechanical behaviors of SMAs are investigated by macroscopic energy functions that are dependent on internal variables. Among the available macroscopic models, some (Boyd and Lagoudas, 1996; Arghavani et al., 2010; Lagoudas et al., 2012; Mehrabi et al., 2012; Oliveira et al., 2010; Panico and Brinson, 2007; Saleeb et al., 2011; Zaki, 1

Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran. 2 Dynamic and Smart Systems Laboratory, Mechanical, Industrial, and Manufacturing Engineering Department, University of Toledo, Toledo, OH, USA. 3 Department 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]


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Journal of Intelligent Material Systems and Structures

2012) are developed for three-dimensional (3D) multiaxial loadings. Another phenomenological approach for modeling the SMAs is ‘‘microplane theory.’’ Microplane method was first introduced by Bazant (1984), Bazant and Prat (1988a, 1988b), and Carol and Bazant (1997) to study the behavior of quasi-brittle materials, such as concrete, soil, fiber composite, and stiff foams. In this approach, one-dimensional (1D) constitutive law is considered for associated normal and tangential stress/strain components on any arbitrary plane, called microplane, at each material point. Then a homogenization process is employed to generalize the 1D equation to obtain a 3D macroscopic model. Brocca et al. (2002) proposed the first SMA model based on microplane theory. In their model, shear stress on each microplane was divided into two perpendicular components within the plane. In the micro-level constitutive relations, shear and normal moduli were equal, and the evolution equations for shear directions on all microplanes at a point were obtained from the same phase diagram. Kadkhodaei et al. (2007, 2008) proposed the idea of utilizing one resultant shear component within each plane and applying the volumetric–deviatoric split for constitutive equations. They showed that microplane formulations with two shear components have a directional bias nature and may result in prediction of unfeasible behaviors, such as producing shear strain during pure axial loading or axial strain during pure shear loading. They also showed that the use of identical evolution equations for all microplanes at a point does not coincide with the physical principles relating the martensite volume fraction. Mehrabi and Kadkhodaei (2013) proposed a 3D phenomenological model based on microplane theory to show the ability of this approach in predicting martensite reorientation in nonproportional loadings. Almost all the previous works published on the microplane theory lack a robust evaluation of experimental data for loading conditions more complicated than simple tension. This issue is thoroughly addressed in this work. Some experimental studies under tension, torsion, proportional, and nonproportional loadings have been performed on SMA tubes to assess the mathematical models. Lim and McDowell (1995) carried out experiments on superelastic NiTi tubes to investigate their response under tension/torsion loading modes. They reported a rate-dependent behavior due to heat generation during the stress-induced phase transformation. Sittner et al. (1995) performed combined tension– torsion experiments on superelastic CuAlZnMn polycrystalline thin-wall tubes. They showed that the inelastic strains in proportional and nonproportional combined loadings are fully reversible if the axial strain is reversible. Sun and Li (2002) reported an experimental study on the behavior of polycrystalline NiTi microtubes under tension, torsion, and tension–torsion combined loadings. They found that during torsion, the

stress–strain curve exhibits monotonic hardening, the stress-induced transformation is axially homogeneous throughout the whole tube, and the transformation strain in torsion is smaller than that under tension. Wang et al. (2010) and Grabe and Bruhns (2008, 2009) experimentally investigated the superelastic stress–strain response of NiTi under combined tension–torsion loading conditions and showed that the loading path significantly affects the mechanical responses of the material. Investigation of the NiTi thin-walled tubes’ behavior under tension, torsion, and proportional tension–torsion is the focus of this study. To this end, microplane theory is utilized to predict superelasticity (SE) and shape memory effect (SME). 1D constitutive relations are considered for normal and tangential components on any generic plane passing through a material point. The integral form of the principle of complementary virtual work is then used to generalize 1D model and obtain the macroscopic 3D constitutive equations. Since all the material parameters needed for the proposed model are not measured and reported in the existing experimental works, conducting new tests on tension and torsion for material parameter derivation is inevitable. The proposed model is verified against the new experimental data obtained for a NiTi thin-walled tube under uniaxial tension, pure torsion, and proportional tension–torsion in stress- and strain-controlled loading/unloading. The numerical results show that the developed microplane formulation successfully reproduces the experimental results. This model is further used to predict the torque– angle and shear stress–strain responses of thin-walled tubes with different thicknesses below the austenite start temperature, when shape memory phenomenon takes place. The numerical results indicate the ability of the closed-form microplane model in capturing the SE and SME behavior of SMAs.

Microplane model The SMA microplane model considers the possibility of martensitic transformation on several planes with different orientations, and it generally obtains the transformation strain as a superposition of shear-induced transformation strains to reproduce the actual physical behavior of SMAs. Three main steps of this theory can be 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 point to generalize the 1D relations to 3D ones. These three main steps are schematically shown in Figure 1. According to Figure 2, the normal and shear stress vectors on any microplane passing through a material point are considered as the projection of macroscopic stress tensor in the form of sN = N : s,


sT = T : s

ð1Þ

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

in which ni represents the components of the unit normal vector (n) on the plane. The normal stress is divided into the volumetric stress, sV , and the deviatoric stress, sD , in the form of sV = V : s =

smm , 3

sD = s N s V = D : s

ð3Þ

where the tensors V and D have the Cartesian components as Vij =

dij , 3

Dij = ni nj

dij 3

ð4Þ

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

ð e : dsdV =

Figure 2. Stress components on a microplane. V

where sN is the normal stress, sT is the shear stress, and the tensors N and T have the Cartesian components as Nij = ni nj ,

Tij =

(ni tj + nj ti ) , 2

sik nk sN ni ti = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sjr sjs nr ns s2N ð2Þ

(eN dsN + eT dsT )dV

ð5Þ

V

where V is the surface of a unit sphere representing all possible microplane orientations passing through a material point. The left term in equation (5) is integration on a unit sphere and can be simplified as 4p=3e : ds By substituting equations (1) and (3) into equation (5) and considering the independence of individual


4


components of the virtual stress tensor, the macroscopic strain is stated in the following close form e = eV 1 +

ð 3 (eD N + eT T)dV 4p

ð6Þ

V

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 stresses and the corresponding strains are considered. Deviatoric part of the normal stress on a microplane is projected from macroscopic stress by equation (3) while tangential stress is projected by equation (1). Since martensitic transformation induces shear deformations, shear is assumed to be the only source of inelastic behavior. Consequently, a 1D SMA constitutive law is used for the shear component while a linear elastic stress– strain relation is used for the normal components, that is, the deviatoric part has no inelastic response. The local constitutive equations for the volumetric and the deviatoric parts of the normal strain as well as the elastic part of the tangential strain are defined as eV =

sV , EV0

eD =

sD , ED0

eeT =

sT ET0

ð7Þ

where EV0 , ED0 , and ET0 are local components of the linear elastic stiffness tensor. By substituting equation (7) into equation (6) and evaluating this integral followed by comparing it with constitutive equations of linear elasticity, the relations between local and global components of the modulus are derived as (Kadkhodaei et al., 2007) EV0 =

E(j) , (1 2y)

ED0 =

E(j) , (1 + y)

ET0 =

E(j) ð8Þ (1 + y)

where E and y are Young’s modulus and Poisson’s ratio, respectively. So the final local constitutive equations are obtained by substituting equation (8) into equation (7) eV =

(1 2y)sV , E(j)

eD =

(1 + y)sD , E(j)

eeT =

(1 + y)sT E(j) ð9Þ

In fact, decomposition of normal microplane strain is defined as eN = eV + eD , and tangential microplane strain is decomposed as eT = eeT + etrT . Moreover, the inelastic tangential strain is considered to be in the form of etrT = e j( s, T )

ð10Þ

where e is the axial maximum recoverable strain of an SMA and j( s, T ) is the martensite volume fraction as a function of the effective stress, s , and temperature. In

Figure 3. Critical stress–temperature phase diagram for transformation.

this article, for a thin-walled tube, the effective stress is macroscopically stated as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s = s2 + 3t 2

ð11Þ

where s is the macroscopic tensile stress and t is the macroscopic shear stress. Here, the relationships suggested by Brinson (1993) are utilized for j( s, T).Referring to Figure 3, the evolution equations for j( s, T) at different regions in the phase diagram are as follows: Conversion to detwinned martensite For T.TsM CM (T TsM )

M scr s\scr s + CM (T Ts )\ f +

and

1 j0 j( s, T ) = 2 ( ) h i p cr M cos cr 3 s sf CM (T Ts ) ss scr f +

ð12Þ

1 + j0 2

For T \TsM and scr s\scr s \ f ( ) 1 j0 p 1 + j0 cr cos cr j( s, T ) = 3 ½ s sf + cr ss s f 2 2 ð13Þ

Conversion to austenite s\CA (T TsA ) For T .TsA and CA (T TfA )\ j j( s, T ) = 0 2

(

#) p s A 1 + cos A T Ts ð14Þ CA Tf TsA


"

Mehrabi et al.

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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 3 3. Calculate the elastic strain: ee , equation (16b) 4. Calculate strain e = ee + etr , equation (16a) 5. Calculate Jacobian matrix ð ∂e y 1+y 3 1 = dij dpq + (Nij Npq + Tij Tpq )dV Cet = ∂s E E 4p V ð ð ð ∂Tij 1+y 3 ∂Trs 9 Spq dj( s, T) 3 ∂Tij dV + srs e Tij + Trs Tij dV + e j( s, T) dV + E 4p 8p s d s 4p ∂Tpq ∂Tpq ∂Tpq V

V

V

6. Calculate stress incremental tensor Ds = Cet : De 7. Update stress sn + 1 = sn + Ds 8. End the program

in which j0 , TfM , TsM , TsA , and TfA are the initial martensite fraction, martensite finish, martensite start, austenite start, and austenite finish temperatures, respectively. Also, CM and CA represent the slope of martensite and austenite strips in the stress–temperature phase diagram, cr and scr s and sf are the start critical stress and final critical stress in the stress–temperature phase diagram in which the forward and reverse transformations take place, as shown in Figure 3. The effective elastic modulus in transformation step is calculated in terms of the martensite volume fraction using the Reuss model for SMAs (Brinson and Huang, 1996) 1 (1 j) j = + E(j) EA EM

ð15Þ

where EA and EM are the elastic moduli of pure austenite and pure martensite phases, respectively. Substituting equations (3), (4), (9), and (10) in equation (6), the total strain is eij = eeij + etrij eeij =

Algorithm and formulation used in user material subroutine The computational algorithm in user material subroutine (UMAT) is outlined in Table 1. Microplane model provides closed-form relationships to calculate strains in terms of stresses. Calculation of equation (16) is straightforward because the magnitude of martensite volume fraction can be determined in closed form and the derivation of the material Jacobian is straightforward. The implicit solver of ABAQUS (UMAT) implements these formulations. So the material Jacobian (DDSDDE) is calculated, and then stress is updated. Moreover, since the proposed model is a quasi-static model, there is no time-discrete equation and no trial value for the parameters.

ð16aÞ ð

y 1+y 3 sss dij + srs (Nrs Nij + Trs Tij )dV E(j) E(j) 4p V

ð16bÞ etrij = e j( s, T )

In calculation of the strain, the above integrals are evaluated numerically by using a 42-point Gaussian integration formula for a sphere surface (Bazant and Oh, 1986).

ð

3 Tij dV 4p

ð16cÞ

V

(16)

Experiment and material details All mechanical tests were performed using a Bose ElectroForce machine (Figure 4). All experiments were controlled by axial force and torsional torque. The deformation was recorded by axial displacement and rotation. Johnson Matthey supplied Nitinol thin-wall tubes with an outer diameter of 4.5 mm, a thickness of 0.3 mm, and a gage length of 14 mm. The transformation temperatures of the samples were characterized by


6


Figure 4. Experimental setup.

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

Values

Units

EA EM yA = yM TfM TsM TsA TfA scrs scrf CM CA e

20,000 13,300 0.33 241 258 268 288 20 100 6 8.2 0.038

MPa MPa K K K K MPa MPa MPa/K MPa/K

the differential scanning calorimetry (DSC) and are shown in Table 2. Samples were trained prior to testing at the constant temperature of 323 K to obtain a stable response and remove residual strains possibly generated through the manufacturing processes. Stable response was observed in specimens after about 30 loading/unloading cycles. The loading rate was under 103 s1 to be near the isothermal boundary conditions that are used in the present simulations. Three uniaxial characterization tests were performed at T = 296 K, T = 313 K, and T = 323 K to calibrate the material parameters. The material parameters calibrated from these uniaxial experiments are listed in Table 2. The uniaxial tension, pure

Figure 5. Schematic representation of the finite element model 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.

torsion, and proportional tests were carried out at the constant 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 tension and torsion is investigated. In the following, with the same material parameters, microplane predictions are compared with experimental data obtained in


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

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

proportional tension–torsion loadings. In the last part, stress-induced martensite at TsM \T \TsA is investigated. Figure 5(a) shows an undeformed thin-walled SMA tube with the outer diameter of 4.5 mm, a thickness of 0.3 mm, and a length of 14 mm subjected to torsion at one end, while the other end is fixed. Figure 5(b) shows shear stress contour in deformed tube under a 30° rotation. It is clear that shear stress on the outer diameter is nearly constant. Shear stress–strain curves in three points at different places are shown in Figure 5(c). As shown, results of these three different places on the cross 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 in the whole finite element (FE) model. To make sure that the quantity of the elements is adequate, some sets of elements are assigned to the model in which similar results are obtained by increasing the number of elements. In numerical simulations produced by the developed UMAT, the time step for each loading and unloading step is fixed at 0.01, and each step is divided into 100 increments. In each increment with a specific strain increment, strain is calculated by equation (16), and then the Jacobian matrix and stress incremental are calculated. All the reported results belong to the outer diameter of the thin-walled tube.

the reasonable accuracy of the obtained material parameters. It has been reported that the mechanical characteristics are significantly different in tension and torsion (Wang et al., 2010). In other words, an SMA tube behaves differently under torsion in contrast with tension (Sun and Li, 2002). According to these results, the same material parameters are used in the corresponding model for the study of pure torsion of NiTi torque tube except axial maximum recoverable strain, which is pffiffiffi defined as e = 3 = 0:0219. Although this assumption solved the discrepancy issues in torsional modeling, it needs to be investigated further in a more fundamental approach. Another possible assumption is the von Mises behavior of the material. The other possible assumption is dominance of shear mode in transformation. Figure 7 shows the comparison of microplane model with experimental data at the temperature of 296 K. The p shear ffiffiffi strain transformation in pure torsion should be 3e j (Kadkhodaei et al., 2008), which according to equation (16c) is ð tr tr 3 g = 2e12 = 2e j T12 dV = 2 3 0:0219 4p

Investigation of SMA superelastic behavior in tension and torsion The ability of the proposed microplane model in capturing SE is investigated here. The uniaxial stress–strain response obtained from the microplane approach is compared with experimental result at the temperature of 296 K and is shown in Figure 6. The agreement between the simulation results and experiment indicates

V

3 1 3 0:895 = 0:039

ð17Þ

There is a 3.3% error in microplane prediction that is due to the numerical evolution of the integrals. The discrepancy between the experimental data and microplane results in forward and reverse transformation regions, which may be due to the kinetics formulas that are utilized to estimate j. Also, some discrepancies during the elastic loading and unloading are mainly due to the assumption of constant modulus of elasticity for pure austenite (EA ) and pure martensite (EM ), which is not the exact case in reality. Moreover, it is seen that shear strain in forward transformation begins from


8


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

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

about 0.02 and finishes at 0.074, which approximately correspond to 8° and 28° of rotation, respectively. The proposed formulation is based on small strain theory, and hence, such large angles of rotations may impose artificial strains and is considered as another source of discrepancy observed in Figure 7. Therefore, small rotation is applied in pure torsion so that the proposed model would be applicable without error. However, extension of the proposed model to large strain domain needs to be more thoroughly investigated in future studies. The effect of wall thickness of the tube on the torque–angle response is depicted in Figure 8. Three tubes are simulated while subjected to a same amount of rotation. As shown, by increasing the thickness, the torque required for full transformation is increased, and it shows that thicker tubes need more torque to complete the transformation.

equation (11) for the effective stress and the following equation for the effective strain rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 e = e2 + 3

ð18Þ

where e is the macroscopic tensile strain and g is the macroscopic shear strain. There is a reasonable agreement between microplane predictions and experimental results. One more proportional loading case in a straincontrolled manner in the form of Figure 11 is considered, where maximum displacement is 0.5 mm and maximum rotation is reached to 0.26 rad. Again, two repetitions are carried out for each step and experimental results are shown in Figure 12. Referring to the discussion provided for Figure 7, some differences between the numerical and experimental results may be attributed to the utilized small strain formulation in this work.

Superelastic proportional tension–torsion To demonstrate aspects of the proposed model, proportional loading in stress- and strain-controlled manners is experimentally performed, and the findings are compared with the obtained numerical results using the same material parameters extracted in uniaxial loading. The studied proportional loading paths in stress- and strain-controlled cases are shown in Figures 9 and 11, respectively. In the stress-controlled case, maximum axial stress and shear stress are 230 and 101 MPa, respectively. At least two repetitions are carried out for each loading path. The proposed model predictions are qualitatively similar to the experimental results, as shown in Figure 10. Uniaxial stress–strain and shear stress–strain responses are shown in Figure 10(a) and (b). Figure 10(c) shows effective stress versus effective strain using

Stress-induced martensite at TsM \T \TsA In this part, torsion of thin-walled NiTi tubes with different thicknesses is studied by considering a constant loading/unloading cycle at T = 265 K, which is under TsA . These tubes are initially in the austenite phase, and the material parameters are the same as those shown in Table 2. In the loading step, a portion of austenite is transformed to martensite. Then, in the unloading part, martensite fraction remains constant and a residual strain appears in the tube. This residual strain can be recovered by increasing the temperature to the austenite finish temperature. The corresponding torque–twist angle curves are shown in Figure 13(a) using microplane model. It is shown that by increasing the wall thickness, the torque required for the full


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

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

transformation increases so that the martensitic transformation is not fully completed for the thickness of 0.4 mm. Shear stress–strain diagrams corresponding to Figure 13(a) are shown in Figure 13(b). A tube with 0.2 mm thickness shows the complete transformation while the thickness of 0.4 mm does not. The results shown in Figure 13(a) and (b) indicate that at a relatively small thickness, the martensitic transformation is completed. This, however, is not the case at a greater thickness. In Figure 14, the ability of the microplane model in the simulation of SME at T = 265 K for a tube with the thickness of 0.2 mm is depicted. This figure shows the ability of the proposed microplane model in predicting strain recovery after the removal of stress in the heating process.


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Figure 12. Comparison of the microplane model with the experimental data in proportional loading, strain-controlled: (a) axial stress–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 at T = 265 K obtained from microplane model.


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11 Author’s note R.M. is currently a visiting PhD student at the Dynamic and Smart Systems Laboratory, Mechanical, Industrial, and Manufacturing Engineering Department, University of Toledo, 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 Ministry of Science, Research and Technology (to R.M.).

References

Figure 14. Numerical simulation of shape memory effect at T = 265 K by microplane model.

Conclusion In this article, behavior of NiTi tubes under tension, torsion, and proportional loading is studied. For this purpose, a 3D constitutive model is developed by utilizing the microplane theory, in which 1D constitutive relations for normal and shear directions of all microplanes at a material point are generalized to 3D macroscopic constitutive equations by employing a homogenization technique. The results of the proposed microplane approach are compared with the experimental results for a superelastic NiTi tube. Results show that an increase in the wall thickness increases the necessary torque to induce a complete transformation in SMA torque tubes with constant outer diameters. Microplane formulation is also able to demonstrate the characteristics of proportional loading paths. The proposed model predicts an SME at temperatures below the austenite start temperature. This indicates the capability of the proposed model in predicting the SMA behavior as a simple approach. This proposed model provides closed-form relations for the inelastic strain in terms of the applied stresses. An advantage of this approach is the simplicity of deriving the material properties in uniaxial tests and using them for torsion, as well as other loading conditions. One of the future steps of this modeling approach is an investigation of the maximum recoverable strain in torsion in order to find a fundamental solution. Acknowledgements The authors would like to acknowledge collaboration of Professor Arbab Chirani at Laboratoire Brestois de Mećanique et des Syste`mes, ENSTA Bretagne/UBO/ENIB, Technopoˆle Brest-Iroise, Plouzane´, France, for the DSC tests.

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Structures Journal of Intelligent Material Systems and

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