A Comparative Study of Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 Tube Actuators J.S. Owusu-Danquah, A.F. Saleeb, B. Dhakal, and S.A.

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1 JMEPEG (2015) 24: ÓASM International DOI: /s /$19.00 A Comparative Study of Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 Tube Actuators J.S. Owusu-Danquah, A.F. Saleeb, B. Dhakal, and S.A. Padula II (Submitted November 19, 2014; in revised form January 21, 2015; published online February 18, 2015) A shape memory alloy (SMA) actuator typically has to operate for a large number of thermomechanical cycles due to its application requirements. Therefore, it is necessary to understand the cyclic behavioral response of the SMA actuation material and the devices into which they are incorporated under extended cycling conditions. The present work is focused on the nature of the cyclic, evolutionary behavior of two widely used SMA actuator material systems: (1) a commercially available Ni 49.9 Ti 50.1, and (2) a developmental high-temperature Ni 50.3 Ti 29.7 Hf 20 alloy. Using a recently developed general SMA modeling framework that utilizes multiple inelastic mechanisms, differences and similarities between the two classes of materials are studied, accounting for extended number of thermal cycles under a constant applied tensile/compressive force and under constant applied torque loading. From the detailed results of the simulations, there were significant qualitative differences in the evolution of deformation responses for the two different materials. In particular, the Ni 49.9 Ti 50.1 tube showed significant evolution of the deformation response, whereas the Ni 50.3 Ti 29.7 Hf 20 tube stabilized quickly. Moreover, there were significant differences in the tension-compression-shear asymmetry properties in the two materials. More specifically, the Ni Ti 29.7 Hf 20 tube exhibited much higher asymmetry effects, especially at low stress levels, compared to the Ni 49.9 Ti For both SMA tubes, the evolution of the deformation response under thermal cycling typically exhibited regions of initial transients, and subsequent evolution. Keywords asymmetry in tension-compression-shear, Ni 49.9 Ti 50.1, Ni 50.3 Ti 29.7 Hf 20, thermomechanical cycles, tube actuators 1. Introduction In recent years, SMA-based, solid-state actuators have attracted considerable attention (i.e., in comparison to other conventional actuation counterparts) in many engineering fields such as aerospace, automotive, and energy. This is mainly attributed to their high energy density which offers more efficient actuation, providing high strokes with desirable features of reduced weight, size, and complexity (Ref 1-4). The design of SMA actuators is typically based on the shape memory effect triggered through the phase transformation between a high-symmetry Austenite (A parent) phase and low-symmetry Martensite (M daughter) phase of the SMA material systems. These actuators operate such that under a constant applied system of mechanical loads (axial force, bending moment, and/or torque), cooling of the SMA actuator below the martensite finish (M f ) temperature produces a stroke (i.e., change in deformation), which is recovered upon heating above the austenite finish (A f ) temperature. The mode by which the mechanical load(s) is/are applied determines the type(s) of deformation (i.e., axial, rotational, or both) produced in the actuation system. J.S. Owusu-Danquah, A.F. Saleeb, and B. Dhakal, Department of Civil Engineering, The University of Akron, 302 Buchtel Common, Akron, OH ; and S.A. Padula II, N.A.S.A. Glenn Research Center, Brookpark Rd., Cleveland, OH Contact saleeb@uakron.edu. The magnitude of the actuation stroke produced upon the first forward phase transformation by an SMA actuator is a fundamental factor in assessing the capabilities of the SMA actuator system. The NiTi-based SMAs, especially the commercially available Ni 49.9 Ti 50.1 (at.%), have proved to be viable materials for the design of solid-state actuator systems. Their ability to produce large recoverable work output at a lower cost is a major reason for their popularity (Ref 5, 6). However, there are a number of limitations associated with the use of these materials, such as their moderate transformation temperature range ( 90 C to +100 C), and their higher degree of dimensional instability (i.e., the continual increase in the accumulated strains over the extended number of thermal cycles). This will present significant difficulties in the field of aeronautics where many actuator systems operate at temperature range higher than 100 C, and where there are severe operational restrictions on the actuator space. Many efforts have been made to overcome these undesirable limitations in the Ni 49.9 Ti 50.1, including the development of several alternative training procedures and addition of other alloys. For instance, thermal cycling and different aging treatments (Ref 7) have been proved to influence the dimensional stability and transformation temperature of this material system. The inclusion of additional elements such as Hf, Zr, Pd, Pt, and Au in the amounts of 8-20% (at.%) increases the transformation temperature of the NiTi alloy by more than 70 C (Ref 8, 9). The NiTiHf alloy, relative to the other NiTiX (X = Pd, Pt, Zr etc.) ternary alloys, is considered to be a more favorable actuation alloy than the other high-temperature shape memory alloys (HTSMAs). Its benefits include the relatively lower cost, reasonable ductility, and unique ability to withstand higher levels of stresses (Ref 10). In addition to the SMA material properties, the geometrical configuration of the actuator essentially affects the output of the actuator system. In recent years, a wide variety of geometrical 1726 Volume 24(4) April 2015

2 structures (wire, plate, helical spring, rods, tube, honeycombs, etc.) have been experimentally studied to improve the performance of SMA actuators (Ref 11). Also there are more recent applications involving mini/micro SMA actuators utilizing the unusual snake-like wire geometries (Ref 12). SMA actuators existing in the forms of wires, beams, cables, and coils have been found very useful for specific applications. Although the simple wire geometry actuators are efficiently able to provide linear actuation, applications (especially in the field of aerospace) requiring complex, high actuation forces rely upon the tubularconfiguration actuators to provide such high strokes. In particular, the tubular geometry is aerodynamically favorable for applications requiring large recoverable torsional deformations (Ref 1). These actuators are designed such that high magnitudes of actuation twists or strokes are produced when the tube is thermally cycled between temperatures above the Austenite finish and temperatures below the Martensite finish phases of the SMA material. Among these, the tubular solid-state actuators have been shown to be well suited for the design of propellers (Ref 13) reconfigurable rotor blade, and airplane wing morphing systems (Ref 14). In such tubular actuation systems, the thickness and length of the actuator tube determines the magnitude of force or torque required to produce a desirable amount of stroke in the SMA actuator. In particular, a thicker tube would require a larger amount of torque or force to generate the same amount of angular or axial deformation produced in a relatively thinner tube (Ref 15). Over the past decade, significant progress has been made toward the development of micromechanical (Ref 16-18) and phenomenological (Ref 19-21) material models, to expound on the underlying mechanisms responsible for the unique features that are exploited in the SMA materialõs application (Ref 22). The systematic investigations carried out using a suitable SMA model can provide a simplified and a guided means of studying the global thermomechanical responses of different SMA materials and structures without resorting to an extensive set of empirical experiments. In our present study, a thin, tubular, solid-state actuator was analyzed using two different SMA material systems (i.e., a commercially available Ni 49.9 Ti 50.1 and a high-temperature Ni Ti 49.7 Hf 20 ), to determine their performance under the iso-force and iso-torque loading conditions. A general 3D SMA model developed by Saleeb et al. (Ref 23), which has been utilized to calibrate the Ni 49.9 Ti 50.1 material system and has been implemented in the large scale simulation of a Ni 49.9 Ti 50.1 helical spring actuator (Ref 24)is used here for the following purposes. First, the modeling framework was re-parameterized to account for the behavior observed in the HTSMA Ni 50.3 Ti 49.7 Hf 20 material. Subsequently, the calibrated SMA model for these two different actuator materials is implemented for the simulation of the Ni 49.9 Ti 50.1 and the Ni 50.3 Ti 49.7 Hf 20 SMA tube actuators. Detailed qualitative and quantitative comparisons (i.e., the similarities and differences) in the cyclic deformation response of these two different tube actuators, under different mechanical loading conditions (axial tensile/compressive, torque), constitute the major effort in the present study. on the balance of the energy dissipated and the energy stored during the thermomechanical deformation process of the SMA material system. The model is designed to comprehensively capture the unique responses observed in ordinary SMAs as well as the high-temperature SMAs. In the formulation, the total strain tensor e ij (and its rate, _e) is compactly given as e ij ¼ e e ij þ ei ij where ee ij, i.e., the reversible or elastic component implicitly accounts for all possible rate dependencies of individual austenite and martensite phases and e I ij, the irreversible or inelastic component, accounts for all transformation-induced strains. Due to the insignificant magnitude of the thermal strain in comparison to the transformation-induced strain, the thermal strain contribution to the total strain is neglected in the formulation. The corresponding stress tensor, r ij, is also decomposed into effective stress, ðr ij a ij Þ, and an internal state tensorial variable a ij ¼ PN a ðbþ ij, where N indicates the number of inelastic mechanisms b 1 (denoted by superscript b ), with the stress-like, and its conjugate strain-like, internal variables defined as a (b) ij and c (b) ij, respectively. More specifically, for the present applications of the model, six inelastic mechanisms were utilized (i.e., b = 1-6) for the two SMA materials (Ni 49.9 Ti 50.1 and Ni 50.3 Ti 49.7 Hf 20 ). In particular, the mechanisms, b = 1-3 are devoted to regulate the energy stored, and those for b = 4-6 correspondingly regulate the energy dissipated during the materialõs transformation process. These mechanisms in the model (to different degrees) affect the non-linear response of the material; as such their magnitudes are selected based on the transformation characteristics of a particular SMA material system. For instance, for the superelastic stress-versus-strain curve shown in Fig. 1, the mechanism b = 1 primarily dictates the rapid development of the transformation strains (i.e., during the loading) toward the critical state denoted as a in Fig. 1, which is the onset of the limited stress transformation surface. The gradual hardening in the transformation regime (i.e., from a to b in Fig. 1) is also controlled by the mechanism, b = 2. These two mechanisms, i.e., b = 1, and 2 also together determine the pattern and magnitude of strains developed at latter part of the unloading stage (i.e., e - f in Fig. 1) as well as the amount of residual strains attained at the end of the unloading branch (i.e., stage f in Fig. 1). Moreover, the storage mechanism, b = 3, also provides an increasing hardening function that determines the limiting internal forces regulating the completion of all phase transformations; i.e., the rehardening regions ÔÔb-c-d in Fig. 1. The energy dissipation mechanisms b = 4, 5, and 6 mainly 2. Summary of the 3D multi-mechanism SMA model The SMA model framework, developed by Saleeb et al. (Ref 23), is a general constitutive model whose formulation is based Fig. 1 A schematic for a typical pseudoelastic stress vs. strain response of an SMA material Volume 24(4) April

3 control the height of the hysteresis loop and they play a very important role in also regulating the evolutionary character of the SMA; i.e., changes in the details of such hysteresis loops with repeated thermomechanical cycles. The two fundamental energy potentials used in the model formulation are the GibbÕs complementary function, U, and dissipation function, X. These thermodynamic/energy functions (stored energy and dissipation) are formulated in terms of the strain-and-stress contributions from each of the six inelastic mechanisms. These tensorial variables in the model, including the stress tensor, as well as the inelastic strain- and internal stress-state variables are the key quantities responsible for energy storage-dissipation partitioning that are needed to capture such important SMA characteristics, such as superelasticity, pseudoplasticity, shape memory effect, and the cyclic evolutionary response under thermomechanical loads. The resulting sets of mathematical equations in the model formulation, confined to our present study are briefly stated in Tables 1 (basic equations), and 2 (transformations and hardening functions). Note that all tensorial quantities in the model are formulated in the rotated configuration in order to accommodate more complicated boundary-value problems, which consider the effect of large deformations (Ref 25). For the purpose of model calibration in section 3, as well as its application in the numerical simulations of section 4, the present SMA model was implemented in ABAQUS (Ref 26) via the user material subroutine (UMAT) option. 3. Parameterization of the SMA model In order to parameterize the current SMA model for the two selected materials, i.e., the Ni 49.9 Ti 50.1 and the Ni 50.3 Ti 49.7 Hf 20 alloys, a total of 25 material constants needed to be determined. These included (a) (b) 2 empirical (handbook value quantities; elastic modulus (E) and PoissonÕs ratio (t) (see Table 3)). 5 material parameters to define the rate equation governing the transformation (inelastic) strain. Among these, two parameters, i.e., exponent ÔnÕ and modulus Ôl,Õ account for rate-dependency of the material. The numerical values were selected to be the same for the two (c) different SMA materials, and reflect their well-known weak rate-dependency, i.e., relatively large ÔnÕ and small ÔlÕ, (see Table 3). The remaining 3 parameters comprised two material constants (i.e., ÔcÕ and ÔdÕ), dedicated to represent the degree of ATC, and the threshold ÔjÕ, that defines the critical onset stresses for A fi detwinned M (at T >A f ), and twinned M fi detwinned M variants (at T < M f ). 18 non-linear hardening parameters distributed among the 6 inelastic mechanisms (3 for each mechanism), i.e., threshold Ôj (b) Õ, exponent Ôb (b) Õ and modulus ÔH (b) Õ for each of the b = 1-6. In the specific parameterization of the model for each of the two materials studied here, the 13 material parameters (ÔdÕ, ÔH (b) Õ and Ôb (b) Õ for b = 1to6) were taken to be fixed as given in Table 3. However, the set of 8 remaining material constants (Ô jõ, ÔcÕ, and Ôj (b) Õfor b = 1-6) is taken to be functionally dependent, to account for the possible temperature and/or stress-state dependencies of the thermomechanical SMA responses such as strain evolution during thermal cycling, ATC, transformation temperature-shifts, and any other unique behaviors observed experimentally for a particular SMA material. In particular, in such cases, the temperature and stress-state dependency will be assumed in a multiplicatively decoupled form written as follows: j ¼ j ðrþ ðtþgðr e Þ and j ðbþ ¼ j ðrþ ðbþ ðtþg ðbþðr e Þ, where Ô j ðrþ Õ and Ô j ðrþ ðbþõ are temperature dependent reference threshold functions; Ô gõor Ô g ðbþ Õ is a non-dimensional factor dependent on the intensity of stress; Ô TÕ is the temperature, and Ô r e Õ is the effective (also known p as von Misses/multiaxial stress intensity), given by r e ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ðr ij M ijkl r kl Þ= Ni 49.9 Ti 50.1 SMA Material Data collected from isobaric experiments, conducted in tension on Ni 49.9 Ti 50.1 specimens at different bias-stress levels (i.e. from 0 to 300 MPa), for 100 thermal cycles between a lower cycle temperature (LCT) = 30 C and an upper cycle temperature (UCT) = 165 C, were used for the model parameterization (Ref 27-29).The specimens were 10-mm diameter rods. These circular rods have varying lengths in the hot-rolled/hot-drawn and hot-straightened condition (Ref 27). From the differential scanning calorimetry (DSC), the Ni 49.9 Ti 50.1 material had an austenite start temperature, Table 1 Summary of basic equations of the multi-mechanism SMA material model Equation Set 1: Decomposition of stress and strain e ij ¼ e e ij þ ei ij ; a ij ¼ PN b¼1 a ðþ b ij Equation Set 2: Specific functional forms for stored energy and dissipation potentials U R r ij ¼ 1 2 r ije 1 ijkl r kl; U IR r ij ; a ðbþ ij ¼ r ij e I ij þ PN H ðbþ ; b¼1 ðbþ X r ij a ij ; a ij ¼ R j 2 F n 2l df; Equation Set 3: Evolutionary laws _e ij _e I ij ¼ R ij ¼ E 1 ijkl _r kl; _e I ij ; 1 _a ðþ b _c ðþ b ðþ ðþ ij ; _c ðþ b ij ; ðþ kl U b b b ij Where j, l, and n are material constants, F is the transformation function, and H ðbþ are hardening functions (see Table 2). E ijkl is the isotropic fourth-order tensor of elastic moduli (YoungÕs Modulus, E and PoissonÕs ratio m) 1728 Volume 24(4) April 2015

4 Table 2 Summary of transformation and hardening functions used in the multi-mechanism SMA material model Equation Set 4: Transformation and hardening functions 1 1 F r ij a ij ¼ j 2 2q 2 r ij a ij Mijkl ðr kl a kl Þ ; 8 R < j 2 1 ðbþ hðg H ðþ b ¼ ðþ b Þ dgðþ b ; for b ¼ 1; 2; 3; R : j 2 1 ðbþ hg ð ðbþ Þ dgðb Þ ; for b 4; 8 pffiffiffiffiffið b q ðbþ j ðþ b H ðþ b g ðþ b ðþ b 1Þ >< pffiffiffiffiffi b h g ðbþ j ðþ b þh ðbþ g ¼ ðbþ ðbþ ; for b ¼ 1; 2; " pffiffiffiffiffi # bðþ b g >: q ðbþ H ðbþ 1 þ ðbþ j ðbþ =H ðþ b ; for b ¼ 3; 2 pffiffiffiffiffiffiffiffi! 3 b h G ðbþ G ¼ H ðbþ 1 ðbþ b 4 ðþ^h ðlþ 5; for ; b 4; where G ðþ b a ðbþ ij ¼ 1 2j 2 ðþ b a ðþ b ij q ðbþ M ijkl a ðbþ ; g ðbþ kl c ðbþ ij ¼ c b ð Þ ij c ðbþ p ij ; q ¼ ffiffi 1þc p ffiffiffiffiffiffiffi d ; q ðbþ ¼ 1; k 3 ¼ cos 3h ; where h is LodeÕs angle calculated from the invariants of the effective stress ( r ij a ij ) (Ref 33) ^hðlþ = the Heaviside function with argument being the loading index L ¼ a ðbþ C ij ; where C r ij a ij M ijkl ¼ 1 2 d ikd jl þ d il d jk 1 3 d ijd kl ; With d ij = Kronecker delta 1þc dþk 3 ij H ðbþ, b ðbþ, and j ðbþ = Material parameters for the individual hardening mechanism c, d; = Material parameters for tension/compression asymmetry (Ref 23) Table 3 Set of fixed material parameters used for simulated test cases Parameters Units Material System Ni 49.9 Ti 50.1 Ni 50.3 Ti 29.7 Hf 20 (EXT124) Deflated Elastic stiffness MPa modulus, E PoissonÕs ratio, m 0.3 n 5 l MPa s 1.00E + 05 Number of inelastic 6 mechanisms, b H(b), b(b) b = 1 MPa, Non-dimensional , ,1 b = , ,1 b = 3 200, , 1 b = , ,1 b = , ,1 b = 6 600, , 5 A s =95± 5 C, and austenite finish temperature, A f =115 C under stress-free conditions (Ref 30). A typical tensile, isobaric experimental test result, showing the strain-versus-temperature response of the Ni 49.9 Ti 50.1 SMA material at 150 MPa bias-stress, is shown in Fig. 2(a). In calibrating the model for the Ni 49.9 Ti 50.1 material, the intrinsic ATC parameters (i.e., ÔcÕ, and ÔdÕ) were deactivated since only tensile results (but not their compressive counterparts) from isobaric tests were available at the time of calibration. However, under effects of large deformation, there will be differences in the magnitudes of displacement measured under same intensity of compressive and tensile loading (see Fig. 4f, in Ref 28). The key features observed in the experimental test response that served as a benchmark for comparing the calibrated model results were (a) the initial transient behavior occurring in the cooling branch of the first thermal cycle, and (b) the strain evolution in the Ni 49.9 Ti 50.1 material with increasing number of thermal cycles. The reader is referred to our previous papers (Ref 28, 29) for more elaborate information on the procedures involved in the parameterization of the Ni 49.9 Ti 50.1 SMA material, and extensive comparisons between the model responses and the experimental measurements. For our purpose here, the model predicted strain versus temperature responses of the Ni 49.9 Ti 50.1 material at 150 MPa bias-stress level is shown in Fig. 2 in comparison to the experimental results. The values of the material parameters used in this model simulation are summarized in Tables 3, 4, and 5. Volume 24(4) April

5 Fig. 2 Strain vs. temperature response of Ni 49.9 Ti 50.1 for (a) Experimental results and (b) SMA model prediction at a constant bias stress of 150 MPa for 100 thermal cycles between LCT = 30 C and UCT = 165 C (Ref 19) Table 4 Temperature-dependency of material parameters j and j (b), for b =1,2,, 6 Material Parameters (MPa) Material system Temperature ( C) j j (b), b =1,2 j (b), b =3 j (b), b =4 j (b), b =5 j (b), b =6 Ni 49.9 Ti 50.1 T 1 = E T 2 = 65 (50 C for b = 4) T 3 = 115 (120 C for b = 4) T 4 = Ni 50.3 Ti 29.7 Hf 20 T 1 = E + 21 T 2 = T 3 = T 4 = T 5 = (For each parameters, values are interpolated linearly between the values given in the table at characteristic temperatures T 1, T 2, T 3, and T 4 ) Table 5 Stress-dependency of the non-dimensional factors g and g (b).values are linearly interpolated between shown stress levels Scale factors Material system Stress levels (MPa) g g (b), b =1,2 g (b), b =3 g (b), b =4 g (b), b =5 g (b), b =6 Ni 49.9 Ti Ni 50.3 Ti 29.7 Hf Ni 50.3 Ti 29.7 Hf 20 SMA Material The experimental results from a series of isobaric tension and compression tests carried out on 5 mm diameter, 17.8 mm gage length-ni 50.3 Ti 29.7 Hf 20 dog-bone tensile specimens, (machined from extrusion-124) were used for the Ni 50.3 Ti 29.7 Hf 20 SMA material model calibration. DSC results for 1730 Volume 24(4) April 2015

6 Fig. 3 (a) Experimental results and (b) SMA model predictions at various stress levels in tension and compression for Ni 50.3 Ti 29.7 Hf 20. The results taken from the 2nd thermal cycle at each stress level indicate significant stress-dependency and ATC Table 6 Stress-dependency of distortion constant, ÔcÕ and temperature-shift factor, ÔT shift Õ Material system Stress levels (MPa) Distortion constants c Temperature-shift (in C) T shift Ni 50.3 Ti 29.7 Hf the Ni 50.3 Ti 29.7 Hf 20 SMA material showed approximate transformation temperatures of M f = 129 C, M s = 136 C, A s = 156 C and A f = 165 C, under stress-free conditions (Ref 8). As no extended thermal-cycling data were reported for this extrusion-124, an assumption was made regarding the amount of cyclic strain evolution that would be allowed in the calibrated model based upon the limited test data of another extrusion (extrusion-146) of the same target composition, which showed a rather small amount of accumulated open-loop strains during 100 thermal cycles (Ref 31). The criteria for comparison between the Ni 50.3 Ti 29.7 Hf 20 model calibrated results and the experimental response under uniaxial tension and compression tests were (a) the significant differences in the magnitudes of actuation strains at all the different bias-stress levels, (b) the marked shifts in transformation temperature regions with the increase of the stress levels both in tension as well as in compression, and (c) the strong asymmetry (ATC) in the material under tension-versus-compression load-biased stresses (Ref 8). Moreover, with regard to the evolutionary response with thermal cycling, the model parameterization was forced to exhibit a limited amount of strain evolution during 100 thermal cycles under each of the individual bias stresses (tension and compression), in conformity with the discussion alluded to at the beginning of this subsection. To account for the significant ATC effects observed experimentally in the Ni 50.3 Ti 29.7 Hf 20 material, especially at lower stress levels (see Fig. 3), the intrinsic ATC parameters, ÔcÕ, and ÔdÕ, in the material model had to be activated. In particular, a constant value of ÔdÕ = 1.05 and stress-dependent values of ÔcÕ was used (see Table 6). Furthermore, the experimentally observed temperature-shifts were accounted for in the model through the temperature-shift factors (T shift ) stated in Table 6; i.e., this simply amounts to the replacement of the functional temperature-dependency of jðtþ and j ðbþ ðtþ by jðt T shift Þ and j ðbþ ðt T shift Þ, respectively. The final values of the material parameters in this case are stated in Tables 3, 4, 5, and 6. A comparison of the experimental and model results, showing the variation of strain with temperature at varying stress levels from 100 to 500 MPa (for tension) and 100 to 500 MPa (for compression) at increments of 100 MPa is shown in Fig Finite Element Simulation of the Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 SMA Tube Actuators 4.1 Geometry, Boundary Conditions, and Mesh Convergence A thin-walled tube cylinder (i.e., t/d = ) of external diameter, D = 5.08 mm, internal diameter, d = 4.60 mm, length, L = 50 mm and thickness, t = 0.24 mm with the associated boundary conditions as shown in Fig. 4, was used for the present study. During the analysis, one end of the tube was fixed against translational and rotational degrees of freedom and the other end was made free to deform. Volume 24(4) April

7 Fig. 4 Geometric details and load conditions for the SMA tube being utilized in the study The generalized 4-node, bilinear, axisymmetric stress element with twists (CGAX4 in ABAQUS) which provides radial translation, R, axial translation, Z, and twist angle, U degrees of freedom was selected for mesh generation and large scale simulation of the tube actuator. Using the CGAX4 full integration elements allowed the tube to be modeled and analyzed at a more reduced complexity and computational expense. The nodal degrees of freedom associated with the CGAX4 element enabled any of the mechanical load controls; i.e., tensile, compressive forces and more especially the torque, to be directly applied at the free-end nodes. A mesh convergence study was conducted to determine the suitable mesh size needed for accurate simulation of the tube. Based on the convergence of the magnitude of angle of twist with increasing number of elements, the axisymmetric section was discretized by elements; with 1 element in the thickness (due to the thinness of the wall of the tube), and 50 elements along the length of the tube (to enable the handling of the anticipated large spatial twist angles in the simulation of the tube actuators). In each analysis case, the same tube geometry with the fixed-free boundary condition was used. Note however, that there are differences in the magnitude of applied mechanical and thermal load corresponding to the selected Ni 49.9 Ti 50.1 or Ni 50.3 Ti 29.7 Hf 20 tube actuators. Both SMA tube actuators were studied under the iso-force and iso-torque conditions. 4.2 Loading Controls The geometrical configuration of the thin tube gives more accessible options for it to be mechanically loaded in order to experience the individual or combined stress states of tension, compression, shear, or internal pressure. By their very nature (i.e., of small thicknesses), stress distribution in thin tubes are more uniform than in thick tubes. The type of load applied to the tube dictates the final state of stress developed in the tube. In our present study, we restrict ourselves to the individual stress states of (i) shear (by applying torque), (ii) tension (by applying tensile axial force), and (iii) compression (by applying compressive axial force) in the thin SMA tube (see Fig. 5a). Loading case (i) involving the torque is referred to here as isotorque loading condition, while cases (ii) and (iii) involving tension and compression respectively, are for brevity termed together as the iso-force loading condition. An important remark is made here to acknowledge that although the present paper is focused on the individual stress states of tension, compression, and shear, our future work will investigate the thermomechanical behavior of this same SMA tube actuator under more general conditions of combined state of stresses (i.e., proportional and non-proportional loads of torsion combined with tension or compression) Thermomechanical Loading Procedure. A torque or axial force was applied at the free end of the tube to initiate a torsional or an axial deformation at the initial lower cycle temperature (LCT) of 30 C. For the two SMA tube actuators, a different magnitude of effective stress was targeted at the end of the load up during the mechanical phase of the loading procedure. This was made mainly to achieve a similar state of deformation of the two different SMA tubes in the twist actuation mode at the end of the first thermal cycle, where the SMA materials are known to exhibit the significant transient actuation strokes (starting the load up at the LCT and ending the first cooling also at the LCT). This in turn will enable the focus of the present study to be placed on the comparison of the evolutionary aspects (i.e., occurring between the 2nd and 50th thermal cycles) affecting the dimensional stability of the two different SMA tubes. The specific selected values of the bias torques and/or axial forces (tension/compression) in the present study were obtained from a prior study (not reported here) involving the following steps. First, choosing as a reference the value of 73 twist rotation (reached at the end of the first thermal cycle) for the two SMA tubes, two different magnitudes torques required to achieve this rotation were determined for the two SMA tubes. The resulting levels of the effective stresses were 150 and 400 MPa, in the Ni 49.9 Ti 50.1 tube and Ni 50.3 Ti 29.7 Hf 20 tube actuators, respectively. Typical distribu Volume 24(4) April 2015

8 Fig. 5 Details of the isobaric loading programs considered in the study highlighting (a) a description of the loading paths taken for the different tests (i.e., tension, compression and/or shear), (b) the nature of the isothermal loading phase, and (c) aspects of the subsequent thermalcycling phase Fig. 6 Distribution of the effective and shear stresses in (a) the Ni 49.9 Ti 50.1, and (b) the Ni 50.3 Ti 29.7 Hf 20 SMA tube at the end of the isothermal, torsional loading phase. Note the small nonuniformity/inhomogeneity in the distribution of the stress showing small stress gradient over the thin wall of the tube (here, the axisymmetric tube model is swept about 360 to show the full view of tube) Volume 24(4) April

9 Table 7 Applied mechanical loads and the corresponding displacements at the ends of load up (i.e., d o, U o ), first heating branch (i.e., d A, U A ) and martensite displacement (d M ) or twist (U M ) occurring at the end of cooling branch of 1st and 50th thermal cycle Response at end of Response at martensite Material system Load case Applied load Load up 1st heating 1st cycle 50th cycle Ni 49.9 Ti 50.1 (LCT = 30 C, UCT = 165 C) Tension N d A = d A = d M = 2.81 d M = 5.13 Compression N d A = d A = d M = 2.26 d M = 4.01 Torque N mm U A = 7.39 U A = 5.14 U M = U M = Ni 50.3 Ti 29.7 Hf 20 (LCT = 30 C, UCT = 300 C) Tension N d A = d A = d M = 1.94 d M = 1.99 Compression N d A = d A = 0.30 d M = 1.45 d M = 1.50 Torque N mm U A = U A = U M = U M = Table 8 The accumulated open-loop displacement (d M ) or twist (U M ) between selected thermal cycles Ni 49.9 Ti 50.1 Ni 50.3 Ti 29.7 Hf 20 Iso-force (d M ) Iso-torque (U M ) Iso-force (d M ) Iso-torque (U M ) Cycles Tension Compression Torque Tension Compression Torque 2nd-25th th-50th Evolutionary behavior/dimensional stability Highly evolving More dimensionally stable Fig. 7 Evolutionary response under tensile, iso-force for: (a, b) the Ni 49.9 Ti 50.1 and (c, d) the Ni 50.3 Ti 29.7 Hf 20 tube actuators, indicating the axial displacement vs. time and the axial displacement vs. temperature variations over 50 thermal cycles tion of the effective as well as the shear stress components, showing the very small stress gradient over the thin wall of each SMA tube is depicted in Fig. 6(a) and (b). Next, when the axial actuation mode was subsequently considered, similar effective stress values were targeted at the end of the load up at LCT under either tensile or compressive axial forces. Finally, keeping the applied torque/force constant after reaching the targeted stress, the tubes were thermally cycled between the LCT and UCT range that was appropriate for each of the individual SMA material systems. The heating 1734 Volume 24(4) April 2015

10 Fig. 8 Evolutionary response under compressive, iso-force for: (a, b) the Ni 49.9 Ti 50.1 and (c, d) the Ni 50.3 Ti 29.7 Hf 20 tube actuators, indicating the axial displacement vs. time and the axial displacement vs. temperature variations over 50 thermal cycles and cooling rates of 0.25 and C/s, respectively, were utilized. The magnitudes of applied forces or torques as well as the corresponding LCT and UCT used for the different load cases with respect to (w.r.t) the two SMA tube actuators are stated in Tables 7 and 8. Typical mechanical and thermal load control histories used for the simulation are also shown in Fig. 5(b), and (c), respectively. Under the iso-torque loading condition, torques of 700 N-mm (at a rate of N-mm/s) and 2030 N-mm (at a rate of N mm/s) were applied at the free end of the Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 tube actuators, respectively. Similarly, for the case of iso-force loading conditions, tensile or compressive forces of ±547.6 N (at a rate of ±0.684 N/s) and ±1430 N (at a rate of ±1.788 N/s) were applied to the Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 tube actuators, respectively. 4.3 Thermomechanical Response of the SMA Tube Actuator The detailed thermomechanical response of the Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 SMA tube actuators, i.e., the state of deformation at the end of load up, and the similarities and differences in their response patterns during thermal cycling (cycles 1-50) under the iso-force and iso-torque conditions are discussed in this section. For clarity and elaborate discussion of the results, the responses under the iso-force loading conditions are described first in section followed by that under the iso-torque condition in section To facilitate the subsequent discussion, we denoted the axial displacements at LCT (martensite) and UCT (austenite) as Ô d M Õand Ô d A Õ, respectively, and the corresponding angles of twist at the LCT and UCT as Ô U M Õand Ô U A Õ. The axial and twist actuation strokes, which determine the actuation capability of the SMA tube under the selected loading mode, were represented as Ô d ACT Õ and Ô U ACT Õ, respectively. Mathematically, d ACT was simply defined as the difference in the displacements measured at the end of heating and cooling for each thermal cycle, i.e., d ðnþ ACT ¼ dðnþ M dðnþ A, where N is the thermal cycle number. Similarly, the U ACT was also designated as U ðnþ ACT ¼ UðNÞ M UðNÞ A Response Under Iso-force Condition. A comparison of the variation of the axial displacement with time and temperature for the Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 tube actuators, under the tensile and compressive forces is shown in Fig. 7 and 9, respectively. The two tube actuators under the iso-force (pure tension or compression) loading conditions produced different magnitudes of axial displacements at the end of the cooling branch of the 1st thermal cycle. Although higher magnitudes of axial forces were applied to the Ni 50.3 Ti 29.7 Hf 20 tube actuator, the amount of axial displacement, and hence the magnitude of actuation, produced by the Ni 50.3 Ti 29.7 Hf 20 tube actuator was less than that observed in the Ni 49.9 Ti 50.1 tube actuator. In particular, the 1430 KN tensile force applied to the Ni 50.3 Ti 29.7 Hf 20 tube generated a d M of 1.94 mm (3.8% w.r.t tube length) at the end of the 1st thermal cycle. In contrast, a lesser magnitude of N tensile force applied to the Ni 49.9 Ti 50.1 tube produced a d M, of 2.81 mm (5.6% w.r.t tube length) after the 1st thermal cycle cooling. These values (i.e., the percentages) of displacements were inherently correspondent to the strains measured in the simple, homogeneous, uniaxial, tensile, isobaric experiment, and model test responses at stress levels of 150 MPa in Ni 49.9 Ti 50.1 and 400 MPa in Ni 50.3 Ti 29.7 Hf 20. Considering the transient behavior of the two materials during the 1st thermal cycle, both exhibited qualitatively similar patterns, i.e., very small changes in displacements during the first heating branch, followed by a significant increase in displacement during the subsequent cooling branch. To facilitate the quantitative comparisons, the measured displacements at the ends of the isothermal load-up stage (at room temperature for both tubes), first thermal heating, and subsequent cooling are reported in Table 7. In reference to the loading case involving the tubes under tension (see Fig. 7c and d), it is observed that the displacements of mm in the Ni 49.9 Ti 50.1 tube and mm in the Ni 50.3 Ti 29.7 Hf 20 tube developed at the end of the load-up stage reduced to and mm, respectively, after the first heating branch. These magnitudes of displacements markedly increased after the completion of the first cooling branch; i.e., to values of 2.81 and 1.94 mm in the Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 tube actuators, respectively. This signifies quantitatively distinct transient responses in the two actuator tubes. Although these materials share qualitatively similar behavior in their initial (transient) response, there is a striking disparity in the displacement response occurring between the 2nd and 50th thermal cycles in the Ni 49.9 Ti 50.1 tube compared to its Ni 50.3 Ti 29.7 Hf 20 counterpart. Particularly, under the compression stress state, the Ni 49.9 Ti 50.1 tube actuator attained an initial axial displacement of 2.26 mm after the 1st thermal cycle cooling; yet the measured displacement at the end of the Volume 24(4) April

11 50th cycle cooling was 4.00 mm, reflecting about 77% (i.e., d ð50þ M dð1þ M 100) increase in axial displacement. This observed d ð1þ M difference in the magnitudes of displacements occurring between the end of the 2nd and 50th thermal cycles is also a measure of the integrated open-loop displacement occurring between successive thermal cycles. Here, the open-loop displacement is defined as the difference in the axial displacement at the end of cooling between two successive thermal cycles i.e., d ðnþ OL ¼ dðnþ M dðn 1Þ M for Nth cycle (with N = 2-50). This clearly demonstrates the highly evolutionary character of the Ni 49.9 Ti 50.1 with thermal cycling. On the other hand, the Ni 50.3 Ti 29.7 Hf 20 tube actuator under similar loading condition (i.e., compression) showed a rather limited amount of axial displacement increase of approximately 3.27% between the 1st and 50th thermal cycle cooling, which is remarkably less than that occurring in the Ni 49.9 Ti 50.1 tube actuator. The open-loop displacements between cycles kept reducing for higher number of thermal cycles in both the Ni 49.9 Ti 50.1 and the Ni 50.3 Ti 29.7 Hf 20 tube actuators. The recorded magnitudes of the accumulated open-loop displacement between 2nd and 25th thermal cycles and between 25th and 50th thermal cycles are presented in Table 8. Moreover, although the same intensity of force was applied to each SMA tube under the two iso-force conditions, smaller displacements were observed for the SMA tube actuators under compression compared to those under tension (refer to Fig. 7 and 8). This was noticeable in the Ni 49.9 Ti 50.1 tube actuator as well as Ni 50.3 Ti 29.7 Hf 20 tube actuator, hence indicating the degree of ATC present in both actuation systems. In particular, Fig. 9 Summary plots showing the cyclic evolution of axial displacement (in mm) at martensite (d M ), austenite (d A ) and the corresponding actuation stroke d ACT = d M d A for: (a) the Ni 49.9 Ti 50.1, and (b) the Ni 50.3 Ti 29.7 Hf 20 tube actuator under tensile, iso-force thermal cycling Fig. 10 Summary plots showing the cyclic evolution of axial displacement (in mm) at martensite (d M ), austenite (d A ) and the corresponding actuation stroke d ACT = d M d A for: (a) the Ni 49.9 Ti 50.1, and (b) the Ni 50.3 Ti 29.7 Hf 20 tube actuator under compressive, iso-force thermal cycling 1736 Volume 24(4) April 2015

12 Fig. 11 Evolutionary response under iso-torque for: (a, b) the Ni 49.9 Ti 50.1 and (c, d) the Ni 50.3 Ti 29.7 Hf 20 tube actuators, indicating the angle of twist vs. time and angle of twist vs. temperature variations over 50 thermal cycles Fig. 12 Summary plots showing the cyclic evolution of angle of twist (in degrees) at martensite (U M ),austenite (U A ) and the corresponding actuation stroke U ACT = U M U A for: (a) the Ni 49.9 Ti 50.1, and (b) the Ni 50.3 Ti 29.7 Hf 20 tube actuator under iso-torque thermal cycling ratios of 1.24 for the Ni 49.9 Ti 50.1 actuator and 1.34 for the Ni 50.3 Ti 29.7 Hf 20 actuator were produced (when comparing tension displacements to compression displacements) at the end of the 1st thermal cycle cooling. This signifies higher ATC effect in the Ni 50.3 Ti 29.7 Hf 20 tube actuator than its Ni 49.9 Ti 50.1 counterpart. It is important to recall that the ATC occurring in the Ni 49.9 Ti 50.1 tube actuator is as result of large deformation effects only (alluded to the above statement in section 3.1). However, the ATC observed in the Ni 50.3 Ti 29.7 Hf 20 tube actuator comprised both the effects of large deformation and the intrinsic material ATC effect which was accounted for explicitly in the activated ÔcÕ and ÔdÕ model parameters for the Ni 50.3 Ti 29.7 Hf 20 material case. Also, the degree of the ATC in the Ni 50.3 Ti 29.7 Hf 20 tube actuator at the higher stress of 400 MPa is relatively lesser compared to its counterpart at the lower stress level of 100 MPa (contrast case in Fig. 7c, d for tension to their counterparts Fig. 8c, d for compression). This is evident in the experimental response (see Fig. 3a) used for the model calibration; the ATC was lesser for higher levels of stresses. Further elaborations on the ATC aspects of the Ni 50.3 Ti 29.7 Hf 20 tube are given in section The cyclic variation of d M, d A, and d ACT, as shown in Fig. 9 and 10 shows that a stable axial displacement actuation i.e., d ACT was achieved by the Ni 50.3 Ti 29.7 Hf 20 and the Ni 49.9 Ti 50.1 tube actuators, after the 10th thermal cycle. Nevertheless, the high evolution nature of the Ni 49.9 Ti 50.1 material is reflective in the slopes of the curves displaying the variation of d M and d A with number of cycles. It is clearly seen that while the d M and Volume 24(4) April

13 Fig. 13 Plots of the secondary axial displacement response vs. the associated primary tangential displacement for: (a) the Ni 49.9 Ti 50.1 and (b) the Ni 50.3 Ti 29.7 Hf 20 tube actuators under iso-torque thermal cycling. Note that the scale of secondary axial displacement response is one-tenth of the scale of the primary tangential displacement response. Also note that the tangential displacement defined in these figures is calculated as ðr m UÞ p ffiffiffi 3, where U is the angle of twist (in radians) at the free-end of the tube and Rm is the mean radius of its section Fig. 14 Comparisons of the evolution of the effective displacement responses in (a, b) compression, (c, d) torsion, and (e, f) tension loading conditions, showing marked intrinsic ATC effect in the Ni 50.3 Ti 29.7 Hf 20 tube actuator. Note that the effective displacement is physically axial displacement in case of iso-force (tension/compression) cases whereas it is the tangential displacement for iso-torque loading condition (defined as R m U p ffiffiffi 3, where U is the angle of twist, in radians, at the free-end of the tube and Rm is the mean radius of its section) d A produced by the Ni 50.3 Ti 29.7 Hf 20 tube actuator remained almost constant after 10th thermal cycle, signifying dimensional stability in the Ni 50.3 Ti 29.7 Hf 20 material, the d M and d A generated by its Ni 49.9 Ti 50.1 counterpart would require extended (i.e., more than the 50) number of thermal cycles to observe meaningful stability or saturation Response Under Iso-torque Condition. Similar to the iso-force loading cases in section 4.3.1, the plot of angle of twist-versus-temperature shows an initial transient response after the 1st cooling under torque, followed by cyclic evolution during the subsequent thermal cycles. Although the two SMA tubes had initiated almost the same initial angle, U ð1þ M,of73, the final angles of twist at martensite, U ð50þ M, attained at the end of the 50th thermal cycle were (i.e., 139% increase) and (i.e., 4.13% increase) in the Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 tube actuators, respectively. Typical plots of the angle of twist-versustime and angle of twist-versus-temperature for the Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 tube actuators are shown in Fig. 11(a) and (b), respectively. The values of integrated open-loop angle of twists, U ðnþ OL ¼ UðNÞ M UðN 1Þ M, for the Ni 49.9 Ti 50.1 and Ni Ti 29.7 Hf 20 SMA tube actuators between the 1st and 25th cycles, and between the 25th and 50th cycles, are depicted in Table 8. Cyclic behavior of the angle of twist at LCT, U M, and UCT, U A, and the corresponding actuation angle of twist, U ACT for the Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 SMA tube actuators are shown in Fig. 12(a), and (b), respectively. From the plots, it is noticed that although the angles of twist at LCT, U M, and UCT, U A, kept increasing with thermal cycles in the Ni 49.9 Ti 50.1 tube, the amount of angular actuation, U ACT, remained stable after the 10th thermal cycle (Fig. 12a) Volume 24(4) April 2015

14 Furthermore, together with the targeted angle of twist, termed here as the primary deformation response, an axial displacement was observed in the Ni 49.9 Ti 50.1 and Ni Ti 29.7 Hf 20 SMA tube actuators under the iso-torque loading condition, referred to here as the secondary deformation response. However, the magnitudes of the secondary deformation in comparison to the primary deformation were not significant under the pure torque loading condition (see Fig. 13). The axial deformation generated under the so-called pure iso-torque loading condition could be due to the effects of loading history, path dependence (Ref 32), and/or the ÔPoynting effectõ (Ref 15) in the material system caused by large shear deformation of the SMA materials Additional Model Simulations for Ni 50.3 Ti 29.7 Hf 20 Tube. The degrees of asymmetry in the magnitudes of strains developed under isobaric tensile-verses-compression experimental tests are more acute in the Ni 50.3 Ti 29.7 Hf 20 material than its Ni 49.9 Ti 50.1 counterpart. In order to provide further insight into the effect of the intrinsic ATC parameters used in the Ni 50.3 Ti 29.7 Hf 20 material model (as alluded to in section above), additional simulations of the tube under pure tension, pure compression, and pure torque loading corresponding to an effective stress value (i.e., engineering biasstress) of 100 MPa were carried out. This stress level was selected following the observed severity in the degree of ATC at such lower stress levels (compare the ratio of strains measured in tension and compression at lower stress of 100 MPa to that at a higher stress of 500 MPa in Fig. 3a). It is worth mentioning that, under this relatively low stress magnitude, significant portion of the ATC effect is accounted for, primarily by the intrinsic material parameters, ÔcÕ and ÔdÕ used in the model. The corresponding results for this supplemental study are illustrated in Fig. 14, following identical formats as for the counterpart figures in sections and Comparing the results in parts (a), (b), and (c) of Fig. 14, one can clearly see the marked differences in the resulting actuation strokes depending on the mode of the mechanical loading, i.e., compressive-versus-shear-versus-tension. In particular, when the compression case is taken as a reference, the associated effective displacement in this case is approximately 0.53 mm at the end of the 1st thermal cycle. However, for the counterpart shear (torsion) case, the effective displacement is 0.66 mm; i.e., ratio of nearly 1.24 compared to the compressive loading mode. Finally, for the tensile loading condition, the corresponding value is approximately 0.83 mm, which is about 1.57 times of the compression reference case. Mathematically, these results can be represented as d ðtensionþ ACT > d ðtorsionþ ACT > d ðcompressionþ ACT in conformity with the original SMA model development (see Fig. 13 and 14 of Ref 23). Note the higher ATC effect under the 100 MPa stress level as compared to that evaluated at the reference value of 400 MPa stress level in section Summary and Conclusion Comparative studies of the thermomechanical behavior of Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 SMA thin-walled tube actuators, under iso-force (tension, compression), and iso-torque loading conditions for extended thermal cycles were performed using a general 3D SMA modeling framework. In all cases of loading, effective stresses of 150 and 400 MPa were targeted for the Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20 tube actuators, respectively. Against the background and simulation results obtained in this investigation, the following conclusions can be made: (a) (b) (c) (d) (e) The SMA tube actuators under the different loading modes demonstrated significant variation in their thermomechanical response. The amount of actuation stroke and the degree of dimensional stability significantly depends on the selected SMA material system as well as the applied load mode (tension, compression or torque). The Ni 49.9 Ti 50.1 tube was able to produce more axial and angular actuation with less energy input (i.e., using less mechanical force and lower upper cycle temperature) than its Ni 50.3 Ti 29.7 Hf 20 counterpart. However, the Ni 50.3 Ti 29.7 Hf 20 appears to be more dimensionally stable in all cases investigated here. Thus, in actuation systems where working space is an important design limitation, the Ni 50.3 Ti 29.7 Hf 20 tube actuator provides some benefits over the Ni 49.9 Ti 50.1 tube actuator. Both materials exhibit ATC response characteristics due to the effect of large deformations. Furthermore, for the case of the Ni 50.3 Ti 29.7 Hf 20 tube, the intrinsic ATC effects are accounted for in the model calibration, and this triggered marked influence on the response of the Ni Ti 29.7 Hf 20 tube under the different mechanical loading modes (compression, shear, tension), especially at low stress levels (contrast Fig. 7d and 8d versus the results of Fig. 14f and b). In particular, considering a value of 100 MPa of the effective engineering bias-stress, the Ni 50.3 Ti 29.7 Hf 20 tube showed (see Fig. 14) that the ratios of the transformation effective displacements in compression, torsion, and tension are 1:1.24:1.57, respectively. This indicates that the ordering of the resulting actuation strokes under the same load intensity but different loading modes is as follows d ðtensionþ ACT > d ðtorsionþ ACT > d ðcompressionþ ACT. The model predictions obtained from simulating the two SMA tubes under the different stress states showed response patterns qualitatively similar to that resulting from their respective simple homogeneous, isobaric, uniaxial, experimental, and model material-point tests responses. For instance, compare the results in Fig. 2 versus 7(b), and 3 versus 7(d), for the cases of Ni 49.9 Ti 50.1 and Ni 50.3 Ti 29.7 Hf 20, respectively. Under the applied iso-torque loading condition, the Ni 49.9 Ti 50.1 tube and Ni 50.3 Ti 29.7 Hf 20 tube produced a secondary (axial displacement) deformation response (in addition to the primary response). The magnitudes of the secondary response, relative to the primary (angle of twist) deformation response, under the case of pure torque loading is insignificant. This, together with the observations made in item (a) above may hint on the more significant combined load effects (i.e., axial plus torque) in the case of Ni 50.3 Ti 29.7 Hf 20 compared to the Ni 49.9 Ti This will be a topic for our future studies. Acknowledgments This work was supported by NASA GRC, the Fundamental Aeronautics Program, Subsonic, Fixed-Wing, Project No. Volume 24(4) April