# Hysteretic Behavior and Ultimate Energy Dissipation Capacity of Large Diameter Bars Made of Shape Memory Alloys under Seismic Loadings

^{*}

## Abstract

**:**

## 1. Introduction

_{f}, M

_{s}, A

_{s}and A

_{f}(ordered from lowest to highest). During the forward transformation, under zero load, austenite begins to transform to twinned martensite at the martensitic start temperature M

_{s}. This transformation completes to martensite at the martensitic finish temperature M

_{f}. At this stage, the material is fully in the twinned martensitic phase. During heating, the reverse transformation initiates at the austenitic start temperature A

_{s}and the transformation is completed at the austenitic finish temperature A

_{f}[9]. Figure 1 shows a typical phase diagram and stress-strain-temperature curve of a NiTi SMA [10].

_{f}, with the material in its austenite phase. When sufficiently high stress is applied to the material in the austenite phase, the SMA transforms into so-called “detwinned” martensite. When the load is released, a reverse transformation to the austenite state occurs, resulting in nearly complete shape recovery and a substantial hysteretic loop. The shape recovery is known as the superelastic effect, and it provides the structures equipped with SMAs with recentering properties. The hysteretic loop is a source of energy dissipation. The mechanical behavior of superelastic SMAs fits perfectly with the requirements of a seismic control device [2]. The main benefits can be summarized as follows: (i) reduction or even nullification of the residual deformation on the main structure after the earthquake due to the self-centering property, (ii) increase in the energy dissipation capacity of the overall structure, (iii) limitation of the forces imparted to the main structure because of the stress plateau present in strain levels up to 6–8%, (iv) reduction of lateral displacements and, as a result, limitation of the P-Δ effects, and (v) excellent resistance to corrosion and high-cycle resistance. The P-Δ the effect is a destabilizing moment that takes place when the structure deforms laterally (e.g., due to earthquake or wind loads), which equals the force of the gravity loads multiplied by the horizontal displacement of the structure. The most commonly used SMAs are those based on nickel-titanium- (NiTi) and copper- (Cu) based alloys [2,3,7]. For engineering applications, the almost equiatomic system of NiTi alloys is found to be the best combination, owing to their temperature variation stability and higher resistance to corrosion and fatigue [1,3].

_{R}, loading transformation stresses at start σ

_{Ls}and finish σ

_{Lf}of the phase, unloading transformation stresses at start σ

_{U}

_{Ls}and finish σ

_{U}

_{Lf}of the phase, initial elastic modulus E

_{A}, loading phase transformation elastic modulus E

_{A-M}, unloading phase transformation elastic modulus E

_{M-A}, and modulus in the full martensite phase E

_{M}. The shaded area in Figure 2 represents the energy dissipated in one loop of hysteresis E

_{D}. To determine σ

_{Ls}, σ

_{Lf}, σ

_{ULs}, and σ

_{ULf}, five lines tangent to the relevant parts of the ε-σ curve are drawn, as shown with dot lines in Figure 2. The slopes of these lines are E

_{A}, E

_{A-M}, E

_{M}, E

_{M-A,}and E

_{A}, respectively. σ

_{Ls}, σ

_{Lf}, σ

_{ULs}, and σ

_{ULf}are the ordinates of the intersection points of these lines.

_{A}is referred to as the austenite stable phase and it provides valuable information to calculate the initial stiffness of the device. Two additional parameters that characterize the mechanical behavior of bars made of NiTi alloys are the equivalent viscous damping ratio associated with a given cycle ζ

_{eq}and number of cycles to failure. For a given cycle, ζ

_{eq}is defined by [11]:

_{max}and σ

_{max}are the maximum strain and stress, respectively, in the cycle, and V is the volume of the bar. The number of cycles to failure is an indicator of the life of the material.

_{Ls}and σ

_{Lf}, while σ

_{ULs}and σ

_{Ulf}remain approximately constant. This results in a decrease of the energy E

_{D}dissipated in the cycle and in the corresponding ζ

_{eq}. However, the response stabilizes as the number of cycles increases. Under cycles of increasing amplitude, ε

_{R}remains constant as well as σ

_{Ls}and σ

_{Lf}, whereas σ

_{ULs}and σ

_{Ulf}decrease. Consequently, the hysteresis shape involves greater energy dissipated, resulting in higher equivalent viscous damping. When the loading stress plateau (i.e., the segment with slope E

_{A-M}in Figure 2) is overcome (onset of the pure martensite phase), the strain-stress curve exhibits a strain hardening effect [1,2,3,4,5].

_{R}but does indeed have a significant influence on the shape of the hysteresis loops. Increasing the strain rate results in a vertical displacement (i.e., both loading and unloading transformation stresses increase) and narrowing of the hysteretic loops. The narrowing implies a loss of energy dissipated per cycle. This reduction of dissipated energy, together with the vertical displacement of the loops, reduces the equivalent viscous damping. The reason for this behavior is the self-heating of the material associated with an increasing difficulty to transfer the heat generated between phase transformations at high strain rates [1,2,3].

_{f}, thus keeping the recentering capability unaffected. However, increasing temperature above A

_{f}causes a vertical displacement of the hysteresis loop (i.e., higher loading and unloading transformation stresses), while the energy dissipated per cycle remains almost the same. The vertical displacement of the loops, despite the fact that E

_{D}remains unchanged, results in a reduction of ζ

_{eq}. Finally, regarding the amount of energy that the SMA can dissipate up to failure (ultimate energy dissipation capacity), most studies to date address this issue as a problem of high-cycle fatigue [12]. Yet earthquakes impose on the structures a relatively low number of cycles (in comparison with wind or traffic loads) having high stress levels that involve plastic deformations. These are the two common factors attributed to low-cycle fatigue. Some recent investigations on SMA low-cycle fatigue have been carried out on small diameter wires and micro-tubes [13,14], whose conclusions may not be consistent with large diameter bars. Studies on the ultimate energy dissipation capacity of SMA large diameter bars under low cyclic fatigue are almost inexistent.

## 2. Cyclic Tests on NiTi Bars

#### 2.1. Test Specimens

_{s}) between −30 and −10 °C. The transition temperatures are M

_{f}= −37.66 °C, M

_{s}= −31.36 °C, A

_{s}= −16.13 °C and A

_{f}= −5.16 °C. Figure 3 shows the results of the (Differential Scanning Calorimetry) DSC experiment. The NiTi bars were manufactured by the company SAES Smart Materials (New Hartford, NY, USA). Unfortunately, no more specific material-related information was available from the manufacturer.

#### 2.2. Loading Set up and Loading Protocol

#### 2.2.1. Quasi-Static Cyclic Tests

_{11}to S

_{16}, where the letter refers to Static, the first sub index identifies the loading protocol, and the second sub index the number of the specimen. The results of the SMA under constant amplitude and low frequency test (protocol 1) were used as “benchmark response”. A relatively large number of specimens were tested with this protocol 1 in order to assess the repeatability of the results, particularity in terms of (i) number of cycles required to stabilize the shape of the hysteresis loops, (ii) the maximum stress attained in the cycle, and (iii) the residual strain. The coefficients of variation (ratio of standard deviation to the mean) obtained for these variables were 0, 0.04 and 0.10, respectively. Two specimens (referred to as S

_{21}to S

_{22}) were subjected to cyclic loads following the loading protocol 2 shown in Figure 5, applied at two different frequencies of 0.02 Hz (in specimen S

_{21}) and 0.04 Hz (in specimen S

_{22}). Two specimens (referred to as S

_{31}and S

_{32}) were subjected to cyclic displacements until failure, following the multiple-step loading protocol 3 shown in Figure 6 at a frequency of 0.02 Hz (quasi-static). One specimen (referred to as S

_{41}) was subjected to the cyclic displacements until failure following the loading pattern shown in Figure 7 applied at a frequency of 0.02 Hz (quasi-static). All tests were conducted in ambient conditions (20–25 °C) with a universal testing machine, SAXEWAY T1000 (MOOG Inc., East Aurora, NY, USA). The experimental set up involved a pair of transducers for the displacement control plus the internal load cell of the actuator that measured the applied force.

#### 2.2.2. Dynamic Cyclic Tests

_{21}and D

_{22}, respectively, where the letter indicates Dynamic. The tests were conducted in ambient conditions (20–25 °C) with an INSTRON 8803 fatigue testing system (INSTRON, Norwood, MA, USA).

#### 2.3. Test Results and Discussion

#### 2.3.1. Hysteretic Behavior

_{11}. Specimens S

_{12}to S

_{16}exhibited similar behavior. Three relevant features should be noted. First, in the initial cycles, the loading and unloading transformation stresses tend to diminish, which results in a reduction of the energy dissipated in each cycle. This phenomenon is called in the literature “functional fatigue” [15]. In successive cycles, the shape of the hysteresis loops tends to quickly stabilize, becoming almost identical. Second, the maximum stress is practically the same in all cycles. Third, the residual strain ε

_{R}remains approximately constant.

_{21}, S

_{22}, D

_{21}, and D

_{22}, subjected to cycles of increasing amplitude following protocol 2 at frequencies of 0.02, 0.04 (quasi-static) and 0.2, 1.0 Hz (dynamic), respectively. The first frequency (0.2 Hz) is approximately the fundamental frequency of a high-rise building vibrating in the fundamental mode. The second frequency (1 Hz) is typical of low to moderate rise buildings. Two relevant features should be noted. First, for a fixed frequency, the shape of the loops at different amplitudes is seen to follow basically the same pattern, that is, the loading and unloading paths for a given amplitude overlap the loading and unloading paths obtained in cycles of lower amplitude. Kimiecik et al. [16] studied the configurations of transforming martensite during ambient temperature cyclic deformations of superelastic NiTi and found that local transformation history is responsible for this macroscopically observed performance.

_{eq}at ε = 6%. Yet specimen D22 failed during the second cycle at ε = 6%, therefore there is only one point at this strain amplitude. It can be observed that, for the same frequency, ζ

_{eq}tends to increase with ε, and becomes approximately constant beyond ε = 4%. Under cycles of constant amplitude at ε = 6%, ζ

_{eq}tends to decrease with the number of cycles applied.

_{21}and D

_{22}were compared with the tests conducted by other researchers [2,3,4,17] on wires and bars subjected to one cycle of amplitude ε = 6% at frequencies of 0.02 Hz and 1.0 Hz. The information regarding the SMA materials used in these studies can be summarized as follows. In [2]: NiTi 50%Ni, cold working and annealing. In [3] NiTi 56%Ni, cold drawn with 30% cold working and annealing. In [4]: NiTi near equiatomic, cold drawn 30% and cold worked prior to annealing. In [17]: Nitinol.

_{D}dissipated in one cycle was normalized by the product of the yield force F

_{y}and yield displacement δ

_{y}determined as follows. The loading branch of the force-deformation curve F-δ obtained experimentally was idealized with two segments as shown in Figure 12. The slope and position of these segments were determined so that: (i) the slope of the second segment of the bilinear approximation closely fits the path of the loading transformation phase, and (ii) the area under the real curve and the bilinear approximation was the same. The results are shown in Table 1. In this Table, ϕ is the diameter of the wire or bar, σ

_{y}is the yield stress obtained dividing F

_{y}by π(ϕ/2)

^{2}, ε

_{y}is the yield strain obtained dividing δ

_{y}by the initial length, and ${\overline{E}}_{\mathrm{D}}$ is the dissipated energy E

_{D}normalized by Fyδy. The values of σ

_{y}, ε

_{y}, E

_{A}, and ${\overline{E}}_{\mathrm{D}}$ vary depending on the frequency applied. The last column of Table 1 shows the ratio between ${\overline{E}}_{\mathrm{D}}$ obtained for quasi-static loading (0.02Hz), ${\overline{E}}_{\mathrm{D},\mathrm{static}}$, and the corresponding value obtained for dynamic loading (1.0 Hz), ${\overline{E}}_{\mathrm{D},\mathrm{dynamic}}$. The specimens with ϕ < 2 mm are referred to as wires hereafter, and those with ϕ > 6 mm as bars.

_{y}increases with the frequency. This increase is larger in bars than in wires. As for the yielding strain, it is reduced in wires when the load is applied dynamically in comparison with the static case. The tendency is the opposite for bars, however. This is possibly due to the sample size that may produce different deformation mechanisms within the superelastic strain range [20].

#### 2.3.2. Ultimate Energy Dissipation Capacity

_{31}, S

_{32}, and S

_{41}subjected to cycles of constant amplitude were used. The total energy dissipated by these specimens and accumulated in successive cycles until failure, $\sum {\overline{E}}_{\mathrm{D},\mathrm{static}}$, was normalized by F

_{y}δ

_{y}(determined as described in Section 2.3.1), i.e., $\sum {\overline{E}}_{\mathrm{D},\mathrm{static}}}={\displaystyle \sum {E}_{\mathrm{D},\mathrm{static}}}/({F}_{y}{\mathsf{\delta}}_{y})$, and is shown in the second column of Table 2. The normalized energy dissipated in the first cycle was also calculated for each specimen, i.e., ${\overline{E}}_{\mathrm{D},\mathrm{static}}={E}_{\mathrm{D},\mathrm{static}}/({F}_{y}{\mathsf{\delta}}_{y})$, and is shown in the third column of Table 2. Further, the total amount of dissipated energy was expressed in terms of the equivalent number of cycles N

_{f}defined by N

_{f}= $\sum {\overline{E}}_{\mathrm{D},\mathrm{static}}}/{\overline{E}}_{\mathrm{D},\mathrm{static}$ and is shown in the fourth column of Table 2. It is worth recalling that the loads applied to specimens S

_{31}, S

_{32}and S

_{41}were quasi-static, and it was shown in Section 2.3.1 that the amount of energy dissipated under dynamic loading is smaller than under static loads. More precisely, for the 12.7 mm diameter bars tested in this study, the ratio ${\overline{E}}_{\mathrm{D},\mathrm{dynamic}}/{\overline{E}}_{\mathrm{D},\mathrm{static}}$ is 0.62 (last row in Table 1). Therefore, the normalized energy dissipated in a single cycle under dynamic loading ${\overline{E}}_{\mathrm{D},\mathrm{dynamic}}$ can be estimated by multiplying ${\overline{E}}_{\mathrm{D},\mathrm{static}}$ by 0.62, and it is indicated in the last column of Table 2. The pairs of values (N

_{f}, ${\overline{E}}_{\mathrm{D},\mathrm{dynamic}}$) obtained in this way for specimens S

_{31}, S

_{32}and S

_{41}are plotted with circles in Figure 13 and compared with those obtained by [24] (square symbols) for 6.2 mm diameter NiTi bars tested under dynamic (0.3 Hz) loads. Since the ratio ${\overline{E}}_{\mathrm{D},\mathrm{dynamic}}$ increases with the amplitude of the cycle, ${\overline{E}}_{\mathrm{D},\mathrm{dynamic}}$ is directly related to the amplitude of the cyclic loading. The ultimate energy dissipation capacity corresponding to each point, shown in Figure 13, can be simply obtained by multiplying its abscissa (N

_{f}) by its ordinate ${\overline{E}}_{\mathrm{D},\mathrm{dynamic}}$.

_{f}and ${\overline{E}}_{\mathrm{D},\mathrm{dynamic}}$, can be approximated through the following expression:

## 3. Numerical Characterization of the Hysteretic Behavior of NiTi Bars

_{1}, loading phase transformation stiffness k

_{2L}, unloading phase transformation stiffness k

_{2UL}, the strain hardening stiffness k

_{3}, loading transformation strength at start F

_{Ls}, loading transformation deformation at finish δ

_{Lf}, and ratio of loading transformation strength at start β. Here, F

_{Ls}and k

_{1}can be easily determined from the geometry (cross area A and length L) of the bar, and the mechanical properties of the material (Young’s modulus E and yield stress σ

_{LS}), i.e., F

_{LS}= σ

_{Ls}A and k

_{1}= EA/L. The simplicity reduces computational efforts substantially when performing complex time history nonlinear analysis of structures subjected to seismic loadings. The conventional flag-shape model cannot, however, capture the residual deformation associated with the residual strain ε

_{R}typically exhibited by all hysteresis loops, as seen in Figure 10. If the NiTi bar is subjected to just a few cycles of large amplitude (e.g., far beyond ε = σ

_{LS}/E

_{A}in Figure 2), the amount of dissipated energy associated with the residual strain ε

_{R}(represented by the area σ

_{LS}ε

_{R}in Figure 2) is negligible in comparison with the energy dissipated in a complete cycle (represented by the shaded area of the complete loop in Figure 2). In such a case, the conventional flag-shape model captures the actual amount of energy dissipated by the NiTi bar reasonably well. Yet if the loading history consists of a combination of few cycles of large amplitude and a large number of cycles of small amplitude, i.e., below ε = σ

_{LS}/E

_{A}in Figure 2, the amount of energy dissipated by the small amplitude cycles can be comparatively large. The latter is the typical displacement pattern imposed by earthquakes on structural members. In this case, the conventional flag-type model can lead to a wrong prediction of the energy accumulated on the NiTi bars and to an unsafe estimation of failure.

_{LS}and elastic stiffness k

_{EPP}, as shown in Figure 15b. The sum of the restoring forces provided by each spring at a given displacement δ gives the complete hysteretic model depicted in Figure 15c. Worth noting in Figure 15c is that the secant stiffness at δ = δ

_{LS}gives the stiffness k

_{1}= EA/L.

_{Ls}and k

_{1}are determined from the geometry of the bar and the mechanical properties of the material, as indicated above (i.e., F

_{LS}= σ

_{Ls}A and k

_{1}= EA/L). The rest of the parameters were calibrated with the results of the dynamic cyclic tests described in Section 2, giving: k

_{2L}= k

_{1}/15, k

_{2UL}= 3k

_{1}/50, k

_{3}= k

_{1}, δ

_{Lf}= 4F

_{Ls}/k

_{1}, β = 0.1, k

_{EPP}= 4k

_{1}, and γ = 0.86. Figure 16 compares the shape of the hysteresis loops obtained with the proposed model and the results of the dynamic cyclic tests (specimen D

_{22}). Comparison in terms of dissipated energy gives E

_{D}= 130 kN·mm for the numerical model and E

_{D}= 119 kN·mm for the test, the difference being less than 10%.

## 4. Shake Table Tests of a Structure with NiTi Bars

#### 4.1. Brace-Type NiTi Damper

#### 4.2. Test Specimen and Experimental Set up

^{2}. Second, a partial structural model having three columns and the height of one story and a half was selected from this prototype structure. Third, a test specimen was defined from the partial structural model by applying scale factors of λ

_{L}= 2/5 for length. The test specimen was built in Laboratory and three brace-type hysteretic dampers consisting of NiTi bars and steel tubes assembled as shown in Figure 17 were installed in each story as diagonal elements. Additional steel blocks were attached at the top of the RC plate and at the top half of the columns of the second story to represent the gravity loads acting on the floors. Finally, the test specimen with the brace-type NiTi dampers was mounted on a bidirectional 3 × 3 m

^{2}shake table forming the experimental set up shown in Figure 18 and Figure 19. The brace-type NiTi dampers were instrumented with displacement transducers and strain gauges.

#### 4.3. Seismic Tests and Results

_{D}= 2699 kN·mm) is very similar to the actual value (E

_{D}= 2563 kN·mm) measured during the tests. Finally, the response of the structure equipped with the SMA dampers is compared in Figure 21 with that of a counterpart structure without dampers that were subjected to the same earthquake in a previous study [28]. The response is compared in terms of maximum inter-story drift for the horizontal X direction, IDx, and in the Y direction IDy. The inter-story drift is defined as the relative lateral displacement between the top and bottom parts of each story, divided by the story height. It can be seen, that the SMA dampers reduced to less than one fourth the maximum inter-story drifts.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Dolce, M.; Cardone, D. Mechanical behaviour of shape memory alloys for seismic applications 1. Martensite and austenite NiTi bars subjected to torsion. Int. J. Mech. Sci.
**2001**, 43, 2631–2656. [Google Scholar] [CrossRef] - Dolce, M.; Cardone, D. Mechanical behaviour of shape memory alloys for seismic applications 2. Austenite NiTi wires subjected to tension. Int. J. Mech. Sci.
**2001**, 43, 2657–2677. [Google Scholar] [CrossRef] - DesRoches, R.; McCormick, J.; Delemont, M. Cyclic properties of superelastic shape memory alloy wires and bars. J. Struct. Eng.-ASCE
**2004**, 130, 38–46. [Google Scholar] [CrossRef] - McCormick, J.; DesRoches, R.; Fugazza, D.; Auricchio, F. Seismic vibration control using superelastic shape memory alloys. J. Eng. Mater. Technol.-Trans. ASME.
**2006**, 128, 294–301. [Google Scholar] [CrossRef] - McCormick, J.; Tyber, J.; DesRoches, R.; Gall, K.; Maier, H.J. Structural engineering with NiTi. II: Mechanical behavior and scaling. J. Eng. Mech.-ASCE
**2007**, 133, 1019–1029. [Google Scholar] [CrossRef] - Tyber, J.; McCormick, J.; Gall, K.; DesRoches, R.; Maier, H.J.; Maksoud, A.E.A. Structural engineering with NiTi. 1: Basic materials characterization. J. Eng. Mech.-ASCE
**2007**, 133, 1009–1018. [Google Scholar] [CrossRef] - Ozbulut, O.E.; Hurlebaus, S.; Desroches, R. Seismic response control using shape memory alloys: A review. J. Intell. Mater. Syst. Struct.
**2011**, 22, 1531–1549. [Google Scholar] [CrossRef] - Wang, J.; Zhao, H. High performance damage-resistant seismic resistant structural systems for sustainable and resilient city: A review. Shock Vibrat.
**2018**, 2018, 8703697. [Google Scholar] [CrossRef] - Lagoudas, D.C. Shape Memory Alloys: Modeling and Engineering Applications; Springer: New York, NY, USA, 2008; pp. 6–7. [Google Scholar]
- Hartl, D.J.; Lagoudas, D.C. Aerospace applications of shape memory allows. J. Aerosp. Eng.
**2007**, 221, 535–552. [Google Scholar] [CrossRef] - Chopra, A.K. Dynamics of Structures; Prentice Hall: Upper Saddle River, NJ, USA, 1995; Volume 3. [Google Scholar]
- Mahtabi, M.J.; Shamsaei, N.; Mitchell, M.R. Fatigue of nitinol: The state-of-the-art and ongoing challenges. J. Mech. Behav. Biomed. Mater.
**2015**, 50, 228–254. [Google Scholar] [CrossRef] - Song, D.; Kang, G.; Kang, Q.; Yu, C.; Zhang, C. Experimental observations on uniaxial whole-life transformation ratchetting and low-cycle stress fatigue of super-elastic NiTi shape memory alloy micro-tubes. Smart Mater. Struct.
**2015**, 24, 075004. [Google Scholar] [CrossRef] - Zhang, Y.; You, Y.; Moumni, Z.; Anlas, G.; Zhu, J.; Zhang, W. Experimental and theoretical investigation of the frequency effect on low cycle fatigue of shape memory alloys. Int. J. Plast.
**2017**, 90, 1–30. [Google Scholar] [CrossRef] - Eggeler, G.; Hornbogen, E.; Yawny, A.; Heckmann, A.; Wagner, M. Structural and functional fatigue of NiTi shape memory alloys. Mater. Sci. Eng. A-Struct. Mater. Prop. Microstruct. Process.
**2004**, 378, 24–33. [Google Scholar] [CrossRef] - Kimiecik, M.; Jones, J.W.; Daly, S. The effect of microstructure on stress-induced martensitic transformation under cyclic loading in the SMA nickel-titanium. J. Mech. Phys. Solids
**2016**, 89, 16–30. [Google Scholar] [CrossRef] - Zhu, S.; Zhang, Y. Loading rate effect on superelastic SMA-based seismic response modification devices. Earthq. Struct.
**2013**, 4, 607–627. [Google Scholar] [CrossRef] - Kan, Q.; Yu, C.; Kang, G.; Li, J.; Yan, W. Experimental observations on rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy. Mech. Mater.
**2016**, 97, 48–58. [Google Scholar] [CrossRef] - Ammar, O.; Haddar, N.; Dieng, L. Experimental investigation of the pseudoelastic behaviour of NiTi wires under strain- and stress-controlled cyclic tensile loadings. Intermetallics
**2017**, 81, 52–61. [Google Scholar] [CrossRef][Green Version] - Nemat-Nasser, S.; Choi, J.-Y.; Guo, W.-G.; Isaacs, J.B. Very high strain-rate response of a NiTi shape-memory alloy. Mech. Mater.
**2005**, 37, 287–298. [Google Scholar] [CrossRef] - Treadway, J.; Abolmaali, A.; Lu, F.; Aswath, P. Tensile and fatigue behavior of superelastic shape memory rods. Mater. Des.
**2015**, 86, 105–113. [Google Scholar] [CrossRef][Green Version] - Kang, G.; Song, D. Review on structural fatigue of NiTi shape memory alloys: Pure mechanical and thermo-mechanical ones. Theor. Appl. Mech. Lett.
**2015**, 5, 245–254. [Google Scholar] [CrossRef][Green Version] - Zhang, Y.; Zhu, J.; Moumni, Z.; Van Herpen, A.; Zhang, W. Energy-based fatigue model for shape memory alloys including thermomechanical coupling. Smart Mater. Struct.
**2016**, 25, 035042. [Google Scholar] [CrossRef][Green Version] - Moumni, Z.; Van Herpen, A.; Riberty, P. Fatigue analysis of shape memory alloys: Energy approach. Smart Mater. Struct.
**2005**, 14, S287–S292. [Google Scholar] [CrossRef] - Ikeda, T.; Nae, F.A.; Naito, H.; Matsuzaki, Y. Constitutive model of shape memory alloys for unidirectional loading considering inner hysteresis loops. Smart Mater. Struct.
**2004**, 13, 916–925. [Google Scholar] [CrossRef] - Saleeb, A.F.; Padula II, S.A.; Kumar, A. A multi-axial, multimechanism based constitutive model for the comprehensive representation of the evolutionary response of SMAs under general thermomechanical loading conditions. Int. J. Plast.
**2011**, 27, 655–687. [Google Scholar] [CrossRef] - Karakalas, A.A.; Machairas, T.T.; Solomou, A.G.; Saravanos, D.A. Modeling of partial transformation cycles of SMAs with a modified hardening function. Smart Mater. Struct.
**2019**, 28, 1–20. [Google Scholar] [CrossRef] - Benavent-Climent, A.; Galé-Lamuela, D.; Donaire-Avila, J. Energy capacity and seismic performance of RC waffle-flat plate structures under two components of far-field ground motions: Shake table tests. Earthquake Eng. Struct. Dyn.
**2019**, 48, 949–969. [Google Scholar] [CrossRef]

**Figure 5.**Loading protocol 2 applied at frequencies 0.02, 0.04, 0.2, 1.0 Hz used for S

_{21}, S

_{22}, D

_{21}, D

_{22}.

**Figure 15.**Proposed hysteretic model: (

**a**) flag-shape component, (

**b**) elastic-perfectly plastic component, (

**c**) complete model.

**Figure 17.**Assemblage of NiTi bars to form a hysteretic damper: (

**a**) elevation, (

**b**) sections, (

**c**) detail of the anchorage of the NiTi bars.

**Figure 19.**Details of the experimental set up for the shake table tests: (

**a**) elevation view A-A’, (

**b**) elevation, (

**c**) plan (first storey), (

**d**) plan (second storey).

Reference | Frequency (Hz): ϕ (mm) | σ_{y} (MPa) | ε_{y} (%) | E_{A} (MPa) | ${\overline{\mathit{E}}}_{\mathit{D}}={\mathit{E}}_{\mathit{D}}/({\mathit{F}}_{\mathit{y}}{\mathit{\delta}}_{\mathit{y}})$ | $\frac{{\overline{\mathit{E}}}_{D,\mathrm{dynamic}}}{{\overline{\mathit{E}}}_{D,\mathrm{static}}}$ | ||||
---|---|---|---|---|---|---|---|---|---|---|

0.02 | 1.0 | 0.02 | 1.0 | 0.02 | 1.0 | 0.02 | 1.0 | |||

McCormick [4] | 0.25 | 504 | 553 | 1.83 | 1.79 | 275 | 309 | 1.12 | 1.00 | 0.89 |

Zhu [17] | 0.58 | 305 | 309 | 1.22 | 1.12 | 250 | 275 | 2.10 | 1.48 | 0.71 |

Dolce [2] | 1.84 | 390 | 415 | 1.40 | 1.25 | 279 | 332 | 1.00 | 0.83 | 0.84 |

DesRoches [3] | 7.10 | 315 | 374 | 1.33 | 1.40 | 237 | 267 | 1.67 | 0.71 | 0.42 |

McCormick [4] | 12.70 | 328 | 414 | 1.33 | 1.54 | 247 | 269 | 1.41 | 0.78 | 0.56 |

This study | 12.70 | 245 | 359 | 1.12 | 1.39 | 219 | 258 | 1.78 | 1.11 | 0.62 |

ε (%) | $\sum {\overline{\mathit{E}}}_{D,\mathrm{static}}$ | ${\overline{\mathit{E}}}_{D,\mathrm{static}}$ | N_{f} | ${\overline{\mathit{E}}}_{D,\mathrm{dynamic}}$ |
---|---|---|---|---|

4.0 | 124 | 1.17 | 106 | 0.73 |

4.0 | 112 | 1.17 | 96 | 0.73 |

2.5 | 260 | 0.52 | 500 | 0.32 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

González-Sanz, G.; Galé-Lamuela, D.; Escolano-Margarit, D.; Benavent-Climent, A. Hysteretic Behavior and Ultimate Energy Dissipation Capacity of Large Diameter Bars Made of Shape Memory Alloys under Seismic Loadings. *Metals* **2019**, *9*, 1099.
https://doi.org/10.3390/met9101099

**AMA Style**

González-Sanz G, Galé-Lamuela D, Escolano-Margarit D, Benavent-Climent A. Hysteretic Behavior and Ultimate Energy Dissipation Capacity of Large Diameter Bars Made of Shape Memory Alloys under Seismic Loadings. *Metals*. 2019; 9(10):1099.
https://doi.org/10.3390/met9101099

**Chicago/Turabian Style**

González-Sanz, Guillermo, David Galé-Lamuela, David Escolano-Margarit, and Amadeo Benavent-Climent. 2019. "Hysteretic Behavior and Ultimate Energy Dissipation Capacity of Large Diameter Bars Made of Shape Memory Alloys under Seismic Loadings" *Metals* 9, no. 10: 1099.
https://doi.org/10.3390/met9101099