Torsional behavior of nitinol : modeling and experimental evaluation

Transcription

1 The University of Toledo The University of Toledo Digital Repository Theses and Dissertations 2012 Torsional behavior of nitinol : modeling and experimental evaluation Zohreh Karbaschi The University of Toledo Follow this and additional works at: Recommended Citation Karbaschi, Zohreh, "Torsional behavior of nitinol : modeling and experimental evaluation" (2012). Theses and Dissertations This Thesis is brought to you for free and open access by The University of Toledo Digital Repository. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of The University of Toledo Digital Repository. For more information, please see the repository's About page.

2 A Thesis entitled Torsional Behavior of Nitinol: Modeling and Experimental Evaluation by Zohreh Karbaschi Submitted to the Graduate Faculty as partial fulfillment for the requirements of the Master of Science Degree in Bioengineering Dr. Mohammad H. Elahinia, Committee Chair Dr. Arran Nadarajah, Committee Member Dr. Scott Molitor, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo August 2012

3 Copyright 2012 Zohreh Karbaschi This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author.

4 An Abstract of Torsional Behavior of Nitinol: Modeling and Experimental Evaluation by Zohreh Karbaschi Submitted to the Graduate Faculty as partial fulfillment for the requirements of the Master of Science Degree in Bioengineering The University of Toledo August 2012 Smart materials have been gaining the researchers attention in the past several decades because of their distinct properties. A group of these materials, Shape Memory Alloys (SMA) are metallic alloys that possess different properties compared to conventional metals. This type of alloy is capable of remembering its shape following a deformation. Relatively large recoverable strain of around 8% is one of the SMA s distinct properties, which makes it a favorable candidate for applications involving large deforming loads. Nitinol (NiTi), one of the most well-known shape memory alloys, is biocompatible and desirable for biomedical applications. These intrinsic properties are the result of solid state phase transformation, which leads to complex theromechanical behavior. Since experimental testing on SMAs is iii

5 neither time- nor cost-effective, modeling the behavior of SMAs is necessary. Having a model facilitates the prediction of SMAs response in various loading conditions. A great deal of studies has been carried out on modeling the behavior of SMAs under uniaxial loading conditions. However, no study has yet suggested a simple and user-friendly model to predict the SMA s torsional behavior. The model developed and presented in this study is fairly simple and can be utilized for optimization purposes when dealing with specific applications such as minimally invasive medical devices. This model is based on thermodynamics of these alloys and has been verified against experimental results. iv

6 Acknowledgements I would like to first thank my advisor Dr. Mohammad Elahinia for providing me with such a great opportunity to work on this project. Without his abundant help and knowledge, this project would have not been successful. I would like to express my gratitude to him for his excellent guidance and support throughout the completion of this project. Furthermore, I would like to thank my sister and her family because of her wonderful support and help. In these past few years of my study, my sister was incredibly patient and supportive. I am so honored to have you as my sister. I would also like to express my love and gratitude to my beloved parents back in Iran who were always supportive and understanding during these past two years. Their endless love and prayers kept me go on during the tough times. If it were not for their abundant help and support, I would not have achieved what I have in my life. I am extremely blessed for having them as my parents. My special thanks goes to my loving, supportive, encouraging husband Hadi because of his persistence support during the completion of my thesis. His faithful love always made me believe in myself. My life would mean nothing without him and I am so grateful to have him beside me in my life. Finally I would like to dedicate this thesis to my parents and my husband for always being there for me. v

7 Table of Contents Abstract... iii Acknowledgements... v Table of Contents... vii List of Tables... ix List of Figures... x List of Symbols... xiv Chapter One... 1 Introduction Shape Memory Alloys Outline Approach Contribution... 3 Chapter Two... 5 Unique Properties of Shape Memory Alloys Unique Properties of SMAs Modeling Shape Memory Alloy under Torsion Uniaxial Modeling of Shape Memory Alloys Torsional Modeling of Shape Memory Alloys vii

8 2.3. Some Applications of SMAs Necessity of a Torsional Model Chapter Three Torsional Modeling of SMA Wire MATLAB Program for 1 Dimensional Uniaxial Model Calculating Normal Stress from Stress-Strain MATLAB Program for Obtaining Material Properties Chapter Four Tensile Test of an SMA Wire Sample Determination of SMA wire Material Properties Torsional Test of SMA Samples Chapter Five Uniaxial Model s Prediction vs. Experimental Data Torsional Model Validation Model s Prediction vs. Other Models Prediction Effect of Different Parameters on the Torsional Profile Effect of Wire Diameter on the Torsional Profile Effect of Wire Length on the Torque-Angle Profile Effect of Temperature on the Torque-Angle Profile Chapter Six Conclusion Future Work References viii

9 List of Tables 3-1 The list of material properties Material properties of the SMA wire sample Material properties obtained from Mirzaeefar et all, work. These parameters are input into the model and the results of both models are compared against each other [24] ix

10 List of Figures 2-1 Crystalline structure of SMAs in two different states of the material, inducing enough stress takes the material from austenite to detwinned martensite Crystalline structure of SMAs during temperature induced phase transformation under no mechanical load Crystalline structure of a superelastic SMA during stress-induced phase transformation The phase transformation diagram of a typical SMA showing the shape memory effect [1] Phase diagram showing different state of the SMA [9] The loading/unloading curve of a typical SMA under axial tension presenting shape memory effect [10] The loading/unloading curve of a typical SMA under axial tension presenting superelasticity [10] Shape memory effect of SMA wire under torsion [10] Superelasticity of SMA wire under torsion [19] SMA beam element used in chevron engine [28] Commercially available NiTi bone stable [32] Orthodontic SMA wire treatment [1][41] x

11 2-13 Schematic of NiTi stent for support of the blood vessels [1][43] Shear stress distribution over the wire/rod cross sectional area, depending on the radial location, shear stress increases linear or nonlinearly Superelastic loading/unloading curve under uniaxial tension. The uniaxial profile is used to obtain the equivalent stress for each given strain Linear interpolation to obtain the relative stress for each calculated strain An illustration of determination of normal stress associated with normal strain using interpolation Tangent lines which are drawn in order to obtain the start and finish martensite/austenite stresses The stress-temperature profile of the SMA sample, this profile is used to calculate CA and CM and the transformation temperatures Illustration of BOSE machine and the environmental chamber Stress-Strain profile of the NiTi sample at different centigrade temperatures Tangent lines are generated using the program for all test temperature. The slope of the tangent lines is calculated to obtain the austenite and martensite modulus of elasticity. This is shown for only two temperatures of 70 (top) and 40 (bottom) degrees of Celsius respectively Transformation stresses at 70 degree Celsius are obtained using the stress-strain profile The phase diagram of the NiTi wire. This diagram is used to obtain the transformation temperatures xi

12 4-6 Calculation of residual strain, the tangent line to the martensite region is extended to cross the strain axis An illustration of Micro-Torsion MT1 testing machine The load to fracture curve of the three samples. This test is performed to obtain the maximum rotation failure angle for each sample The torsional profile comparison of the SMA wire samples. The torque value for the start of the transformation increases as the wire s radius increases A flowchart illustration of the uniaxial model. The dashed lines are not generated in the steps; they just show the overall path Uniaxial model's prediction vs experimental result, the experiment is performed at 40 degree Uniaxial model's prediction vs. experimental result, the experiment is performed at 50 degree Uniaxial model's prediction vs. experimental result, the experiment is performed at 70 degree An illustration of the torsional model steps Comparison of the torsional model's prediction with the experimental torque-angle data for the wire with inch diameter (sample one) Torsional model versus the data obtained from the torsion experiment for wire possessing 0.02 inch diameter (sample two) Model's prediction against the torque-angle curve obtained from experiment for inch diameter wire(sample three) The model's prediction vs Mirzaeefar exact solution [24] xii

13 5-10 Effect of diameter on the torsional profile. Thicker wires require more torque for transformation The effect of varying wire lengths on its torsional behavior. As the length decreases, the required torque value for the start and finish of the transformation increase Effect of different temperature. Higher temperatures shift the torque-angle profile upwards xiii

14 List of Symbols A f... Austenite transformation finish temperature A s... Austenite transformation start temperature M f... Martensite transformation finish temperature M s... Martensite transformation start temperature D... Modulus of the material D a... Austenite modulus of elasticity D m... Martensite modulus of elasticity M d... Detwinned martensite E... Young s modulus E A... Elastic modulus of austenite E M... Elastic modulus of martensite υ... Passion ratio ρ... Density H... Maximum residual strain b A... Model parameter for polynomial model b M... Model parameter for polynomial model C A... Slope of the stress-temperature austenite transformation line C M... Slope of the stress-temperature martensite transformation line A... Area D i... Inner diameter D o... Outer diameter G... Shear modulus L... Length r... Radius r o... Outer radius T... Temperature T Applied... Applied torque θ... Angle of deflection xiv

15 ε... Normal strain ε l... Residual strain γ... Shear strain S...Compliance tensor S A... Compliance tensor of austenite S M... Compliance tensor of martensite ξ... Martensite fraction σ max... Maximum stress σ... Normal stress σ f cr... Critical transformation finish stress σ s cr... Critical transformation start stress σ Mf... Completion stress for forward transformation into martensite σ Ms... Initiation stress for forward transformation into martensite σ Af... Completion stress for reverse transformation into austenite σ As... Initiation stress for reverse transformation into austenite [M]... Martensite transformation region [A]... Austenite transformation region [d]... Detwinning transformation region τ... Shear stress ε t... Transformation strain xv

16 Chapter One Introduction 1.1. Shape Memory Alloys Shape memory alloys (SMA) have gained the attention of a lot of researchers due to their very distinct properties. These distinct properties have made them popular enough to be employed in a wide variety of applications, from aerospace to biomedical devices. As the name implies, shape memory alloys have the ability to recover their shape following subjection to an applied load [1]. The shape recovery can be achieved by increasing the temperature to a certain threshold. This property makes them desirable candidates for actuation and damping applications. Moreover, these alloys can be utilized in different applications in order to absorb and dissipate energy. This is due to their unique characteristics of a reversible hysteretic behavior [1]. Common modes of actuation and therefore common ways of testing of SMAs are tensile, torsional or combined tensile-torsional tests. The response of the alloy under each different loading condition is different. Thus predicting the behavior of these alloys under various conditions is essential for predicting different modes of motion and actuation of theses alloys. Different uniaxial and torsional models have been developed by researches 1

17 that can predict the SMA s behavior under specific loading conditions. The focus of this work is developing a torsional model for SMAs. This model will be used for developing biomedical devices Outline This thesis provides the fundamental basis for modeling shape memory alloys under torsion. Chapter 2 presents the literature review on shape memory alloys. This chapter explains the distinct properties of SMAs in details. This chapter also provides a review on modeling the behavior of SMAs. Both uniaxial and torsional models developed by previous studies are discussed. Chapter 3 presents the approach that is chosen to model the torsional behavior of shape memory alloys in details. This chapter also includes the details about obtaining the material properties using the MATLAB-based program. Chapter 4 presents the result of the experimental tests. In this chapter, tensile and torsional experiments are discussed in details. The testing machines for both set of tests are also explained. The calculated material properties are shown. Chapter 5 presents the comparison of the results of the model versus the experimental data for both the tensile and torsional experiments. Moreover, for confirmation of the model s result, the model s prediction is also compared against the previous models from literature. Included in this chapter is also the investigation of the effect of different parameters on the torsional profile of SMA. Chapter 6 presents the potential concepts for future work and further steps to improve the developed model. 2

18 1.3. Approach In the presented study, the torsional behavior of shape memory alloys is investigated. A MATLAB-based torsional model is developed to capture the torque-angle profile of shape memory alloys in order to be utilized in different kinds of applications such as biomedical devices. The model is developed based on a uniaxial model of SMAs. Tensile and torsional experiments are performed on a specific alloy (NiTi#1 wire by Fort Wayne Metals). NiTi is the most well-known among shape memory alloys. It should be noted that large recoverable deformation, good fatigue life, flexibility and crush resistance are NiTi s outstanding properties, which help to develop minimally invasive medical devices [2]. Tests are conducted on various samples possessing different diameters and lengths. The data from the tensile tests are used to obtain the wire sample s material properties. A MATLAB-based program is generated to calculate the material properties of given SMA wire sample using the tensile test data. The tensile and torsional experiment results are compared against the model s prediction to validate the model. Moreover, the model s prediction is compared with other model s results from literature to confirm the validity of the model. The effect of different parameters on the behavior of shape memory alloys is studied using the model. Each affecting parameter is investigated independent from the others and the results are explained in details Contribution The presented study resulted in some publications which are listed below: 3

19 Karbaschi, Z., Elahinia M., Modeling the Torsional Behavior of Superelastic Wires, Proceedings of the ASME 2011 Conference on Smart Materials, Adaptive Structures and Intelligent Systems SMASIS2011. Karbaschi, Z., Elahinia M., Modeling the Torsional behavior of shape memory alloys, 2 nd Annual Midwest graduate Research Symposium 2011, The University of Toledo. Chapman, C., Eshghiinejad, A., Karbaschi, Z., Elahinia, M., Torsional Behavior of NiTi Wires and Tubes: Modeling and Experimentation, Journal of Intelligent material Systems and Structure (JIM), Chapman, C., Karbaschi, Z., Elahinia, M., Torsional behavior of shape memory alloys tubes for biomedical applications, Proceedings of the ASME 2010 Conference on Smart Materials, Adaptive Structures and Intelligent Systems SMASIS

20 Chapter Two Unique Properties of Shape Memory Alloys 2.1. Unique Properties of SMAs The two distinct properties of SMAs, which make them unique among the conventional metals, are shape memory effect and superelasticity. Shape-memory effect is the ability of the alloy to recover its original shape upon heating [1]. When the material is deformed under an applied load and is performing its shape memory effect, the alloy can recover the strain (original shape) by increasing the temperature above a certain threshold known as austenite finish temperature, A f. This behavior is because of a thermo-mechanical coupling of the metal which occurs due to the alloy crystalline structure[3][4]. Here, it is essential to understand how this crystalline structure changes such that SMAs are capable of performing their distinct behavior. The change in crystalline structure stems from the interaction of the neighboring atoms and reorientation of the atomic structure of the material [5]. It should be noted that the material has two different states, the austenitic and martensitic. In its austenitic state, the alloy has its original shape. Under a mechanical load, however, the alloy is detwinned martensite. The crystalline structure of the two states is shown in Figure

21 Figure 2-2: Crystalline structure of SMAs in two different states of the material, inducing enough stress takes the material from austenite to detwinned martensite. It can be observed from Figure 2-3 that an alloy transformed from austenite to detwinned martensite by a mechanical load can recover when heated above A f [1]. Since the phase transformation takes place due to a mechanical load this is usually referred to as a stress-induced phase transformation [1]. In addition, under a cooling process, if the temperature decreases below a specific value, a transition from austenite to twinned martensite (also referred to as a soft martensite will take place [6]. This specific temperature is known as M f, the martensite finish temperature. This process, when the transformation occurs with increasing/decreasing temperature under no mechanical load, is called temperature induced phase transformation [1]. Figure 2-4 shows an illustration of the crystalline structure of the temperature induced phase transformation of the SMAs. Figure 2-5 Crystalline structure of SMAs during temperature induced phase transformation under no mechanical load. 6

22 Superelasticity i.e., the ability of the metal to undergo large elastic deformations, is also one the distinct properties of SMAs [6]. This property makes this alloy a good candidate for actuation purposes [7][8]. The material exhibits this behavior when it is completely above A f temperature. In such a case the material is fully austenite. When subjected to a load, the material deforms, i.e., a phase transformation from austenite to detwinned martensite occurs [1]. However, since the material is austenite, it will recover its original shape, once the load is released. In other words, if the load is repeatedly removed and applied, a cyclic phase transformation occurs. The crystalline structure of this phase transformation is illustrated in Figure 2-6. Figure 2-7 Crystalline structure of a superelastic SMA during stress-induced phase transformation. A phase diagram can well illustrate both the stress and temperature induced transformation. Figure 2-8 shows the phase diagram for a typical shape memory alloy. It should be noted that the transformation from austenite to martensite is known as the forward transformation and the transition from martensite to austenite is called the reverse transformation [10][1]. 7

23 Figure 2-9 The phase transformation diagram of a typical SMA showing the shape memory effect [1]. The transformation temperatures are shown in figure 4. M f, M s, A s and A f are martensite finish temperature, martensite start temperature, austenite start temperature and austenite finish temperature respectively. The shaded areas show the regions where the phase transformation takes place. Moreover, the forward and reverse transformations are shown with red and blue arrows respectively. Figure 2-4 schematically demonstrates the two properties of SMAs. As it can be observed from Figure 2-4, when the material is at high temperature, it is stable and austenite. Following a cooling process, under zero stress the crystalline structure of the alloy can be rearranged to form a twinned martensite at a temperature below M f. A subsequent mechanical load can take the alloy from twinned to a detwinned martensite at which the material is deformed. However, it should be noted that the applied stress needs to be significantly high enough to deform the alloy. The minimum essential stress to detwinned the SMA is called the detwinning start stress as shown in the Figure

24 Moreover, the stress level at which the detwinning process is completed is called the detwinning finish stress. At this state, the alloy stays deformed when the load is released. An austenite state can be retrieved by heating the material above A f. This process represents the shape memory effect (shown in solid arrows) and can be repeated as many times. Figure 4 also illustrates the superelasticity. If the material is austenite at high temperature and is loaded to a minimum required stress level, the crystalline structure can reorient to form a detwinned martensite, M d. The dotted arrows represent this process. The alloy maintains this state until a sufficient load is applied. Once the load is released, during unloading, SMA will recover its original shape. Figure 2-5 shows another common illustration of a phase diagram in which the twinned and detwinned martensitic states are distinguished. The arrows show the occurrence of transformation between different states of the material. Figure 2-10 Phase diagram showing different state of the SMA [9]. 9

25 Both of shape memory and superelasticity properties produce hysteretic behaviors. The hysteretic cycling as mentioned earlier enables the alloy to be employed in damping application where for instance vibration energy should be dissipated as heat. These two properties enable shape memory alloys to recover the axial strain up to 8% which is a far larger than the recoverable strain of the conventional material used in different applications [1]. The recoverable strain of the latter is around 1%. A loading/unloading curve for a typical SMA is illustrated in Figure 2-6 and 2-7, where shape memory effect and superelasticity are shown in part respectively. Figure 2-11 The loading/unloading curve of a typical SMA under axial tension presenting shape memory effect [10]. 10

26 Figure 2-12 The loading/unloading curve of a typical SMA under axial tension presenting superelasticity [10]. The explanation in this section covers the behavior of SMAs under uniaxial tension. The same properties are expected when SMAs are applied to torsion loading. In different kinds of applications, these torsional properties can be utilized in developing or improving devices such as biomedical implants Therefore, the torsional behavior of shape memory alloys need to be significantly understood to further enhance or develop the design of different devices. The following section will explain the previous work that has been done in modeling the torsional behavior of SMAs Modeling Shape Memory Alloy under Torsion Modeling the behavior of shape memory alloys has attracted many of the researcher s attention due to their distinct behavior. Since SMA s behavior is dependent on various factors such as stress, temperature, the rate at with the stress is applied, the 11

27 boundary conditions and etc, predicting the final stress-strain (in the case of axial load) and torque angle (in the case of torsion) profile becomes more complicated compared to conventional metals. On the other hand, the complications associated with the manufacturing the alloy and the manufacturing cost motivate one to developing a model which can be used to predict the SMA distinct behavior under varying conditions [11] Uniaxial Modeling of Shape Memory Alloys Extensive studies have been done in order to model the complex phenomena of shape memory alloys. Several constitutive equations were developed to model their behavior under different conditions. Generally these models can fall into two different classes. One class of models is based on macroscopic observation. These models take the measurable variables such as stress, temperature as the state variables and obtain the immeasurable variables, also known as internal variables such as martensite fraction based on the state variables. These models are called phenomenological. The second class of constitutive models is built upon micromechanics of single crystals. These models are called micromechanical and are usually more difficult to be implemented [12]. An example of a micromechanical is the constitutive model developed by Peng. X. or Blanc and L Excellent [13][14]. The constitutive equations developed by Tanaka falls into phenomenological category. His constitutive equations relate the applied stress, the temperature, strain and the martensite fraction [15][16]. In order to relate the variables various experimental test were carried out on SMA samples at different stress level and temperatures. Liang and Rogers modified Tanaka s model and developed more unified constitutive equations [17]. In addition, Tanaka s work was further improved to a more comprehensive constitutive 12

28 law to predict the thermomechanical behavior of shape memory alloys by Brinson, She introduced a cosine function to capture transition region in the stress-strain profile and the martensite fraction was separated into the temperature-induced and stress-induced [10][18]. Elahinia and Ahmadian developed a unified approach for onset conditions of phase transformation [3][4]. Another phenomenological model in which the constitutive equations relate the internal and the state variables was developed by Lagoudas [1]. This model will be further discussed in chapter three as it has been used extensively in this work. All the aforementioned models are capable of predicting the behavior of shape memory alloys under axial loading. However, they are not capable of predicting the torsional behavior of SMAs. In the next section, the works in literatures that have been dedicated on torsional modeling is discussed Torsional Modeling of Shape Memory Alloys Since the torsional SMA actuators are being used in a vast variety of applications, many researchers have contributed to the literature on developing models of which the torsional behavior of SMAs can be predicted. SMAs show the same superelastic and shape memory effect under torsion. Figure 2-8 and 2-9 show the shape memory and superelastic behavior of SMAs under torsion. 13

29 Figure 2-13 Shape memory effect of SMA wire under torsion [10]. Figure 2-14 Superelasticity of SMA wire under torsion [19]. 14

30 Researchers have chosen different approaches to develop these torsional models. In some approaches a uniaxial constitutive model was extended to a torsional domain to capture the torsional behavior of SMAs, in different approaches, however, an exact solution was obtain to describe SMAs torsional behavior. Analytical models have also been developed in order to predict the torsional response of SMAs. Finite element modeling (FEM) of SMAs under torion also has been conducted in ABAQUS or COMSUL such as FEM developed by Tabesh et al. to predict the behavior of NiTi superelastic-shape memory beams [20]. Analytical models however are simpler as oppose to FEMs and can be easily used by practicing engineers. An analytical torsional model for SMA tube actuator was developed by Keefe et al [21]. This model was based on simple nonlinear constitutive relationships to predict the torsional behavior of the SMA tube actuator. The displacement distribution was assumed to be nonlinear through the cross sectional area of the SMA bar [21]. It should be noted that the nonlinear constitutive relationships were first developed in an effort by Shishkin who investigated the interrelationship between the thermomechanical diagrams of SMAs in tension, compression and torsion [22]. In a work by Plahand and Chopra, the torsional model was developed based on a uniaxial extension model [23]. In their study, they used Brinson uniaxial phenomology model and extended in to a torsional domain to describe the torsional behavior of shape memory alloys [10][18][23]. This approach will be discussed in details in chapter three. In addition to Chapra s work. A close-form solution (Exact solution) for the torsional behavior of SMAs was introduced by Mirzaeifar where a three-dimensional phenomological model was reduced to a one dimensional pure shear [24]. His one 15

31 dimensional model was then solved in order to obtain an exact solution for the torsional behavior of an SMA circular bar during both loading and unloading in pure torsion [24]. Other three-dimensional models also can theoretically describe the behavior of SMAs under any arbitarary loading consitions including pure torsion or combined tentional-torsional loading. For example a three-dimensional model was developed by Boyd and Lagoudas [25]. Lexcellent and Rejzner have also developed a threedimensional model that can predict the behavior of SMAs under different loading conditions [26]. However three-dimensional model are usually more difficult to be implemented and are not very user-friendly for a practicing engineers. In the investigation presented herein, a torsional model was developed based on Lagoudas uniaxial model. Furthermore a similar approach to Chapra s was taken to extend the axial model to a torsional domain [23]. The model can be utilized to describe the superelasticity behavior of shape memory alloys under torsion in both loading and unloading. The model can accommodate different SMA bar geometries such as length and radius as well as varying temperatures and stress level Some Applications of SMAs The two distinct characteristics of SMAs have led to their wide use in various applications. This section prides a brief introduction of some applications that utilize SMA in their design. SMAs have been used in a number of automotive and aerospace applications [27][7][8]. For an instance, in a work by Hartl. D., et al, NiTi SMA beam was utilized for active jet engine chevron. In this work the authors benefit from the thermomechanical characteristics of NiTi SMAs. The NiTi SMA beam was employed as an actuator in the 16

32 jet chevron to provide bending force on the laminate substrate to reduce the engine noise during take-off [28]. Figure 2-10 shows the SMA elements employed in chevron engine. Figure 2-15 SMA beam element used in chevron engine [28]. In another application torsional SMA tube actuation were used in helicopter rotor blade by the work of Keneddy et al. order to reduce the maintenance cost and the helicopter vibration and also increase the helicopter performance [29]. Deploying NiTi SMAs in medical application also has pretty long history. Tarkesh and Elahinia investigated use of these alloys in assistive and orthotic devices [30][31]. NiTi bone staples for bone fracture fixation are one of the first medical applications NiTi SMAs [32]. These staples are used to fuse the two part of the fractured bone together by applying a compressive force to the bone. The NiTi bone stable benefits either from the superelasticity or shape memory effect to generate the compressive force. Figure 2-11 shows some of the commercially available bone stables. 17

33 Figure 2-16 Commercially available NiTi bone stable [32]. Intervertebral fusion implants were also manufactured out of porous nitinol. Intervertebral fusion implants are the common solution to the disk diseases such as degenerative disk disease. Based on a study by Assad M. et al., porous nitinol implants should better bone fusion result compared to conventional titanium implants [33]. It should be noted that in order to utilize NiTi implants inside the body, the biocompatibility of this material needs to be investigated. A lot of researches have been dedicated to test the biocompatibility of NiTi within the animal specimens or simulated body fluids [34][35][36][37][38][39]. The results have shown that NiTi is biocompatible and is not toxic to the body. Another use of nitinol SMA wires is in orthodontic treatment. Superelastic SMA wires have been used for the correction of tooth position from 1970s instead of stainless steel. Superelastic SMA wires provide a relatively constant force (in the plateau region) during a long time of treatment to correct the position of the tooth. However, stainless steel provides a large stress for a small increment of the strain which causes pain for the patients [1][40]. Thus the relatively constant stress during large strain incrimination is the unique property of SMAs that had been implemented in orthodontic treatment. Figure 2-12 illustrate the superelastic SMA wires in orthodontic treatment. 18

34 Figure 2-17 Orthodontic SMA wire treatment [1][41]. Other biomedical applications that use SMAs are actuators for endoscopic tools skin-wound closure devices and in dental applications, drills for root canals [1][42]. All of these applications benefit from the unique properties of SMAs. Another example of such devices is an expandable NiTi stent in cardiovascular applications. One application of the stents is used to support the walls of the body s tubular passages such as blood vessels [1]. The stent is shape set such that it can hold the blood vessels open, however it is collapsed into a smaller form when inserting into the body. This is to keep the insertion minimally invasive. NiTi stent benefits from the large recoverable strain of around 8% and NiTi transformation temperature As which is below body temperature. When inserted into the body, due to the increase in temperature, the stet will recover its original shape and it will 19

35 expand to hold the inner circumference of the blood vessel. An schematic of NiTi stent is shown in Figure 2-13 [1][43]. Figure 2-18 Schematic of NiTi stent for support of the blood vessels [1][43] Necessity of a Torsional Model As previously mentioned, the behavior of SMA under variable conditions is Complex. This complication stems from SMA s distinct properties. Moreover, due to the same properties, SMAs are being widely utilized in various applications such as development of biomedical devices. Especially, in development of minimally invasive medical devices which were impossible to be created with conventional metals [44]. This paragraph is off in margin, fix. It should be noted that torsional behavior of SMAs can be utilized in development or improvement of minimally invasive medical devices. This creates the demand for a model that can predict the SMA s response due to torsion. In addition, a torsional model 20

36 can be used for optimization in the development of the device. Although studies have been done in the development of a torsional model, a simple and user-friendly model that can be easily utilized is still not available. To this end, in this study, a simple torsional model is presented and implemented with MATLAB. In the presented study the behavior of superelastic SMA wires under torsion is investigated because superelastic SMAs can be used as self-actuators. No source of energy is needed to actuate for shape-set superelastic SMAs. 21

37 Chapter Three Computer Modeling 3.1. Torsional Modeling of SMA Wire Since the torsional behavior of shape memory alloys can be employed in different application such as various biomedical applications and devices, modeling the behavior of these alloys under torsion should be significantly understood. The focus of this study is on the behavior of superelastic SMA wires under torsion. It should be noted that conducting experimental testing on various SMA wires in order to observe their behavior is neither cost- nor time-effective. This is because the behavior of SMAs significantly changes not only with different wire s geometries but also under different loading conditions as well as other affecting parameters. Thus, having a model eases the way for predicting SMAs behavior under various applied load conditions as well as investigating the role of wire s geometry on the torque-angle profile. The goal of this analysis and modeling is to develop a model to relate the applied torque to angular deflection of superelastic wires. To this end, an approach similar to Prahland et al. model is used. In that study, a torsional model is developed based on extension of a SMA uniaxial model to a torsional 22

38 domain [23]. In the presented study, in the same manner the model is first created to predict the behavior of SMAs under uniaxial tension loading i.e., The uniaxial model predicts the thermomecahnical behavior of the SMA. Following this, the model is extended to a torsional domain to capture the torsional behavior of SMA wire/rod. The torsional model can include various wire cross sections, lengths and other affecting parameters as the inputs and predicts the torque-angle relationship of the SMA wire/rod as the output. To predict the behavior of SMAs under a uniaxial tension loading, a uniaxial constitutive model is used. In the work by Prahlan et al., Brinson uniaxial model is used, however, in the presented work, Lagoudas one-dimensional reduction of the uniaxial constitutive model is implemented which will be comprehensively explained later on in this chapter [1][10][23]. This is one of the main adavantages of the current model over previous work by Chopra et al. The fact that the model is based on a multi-axial model makes the presented model more versatile for applications where multi-axial loading is involved. Even though in this thesis the focus is on torsion only having the foundation of the multi-axial constitutive model is essential for future expansion and application of this model for applications that involve axial loading along with torsional loading. The other advantage of this work over the previous research is that in the model used the dependency of residual strain to applied stress is considered. In other words this model can capture thermomechanical behavior based on partial phase transformations In summary, the difference in the presented torsional model from the model developed by Prahland is in choosing a different uniaxial model. In this study, Lagoudas uniaxial model is chosen versus Brinson s uniaxial model [1][10]. Additionally, in 23

39 Lagoudas s 1D uniaxial model, the martensite fraction is considered to be only one internal variable as appose to two internal variables for martensite fraction in Brinson s model; where the martensite fraction is divided into stress-induced and temperatureinduced in Brinson s model which makes it more complicated to implement [10]. Since in the applications that this model is going to be used, the temperatureinduced martensite fraction is not the concern, the Lagoudas model is chosen as a simpler approach to predict the uniaxial behavior of the SMA wires [1]. This works compares the results of the two model in later chapters. Another advantage of this modeling approach is that the material properties can be obtained from the tensile tests and input into the uniaxial model as appose to obtaining the material properties directly from the torsional profile (torque-angle) of the SMA. This will in return gives us a more accurate representation of the model s parameters due to the uniformity across the material [23]. When developing a torsional model one should know that the behavior of the SAM is tightly coupled with the current state of the material and is different from its extensional properties. In order to develop the model some assumptions need to be made. Firstly, the shear stress and shear strain are considered to be functions of the radial location and are not constant on the cross sectional area of the wire. As can be seen in figure 14 the shear stress changes nonlinearly throughout the cross-sectional area of the SMA wire/rod [24][45]. When the wire is subjected to torsion, it is assumed that the cross sectional area can be in three different states. 24

40 As shown in figure 14, under an applied torque, austenite to martensite phase transformation may or may not take place across the whole surface of the wire s cross section. At a level of stress that is not high enough for complete transformation, the distribution is such that the inner area is in austenite state with no phase transformation taking place. The middle area is assumed to be under phase transformation from austenite to martensite and the outer region is in fully martensite state. This is shown by the arrows in Figure 3-1. Figure 3-1 Shear stress distribution over the wire/rod cross sectional area, depending on the radial location, shear stress increases linear or nonlinearly. The temperature is assumed to be constant throughout the material. Moreover, a pure torsional load is considered i.e. no axial force is applied to the SMA wire. We should keep in mind that the purpose of a torsional model is to relate the angular deflection to the relative internal torque to create the angle-torque profile of the given SMA wire. This approach starts with assuming an angular deflection and 25

41 calculating the developed torque through a few steps, which are to be explained. The assumption is that the shear strain is a function of the radial location, for an angular deflection of θ the shear strain, can be defined as: (1) Where and are the length and the radius of the SMA wire/rod. Shear strain is a function of normal strain according to the following relationship (Gere and Timosenko, 1984 [46]): (2) Therefore, for each radial location the shear strain and the equivalent normal strain are calculated throughout the cross sectional area. Based on the aforementioned equation we can see that the outer region of the wire/rod experience higher shear strain. This could also be observed in figure 13. Now that we converted the shear strain to normal strain we can calculate the normal stress using the uniaxial model and then convert the normal stress back to shear stress to stay in a torsional domain. The uniaxial model is used to find the normal stresses for the calculated normal strains in each radial location. It is worth noting that stress-strain behavior of shape memory material as shown in previous chapter is hysteretic and path-dependent. In other words stress-strain relationship for loading is different than that of unloading. The uniaxial model is created based on the 1D constitutive models developed by Lagoudas, which will be explained comprehensively 26

42 later in this chapter [1]. For now, let us assume that the uniaxial model is already developed which means we have the stress-strain relationship of the SMA wire. Fix this paragraph and check the entire document for formatting. From the stressstrain curve, the normal stress is calculated for each radial location as shown in Figure 3-2. Equation 3 also explains that given the uniaxial model, the normal stresses can be obtained for the normal strain. (3) Figure 3-2 Superelastic loading/unloading curve under uniaxial tension. The uniaxial profile is used to obtain the equivalent stress for each given strain. The approach which is used to obtain the normal stress form the stress-strain profile will be explained in the next subsection. 27

43 It should be reminded that the objective of the presented model is to relate the angular deflection to the applied torque. Since the applied torque can be calculated by an integration of the shear stresses over the cross sectional area, the shear stresses corresponding to the associated shear strains need to be calculated. The equivalent shear stress, for each radial location is determined using the calculated normal stress and Poisson s ratio υ as shown in the following equation [23] (4) Finally, the applied torque T, causing the original angular deflection, is calculated by integrating the shear stresses over the cross sectional area [23]: (5) Where is the outer radius of the SMA wire. In the case of an SMA tube, equation 5 can also be used by changing the integration limits, from inner radii instead of zero to the outer radii. The method used in this approach related the torque and the angle of an SMA wire/rod under torsion. In summary, first the angular deflection in converted to the normal strain. Following this, the relative normal stresses of the calculated normal strains are obtained by using the extensional characteristics of the SMA wire. The normal stresses are then converted to the shear stresses and finally the internal torque is calculated by an integration of shear stresses over the cross section of the wire/rod. In the following section, the approach used to obtain the extensional behavior of the SMA rod will be discussed. 28

44 3.2. MATLAB Program for 1 Dimensional Uniaxial Model Since the model is based upon a one-dimensional axial model, a MATLAB-base one dimensional axial model is created which to capture the stress-strain relationship of a SMA wire/rod. Here the constitutive equations following Lagoudas that relate the normal strain to the normal stress is explained in details [1]. Lagoudas model is a three-dimensional model. This section explains how this model is simplified and reduced to one dimension. In the one-dimensional case, a uniaxial load is considered which reduces the stress tensor to have only one non-zero element: (6) Where is the normal stress and the strain tensor is reduced to: (7) Where is the axial transformation strain and is the normal strain. To develop the stress-strain relationship, each region of the axial profile is created separately. Also a pseudeoelastic loading-unloading path in a constant temperature is considered. Based on Lagudas constitutive equations, the equations for the elastic and the transformation paths given are as the following [1]. When the stress, is below the Martensite start, <, the elastic austenitic region equations are given as: 29

45 (8) ; ; Where is the martensitic fraction which changes with respect to the change of stress level and temperature. is the austenitic compliance. When the stress goes pass the martensitic start, the transformation takes place. The equations that govern the forward martensitic transformation when the stress level is < are: ; (9) ; ; Where H is the equivalent to the maximum transformation strain and is the Martensitic compliance. Sgn( ) and are calculated as: (10) ; respectively. Where and are the martensitic start and finish temperatures at zero 30

46 When the stress level reaches above, the second elastic region occurs. This martensitic phase continues until the stress level reaches the maximum stress. The equations for the second elastic region are when < : ; ; (11) The same equations 11 apply for the martensitic region when the stress level is decreasing during unloading, where we have > >. The following equations are used for the reverse martensitic transformation when the stress level is > > : ; (12) ; ; Where is calculated as: ; (13) and are the transformation temperatures at zero stress and are the material properties. Finally, the elastic response for the austenitic region when the stress level is decreasing to zero, > > 0, is predicted utilizing the following equations: formatting for formula 31

47 (14) ; ; The aforementioned constitutive equations are used to predict the stress-strain relationship of the SMA wire. A part of the code for the austenitic if(sigma<sigma_ms) % if the transformation dose not occure % loading curve for i=1:length(sigma) zeta(i)=0; S(i)=Sa; Strain(i)=Sigma(i)*Sa; % Martensitic elastic portion % Set martensite fraction to zero % Compliance calculation % Strain calculation end End if (Sigma (i)<=max(sigma) && Sigma (i)>=sigma_ms) % Forward transformation zeta(i)=((1/(ro_bm))*[sigma(i)*h+.5*delta_s*sigma(i)^2+ro_deltas0*(t-ms)]); % Martensite fraction calculation S(i)= Sa+zeta(i)*(Sm-Sa); % Compliance calculation 32

48 Strain (i)=s(i)*sigma(i)+h*zeta(i); % Strain calculation if(zeta(i)>1) % If we go pass transformation region we enter martensite elastic region thus martensite fraction should equal one zeta(i)=1; % Martensite fraction set to one S(i)=Sm; Strain (i)=sm*sigma (i)+h*zeta(i); % Compliance calculation % Strain calculation Of course the above code does not show the complete code. The material properties are obtained from several extensional testing on an NiTi wire sample and are then fed into the model as the inputs along with the wire/rod geometry, temperature and maximum angular deflection Calculating Normal Stress from Stress-Strain As stated in the section 3.1, the normal stress for each calculated normal strain needs to be obtained from a uniaxial SMA profile. Generally in uniaxial models, the stress is known and is increased in a stepwise fashion and the relative strain is calculated in order to generate the stress-stress profile. Here it is reversed; the strain is known and the relative stress should to be calculated. 33

49 Instead of solving the coupled multi-variable constitutive and transformation kinetics equations to find the stress based on the stain, here we assume that we have a predefined uni-axial stress profile. Using the constitutive equations a stress-strain profile is generated in advanced. This is done by increasing the stress value in a step-wise fashion and fining the relative strain from the constitutive equations. The stress profile is such that the wire is loaded until complete transformation to stress induced martensite followed by a complete reversal of the stress and recovery of strain. The result is a two dimensional array including the stress values and their associated strains.. To this end, a MATLAB program is generated, the strain value and the axial profile is input to the program and the associated stress is calculated. The uniaxial model explained in the previous section is used here. Given the pre-defined stress-strain profile, linear interpolation as shown graphically in figure 16 is used for values of stress and strain that are not included in the predetermined stress and strain array. The MATLAB program breaks the stress-strain profile into two parts, the forward and the reverse transformation. This is achieved by using the stress derivative & σ which is positive in the forward transformation and negative in the reverse transformation. This is because for each strain value there exists two different stress values as shown in Figure 3-3. (15) 34

50 Figure 3-3 Linear interpolation to obtain the relative stress for each calculated strain. Figure 3-4 shows the determination of normal stresses associated with normal strains using interpolation: 35

51 Figure 3-4 An illustration of determination of normal stress associated with normal strain using interpolation. Interpolation is also shown below; this was generated in a completely different m- file as a separate function: function stressreturn=search(strain,stress,givenstrain,dgivenstrain) stressreturn=0; if(dgivenstrain>0) % separating the stress-strain profile into two different curve, the forward and the reverse transformation by checking the value of the stress derivative for(i=1:length(strain)-1) % check all strain value in the strain array 36

52 if(strain(i)<strain(i+1) && strain(i)<givenstrain && strain(i+1)>givenstrain) % find the strain value closest to the given strain stressreturn= ((stress(i+1)-stress(i))/(strain(i+1)-strain(i))*(givenstrainstrain(i)))+stress(i); % calculates the stress value using interpolation end end end The function receives the stress-strain profile, the given strain and the value of the stress derivative. It should be noted that the above code only shows the forward transformation MATLAB Program for Obtaining Material Properties As shape memory alloys have gained attention and have become widely used in different applications both in medical and non-medical areas, ASTM international has introduced standardized test methods under which shape memory alloys should be testes. These standards become even more important when dealing with medical applications [1]. The experiments performed on SMAs in order to obtain their material properties also should follow the same standards. Since the tensile tests that should be performed on an SMA wire to obtain the material properties are standard and mostly identical, a MATLAB-based program is generated which is able to calculate the material properties of a given SMA wire sample. This program can further be used for other application other 37

53 than the scope of this project as an easy way to receive the experiment data and calculate the material properties. To obtain the material properties which are used in modeling the behavior of SMAs, the wire should first be stabilized to be ready for the next set of tests, this will be comprehensively explained the next chapter. The material properties that are obtained by the program are listed in Table 3-1. transformation temperatures, M s, M f, A s, A f, the martensitic and austenitic modulus of elasticity, E m and E a, and also other material properties such as C m and C a, and the residual strain ε l (H), which are required in order to model the SMA behavior, are calculated using this program. Material Properties (Symbol) Table 3-1 The list of material properties. M s M f A s A f E m E a C M C A ε l (H) Martensite transformation start temperature Austenite transformation start temperature Austenite transformation start temperature Austenite transformation finish temperature Elastic modulus of austenite Elastic modulus of austenite Slope of the stress-temperature martensite transformation line Slope of the stress-temperature austenite transformation line Maximum residual strain 38

54 In most cases, the tensile tests are given in excel sheets based on force and displacement. This code is generated in order to ease the way to convert the generated force-displacement data to their equivalent stress-strain. The code requires the length and the radius of the wire sample and it automatically uses them to convert the forces to their equivalent stresses and the displacement to its equivalent strain. Given the stress-strain profile of the SMA sample, the program calculates the tangent lines to both the elastic and transformation regions and obtains the points where they cross to find the critical start/finish martensite and austenite stresses. These points will then be used to calculate the transformation temperatures, C A and C M. Figure 3-5 shows how these tangent lines are generated and the critical stresses are obtained: Figure 3-5 Tangent lines which are drawn in order to obtain the start and finish martensite/austenite stresses. 39

55 It should be noted that in figure 3-5 the initiation stress for martensite/austenite forward and reverse ( transformation are shown. However, the completion stress for both martensite/austenite forward and reverse transformation ( are also calculated which are not shown in the figure. Moreover, that the maximum residual strain is calculated by extending the tangent line of the austenite modulus of elasticity until it crosses with the strain axis as shown in the figure 18. The program calculates the start and finish martensite/austenite stresses using the tangent lines for different test temperatures to find the transformation stresses, M s, M f, A s, A f, C A and C M. To calculate the said properties, the program generates a graph based on transformation stresses and their associate test temperature and creates a line which fits the best through all the obtained points. This can be seen in Figure 3-6. It is worth noting that for generating this figure uni-axial tensile tests are performed in four different temperatures as shown on the figure. 40

56 Figure 3-6 The stress-temperature profile of the SMA sample, this profile is used to calculate CA and CM and the transformation temperatures. As an example in Figure 3-7, line 1 is fit through all the completion stresses of the forward transformation at different temperatures (the points are generated from the stressstrain profiles at different temperatures). The slope of this line which is C M is also calculated. C A is also calculated in the same manner. However, it would be the slop of the line which fits though the completion stresses of the reverse transformation at different test temperatures. Moreover, as shown in the Figure 3-8, the point where the line 1 crosses the temperature axis is calculated and is considered as the martensite finish transformation temperature M f. All other transformation temperatures are calculated in the same manner. In the next chapter, the material properties of a NiTi wire sample obtained using the program will be completely explained and the generated graphs will be show. 41

57 Chapter Four Experimental Evaluation of Shape Memory Alloys In order to verify the model, the predictions need to be compared against the experimental results. To this end, various tensile and torsional tests are performed on SMA wires with different lengths and diameters under different test conditions. It should be noted that both tensile and torsional tests are carried out on the same identical SMA samples. The uniaxial tensile tests are performed to be compared against the result from the uniaxial model as well as calculating the SMA wire material properties. The torsional tests are also conducted on the same SMA wire samples to validate the torsional model prediction Tensile Test of an SMA Wire Sample The uniaxial tensile test is performed on SMA wires to verify the uniaxial model prediction. NiTi#1 from Fort Wayne Metall (Fort Wayne, IN) is chosen as the SMA wire sample. The composition of this alloy is 54.5 to 57 percent nickel and balanced out titanium. With this composition this alloy behaves superelasticlly. The tests are done using BOSE ElectroForce 3330, a mechanical testing machine with the possibility of controlling the test temperature. The wire sample is placed in between two grippers 42

58 which could grab it very tightly in place to insure the minimum movement of the wire. The upper gripper pulls the wire upwards and the lower gripper is stationary and sits on a load cell which records the applied force. The mechanical testing machine is set on a displacement-control i.e., the machine would stop the tensile force when reaches to a given maximum displacement which is equivalent to maximum strain (the maximum displacement is input into the machine prior to the test). The gauge length between the two grippers could be also changed selectively. The tensile tests are performed on NiTi wires at different temperatures. The mechanical testing machine is set up with an environmental chamber in which the temperature could be selectively controlled. The test temperatures are at 40, 50, 60 and 70 degree Celsius. The applied force and relative displacement are recorded with an interface software. The stress-strain profile of each sample at different temperature is then obtained using the collected data. The material properties are also obtained using the same set of stress-strain profiles at different temperatures. This will be explained in details later on in this chapter. An illustration of the BOSE machine and the environmental chamber is shown in Figure

59 Figure 4-2 Illustration of BOSE machine and the environmental chamber. Prior to the tensile tests, in order to obtain the stress-strain profile of the samples, cyclic uniaxial tensile testing is performed on each SMA sample for stabilization purposes. The axial and torsional profile of the SMA wires is repeatable if only the SMA wire is stabilized in advance through cyclic testing. If the wires are not stabilized the loading and unloading plateaus may not sit perfectly on top of one another. To stabilize the SMA wires, a series of cyclic loading-unloading uniaxial tensile tests are carried out on each NiTi wire until the loading-loading plateaus sit almost perfectly on top of each other. The samples undergo 50 complete cycles of loading and unloading Determination of SMA wire Material Properties It should be noted that only one NiTi sample is required for obtaining the material properties. The stress-strain profiles of a NiTi sample at different temperatures is shown 44

60 is Figure 4-2. The length of the sample is 3.74 inch and the wire s diameter is inch. The displacement rate is 0.02 inch (either English or metric)per second. The data collected from Bose machine is based on force-displacement which then are converted to stress-strain using the length and diameter of the wire sample: Stress = Force / A (Cross sectional area) Strain = Displacement / L (Length) Figure 4-3 Stress-Strain profile of the NiTi sample at different centigrade temperatures. As mentioned in the modeling chapter, a MATLAB code is generated to calculate the material properties using the stress-strain profile of the NiTi sample at different temperatures. Determination of accurate material properties is essential for both the uniaxial and torsional model in order to result in an acceptable prediction. 45

61 After the force-displacement profile of the sample is converted to the stress-strain profile, the austenite and martensite modulus of elasticity are obtain by calculating the slope of the tangent lines to the fully elastic austenite and martensite regions of the stressstrain curve. The tangent lines are shown in Figure 4-3 for two different temperatures. 46

62 Figure 4-4 Tangent lines are generated using the program for all test temperature. The slope of the tangent lines is calculated to obtain the austenite and martensite modulus of elasticity. This is shown for only two temperatures of 70 (top) and 40 (bottom) degrees of Celsius respectively. 47

63 The tangent lines are calculated for all temperatures and the slopes of the lines are averaged to obtain the austenite and martensite modulus of elasticity (the modulus of elasticity may slightly differ between the different stress-strain profiles at different temperatures). The tangent line of the elastic region and the tangent line of the transformation region are extended to cross one another as shown in Figure 4-5. The points where the tangent lines meet are considered as the transformation stresses for each temperature. Figure 4-6 Transformation stresses at 70 degree Celsius are obtained using the stressstrain profile. The transformation temperatures for other stress level are also found by assuming a linear dependency between transformation temperatures and stress and generating a phase diagram using the calculated transformation stresses of each temperature. As it can be observed from Figure 4-5, in the phase diagram a line is fit through the transformation 48

64 stresses associated to each temperature. It should be noted that the figure only shows the austenite finish stresses at different temperatures. The transformation temperatures are obtained by extending these lines to cross the temperature axis. The slopes of the lines which represent C A and C M are also calculated. Figure 4-7 The phase diagram of the NiTi wire. This diagram is used to obtain the transformation temperatures. As shown in Figure 4-5, the transformation temperatures are the temperatures at zero stress. The residual strain is obtained by extending the tangent line to the fully martensite region to cross the strain axis as shown in Figure 4-6. This point is considered as the residual strain. 49

65 Figure 4-8 Calculation of residual strain, the tangent line to the martensite region is extended to cross the strain axis. Finally, the material properties of the NiTi sample are calculated with a fair approximation and are listed in Table

66 Table 4-2 Material properties of the SMA wire sample. Property Value(Unit) M s M f A s A f C A C M E A E M -21 o C -28 o C -8 o C -1 o C 6.5 Mpa/ o C 6.5 Mpa/ o C MPa MPa ε l Torsional Test of SMA Samples The torsional test is performed on three NiTi wires possessing identical length but different diameters. The diameters of the wires are 0.018, 0.02 and inch (referred to as sample one, two and three respectively) and the length of the wires is 0.4 inch. These tests are performed in collaboration with Fort Wayne Metals Research Products Corporation (FWM). Using an Instron Micro-Torsion MT1 testing machine the data is collected during different trials. The machine is equipped with two grippers which would hold the wire in place. One gripper is stationary and attached to a torque cell and an encoder which collects the torque and angular rotation data. The other gripper is attached to a rotary motor and could rotate the wire to a given maximum angle with selective rates of rotation. The test is carried out such that a pure torsion is achieved. An Illustration of the torsional testing machine is shown in Figure

67 Grippers Torque cell Encoder for measuring the displacement Wire sample. Figure 4-10 An illustration of Micro-Torsion MT1 testing machine. The torque is collected as the angle is increasing. No axial force is applied to the wire (pure torsion). This condition is achievable as the torque cell is on low friction linear bearings and it is free to move. The angle is increasing with a rate of one rotation per minute. The test is conducted at the room temperature. Each sample is torqued until a full transformation is achieved i.e., the plateau entered the fully martensite elastic region. The test continues for several trials until the torque-angle profile is completely settled. Figure 4-8 illustrates the load to fracture curve of the three different samples. This test is conducted in order to obtain the maximum angle of rotation before the sample goes under plastic deformation. The maximum angles of rotation are 184o, 228 o, and 317 o for the 0.018, 0.020, and wire diameters respectively. This can be seen in figure 27. Figure 4-9 shows the comparison of the three sample s torsional profile. It can be seen from figure 28 that the hysteretic curves are settled after almost 2 cycles and for the rest of the cycles the hysteretic paths are stable and sit perfectly on top of each other. The 52