Luận Văn Design And Simulation Of A Single-Hinge And Adaptive Ankle Foot Orthoses Based On Superelasticity Of Shape Memory Alloys.pdf

Microsoft Word ThesisMorteza A Thesis entitled Design and Simulation of a Single Hinge and Adaptive Ankle Foot Orthoses Based on Superelasticity of Shape Memory Alloys by Morteza Gorzin Mataee Submitt[.]

A Thesis entitled Design and Simulation of a Single-Hinge and Adaptive Ankle Foot Orthoses Based on

Superelasticity of Shape Memory Alloys

by Morteza Gorzin Mataee Submitted to the Graduate Faculty as partial fulfillment of the requirements for the

Master of Science Degree in Mechanical Engineering

An Abstract of Design and Simulation of a Single-Hinge and Adaptive Ankle Foot Orthoses Based on

Superelasticity of Shape Memory Alloys

by Morteza Gorzin Mataee Submitted to the Graduate Faculty as partial fulfillment of the requirements for the

Master of Science Degree in Mechanical Engineering

The University of Toledo December 2013 The goal of this thesis is to propose and develop new designs of Ankle Foot Orthosis (AFO) based on superelastic characteristics of shape memory alloys (SMAs) The problem investigated in this research is a human gait abnormality called drop foot caused by the paralysis of the muscles which allow the ankle to dorsiflex This neuromuscular disorder results in foot slap after heel strike and toe drag during leg swing

As the most common solution, drop foot patients use an orthotic device called AFO add support and improve their gait However, development of a more compact assistive device, which could passively or actively secure the normal gait requirements, is still a need by both patients and clinicians

Based on investigations and experimentations performed in Dynamic and Smart Systems Laboratory at University of Toledo, SMA is a potential solution due to its unique stiffness behavior and hysteretic characteristics In this work superelastic characteristics of SMAs is considered as an enabler in the development of new generation

of AFOs

Within this work a passive AFO design is proposed which employs a superelastic SMA element as the hinge of the device This SMA hinge controls the ankle motion by storing and releasing energy during walking The superelastic element enables the AFO

to provide sufficient torque during dorsiflexion to raise the foot in the swing phase of the gait In order to evaluate the design performance a comprehensive gait analysis study is performed to extract the requirements of motion, understand the critical loads, and calibrate the desired stiffness profiles for the ankle and the superelastic element

A Finite Element Analysis is performed to realize an optimum design for the SMA hinge Preliminary simulations are carried out in the sagittal plane of the body to verify the functionality of the design in providing the motion requirements Unlike existing AFOs with two hinges, the proposed design uses only one hinge The multi-axial loading of the ground reaction in 3D is then simulated to estimate lateral response of the hinge in preventing hypermobility and securing the walking stability. To maintain stability the hinge should limit the motion in directions other than rotation in the sagittal plane

In addition to this passive hinge, superelastic SMA is also envisioned to realize an active AFO To this end, an active superelastic SMA element which is adjusted structurally and dynamically is used to reproduce the stiffness variation of a healthy ankle This concept could produce a controlled stiffness profile desired for different walking conditions such as various speeds Actuation mechanism design for this concept

is also discussed

Although, the major contribution of this study is developing a reliable passive AFO design, experimental and numerical analyses confirm the functionality of both passive and active SMA AFOs

Acknowledgements

I would like to express my gratitude to my advisor, Dr Mohammad Elahinia, for his understanding, encouraging and personal guidance in this research and throughout my studies at University of Toledo Without his guidance and persistent help this research would not have been possible. In addition, I would like to thank my committee members

Dr Berhan and Dr Hefzy, for the useful comments, directions and engagement to complete this master thesis

I also thank to all my lab mates at the Dynamic and Smart System Lab, who provided me support, motivation and wishes for the successful completion of this project

I would like to thank especially to Reza Mehrabi for his incredible help to me and sharing the knowledge during this work

Finally, I would like to express my heartfelt thanks to my parents for their blessings and emotional support throughout my life

Table of Contents

Abstract iii

  Acknowledgements vi

  Table of Contents vii

List of Tables x

  List of Figures xi

  List of Abbreviations xv

  List of Symbols xvi

  1 Introduction 1

1.1 Ankle foot orthosis 2 

1.1.1 Passive ankle foot orthosis 3 

1.1.2 Active ankle foot orthosis 7

1.2 Problem statement 10

1.3 Objective 11 

1.4 Approach 12

1.5 Contributions 13

1.6 Outline 14

1.7 Publications 15 

  2 Shape memory alloys 16

2.1 SMA phase diagram and transformation 17

2.2 Shape memory effect 18

2.3 Superelasticity 20 

2.4 Stiffness variation property 22 

  3 Gait analysis 25

3.1 Gait cycle and its phases 25

3.2 Gait parameters and analysis techniques 27

3.3 Ankle stiffness behavior 30 

3.4 Ankle range of motion 36 

3.5 Multi-axial loading of ankle-foot complex during the gait 40 

  4 Design of a passive AFO with a one-sided SMA hinge 47

4.1 Concept development 47

4.2 SMA hinge desing 49

4.3 Sagittal plane simulation 51 

4.4 Multi-axial loading simulation 55 

4.5 Design optimization 61 

5.1 Adjustable compliance concept 65

5.2 Actuation mechanism 67

5.3 Mechanical adjustment design 68 

5.4 Structural adjustment design 70 

5.5 Modeling and simulation 76 

  6 Results and discussion 79

6.1 Finite element analysis for the passive one-sided hinge 79

6.2 Stiffness behavior evaluation for active SMA element 88

7 Conclusions and future works 95

6.1 Conclusions 95

6.2 Future works 96

  References 98

List of Tables

2.1: Summary of austenite elastic modulus EA and martensite elastic modulus EM

reported in literatures [41, 48] 23

4.1: Material properties for the superelastic SMA hinge [41] 51

4.2: Four-step loading-unloading rotation of the ankle 53

4.3: Simplified resultant rotation of two phases 53

4.4: Critical conditions in the 3D loading 58

5.1: Dimensions of tubes in telescopic tube configuration 73

5.2: Optimized dimension of the adjustable SMA hinge 75

6.1: Material properties for the SMA rod [41] 89

6.2: Parameters to control uni-axial loading 90

List of Figures

1-1: Passive thermoplastic AFOs [17] 4

1-2: Tamarack Flexure Joint for thermoplastic and carbon laminate bracing [18] 5

1-3: Passive hybrid AFOs [19, 20] 6

1-4: University of Toledo SMA AFO device with superelastic wires [10, 21] 7

1-5: Active AFOs [14, 22-27] 9

2-1: Stress-temperature-transformation plot of a shape memory material [41] 18

2-2: Shape memory effect temperature-load phase diagram [43, 44] 19

2-3: Shape memory effect in 3D stress–strain-temperature space [42] 20

2-4: Stress-temperature diagram for pseudo-elastic effect [42] 21

2-5: Peudoelastic effect stress-strain diagram [45] 22

2-6: Stress-strain diagram shows stiffness variation 24

3-1: Human gait cycle and phases [49] 25

3-2: Segmented cycle diagram of human gait main events and phases [50] 27

3-3: Reference planes of the human body in the standard anatomical position [50] 28

3-4: Step length and stride length in the walking 29

3-5: Ankle stiffness behavior for slow, normal and fast speeds of gait [55] 31

3-6: Rotation-moment ankle behavior for a healthy foot in normal condition 32

3-7: Ankle stiffness variation both dorsi-flexion (DF) and plantar-flexion (PF) during stance phase [57] 33

3-8: Ankle torque vs rotation for fast walking speed in swing phase of the gait 34

3-9: Ankle torque vs rotation for normal walking speed in swing phase 35

3-10: Ankle torque vs rotation for slow walking speed in swing phase 35

3-11: Comparisons of the 2nd order polynomial ankle stiffness in various speeds 36

3-12: Body advancement due to the heel, ankle and forefoot rockers used [59] 37

3-13: Ankle range of motion during a gait cycle 39

3-14: Ankle rotation for a drop foot subject with hinged/non-hinged AFOs [10] 40

3-15: Foot subjected to the mlti-axial loading [50] 41

3-16: The 3D ground reaction forces as a function of time [50] 42

3-17: Foot pressure patterns and CoP path for left and right side foots [65] 43

3-18: Center of pressure pattern for normal and neuropathic subjects [66] 44

3-19: The GRF components over the entire stance phase for a representative subject walking at normal speed [67] 45

3-20: The cop displacements in anterior-posterior direction and medial-lateral direction over the entire stance phase for a representative subject walking at normal speed [67] 46

4-1: The superelastic hinge stores and releases energy during the gait 48

4-2: CAD image of the one-sided SMA hinge AFO 49

4-3: SolidWorks CAD image of the hinge element 50

4-4: Main ankle rotation happens in sagittal plane [50, 68] 52

4-5: Ankle rotation profile for four-step and two-step loading-unloading behavior 53

4-6: Stiffness profile of the ankle for the normal condition in swing phase 54

4-9: CoP distance variations with respect to the hinge position 56

4-10: Resultant applied moments in the transverse plane 57

4-11: Resultant applied moments in the frontal plane 57

4-12: Loading condition and CoP location at the 1st critical point 59

4-13: Loading and CoP at the 2nd and 3rd critical points 59

4-14: Loading and CoP at the 4th critical point 60

4-15: The hinge deflections in frontal and transverse planes after deformation 60

4-16: Optimized dimensions of the SMA hinge element 62

4-17: Appointed position of the hinge after optimization [68] 63

4-18: Hinge element meshing with two different qualities 64

5-1: Actuation mechanism embodiment 67

5-2: 4-Step multi-axial loading conditions 69

5-3: Modeled 4-link transmission mechanism in ADAMS 70

5-4: Bending moment for the deflection of the beam [74] 70

5-5: Torsional moment for the tube element [75] 71

5-6: Telescopic concept actuation mechanism 72

5-7: Telescopic tubes configuration 73

5-8: Tubes engagement/disengagement mechanism 74

5-9: Adjustable hinge activation concept 75

5-10: Adjustable hinge CAD design 76

6-1: Rotation-Moment profile for SMA hinge simulation in sagittal plane for 2-step loading in comparison with experimental ankle stiffness in the swing 80

6-2: Rotation-Moment profile for SMA hinge simulation in sagittal plane for 4-step loading in comparison with experimental ankle stiffness in the swing 81 6-3: Stress and strain distribution for the deformed shape of the hinge 82 6-4: Stress and strain profile for four nodes at a critical element of the hinge (element in the edge of the sides) in 2-step loading 83 6-5: Stress and strain profile for four nodes at a critical element of the hinge (element in the edge of the sides) in 4-step loading 84 6-6: The 3D deflection profiles at four critical points during stance phase of the gait 86 6-7: Stress distributions and strain vector plot for the deformed shape of the hinge 88 6-8: Actuator loading parameters: pre-tension vs steps and PAR 90 6-9: Element stiffness from simulation results for SMA rod underlying of multi-axial loading in comparison with experimental ankle in normal, slow and fast walking 91 6-10: Element stiffness from simulation results for SMA telescopic tubes in comparison with experimental ankle in normal, slow and fast walking 92 6-11: Element stiffness from simulation results for SMA telescopic tubes in comparison with experimental ankle in normal, slow and fast walking 93 6-12: Hinge stiffness vs slider position (active length of the hinge) 94

List of Abbreviations

2D Two dimensional

3D Three dimensional

AFO Ankle foot orthosis

CoP Center of pressure

DACS Dorsiflexion assist controlled by spring

DF Dorsi-flexion

EMG Electromyography

FEA Finite Element Analysis

GRF Ground reaction force

SMA Shape memory alloy

UMAT User Material

List of Symbols

 Cauchy stress tensor

Ms Martensite start transformation stress

Mf Martensite finish transformation stress

As Austenite start transformation stress

Af Austenite finish transformation stress

ε Total strain tensor

Af .Austenite finish temperature at zero stress

As .Austenite start temperature at zero stress

K Ankle stiffness

Ke .Desired ankle stiffness

Kda Ankle stiffness for the drop foot

Kha Ankle stiffness for the healthy foot

H Maximum axial transformation strain

α Effective thermal expansion tensor

εe .Elastic strain tensor

εt .Transformation strain tensor

Λ Transformation tensor

Γ Transformation identity tensor

ks .Effective stress coefficient

ke .Effective strain coefficient

Ω Effective stress identity tensor

Φ Transformation function

σ Deviatoric stress tensor

' Effective stress

Drop foot is a neuromuscular disorder that deteriorates patients' walking ability by preventing them from raising their feet during the swing [3] This issue happens due to paralysis of the musclesand/or their malfunction at the ankle joint [4] This issue causes two main problems during walking which include: uncontrolled falling of the forefoot after heel strike which consequently makes foot slap and dragging of the toes when the affected leg starts swinging [5]

causing a permanent abnormal gait There are several treatment options including physical therapy, assistive devices and surgery Each of these techniques could be a temporary or permanent solution depending on the specific conditions of the patient [6] Orthotic devices are the most common treatment for drop foot patients, in which a controlled force prevents the foot drop and foot slap during the stance and swing phases

of the gait, respectively these devices are intended to improve patients’ walking pattern and normalize the gait [7, 8]

1.1 Ankle foot orthosis

An Ankle Foot Orthosis (AFO) is a mechanical device used by drop foot patients with paretic ankle dorsiflexor muscles, to support and improve the functionality of the foot and ankle joint [9] Although, the aim of AFO is preventing the forefoot to drop in swing by inhibiting the ankle movement, it also improves the ankle capability to support body weight, provides progression and secures push-off ability during stance phase of walking [10]

Many current AFO designs used for foot drop treatment while capable of controlling the foot during swing, restrict the ankle motion during the stance phase of gait This restriction of movement causes abnormal gait pattern, disuse atrophy of the ankle flexor muscles, harmful effect on other joints and ligaments and further energy consumption [11, 12]

AFOs are mainly divided in three groups: passive, semi-active, and active Passive AFOs include a mechanical element such as spring or damper to provide motion control of the ankle joint during gait Semi-active AFOs may consist of a control system

to adjust joint compliance or damping In fully active devices, an actuator connected to a source of power provides motion at the joint [13]

1.1.1 Passive ankle foot orthosis

A passive AFO assists the patient by preventing undesirable foot motion during the swing phase of the gait Essentially, acts like a torsional spring, limiting the foot deflection by providing an external torque

Passive AFOs are commercially available devices commonly used by patient in daily walking Commercialized AFOs must be lightweight, durable, compact, and relatively inexpensive For commercial success and acceptance, AFOs could be either articulated or non-articulated devices and are also categorized based on their constituent material which might be metal, leather, thermoplastic, composite, or a combination of materials in hybrid AFOs [14, 15]

Thermoplastic AFOs, as shown in Figure 1-1, are the most common and are made

as L-shaped polypropylene plastic braces, with the upright portion behind the calf and the lower portion under the foot It is attached to the calf with a strap, and is made to fit inside the accommodative shoes In a thermoplastic AFO also called posterior leaf spring AFO, motion control characteristics is specified by the material properties and the geometry [16, 17]

Figure 1-2: Tamarack Flexure Joint for thermoplastic and carbon laminate bracing [18]

Hybrid AFOs are developed in order to provide motion control in swing without unrestricted range of motion during stance These devices consist of lightweight thermoplastic or carbon braces with articulated joints and passive elements aimed at storing and releasing for motion control [14] A passive hybrid AFO called Dorsiflexion Assist Controlled by Spring (DACS) [19] is developed at University of Health and Welfare in Japan for drop foot patients (Figure 1-3 (a)) Also another hybrid device developed in Japan by researchers at Osaka University[20], employing a passive pneumatic element to control the ankle joint (Figure 1-3 (b)) These hybrid AFOs enable the patient to use a variety of shoes, and also provide biomechanical options for adjustability of the ankle joint

(a) (b) Figure 1-3: Passive hybrid AFOs: (a) The DACS’s hybrid AFO, (b) The Osaka

University AFO [19, 20]

Dynamic and Smart Systems Laboratory at University of Toledo proposed a novel design for passive AFOs based on shape memory alloy materials This device consists of parallel superelastic wires subjected to tensile loading In this design, SMA wires elongate during the powered plantarflexion, which assist the foot in dorsiflexion in swing phase Although, this design is able to provide the required motion during a gait cycle and successfully prevents the drop foot, some problems such as structure durability, oversized wires architecture and noise of the actuation mechanism existed The first generation of passive SMA AFOs, was proposed and fabricated by Bhadane M [10] as shown in the Figure 1-4 (a), which consists of a combination of eight parallel superelastic wires wrapped around fourteen plastic freely rotating pulleys and the arrangement is assembled on a polypropylene hinged AFO Deberg L [21] developed the second generation of these AFOs by modifying the arrangements of the wires and designing a

guide and carriage mechanism to transfer the linear motion to the hind-foot of the AFO structure This is shown in the Figure 1-4 (b)

Figure 1-4: University of Toledo SMA AFOs with superelastic wires: (a) Bhadane’s

SMA AFO, (b) Deberg’s SMA AFO [10, 21]

1.1.2 Active ankle foot orthosis

As discussed in the previous section, passive AFOs produce excessive resistance

to plantarflexion in stance phase Thus, ankle motion becomes restricted through loading response and the stability of the leg during walking is affected Due to this fact, passive AFOs are unable to completely normalize the entire gait functionality

Therefore researchers have been interested in active AFOs, intended to adjust

Blaya et al at Massachusetts Institute of Technology developed a powered foot orthosis based on Series Elastic Actuators [22] This device was aimed at changing the orthosis impedance (stiffness) actively, thus eliminates the foot slap The MIT group developed a Series Elastic Actuator (SEA) for the active AFO to realize variable stiffness This SEA is comprised of a DC motor, a helical spring and a ball screw mechanism Compliance of the system is controlled by driving the lead screw and adjusting the spring height Additionally a control algorithm was developed for the actuator to provide proper stiffness for different events during the walking cycle Although this active AFO shows promising results in a lab environment, the actuator is not practical for daily-wear application as it weighs 2.6 kg and has a bulky structure [23]

ankle-In the Human Neuromechanics Laboratory at the University of Michigan an active AFO was developed with pneumatically powered actuators called artificial pneumatic muscles (McKibben Muscles) which are providing both dorsiflexion and plantar flexion motion Also in this device a control algorithm adjusts air pressure of each actuator The total weight of the device excluding the off-board computer and air compressor is 1.6 kg [24]

At Arizona State University, researchers proposed and fabricated another active AFO with highly compliant actuator knows as robotic tendon This device also includes a motor and screw mechanism and an adjustable spring Robotic tendon act as SEA in harvesting energy from the gait cycle by utilizing increased elasticity The modified version of this AFO weighs 1.75 [25-27]

All the aforementioned active AFOs (Figure 1-5), need to be connected to the external power supplies and computers for operation Therefore, the applicability of these

orthoses is currently limited to laboratory studies A portable powered ankle-foot orthosis (PPAFO) was developed at the University of Illinois [14], using a pneumatic actuator, a

CO2 power source, and an onboard controller Although, this AFO presents an untethered controllable device, performance of the device as a sustained rehabilitation tool for daily wearing depends on future studies and improving the endurance of the AFO

SMA based active AFOs have been studied by several research groups recently [10, 28-30] Two generations of innovative actuation design based on thermo-mechanical properties of shape memory alloys was proposed at the University of Toledo [10, 21, 28] Combinations of superelastic and shape memory wires were investigated to develop an active actuator for AFOs The designed SMA AFO although could provide motion requirements, was not successful in practice The limitation of the first generation design was heating and cooling of the shape memory element due to the limited response time of the actuator in comparison to the walking cycle time The second generation solved some

of the issues of the previous generation by using superelastic elements to store and release energy

1.2 Problem statement

Drop foot patients suffer from paralysis of their muscles in the anterior portion of the lower leg which disable them to lift their foot at ankle This prevents proper dorsiflexion in swing and makes foot slap or toes drag during walking This problem also causes instability in walking and leads to an abnormal gait

Conventional passive AFOs as the most common treatment option for drop foot patients although could prevent the drop foot in swing, they restrict the ankle movement Also they do not produce normal ankle stiffness behavior during the whole gait cycle Additionally passive AFOs are unable to provide adaption to various walking conditions

Active AFOs are expected to provide more natural stiffness behavior This is possible through adjusting the device compliance during walking which stabilizes the gait However, developed active AFOs at the present time, are limited in their practical

assistive applicability due to the design issues and the factors such as durability, efficiency and portability

SMA based AFOs are studied as possible candidates capable of promoting sufficient ankle dorsiflexion and normalizing the gait Although thermo-mechanical characteristics of SMA in active AFO leads to some limitations due to long cooling rate

of the material, the SMA superelasticity provides appropriate stiffness and stability in ankle assistance There is however a need to AFOs with smaller profile that would allow the patients to wear them on daily basis Such as AFO should allow for wearing regular shoes and should have a single hinge for a minimalist profile

1.3 Objective

The main objective of this study is to design and develop an articulated portable ankle foot orthoses device based on superelastic characteristics of SMA to address drop foot neuromuscular disorder The device is aimed to provide desired controlled dorsiflexion motion in the sagittal plane during leg swinging without disturbing the entire gait movement Hinged joint provides flexibility in ankle movement and lateral loading response is controlled to inhibit hypermobility and improve lateral stability The second goal of this study is investigating different concepts of active mechanism in order to control stiffness variations at the ankle joint due to the various walking conditions such as various walking speeds

1.4 Approach

In this thesis, a novel compact design is proposed for an articulated passive AFO using the superelastic behavior of SMA This design is aimed to provide a low weight, flexible and efficient device for daily walking rehabilitation purposes This AFO secures desired motion in the regular direction and prevents unwanted motion in other directions Several active concepts for the compliance adaptation have then investigated to extend the performance of SMA AFO in supporting various walking conditions

To achieve a reliable design for the proposed AFO, a comprehensive gait analysis

is performed to extract and evaluate all the effects of loading components during a normal gait Based on desired stiffness and motion requirements in the sagittal plane, a Finite Element Analysis (FEA) is developed for the proposed design of the passive SMA AFO Lateral loading response is also evaluated by corresponding multi-axial loading simulation in 3D space The design is optimized and updated several times to meet all stiffness, motion and loading requirements

For the proposed active AFO concepts, numerical simulations are carried out to evaluate the stiffness properties of the active component under different walking speeds

It is been tried to achieve experimental ankle stiffness profiles by tuning the design variables during the simulation process Since the SMA elements in both passive and active concepts are made of superelastic material, a particular simulation method is employed to assign the required material properties

1.5 Contributions

The main contributions of this dissertation include:

 Develop a concept of lightweight, comfortable and efficient one-sided SMA hinge for AFO, based on superelastic characteristics of shape memory alloy

 Simulation of the SMA hinge in both sagittal plane of motion and frontal/transverse planes by evaluating the behavior of the element in satisfying the uni-axial ankle rotation in the regular plane of motion and resisting against deflections in the 3D multi-axial loading model

 Gait analysis or extracting ankle-foot complex requirements during the whole gait

 Calculation of 3D loading components of the ground reaction on the defined foot plate during stance phase of the gait

 Estimate desired ankle stiffness and moment in various walking conditions during the swing phase of the gait

 Design and optimization of the SMA hinge for the passive AFO concept

 Develop the design of actuation techniques based on mechanical and structural stiffness adjustment of SMA element in order to control the stiffness of the ankle for various walking conditions

 Perform the simulation for the active element based on stiffness and rotation requirements in the main plan (sagittal plane) of motion

1.6 Outline

Chapter one of this thesis concludes with the outlines of the intended contributions Chapter 2 provides required backgrounds on the special behavior of shape memory alloys In Chapter 3, gait analysis parameters and techniques are presented at the beginning Then, ankle stiffness behavior and range of motion are reviewed in Sections 3.3 and 3.4 Section 3.5, in particular focuses on multi axial loading of ankle-foot complex during the gait

Design and simulation of a novel passive AFO including one-sided superelastic hinge is presented in Chapter 4 In the first two sections of this chapter, concept development and design of the hinge are discussed Detailed design study and simulation

in the regular plane of the motion is demonstrated in Section 4.3 Simulation for multi axial loading with estimation of 3D loading conditions is developed in Section 4.4 Sections 4.5 and 4.6 concentrate on optimization process and mesh study for the proposed SMA hinge

In Chapter 5, an investigation for active mechanism of AFO based on the adjustable compliance concept is presented The basic concept is introduced in Section 5.1 Section 5.2 describes the actuation mechanisms Design and development of the active elements for the proposed mechanically and structurally stiffness control techniques are discussed respectively in Sections 5.3 and 5.4 Section 5.5 provides information about the modeling and simulation method

Chapter 6 contains all the simulation results in comparison with experimental data and finally Chapter 7, summarizes the conclusions of this thesis and comprises recommendations for further research

1.7 Publications

 GorzinMataee, M., Taheri Andani, M and Elahinia, M.,” A compliant ankle-foot orthoses based on multi-axial loading of superelastic shape memory alloys", ASME 2013 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, September 2013, Snowbird, Utah.

Chapter 2

Shape memory alloys

Shape memory alloys (SMAs) consist of a group of metallic materials that have the ability to recover large amounts of deformation when subjected to specific thermo-mechanical conditions [31] This unique behavior provides capability of undergoing large strains (about 10% strain) and eventually recovering these strains through removing the load or heating the material These make SMAs as one of the most widely smart materials currently used in wide variety of applications Some examples of these alloys are AgCd, AuCd, CuAlNi, CuSn, CuZn(X), InTi, NiAl, NiTi, FePt, MnCu, and FeMnSi [32-36]

Industrial applications of these alloys are developing fast, which include automotive, aerospace and medical devices [37] Nickel-Titanium (NiTi) as the most flexible and beneficial shape memory alloy has various applications in engineering and has been developed as couplings, medical devices, toys, actuators, heat Engines, and sensors [38, 39]

2.1 SMA phase transformation

Shape memory alloys demonstrate their unique properties due to a solid state phase change known as martensitic transformation This means by deformation in their crystal structure could be caused by stress or temperature changes, leading to a solid-to-solid phase transformation [40] A SMA stress-temperature-transformation diagram is depicted in Figure 2-1 Austenite phase represents the high temperature and martensite is defined for the low temperature phase The reversible transformation from austenite phase to martensite phase creates the thermo-mechanical behavior of shape memory alloys Martensitic transformation starts and ends at transition temperatures of Ms and Mfrespectively Correspondingly, As and Af are the temperatures at which austenite phase transformation begins and finishes [41] The temperature difference between Ms and

Mf is an important factor in characterizing shape memory behavior The stress-strain response of SMAs is strongly nonlinear, hysteretic and include a large reversible strain due to the martensitic phase transformation Besides temperature, this behavior is influenced significantly by cyclic thermo-mechanical loading Moreover, micro-structural aspects have a prominent role in determination of stress-strain-temperature curves [42]

Chemical composition of the alloy, loading path, thermo-mechanical history and ambient temperature could cause either of the two specific phenomena which are pseudo-elasticity (superelastic) and shape memory effect

Figure 2-1: Stress-temperature-transformation plot of a shape memory material [41]

2.2 Shape memory effect

Shape memory effect is a specific feature of these alloys to recover a certain amount of unrecovered strain through heating This phenomenon is observed when a shape memory alloy element is cooled to below the temperature Mf At this stage the alloy is completely composed of martensite which can be easily deformed By applying load to the material structure, it reaches to the detwinned martensite or deformed martensite phase After deformation, the strain can be recovered by heating the material above the temperature Af, that results in the SMA material go back to its original shape

In Figure 2-2, temperature-load phase diagram for shape memory effect is shown [43, 44]

Figure 2-2: Shape memory effect temperature-load phase diagram [43, 44]

A stress-strain-temperature diagram as shown in Figure 2-3, can capture the complete process of deformation and shape recover The process establishes by stress-free cooling of austenite provides a complex arrangement of martensitic phase variants Thanks to mobility feature for boundaries between the martensite variants and twinning interfaces accompanied by detwinning to obtain a stress level much lower than the plastic yield limit of martensite This provides a mode of deformation named as reorientation of variants Loading the material at first stage leads to the development of the self-accommodated martensitic structure During the second stage of loading, the martensitic phase induces reorientation of the variants and results in a large inelastic strain to

material Finally, during the last step a reverse transformation caused by heating recovers the residual strain Since martensite variants have been reoriented, a large transformation strain path with same amplitude but in opposite direction is produced while going back to the austenitic phase

Figure 2-3: Shape memory effect in 3D stress–strain-temperature space [42]

Shape memory effect is used mostly in actuation and sensing applications Some

of these applications currently developed are coffeepots, thermostats and various actuators

2.3 Superelasticity

The other unique feature of SMAs is recovering a large amount of strain through pure mechanical loading and unloading Unlike the shape memory effect, pseudo-

elasticity occurs without a change in temperature This process happens when the loading and unloading occur at a temperature greater than Af Initially, the material is in the austenitic phase By applying the load, the simultaneous transformation from austenite to martensite and detwinning of the martensitic variants happens and results in fully transformed and detwinned martensite Immediately when loading is removed, the martensite begins to transform back to austenite and strain fully recovers while remaining

at the same temperature which is still above Af This operation is shown with the temperature diagram in Figure 2-4 [45-47]

stress-Figure 2-4: Stress-temperature diagram for pseudo-elastic effect [42]

differences between Mf and As and between Ms and Af These four stresses define the start and finish of the two stress-caused transformation to martensite and two austenite Changing the temperature may changes the values of critical transformation stresses but doesn’t affect the general shape of the hysteresis loop [46]

Figure 2-5: Peudoelastic effect stress-strain diagram [45]

Pseudo-elastic characteristics of SMA is implemented in a wide range of applications where recovering a large deformations is required Some of this applications are eyeglass frames, cardiovascular stents, cellular phone antennae and orthodontic arches

2.4 Stiffness variation property

A special behavior of SMAs is the stiffness variation pattern provided by mechanical response In psuedoelastic materials, strain is produced and recovered by

thermo-mechanically loading-unload of the element in the austenitic phase however in shape memory recovering a certain amount of strain provided due to the thermal loading In both processes a particular hysteresis loop is produced and the stiffness profile changes due to phase transformation These variations can be used to achieve desired behavior for various applications

Stiffness of the SMA depends on elastic modulus of each phase (EA and EM), and total martensitic volume fraction ( ) that represents hysteresis curve There is a significant difference between the elastic moduli of the austenite and martensite From published experimental results and the tests previously conducted in our group, the ratio

of the martensite elastic modulus to that of the austenite varies from 0.36 to 0.65 Some

of this published values are summarized in Table 2.1 [48]

Table 2.1: Summary of austenite elastic modulus EA and martensite elastic

modulus EM reported in literatures [41, 48]

EA (GPa) EM (GPa) EM/EA Reference