Preliminary Investigation of Shape Memory Alloys for Space Applications

UNIVERSITÀ DEGLI STUDI DI PISA

Dipartimento di Ingegneria Civile E Industriale

Corso di Laurea Magistrale in Ingegneria

Aerospaziale

Tesi di Laurea Magistrale

PRELIMINARY INVESTIGATION OF SHAPE

MEMORY ALLOYS FOR SPACE

APPLICATIONS

Relatore:

Prof. MARIO ROSARIO CHIARELLI

Candidato:

DEVA HARSHA YARRAMSHETTI

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Acknowledgements

First of all, I would like to thank my thesis advisor Prof. Mario Rosario Chiarelli, for his support and motivation during the course of this thesis. Whatever the problem or the situation is, and how much ever high the hurdles are, the strong motivation one has can only be brought out, if and only if there is a strong encouragement and a healthy support from the advisor. In my case, my solid pillar of support was my professor. His humble nature and positive approach towards his students, has what aided me to complete this thesis successfully.

Secondly, the most important part of my life, my family. I cannot imagine myself, being in a tough situation without their words and inspiration. They helped me to be courageous whenever I underestimated myself. Also, I look up to my parents as the core inspiration for my entire life. I am indebted to them until the last breath of my life. All of this wouldn’t be possible without them and they are the sole reason for what I am today.

Last but not the least, my friends. My journey towards achieving the Masters’ degree began along with them. Their flawless motivation and inspirational hard-work always obliged to challenge myself. The constant bond I share with them, has helped me to share the knowledge effectively. Consistency and hard work that I have, reached new heights because of them. All of these has happily led me to the point of considering them also as my family without any hesitation.

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ABSTRACT

It has been recognised that SMA materials have a significant potential for deployment actuators, the number of applications of SMA-based actuators to the present day is quite small, since a deeper understanding of Thermomechanical behaviour of SMA and how it might be exploited in the design of working actuators is necessary.

This Thesis Investigates structural behaviour of Ni-Ti alloy and by using a Ni-Ti Shape Memory spring Actuator, we develop a space application, i.e. Satellite solar panel deployment.

Even though many constitutive models have been developed the Thermomechanical behaviour of Shape Memory Alloys, we have taken a reference of L.C. Brinson’s phenomenological model (1993) for deeper understanding of Shape Memory Effect in all Transformation phases, i.e. Austenite, Twinned and De-Twinned Martensite.

After a clear understanding of Shape Memory Effect, we have Implemented the Brinson’s Mathematical model and compared with the standard results of Brinson’s approach and we have shown our Implemented model can reproduce accurately the Stress-Strain-Temperature plots.

Finally, in the application part we have used a Nitinol (Ni-Ti) wire and developed a One-way Spring actuator of length 152.4 cm, which can eventually deploy the satellite solar panel of area 0.3 square meter segmented plates, with the help of circular hinge mechanism of radius 6mm between them, by giving an input of 1 ampere current, and it takes 10.45 seconds deploy the panels.

Keywords: Shape Memory Alloys, Ni-Ti behaviour, quasi one-dimensional constitutive

model for SMA, Solar panel deployment, Brinson’s model for designing SMA, Thermomechanical model of SMA, SMA actuator.

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Contents

List of Figures

: ... vi

List of Tables

: ... vii

Nomenclature

: ... viii

CHAPTER 1: STATE OF ART ...1

1.1 Introduction: ... 1

1.2 Motivation: ... 1

1.3 General Aspects of SMA: ... 1

1.4 Application fields: ... 2

1.5 Deployment Actuator: ... 2

1.6 Aims of the Dissertation: ... 2

1.7 Layout of the Dissertation: ... 3

CHAPTER 2: RESEARCH AND REVIEW ON BEHAVIOUR

OF SHAPE MEMORY ALLOYS ...5

2.1 A Little bit History of SMA: ... 5

2.2 Shape Memory Materials and Thermal conditions: ... 5

2.3 Shape Memory Alloy Phase Transformations: ... 6

2.3.1 One-way shape memory effect (OWSME):... 8

2.3.2 Two-way shape memory effect (TWSME) or Reversible SME: ... 8

2.3.3 Pseudo Elasticity (PE) or Super Elasticity (SE): ... 8

2.4 Basic Type of SMA-Based Actuators: ... 8

2.5 Heating methods: ... 11

2.6 Space Applications of Shape Memory Alloys: ... 12

2.7 Advantages of Shape Memory Alloys: ... 12

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CHAPTER 3: MATHAMETICAL MODELLING OF SHAPE

MEMORY ALLOYS ...14

3.1 Various Models

: ...14

3.2 Development of the Constitutive law from the Thermodynamics: ... 15

3.3 Tanaka’s Model: ... 16

3.4 Liang and Rogers Model: ... 18

3.5 Brinson’s Model:... 19

CHAPTER 4: L. C. BRINSON’S MATHEMATICAL MODEL

IMPLEMENTATION AND COMPARISON ...22

4.1 Modification to the constitutive law with constant material functions: ... 22

4.2 Modification to the constitutive law with non-constant material functions: 23

4.3 Numerical Examples: ... 25

4.4 Comparing the Brinson’s Mathematical model and Implemented model: ... 28

4.4.1 Austenite/Twinned martensite ⟹ De-twinned martensite: ... 28

4.4.2 De-twinned martensite ⟹ Austenite/Twinned martensite:... 32

CHAPTER 5: DESIGN AND DEPLOYMENT PHASE ...35

5.1 Designing of a Spring Actuator: ... 35

5.1.1 Geometrical parameters: ... 35

5.2 Designing of the solar panels: ... 37

5.3 Fixing the actuator to the solar panels: ... 39

5.4 Deployment of the panels: ... 40

CHAPTER 6: CONCLUSIONS ...45

6.1 BEHAVIOUR OF NITINOL: ... 45

6.1.1 Constitutive models: ... 45

6.2 NITINOL-BASED ACTUATOR: ... 46

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6.2.2 Deploy: ... 46

6.2.3 Panel lock: ... 46

6.3 FUTURE WORK: ... 47

BIBLIOGRAPHY ... 48

APPENDIX ... 50

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List of Figures:

Figure 2.1: NiTi SMA Phase Transformation Figure 2.2: SMA Phases and Crystal structures

Figure 2.3: Basic type of SMA Actuators using One-way SMAs

Figure 2.4: SMA Satellite Antenna (from Gandhi and Thompson 1992) Figure 2.5: SMA Satellite Antenna (from yang et al.1985)

Figure 2.6: Heating methods

Figure 2.7: Power/weight ratio vs. weight of different actuators Figure 3.1: Critical stress-temperature profiles used in Tanaka model

Figure 3.2: Stress-Strain-Temperature curve illustrating the shape memory effect Figure 3.3: Critical stress-temperature profiles used in Brinson model

Figure 4.1: Stress-Strain curves illustrating the shape memory effect Figure 4.2: Stress-Strain curves to maximum residual strain, 𝜀𝐿 Figure 4.3: Strain vs Temperature: 𝜉_𝑆0 = 0.02

𝜀𝐿 , 𝜉𝑇0 = 0.5

Figure 4.4: stress-strain curve illustrating elastic, loading and un loading sections Figure 4.5: Implemented stress-strain curve illustrating shape memory effect Figure 4.6: comparing the stress-strain Brinson’s data with Implemented data Figure 4.7: Implemented stress-strain curves to maximum residual strain,

𝜀

0.02

𝜀𝐿 , 𝜉𝑇0 = 0.5

Figure 4.9: comparing the strain vs temperature Brinson’s data with Implemented data Figure 5.1: SMA spring actuator concept

Figure 5.2: 0.3 square meter segmented solar panels

Figure 5.3: the circular hinge (segmented panels zoom in image) Figure 5.4: fixed actuator to the segmented panels

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Figure 5.6: zoom in image of the hinge when deploying the panel Figure 5.7: completely deployed panels (180° position)

List of Tables:

Table 2.1: Thermal conditions for SMA materials

Table 4.1: Material properties for Nitinol alloy used in the following examples [Dye, 1990; Liang, 1990]

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Nomenclature:

𝐴𝐴 – Material constant of Austenite phase. 𝐴𝑀 – Material constant of Martensite phase. 𝐴_𝑓 – Austenite final temperature.

𝐴𝑠 – Austenite starting temperature.

𝐵𝐴 – Ratio of material constant to the stress influence coefficient in Austenite phase. 𝐵_𝑀 – Ratio of material constant to the stress influence coefficient in Martensite phase. C – Spring index

𝐶_𝐴 – Stress influence coefficient of Austenite phase. 𝐶_𝑀 – Stress influence coefficient of Martensite phase. d – wire diameter.

D – spring diameter.

𝐷_𝑎 – Youngs modulus value of SMA 100% austenite. 𝐷𝑚 – Youngs modulus value of SMA 100% martensite. f – Frictional coefficient.

F – Force generated by spring recovery. g – Acceleration due to gravity.

G – Shear modulus. I – Current.

K – Arbitrary constant. 𝐿𝑓 – Final length of the spring. 𝐿₀ – Free spring length. 𝐿_𝑠 – Solid spring length. m – Mass of the solar panel. 𝑀𝑎 – Axial momentum.

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𝑀_𝑓 – Martensite final temperature. n – Number of turns in the spring.

p – Normal forces acting on the circular hinge. q – Heat production term.

𝑞_𝑠𝑢𝑟 – Heat flux. Q – Heat. S – Entropy density. t – Time. 𝑡𝑎 – Actuation time. T – Temperature. 𝑇0 – Initial Temperature. 𝑇0 _{– Ambient Temperature.} 𝑇_𝑓 – Final Temperature. U – Internal energy density. X – Coordinate axis.

Y – Coordinate axis. Z – Coordinate axis. 𝜎 – Stress.

𝜎₀ – Initial stress.

𝜎𝑠𝑐𝑟 – Critical stress at starting. 𝜎_𝑓𝑐𝑟 – Critical stress at final. 𝜎̂ – Cauchy stress.

𝜀 – Strain.

𝜀₀ – Initial strain. 𝜀𝑒

–

x

𝜀_𝐿 – Maximum residual strain. 𝜀_𝑟𝑒𝑠 – Residual strain.

Δ𝑡 – Change in time.

Δ𝑇 – Change in temperature. 𝜉 – Transformation fraction. 𝜉₀ – Initial transformation fraction.

𝜉𝑆 – Stress- induced transformation fraction. 𝜉_𝑇 – Temperature-induced transformation fraction. 𝜉_𝑆0 – Initial stress-induced transformation fraction. 𝜉_𝑇0 – Initial temperature-induced transformation fraction.

𝜉_𝐴→𝑀 – Transformation from Austenite phase to Martensite phase. 𝜉_𝑀→𝐴 – Transformation from Martensite phase to Austenite phase. Ω – Transformation coefficient.

Ω𝑆 – Stress-induced transformation coefficient.

Ω_𝑇 – Temperature-induced transformation coefficient. Θ – Thermal expansion coefficient.

Φ – Helmholtz free energy.

𝜚 – Density in the constitutive model configuration.

𝜚₀ – Initial state of Density in the constitutive model configuration. 𝛿 – Linear displacement.

𝛿𝐿 – spring displacement 𝜇 – Frictional coefficient. 𝜂 – Packing density.

𝜌_𝐷 – Density of the actuator.

𝜌𝑅 – Density of the Electrical resistivity. 𝜏 – Torque on the circular hinge

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CHAPTER 1: STATE OF ART

1.1 Introduction:

Shape memory alloys (SMAs) belong to a class of Shape memory materials (SMMs), which have the ability to ‘Memorise’ or retain their previous shape, after it has been deformed by heating over its transformation temperature. This unique effect of returning to an original geometry after a large inelastic deformation (near 10%) Is known as the Shape memory effect (SME).

1.2 Motivation:

The technology push, towards “smart” systems with adaptive and/or intelligent functions and features, necessitates the increased use of sensors, actuators and controllers. Thereby resulting in an un desirable increase in weight and volume of components. The development of high functional density and smart applications must overcome technical and commercial restrictions, such as available space, operating environment, response time and allowable cost. In particularly for space related construction and design, increased mass directly results in increased fuel consumption, also those suppliers are highly cost-constrained. Research on the application of smart technologies must concentrate on ensuring that these smart systems are compatible with space environment and existing technologies.

1.3 General Aspects of SMA:

SMA belong to the category of the so-called smart materials, or responsive materials. The first problem encountered with these unusual materials is defining what the word "smart" actually means. This word describes something which is astute or 'operating as if by human intelligence' and this is what they are. From dictionary definition they are "materials that can significantly change their mechanical properties (such as shape, stiffness, and viscosity), or their thermal, optical, or electromagnetic properties, in a predictable or controllable manner in response to their environment". They respond to environmental stimuli with particular changes in some variables, e.g., temperature or magnetic fields and mechanical stress. SMA are materials in which large deformation can be induced and recovered through temperature changes (shape memory effect) or stress changes (pseudo-elasticity). Thanks to these innovative and potential properties, they are useful not only as structural elements, appreciable for their mechanical toughness, but they are also capable of fulfilling sensing and/or actuation functions.

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1.4 Application fields:

• Medical technology: stents, orthodontic wires, guidewires and automation systems, clinical instruments.

• Electronic Engineering: ice braking and weakening control of power cable, circuit-breakers and fixable fuses for transformers, electromechanical cables.

• Aerospace Engineering: Solar panel deployment, Actuator controllers in spacecrafts, satellite Antennas, couplings and flaps control on the trailing edge of aircraft wings, light aircraft control cables and astronaut tethering.

• Building security: fire-stop valve, fire detection systems and automation systems; • Mechanical Engineering: mechanical shock absorber, activators in several possible

automotive applications, industrial piping tips and washers, lifting crane rope for lifting and moving material and equipment, indoor cranes and gantry cranes.

• Civil Engineering structures: for passive, active and semi-active control of civil structures, for seismic protection devices, vibration control and pre-stressing or post-tensioning of structures with fibre and tendons, actuation and information processes essential to monitoring, self-adapting and healing of structures, power cables, bridge stays and mine shafts.

• Marine and naval structures: salvage/recovery, towing, vessel mooring, yacht rigging and oil platforms.

1.5 Deployment Actuator:

From all the application fields we have seen above, we are going to investigate and develop SMA Deployment actuator for the Aerospace Engineering filed.

Solar panels are the very common using power source to the satellites, at the same time solar panels are being very large to provide required amount of power to the satellites. So, we fold them, fold until we can put them into the launcher. After satellite reach the orbit, we have to deploy the folded solar panel.

For deploying the solar panels, we have been using many techniques like pyros, thermal cutters, and different hinge and deployment mechanisms. These techniques are critical and risk but with the help of SMA, deployment actuators are very simple, easy functioning, easy controlling, and most efficient. So, in this dissertation we design a SMA spring actuator and uses for the solar panel deployment application.

1.6 Aims of the Dissertation:

Although it has been recognised that SMA material have a significant potential for deployment actuators, the number of applications of SMA-based actuators to the present day is still quite small. This is mainly because of following three reasons:

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• First, since SMAs are very different from traditional materials, a deeper understanding of their thermo mechanical behaviour is essential for actuator design.

• Second, the predictive methods based on the available constitutive thermo mechanical models for SMAs are, in most cases far from reliable. Therefore, the development of a robust predictive methodology to support the design process is required.

• Third, the design procedure for SMA-based actuators is different from ordinary design in many respects and is insufficiently well understood.

Although some of the modelling work presented in this Dissertation is valid more generally, the particular SMA that is studied is Nitinol because its properties are much better than other commercially available SMAs for our type of application.

Our aims of the study in this Dissertation were twofold.

The first aim was to investigate the thermomechanical behaviour of Nitinol theoretically, and second, we develop an actuator for solar panel deployment.

1.7 Layout of the Dissertation:

The Dissertation is presented as follows:

Chapter 1: State of Art: In this present chapter, firstly, we have started with the

motivation, to give an answer for the very usual question i.e. why Smart systems? And given a brief idea about how many application fields that can use Shape memory alloys, after we discussed about our Application that we are going to do and mentioned the aims of this Dissertation.

Chapter 2: Research and Review on behaviour of Shape Memory Alloys: In this

chapter, we started with a small history and the Thermal conditions for all SMA materials and we have introduced the concept of Shape memory behaviour and given a brief review of its applications. We have also summarised the basic type of SMA-based actuators and their advantages. Heating methods also summarised.

Chapter 3: Mathematical modelling of Shape Memory Alloys: In this chapter, we

have roughly summarised the Constitutive models have been developed to describe the Thermo mechanical behaviour of Shape Memory Alloys. From that we have chosen a phenomenological model based on uniaxial stress-strain-temperature data (Tanaka 1985,1986, Liang and Rogers 1990, Brinson 1993), and presented that data.

Chapter 4: Brinson’s Mathematical model Implementation and Comparison:

from the data of Tanaka’s, Liang’s, Roger’s and Brinson’s Mathematical modelling, in this chapter, we have chosen the promising model i.e. Brinson’s model. We have implemented the Brinson’s method properly by referring the examples of Brinson’s

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work (L.c. Brinson,1993). And we have compared our implementation results with his data.

Chapter 5: Design and Deployment phase: From the experience we got till here, in

this chapter, we have designed a one-dimensional SMA actuator and we fix it to a solar panel, which is designed by us. Then we calculate the forces and time to get panels deploy when we give electrical resistance. After the complete investigation of SMA, we designed this application.

Chapter 6: Conclusions: In this chapter, we have given the conclusions of our

dissertation. Mainly, we have discussed about the conclusions of Nitinol behaviour and Nitinol Actuator. Finally, we have mentioned some of Future works, which we have to do in the next phase of the dissertation.

Bibliography: Here, we have given the information of all the references, which we

taken in the time of preparing the dissertation.

Appendix: Here, we have given the information of all the results and MATLAB program codes, which we were used in the dissertation.

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CHAPTER 2: RESEARCH AND REVIEW ON BEHAVIOUR

OF SHAPE MEMORY ALLOYS

2.1 A Little bit History of SMA:

Shape memory alloy (SMA) or “Smart alloy” was first discovered by Arne Ölander in 1932, and the term “Shape memory” was first described by Vernon in 1941 for his Polymeric dental material. The importance of Shape memory materials (SMMs) was not recognised until William Buehler and Frederick Wang revealed the Shape memory effect (SME) in Nickel-Titanium (NiTi) alloy in 1962, which is also known as Nitinol (derived from the material composition and the place of discovery, i.e. a combination of NiTi and Naval Ordnance Laboratory).

Although Iron-based and Copper-based SMAs, such as Fe-Mn-Si, Cu-Zn-Al and Cu-Al-Ni, are low-cost and commercially available. Due to their Instability (e.g. Brittleness) and poor Thermo-Mechanic performance, NiTi- based SMAs are much more preferable for most applications. However, each material has their own advantage for particular requirements or applications.

2.2 Shape Memory Materials and Thermal conditions:

ALLOY COMPOSITION RANGE OF

TRANSFORMATION TEMPERATURES, ℃ TRANSFORMATION HYSTERESIS, °C AgCd 44 ~49 at %Cd -190 ~50 ~15 AuCd 46.5 ~50 at %Cd 30 ~100 ~15 CuAlNi 14 ~14.5 wt %Al 3 ~4.5 wt %Ni -140 ~100 ~35 CuSn ~15 at %Sn -120 ~30 ~10 CuZn 38.5 ~41.5 wt %Zn -180 ~ -10 ~10 CuZn X (X= Si, Sn, Al) Small wt %X -180 ~ 200 ~10 InTl 18 ~23 at %Tl 60 ~100 ~4 NiAl 36 ~38 at %Al -180 ~100 ~10 TiNi 46.2 ~51 at %Ti -50 ~110 ~30 TiNi X (X=Pd,Pt) 50 at %Ni +X 5 ~50 at %X -200 ~700 ~100 TiNiCu ~15 at %Cu -150 ~100 ~50 TiNiNb ~15 at %Nb -200 ~50 ~125 TiNiAu 50 at % Ni+Au 20 ~610 ~50

6 TiPd X (X=Cr, Fe) 50 at % Pd +X ~15 at %X 0 ~600 ~50 MnCu 5 ~35 at %Cu -250 ~180 ~25 FeMnSi 32 wt %Mn 6 at %Si -200 ~150 ~100 FePt ~25 at %Pt ~ -130 ~4 FePd ~30 at %Pd ~ 50 ~4 FeNi X (X= C, Co, Cr) Small wt %X ~ 50 ~4

Table 2.1: Thermal conditions for SMA materials

2.3 Shape Memory Alloy Phase Transformations:

SMAs are Inter metallic compounds able to recover, in a continuous and reversible way, a predetermined shape during a heating/cooling cycle. From a Microscopic point of view, SMAs present two solid phases stable at two different temperatures. The Martensite stabilizes at low temperatures and the Austenite stabilizes at high temperatures. The forward and reverse transformations between the two phases show a Thermal Hysteresis.

Hysteresis is a measure of the difference in the transition temperatures between heating and cooling (i.e. ∆T = 𝐴𝑓 – 𝑀𝑠 ), which is generally defined between the temperatures at which the material is in 50% transformed to Austenite upon heating and in 50% transformed to Martensite upon cooling. This property is important and requires careful consideration during SMA material selection for targeted technical applications; e.g. a small Hysteresis is required for fast actuation applications (such as Robotics), larger Hysteresis is required to retain the predefined shape within a large temperature range (such as in deployable structures).

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Practically, SMAs can exist in two different Phases with three different crystal structures (i.e. Twinned Martensite, Detwinned Martensite and Austenite) and six possible transformations. When SMA is heated, Martensite begins to transform into the Austenite phase. The Austenite-start-temperature (𝐴𝑠) is the temperature where this transformation starts and the Austenite-finish-temperature (𝐴_𝑓) is the temperature where this transformation is complete. Once a SMA is heated beyond 𝐴_𝑠 (austenite starts temperature) it begins to contract and transform into the Austenite structure, i.e. to recover into its original form. This transformation is possible even under high applied loads, and therefore, results in high actuation energy densities. During the cooling process, the transformation starts to revert to the Martensite at Martensite-start-temperature (𝑀_𝑠) and is complete when it reaches the Martensite-finish-temperature (𝑀_𝑓). The highest temperature at which Martensite can no longer be stress induced is called (𝑀_𝑑), and above this temperature the SMA is permanently deformed like any ordinary metallic material. These shapes change effects, which are known as the SME and Pseudo elasticity (or Super elasticity), can be categorised into three shape memory characteristics as follows:

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2.3.1 One-way shape memory effect (OWSME):

The one-way SMA (OWSMA) retains a deformed state after the removal of an external force, and then recovers to its original shape upon heating.

2.3.2 Two-way shape memory effect (TWSME) or Reversible SME:

In addition to the one-way effect, a two-way SMA (TWSMA) can remember its shape at both high and low temperatures. However, TWSMA is less applied commercially due to the ‘Training’ requirements and to the fact that is usually produces about half of the recovery strain provided by OWSMA for the same material and it strain tends to deteriorate quickly, especially at high temperatures. Therefore, OWSMA provides more reliable and economical solution. Various training methods have been proposed, and two of them are: Spontaneous and External load-assisted induction.

2.3.3 Pseudo Elasticity (PE) or Super Elasticity (SE):

The SMA reverts to its original shape after applying mechanical loading at temperatures between 𝐴_𝑓 and 𝑀_𝑑, without the need for any Thermal activation.

2.4 Basic Type of SMA-Based Actuators:

Both One-way SMA and Two-way SMA may be used for deployable structures Although Two-way SMA can perform in two directions due to its two-way shape memory mechanism, transformation strain associated with it is normally only half of that in one-way SMA.

An alternative solution is to put two one-way SMA based actuators one against another to generate mechanical two-way performance, i.e. heating SMA in one actuator to get forward motion, and heating SMA in another actuator to reverse. The advantage of mechanical two-way actuator is higher motion and higher force than in material two-two-way actuator, while the advantage of material two-way actuator is simpler, compacter and much less elements involved.

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Figure 2.3: Basic type of SMA Actuators using One-way SMAs. (a) One-way Actuator (b) Biased Actuator (c) Two-way Actuator

• Figure 2.3(a) shows a One-way Actuator. The SMA element is elongated initially, at low temperature, and is then heated to move element P in the direction of the arrow. • Figure 2.3(b) shows a Biased Actuator, which is capable of moving the element P

back and forth. The SMA element is deformed at low temperature, before being connected to the spring. When it is heated, the recovery force which is generated pulls the spring, thus storing energy on it. When the SMA element is cooled, the energy stored in the spring is released and the SMA element deforms back, thus completing the cycle.

• Figure 2.3(c) shows a Two-way Actuator, which includes two SMA elements. Two opposing SMA elements are used, instead of the SMA element and Bias spring of the Biased actuator. Any motion can be obtained by appropriately cooling or heating the two SMA elements.

• Two-way SMA based actuator is similar to Fig 2.3(a), one-way actuator, in shape, while its behaviour is more similar to Fig 2.3(b), biased actuator.

To deploy a structure, it is necessary that a relative moment, i.e. a rotation or linear motion, occurs between different parts of that structure. Such movement can be generated either by a specially designed actuator or simply by a part of the structure changing its shape.

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Figure 2.4: SMA Satellite Antenna (from Gandhi and Thompson 1992)

In the 1960’s, Nitinol sheets and rods had been considered to unfurl satellite Antennas upon exposure to solar heating, see Funakubo (1987) and Fig 2.4 and Fig 2.5 shows another simple SMA satellite Antenna made of Nitinol wire, which was designed and tested in China.

Figure 2.5: SMA Satellite Antenna (from yang et al.1985)

The SMA elements have been made in the following shapes: 1. Straight wires in tension, for small linear motion/high force.

2. Helical elements, for large linear motion/small force, or large rotation/small torque. 3. Torsion bar/tube, for large rotation/small torque.

4. Cantilever strips, for large displacement/small force. 5. Belleville-type discs, for small linear motion/high force.

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Generally speaking, only large deformations or large forces can be obtained from SMA, but not both.

2.5 Heating methods:

Basically, SMAs can be heated by the following three different methods,

Figure 2.6: Heating methods. (a) passing current through; (b) external heating by wire; and (c) thermal radiation.

• Fig 2.6(a) shows by passing an electrical current through them, this method is only applicable where a small diameter SMA wire or spring is used, otherwise the electrical resistance is too small to produce enough heating. The main advantage is simplicity, while the big disadvantage is that the SMA element needs to be electrically insulated.

• Fig 2.6(b) shows by passing an external current though a high resistance wire or tape wrapped around the SMA element, this method is available for SMA bars or tubes. The electrical wire needs to be electrically insulated, but the insulator should have good thermal conductivity to let the heat flow to the SMA element.

• Fig 2.6(c) shows by exposing the SMA component to thermal radiation, this may be the simplest way in space, since a component exposed to the light of the sun will heat to temperature of 150°C or more. No additional heating system is required. The main disadvantage of this method is that it is inflexible, and it could be very difficult to retract the structure.

An advantage of using heating by electrical current is that by controlling the electrical current, deployment and retraction can be made controllable.

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2.6 Space Applications of Shape Memory Alloys:

• One of the first suggestions as to the use of NiTi in space engineering were deployable constructions.

• The next stage in the development of applications of materials with shape memory were technical solutions where the SMA was used only as a functional material. One of the first examples of such application can be considered a model of NiTi wire actuators of the flaps of the instrument container for the satellite Nimbus.

• One of the first products was a drive for transforming a parabolic antenna from its transportation configuration into the operation shape. The drive had three working elements of NiTi with the diameter of 2mm that operated by torsion.

• Another wide area of application of SMAs in space technique is suggested by the necessity to damp vibration in constructions. Excellent damping ability of these alloys has always attracted designers.

• SMAs have future also in production of locking and release devices, Substitution of pyrotechnical materials, that have so far been commonly used in these devices, with shape memory materials would eliminate shock load during release, allow less strict requirements for storage and performance check of these devices, as well as repeated use of the device and the working element.

• The first launch of NiTi to an Earth orbit can be considered to have happened in 1982, when Thermo mechanical pipe couplings on the orbital station Salyut-7 were tested for their hermetic durability.

• The aim of another space experiment, called Sofora, was to approbate the developed technique of assembling truss constructions in open space with the help of Thermo mechanical couplings, the main element of which is a sleeve made of a shape memory alloy. This technique allows assembling trusses of large sizes by automatic manipulators as well as manually.

• In December 1996 SMAs were tested outside the Earth’s orbit, when the automatic Interplanetary station Pathfinder carrying a Mars Rover Sojourner set off to Mars. The rover featured a SMA drive which was a NiTi wire of 30mm in length and 0.15mm in diameter, operating by bend. In July 1997 the drive removed a thin glass panel with mars dust, un shielding a photo element in the device that determined dust content in the planet’s atmosphere by the difference in sun radiation intensity.

2.7 Advantages of Shape Memory Alloys:

In recent years, the use of SMAs has been proposed as an alternative to electrical motors. The advantages of SMA structures over existing solutions are reported to be:

• Mass and Volume savings. Compared the power/weight ratio vs. weight of a particular form of SMA actuator with many other conventional motors. Below figure

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suggests that SMA actuators have the potential capability to achieve an output/weight ratio which can’t be realised by traditional actuators.

Figure 2.7: Power/weight ratio vs. weight of different actuators.

• Avoidance of end-of-deployment shock loadings, which are always associated with spring- deployed structures. Therefore, no more need for dampers, and hence overall system complexity can be reduced.

• Sensing capability, both actuating and sensing functions can be combined by measuring changes in electrical resistance associated with the phase transformation. • Large recoverable strains which permit extremely long strokes and the application of

large forces when deformation is restrained.

• High electrical resistivity means that the shape transformation can be activated by passing an electrical current through a SMA element, thus avoiding the need for separate heaters.

• Design flexibility, Shape memory alloy actuators can be linear, rotary, or some combination of the two, as required, and can form an integral part of a component. • There are many further advantages to the use of SMA materials in actuators, the most

relevant being the simplicity of mechanism, cleanliness, silent actuation, remote ability, sensing ability, low driving voltage.

2.8 disadvantages of Shape Memory Alloys:

• A strong relationship between the strain operation range and fatigue life. • Quite a low response speed.

14

CHAPTER 3: MATHAMETICAL MODELLING OF SHAPE

MEMORY ALLOYS

3.1 Various Models:

Many scientists and engineers have contributed to the vast available literature on the experimental behaviour of Shape memory alloys. Many constitutive models have been developed to describe the Thermo mechanical behaviour of shape memory alloys. We can roughly summarise them into the following categories:

• Phenomenological models based on uniaxial stress-strain-temperature data. (Tanaka 1985,1986, Liang and Rogers 1990, Brinson 1993);

• A theory of non-equilibrium thermo statics that describes the thermodynamic paths of SMA (Cory and McNichol 1985);

• Models based on the interaction of the different sets of atoms in the alloy (Kafka 1994);

• Models derived from a special free energy formulation (Achenbach and Muller 1985, Leclercq and Lexcellent 1996);

• Models based on the thermodynamics laws (Ortin and planes 1991, Moumni and Nguyen 1996);

• Constitutive laws based on a model of hysteresis, e.g. Preisach model (Ortin 1992); • Models based on nonlinear thermo-plasticity theory, generalised plasticity. Or visco-

plastic theory (Boyd and Lagoudas 1994, Lubiner and Auricchio 1996);

• Mathematical models for the dynamics of phase boundary motion (Abeyaratne and Knowles 1993, Bhattacharya and James 1996);

• Models derived from the deformation of crystal structure during phase transformation (Falk 1989, Fischer and Tanaka 1992);

• Constitutive laws that allows for micro- structural deformation during phase transformation and the free energy concept, using an energy dissipation or energy balance approach (Sun and Hwang 1993a, 1993b, Patoor et al. 1994, Huang and Brinson 1997).

Each of these models aims to describe the behaviour of SMAs from a certain aspect and on a different scale. “In spite of all these efforts and qualitatively very interesting results, no single model exists that is able to describe quantitatively the shape memory performance of a real material. The reason for this is related to the very strong influence of micro-structure and processing on the mechanical properties described in the (σ, ε, T)-space.”

To fully understand the unique behaviour of SMAs from a theoretical point of view, we must start from the basics: the micro-structural changes that occurs during phase transformations. On the other hand, from an engineering application point of view, the most practical and productive approach is based on phenomenological models, which fit the uniaxial

15

experimental data, without attempting to capture the detailed underlying thermomechanical behaviour. Historically, the first such model was proposed by Tanaka (1985).

3.2 Development of the Constitutive law from the Thermodynamics:

Here we review Tanaka’s approach to the derivation of the constitutive law of shape memory materials. Considering a one-dimensional SMA material undergoing transformation, from principles of thermodynamics the energy balance and Clausius-Duhem inequality can be expressed as 𝜚𝑈̇ − 𝜎̂𝐿 +𝜕𝑞𝑠𝑢𝑟 𝜕𝑥 − 𝜚𝑞 = 0 (3.1a) 𝜚𝑆̇ − 𝜚𝑞 𝑇+ 𝜕 𝜕𝑥( 𝑞𝑠𝑢𝑟 𝑇 ) ≥ 0 (3.1b)

In the current configuration (Tanaka,1986) where U, 𝜎̂, q and 𝑞_𝑠𝑢𝑟 represent the internal energy density, the Cauchy stress, the heat production term and the heat flux, respectively, and S, T, x and 𝜚 represent the entropy density, temperature, the material coordinate and the density in the current configuration, respectively.

It is assumed that the thermo mechanics of an SMA material are fully described by the set of variables (𝜀, 𝑇, 𝜉), where 𝜀 is the green strain and 𝜉 is an internal variable representing the stage of transformation. The definition of 𝜉 is the martensite fraction of material, which varies from zero to one with unity representing 100% martensite, and its value is governed by temperature and stress, this relationship given in the transformation kinetics section.

By introducing the Helmholtz free energy Φ = 𝑈 − 𝑇𝑆, in equality can be rewritten in the reference configuration as (𝜎 − 𝜚0 𝜕Φ 𝜕𝜀)𝜀̇ − (𝑆 + 𝜕Φ 𝜕𝑇) 𝑇̇ − 𝜕Φ 𝜕𝜉𝜉̇ − 1 𝜚0𝑇 𝜚0 𝜚 𝑞𝑠𝑢𝑟𝐹 −1 𝜕𝑇 𝜕𝑋≥ 0 (3.2)

Where 𝜎 is the second piola–kirchhoff stress, F is the deformation gradient, and 𝜚0 the density and X the material coordinate in the reference configuration.

A sufficient condition for above equation to hold for every choice of 𝜀̇, 𝑇̇, their respective coefficients must vanish, thus yielding

𝜎 = 𝜚0

𝜕Φ(𝜀,𝜉,𝑇)

𝜕𝜀 = 𝜎(𝜀, 𝜉, 𝑇) (3.3a) 𝑆 = −𝜕Φ

𝜕𝑇 (3.3b) Equation (3.3a) is then the mechanical constitutive equation of the material.

Constitutive model of Shape memory alloys derived from above equations by differential calculus, one can write equation (3.3a) as

16 𝑑𝜎 = 𝜕𝜎 𝜕𝜀𝑑𝜀 + 𝜕𝜎 𝜕𝜉𝑑𝜉 + 𝜕𝜎 𝜕𝑇𝑑𝑇 (3.4) Leading to the most general equivalent expression

𝑑𝜎 = 𝐷(𝜀, 𝜉, 𝑇)𝑑𝜀 + Ω(𝜀, 𝜉, 𝑇)𝑑𝜉 + Θ(𝜀, 𝜉, 𝑇)𝑑𝑇 (3.5) Where the material functions are defined by

𝐷(𝜀, 𝜉, 𝑇) = 𝜚₀𝜕2Φ 𝜕𝜀2 , Ω(𝜀, 𝜉, 𝑇) = 𝜚0 𝜕2Φ 𝜕𝜀𝜕𝜉 , Θ(𝜀, 𝜉, 𝑇) = 𝜚0 𝜕2Φ 𝜕𝜀𝜕𝑇 (3.6)

From the form of the incremental constitutive law (equation (3.5)), the function 𝐷(𝜀, 𝜉, 𝑇) is representative of the “modulus” of the SMA material, Ω(𝜀, 𝜉, 𝑇) can be considered the “transformation tensor”, and Θ(𝜀, 𝜉, 𝑇) is related to the “thermal coefficient” of expansion for the SMA material.

If these material functions are all assumed to be ‘constants’, then the constitutive relation can be easily derived as

𝜎 − 𝜎₀ = 𝐷(𝜀 − 𝜀₀) + Ω(𝜉 − 𝜉0) + Θ(𝑇 − 𝑇0) (3.7)

If these material functions are all assumed to be ‘non-constants’, then the constitutive relation can be derived as

𝜎 − 𝜎₀ = 𝐷(𝜉)(𝜀 − 𝜀₀) + Ω(ξ)(𝜉 − 𝜉0) + Θ(𝑇 − 𝑇0) (3.8) Where (𝜎0, 𝜀0, 𝜉0, 𝑇0) represent the initial state or original condition of the material.

3.3 Tanaka’s Model:

Tanaka’s model is one of the first constitutive model for SMAs. This formulation assumes that Strain, Temperature and Martensite volume fraction are the only state variables for this model and the Stress is determined based on these variables. Also phase transformation kinetics is expressed in an exponential form and is a function of stress and temperature. Tanaka proposed a unified one- dimensional martensitic phase transformation model. This formulation is actually limited to the stress-induced martensite phase transformation only. Tanaka considered a one-dimensional element of SMA of length L, which is undergoing either the martensitic transformation or its reverse transformation. The constitutive equation (3.8) related to the state variable stress(σ), strain(ε) and temperature(T) in the terms of the martensite volume fraction (𝝃), is:

𝜎 − 𝜎₀ = 𝐷(𝜉)(𝜀 − 𝜀₀) + Ω(ξ)(𝜉 − 𝜉0) + Θ(𝑇 − 𝑇0)

Where (𝜎0, 𝜀0, 𝜉0, 𝑇0) represent the initial state or original condition of the material. D is the module of elasticity and assumed to be a linear function of the martensite volume fraction:

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𝐷(𝜉) = 𝐷_𝐴 + 𝜉(𝐷_𝑀 − 𝐷_𝐴) (3.9) Ω is called the phase transformation coefficient and is defined as:

Ω(ξ) = −𝜀𝐿𝐷(𝜉), (3.10)

Where 𝜀𝐿 is the maximum recoverable strain, the kinetics equation describing the martensite volume fraction as an exponential function of stress and temperature are:

𝜉_𝐴→𝑀= 1 − exp[𝐴_𝑀(𝑇 − 𝑀_𝑠) + 𝐵_𝑀𝜎], (3.11) For 𝑇 > 𝑀_𝑓 And 𝐶_𝑀(𝑇 − 𝑀𝑠) < 𝜎 < 𝐶𝑀(𝑇 − 𝑀𝑓), 𝜉_𝑀→𝐴= exp[𝐴_𝐴(𝑇 − 𝐴𝑠) + 𝐵𝐴𝜎], (3.12) For 𝑇 > 𝐴𝑠 And 𝐶𝐴(𝑇 − 𝐴𝑓) < 𝜎 < 𝐶𝐴(𝑇 − 𝐴𝑠),

Figure 3.1: Critical stress-temperature profiles used in Tanaka model

Where 𝐴𝐴, 𝐴𝑀, 𝐵𝐴, and 𝐵𝑀 are material constants in terms of the transition temperatures 𝐴_𝑠, 𝐴_𝑓, 𝑀_𝑠 and 𝑀_𝑓.

As usual in metallurgy, a transformation is regarded as complete when 𝜉𝑀 or 𝜉𝐴 are 0.99. substituting this value and its corresponding temperature in to equations, the constant 𝐴_𝐴 and 𝐴_𝑀 can be expressed as

18 𝐴𝑀 = − −2 ln 10 𝑀_𝑠− 𝑀_𝑓 𝐴_𝐴 = −−2 ln 10 𝐴_𝑓− 𝐴_𝑠 The constants 𝐵_𝐴 and 𝐵_𝑀 can be expressed as

𝐵_𝐴 = 𝐴𝐴 𝐶_𝐴 𝐵_𝑀 =𝐴𝑀 𝐶𝑀

The two more material constants 𝐶_𝐴 and 𝐶_𝑀 are called the stress-influence coefficients (in Fig 3.1), which indicate the influence of stress on the transition transformation, are obtained from the experimental tests.

Tanaka’s model is simple and based on material parameters that can be measured easily. It has been used in a variety of studies (Tanaka et al. 1995).

3.4 Liang and Rogers Model:

This model has almost the same form of the constitutive equation (3.8) as proposed in Tanaka model. However, for phase kinetics, a cosine function to describe the martensite volume fraction as a function of stress and temperature is supposed, respectively. The kinetic equations describe the martensite volume fraction as a cosine function of stress and temperature.

Liang (1990) assumed D, Θ and Ω to be constants.

𝜎 − 𝜎₀ = 𝐷(𝜉)(𝜀 − 𝜀₀) + Ω(ξ)(𝜉 − 𝜉₀) + Θ(𝑇 − 𝑇₀)

Where (𝜎₀, 𝜀₀, 𝜉₀, 𝑇₀) represent the initial state or original condition of the material. Liang assumed ξ to be a cosine function, and hence the transformation from austenite to martensite could be described by ξ_𝐴→𝑀 = 1−ξ0 2 cos [𝐴𝑀(𝑇 − 𝑀𝑓− 𝜎 𝐶𝑀)] + 1+ξ0 2 (3.13) For 𝑇 > 𝑀_𝑓 And 𝐶_𝑀(𝑇 − 𝑀_𝑠) < 𝜎 < 𝐶_𝑀(𝑇 − 𝑀_𝑓),

While the reverse transformation is described by ξ_𝑀→𝐴 = ξ0 2 cos [𝐴𝐴(𝑇 − 𝐴𝑠 − 𝜎 𝐶𝐴)] + ξ0 2 (3.14)

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For 𝑇 > 𝐴_𝑠

And 𝐶_𝐴(𝑇 − 𝐴_𝑓) < 𝜎 < 𝐶_𝐴(𝑇 − 𝐴_𝑠), Where 𝐴_𝑀 and 𝐴_𝐴 are given by

𝐴_𝑀 = 𝜋

𝑀_𝑠− 𝑀_𝑓 𝐴_𝐴 = 𝜋

𝐴_𝑓− 𝐴_𝑠 And 𝐶𝑀 and 𝐶𝐴 are the same as in Tanaka’s model.

3.5 Brinson’s Model:

In a stress-free state, an SMA material at high temperature exists in the parent phase (usually a body-centred cubic crystal structure, i.e. Austenite phase) and upon decreasing the material temperature, the crystal structure undergoes a self-accommodating crystal transformation into martensite (usually a face-centred cubic structure). The phase change in the unstressed formation of martensite from austenite is referred to as “self-accommodating” due to the formation of multiple martensitic variants and “twins” which prohibits the incurrence of a transformation strain.

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The martensite variants, evenly distributed throughout the material, are all crystallographically equivalent, differing only by habit plane indices, and each variant consists of two twin-related martensite’s. It is exactly this effect of self-accommodation by twinning which subsequently allows shape memory alloys to exhibit the large reversible strains with stress (Funakubo, 1987; wayman and duerig,1990).

When unidirectional stress is applied to the SMA material, there is a critical value, dependent upon temperature, at which the martensite variants begin a “Detwinning” process the results ultimately in the material consisting of a single variant of martensite aligned with the axis of loading. Additionally, for material in the austenite phase prior to loading, there is likewise a critical stress value, dependent upon temperature, at which the austenite undergoes a crystalline transformation to martensite and in fact, because of presence of stress, to a single variant of detwinned martensite. In the transformation process to detwinned martensite with application of load, the stress raises only slightly, and a large, apparently plastic strain is achieved.

The major shortcoming of both Tanaka and Liang and Rogers models is that they can only explain the phase transformation from martensite to austenite and its reverse transformation. since the shape memory effect (SME) at lower temperatures caused by the conversion between stress-induced martensite and temperature-induced martensite, these models cannot be implemented to the detwinning of martensite, which is responsible for SME. This problem was solved by the Brinson model. In this model the martensite volume fraction (𝜉) is separated into stress-induced (𝜉𝑆) and temperature-induced (𝜉𝑇) components:

𝜉 = 𝜉_𝑆+ 𝜉_𝑇 (3.16) The original form of constitutive equation in Brinson model is as follows:

𝜎 − 𝜎₀ = 𝐷(𝜉)𝜀 − 𝐷(𝜉₀)𝜀₀+ Ω(ξ)𝜉_𝑆− Ω(𝜉₀)𝜉_𝑆0+ Θ(𝑇 − 𝑇₀) (3.17) Where (𝜎₀, 𝜀₀, 𝜉₀, 𝑇₀, 𝜉_𝑆0) represent the initial state or original condition of the material.

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Conversion to detwinned martensite:

For 𝑇 > 𝑀_𝑠 and 𝜎_𝑠𝑐𝑟+ 𝐶_𝑀(𝑇 − 𝑀_𝑠) < 𝜎 < 𝜎_𝑓𝑐𝑟 _{+ 𝐶} 𝑀(𝑇 − 𝑀𝑠): 𝜉𝑆 = 1−𝜉𝑆0 2 cos { 𝜋 𝜎𝑠𝑐𝑟−𝜎𝑓𝑐𝑟 [𝜎 − 𝜎_𝑓𝑐𝑟− 𝐶𝑀(𝑇 − 𝑀𝑠)]} + 1+𝜉𝑆0 2 (3.18a) 𝜉𝑇 = 𝜉𝑇0− 𝜉𝑇0 1−𝜉𝑆0(𝜉𝑆− 𝜉𝑆0) (3.18b) For 𝑇 < 𝑀𝑠 and 𝜎𝑠𝑐𝑟 < 𝜎 < 𝜎𝑓𝑐𝑟: 𝜉_𝑆 =1−𝜉𝑆0 2 cos [ 𝜋 𝜎𝑠𝑐𝑟−𝜎𝑓𝑐𝑟 (𝜎 − 𝜎_𝑓𝑐𝑟)] +1+𝜉𝑆0 2 (3.19a) 𝜉_𝑇 = 𝜉_𝑇0− 𝜉𝑇0 1−𝜉𝑆0(𝜉𝑆− 𝜉𝑆0) + Δ𝑇𝜀 (3.19b) Where, if 𝑀_𝑓< 𝑇 < 𝑀_𝑠 and 𝑇 < 𝑇₀, Δ_𝑇𝜀 = 1−𝜉𝑇0 2 {cos[𝐴𝑀(𝑇 − 𝑀𝑓)] + 1} (3.20a) Else, Δ𝑇𝜀 = 0 (3.20b) Conversion to Austenite: For 𝑇 > 𝐴_𝑠 and 𝐶_𝐴(𝑇 − 𝐴_𝑓) < 𝜎 < 𝐶_𝐴(𝑇 − 𝐴𝑠): 𝜉 = 𝜉0 2 {cos [𝐴𝐴(𝑇 − 𝐴𝑠− 𝜎 𝐶𝐴)] + 1} (3.21a) 𝜉𝑆 = 𝜉𝑆0− 𝜉𝑆0 𝜉0 (𝜉0− 𝜉) (3.21b) 𝜉_𝑇 = 𝜉_𝑇0−𝜉𝑇0 𝜉0 (𝜉0− 𝜉) (3.21c)

Where 𝐴_𝑀 and 𝐴_𝐴 are given by

𝐴𝑀 = 𝜋 𝑀𝑠− 𝑀𝑓 𝐴𝐴 = 𝜋 𝐴𝑓− 𝐴𝑠

The two more material constants 𝐶_𝐴 and 𝐶_𝑀 are called the stress-influence coefficients (in Fig 3.3), which indicate the influence of stress on the transition transformation, are obtained from the experimental tests. The critical stress values below 𝑀_𝑠 to be constant and denoted by 𝜎_𝑠𝑐𝑟 and 𝜎_𝑓𝑐𝑟 for the critical stresses at the start and finish of the conversion of the martensitic variants (in Fig 3.3).

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CHAPTER 4: L. C. BRINSON’S MATHEMATICAL MODEL

IMPLEMENTATION AND COMPARISON

In Tanaka’s, Liang’s and Roger’s models described above can be used only at temperatures higher than Martensite starting temperature (𝑀_𝑠), because they assume that there is no twinned martensite. Above all Brinson’s model is promising and suggests a one-dimensional model that can describe Shape Memory Effect (SME) and Super Elasticity (SE) simultaneously.

The Brinson’s model separates the martensite volume fraction (𝜉) into two parts to represent the Shape memory effect:

• A twinned martensite 𝜉_𝑇 (Temperature-induced) • A detwinned martensite 𝜉_𝑆 (stress-induced) As mentioned in equation (3.16)

𝜉 = 𝜉_𝑆+ 𝜉_𝑇

According to the Brinson’s journal of “one-dimensional constitutive behaviour of Shape memory alloys”, he mentioned the constitutive law with, constant and non-constant material functions.

4.1 Modification to the constitutive law with constant material functions:

With the introduction of 𝜉 = 𝜉_𝑆+ 𝜉_𝑇 into the constitutive equation (3.3a), it follows immediately from differential calculus that

𝑑𝜎 = 𝜕𝜎 𝜕𝜀𝑑𝜀 + 𝜕𝜎 𝜕𝜉𝑆𝑑𝜉𝑆+ 𝜕𝜎 𝜕𝜉𝑇𝑑𝜉𝑇+ 𝜕𝜎 𝜕𝑇𝑑𝑇 (4.1) Which can be written as

𝑑𝜎 = 𝐷𝑑𝜀 + Ω_𝑆𝑑𝜉_𝑆+ Ω_𝑇𝑑𝜉_𝑇+ ΘdT (4.2)

Assuming the material functions, D, Ω_𝑆, Ω_𝑇, Θ, to be constants. With the initial conditions of (𝜎0, 𝜀0, 𝜉0, 𝑇0, 𝜉𝑆0), solving the differential form of the constitutive equation yields

𝜎 − 𝜎₀ = 𝐷(𝜀 − 𝜀₀) + Ω_𝑆(𝜉_𝑆− 𝜉_𝑆0) + Ω_𝑇(𝜉_𝑇− 𝜉_𝑇0) + Θ(𝑇 − 𝑇₀) (4.3)

Application of the material restriction of the case of maximum residual strain with the material initially 100% austenite, 𝜉_𝑆0 = 0 and 𝜉_𝑇0 = 0, and the remaining conditions (𝜎₀ = 𝜀₀ = 0), (𝜎 = 0, 𝜀 = 𝜀_𝐿, 𝜉_𝑆 = 1, 𝜉_𝑇 = 0) and 𝑇 = 𝑇₀ (𝑀_𝑆 < 𝑇 < 𝐴_𝑆) provides the relationship

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Considering the case of maximum residual strain with the material initially 100% undeformed martensite, 𝜉_𝑆0 = 0 and 𝜉_𝑇0 = 1, and remaining conditions identical to the previous case (except here the temperature constraint can be relaxed to 𝑇 < 𝐴_𝑆) results in the restriction that

Ω_𝑇 = 0 (4.5)

Thus, the thermomechanical constitutive law with constant material functions consistent with the separation of the stress-induced and temperature-induced martensite fractions becomes

𝜎 − 𝜎₀ = 𝐷(𝜀 − 𝜀₀) + Ω(𝜉_𝑆− 𝜉_𝑆0) + Θ(𝑇 − 𝑇₀) (4.6)

Dropping the subscript on Ω_𝑆 so that again Ω = −𝜀_𝐿𝐷. Obviously, this equation is capable of capturing the shape memory effect at all temperatures and with any percentage of initial twinned martensite.

4.2 Modification to the constitutive law with non-constant material

functions:

Assuming the material functions, D, Ω_𝑆, Ω_𝑇, Θ, to be non-constants. Solving the differential form of the constitutive equation yields

𝜎 − 𝜎₀ = 𝐷(𝜀 − 𝜀₀) + Ω_𝑆(𝜉_𝑆− 𝜉_𝑆0) + Ω_𝑇(𝜉_𝑇− 𝜉_𝑇0) + Θ(𝑇 − 𝑇₀)

The constitutive equation with constant material functions is quite trivial to derive from the incremental constitutive equation (4.3). However, the modulus of SMA materials indicates clearly that the young’s modulus, D, has a strong dependence on the martensite fraction of the material, 𝜉. A reasonable assumption for the modulus fraction of an SMA material is

𝐷(𝜀, 𝜉, 𝑇) = 𝐷(𝜉) = 𝐷_𝑎+ 𝜉(𝐷_𝑚− 𝐷_𝑎) (4.7)

Where 𝐷_𝑚 is the modulus value for the SMA as 100% Martensite and 𝐷_𝑎 is the modulus value for the SMA as 100% Austenite. The ratio of the magnitudes of 𝐷𝑎 to 𝐷𝑚 usually have a value of 3 or greater.

Based on 𝐷(𝜉), to obtain a form for the transformation tensor, expand Ω in a Taylor series about 𝜉₀ and neglect higher order terms:

Ω(𝜉) = Ω(𝜉₀) + (𝜉 − 𝜉₀)Ω′_(𝜉

0) (4.8)

Where Ω(𝜉₀) = −𝜀𝐿𝐷(𝜉0), substituting this relationship and equation (4.7) into equation (4.8) and simplifying, one obtains

Ω(𝜉) = −𝜀𝐿𝐷(𝜉0) + (𝜉 − 𝜉0)[−𝜀𝐿𝐷′(𝜉0)] (4.9) Which upon expansion and cancellations of terms reduces to

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Ω(𝜉) = −𝜀_𝐿𝐷_𝑎− 𝜀_𝐿𝜉(𝐷_𝑚− 𝐷_𝑎) (4.10) Or equivalently

Ω(𝜉) = −𝜀_𝐿𝐷(𝜉) (4.11)

The material function Θ(𝜀, 𝜉, 𝑇) is assumed to remain a constant due to its necessarily relatively small value [five orders of magnitude less than D(𝜉). Also, according to the derivation for constant material functions, we assume that Ω_𝑇 = 0.

Utilizing these material function definitions, the general differential form of the constitutive equation analogous to equation (4.2) can be rewritten as

𝑑𝜎 = 𝐷(𝜉)𝑑𝜀 + Ω(𝜉)𝑑𝜉𝑆+ Θ𝑑𝑇 (4.12) Performing a partial integration solution to this differential equation

∫ 𝑑𝜎 = ∫[𝐷𝑎+ 𝜉(𝐷𝑚− 𝐷𝑎)]𝑑𝜀 − ∫ 𝜀𝐿[𝐷𝑎+ 𝜉(𝐷𝑚− 𝐷𝑎)]𝑑𝜉𝑆+ ∫ ΘdT (4.13a) One obtains

𝜎 + 𝐾 = 𝐷_𝑎𝜀 + (𝐷_𝑚− 𝐷_𝑎)𝜉𝜀 + 𝐶(𝜉) − 𝜀𝐿𝐷𝑎𝜉𝑆− 𝜀𝐿(𝐷𝑚− 𝐷𝑎) [ 𝜉𝑆2

2 + 𝜉𝑇𝜉𝑆] + Θ𝑇 (4.13b) Where K is an arbitrary constant and C (𝜉) is an arbitrary function of 𝜉. Rearranging terms and recognizing the expansion

𝐷(𝜉)𝜉𝑆= 𝐷𝑎𝜉𝑆+ (𝐷𝑚− 𝐷𝑎)𝜉𝑆2+ (𝐷𝑚− 𝐷𝑎)𝜉𝑇𝜉𝑆 (4.13c) Equation (4.13b) simplifies to

𝜎 + 𝐾 = 𝐷(𝜉)𝜀 + Ω(𝜉)𝜉_𝑆+ Θ𝑇 + 𝜀_𝐿(𝐷𝑚− 𝐷𝑎) 𝜉𝑆2

2 + 𝐶(𝜉) (4.13d)

Which is the general solution to the governing differential equation (4.12). To obtain a particular solution, first apply initial conditions: equation (4.13d) must hold at the initial state (𝜎₀, 𝜀₀, 𝜉₀, 𝑇₀). Thus, the unknown constant, K, is determined

𝐾 = 𝐷(𝜉0)𝜀0+ Ω(𝜉0)𝜉𝑆0+ Θ𝑇0+ 𝜀𝐿(𝐷𝑚− 𝐷𝑎) 𝜉𝑆2

2 + 𝐶(𝜉0) − 𝜎0 (4.14) And equation (4.13d) becomes

𝜎 − 𝜎₀ = 𝐷(𝜉)𝜀 − 𝐷(𝜉₀)𝜀₀+ Ω(𝜉)𝜉_𝑆− Ω(𝜉₀)𝜉_𝑆0+ Θ(𝑇 − 𝑇₀) + 𝜀_𝐿(𝐷_𝑚− 𝐷_𝑎) [𝜉𝑆2

2 − 𝜉_𝑆02

2 ] + 𝐶(𝜉) − 𝐶(𝜉0) (4.15)

To determine the unknown function 𝐶(𝜉), consider again the specific material restriction of residual strain in shape memory alloys, here we consider the general case of residual strain such that 𝜀_𝑟𝑒𝑠 < 𝜀_𝐿 upon unloading to zero stress. (note that 𝜀_𝑟𝑒𝑠 < 𝜀_𝐿 can also be achieved by unloading the material before 𝜉 achieves a value of 1.)

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Using the definition of maximum residual strain, and the micromechanics concept governing the value of the martensite fraction, it follows that

𝜀𝑟𝑒𝑠= 𝜀𝐿𝜉𝑆 (4.16)

Where the subscript ‘res’ indicates the residual value of strain after unloading to 𝜎 = 0. Applying the simplest case of residual strain to equation (4.15), take an initial state of (𝜎₀ = 𝜀0 = 𝜉𝑆0 = 0) and a final state of (𝜎 = 0, 𝜀 = 𝜀𝑟𝑒𝑠 = 𝜀𝐿𝜉𝑆, 𝜉𝑆) with 𝑇 = 𝑇0 (and consequently 𝜉𝑇 = 𝜉𝑇0). This yield

0 = 𝐷(𝜉)𝜀_𝐿𝜉_𝑆+ Ω(𝜉)𝜉_𝑆+ 𝜀_𝐿(𝐷_𝑚− 𝐷_𝑎)𝜉𝑆 2

2 + 𝐶(𝜉) − 𝐶(0) Which, upon recalling the definition [equation 4.11], implies

𝐶(𝜉) = −𝜀_𝐿(𝐷_𝑚− 𝐷_𝑎)𝜉𝑆2

2 (4.17)

And consequently, the final constitutive equation for shape memory alloy behaviour with material functions that are linear in 𝜉 is

𝜎 − 𝜎0 = 𝐷(𝜉)𝜀 − 𝐷(𝜉0)𝜀0+ Ω(𝜉)𝜉𝑆− Ω(𝜉0)𝜉𝑆0 + Θ(𝑇 − 𝑇0) (4.18)

4.3 Numerical Examples:

The above final constitutive equation (4.18) coupled with Brinson’s transformation equations (3.18 to 3.21) whose were mentioned in the previous chapter, is utilized to calculate the thermomechanical response of shape memory alloys. Stress-strain curves representative of the shape memory effect are given. Since this Brinson’s model is closely based on previous work by Liang and Rogers, the effects are clearly illustrated in their work. In all cases, the numerical results agree well with experimental observations.

The material properties for the shape memory alloy in the following examples are taken from data given by Dye (1990) and Liang (1990) on a Nitinol alloy (𝑁𝑖₅₅𝑇𝑖). The values for the necessary material properties are listed in below table.

Moduli Transformation Temperatures Transformation Constants Maximum Residual Strain 𝐷𝑎 = 67 × 103𝑀𝑃𝑎 𝐷_𝑚 = 26.3 × 103𝑀𝑃𝑎 Θ = 0.55 𝑀𝑃𝑎/℃ 𝑀_𝑓 = 9℃ 𝑀𝑠 = 18.4℃ 𝐴𝑠 = 34.5℃ 𝐴_𝑓 = 49℃ 𝐶_𝑀 = 8 𝑀𝑃𝑎/℃ 𝐶𝐴 = 13.8 𝑀𝑃𝑎/℃ 𝜎𝑠𝑐𝑟 = 100𝑀𝑝𝑎 𝜎_𝑓𝑐𝑟 = 170𝑀𝑝𝑎 𝜀_𝐿 = 0.067

Table 4.1: Material properties for Nitinol alloy used in the following examples [Dye, 1990; Liang, 1990].

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Note that the value for 𝐶_𝑀 is taken from the experimental curves for critical transformation stress given by Dye, which correspond well with figure 3.2, i.e. the curve does not pass through the Martensite start temperature at zero stress. The experimental data indicate a slight increase in the values of critical transformation stress at temperatures below 𝑀_𝑠, but 𝜎_𝑠𝑐𝑟 and 𝜎_𝑓𝑐𝑟 are taken to be constants here. Additionally, although the experiments show a decrease in the maximum residual strain at temperatures above 𝐴_𝑓, this decrease is not considered in these examples.

Figure 4.1: Stress-Strain curves illustrating the shape memory effect.

Constitutive equation (4.18), transformation equations (3.18 to 3.21) and the data from table 4.1 were utilized to calculate the Stress-Strain curves of the shape memory alloy at various temperatures. The results for temperature less than 𝑀𝑠 are grouped together in above figure 4.1. For all of these curves, the initial value of the stress-induced martensite variable is clearly zero.

For temperatures above 𝑀_𝑠, the initial value of the temperature-induced martensite variable was taken to be zero and for temperature less than 𝑀𝑠, the initial values of 𝜉𝑇 were proportional to temperature as indicated by equation (3.20a). with these initial conditions, only the curve for 𝑇 = 5℃ is representative of a fully martensitic specimen before loading. At 𝑇 = 12℃ and 𝑇 = 15℃, the material is partially martensite and partially austenite prior to application of stress, and at all higher temperatures the material is fully austenite.

The subsequent different initial values of the modulus functions are manifested in the difference in slope of the linear loading portion of the stress-strain curves. The slight variation of slope of the unloading portion of the curves in figure 4.1 arises from small

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austenite contributions to the modulus at 𝑇 = 12℃ and 𝑇 = 15℃, since the material has not completely converted to detwinned martensite at the final strain shown here.

Figure 4.2: Stress-Strain curves to maximum residual strain, 𝜀_𝐿.

See figure 4.2 for comparison, where stress-strain curves extending to 100% transformation for two temperatures are given. In figure 4.2, the material transforms completely to detwinned martensite at both temperatures, after which the stress-strain curve again becomes linear with slope of 𝐷_𝑚. Upon unloading the maximum residual strain 𝜀_𝐿 is achieved. Due to the assumed constant values of 𝜎_𝑠𝑐𝑟 and 𝜎_𝑓𝑐𝑟 in this example, the stress-strain curves for materials with 𝜉_𝑇 = 1 prior to loading will be identical regardless of temperature and will coincide with the curve for 𝑇 = 5℃.

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Figure 4.3 demonstrate free strain recovery of the material, in which the material recovers an initial residual strain at zero stress by raising the temperature above 𝐴_𝑓. This is case is then illustrative of the completion of the SME for curves in figure 4.1. the example shown here is of a specimen at 𝑇₀ = 20℃ with 𝜉_𝑇0 = 0.5 and 𝜀₀ = 0.02; 𝜉_𝑆0 is then defined by 𝜀₀/𝜀_𝐿. As the temperature is raised from 𝑇₀, the inverse transformation to austenite begins and the material starts to recover the residual strain at 𝐴𝑠. At 𝐴𝑓 this transformation is complete, and the strain and both martensite fraction variables are zero.

4.4 Comparing the Brinson’s Mathematical model and Implemented

model:

4.4.1 Austenite/Twinned martensite

⟹ De-twinned martensite:

For comparing figure 4.1, first we need to utilize final constitutive equation (4.18) coupled with Brinson’s transformation equations (3.18 to 3.21) whose were mentioned in the previous chapter with numerical data table 4.1.

The stress-strain curves figure 4.1 illustrating the Shape Memory Effect, that means how the strain varies when stress load acts from Austenite to twinned martensite/de-twinned martensite. And this figure 4.1 also represented curves at three different Temperatures, i.e. temperature T at 5℃, 12℃ and 15℃.

So, the conditions 𝑇 < 𝑀_𝑠 and 𝑀_𝑓 < 𝑇 < 𝑀_𝑠 are satisfied for these temperatures, the values for 𝑀𝑠 and 𝑀𝑓 are mentioned in numerical data table 4.1. Now for these temperature curves we should use the 3.19a, 3.19b, 3.20a and 3.20b Transformation equations whose are mentioned in previous chapter and rewritten below.

For 𝑇 < 𝑀_𝑠 and 𝜎_𝑠𝑐𝑟 _{< 𝜎 < 𝜎} 𝑓𝑐𝑟: 𝜉_𝑆 =1−𝜉𝑆0 2 cos [ 𝜋 𝜎𝑠𝑐𝑟−𝜎𝑓𝑐𝑟 (𝜎 − 𝜎_𝑓𝑐𝑟)] +1+𝜉𝑆0 2 𝜉_𝑇 = 𝜉_𝑇0− 𝜉𝑇0 1−𝜉𝑆0(𝜉𝑆− 𝜉𝑆0) + Δ𝑇𝜀 Where, if 𝑀_𝑓< 𝑇 < 𝑀_𝑠 and 𝑇 < 𝑇₀, Δ_𝑇𝜀 = 1−𝜉𝑇0 2 {cos[𝐴𝑀(𝑇 − 𝑀𝑓)] + 1} Else, Δ_𝑇𝜀 = 0

For temperature T= 5℃ curve we can directly calculate 𝜉𝑆 and 𝜉𝑇 where Δ𝑇𝜀 = 0, for T= 12℃ and T=15℃, we should mention the value of Δ_𝑇𝜀.