CHARACTERIZATION AND MODELING OF THE

FERROMAGNETIC SHAPE MEMORY ALLOY Ni-Mn-Ga

FOR SENSING AND ACTUATION

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

By

Neelesh Nandkumar Sarawate, B.E., M.S.

* * * * *

The Ohio State University

2008

Dissertation Committee:

Marcelo Dapino, Adviser

Rajendra Singh

Stephen Bechtel

Rebecca Dupaix

Approved by

Adviser

Graduate Program inMechanical Engineering

c© Copyright by

Neelesh Nandkumar Sarawate

2008

ABSTRACT

Ferromagnetic Shape Memory Alloys (FSMAs) in the Ni-Mn-Ga system are a

recent class of active materials that can generate magnetic field induced strains of

10% by twin-variant rearrangement. This work details an extensive analytical and

experimental investigation of commercial single-crystal Ni-Mn-Ga under quasi-static

and dynamic conditions with a view to exploring the material’s sensing and actua-

tion applications. The sensing effect of Ni-Mn-Ga is experimentally characterized by

measuring the flux density and stress as a function of quasi-static strain loading at

various fixed magnetic fields. A bias magnetic field of 368 kA/m is shown to mark

the transition from irreversible to reversible (pseudoelastic) stress-strain behavior.

At this bias field, a reversible flux-density change of 0.15 Tesla is observed over a

range of 5.8% strain. A constitutive model based on continuum thermodynamics is

developed to describe the coupled magnetomechanical behavior of Ni-Mn-Ga. Me-

chanical dissipation and the microstructure of Ni-Mn-Ga are incorporated through

internal state variables. The constitutive response of the material is obtained by

restricting the process through the second law of thermodynamics. The model is

further modified to describe the actuation and blocked-force behavior under a unified

framework. Blocked-force characterization shows that Ni-Mn-Ga exhibits a block-

ing stress of 1.47 MPa and work capacity of 72.4 kJ/m3. The model requires only

ii

seven parameters which can be obtained from two simple experiments. The model is

physics-based, low-order and is therefore suitable for device and control design.

The behavior of Ni-Mn-Ga under dynamic mechanical and magnetic excitations is

addressed. First, a new approach is presented for modeling dynamic actuators with

Ni-Mn-Ga as a drive element. The constitutive material model is used in conjunction

with models for eddy current loss and lumped actuator dynamics to quantify the

frequency dependent strain-field hysteresis. Second, the magnetization response of

Ni-Mn-Ga to dynamic strain loading of up to 160 Hz is characterized, which shows

the response of Ni-Mn-Ga as a broadband sensor. A linear constitutive equation is

used along with magnetic diffusion to model the dynamic behavior.

Finally, the effect of changing magnetic field on the resonance frequency of Ni-Mn-

Ga is characterized by conducting mechanical base excitation tests. The measured

field induced resonance frequency shift of 35% indicates that Ni-Mn-Ga is well suited

for vibration absorption applications requiring electrically-tunable stiffness.

Ferromagnetic shape memory Ni-Mn-Ga is thus demonstrated as a multi-functional

smart material with possible applications in sensing, actuation, and vibration control

which require large deformation, low force, tunable stiffness and fast response. Other

applications being investigated elsewhere such as energy harvesting further expand

the application potential of Ni-Mn-Ga. The physics-based constitutive model along

with the models for dynamic magnetic and mechanical processes provide a thorough

understanding of the complex magnetomechanical behavior.

iii

ACKNOWLEDGMENTS

I would like to express my sincere gratitude towards my advisor Prof. Marcelo

Dapino, for his continuous guidance, understanding, and patience during my Ph.D.

study. I have thoroughly enjoyed interacting with him during my stay at OSU. This

research would not have been possible without his insightful suggestions, enthusiasm,

and trust in me.

I would also like to thank my dissertation committee, Prof. Rajendra Singh, Prof.

Stephen Bechtel, and Prof. Rebecca Dupaix for their assistance in addressing several

technical issues, thoroughly reviewing my proposal and providing valuable sugges-

tions. The knowledge acquired through their courses has been invaluable towards my

research.

I am grateful to all the colleagues in Smart Materials and Structures Lab, espe-

cially LeAnn Faidley, Xiang Wang, and Phillip Evans for their help in addressing

various experimental and theoretical issues. I am thankful to the Mechanical Depart-

ment staff for their cooperation. I would like to thank the machine shop supervisor,

Gary Gardner, for his help in completing the test setups.

Finally, I would like to thank my parents and brother for their continuous love

and encouragement.

iv

VITA

November 20, 1979 . . . . . . . . . . . . . . . . . . . . . . . . . Born - Pune, India

2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .B.E. Mechanical Engineering,University of Pune, India

2001-2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design EngineerHodek Vibration Technologies, India

2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .M.S. Mechanical EngineeringUniversity of Missouri-Rolla,Rolla MO

2004-present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Graduate Research Associate,The Ohio State UniversityColumbus OH

PUBLICATIONS

Journal Publications

N. Sarawate and M. Dapino, “Characterization and modeling of the dynamic sensingbehavior of Ni-Mn-Ga”, Smart Materials and Structures, Draft in preparation.

N. Sarawate and M. Dapino, “Magneto-mechanical energy model for nonlinear andhysteretic quasi-static behavior of Ni-Mn-Ga”, Journal of Intelligent Material Systemsand Structures, in review.

N. Sarawate and M. Dapino, “Dynamic actuation model for magnetostrictive mate-rials,” Smart Materials and Structures, in review.

N. Sarawate and M. Dapino, “Stiffness tuning using bias fields in ferromagnetic shapememory alloys,” Journal of Intelligent Material Systems and Structures, in review.

v

N. Sarawate and M. Dapino, “Magnetization dependence on dynamic strain in ferro-magnetic shape memory Ni-Mn-Ga,” Applied Physics Letters, Vol. 93(6), p. 062501,2008.

N. Sarawate and M. Dapino, “Magnetic field induced stress and magnetization inmechanically blocked Ni-Mn-Ga,” Journal of Applied Physics. Vol. 103(1), p. 083902,2008.

N. Sarawate and M. Dapino, “Frequency dependent strain-field hysteresis model forferromagnetic shape memory Ni-Mn-Ga,” IEEE Transactions on Magnetics, Vol. 44(5),pp. 566-575, 2008.

N. Sarawate and M. Dapino, “Continuum thermodynamics model for the sensing ef-fect in ferromagnetic shape memory Ni-Mn-Ga,” Journal of Applied Physics, Vol. 101(12), p. 123522, 2007.

N. Sarawate and M. Dapino, “Experimental characterization of the sensor effect in fer-romagnetic shape memory Ni-Mn-Ga,” Applied Physics Letters, Vol. 88(1), p. 121923,2006.

Conference Publications

N. Sarawate, and M. Dapino, “Characterization and modeling of dynamic sensingbehavior of ferromagnetic shape memory alloys,” Proceedings of ASME Conference onSmart Materials, Adaptive Structures and Intelligent Systems, Paper #656, EllicottCity, MD, October 2008.

N. Sarawate, and M. Dapino, “Dynamic strain-field hysteresis model for ferromagneticshape memory Ni-Mn-Ga,” Proceedings of SPIE Smart Structures and Materials,Vol. 6929, p. 69291R, San Diego, CA, March 2008.

N. Sarawate, and M. Dapino, “Electrical stiffness tuning in ferromagnetic shape mem-ory Ni-Mn-Ga,” Proceedings of SPIE Smart Structures and Materials, Vol. 6529,p. 652916, San Diego, CA, March 2007.

N. Sarawate, and M. Dapino, “Magnetomechanical characterization and unified mod-eling of Ni-Mn-Ga,” Proceedings of SPIE Smart Structures and Materials, Vol. 6526,p. 652629, San Diego, CA, March 2007.

vi

N. Sarawate, and M. Dapino, “A thermodynamic model for the sensing behavior of fer-romagnetic shape memory Ni-Mn-Ga,” Proceedings of ASME IMECE, Paper #14555,Chicago, IL, November 2006.

N. Sarawate, and M. Dapino, “Sensing behavior of ferromagnetic shape memoryNi-Mn-Ga,” Proceedings of SPIE Smart Structures and Materials,” Vol. 6170, pp.61701B, San Diego, CA, February 2006.

FIELDS OF STUDY

Major Field: Mechanical Engineering

Studies in:

Smart Materials and Structures Prof. DapinoApplied Mechanics Prof. Dapino, Prof. Bechtel, Prof. DupaixSystem Dynamics and Vibrations Prof. Dapino, Prof. Singh

vii

TABLE OF CONTENTS

Page

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Chapters:

1. Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . 11.2 Overview of Smart Materials . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Ferroelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Magnetostrictives . . . . . . . . . . . . . . . . . . . . . . . . 81.2.3 Shape Memory Alloys . . . . . . . . . . . . . . . . . . . . . 9

1.3 Ferromagnetic Shape Memory Alloys . . . . . . . . . . . . . . . . . 141.3.1 Early Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.2 Properties and Crystal Structure . . . . . . . . . . . . . . . 181.3.3 Magnetocrystalline Anisotropy . . . . . . . . . . . . . . . . 191.3.4 Strain Mechanism . . . . . . . . . . . . . . . . . . . . . . . 20

1.4 Literature Review on Ni-Mn-Ga . . . . . . . . . . . . . . . . . . . . 221.4.1 Sensing Behavior . . . . . . . . . . . . . . . . . . . . . . . . 231.4.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.4.3 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . 30

1.5 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 331.6 Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 33

viii

1.6.1 Quasi-static Behavior . . . . . . . . . . . . . . . . . . . . . 341.6.2 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . 35

2. Characterization of the Sensing Effect . . . . . . . . . . . . . . . . . . . 37

2.1 Electromagnet Design and Construction . . . . . . . . . . . . . . . 382.1.1 Magnetic Circuit . . . . . . . . . . . . . . . . . . . . . . . . 392.1.2 Electromagnet Construction and Calibration . . . . . . . . . 42

2.2 Experimental Characterization . . . . . . . . . . . . . . . . . . . . 452.2.1 Stress-Strain Behavior . . . . . . . . . . . . . . . . . . . . . 462.2.2 Flux Density Behavior . . . . . . . . . . . . . . . . . . . . . 50

2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.3.1 Magnetic Field Induced Stress and Flux Density Recovery . 532.3.2 Optimum Bias Field for Sensing . . . . . . . . . . . . . . . 57

3. Constitutive Model for Coupled Magnetomechanical Behavior of SingleCrystal Ni-Mn-Ga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1 Thermodynamic Framework . . . . . . . . . . . . . . . . . . . . . . 623.2 Incorporation of the Ni-Mn-Ga Microstructure in the Thermody-

namic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3 Energy Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.3.1 Magnetic Energy . . . . . . . . . . . . . . . . . . . . . . . . 703.3.2 Mechanical Energy . . . . . . . . . . . . . . . . . . . . . . . 73

3.4 Evolution of Domain Fraction and Magnetization Rotation Angle . 753.5 Evolution of Volume Fraction . . . . . . . . . . . . . . . . . . . . . 783.6 Sensing Model Results . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.6.1 Stress-Strain Results . . . . . . . . . . . . . . . . . . . . . . 803.6.2 Flux Density Results . . . . . . . . . . . . . . . . . . . . . . 823.6.3 Thermodynamic Driving Force and Volume Fraction . . . . 87

3.7 Extension to Actuation Model . . . . . . . . . . . . . . . . . . . . . 893.7.1 Actuation Model . . . . . . . . . . . . . . . . . . . . . . . . 923.7.2 Actuation Model Results . . . . . . . . . . . . . . . . . . . 95

3.8 Blocked Force Model . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.8.1 Results of Blocked-Force Behavior . . . . . . . . . . . . . . 105

3.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4. Dynamic Actuator Model for Frequency Dependent Strain-Field Hysteresis 112

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.2 Magnetic Field Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2.1 Diffused Average Field . . . . . . . . . . . . . . . . . . . . . 120

ix

4.3 Quasistatic Strain-Field Hysteresis Model . . . . . . . . . . . . . . 1224.4 Dynamic Actuator Model . . . . . . . . . . . . . . . . . . . . . . . 126

4.4.1 Discrete Actuator Model . . . . . . . . . . . . . . . . . . . . 1284.4.2 Fourier Series Expansion of Volume Fraction . . . . . . . . . 1304.4.3 Results of Dynamic Actuation Model . . . . . . . . . . . . . 1344.4.4 Frequency Domain Analysis . . . . . . . . . . . . . . . . . . 138

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1444.6 Dynamic Actuation Model for Magnetostrictive Materials . . . . . 145

5. Dynamic Sensing Behavior: Frequency Dependent Magnetization-StrainHysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.1 Experimental Characterization of Dynamic Sensing Behavior . . . . 1575.2 Model for Frequency Dependent Magnetization-Strain Hysteresis . 1645.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6. Stiffness and Resonance Tuning With Bias Magnetic Fields . . . . . . . . 171

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.2 Experimental Setup and Procedure . . . . . . . . . . . . . . . . . . 1736.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.4.1 Longitudinal Field Tests . . . . . . . . . . . . . . . . . . . . 1826.4.2 Transverse field Tests . . . . . . . . . . . . . . . . . . . . . 185

6.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1967.1.1 Quasi-static Behavior . . . . . . . . . . . . . . . . . . . . . 1967.1.2 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . 1997.1.3 Characterization Map . . . . . . . . . . . . . . . . . . . . . 201

7.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2027.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

7.3.1 Possible Improvements . . . . . . . . . . . . . . . . . . . . . 2047.3.2 Future Research Opportunities . . . . . . . . . . . . . . . . 205

Appendices:

A. Miscellaneous Issues with Quasi-static Characterization and Modeling . . 206

A.1 Electromagnet Design and Calibration . . . . . . . . . . . . . . . . 206A.1.1 Effect of Dimensions on Field . . . . . . . . . . . . . . . . . 206

x

A.1.2 Electromagnet Calibration with Sample . . . . . . . . . . . 214A.2 Verification of Demagnetization Factor . . . . . . . . . . . . . . . . 218A.3 Damping Properties of Ni-Mn-Ga . . . . . . . . . . . . . . . . . . . 225A.4 Magnetization Angles . . . . . . . . . . . . . . . . . . . . . . . . . 229

B. Miscellaneous Issues with Dynamic Characterization and Modeling . . . 231

B.1 Jiles-Atherton Model . . . . . . . . . . . . . . . . . . . . . . . . . . 231B.2 Kelvin Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235B.3 Prototype Device for Ni-Mn-Ga Sensor . . . . . . . . . . . . . . . . 236

C. Model Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

C.1 Quasi-static Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 238C.1.1 Model Flowchart . . . . . . . . . . . . . . . . . . . . . . . . 238C.1.2 Sensing Model Code . . . . . . . . . . . . . . . . . . . . . . 239C.1.3 Actuation Model Code . . . . . . . . . . . . . . . . . . . . . 244

C.2 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247C.2.1 Dynamic Actuator Model . . . . . . . . . . . . . . . . . . . 247C.2.2 Dynamic Sensing Model . . . . . . . . . . . . . . . . . . . . 254C.2.3 Jiles-Atherton Model . . . . . . . . . . . . . . . . . . . . . . 257

D. Test Setup Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

D.1 Electromagnet Drawings (Figures D.1-D.6) . . . . . . . . . . . . . . 260D.2 Dynamic Sensing Device Drawings (Figures D.7-D.15) . . . . . . . 260

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

xi

LIST OF TABLES

Table Page

1.1 Overview of transduction principles in smart materials. . . . . . . . . 6

6.1 Summary of longitudinal field test results. Units: fn: (Hz), Ks: (N/m) 185

6.2 Summary of transverse field test results. Units: fn: (Hz), Ks: (N/m) . 188

xii

LIST OF FIGURES

Figure Page

1.1 Comparison of FSMAs with other classes of smart materials. . . . . . 2

1.2 Joule magnetostriction produced by a magnetic field H. (a) H is ap-proximately proportional to the current i that passes through thesolenoid when a voltage is applied to it, (b) the rotation of magneticdipoles changes the length of the sample, (c) and (d) curves M vs. Hand ∆L/L vs. H obtained by varying the field sinusoidally [20]. . . . 10

1.3 SMA transformation between high and low temperature phases. . . . 11

1.4 Schematic of phase transformation. . . . . . . . . . . . . . . . . . . . 12

1.5 Stress-strain behavior of shape memory alloys (a) below Mf , (b) aboveAf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 (a) Relative orientation of sample, strain gauge, and applied field formeasurements shown in (b) and (c). (b) Strain vs applied field in theL21 (austenite) phase at 283 K. (c) Same as (b) but data taken at265 K in the martensitic phase [128]. . . . . . . . . . . . . . . . . . . 16

1.7 Ni-Mn-Ga crystal structure (a) Cubic Heusler structure, (b) Tetrag-onal structure, under the martensite finish temperature. Blue: Ni,Red: Mn, Green: Ga. . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.8 Schematic of strain mechanism in Ni-Mn-Ga FSMA under transversefield and longitudinal stress. . . . . . . . . . . . . . . . . . . . . . . . 21

2.1 Schematic of the electromagnet. Two E-shaped legs form the flux pathindicated by arrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

xiii

2.2 Magnetic circuit of the electromagnet. . . . . . . . . . . . . . . . . . 42

2.3 Finite element analysis of the electromagnet. . . . . . . . . . . . . . . 43

2.4 Electromagnet calibration curve. . . . . . . . . . . . . . . . . . . . . . 44

2.5 Experimental setup for quasi-static sensing characterization. . . . . . 47

2.6 Stress vs. strain plots at varied bias fields. . . . . . . . . . . . . . . . 48

2.7 Flux density vs. strain at varied bias fields. . . . . . . . . . . . . . . . 52

2.8 Flux density vs. stress at varied bias fields. . . . . . . . . . . . . . . . 52

2.9 Schematic of loading and unloading at low magnetic fields. . . . . . . 55

2.10 Schematic of loading and unloading at high magnetic fields. . . . . . 56

2.11 Variation of flux-density change with bias field. . . . . . . . . . . . . 58

2.12 Easy and hard-axis flux-density curves of Ni-Mn-Ga. . . . . . . . . . 59

3.1 Simplified two-variant microstructure of Ni-Mn-Ga. . . . . . . . . . . 67

3.2 Image of twin-variant Ni-Mn-Ga microstructure by Scanning electronmicroscope [39]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3 Schematic of stress-strain curve at zero bias field. . . . . . . . . . . . 75

3.4 Variation of (a) domain fraction, and (b) rotation angle with appliedfield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.5 Stress vs strain plots at varied bias fields. Dotted line: experiment;solid line: calculated (loading); dashed line: calculated (unloading). . 81

3.6 Variation of twinning stress with applied bias field. . . . . . . . . . . 83

3.7 Model results for (a) flux density-strain and (b) flux density-stresscurves. Dotted line: experiment; solid line: calculated (loading);dashed line: calculated (unloading). . . . . . . . . . . . . . . . . . . . 85

xiv

3.8 Variation of sensitivity factor with applied bias field. . . . . . . . . . 86

3.9 Model results for easy and hard axis curves. (a) flux-density vs. field(b) magnetization vs. field. . . . . . . . . . . . . . . . . . . . . . . . 88

3.10 Evolution of thermodynamic driving forces. . . . . . . . . . . . . . . 90

3.11 Evolution of volume fraction. . . . . . . . . . . . . . . . . . . . . . . 91

3.12 Variation of residual strain with applied bias field. . . . . . . . . . . 92

3.13 Strain vs applied field at varied bias stresses. Dotted line: experiment;solid line: calculated (loading); dashed line: calculated (unloading). . 97

3.14 Variation of maximum MFIS with bias stress. . . . . . . . . . . . . . 98

3.15 Variation of the coercive field with bias stress. . . . . . . . . . . . . . 100

3.16 Magnetization vs applied field at varied bias stresses. Dotted line:experiment; solid line: calculated (loading); dashed line: calculated(unloading). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.17 Stress vs field at varied blocked strains. Dotted: experiment; solid line:model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.18 Magnetization vs field at varied blocked strains. Dashed line: experi-ment; solid line: model. . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.19 Variation of initial susceptibility with biased blocked strain. . . . . . 107

3.20 Experimental blocking stress σbl, minimum stress σ0, and availableblocking stress σbl − σ0 vs. bias strain. . . . . . . . . . . . . . . . . . 109

4.1 Flow chart for modeling of dynamic Ni-Mn-Ga actuators. . . . . . . 116

4.2 Dynamic actuation data by Henry [48] for (a) 2 − 100 Hz (fa = 1 −50 Hz) and (b) 100− 500 Hz (fa = 50− 250 Hz). . . . . . . . . . . . 117

4.3 Magnetic field variation inside the sample at varied depths for (a) si-nusoidal input and (b) triangular input. x = d represents the edge ofthe sample, x = 0 represents the center. . . . . . . . . . . . . . . . . 121

xv

4.4 Average field waveforms with increasing actuation frequency for (a)sinusoidal input and (b) triangular input. . . . . . . . . . . . . . . . 123

4.5 Dependence of normalized field amplitude on position with increasingactuation frequency for (a) sinusoidal input and (b) triangular input. 124

4.6 Model result for quasistatic strain vs. magnetic field. The circlesdenote experimental data points (1 Hz line in Figure 4.2) while the solidand dashed lines denote model simulations for ˙|H| > 0 and ˙|H| < 0,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.7 Dynamic Ni-Mn-Ga actuator consisting of an active sample (spring)connected in mechanical parallel with an external spring and damper.The mass includes the dynamic mass of the sample and the actuator’soutput pushrod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.8 Volume fraction profile vs. time (fa = 1 Hz). . . . . . . . . . . . . . 131

4.9 Single sided frequency spectrum of volume fraction (fa = 1 Hz). . . 132

4.10 Model results for strain vs. applied field at different frequencies for (a)sinusoidal, (b) triangular input waveforms. Dotted line: experimental,solid line: model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.11 (a) Normalized maximum strain vs. Frequency (b) Hysteresis loop areavs. Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.12 Model results for strain vs. applied field in frequency domain for tri-angular input waveform for (a) fa = 50 Hz, (b) fa = 100 Hz, (c) fa =150 Hz, (d) fa = 175 Hz, (e) fa = 200 Hz, (e) fa = 250 Hz. Dottedline: experimental, solid line: model. . . . . . . . . . . . . . . . . . . 139

4.13 (a) Strain magnitude vs. harmonic order, (b) Phase angle vs. harmonicorder at varied actuation frequencies. . . . . . . . . . . . . . . . . . . 141

4.14 (a) Strain magnitude vs. actuation frequency, (b) Phase angle vs.actuation frequency at varied harmonic orders. . . . . . . . . . . . . . 143

4.15 Variation of maximum strain and field with actuation frequency. . . . 144

xvi

4.16 Normalized average field vs. non-dimensional time. . . . . . . . . . . 149

4.17 Dynamic magnetostrictive actuator. . . . . . . . . . . . . . . . . . . 150

4.18 Strain vs. applied field at varied actuation frequencies. Dashed line:experimental, solid line: model. . . . . . . . . . . . . . . . . . . . . . 152

4.19 Frequency domain strain magnitudes at varied actuation frequencies.Dashed line: experimental, solid line: model. . . . . . . . . . . . . . 154

4.20 Variation of (a) magnitude and (b) phase of the first harmonic. . . . 156

5.1 Experimental setup for dynamic magnetization measurements. . . . 158

5.2 (a) Stress vs. strain and (b) flux-density vs. strain measurements forfrequencies of up to 160 Hz. . . . . . . . . . . . . . . . . . . . . . . . 161

5.3 Hysteresis loss with frequency for stress-strain and flux-density strainplots. The plots are normalized with respect to the strain amplitudeat a given frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.4 Scheme for modeling the frequency dependencies in magnetization-strain hysteresis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5 Model results: (a) Internal magnetic field vs. time at varying depthfor the case of 140 Hz strain loading (sample dim:±d), (b) Averagemagnetic field vs. time at varying frequencies, and (c) Flux-density vs.strain at varying frequencies. . . . . . . . . . . . . . . . . . . . . . . . 168

6.1 Left: simplified 2-D twin variant microstructure of Ni-Mn-Ga. Center:microstructure after application of a sufficiently high transverse mag-netic field. Right: after application of a sufficiently high longitudinalfield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.2 Schematic of the longitudinal field test setup. . . . . . . . . . . . . . 176

6.3 Schematic of the transverse field test setup. . . . . . . . . . . . . . . 176

6.4 DOF spring-mass-damper model used for characterization of the Ni-Mn-Ga material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

xvii

6.5 Experimentally obtained acceleration PSDs. . . . . . . . . . . . . . . 180

6.6 Transfer function between top and base accelerations. . . . . . . . . . 181

6.7 Acceleration transmissibility with longitudinal field. . . . . . . . . . . 184

6.8 Longitudinal field test model results and repeated measurements underthe same field inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.9 Transmissibility ratio measurements with transverse field configuration. 186

6.10 Additional measurements of transmissibility ratio with transverse fieldconfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.11 Variation of damping ratio with initial transverse bias field. . . . . . . 189

6.12 Variation of viscous damping coefficient with initial transverse bias field.190

6.13 Variation of resonance frequency with initial transverse bias field. . . 190

6.14 Variation of stiffness with initial bias field. . . . . . . . . . . . . . . . 192

7.1 Characterization map of Ni-Mn-Ga. Plain blocks in “Experiment” and“Modeling” rows show the new contribution of the work; Light grayblocks show that a limited prior work existed, which was completelyaddressed in this research; Dark gray blocks indicate that prior workwas available, and no new contribution was made. . . . . . . . . . . . 201

A.1 Schematic of the Electromagnet. . . . . . . . . . . . . . . . . . . . . . 207

A.2 Effect of ratio (d/D) on field. . . . . . . . . . . . . . . . . . . . . . . 208

A.3 Effect of angle (Φ) on field. . . . . . . . . . . . . . . . . . . . . . . . . 208

A.4 Variation of current density with field. . . . . . . . . . . . . . . . . . 209

A.5 Comparison of various wire sizes. . . . . . . . . . . . . . . . . . . . . 211

A.6 Comparison of current carrying capacity, possible turns and MMF pro-duced by various wires (The current and turns are multiplied by scalingfactors) Wire size AWG 16 is seen as an optimum size. . . . . . . . . 213

xviii

A.7 Picture of the assembled electromagnet. . . . . . . . . . . . . . . . . 214

A.8 Electromagnet calibration curve in presence of sample, the easy axiscurve shows maximum variation. . . . . . . . . . . . . . . . . . . . . 216

A.9 Schematic of the demagnetization field inside the sample. The appliedfield (H) creates a magnetization (M) inside the sample, which resultsin north and south poles on its surface. H and M are shown by solidarrows. The demagnetization field (Hd = NxM) is directed from northto south poles as shown by dashed arrows. Although inside the sam-ple, the demagnetization field opposes the applied field, it adds to theapplied field outside the sample. Therefore, the net field inside thesample is given as H −NxM , whereas the net field outside the sampleis given as H + NxM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

A.10 A snapshot from COMSOL simulation. . . . . . . . . . . . . . . . . . 220

A.11 Magnetic field vs distance. Solid: COMSOL, Dashed: recalculated. . 222

A.12 Flux density vs distance. Solid: COMSOL, Dashed: recalculated. . . 222

A.13 Magnetization. Solid: COMSOL, Dashed: recalculated. . . . . . . . . 223

A.14 Energy absorbed in the stress-strain curves of Ni-Mn-Ga. . . . . . . . 227

A.15 Damping capacity as a function of bias field. . . . . . . . . . . . . . . 227

A.16 Variation of tan δ with magnetic bias field. . . . . . . . . . . . . . . . 228

A.17 Schematic of Ni-Mn-Ga microstructure assuming four different anglesin the four regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

B.1 Magnetization vs. field using Jiles model. . . . . . . . . . . . . . . . . 234

B.2 Magnetostriction vs. field using Jiles model. . . . . . . . . . . . . . . 234

B.3 Kelvin functions (a) ber(x) and bei(x). . . . . . . . . . . . . . . . . . 235

B.4 Prototype device for Ni-Mn-Ga sensor. . . . . . . . . . . . . . . . . . 237

xix

C.1 Flowchart of the sensing model for loading case (ξ < 0). . . . . . . . . 238

D.1 E-shaped laminates for electromagnet. . . . . . . . . . . . . . . . . . 261

D.2 Plate for mounting electromagnet. . . . . . . . . . . . . . . . . . . . . 262

D.3 Holding plates for electromagnet. . . . . . . . . . . . . . . . . . . . . 263

D.4 Base channels for mounting electromagnet. . . . . . . . . . . . . . . . 264

D.5 Bottom pushrod for applying compression using MTS machine. . . . 265

D.6 Top pushrod for applying compression using MTS machine. . . . . . . 266

D.7 2-D view of the assembled device. . . . . . . . . . . . . . . . . . . . . 267

D.8 Bottom plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

D.9 Top plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

D.10 Side plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

D.11 Support disc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

D.12 Disc to adjust the compression of spring. . . . . . . . . . . . . . . . . 271

D.13 Seismic mass (material: brass). . . . . . . . . . . . . . . . . . . . . . 272

D.14 Plate to secure magnets (2 nos). . . . . . . . . . . . . . . . . . . . . . 272

D.15 Grip to hold the sample (2 nos). . . . . . . . . . . . . . . . . . . . . . 273

xx

CHAPTER 1

INTRODUCTION AND LITERATURE REVIEW

1.1 Introduction and Motivation

Ferromagnetic Shape Memory Alloys (FSMAs) in the nickel-manganese-gallium

(Ni-Mn-Ga) system are a recent class of smart materials that have generated great

research interest because of their ability to produce large strains of up to 10% in the

presence of magnetic fields. This strain magnitude is around 100 times larger than

that exhibited by other smart materials such as piezoelectrics and magnetostrictives.

Due to the magnetic field activation, FSMAs exhibit faster response than the ther-

mally activated Shape Memory Alloys (SMAs). The combination of large strains and

fast response gives FSMAs a unique advantage over other smart materials. As seen in

Figure 1.1, FSMAs bridge the gap between various existing classes of smart materials.

Ni-Mn-Ga FSMAs therefore open up opportunities for various possible applications

such as sonar transducers, structural morphing, energy harvesting, motion/force sens-

ing, vibration control, etc. However, Ni-Mn-Ga FSMAs are still relatively new, and

their behavior and mechanics are not fully understood. Also, most of the aforemen-

tioned applications can be classified into two fundamental behaviors: sensing and

1

Figure 1.1: Comparison of FSMAs with other classes of smart materials.

actuation. Actuation refers to the application of magnetic field to generate deforma-

tion (strain), whereas sensing refers to the application of mechanical input (stress or

strain) to alter the magnetization of the material. If these two behaviors are thor-

oughly studied in various static and dynamic conditions, it will lead to a significant

advancement in the state of the art of this technology.

Because of the ability of Ni-Mn-Ga to generate large strains under magnetic fields,

most of the prior work has been focused on the experimental characterization and

modeling of the actuation effect. The characterization of the actuator effect is usually

conducted by subjecting the material to magnetic fields created with an electromag-

net, which results in the generation of displacement that can be measured by a suit-

able sensor. Early challenges in conducting the actuation characterization involved

2

construction of electromagnets that can apply the large magnetic field required to

saturate the material to generate maximum strain. Typically the magnetic field is

applied in the presence of a constant compressive (bias) stress. This bias stress is

used to restore the original configuration of the material when the magnetic field is

removed. Ni-Mn-Ga exhibits a low-blocking stress (≈ 3 MPa), which can limit its

actuation authority. Investigation of applications other than actuation is necessary

to fully understand the capabilities of Ni-Mn-Ga FSMAs.

The sensing effect has received limited attention. Although a few prior studies

have shown the ability of Ni-Mn-Ga to respond to mechanical inputs by magnetiza-

tion change, a comprehensive characterization under a wide range of inputs and bias

variables is lacking. Development of models that can describe the macroscopic behav-

ior of the material in sensing mode is also required. The presented work will provide

a physics-based model that describes the coupled magnetomechanical behavior of the

material in sensing mode.

A major advantage of FSMAs over thermal SMAs is their fast response, or high

operating frequency. Even so, most of the prior work on Ni-Mn-Ga is focused on quasi-

static behavior. While experimental work on dynamic actuation does exist, there are

no models to describe the frequency-dependent behavior of a dynamic Ni-Mn-Ga

actuator. The applications of Ni-Mn-Ga as a dynamic sensor and as a vibration

absorber have not been fully explored. Understanding the dynamic behavior of Ni-

Mn-Ga is required to realize its potential as a dynamic actuator, sensor or a vibration

absorber.

The presented research addresses various unresolved aspects with the modeling

and characterization of commercial quality single crystal Ni-Mn-Ga in quasi-static

3

and dynamic conditions. In the quasi-static part, experimental characterization of

the sensing effect of Ni-Mn-Ga is conducted. A magnetomechanical test setup is

developed to conduct the characterization. Further, a continuum thermodynamics

based energy model is developed to describe the sensing behavior of Ni-Mn-Ga. The

thermodynamic framework is extended to also describe the actuation and blocked-

force behaviors, thus fully describing the non-linear and hysteretic constitutive re-

lationships in Ni-Mn-Ga. In the dynamic part, study of Ni-Mn-Ga under dynamic

mechanical and magnetic excitation is conducted. To model the strain dependence on

dynamic fields (magnetic excitation), the constitutive actuation model is augmented

with magnetic field diffusion and system-level structural dynamics. The dynamic

mechanical excitation includes two characterizations: dynamic sensing and tunable

stiffness. Dynamic sensing characterization is conducted by altering the magneti-

zation of Ni-Mn-Ga by subjecting it to cyclic strain loading at frequencies of up

to 160 Hz. The stiffness of Ni-Mn-Ga is characterized under varied collinear and

transverse magnetic field drive configurations, to illustrate its viability for tunable

vibrations absorption applications.

This chapter reviews existing state of the art on Ni-Mn-Ga. An overview of

various smart materials is presented. Properties of Ni-Mn-Ga FSMAs are discussed

and the active strain mechanism is introduced. The details of prior experimental work

on the sensing behavior of Ni-Mn-Ga are reviewed, followed by a review of various

approaches to model the coupled magnetomechanical quasi-static behavior. Finally,

the characterization and modeling of the dynamic behavior of Ni-Mn-Ga is reviewed,

which includes dynamic actuation (frequency dependent strain-field hysteresis) and

stiffness tuning under varied bias fields.

4

1.2 Overview of Smart Materials

A smart material is an engineered substance that converts one form of input en-

ergy into different form of output energy. These “active” or “smart” materials can

react with a change in dimensional, electrical, elastic, magnetic, thermal or rheological

properties to external stimuli such as heat, electric or magnetic field, stress and light.

In most operating regimes, smart materials have the ability to recover the original

shape and properties when the external driving input is removed which makes them

suitable candidates for use in actuator and sensor applications. Smart materials can

be broadly categorized into several classes based on the type of driving input and

the phenomenon by which the response is produced: piezoelectric, electrostrictive,

magnetostrictive, electrorheological and magnetorheological, shape memory, and fer-

romagnetic shape memory. In general, all of these smart materials are transducers,

and they convert energy from one form to another. The smart materials have poten-

tial to replace conventional hydraulic and pneumatic actuators. Table 1.1 shows the

transduction principles or the effects that couple one domain to another.

Smart materials have been widely utilized in various commercial sensors and actu-

ators. Major advantages of smart material actuators and sensors include high energy

density, fast response, compact size, and less-moving parts. The disadvantages of

these materials are limited strain outputs, limited blocking forces, high cost and sen-

sitivity to harsh environmental conditions.

5

Output/Input

Charge,Current

Magneticfield

Strain Temperature Light

Electricfield

Permittivity,Conductivity

Electro-magnetism

ConversePiezo Effect

ElectroCaloricEffect

Electro Op-tic Effect

Magneticfield

Mag-electEffect

Permeability Magneto-striction

MagnetoCaloric effect

Magneto Op-tic Effect

Stress Piezo-electricEffect

Piezo-magneticEffect

Compliance - Photo Elas-tic Effect

Heat PyroelectricEffect

- Thermal Ex-pansion

Specific Heat -

Light PhotovoltaicEffect

- Photostriction - RefractiveIndex

Table 1.1: Overview of transduction principles in smart materials.

1.2.1 Ferroelectrics

Ferroelectric materials constitute a class of smart materials that exhibit coupling

between the mechanical and electrical domains. Piezoelectrics are the most com-

monly known examples of ferroelectric class. Piezoelectric materials produce strains

of up to 0.1% (PZT) and 0.07% (PVDF) when exposed to an electric field [112]

and also produce a voltage when subjected to an applied stress. They have found

numerous applications as both actuators and sensors. Piezoelectric devices are also

known for their high frequency capability; this technology is often used in ultrasonic

applications [70]. Microscopically, piezoelectric materials are characterized by hav-

ing an off-center charged ion in a tetragonal unit cell which can be moved from one

axis to another through the application of an electric field or stress [112]. As the

ion changes position, it causes strain in the material due to the electromechanical

coupling. In order for bulk strain to occur, these materials are generally polarized.

Typical piezoelectric materials, PZT and PVDF, are generally employed in stacks,

6

where the strain amplitude is amplified by placing many devices in series and in bi-

morphs and THUNDER actuators where the strain is amplified through the elastic

structure to which the active material is attached. In general, piezoelectrics are char-

acterized as a moderate force, low stroke, solid state device. For actuation, excitation

voltages required to energize these materials can be as high as 1-2 kV, although 100 V

is typical. Because piezoelectrics have high energy density, operate over wide band-

widths, and are easy to incorporate into structures, they are a good candidate for

smart actuation. Piezoelectric materials also find wide applications as sensors, for

example in accelerometers.

Electrostrictive materials are similar to piezoelectrics in terms of the operating

principle, but they typically generate larger strains (0.1%), and are highly nonlinear

and hysteretic. They require higher fields to generate the saturating strain, and

have stringent temperature requirements. Furthermore, only a unidirectional strain

is possible as the strain depends on the magnitude of the electric field, and not

the polarity. All ferroelectric materials typically exhibit a domain structure and

a spontaneous polarization, when cooled below the Curie temperature. When an

electric field is applied to the material, the domains tend to align along the direction

of applied field, resulting in the strain generation. Single crystal materials exhibit

higher energy density and large strain, whereas polycrystalline materials exhibit lesser

strain and higher hysteresis. But, polycrystalline materials are significantly cheaper

and easy to manufacture than the single crystal materials.

7

1.2.2 Magnetostrictives

Magnetostrictive materials are similar to Ferromagnetic Shape Memory Alloys

(FSMAs) in that they both strain when exposed to a magnetic field and both pro-

duce a change in magnetization when a stress is applied. However, the mechanism

responsible for these phenomena is distinctly different for the two materials. Giant-

magnetostrictive materials such as Terfenol-D and Galfenol have strong spin-orbit

coupling. Thus, when an applied magnetic field rotates the spins, the orbital mo-

ments rotate and considerable distortion of the crystal lattice occurs resulting in

large macroscopic strains [20]. A diagram of this strain mechanism is shown in Fig-

ure 1.2. The magnetostrictive material is usually pre-compressed in order to orient

the magnetic moments perpendicular to its longitudinal axis. When a longitudinal

magnetic field is applied to the material, the magnetic moments tend to align along

the direction of the field. This results in orientation of the domains in longitudinal

direction, which results in the strain generation. The strain is approximately pro-

portional to the square of magnetization, which results in butterfly curves that give

two strain cycles per magnetization and field cycle. The magnetostrictive materials

respond to the applied mechanical stress by producing a change in their magnetiza-

tion, which can be detected by measuring the induced voltage in a pickup coil or a

suitable magnetic sensor. This ‘inverse’ phenomenon is termed as Villari effect.

Since the magnetostriction of Terfenol-D is dependent on the magnetization vec-

tors turning away from their preferred direction, it can be understood that mag-

netostriction depends on a relatively low value of magnetic anisotropy whereas the

opposite is a requirement for FSMAs. Terfenol-D achieves maximum strains of around

0.12% and can be operated for frequencies of up to 10 kHz [43] including a Delta-E

8

effect [63] similar to that discussed for Ni-Mn-Ga in Chapter 6. Some of the disadvan-

tages of Terfenol-D are that it is relatively expensive to produce and is highly brittle.

A similar material, Galfenol, which is easier to produce and has higher strength is

gaining in popularity. Galfenol can produce 0.03% strain [64] and is machinable with

common techniques [13]. Both of these materials are commonly employed in solenoid

based actuators as opposed to Ni-Mn-Ga actuators that consist of an electromagnet.

Magnetostrictives have found applications as actuators and sensors in a broad range

of fields including industry, bio-medicine, and defense [20].

1.2.3 Shape Memory Alloys

Shape Memory Alloys (SMAs) are alloys that undergo significant deformation

at low temperatures and retain this deformation until they are heated [130]. In

comparison to piezoelectric and magnetostrictive materials, SMAs have the advantage

of generating significantly large strains of around 10%. SMAs produce strain by a

similar mechanism as that in the FSMAs. Thus, an in-depth review of these materials

is useful from the viewpoint of understanding the behavior of Ni-Mn-Ga FSMAs.

At high temperatures, SMAs such as Nickel-Titanium (Ni-Ti) alloy exhibits a body

centered cubic austenite phase. At low temperatures, the material exhibits martensite

phase, which has a monoclinic crystal structure. The transformation between the

low and high temperature phases is shown in Figure 1.3. When the material is

cooled from the high temperature austenite phase, a “twinned” martensite structure is

formed. This twinned structure consists of alternating rows of atoms tilted in opposite

direction. The atoms form twins of themselves with respect to a plane of symmetry

called as a twinning plane, or twin boundary. When a stress is applied to the material,

9

(a) (b)

(c) (d)

Figure 1.2: Joule magnetostriction produced by a magnetic field H. (a) H is ap-proximately proportional to the current i that passes through the solenoid when avoltage is applied to it, (b) the rotation of magnetic dipoles changes the length of thesample, (c) and (d) curves M vs. H and ∆L/L vs. H obtained by varying the fieldsinusoidally [20].

10

Figure 1.3: SMA transformation between high and low temperature phases.

the twins are reoriented so that they all lie in the same direction. This process is called

as “detwinning”. When the material is heated, the deformed martensite reverts to the

cubic austenite form, and the original shape of the component is restored. Therefore

this behavior is called as “shape memory effect” as the material remembers its original

shape. This entire process is shown in Figure 1.3.

This process is highly hysteretic. The hysteresis associated with temperature is

shown in Figure 1.4. The amount of martensite in the material is quantified by the

martensite volume fraction (ξ). Naturally the austenite volume fraction is (1-ξ).

Referring to Figure 1.4, at a temperature below Mf , the material is 100% martensite.

When heated, the material does not transform to the austenite phase until a temper-

ature As is reached, after which the material starts transforming to austenite. The

11

Figure 1.4: Schematic of phase transformation.

material consists of 100% austenite when a temperature Af is exceeded. When the

material is cooled below Af , it does not start transforming to the martensite phase

until a temperature Ms is reached. The martensite transformation is completed when

the temperature reaches Mf . The values of the four critical temperatures (Mf ,Ms,Af ,

and As) depend on the composition of alloy, with typical width of hysteresis loop being

10-50C.

The temperature and associated phase transformations also significantly affect the

stress-strain behavior of the material. Figure 1.5 shows the stress-strain behavior of

SMAs at two constant temperatures, namely below Mf and above Af . At temperature

below Mf (Figure 1.5(a)), the material consists of a complete martensite phase, and in

absence of load, the material consists of a twinned structure. The elastic region (o →

12

a) corresponds to the elastic compression of the material until the stress level is

sufficient to start detwinning. In the detwinning region (a → b), the twins reorient

themselves until they all lie in the same crystallographic region. The amount of

stress needed is relatively small (beyond the elastic region) to cause detwinning, which

corresponds to a low slope region. The material again gets compressed elastically (b →

c) after the detwinning is completed. In the plastic region (beyond c), the subsequent

shape memory effect is destroyed. In the unloading region (c → d), the material

does not come back to its original shape because the material is deformed when it is

detwinned. Only the elastic deformation is recovered. The residual strains can only

be recovered if the material is heated to Af .

The second configuration in the Figure 1.5(b) is at temperature above Af . The ini-

tial microstructure consists of randomly oriented austenite. The elastic region (o → a)

is followed by the transformation region (a → b), where the stress-induced marten-

site is formed upon loading, which is again followed by an elastic region (b → c).

Upon unloading (c → e), the stress induced martensite goes through elastic unload-

ing, which is followed by the transformation back to the austenite phase. Thus the

shape memory behavior is seen in the stress-strain curves also, where the stress and

temperature are both responsible for the phase change. Later (in Chapter 2) it will

be seen that in case of FSMAs, the magnetic field acts in an analogous manner to

the temperature: at high magnetic fields, the stress-strain plots of FSMAs exhibit

pseudoelastic or reversible behavior, whereas at low magnetic fields, the stress-strain

behavior is irreversible.

The major advantage of SMAs is that they generate large strain of around 10%.

Also, their Young’s modulus changes by about 3-5 times when constrained. The major

13

(a) (b)

Figure 1.5: Stress-strain behavior of shape memory alloys (a) below Mf , (b) aboveAf .

disadvantage of SMAs is their limited bandwidth due to the slow heating process,

which limits their use when fast actuation response is required. They have found

numerous applications in the aerospace, medical, safety devices, robotics, etc. Some

of their applications include couplers in fighter planes, tweezers, orthodontic wires,

eyeglass frames, fire-sprinklers, and micromanipulators to simulate human muscle

motion.

1.3 Ferromagnetic Shape Memory Alloys

Ferromagnetic Shape Memory Alloys (FSMAs), which are also called Magnetic

Shape Memory Alloys (MSM-Alloys), were first identified by Ullakko at MIT in

1996 [128]. This new class of materials, which generates strain when subjected to

a magnetic field, showed promise of relatively high strain and high operating fre-

quency of several hundred Hz [126]. Therefore, they have been the subject of much

14

research over the past 10 years. This section provides an overview of the work done

by key contributors to the field and motivates the importance of the investigations

performed for this dissertation.

1.3.1 Early Work

Ferromagnetic shape memory effect occurs in various alloys such as nickel-manganese-

gallium (Ni-Mn-Ga), iron-palladium (Fe-Pd), and cobalt-nickel-aluminum (Co-Ni-

Al). The problem of slow thermally-induced phase transformation response exhibited

by the nickel-titanium (Ni-Ti) alloys has been addressed with the discovery of ferro-

magnetic shape memory alloys. Of these, Ni-Mn-Ga is the most commonly studied

FSMA, which is also commercially available [1].

The first report of the significant magnetic field induced strain in Heusler type

non-stoichiometric Ni2MnGa alloys was presented in 1996 by Ullakko et al. [128]. This

phenomenon was further validated through a series of publications by Ullakko [126,

127, 129]. The experimental results for unstressed crystals of Ni2MnGa at 77 K

showed strains of 0.2% under a 8 kOe magnetic field. This original data is repro-

duced in Figure 1.6. The tests are conducted with two directions of applied field,

namely along [001] and [110] direction with respect to the bcc parent phase. The

strain is measured in the direction along the field and perpendicular to it. In the ini-

tial years of research, the magnetic field induced strain was assumed because of the

magnetostriction, and was reported as λs = 133× 10−6, with e|| − e⊥ = 0.20× 10−3.

Experimental advancement continued with testing of off-stoichiometric Ni-Mn-

Ga that demonstrated larger strains at higher temperatures. Tickle and James pre-

sented several results in their publications [123, 124, 57]. The measurements on

15

Figure 1.6: (a) Relative orientation of sample, strain gauge, and applied field formeasurements shown in (b) and (c). (b) Strain vs applied field in the L21 (austenite)phase at 283 K. (c) Same as (b) but data taken at 265 K in the martensitic phase [128].

16

Ni51.3Mn24.0Ga24.7 at -15C exposed to fields of less than 10 kOe were presented, which

showed strains of up to 0.2% due to cyclic application of an axial magnetic field and

strains of 1.3% when fields were applied transverse to the sample that started from

a stress biased state. This finding shifted the focus of Ni-Mn-Ga research towards

the orthogonal stress-field orientation. The transverse field tends to oppose the ef-

fect of the collinear compressive stress, and therefore this configuration provides the

opportunity to obtain maximum possible strain.

Further work was focused on compositional dependence on the strain generation

ability. Murray et al. [88] reported compositional and temperature dependence on the

performance of polycrystalline Ni-Mn-Ga alloys. Jin et al. [60] studied the empirical

mapping of Ni-Mn-Ga properties with composition and valence electron concentra-

tion. A range between Ni52.5Mn24.0Ga23.5 and Ni49.4Mn29.2Ga21.4 was identified, in

which the martensitic transformation temperature, Tm, is higher than room temper-

ature and lower than the Curie temperature, Tc, and the saturation magnetization

is larger than 60 emu/g. These conditions are suggested as optimum for creating

samples with the best capability for large, room temperature strains.

Large strains of 6% in Ni-Mn-Ga single crystals were reported in numerous pub-

lications by Murray et al. [89, 90, 88], Heczko et al. [47], and Likhachev [79]. The

alloys used in these measurements consisted of tetragonal martensite structure with a

five-layer (5M) shuffle type modulation. Strains of 9.5% were reported by Sozinov et

al. [114] having seven-layer (7M) modulation, which is the most promising result in

Heusler type of ferromagnetic shape memory alloys of the family Ni2+x+yMn1−xGa1−y.

17

1.3.2 Properties and Crystal Structure

Currently, the ferromagnetic shape memory alloys are grown by conventional single

crystal growth techniques such as Bridgman [113]. After producing the single crystal

bars, the materials are homogenized at about 1000C for 24 hours and ordered at

800C for another 20 hours. The material is then oriented using X-ray techniques

to produce the desired crystallographic structure for the MSM effect. Following the

crystal orientation, the material is cut and thermomechanically treated. The key to

obtaining high strains is to cut the samples so that the twin boundaries are aligned

at 45 to the sample axis (when magnetic field is applied transverse to the bar).

Ni2MnGa is an intermetallic compound that exhibits Heusler Structure. At high

temperatures, it exhibits cubic austenite (L21, Fm3m) structure as shown in Fig-

ure 1.7(a) [98, 29]. Ni-Mn-Ga exhibits a paramagnetic/ferromagnetic transition with

a Curie temperature of about 373 K. When cooled below the Curie temperature, the

material undergoes a phase change to a martensite, tetragonal (l4/mmm) structure

as shown in Figure 1.7(b). The unique c-axis of the tetragonal unit cell is shorter

than the a-axis, c/a < 1 [98]. Therefore, the theoretical maximum strain can be given

as,

εmax = 1− c/a (1.1)

Most commonly observed value of the c/a ratio is 0.94, and therefore a strain of

around 6% is typically observed.

The self accommodating twin-variant martensite structure is similar to the marten-

site structure in SMAs. Because of the tetragonal nature of the martensitic phase,

three twin orientations are possible of which two are identical relative to the axis of

the sample. The variants with their c-axis aligned with the sample axis are referred

18

(a) (b)

Figure 1.7: Ni-Mn-Ga crystal structure (a) Cubic Heusler structure, (b) Tetragonalstructure, under the martensite finish temperature. Blue: Ni, Red: Mn, Green: Ga.

to as the “axial” or “stress-preferred” variants while those with one of their a-axes

aligned with the samples axis are the “transverse” or “field-preferred” variants.

1.3.3 Magnetocrystalline Anisotropy

The key factor responsible for the ferromagnetic shape memory effect is the

large magnetic anisotropy associated with these alloys [90]. The magnetocrystalline

anisotropy is one form of the magnetic anisotropy, which introduces a preferential

crystal direction for the magnetization. In simplest terms, it means that the mate-

rial exhibits different magnetic properties in different directions. It arises from the

spin-orbit coupling between the spins and the lattice of the material. The simplest

and most commonly observed form of the anisotropy is the uniaxial anisotropy, which

19

means that there is a certain crystal axis along which the magnetization vectors tend

to align in absence of external fields. Typical form of the uniaxial magnetic anisotropy

energy for the tetragonal martensite is given as [93],

Ua = Ku0 + Ku1 sin2 θ + Ku2 sin4 θ + ... (1.2)

where θ is the angle between the unique axis of the crystal and the magnetization

vector and Kui are experimentally determined coefficients. If this energy is large

enough, the alignment of magnetization vectors with an applied field can change the

physical orientation of the unit cells, thereby creating strain in the material. This

phenomenon, which is of primary importance to the strain mechanism in FSMAs, is

described in more detail in Figure 1.8.

1.3.4 Strain Mechanism

In absence of magnetic field, the material typically consists of two variants, repre-

sented by the volume fraction ξ, that are separated by a twin boundary (panel (a)).

Each variant consists of several distinct magnetic domains divided by 180 walls. The

magnetic domain volume fraction is denoted α. At small transverse fields, H, of the

order of ≈8 kA/m, the magnetic domain walls disappear to form a single domain per

twin variant (panel (b)). Since the behaviors at medium to large fields is of interest,

α = 1 is assumed.

When a transverse field (x-direction) is applied, the variants favored by the field

increase in size through twin reorientation. Alloys in the Ni-Mn-Ga system have large

magnetic anisotropy energies compared to the energy necessary to reorient the unit

cells at the twin boundary, which is usually represented by the twinning stress. Thus,

as the applied magnetic field attracts the unit cell magnetization vectors towards

20

H = 0

(a)

H

a

c

a

a

c

c

(b)

H

a

c

a

a

c

c

(c)

H

c

a

Saturation

(d)

H = 0

c

a

(e)

a

c

a

a

c

c

σ

(f)

Figure 1.8: Schematic of strain mechanism in Ni-Mn-Ga FSMA under transversefield and longitudinal stress.

21

it, the unit cells along the twin boundary switch orientation such that their c-axis

is aligned with the field. This results in the growth of favorable variants at the

expense of unfavorable ones through twin boundary motion resulting in the overall

axial lengthening of the bulk sample (panel (c)). As the field is increased to the point

where no further twin boundary motion is possible and the field energy overcomes

the magnetic anisotropy energy, the local magnetization vectors break away from the

c-axis and aligns with the field. This results in magnetic saturation as shown in panel

(d). When the field is removed (panel (e)) the magnetic anisotropy energy will restore

the local magnetization to the c-axis of the unit cells.

Since both variants are equally favorable from an energy standpoint [89], there is

no restoring force to drive the unit cell reorientation and the size of the sample does

not change upon removal of the field. Twin boundary motion and reversible strain

can be induced by applying an axial field, axial compressive stress, or a transverse

tensile stress, all of which favor the variant with the short c-axis aligned with the axial

direction as shown in panel (f). One common configuration for Ni-Mn-Ga consists

of placing a rectangular sample in an electromagnet such that the field is applied

transversely and a bias axial compressive stress is always present [122] as depicted in

Figure 1.8.

1.4 Literature Review on Ni-Mn-Ga

Because of their ability to produce large strains under magnetic fields, majority

of the prior work on ferromagnetic shape memory Ni-Mn-Ga has been focused on

characterization and modeling of the actuation behavior, i.e., dependence of strain on

magnetic field. A comprehensive summary of the experimental and modeling efforts

22

can be found in the review papers by Kiang and Tong [65], and Soderberg et al. [113].

Some of the significant experimental results, have been discussed in Section 1.3.1.

These studies chiefly focus on the characterization of magnetic field induced strain at

varied bias stresses.

1.4.1 Sensing Behavior

In the context of electrically or magnetically activated smart materials, the term

“sensing behavior” typically refers to the phenomenon of alteration in the electric or

magnetic properties of the material in response to the externally applied mechanical

load. On the contrary to the actuation behavior, the characterization of the sensing

behavior of ferromagnetic shape memory alloys has received only limited attention.

Investigation of the sensing behavior is important to fully understand the coupled

magnetomechanical material behavior and to realize potential applications.

Mullner et al. [87] experimentally studied flux density change in a single crystal

with composition Ni51Mn28Ga21 under external quasistatic strain loading at a con-

stant field of 558 kA/m. This study provided the first experimental evidence that

the magnetization of Ni-Mn-Ga can be changed by applying mechanical compression

in presence of bias magnetic fields. The study also demonstrated that Ni-Mn-Ga ex-

hibits magnetic field induced pseudoelastic behavior, similar to that in SMAs which

is temperature induced. The stress-strain response was hysteretic, whereas the mag-

netization response was almost linear and non-hysteretic. A permanent magnet was

used to apply the bias magnetic field, and therefore the material behavior at other

magnitudes of bias fields was not characterized.

23

Straka and Heczko [115, 116] reported similar measurements, specifically the su-

perelastic or pseudoelastic response of a Ni49.7Mn29.1Ga21.2 single crystal with 5M

martensitic structure for fields higher than 239 kA/m and established the intercon-

nection between magnetization and strain. The earlier publication [115] reported the

stress-strain behavior under different bias fields to demonstrate the reversible behavior

of Ni-Mn-Ga. The effect of the bias field was reported using a term called “sensitivity

of the stress-strain curve to the magnetic field”, which was evaluated as 6.8 MPa/T.

Further, a simple model was proposed based on the earlier work by Likhachev and

Ullakko [78]. The second publication reported magnetization as a function of strain

using vibrating coil magnetometry for different static magnetic fields of up to 1.5 T.

The model in the earlier paper was augmented to describe the magnetization response.

Li et al. [73, 72] reported the effect of magnetic field during martensitic trans-

formation on the magnetic and elastic behavior of Ni50.3Mn28.7Ga21. Similar to the

study by Mullner [87], the tests were conducted by using a permanent magnet and a

mechanical testing machine (Instron). In addition to the major loops of stress-strain

and magnetization-strain similar to those reported by Mullner [87] and Straka [116],

the variation of stress and magnetization under several loading cycles was measured.

The minor loop measurements of stress and magnetization were also reported. It was

demonstrated that behavior in subsequent strain cycles was almost similar to that in

the first, and also the minor loops were overlapping on the major loops. A qualitative

explanation of the observed phenomenon was provided.

Suorsa et al. [118, 117] reported magnetization measurements conducted on stoi-

chiometric Ni-Mn-Ga material for various discrete strain and field intensities ranging

between 0% and 6% and 5 and 120 kA/m, respectively. These measurements were

24

different from the aforementioned characterizations because the magnetization-strain

curves were generated by picking the values from the magnetization-field measure-

ments at different bias strains. Though this study presented interesting observation

that the magnetization-strain relation is linear at high fields and parabolic at lower

fields, the generated data did not provide a true indication of the physical behav-

ior. Suorsa’s other work on the sensing characteristics of Ni-Mn-Ga included voltage

measurements using impulse loading [119], which is more relevant to the dynamic

behavior with possible applications in energy harvesting. Suorsa further presented

measurements of the inductance of an inductor [120], which included a Ni-Mn-Ga

sample in its magnetic circuit. The Ni-Mn-Ga sample was subjected to compressive

loading, which altered its magnetic permeability, and therefore the reluctance of the

air-gap of the inductor was altered. This phenomenon led to the change in the in-

ductance of the inductor which was carrying alternating current at varies frequencies,

from 10 to 200 Hz. This study provided a novel way to demonstrate the practical

implementation of Ni-Mn-Ga as a sensor material. However, the actual flux-density

or magnetization inside the material was not reported.

Though most of the above-mentioned studies provide a demonstration of the sens-

ing behavior of Ni-Mn-Ga under different conditions, a comprehensive investigation

of the simultaneous measurement of stress and magnetization at wide range of mag-

netic bias fields is still lacking. In this work, the experimental measurements on

the dependence of flux density with deformation, stress, and magnetic field in a

commercially-available NiMnGa alloy are presented with a view to determining the

bias field needed for obtaining maximum reversible deformation sensing as well as the

associated strain and stress ranges.

25

1.4.2 Modeling

Several models have been proposed for describing twin variant rearrangement in

FSMAs, with the primary intent of characterizing the magnetic field induced strain

or actuator behavior. Most commonly used approach relies on construction and

minimization of an energy function to obtain the values of stress, strain, and magne-

tization.

James and Wuttig [56] presented a model based on a constrained theory of micro-

magnetics (see also [24, 23]). This theory addresses the challenge of describing the

behavior of FSMAs from a micromechanical approach. The terms contributing to the

free energy in their model are the Zeeman energy, the magnetostatic energy and the

elastic energy. The magnetization is assumed to be fixed to the magnetic easy-axis

of each martensitic variant because of high magnetocrystalline anisotropy. The mi-

crostructural deformations and the resulting macroscopic strain and magnetization

response are predicted by detecting low-energy paths between initial and final con-

figurations. They conclude that the typical strains observed in martensite, together

with the typical easy axes observed in ferromagnetic materials lead to layered do-

main structures that are simultaneously mechanically and magnetically compatible.

Because of the complexity of the model, it has been implemented only for certain

simplified cases [124, 57].

After the discovery of Ni-Mn-Ga, Likhachev and Ullakko proposed one of the

models that has become the basis for much of the subsequent modeling work [79,

78, 80, 74, 75, 76, 77]. In this model, the anisotropy energy difference between the

two variants is identified as the chief driving force. The derivative of the easy-axis

and hard-axis magnetic energy difference is defined as the magnetic field-induced

26

driving force acting on a twin boundary. The magnetization is assumed to be a linear

combination of easy-axis and hard-axis magnetization values related by the volume

fraction. It is argued that regardless of the physical nature of the driving force, twin

boundary motion should be initiated at equivalent load levels. The strain output for

a given magnetic field input can be predicted through an analytical interpolation of

mechanical stress-strain experimental data by replacing the mechanical stress with an

effective force due to the field. A similar model was utilized by Straka and Heczko [116,

45, 46] for describing the stress-strain response at varied bias fields.

OHandley [92] presented a model that quantifies the strain and magnetization de-

pendence on field by energy minimization. The Zeeman energy difference (∆M ·H)

across the twin boundary is determined as the driving force responsible for strain gen-

eration. The contributions of elastic, Zeeman, and anisotropy energy are considered,

with the latter defining three cases depending on its strength being low, medium, or

high. This model does not capture the hysteresis because the technique of energy

minimization results in a reversible behavior. For the intermediate anisotropy case,

a parametric study is conducted showing the influence of varying elastic energy and

anisotropy energy. This model provided a significant advancement towards modeling

of FSMAs by proposing the twin boundary mechanism due to the interaction be-

tween anisotropy and Zeeman energy as the reason behind strain generation. Further

work from the MIT group has been based on this model, with focus on modeling the

strain-field behavior from micromagnetic considerations [90, 94, 95].

A model by Couch and Chopra [15, 16, 18] is based on an approach similar to that

by Brinson [6, 7] for thermal shape memory materials. The stress is assumed to be

a linear combination of strain, volume fraction, and magnetic field. The model was

27

developed to describe the stress-strain behavior at varied magnetic fields to capture

the transition from irreversible to reversible behavior. The model parameters are

obtained in a similar fashion to SMAs, by using the values of slopes that the curves

of critical stress values make when plotted against the bias magnetic field. The critical

stresses are expressed as function of the magnetic field using these slopes. While this

model is tractable, the identification of model parameters requires stress-strain testing

over a range of bias fields in order to obtain the necessary stress profiles as a function

of field.

Glavatska et al. [42] developed a statistical model for MFIS by relating the fer-

romagnetic magnetoelastic interactions to the internal microstress in the martensite.

The probability for the rearrangement of the twins in which the stresses are near the

critical values is described through a statistical distribution. Chernenko et al. [11, 12]

further modified this model to describe the quasiplastic and superelastic stress-strain

response of FSMAs at varied bias fields.

A thermodynamic approach was introduced by Hirsinger and Lexcellent, and was

used in their subsequent publications [53, 52, 50, 51, 19]. Magnetomechanical energy

expressions were developed for the system under consideration. The microstructure

of single-crystal NiMnGa was represented by internal state variables, and evolution of

these variables was used to quantify the strain and magnetization response to applied

magnetic fields. The anisotropy energy effect was not considered in Ref. [52] but was

later considered in Ref. [50, 38] in order to model the magnetization.

Kiefer and Lagoudas [67, 68, 66] employed a similar approach with a more system-

atic thermodynamics treatment. Polynomial and trigonometric hardening functions

were introduced to account for interaction of evolving volume fractions. However, this

28

leads to increased number of parameters in the model. Faidley et al. [32, 33, 28, 31]

used the thermodynamic approach to describe reversible magnetic field induced strain

in research-grade Ni-Mn-Ga. The Gibbs energy potential was constructed for the

case when the twin boundaries are pinned by dislocations, which had been previously

shown by Malla et al. [83] to allow in some cases for reversible twin boundary bowing

when the single crystal is driven with a collinear magnetic field and stress pair. While

similar in concept to the models for MFIS by Hirsinger and Lexcellent [52] and Kiefer

and Lagoudas [67], in this model the energy of a mechanical spring is added to the

Zeeman and elastic energies to account for the internal restoring force supplied by the

pinning sites. The anisotropy energy was assumed to be infinite in Refs. [67] and [32]

and magnetostatic energy was not considered with the argument that it depends on

the geometry of a sample. One tenet of the proposed model is that the magnetostatic

energy is an important component of the magnetization response, which is critical

for the sensing effect. The magnetostatic energy is thus considered as a means to

quantify the demagnetization field in the continuum. While the magnitude of the de-

magnetization field depends on a specimens shape, it can be assumed to be uniform

throughout a continuum.

In this work, a thermodynamic model is presented to describe the sensing behav-

ior. The focus is on modeling the magnetization vs. strain behavior and magnetic

field induced pseudoelasticity in Ni-Mn-Ga FSMAs. Further, this sensing model is

extended to describe the actuation and blocked-force behavior of single crystal Ni-

Mn-Ga.

29

1.4.3 Dynamic Behavior

The dynamic behavior of magnetomechanical materials can refer to several phe-

nomena. The dynamic behavior can be associated with the material itself (dynamics

of magnetization and eddy current losses), as well as the dynamics of the system, for

example, the mechanical load on the actuator. This research addresses three most

commonly occurring behaviors: (i) Dynamic actuation: Strain dependence on field

at varied frequencies of applied field, (ii) Dynamic sensing: Magnetization and stress

dependence on strain at varied frequencies of applied loading, and (iii) Stiffness tun-

ing: Acceleration transmissibility response due to broadband mechanical excitation

under varied bias magnetic fields and resulting resonance and stiffness variation.

The dynamic actuation characterization of magnetomechanical materials is con-

ducted by applying magnetic fields at high frequencies and measuring the resulting

strain by means of a suitable sensor such as a laser sensor. These tests refer to sinu-

soidal application of field at a given frequency to observe the variation of strain-field

hysteresis, and/or a broadband excitation to obtain the strain frequency response.

Achieving the high saturation fields of NiMnGa (around 400 kA/m) requires large

electromagnet coils with high electrical inductance, which limits the effective spectral

bandwidth of the material. For this reason, perhaps, the dynamic characterization

and modeling of FSMAs has received limited attention. The only significant data

of dynamic actuation was presented by Henry et al. [49, 48], who reported the mea-

surements of magnetic field induced strains varied drive frequencies. It was observed

that reversible strain of 3% can be obtained for frequencies of up to 250 Hz. A linear

model was presented which describes the phase lag between strain and field and sys-

tem resonance frequencies. Peterson [97] presented dynamic actuation measurements

30

on piezoelectrically assisted twin boundary motion in NiMnGa. The acoustic stress

waves produced by a piezoelectric actuator complement the externally applied fields

and allow for reduced field strengths. Scoby and Chen [111] presented a preliminary

magnetic diffusion model for cylindrical NiMnGa material with the field applied along

the long axis, but they did not quantify the dynamic strain response. The experimen-

tal evidence of the fast response of Ni-Mn-Ga in time domain was shown by Marioni et

al. [86, 85, 84], who presented the measurements on pulsed magnetic field actuation of

Ni-Mn-Ga for field pulses lasting up to 620 µs. The complete field-induced strain was

observed to occur in 250 µs, indicating the possibility of obtaining cyclic 6% strain for

frequencies of up to 2000 Hz. Magnetization measurements were not reported in these

studies as they are not of great interest for the actuation applications. These studies

are mainly experimental, and attempts to model the frequency dependent strain-field

behavior are lacking. Due to the inherent nonlinear and hysteretic nature, the prob-

lem of modeling dynamic strain-field behavior becomes difficult as the losses due to

eddy currents and structural dynamics of the actuator add to the complexity. A novel

approach for modeling the frequency dependent strain-field hysteresis is presented in

this thesis, by including the magnetic field diffusion and actuator dynamics, along

with the constitutive model.

The dynamic sensing characterization of magnetomechanical materials is con-

ducted by applying mechanical loading, by controlling the force or displacement input

at high frequencies and measuring the resulting change in magnetization. There have

not been any previous attempts of characterizing the dynamic sensing behavior of

Ni-Mn-Ga. One of the reasons could be that unlike magnetostrictive materials, the

sensing behavior of Ni-Mn-Ga can not be characterized by using a shaker. Vibration

31

shaker facilitates the application of high frequency loads with relative ease [10, 8]. In

case of Ni-Mn-Ga, however, the small displacements of shaker are not sufficient to

induce twin variant reorientation and hence the change of magnetization. Recently,

Karaman et al. [62] reported voltage measurements in a pickup coil due to flux den-

sity change under dynamic strain loading of 4.9% at frequencies from 0.5 to 10 Hz

from the viewpoint of energy harvesting. Their experimental setup consisted of MTS

mechanical testing machine along with an electromagnet. Their study presents the

highest frequency of mechanical loading to date (10 Hz) which induces twin bound-

ary motion in Ni-Mn-Ga. However, the dependence of magnetization on strain was

not reported. In the presented study, the dynamic characterization and modeling of

single crystal Ni-Mn-Ga is presented.

Applications of Ni-Mn-Ga other than actuation have received limited attention.

Magnetomechanical materials such as Terfenol-D have shown potential as a tunable vi-

bration absorber [34], and a tunable mechanical resonator [35, 63] because its stiffness

can be altered using magnetic fields in a non-contact manner. Faidley et al. [30, 28]

investigated stiffness changes in a research grade, single crystal Ni-Mn-Ga driven with

magnetic fields applied along the [001] (longitudinal) direction. The material they

used exhibits reversible field induced strain when the longitudinal field is removed,

which is attributed to internal bias stresses associated with pinning sites. The fields

were applied with permanent magnets bonded onto the material, which makes it diffi-

cult to separate resonance frequency changes due to magnetic fields or mass increase.

Analytical models were developed to address this limitation. In the presented work,

the effect of magnetic field on the stiffness of Ni-Mn-Ga is isolated by applying the

32

magnetic fields in a non-contact manner, and the stiffness characteristics under both

longitudinal and transverse magnetic fields are investigated.

1.5 Research Objectives

The objectives of this research are broadly classified as:

1. To conduct experimental characterization of the sensing behavior of Ni-Mn-Ga

2. To develop a model which can describe the nonlinear and hysteretic coupled

magnetomechanical behavior of single crystal Ni-Mn-Ga in quasi-static condi-

tions

3. To study the dynamic behavior of Ni-Mn-Ga and investigate the frequency

dependence of the material’s mechanical and magnetic response

1.6 Outline of Dissertation

This dissertation is divided into seven chapters. Each chapter constitutes the

body of a journal publication. Chapters 2 and 3 focus on the quasi-static behavior

of Ni-Mn-Ga, whereas Chapters 4-6 focus on the dynamic behavior of Ni-Mn-Ga.

The quasi-static part includes experimental characterization of sensing and blocked-

force behavior. A constitutive model is developed that describes sensing, actuation

and blocked-force behavior. The dynamic part includes modeling of the dynamic

actuation and sensing behavior, along with experimental characterization of dynamic

sensing effect and magnetic field induced stiffness tuning.

33

1.6.1 Quasi-static Behavior

In Chapter 2, the characterization of commercial NiMnGa alloy for use as a defor-

mation sensor is addressed. Design and construction of an electromagnet is detailed,

which is used for generating large magnetic fields of around 0.9 Tesla. The sensing

behavior of Ni-Mn-Ga is characterized by measuring the flux density and stress as a

function of strain at various fixed magnetic fields. The bias field is shown to mark

the transition from irreversible quasiplastic to reversible pseudoelastic stress-strain

behavior. The presented measurements indicate that Ni-Mn-Ga shows potential as a

high-compliance, high-displacement deformation sensor.

Chapter 3 presents a continuum thermodynamics based constitutive model to

quantify the coupled magnetomechanical behavior of Ni-Mn-Ga FSMA. A single crys-

tal Ni-Mn-Ga is considered as a continua that deforms under magnetic and mechanical

forces. A continuum thermodynamics framework is presented for a material that re-

sponds to the magnetic, mechanical and thermal stimuli. The microstructure and

mechanical dissipation in the material is included in the continuum framework by

defining internal state variables. Thermodynamic potentials are constructed that

include various magnetic and mechanical energy potentials. Magnetomechanical con-

stitutive equations are derived by restricting the process through the second law of

thermodynamics to describe the relations between strain, stress, magnetization and

magnetic field. Major emphasis of this chapter is to model the sensing behavior,

i.e., the stress-strain and magnetization-strain behavior. The model is extended un-

der a unified framework to also describe the actuation and blocked-force behavior

of Ni-Mn-Ga. Various key parameters of Ni-Mn-Ga, such as the sensing sensitivity,

twinning stress, coercive field, maximum field induced strain, blocking stress, etc., are

34

studied to demonstrate the model performance as well as the rich magnetomechanical

behavior of single crystal Ni-Mn-Ga. The model presented in this chapter is the chief

contribution of the thesis. For dynamic modeling, this model is augmented by adding

frequency dependencies.

1.6.2 Dynamic Behavior

Chapter 4 presents a model to describe the relationship between magnetic field and

strain in dynamic Ni-Mn-Ga actuators. Due to the eddy current losses and structural

dynamics of the actuator, the strain-field relationship changes significantly relative to

the quasistatic response as the magnetic field frequency is increased. The eddy current

losses are modeled using magnetic field diffusion equation. The actuator is represented

as a lumped-parameter, single-degree-of-freedom resonator which is driven by the

applied magnetic field. The variant volume fraction is obtained from the magnetic

field using the constitutive model, and it acts as an equivalent driving force on the

actuator. The total dynamic strain output is therefore obtained after accounting for

the dynamic magnetic losses and the actuator dynamics. The hysteretic strain-field

behavior is analyzed in the frequency domain to view the effect of the actuation

frequency on the macroscopic hysteresis. The application of this new approach is also

demonstrated for a dynamic magnetostrictive actuator to highlight its flexibility.

Chapter 5 addresses the characterization and modeling of the dynamic sensing

behavior of NiMnGa. The flux density is experimentally determined as a function of

cyclic strain loading at frequencies from 0.2 Hz to 160 Hz. With increasing frequency,

the stress-strain response remains almost unchanged whereas the flux density-strain

response shows increasing hysteresis. It indicates that the twin-variant reorientation

35

occurs in concert with the mechanical loading, whereas the rotation of magnetization

vectors occurs with a delay as the loading frequency increases. This phenomenon is

modeled by using the magnetic diffusion along with a linear constitutive equation.

Chapter 6 presents the dynamic characterization of mechanical stiffness changes

under varied bias magnetic fields. Mechanical base excitation is used to measure the

acceleration transmissibility across the sample, from where the resonance frequency

is directly identified. The tests are repeated in the presence of various longitudinal

and transverse bias magnetic fields. Significant stiffness changes of −35% and 61%

are observed for the longitudinal and transverse field tests respectively. The mea-

sured dynamic behaviors make Ni-Mn-Ga well suited for vibration absorbers with

electrically-tunable stiffness.

Chapter 7 provides a summary of the contributions of this research. The presented

work provides a comprehensive understanding of the material behavior in a wide range

of quasi-static and dynamic conditions that has enabled significant advancement of the

state of the art in this technology. Some possible improvements and future research

opportunities in advancement of ferromagnetic shape memory alloys are discussed.

36

CHAPTER 2

CHARACTERIZATION OF THE SENSING EFFECT

In the context of electrically or magnetically activated smart materials, the term

“sensing behavior” typically refers to the phenomenon of alteration in the electric or

magnetic properties of the material in response to the externally applied mechanical

load. The sensing behavior of magnetomechanically coupled materials is character-

ized by subjecting the material to mechanical tension or compression in presence of

a bias magnetic field. Ferromagnetic shape memory Ni-Mn-Ga is operated only un-

der compression because of its brittle nature and low tensile strength. Because of

the magnetomechanical coupling, the permeability of Ni-Mn-Ga changes in response

to the applied mechanical loading. To detect the change in permeability, a finite

magnetization is required to be induced in the material before the start of compres-

sion. Application of a bias magnetic field results in residual magnetization inside the

material, which can be altered by the external mechanical loading. In this chapter,

the characterization of commercial NiMnGa alloy for use as a deformation sensor is

addressed.

Hardware and test rig are developed to conduct uniaxial compression tests in pres-

ence of moderate to high magnetic fields. An electromagnet is designed and built to

generate high magnetic fields of up to 750 kA/m. An MTS frame is used for applying

37

uniaxial compressive loading. The experimental determination of flux density as a

function of strain loading and unloading at various fixed magnetic fields gives the bias

field needed for maximum recoverable flux density change. This bias field is shown to

mark the transition from irreversible quasiplastic to reversible pseudoelastic stress-

strain behavior. A reversible flux density change of 145 mT is observed over a range

of 5.8% strain and 4.4 MPa stress at a bias field of 368 kA/m. The alloy investigated

shows potential as a high-compliance, high-displacement deformation sensor.

2.1 Electromagnet Design and Construction

An electromagnet is a type of magnet in which the magnetic field is produced

by the flow of an electric current. Advantage of an electromagnet over a permanent

magnet is that the magnetic field can be rapidly manipulated by controlling the

flow of the electric current. Electromagnets are widely used in several applications

such as relays, loudspeakers, magnetic tapes, and electromagnetic lifts and locks.

The fundamental property used for most of the applications is the attractive force

that an electromagnet generates on a ferromagnetic material, which is often used to

displace or actuate various mechanisms. However, the force generation properties of

electromagnets are not of interest in characterization of magnetic materials.

For the characterization of magnetic materials, the purpose of the electromagnet

is to generate magnetic field. This magnetic field acts on the material, causing it to

produce magnetization and strain (if the material is active). The material response

to the applied magnetic field produced by the electromagnet is used to determine the

key properties of the material. In fact the force produced by electromagnet is usually

unnecessary, as it can lead to undesirable stresses on the material under study.

38

For the characterization of Ni-Mn-Ga, the electromagnet is required to produce a

magnetic field of around 0.9 Tesla (720 kA/m). This field is significantly higher than

that required for materials such as magnetostrictive Terfenol-D (16 kA/m [8]). Fur-

thermore, the magnetic field application is in the perpendicular (transverse) direction

to the long axis of Ni-Mn-Ga. Application of magnetic field along the long axis is

usually achieved by using a solenoid coil. However, the solenoid coil does not provide

a viable solution in case of transverse field application because of the requirements

of a large inner diameter to span the entire length of sample and the issues with pro-

viding space for mechanical loading arms. Therefore, the simultaneous requirement

of high magnitude of magnetic field and transverse configuration poses a challenging

design problem. A novel electromagnet is designed and built to address this issue.

2.1.1 Magnetic Circuit

Before constructing the electromagnet, it is necessary to study the magnetic cir-

cuit that is responsible for creation of the magnetic field. A magnetic circuit is a

closed path containing a magnetic flux. It generally contains magnetic elements such

as permanent magnets, ferromagnetic materials, coils, and also an air gap or other

materials. Application of a current through the coils of the magnetic circuit creates

a magnetic field in the air gap. The magnetic smart material is placed in this air gap

so that it is subjected to the generated field.

Several iterations are conducted to decide the shape of the electromagnet. A

symmetric design consisting of two E-shaped legs is finalized. Figure 2.1 shows the

schematic of the electromagnet. The two E-shaped cores are constructed by stacking

several layers of laminated transformer steel. The construction using the laminates

39

Laminated core

Air Gap

Coil(s)

Flux path

Figure 2.1: Schematic of the electromagnet. Two E-shaped legs form the flux pathindicated by arrows.

is favorable for reducing the eddy current losses and subsequent heating of the core.

The coils are typically made from AWG copper wire which can carry current of up

to several amperes.

The flux or the magnetic field in the air gap is of interest because the Ni-Mn-Ga

sample is placed in it. The magnetic field or flux flowing through the magnetic circuit

is calculated by using an analogous theory to Kirchhoff’s voltage law. The coils act

as an equivalent voltage source, and generate a magnetomotive force. According to

the Ampere’s law, this magnetomotive force (Vm = MMF) is the product of the of

the current (I) and the number of complete loops (N) made by the coil.

Vm = MMF = NI =

∮ −→H · −→dl (2.1)

40

The magnetomotive force generates magnetic flux (Φ) in the magnetic circuit,

which depends on the net resistance of the magnetic circuit. This resistance is termed

as reluctance (Rm), which depends on the length (l), area (A) and the permeability (µ)

of the material.

Rm =l

µA(2.2)

The magnetic circuit of the electromagnet is shown in Figure 2.2. The net magne-

tomotive force is generated by the two coils, which are connected in parallel. The net

reluctance results from the upper and lower steel legs and the air gap. The air gap is

the chief contributor to the net reluctance as the permeability of air (µr = µ/µ01) is

significantly smaller than that of the laminated steel (µr ≈ 6000). The net magnetic

field intensity (B) or flux density in the central gap is obtained from the magnetic

flux flowing through the center legs.

Φ = BAair =2Vm

Rair + 2Rsteel

(2.3)

Equation (2.3) can be solved to obtain the net magnetomotive force NI required

to be produced by each coil to obtain a given magnetic flux density B. The prod-

uct NI can be achieved in several ways by choosing suitable number of turns and the

magnitude of current through the coil. The number of turns are constrained by the

available space and the diameter of the coil. The current is limited by the available

voltage source, amplifier, and the resistance of the coil. Furthermore, the current

carrying capacity of a given wire is inversely proportional to its diameter. Decreasing

the wire diameter to fit more turns could limit the current capacity of the wire, thus

reducing the MMF.

41

Rsteel

Rsteel

Rair MMF MMF

f f

Figure 2.2: Magnetic circuit of the electromagnet.

2.1.2 Electromagnet Construction and Calibration

Equations (2.1) to (2.3) give an estimate of the magnetomotive force required to

obtain desired flux density. There are several parameters such as the dimensions of

the electromagnet legs, the wire diameters, taper dimensions on the central legs, etc.,

which can not be easily calculated algebraically. Finite element analysis is therefore

used to evaluate the effect of various parameters on the flux density. FEMM, a

commercial 2 − D software, is used to run the simulations. The simulations are

conducted by defining the current density, which is the amount of current flowing

per unit cross-sectional area in the coils. The FEMM simulations also account for

the saturation effects in the laminated steel core, which are not considered in the

algebraic calculations. One example of simulation result is shown in Figure 2.3.

42

Figure 2.3: Finite element analysis of the electromagnet.

The dimensions of the electromagnet are chosen as shown in Appendix A (Sec-

tion A.1). The number of turns in each of the two coils connected in parallel is set

as 550 after accounting for losses due to the packing efficiency and leakage. The coils

are made from AWG 16 magnet wire, which have a current capacity of around 20

Amp. The air gap of 8 mm between the center legs is sufficient for accommodat-

ing the Ni-Mn-Ga sample as well as the Hall probe that is used to measure the flux

density. E-shaped transformer laminates are obtained from Tempel Steel Company.

The laminates are stacked together and are machined by Electrical Discharge Ma-

chining (EDM) to obtain the desired dimensions of the taper and the air-gap. The

coils are wound on rectangular plastic bobbins and are fitted on the center legs of the

stacked E-shaped laminates. The two E-shaped legs, which form the two halves of

43

−20 −15 −10 −5 0 5 10 15 20−800

−600

−400

−200

0

200

400

600

800

Current (Amp)

Fie

ld (

kA/m

)

Figure 2.4: Electromagnet calibration curve.

the electromagnet with coils on them are bolted together to complete the construc-

tion. The electromagnet is powered by an MBDynamics SL500VCF power amplifier

with a power rating of 1000 VA. The electromagnet is calibrated by applying a slowly

alternating sinusoidal voltage to the two coils and by measuring the magnetic field

generated in the central air gap using a Hall probe sensor. Figure 2.4 shows the

dependence of the generated magnetic field in response to the applied current.

The magnetic field varies in a linear fashion with current for a major part of

the calibration curve. The gain in the linear region is around 63.21 (kA/m)/A. The

magnetic field saturates when the current exceeds 10 Amp. The maximum field

produced by the electromagnet is around 750 (kA/m), which is sufficient to saturate

44

Ni-Mn-Ga. The variation of magnetic field in the air gap is less than 2% around the

area of the pole faces.

2.2 Experimental Characterization

As shown in Figure 2.5, the experimental setup consists of the custom built elec-

tromagnet and a uniaxial loading stage. A 6×6×20 mm3 single crystal Ni-Mn-Ga

sample (AdaptaMat Ltd.) is placed in the center gap of the electromagnet. The sam-

ple exhibits a maximum magnetic field induced deformation of 5.8%. The external

uniaxial quasistatic strain is applied using an MTS machine with Instron controller.

The electromagnet is mounted around the loading arms of the MTS machine using a

custom designed fixture (not shown). Two aluminum pushrods are used in series with

the loading arms of the MTS machine to compress the sample. They are designed to

fit in the central gap of the electromagnet and to move smoothly without friction.

Initially, the sample is converted to a single field-preferred variant configuration

by applying a transverse (x direction) DC field of 720 kA/m under zero mechani-

cal loading. This state represents the longest length of the sample, and the reference

configuration with respect to which the strain is calculated. Further, the desired mag-

nitude of bias magnetic field is applied to the material by applying a constant voltage

across the electromagnet coils. In presence of the bias field, the sample is compressed

along longitudinal (y) direction at a fixed displacement rate of 0.001 inch/sec, and

unloaded at the same rate. The applied strain thus varies according to a triangular

waveform with time. The flux density inside the material is measured using a Walker

Scientific MG-4D Gaussmeter with a transverse Hall probe with active area 1×2mm2

placed in the gap between the magnet pole and a face of the sample. The accuracy

45

of the method is confirmed by FEMM software. The small air gap ensures that the

flux density inside the sample and that acting on the Hall probe are equal. The Hall

probe measures the net flux density along the transverse (x) direction, from which the

magnetization along x-direction can be calculated. The compressive force is measured

by a 200 lb load cell, and the displacement is measured by an LVDT. The current

is measured using a monitor on the MBDynamics amplifier. The externally applied

magnetic field is obtained from the measured current using the calibration curve of

the electromagnet. This process of compressive loading and unloading is repeated un-

der varying magnitudes of bias fields ranging from 0-445 kA/m. The measured data

of force, displacement, current in the electromagnet coils and flux density is recorded

using a Dataphysics Dynamic data acquisition system.

2.2.1 Stress-Strain Behavior

Figure 2.6 shows the measured stress vs. strain curves at varied bias fields. Two

key observations are made from these plots: (i) The stress-strain behavior is highly

nonlinear and hysteretic, and (ii) the behavior changes significantly with the bias

magnetic field. The applied transverse field results in orientation of crystals with

their c-axis, i.e., magnetically easy-axis in the transverse direction, which tends to

elongate the sample. This is a consequence of the growth of martensite variants with

their c-axis in the transverse direction, termed as ‘field-preferred variants’. When

the compressive stress is applied to the sample, the twin variants with their c-axis in

longitudinal direction, termed as ‘stress-preferred variants’ tend to grow. Thus, the

compressive stress has an opposing effect to that of the applied field.

46

H

Hall probe

Load cell

ε

Ni-Mn-Ga sample

Electromagnet

Pole piece(s)

Pushrod(s)

Figure 2.5: Experimental setup for quasi-static sensing characterization.

47

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=0 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=55 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=94 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=133 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=173 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=211 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=251 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=291 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=330 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=368 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=407 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a) H=445 kA/m

Figure 2.6: Stress vs. strain plots at varied bias fields.

48

The initial steep region is where the twin variants are not mobile, and this region

indicates elastic compression. The sample exhibits a relatively high stiffness in this

region, until a specific magnitude of stress is reached. This stress is known as ‘twin-

ning stress’, which initiates the growth of the stress-preferred variants. With further

increase in stress, the twin-variant rearrangement occurs, i.e., the stress-preferred

variants grow at the expense of the field preferred variants that corresponds to a low

stiffness region. This rearrangement continues until the sample is converted to one

variant preferred by stress. When the sample is loaded further after the completion

of twin-variant rearrangement, the stress-strain curve follows a steep path indicating

a high stiffness, one-variant configuration.

The stress-strain behavior varies with the applied magnetic field. As the effect of

stress is opposite to that of the applied field, the twinning stress - the stress required

to initiate the growth of stress-preferred variants, increases with increasing bias fields.

The external stress has to do more work at higher applied fields to initiate the twin

rearrangement. The twinning stress is a characteristic of the specimen, and is a key

parameter for the model development [101].

During unloading, the stress-strain curves show reversible or irreversible behavior

depending on the magnitude of bias field. At low fields, the sample does not return to

its original configuration. The stress-induced deformation in the longitudinal direction

remains almost unchanged. This is because the energy due to the magnetic field is not

high enough to initiate the redistribution of twin variants. This irreversible behavior

is also termed as quasiplastic behavior. This effect is analogous to the actuation

behavior under zero or small load, in which the field induced strain in the sample

remains unchanged after the removal of the field, as the bias stress is not strong

49

enough to initiate growth of the stress-preferred variants to bring the sample to its

original length.

At high bias fields, the sample exhibits reversible behavior - known as magnetic

field induced superelasticity or pseudoelasticity. The energy due to the magnetic

field is sufficiently high to initiate the growth of field-preferred variants when the

sample is unloaded. This phenomenon is the ‘magnetic field induced shape memory

effect’ because the magnetic field makes the material remember its original shape

upon removal of the mechanical load. This behavior is analogous to actuation under

moderate stress, in which case the sample returns to its original dimensions after the

removal of magnetic field. For bias fields of intermediate magnitudes, the material

exhibits a partial recovery of its original shape. In this case, the field is strong enough

to initiate the twin variant growth but is not strong enough to achieve a complete

strain recovery.

2.2.2 Flux Density Behavior

Figures 2.7 and 2.8 show the dependence of flux density on strain and stress at

different bias fields. These plots are of interest for sensing applications. First key

observation is that the flux density does change in response to mechanical strain

loading, indicating that Ni-Mn-Ga “can sense”. Similar to the stress-strain behavior,

the flux density behavior changes significantly with the magnitude of the bias field.

The initial value of flux density increases with increasing bias field, which is the

property of a typical ferromagnetic material. As the bias field increases, the angle

between the magnetization vectors and the field direction decreases, which results in

high initial flux density. During loading, the absolute value of flux density decreases

50

with increasing strain and stress. As the sample is compressed from its initial field-

preferred variant state, the stress-preferred variants are nucleated at the expense of

field-preferred variants. Due to the high magnetocrystalline anisotropy of Ni-Mn-

Ga, the magnetization vectors are strongly attached to the c-axis of the crystals.

Thus the nucleation and growth of stress-preferred variants occurs in concert with

rotation of magnetization vectors towards the longitudinal direction. This results in

the reduction of the permeability and flux density in the transverse direction. It is

seen that the magnetic flux density varies almost linearly with increasing compressive

strain. Similar to the stress response, the flux density behavior during unloading

depends on the magnitude of the bias field. At low bias fields, the flux density behavior

is irreversible, whereas at high bias fields, the behavior is reversible. The high range

of strain and significant change in flux density of around 145 mT demonstrate that

the material has potential as a large-strain, low-force displacement sensor. Further

details about the variation of flux density and its relation to the stress response are

given in Section 2.3.

2.3 Discussion

Magnetomechanical characterization detailed in Section 2.2 demonstrates the fea-

sibility of using Ni-Mn-Ga as a sensor. Although an electromagnet is used for the

characterization, the eventual sensor design can be made significantly compact by

employing permanent magnets. In this section, we discuss the experimental results

in detail to understand the material behavior. This understanding is critical from the

viewpoint of model development, and for design of a sensor device.

51

0 0.01 0.02 0.03 0.04 0.05 0.060

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Compressive Strain

Flu

x D

ensi

ty (

Tes

la)

5594

133

445

251291330368

211

407

173

Figure 2.7: Flux density vs. strain at varied bias fields.

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Compressive Stress (MPa)

Flu

x D

ensi

ty (

Tes

la)

5594

133

445

251291330368

211

407

173

Figure 2.8: Flux density vs. stress at varied bias fields.

52

2.3.1 Magnetic Field Induced Stress and Flux Density Re-covery

There is a close correlation between Figure 2.6 and Figures 2.7- 2.8 regarding

the reversibility of the magnetic (flux density) and mechanical (stress) behaviors.

The change in flux density relative to the initial field-preferred single variant is di-

rectly associated with the growth of the stress-preferred variants. Thus, the flux

density value returns to its initial value only if the stress vs. strain curve exhibits

magnetic field induced pseudoelasticity, which occurs for this alloy at bias fields of

368 kA/m and higher. At high bias fields, during unloading, the magnetic energy

is high enough to initiate and complete the redistribution of variants relative to the

single stress-preferred variant formed at maximum compression. During this redis-

tribution the magnetization vectors rotate into the transverse direction, resulting in

recovery of flux density to its original value along with pseudoelastic recovery. At

fields of 133 kA/m or lower, the magnetic field energy is not strong enough to initiate

the redistribution of variants. Hence the flux density remains unchanged while the

sample is unloaded. Correspondingly the stress vs. strain curve also shows irreversible

(quasiplastic) behavior.

Figures 2.9 and 2.10 illustrate this mechanism in more detail. Figure 2.9 illustrates

the compression of a simplified, two-variant FSMA structure at low bias fields. Be-

fore the compression cycle commences, a high transverse field is applied to transform

the sample to a single field-preferred variant. In this configuration, all magnetization

vectors align themselves in the direction of the field. When a low bias field is applied,

the magnetization vectors reorient to form 180-degree stripe magnetic domains which

results in lower net flux density. The magnetization vectors remain in the transverse

53

direction, and since no external stress is yet applied, the field-preferred variant config-

uration remains intact. Note that in these schematics and the subsequent description,

the mechanism of the domain wall motion and rotation of magnetization vectors is not

included. The emphasis of the discussion is to gain a basic understanding of the mag-

netomechanical process under consideration. Additional complex mechanisms such

as domain wall motion and magnetization rotation are included during the model

development.

The compression starts at this maximum sample length, with comparatively low

net flux density, panel (a). With increasing compression, the stress-preferred variants

nucleate and grow. The variant nucleation is associated with rotation of magnetiza-

tion vectors into the longitudinal direction, as they are attached to c-axis due to high

magnetocrystalline anisotropy. This results in the reduction in flux density in trans-

verse direction, panel (b). The sample is entirely converted to stress-preferred state,

but few magnetization vectors remain in the horizontal (hard) direction depending on

the field strength, panel (c). When the sample is unloaded, the magnetic field energy

is not high enough to initiate redistribution of variants into a single field-preferred

variant state, panel (d). Hence, there is little or no change in the flux density value

after unloading, panel (e), which corresponds with the fact that the stress-strain and

flux density plots do not show any recovery for fields lower than 94 kA/m.

Figure 2.10 illustrates the effect of stress loading and unloading at high bias

fields. The initial net flux density is high when the sample is at its maximum length,

panel (a). As in the earlier case, there is a reduction in the transverse flux density

with increasing compression, panel (b). When the sample is converted to single stress-

preferred variant state, some magnetization vectors remain in the transverse direction

54

Hb

ε, σ εmax, σmax

bottom pushrod

top pushrod

ε, σ = 0

(a) (b) (c) (d) (e)

Figure 2.9: Schematic of loading and unloading at low magnetic fields.

as the bias field is large enough to force the magnetic moments to break away from the

c-axis, panel (c). When the unloading starts, the available magnetic energy is high

enough to cause the nucleation and growth of field-preferred variants, while forcing

the magnetization vectors to rotate into the transverse direction. Thus, the sample

starts elongating again, and the expanding sample tries to force on the pushrods

resulting in increasing compressive stress, panel (d). When the sample is near zero

deformation, the field is high enough to induce complete variant rearrangement, the

sample returns to its original structure thus exhibiting pseudoelastic behavior, and

the original value of flux density is also recovered, panel (e). Thus, the magnetic field

induced pseudoelasticity occurs in concert with the recovery of flux density.

55

Hb

ε, σ εmax, σmax

bottom pushrod

top pushrod

ε, σ ε = 0, σ = 0

(a) (b) (c) (d) (e)

Figure 2.10: Schematic of loading and unloading at high magnetic fields.

This correlation can also be realized from the Figure 2.8, where it can be seen that

the flux density-stress curves bear a resemblance to the conventional magnetic field

induced strain curves [78]. Under low stresses, the strain-field plots show irreversible

behavior, whereas at higher stresses the behavior is reversible. If the bias stress

is higher than the blocking force, the material shows no field induced deformation

as the applied stress is too high to allow the twin-variant rearrangement. In an

analogous manner, if the applied bias field is higher than the saturation field, there

will not be any change in flux density even when the sample is completely compressed.

This is because the magnetic field is too high to allow the rotation of magnetization

vectors in a direction perpendicular to it. An optimum compressive stress is needed

to achieve maximum field induced deformation for actuation applications. Similarly,

56

an optimum bias field is required to achieve maximum flux density change for sensing

applications.

2.3.2 Optimum Bias Field for Sensing

The flux density starts changing when the initial twinning stress is reached and

continues to change until the final twinning stress is reached, where the material con-

sists of one variant preferred by stress. The magnitude of total change in flux density

during compression dictates the sensitivity of the material. This net change in flux

density is found to initially increase with increasing bias fields and then decrease after

reaching a maximum at 173 kA/m (Figure 2.11). However, the reversible behavior

required for sensing applications is observed at bias fields of 368 kA/m and higher.

Thus, 368 kA/m can be defined as optimum field for sensing applications for the

sample under consideration.

This behavior can be explained from the easy-axis and hard-axis magnetization

curves for this alloy shown in Figure 2.12. The easy-axis curve refers to magneti-

zation of material along its easy-axis (c-axis). It is obtained by first converting the

sample to a single field-preferred variant and subsequently exposing it to a 0.5 Hz

sinusoidal transverse field while leaving it mechanically unconstrained. The easy-

axis magnetization curve has a steeper slope, and it tends to saturate at low fields,

about 120 kA/m in this case. The hard-axis curve refers to magnetization of material

along its hard-axis (other than the c-axis). To obtain the hard-axis curve, the sample

is first converted to a single stress-preferred variant. Then the sample, in a single

stress-preferred variant state having all crystals with their c-axis in longitudinal di-

rection, is subsequently exposed to a 0.5 Hz sinusoidal field while being prevented

57

0 100 200 300 400 5000

0.05

0.1

0.15

0.2

0.25

0.3

Applied Field (kA/m)

Flu

x D

ensi

ty C

hang

e (T

esla

)

Irreversible/QuasiplasticBehavior

Reversible/PseudoelasticBehavior

PartiallyReversibleBehavior

Figure 2.11: Variation of flux-density change with bias field.

from expanding. This means that the sample is magnetized along an axis other than

the c-axis i.e. the hard-axis. The hard-axis magnetization curve has a lower slope

with higher saturation field, 640 kA/m for this alloy. To magnetize the sample along

hard-axis, the externally applied field has to overcome the anisotropy energy to ro-

tate the magnetization vectors away from the c-axis which is perpendicular to field.

The mechanical constrain on the material ensures that the field-preferred variants do

not nucleate, thus maintaining the material configuration with all the crystals having

their c-axis along the longitudinal direction.

At maximum elongation for a given bias field, the flux density value corresponds

to the easy-axis value for that field, whereas at fully compressed state the flux density

value corresponds to the hard-axis value for that field. When the sample is compressed

58

−1000 −500 0 500 1000−1.5

−1

−0.5

0

0.5

1

1.5

Applied Field (kA/m)

Flu

x D

en

sity

(T

esl

a)

Hard Axis

Easy Axis

Start pointsduringcompression

End pointsduringcompression

Figure 2.12: Easy and hard-axis flux-density curves of Ni-Mn-Ga.

at a constant field, the flux density value changes from the corresponding easy-axis

value to the corresponding hard-axis value. Compression at constant field corresponds

to a straight line starting at easy-axis curve and ending at the hard-axis curve in

Figure 2.12 as shown by the arrows. Therefore at maximum compression, with all

variants being stress-preferred, the hard-axis value is the lowest flux density at given

bias field.

Hence, the maximum flux density change occurs when the two curves are at max-

imum vertical distance from each other. A large flux density change of 230 mT is

observed at a bias field of 173 kA/m. However, the optimum sensing range for re-

versible sensing behavior occurs when the two curves are at a maximum distance from

each other and the sample shows pseudoelastic behavior. At a bias field of 368 kA/m,

59

a reversible flux density change of 145 mT is obtained. Therefore the bias field of

368 kA/m is the optimum bias field for sensing for the sample under consideration.

Characterization of this bias field can enable the design of a compact sensor device

using permanent magnets.

This chapter presents characterization of sensing behavior of single crystal Ni-Mn-

Ga by measuring the dependence of the flux density and stress on strain [99, 100].

A reversible flux density change of 145 mT is observed over a range of 5.8% strain

and 4.4 MPa stress at a bias field of 368 kA/m. By way of comparison, Terfenol-D

exhibits a higher maximum sensitivity of 400 mT at a lower bias field of 16 kA/m and

higher stress range of 20 MPa [63]. However, the associated deformation is only 0.1%

due to higher Terfenol-D stiffness. The Ni-Mn-Ga alloy investigated here therefore

shows potential for high-compliance, high-displacement deformation sensors. The

complex magneto-mechanical behavior observed from the experimental characteriza-

tion is modeled using a continuum thermodynamics approach. It is discussed in the

next section.

60

CHAPTER 3

CONSTITUTIVE MODEL FOR COUPLEDMAGNETOMECHANICAL BEHAVIOR OF SINGLE

CRYSTAL NI-MN-GA

This chapter presents a continuum thermodynamics based constitutive model to

quantify the coupled magnetomechanical behavior of ferromagnetic shape memory

alloys. A single crystal Ni-Mn-Ga is considered as a continua that deforms under

magnetic and mechanical forces. A continuum thermodynamics framework is pre-

sented for a material that responds to the magnetic, mechanical and thermal stimuli.

Three internal state variables are defined to include the magnetic microstructure and

mechanical dissipation of material in the continuum framework. The constitutive

equations are derived such that the associated thermomechanical process satisfies the

restrictions posed by the law of conservation of energy, and the second law of ther-

modynamics. In order to obtain the specific expressions for the macroscopic material

response, a thermodynamic potential is defined which quantifies the contributions due

to various magnetic and mechanical energy components. The evolution equations of

the internal state variables describe the macroscopic behavior of the material, which

are obtained by making certain assumptions that are based on experimental observa-

tions. Majority of this chapter (sections 3.1 to 3.6) discusses the sensing model, which

61

quantifies the stress and magnetization dependence on strain. The model is extended

under a unified framework to quantify the actuation, and blocked-force behavior in

sections 3.7 and 3.8 respectively.

3.1 Thermodynamic Framework

The law of conservation of energy, also known as the 1st law of thermodynamics,

dictates that the rate of change of internal energy of any part S of a body is equal to

the rate of mechanical work of the net external force acting on S plus all other energies

that enter or leave S. For solids, the Lagrangian or referential form is used, where

the reference (unloaded) configuration is known. For a thermo-magneto-mechanical

solid, the conservation law is given in the local form as,

ρε = P · F + µ0

−→H · −→M + ρr −Divq, (3.1)

where ε is the specific internal energy, ρ is the density of the material in referential

coordinates, P is the First Piola-Kirchhoff stress tensor, F is the deformation gradient

tensor, r is the specific heat source inside the system and q is referential heat flux

vector representing the heat going out of the system. The term P · F represents the

stress power, or the rate of work done on the system by external mechanical action.

The term µ0

−→H · −→M represents the energy supplied to the material by a magnetic

field [96], with−→H denoting the resultant applied magnetic field vector and

−→M the net

magnetization vector inside the material. The first law assumes that the mechanical

energy can be changed to heat energy and the converse with no restrictions placed

on the transformation. Experimentally, we know the converse is subject to definite

restrictions. These restrictions in total are called the second law of thermodynamics.

62

One mathematical representation of the second law is the Clausius-Duhem in-

equality. The Clausius-Duhem inequality states that the rate of change of entropy

of part S at time t is greater than or equal to the entropy increase rate due to the

specific heat supply rate r minus the entropy decrease rate due to the heat flux rate h.

Mathematically, it is expressed in local form as,

ρη ≥ ρr

Θ−Div(

q

Θ), (3.2)

where Θ is the absolute temperature, and η is the specific entropy. In other words,

the Clausius-Duhem inequality dictates that mechanical forces and deformation can

only increase the entropy of a part S of the body.

Elimination of r from (3.1) and (3.2) gives

ρΘη − ρε + P · F + µ0H · −→M− 1

Θq ·GradΘ ≥ 0. (3.3)

In the case of the sensing behavior, the material is subjected to a uniaxial strain (ε)

along y-direction in presence of magnetic field (H) along transverse x-direction. This

results in generation of engineering stress (σ) along y-direction and magnetization (M)

along x-direction. Therefore, expression (3.3) is simplified as,

ρΘη − ρε + σε + µ0HM − 1

Θq ·GradΘ ≥ 0. (3.4)

Expression (3.4) represents the Clausius-Duhem inequality for a material that

responds to thermal, mechanical and magnetic stimuli. The quantities involved in

this inequality can be conceptually divided into the following subsets,

Independent variables: ε, M, η (3.5)

Dependent variables: σ,H, ε,q, Θ (3.6)

Balancing terms: r, ρ (3.7)

63

The independent variables or inputs of the model can be arbitrarily specified as

a function of space and time. The dependent variables or outputs are determined

through response functions (constitutive equations) which depend on the history of

the independent variables. Once the dependent variables are determined through

response functions, the balancing terms are assigned the values that are necessary

to satisfy the equations of motion. This conceptual division is chosen based on the

form of the Clausius-Duhem inequality. However, the temperature Θ is a much more

comfortable choice as independent variable instead of entropy as it is easier to measure

and control. To accomplish the change of independent variable from η to Θ, we replace

the independent variable ε with ψ through the Legendre transformation,

ψ = ε−Θη, (3.8)

where ψ is the specific Helmholtz energy potential. It is a free energy potential that

conceptually represents the energy required build a system in presence of temperature

Θ. Equation (3.8) along with (3.4) gives,

−ρψ − ρηΘ + σε + µ0HM − 1

Θq ·GradΘ ≥ 0. (3.9)

We now impose the assumption of isothermal condition. This is because the cou-

pled magnetomechanical behavior of interest in ferromagnetic shape memory Ni-Mn-

Ga occurs in the low-temperature martensite phase. The effect of changing tempera-

ture on the performance of Ni-Mn-Ga is not considered in this study. The isothermal

condition is represented as,

Θ = 0, GradΘ = 0. (3.10)

The Clausius-Duhem inequality (3.9) is reduced to a simplified form given as,

−ρψ + σε + µ0HM ≥ 0. (3.11)

64

Involved in (3.11) are the constitutive assumptions, or constitutive dependencies,

σ =σ(ε,M)

H =H(ε,M)

ψ =ψ(ε,M)

(3.12)

For majority of the applications involving magneto-mechanical materials, such as

sensing and actuation, the magnetic field is chosen as an independent variable be-

cause it is relatively easier to control by monitoring the current through a solenoid or

an electromagnet. Magnetization, on the other hand, represents the response of the

material, which is the amount of magnetic moments per unit volume. It is usually dif-

ficult to control, as it typically requires a feedback control system. To convert the set

of independent variables (ε,M) to (ε,H), we define a new thermodynamic potential

termed as specific magnetic Gibbs energy ϕ through the Legendre transformation,

ρϕ = ρψ − µ0HM. (3.13)

This leads to the inequality,

−ρϕ + σε− µ0MH ≥ 0. (3.14)

Inequality (3.14) is used to arrive at the constitutive response of the material for the

sensing case. The Clausius-Duhem inequality for modeling of the actuation behavior

is discussed in Section 3.7.

65

3.2 Incorporation of the Ni-Mn-Ga Microstructure in theThermodynamic Framework

The framework discussed in Section 3.1 pertains to thermo-magneto-mechanical

materials which have a perfect memory of their reference configuration and tem-

perature. Similar to the thermal shape memory materials, FSMAs have imperfect

memory, i.e., the materials when loaded and unloaded do not necessarily return to

their initial undeformed configuration and temperature. One of the ways to model

such a material is by introducing internal state variables in the argument list [14]. In-

ternal state variables seek to extend the results of thermoelastic theory to dissipative

materials and account for certain microstructural phenomena.

Figure 3.1 shows the microstructure of single crystal Ni-Mn-Ga in low-temperature

martensite phase. This microstructure is represented by three internal state variables:

variant volume fraction ξ, domain fraction α, and magnetization rotation angle θ.

These three variables account for the magnetic microstructure of the material and the

variant volume fraction accounts for the mechanical dissipation. This representation

of the microstructure is motivated from experimental observations of single-crystal

Ni-Mn-Ga [39], which is shown in Figure 3.2.

The applied field is oriented in the x-direction, and the applied strain (or stress)

is oriented in the y-direction. The material is divided into regions which contain the

crystals with their short axis, or magnetically easy c-axis, oriented in perpendicular

directions to each other. These regions are called variants, and their proportion

in the crystal is called as the variant volume fractions. The arrows indicate the

magnetization vectors, and Ms indicates saturation magnetization. The two variants

are separated by a twin boundary which is oriented at around 45 to the crystal axes.

66

θ

Ms

Ms

ξ

1 - ξ

1 - α

α

x

y

e

H

α 1 - α

θ

Figure 3.1: Simplified two-variant microstructure of Ni-Mn-Ga.

Figure 3.2: Image of twin-variant Ni-Mn-Ga microstructure by Scanning electronmicroscope [39].

67

A field-preferred variant, with volume fraction ξ, is one in which the magnetically

easy c-axis is aligned with the x-direction. A stress-preferred variant, with volume

fraction 1−ξ, is one in which the c-axis is aligned in the y direction. The evolution of

the twin variants is termed as twin boundary motion or twin variant rearrangement,

which results in the macroscopic deformation of the material due to the mismatch in

the crystal dimensions. The twin boundary can be driven by either magnetic field or

mechanical stress.

It is assumed that the variant volume fractions are sufficiently large to be sub-

divided into 180-degree magnetic domains with volume fractions α and 1 − α. This

domain structure minimizes the net magnetostatic energy due to finite dimensions of

the sample. In the absence of an external field, the domain fraction α = 1/2 leads

to minimum magnetostatic energy. The high magnetocrystalline anisotropy energy

of Ni-Mn-Ga dictates that the magnetization vectors in the field-preferred variant

are attached to the crystallographic c-axis, i.e., they are oriented in the direction

of the applied field or in the opposite direction. Any rotation of the magnetization

vectors away from the c-axis results in an increase in the anisotropy energy. The

magnetization vectors in the stress-preferred variant are rotated at an angle θ rela-

tive to the c-axis. These conclusions that (i) The vectors in field preferred variants

are aligned with the applied field and (ii) The angles in the two domains of stress

preferred variants are equal and opposite are reached after assuming four different

angles in four different combinations of domains and variants, and applying the same

procedure with consideration of only the magnetic components of energy, and not

mechanical. Energy minimization dictates that this angle is equal and opposite in

the two magnetic domains within a stress-preferred variant (Section A.4).

68

The concept of the thermomechanical process is now different than that described

in the Section 3.1. The independent variables are the strain ε, field H, and the internal

state variables α, θ, ξ. Therefore, the constitutive dependencies for the sensing model

are given as,

ϕ =ϕ(ε,H, α, θ, ξ)

σ =σ(ε, H, α, θ, ξ)

M =M(ε,H, α, θ, ξ).

(3.15)

The rate of magnetic Gibbs energy can be expressed using the chain rule as,

ρϕ =∂(ρϕ)

∂εε +

∂(ρϕ)

∂HH +

∂(ρϕ)

∂αα +

∂(ρϕ)

∂θθ +

∂(ρϕ)

∂ξξ (3.16)

Using (3.16) along with (3.14), we get,

−[∂(ρϕ)

∂εε +

∂(ρϕ)

∂HH +

∂(ρϕ)

∂αα +

∂(ρϕ)

∂θθ +

∂(ρϕ)

∂ξξ

]+ σε− µ0MH ≥ 0 (3.17)

This expression can be expanded as,

[σ − ∂(ρϕ)

∂ε

]ε +

[−µ0M − ∂(ρϕ)

∂H

]H + παα + πθθ + πξ ξ ≥ 0 (3.18)

in which the terms πα, πθ, and πξ represent thermodynamic driving forces respectively

associated with internal state variables α, θ, and ξ. Note that they are defined as,

πα : = −∂(ρϕ)

∂α,

πθ : = −∂(ρϕ)

∂θ,

πξ : = −∂(ρϕ)

∂ξ.

(3.19)

In inequality (3.18), the terms ε and H are independent of each other, and of other

rates. Therefore, for an arbitrary process, the coefficients of ε and H must vanish in

order for the inequality to hold. This leads to the constitutive equations,

σ =∂(ρϕ)

∂ε, (3.20)

69

M = − 1

µ0

∂(ρϕ)

∂H. (3.21)

The Clausius-Duhem inequality is reduced to,

παα + πθθ + πξ ξ ≥ 0. (3.22)

The constitutive equations or response functions for stress and magnetization

are derived. These equations describe the material response under the given set of

independent and dependent variables. Once the specific form of the magnetic Gibbs

energy potential is constructed, the expressions for the stress and magnetization can

be obtained. The energy formulation is discussed in the next section.

3.3 Energy Formulation

The total thermodynamic free energy potential is proposed to consist of the mag-

netic and mechanical components. The energy associated with the conventional mag-

netoelastic coupling is neglected, as the ordinary magnetostriction is around 100 times

lower than the strain produced due to twin variant rearrangement. Also, the energies

associated with the thermal components are neglected as only the isothermal behavior

is of concern.

3.3.1 Magnetic Energy

The total magnetic potential energy of the sample is considered as a summation

of the Zeeman energy, magnetostatic energy and the magnetocrystalline anisotropy

energy. Various magnetic energy components are given as a weighted summation of

the energies of the two variants.

70

The Zeeman energy represents the work done by the external magnetic field

on the material, or the energy available to drive twin boundary motion by magnetic

fields. As seen in (3.13), the net magnetic Gibbs energy consists of the internal or

Helmholtz energy and the Zeeman energy. The Zeeman energy is minimum when the

magnetization vectors inside the material are completely aligned in the direction of the

externally applied field, and is maximum when the magnetization vectors in the sam-

ple are in opposite direction of the externally applied field. For the sensor/actuator

model the Zeeman energy is given as,

ρϕze = ξ[−µ0HMsα + µ0HMs(1− α)] + (1− ξ)[−µ0HMs sin θ]. (3.23)

The magnetostatic energy represents the self energy of the material due to the

magnetization inside the material. It represents the energy opposing the external

work done due to magnetic field, on account of the geometry of the specimen. The

magnetization inside the sample creates a demagnetization field which tends to oppose

the externally applied field. The strength of this demagnetization field depends on

the demagnetization, which depends on the geometry of the sample. A very long

sample magnetized along its length has a very low demagnetization field as compared

to the sample magnetized along its smallest dimension. The associated energy, or

magnetostatic energy, tends to reduce the net magnetization of the material to zero

by forming 180 domain walls. The magnetostatic energy is given as,

ρϕms = ξ[1

2µ0N(Msα−Ms(1− α))2] + (1− ξ)[

1

2µ0NM2

s sin2 θ], (3.24)

where N represents the difference in the demagnetization factors along the x and y

directions [93] and it depends on the geometry of the specimen.

71

The magnetocrystalline anisotropy energy represents the energy needed to

rotate a magnetization vector away from the magnetically easy c-axis. This energy is

minimum (or zero) when the magnetization vectors are aligned along the c-axis and is

maximum when they are rotated 90 degrees away from the c-axis. In Figure 3.1, all the

contribution towards the anisotropy energy comes from the stress preferred variant.

The anisotropy energy is usually given in the form of a trigonometric power series

for uniaxial symmetry. For Ni-Mn-Ga, it has been observed that the approximation

of up to the first term is usually sufficient to express the anisotropy energy, which is

given as,

ρϕan = (1− ξ)[Ku sin2 θ]. (3.25)

The anisotropy constant, Ku, is calculated experimentally as the difference in the

area under the easy and hard axis magnetization-field curves. It represents the energy

associated with pure rotation of the magnetization vectors (hard axis) compared to

the magnetization due to zero rotation of vectors (easy axis). Thus, the parameters

required to calculate the magnetic energy component (Ms and Ku) can be obtained

from one experiment which measures the easy and hard axis magnetization curves.

The expression for contribution of magnetic energy in a given thermodynamic

potential remains unchanged when modeling sensing, actuation and blocked-force

behaviors. The magnetostatic and anisotropy energies represent the magnetic com-

ponent of the internal energy or Helmholtz energy, and the Zeeman energy represents

the work done due to the external magnetic field. Finally, the magnetic component

of the thermodynamic potential is given as,

ρϕmag = ρϕze + ρϕms + ρϕan (3.26)

72

Thus,

ρϕmag =ξ[−µ0HMsα + µ0HMs(1− α) +1

2µ0N(Msα−Ms(1− α))2]

+ (1− ξ)[−µ0HMs sin θ +1

2µ0NM2

s sin2 θ + Ku sin2 θ].

(3.27)

3.3.2 Mechanical Energy

The mechanical energy typically represents the elastic strain energy contribu-

tion towards the internal, or Helmholtz energy. In sensor model, the expression for

the mechanical energy depends on whether the process under consideration is strain

loading (ξ ≤ 0) or unloading (ξ ≥ 0). Similar to the shape memory materials, the

total strain is considered to be composed of an elastic component(εe) and a twinning

component(εtw). Moreover, the twinning strain is proposed to be linearly proportional

to the variant volume fraction,

Loading: εtw = ε0(1− ξ) (ξ ≤)

Unloading: εtw = ε0ξ(ξ ≥)(3.28)

with ε0 being the maximum twinning strain,

ε0 = 1− c/a. (3.29)

The mechanical loading arms are not glued to the sample, and the total strain

depends on the distance of the top loading arm with respect to its initial position.

Therefore, the total strain during unloading case accounts for the irreversible maxi-

mum twinning reorientation (ε0) that occurs after loading.

Loading: ε = εe + εtw

Unloading: ε = εe − εtw + ε0

The discrepancy in the two equations arises because the undeformed or reference

configuration is assumed to be in completely unloaded state, which corresponds to

73

ξ = 1. The mechanical energy equation for both loading, and unloading cases has the

form,

ρϕmech =1

2E(ξ)ε2

e +1

2a(ξ)ε2

tw (3.30)

The first term in (3.30) represents the energy due to elastic strain, and second

term represents the energy due to twinning strain. E(ξ), and a(ξ) represent effective

modulli associated with elastic and twinning strains respectively [99]. The parameters

associated with the mechanical energy component are obtained from experimental

stress-strain curve at zero bias field, shown in Figure 3.3. Modulus a(ξ) is obtained

from the slope of twinning region k by analogy with two stiffnesses in series, having

deformations equivalent to the elastic and twinning strains as,

1

a(ξ)=

1

E(ξ)− 1

k. (3.31)

The compliance (S(ξ)) of the material is considered to be a linear combination of

the compliances at complete field preferred state (S0) and complete stress preferred

state (S1). This linear average for effective material properties has been shown to be

a good approximation for the shape memory alloys by the use of micromechanical

techniques [5, 3]. Therefore, the effective modulus is given as,

E(ξ) =1

S(ξ)=

1

S0 + (1− ξ)(S1 − S0). (3.32)

The parameters (E0 = 1/S0) and (E1 = 1/S1) are obtained from the initial and final

modulli as shown in Figure 3.3.

The total magnetic Gibbs energy potential is the summation of magnetic and

mechanical components.

ρϕ = ρϕmag + ρϕmech (3.33)

74

stw0

E0

E1

k

Figure 3.3: Schematic of stress-strain curve at zero bias field.

From equations (3.20),(3.28),(3.30), and (3.33), the constitutive equation for stress

for both loading and unloading cases is given by,

σ = E(ξ)εe = E(ξ)[ε− ε0(1− ξ)] (3.34)

The constitutive equation for magnetization is obtained from (3.21) and (3.33) as,

M = Ms[2ξα− ξ + sin θ − ξ sin θ] (3.35)

The next step is to obtain the solutions for the evolution of the internal state

variables (α, θ, ξ) so that the macroscopic material response can be obtained from

(4.14) and (4.15).

3.4 Evolution of Domain Fraction and Magnetization Rota-tion Angle

The evolution of domain fraction and rotation angle is associated with the mag-

netization change only, and is not directly related to the mechanical deformation of

the material. The processes associated with the rotation of magnetization vectors

75

and evolution of domain fraction are proposed to be reversible, because the easy-axis

and hard-axis magnetization curves show negligible hysteresis. The easy-axis mag-

netization process involves evolution of domains, which is dictated by the magnitude

of the magnetostatic energy opposing the Zeeman energy due to applied field. The

hard-axis magnetization process involves the rotation of magnetization vectors with

respect to the easy c-axis of the crystals which is dictated by the competition between

the anisotropy energy and Zeeman energy. For reversible processes, the corresponding

driving forces lead to zero increase in entropy. Hence, the driving forces themselves

must be zero,

πα = −∂(ρϕ)

∂α= 0, (3.36)

πθ = −∂(ρϕ)

∂θ= 0 (3.37)

The closed form solutions for domain fraction and magnetization rotation angle

are obtained from (3.27), (3.33), (3.36), and (3.37) as,

α =H

2MsN+

1

2, (3.38)

θ = sin−1

(µ0HMs

µ0NM2s + 2Ku

)(3.39)

with the constraints, 0 ≤ α ≤ 1, and −π/2 ≤ θ ≤ π/2. The variation of domain

fraction and magnetization rotation angle is independent of variant volume fraction,

and hence external strain or deformation. The dependence of these two internal

variables on applied field is shown in Figure 3.4.

76

0 100 200 300 400 500 600 700

0.5

0.6

0.7

0.8

0.9

1

Applied Field (kA/m)

Do

mai

n F

ract

ion

(α)

0 100 200 300 400 500 600 7000

20

40

60

80

100

Applied Field (kA/m)

Ro

tati

on

An

gle

(θ0 )

Figure 3.4: Variation of (a) domain fraction, and (b) rotation angle with applied field.

77

3.5 Evolution of Volume Fraction

From (3.22) and (3.36), (3.37), the Clausius-Duhem inequality is reduced to

πξ ξ ≥ 0. (3.40)

The total thermodynamic driving force associated with the evolution of volume

fraction consists of magnetic and mechanical contributions.

πξ = πξmag + πξ

mech, (3.41)

with the magnetic and mechanical driving forces given by

πξmag = −∂(ρϕmag)

∂ξ=µ0HMsα− µ0HMs(1− α)− 1

2µ0N(Msα−Ms(1− α))2

− µ0HMs sin(θ) +1

2µ0NM2

s sin(θ)2 + Ku sin(θ)2,

(3.42)

Loading : πξmech = −∂(ρϕmech)

∂ξ=− E(ξ)[ε− ε0(1− ξ)]ε0 − 1

2

∂E(ξ)

∂ξ[ε− ε0(1− ξ)]2

+ a(ξ)(1− ξ)ε20 −

1

2

∂a(ξ)

∂ξε20(1− ξ)2.

(3.43)

Unloading : πξmech = −∂(ρϕmech)

∂ξ= −E(ξ)[ε− ε0(1− ξ)]ε0 − 1

2

∂E(ξ)

∂ξ[ε− ε0(1− ξ)]2

− a(ξ)ε20ξ −

1

2

∂a(ξ)

∂ξε20ξ

2.

(3.44)

The mechanical loading process occurs with nucleation and growth of stress-

preferred variants at the expense of field preferred variants, indicating ξ ≤ 0. The

start of the twinning process in shape memory materials and FSMAs requires the

overcoming of a finite energy threshold associated with the twinning stress. This is

78

evident from the stress-strain plots at zero field shown in Figure 3.3, and also from

strain-field plots [67, 52], where a finite threshold field needs to be overcome. The as-

sociated energy or critical driving force (πcr) required for twin variant rearrangement

to start is estimated from the twinning stress at zero field (σtw0) as,

πcr = σtw0ε0. (3.45)

This twinning barrier conceptually represents the work required to rotate a single

crystal, which is therefore the product of the associated force (σtw0) and deforma-

tion (ε0). During loading, the stress preferred variants grow at the expense of field

preferred variants, indicating ξ ≤ 0. Thus the driving force πξ is of negative value to

satisfy the inequality (3.40). The growth of stress preferred variants begins when the

total driving force reaches the negative value of the critical driving force. The value

of ξ is then obtained by numerically solving the relation,

πξ = −πcr. (3.46)

During unloading, the field-preferred variants grow indicating, ξ ≥ 0. Thus, the

driving force πξ has to be positive in order for Clausius-Duhem inequality (3.40) to be

satisfied. When the total force reaches the positive critical driving force, the evolution

of ξ is initiated. The subsequent values of ξ are obtained by numerically solving the

equation,

πξ = πcr. (3.47)

Once α, θ, and ξ are determined, the stress σ and magnetization M are found

through expressions (4.14) and (4.15), respectively. It is noted that ξ is restricted so

that 0 ≤ ξ ≤ 1.

79

3.6 Sensing Model Results

The equations in sections 3.1 to 3.5 are solved using a computation scheme built

in-house (MATLAB). The equations are solved in an interactive manner to check

the twin onset condition at each step. Also, the restrictions are imposed so certain

variables do not exceed their limits.

3.6.1 Stress-Strain Results

Calculated stress-strain plots at bias fields ranging from 94 kA/m to 368 kA/m

are compared with experimental measurements in Figure 3.5. The model parameters

are: E0 = 400 MPa, E1 = 2400 MPa, σtw0 = 0.6 MPa, k = 14 MPa, ε0 = 0.058, Ku

= 1.67E5 J/m3, Ms = 625 kA/m, N = 0.308. The initial high-slope region indicates

the elastic compression of the material, which occurs till a certain critical stress is

reached. Once the critical stress is reached, the twin variant rearrangement starts,

represented by the low-slope region. This low-slope region continues till the twin

variant rearrangement is complete. In final stages, the material again gets compressed

elastically. During unloading, the material follows a similar behavior, i.e., elastic

expansion followed by twin variant rearrangement in the reverse direction. However,

it must be noted that the behavior during unloading depends on the magnitude of

the bias field. At low bias fields, the material does not return to its original shape,

whereas at medium and high bias fields the material respectively shows a partial and

complete recovery of its original shape. Thus, the increasing bias field marks the

transition from irreversible to reversible behavior. For the various applied bias fields,

the model accurately describes the shape of the hysteresis loop and the amount of

pseudoelasticity or residual strain at which the sample returns to zero stress.

80

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a)

ExperimentModel:loadingModel:unloading

94 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a)

ExperimentModel:loadingModel:unloading

133 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a)

ExperimentModel:loadingModel:unloading

211 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a)

ExperimentModel:loadingModel:unloading

251 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a)

ExperimentModel:loadingModel:unloading

291 kA/m

0 0.01 0.02 0.03 0.04 0.05 0.06−1

0

1

2

3

4

5

6

Compressive Strain

Com

pres

sive

Str

ess

(MP

a)

ExperimentModel:loadingModel:unloading

368 kA/m

Figure 3.5: Stress vs strain plots at varied bias fields. Dotted line: experiment; solidline: calculated (loading); dashed line: calculated (unloading).

81

As the bias field is increased, more energy is required for twin variant rearrange-

ment to start, resulting in an increase in the twinning stress. The twinning stress at

a given bias field corresponds to the situation where the net thermodynamic driving

force is equal to the critical driving force (πξ = −πcr) and also the material is in com-

plete field-preferred state (ξ = 1). Therefore, an expression for the twinning stress

can be obtained as detailed below:

πξmag + πξ

mech = −πcr

At start of twinning (ξ=1),

πξmag(H)− σtw(H)ε0 = −σtw0ε0

σtw(H) =πξ

mag(H)

ε0

+ σtw0

(3.48)

Figure 3.6 shows the dependence of the twinning stress on the applied bias field,

and model comparison. The model accurately quantifies the monotonic increase in

twinning stress with increasing bias field. The deficiency of earlier model [101], where

the twinning stress was constant below fields of 195 kA/m creating discontinuity is

now overcome. The twinning stress vs. field curve shows a sigmoid shape, which

eventually saturates at high magnetic fields. This indicates that the stress-strain

behavior will remain unchanged when the magnetic fields are above saturation.

3.6.2 Flux Density Results

The calculated magnetization from the model is obtained from equation (4.15).

However, as seen earlier, the experimental measurements give the values of flux-

density. In order to compare the model results with Hall probe measurements [99],

the magnetic induction or flux-density is calculated by means of the relation,

Bm = µ0(H + NxM), (3.49)

82

0 100 200 300 400 500 600 7000.5

1

1.5

2

2.5

3

3.5

Applied Field (kA/m)

Tw

inni

ng S

tres

s (M

Pa)

Model prediction: πξmag

/ε0+σ

tw0

Experimental values (σtw

(H))

Figure 3.6: Variation of twinning stress with applied bias field.

where Nx is the demagnetization factor in the x direction [58]. It is seen that for

the same magnetization, the measured flux-density depends on the geometry of the

sample.

The flux density plots shown in Figure 3.9 are of interest for sensing applications.

The absolute value of flux density decreases with increasing compressive stress. As

the sample is compressed from its initial field-preferred variant state (ξ = 1), the

stress-preferred variants grow at the expense of field-preferred variants. Due to the

high magnetocrystalline anisotropy of NiMnGa, the nucleation and growth of stress-

preferred variants occurs in concert with the rotation of magnetization vectors into

the longitudinal direction, which causes a reduction of the permeability and flux

density in the transverse direction. The simulated curves show less hysteresis than

83

the measurements and a slight nonlinearity in the relationship between flux density

and strain. This is in agreement with measurements by Straka et al. [116] in which

the magnetization dependence on strain is almost linear with very low hysteresis. As

shown in Figure 3.9(b), the model accurately quantifies the dependence of flux density

on stress. While the tests were conducted in displacement control, the observed

trends should resemble those obtained experimentally with stress as the independent

variable.

The overall change in flux density from the initial state (ξ = 1) to the final

state (ξ = 0) is a function of applied bias field. Because of almost linear nature of

the B − ε curve, the slope of this curve at a given strain is defend as sensitivity,

or a factor similar to piezoelectric coupling coefficient at constant field,∂B

∂ε

H

. This

sensitivity factor is defined as the slope at mid-range (3% strain) in the loading path

of B − ε curve. The variation of this factor with bias field is shown in Figure 3.8.

The experimental values of sensitivity factor are approximated to the ratio of total

flux density change to the associated strain range.

As discussed in Chapter 2, this behavior can be explained from the easy- and

hard-axis flux density curves of this alloy. The easy-axis curve corresponds to a state

of the sample when it is in complete field preferred state, whereas the hard-axis curve

corresponds to the state of the sample when the sample is in complete stress-preferred

state. Therefore, the expressions for the easy-axis (ξ = 1) and hard-axis (ξ = 0)

magnetization as a function of magnetic field can be obtained from (3.38), (3.39) and

84

0 0.01 0.02 0.03 0.04 0.05 0.060

0.2

0.4

0.6

0.8

1

Compressive Strain

Flu

x D

ensi

ty (

Tes

la)

94

173

211

291

368

445Bias H (kA/m)

(a)

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Compressive Stress (MPa)

Flu

x D

ensi

ty (

Tes

la)

94

173

211

291

368

445Bias H (kA/m)

(b)

Figure 3.7: Model results for (a) flux density-strain and (b) flux density-stress curves.Dotted line: experiment; solid line: calculated (loading); dashed line: calculated(unloading).

85

0 100 200 300 400 5000

1

2

3

4

5

Applied Field (kA/m)

(∂ B

/ ∂ε)

H (

Tes

la)

Model resultExperimental values

Irreversible ReversiblePartiallyreversible

Figure 3.8: Variation of sensitivity factor with applied bias field.

(4.15). These expressions are given as,

Measy = M(ξ=1) =H

N

Mhard = M(ξ=0) =µ0HM2

s

µ0NM2s + 2Ku

If M > Ms, M = Ms

If M < −Ms, M = −Ms.

(3.50)

The model results for easy and hard axis magnetization and flux-density are shown

in Figures 3.9(a) and 3.9(b) respectively. When the sample is compressed at a given

constant field, the flux-density changes from the corresponding easy-axis value to

the corresponding hard-axis value. Hence, the optimum sensing range occurs when

the two curves are at the maximum distance from each other and the sample shows

pseudoelastic behavior. At a bias field of 368 kA/m, a reversible flux density change

86

of 145 mT is obtained over a range of 5.8% strain and 4.4 MPa stress. This makes the

magnetic field of 368 kA/m as the optimum bias field to obtain maximum reversible

sensing signal from the material. The NiMnGa alloy investigated here therefore shows

potential for high-compliance, high-displacement deformation sensors.

3.6.3 Thermodynamic Driving Force and Volume Fraction

The volume fraction dictates the deformation of the material. Also, it is the only

variable that is responsible for the coupling between the magnetic and mechanical

domains. Therefore, the evolution of volume fraction and the corresponding thermo-

dynamic driving forces provide a key insight into the material behavior. The driving

forces are calculated from equations (3.42), (3.43) and (3.44), and the volume fraction

is obtained by numerically solving the equations (3.46) and (3.47).

The evolution of the thermodynamic driving forces acting on a twin boundary

with increasing compressive strain is shown in Figure 3.10 for varied bias fields. It is

seen that the driving force due to stress is negative since the stress is compressive, and

more importantly, it opposes the growth of field volume fraction ξ. On the contrary,

the driving force due to magnetic field is positive indicating that the field favors the

growth of volume fraction ξ. During loading, the total force has to overcome the

negative critical driving force (−πcr) for twin variant rearrangement to start. Simi-

larly, during unloading, the total force has to overcome the positive critical driving

force (πcr) for the start of twin variant rearrangement in the opposite direction. The

magnitudes of total driving force during twin variant rearrangement for loading and

unloading are negative and positive, respectively, in order to satisfy Clausius-Duhem

inequality (3.40). Once the twin boundary motion is initiated, the total driving force

87

−200 0 200 400 600 800 1000−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Applied Field (kA/m)

Flu

x D

ensi

ty (

Tes

la)

Experimental (Easy Axis)Experimental (Hard Axis)Model (Easy Axis)Model (Hard Axis)

(a)

0 200 400 600 800−200

−100

0

100

200

300

400

500

600

700

Applied Field (kA/m)

Mag

netiz

atio

n (k

A/m

)

Experimental (Easy Axis)Experimental (Hard Axis)Model (Easy Axis)Model (Hard Axis)

(b)

Figure 3.9: Model results for easy and hard axis curves. (a) flux-density vs. field (b)magnetization vs. field.

88

remains at almost the same value as the critical driving force value. These principles

hold true for the actuation model also. The corresponding variation in the variant

volume fraction is shown in Figure 3.11.

There is a strong correlation between stress-strain (Figure 3.5) and flux density-

strain (Figure 3.9) curves regarding the reversibility of the magnetic and mechanical

behaviors. Because a change in flux density relative to the initial field-preferred

single variant is directly associated with the growth of stress-preferred variants, the

flux density value returns to its initial value only if the stress-strain curve exhibits

magnetic field induced pseudoelasticity. The model calculations accurately reflect this

trend, as seen by the variation of residual strain with bias field shown in Figure 3.12.

3.7 Extension to Actuation Model

In this section, the framework developed for the sensing model is extended to

model the actuation behavior of Ni-Mn-Ga, i.e., dependence of strain and magnetiza-

tion on varying field under bias compressive stress. The actuator model utilizes the

exact same parameters as the sensing model. Further, the actuation model frame-

work is consistent with previous models by Kiefer [67] and Faidley [32]. In a typical

Ni-Mn-Ga actuator, the material is subjected to a bias stress or prestress using a

spring. The initial configuration of the material is usually its shortest length (ξ = 0).

In presence of the bias stress, an external field is applied to generate strain against

the mechanical load. During increasing field ( ˙|H| > 0), the material does the work by

expanding against the prestress and strain is generated. During reverse field applica-

tion ( ˙|H| < 0), if the prestress is sufficiently large, the original length of the sample

is restored to complete one strain cycle.

89

0 0.01 0.02 0.03 0.04 0.05 0.06−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

5

Compressive Strain

The

rmod

ynam

ic D

rivin

g F

orce

(J/

m3 )

H=94 kA/m

±πcr

πmech

πmag

π

0 0.01 0.02 0.03 0.04 0.05 0.06−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

5

Compressive Strain

The

rmod

ynam

ic D

rivin

g F

orce

(J/

m3 )

H=133 kA/m

±πcr

πmech

πmag

π

0 0.01 0.02 0.03 0.04 0.05 0.06−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

5

Compressive Strain

The

rmod

ynam

ic D

rivin

g F

orce

(J/

m3 )

H=211 kA/m

±πcr

πmech

πmag

π

0 0.01 0.02 0.03 0.04 0.05 0.06−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

5

Compressive Strain

The

rmod

ynam

ic D

rivin

g F

orce

(J/

m3 )

H=251 kA/m

±πcr

πmech

πmag

π

0 0.01 0.02 0.03 0.04 0.05 0.06−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

5

Compressive Strain

The

rmod

ynam

ic D

rivin

g F

orce

(J/

m3 )

H=291 kA/m

±πcr

πmech

πmag

π

0 0.01 0.02 0.03 0.04 0.05 0.06−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

5

Compressive Strain

The

rmod

ynam

ic D

rivin

g F

orce

(J/

m3 )

H=368 kA/m

±πcr

πmech

πmag

π

Figure 3.10: Evolution of thermodynamic driving forces.

90

0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

1

Compressive Strain

Vol

ume

Fra

ctio

n

H=94 kA/m

Field preferred ( ξ)

Stress preferred (1− ξ)

0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

1

Compressive Strain

Vol

ume

Fra

ctio

n

H=133 kA/m

Field preferred ( ξ)

Stress preferred (1− ξ)

0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

1

Compressive Strain

Vol

ume

Fra

ctio

n

H=211 kA/m

Field preferred ( ξ)

Stress preferred (1− ξ)

0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

1

Compressive Strain

Vol

ume

Fra

ctio

n

H=251 kA/m

Field preferred ( ξ)

Stress preferred (1− ξ)

0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

1

Compressive Strain

Vol

ume

Fra

ctio

n

H=291 kA/m

Field preferred ( ξ)

Stress preferred (1− ξ)

0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

1

Compressive Strain

Vol

ume

Fra

ctio

n

H=368 kA/m

Field preferred ( ξ)

Stress preferred (1− ξ)

Figure 3.11: Evolution of volume fraction.

91

0 100 200 300 400 5000

0.01

0.02

0.03

0.04

0.05

0.06

Applied Field (kA/m)

Res

idua

l Str

ain

Model resultExperimental values

Figure 3.12: Variation of residual strain with applied bias field.

3.7.1 Actuation Model

In the actuator model, the applied field and bias stress constitute independent

variables, whereas the generated strain and magnetization constitute the dependent

variables. To arrive at the desired set of independent variables (H, σ) from the orig-

inal (M, ε) variables seen in (3.11), the model is formulated by defining the specific

Gibbs energy, φ, as thermodynamic potential via Legendre transform,

ρφ = ρψ − σεe − µ0HM. (3.51)

Gibbs free energy is a thermodynamic potential which conceptually represents the

amount of useful work obtainable from a system. It is obtained by subtracting the

92

work done by external magnetic field and mechanical stress from the Helmholtz en-

ergy. From (3.11) and (3.51), we get the Clausius-Duhem inequality of the form,

−ρφ− σεe − µ0MH + σ ˙εtw ≥ 0. (3.52)

where the twinning strain component is given by,

εtw = ε0ξ. (3.53)

The actuator under consideration has the constitutive dependencies,

φ =φ(σ,H, ξ, α, θ)

ε =ε(σ,H, ξ, α, θ)

M =M(σ,H, ξ, α, θ).

(3.54)

The independent variables are external field and bias stress, and dependent variables

are strain and magnetization. The domain fraction, rotation angle and variant volume

fraction constitute the internal state variables as in the sensing model. Following the

Coleman-Noll procedure similar to that employed to develop the sensing model in

Section 3.2, we arrive at the constitutive equations,

εe = −∂(ρφ)

∂σ, (3.55)

M = − 1

µ0

∂(ρφ)

∂H. (3.56)

The Clausius-Duhem inequality reduces to the form,

(−∂(ρφ)

∂ξ+ σε0

)ξ ≥ 0 (3.57)

πξ∗ ξ ≥ 0 (3.58)

93

where the total thermodynamic driving force πξ∗ is defined as,

πξ∗ = −∂(ρφ)

∂ξ+ σε0 = πξ + σε0 = πξ

mag + πξmech + σε0. (3.59)

The contribution of the magnetic energy to the total Gibbs energy, remains the same

as that given by (3.27). Therefore, the evolution equations for domain fraction (3.38),

rotation angle (3.39), and magnetic driving force (3.42) remain intact. The mechanical

energy contribution in the Gibbs energy is given by,

ρφmech = −1

2Sσ2 +

1

2aε2

0ξ2. (3.60)

The first term represents the elastic Gibbs energy due to bias stress, while the second

term represents the energy due to twinning. Unlike in the sensing model, the me-

chanical energy equation for actuation remains the same during application of both

increasing and decreasing field. The parameters associated with the mechanical en-

ergy are the same as those presented for the sensing model, except compliance. An

average value of compliance (S) is used, which is inverse of average elastic modulus E.

The undeformed configuration for the actuation process represents the sample at

its minimum length (ξ = 0) in the presence of a compressive bias stress σ. This bias

stress compresses the sample elastically, as the sample is already in the complete stress

preferred variant state. When the magnetic field is increased, the driving force due

to the field starts acting opposite to the driving force due to stress. The expression

for net mechanical thermodynamic driving force is

πξ∗mech = −aε2

0ξ + σε0. (3.61)

When the applied field is increasing ( ˙|H| ≥ 0), the volume fraction tends to increase

(ξ ≥ 0). When the total thermodynamic driving force exceeds the positive critical

94

value πcr, twin boundary motion is initiated. The numerical value of volume fraction

ξ can be obtained by solving the relation

πξ∗ = πcr. (3.62)

When the field starts decreasing ( ˙|H| ≤ 0), the stress preferred variants start growing

(ξ ≤ 0) if the field becomes sufficiently low, provided the bias compressive stress is

strong enough to start twin boundary motion in the opposite direction. If the total

driving force becomes lower than negative of critical driving force, the volume fraction

is obtained by solving,

πξ∗ = −πcr. (3.63)

Finally, from the values of α, θ, and ξ, magnetization is obtained from (4.15) and

total strain is obtained by addition of elastic and twinning components,

ε = εe + εtw (3.64)

3.7.2 Actuation Model Results

The model validation and identification of model parameters is conducted by

comparison of model results with experimental data published by Murray [88]. In this,

14×14×6 mm3 single crystal Ni-Mn-Ga sample was subjected to slowly alternating

magnetic fields of amplitude 750 kA/m in presence of compressive bias stresses ranging

from 0 to 2.11 MPa. The magnetic field was applied using an electromagnet, whereas

the bias stresses were applied using dead weights. The initial configuration of the

sample was a complete stress-preferred state, which enabled generation of full 6%

strain under saturating fields. The model parameters required for the actuation model

are same as that for the sensing model, and for the considered data their values are:

95

E = 800 MPa, σtw0 = 0.8 MPa, k = 14 MPa, ε0 = 0.058, Ku = 1.7E5 J/m3, Ms =

520 kA/m, N = 0.239.

Strain vs. Field

The model results of strain dependence on field at varied bias stresses is shown

in Figure 3.13. With increasing field, the material does not start deforming until a

certain critical field is reached, termed as coercive field. Further deformation occurs

with a rapid increase in strain for a relatively smaller range of field. This region

corresponds to the twin boundary motion where the thermodynamic driving force due

to magnetic field exceeds that due to the bias stress. Depending on the magnitude of

the bias field, a saturating strain is reached, after which the material does not deform

with further application of magnetic field. This saturation strain or the maximum

Magnetic Field Induced Strain (MFIS) is a function of the bias stress. When the field

is decreasing, the material does not return to its original shape unless the applied

bias stress is sufficiently large. The increasing bias stress marks the transition from

irreversible to reversible behavior. This effect is analogous to that of the bias field

in case of the sensing model. With increasing bias stress, the total strain produced

decreases monotonically, and the coercive field required to initiate twin boundary

motion increases. For most of the bias stress values, the model results both for the

forward and return path accurately match the measurements.

Maximum Strain

The maximum MFIS is of interest from actuation viewpoint. For the saturating

field, the maximum MFIS is obtained from the set of equations (3.65). The maximum

thermodynamic driving force at saturating field equals to the anisotropy constant, Ku.

96

0 200 400 600 800

0

0.01

0.02

0.03

0.04

0.05

0.06

Applied Field (kA/m)

Str

ain

2.11

1.63

1.43

1.160.89

0.25

Bias Stress(MPa)

Figure 3.13: Strain vs applied field at varied bias stresses. Dotted line: experiment;solid line: calculated (loading); dashed line: calculated (unloading).

The model accurately quantifies the maximum magnetic field-induced deformation at

different bias stresses ranging from 0.25 MPa to 2.11 MPa. According to the model,

the bias stresses of 0.89 MPa and 1.16 MPa can be considered as optimum where the

completely reversible strain is obtained with maximum magnitude. The comparison

with experimental values is shown in Figure 3.14.

πξmag(Hsat) + πξ

mech = πcr

ξmax =Ku + σbε0 − πcr

aε20

ε(Hsat) = Sσb + ε0ξmax

(3.65)

97

0 0.5 1 1.5 2 2.5−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

Bias compressive stress (MPa)

Max

imum

str

ain

Model resultExperimental values

Irreversible Reversible

Figure 3.14: Variation of maximum MFIS with bias stress.

Coercive Field

Coercive field is defined as the field at which the twin-variant motion starts during

forward field application ( ˙|H| ≥ 0). Evaluation of the coercive field is important as

it dictates the strength of magnetic field required to actuate the material. As seen

in Figure 3.13, once the coercive field is exceeded, the subsequent twin-variant rear-

rangement occurs with relatively smaller increase in the magnetic field. Therefore,

accurate evaluation of the coercive field gives an estimate of the magnetic field re-

quirements for the electromagnet design. The coercive field determines the resistance

to the twin boundary motion due to the added contributions of the internal mate-

rial dislocations (twinning stress) and the compressive bias stress. It is an analogous

quantity to the twinning stress in the sensing behavior: the coercive field increases

98

with increasing bias stress in actuation, whereas the twinning stress increases with

increasing bias field in sensing.

When the applied field equals the coercive field, the net thermodynamic driving

force equals the critical value (πξ∗ = πcr), and the material consists of a complete

stress-preferred state (ξ = 0). Under these conditions, the expression for the domain

fraction is reduced to, α = 1, as the magnetic field is assumed to be strong enough

to transform the material in a single domain configuration. The expression for the

magnetization rotation angle θ remains intact as given by (3.39). Using these proper-

ties, the coercive field (Hc) is obtained by solving equation (3.66), which is obtained

from (3.42), (3.45), and (3.62).

− µ0HcMs sin θ + 2µ0HcMsα− µ0HcMs − 2µ0NM2s α2 + 2µ0NM2

s α

+ σbε0 + Ku − 1

2µ0NM2

s cos2 θ −Ku cos2 θ = σtw0ε0

(3.66)

The expression for the coercive field is therefore given as,

Hc =µ0NM2

s + 2Ku −√

2µ0NM2s [Ku + ε0(σb − σtw0)] + 4Kuε0(σb − σtw0) + 4K2

u

µ0Ms

(3.67)

Although the twinning stress at zero field (σtw0) and the bias stress (σb) have

opposite signs in equation (3.67), it must be noted that the twinning stress at zero

field is defined as positive for compression whereas the bias stress is defined as negative

for compression. Thus, the bias stress adds to the resistance offered by the twinning

stress to the twin-variant rearrangement. Therefore, the coercive field increases with

increase in bias stress. Figure 3.15 shows the variation of the coercive field with the

bias stress. The model results are in a good agreement with the experimental data.

The dependence of the coercive field on the bias stress resembles a parabolic pattern,

99

0 0.5 1 1.5 2 2.5100

200

300

400

500

600

700

Bias Stress (MPa)

Coe

rciv

e F

ield

(kA

/m)

ModelExperiment

Figure 3.15: Variation of the coercive field with bias stress.

and it increases rapidly as bias stress increases. Therefore, the optimum bias stress

is desired to be as low as possible in order to keep the coercive field at a reasonably

low value. A lower coercive field facilitates a compact design of the electromagnet.

Magnetization vs. Field

The experimental data of magnetization was not available, however the model

results of magnetization dependence on field are shown in Figure 3.16. The hysteretic

magnetization curves illustrate that the volume fraction varies during the increasing

and decreasing field application. The initial part of M −H curve at all bias stresses

resembles the hard axis curve, as the material consists of only one variant preferred

by stress initially. When the twin boundary motion starts, the curve rapidly goes to

100

0 200 400 600 8000

100

200

300

400

500

600

Applied Field (kA/m)

Mag

netiz

atio

n (k

A/m

)

Bias Stress(MPa)

2.11

1.63

1.43

1.160.89

0.25

Figure 3.16: Magnetization vs applied field at varied bias stresses. Dotted line:experiment; solid line: calculated (loading); dashed line: calculated (unloading).

saturation indicating transition into field preferred variant state. When the field is

decreasing, the curve resembles to that of easy axis curve in case there is zero or very

little evolution of stress preferred variants (0.25 MPa). With increasing bias stresses,

the reverse part of magnetization curve tends to shift away from the easy axis curve.

At bias stress of 2.11 MPa, where twin boundary motion is almost suppressed, the

behavior is similar to the hard axis curve during forward and reverse field applications.

3.8 Blocked Force Model

The force generated by a Ni-Mn-Ga sample in partially blocked conditions dur-

ing actuation measurements was presented by Henry [48] and O’Handley et al. [95].

101

Their measurements suggest the presence of significant magnetoelastic coupling: as

the transverse magnetic field was increased below the field required to initiate twin

boundary motion, the measured stress increased even though the sample and spring

remained undeformed. Because a spring was used to precompress the sample in

the axial direction, some amount of detwinning was allowed and hence the block-

ing stresses were not measured. Further, no model for magnetization was presented.

Force measurements under completely mechanically-blocked conditions at different

bias strains were presented by Jaaskelainen [55] and recently by Couch [17]. Nei-

ther magnetization measurements nor analytical models were included. Likhachev

et al. [76] presented an expression for the thermodynamic driving force induced by

magnetic fields acting on the twin boundary. This force depends on the derivative of

the magnetic energy difference between the hard axis and easy axis configurations.

Although this force is useful in modeling the strain vs. field and stress vs. strain, its

origin is not well understood. This force is independent of the volume fraction, thus

it cannot accurately model the stress vs. field behavior, in which the net generated

stress varies with bias strain (see Fig. 3.20).

The available blocking stress, defined as the maximum field-induced stress relative

to the bias stress, is critical for quantifying the work capacity of an active material.

In this study we characterize and model the magnetic field-induced stress and mag-

netization generated by a commercial Ni-Mn-Ga sample (AdaptaMat Ltd.) when

it is prevented from deforming. We refer to this type of mechanical boundary as

“mechanically-blocked condition.” The material is first compressed from its longest

102

shape to a given bias strain (which requires a corresponding bias stress) and is subse-

quently subjected to a slowly alternating magnetic field while being prevented from

deforming. The tests are repeated for several bias strains.

The experimental setup is the same as that used for sensing characterization, which

consists of a custom-made electromagnet and a uniaxial stress stage. A 6x6x10 mm3

Ni-Mn-Ga sample (AdaptaMat Ltd.) is placed in the center gap of the electromag-

net. The sample exhibits a free magnetic field induced deformation of 5.8% under a

transverse field of 700 kA/m. The material is first converted to a single field-preferred

variant by applying a high transverse field, and is subsequently compressed to a de-

sired bias strain. The sample is then subjected to a sinusoidal transverse field of

amplitude 700 kA/m and frequency of 0.25 Hz. A 1x2 mm2 transverse Hall probe

placed in the gap between a magnet pole and a face of the sample measures the flux

density, from which the magnetization inside the material is obtained after accounting

for demagnetization fields. The compressive force is measured by a 200 pounds of

force (lbf) load cell, and the displacement is measured by a linear variable differential

transducer. This process is repeated for several bias strains ranging between 1% and

5.5%.

Similar to the sensing model, the applied field (H) and blocked bias strain (εb)

constitute the independent variables, whereas the magnetization component in x di-

rection (M) and stress (σ) constitute the dependent variables. The overall model

framework remains the same as in sensing model, with magnetic Gibbs energy as

thermodynamic potential. It is assumed that the volume fraction remains unchanged

103

after initial compression during the field application because of blocked configura-

tion. The value of initial volume fraction before field application is calculated from

the sensing model.

The magnetoelastic coupling is often ignored in the modeling of actuation and

sensing in Ni-Mn-Ga, in which the strains due to variant reorientation are considerably

larger than the magnetostrictive strains. This has been experimentally confirmed by

Heczko [44] and Tickle et al. [123]. The magnetoelastic energy is also ignored in the

calculation of the magnetic parameters through expressions (3.38) and (3.39), as its

contribution is around three orders of magnitude smaller than the other magnetic

energy terms. However, the contribution of the magnetoelastic coupling towards the

generation of stress in mechanically blocked conditions is significant: twin boundary

motion is completely suppressed and the magnetoelastic energy is the sole source

of stress generation when a magnetic field is applied. The magnetoelastic energy is

proposed as

ρϕme = B1εy(1− ξ)(− sin2 θ) + σ0εyξ(− sin2 θ) (3.68)

Here, B1 represents the magnetoelastic coupling coefficient [93] obtained by measuring

the maximum stress generated when the sample is biased by 5.5% (when ξ = 0), and εy

represents the magnetostrictive strain in the y direction. The first term represents the

magnetoelastic energy contribution due to magnetic fields, which contributes only in

the stress preferred variant (1-ξ). The second term represents the energy contribution

due to the initial compressive stress σ0. The applied field leads to increase of energy

in stress preferred variants, whereas the stress leads to increase of energy in field

preferred variants. The stress generated due to magnetoelastic coupling thus has the

104

form

σme = [B1(1− ξ) + σ0ξ](− sin2 θ). (3.69)

The magnetoelastic energy is not considered while evaluating the domain fraction

and magnetization rotation angle because it is around 1000 times smaller than the

Zeeman, magnetostatic, and anisotropy energies. On the other hand, the magnetoe-

lastic energy becomes significant as it is the sole source of stress generation when

field-induced deformations are prevented.

3.8.1 Results of Blocked-Force Behavior

Figure 3.17 shows experimental and calculated stress vs. applied field curves

at varied bias strains. Hysteresis is not included in the model. The significance of

magnetoelastic coupling is evident as the stress starts increasing as soon as the field is

applied, with the rotation of magnetization vectors. The increase in stress is directly

related to the angle of rotation (θ) predicted by the magnetization model. On the

contrary, the variant reorientation process is typically associated with a high amount

of coercive field that increases with increasing bias stress [67, 101]. The absence of

a coercive field, and of discontinuity in stress profiles, confirms the magnetoelastic

coupling rather than twin reorientation as origin of the stress.

Figure 3.18 shows the magnetization dependence on applied field at varied blocked

strains. The negligible hysteresis is typical of single crystal Ni-Mn-Ga when the

volume fraction is approximately constant. Thus, the model assumption of reversible

evolution of α and θ is validated along with the assumption of constant volume

fraction. This is in contrast to Figure 3.16,where the hysteresis occurs in concert with

twin variant rearrangement. The initial susceptibility of Ni-Mn-Ga varies significantly

105

−800 −600 −400 −200 0 200 400 600 8000.5

1

1.5

2

2.5

3

3.5

Applied field

Str

ess

(MP

a)

1 %

2 %

3 %

4 %

5 %Bias strain (%)

Figure 3.17: Stress vs field at varied blocked strains. Dotted: experiment; solid line:model.

with bias strains, as the M −H curve shifts between the two extreme cases of easy

axis and hard axis curves. A 59% change in susceptibility is observed over a range of

4% change in strain experimentally. Figure 3.19 shows the variation of susceptibility

with varied blocked strains. The model parameters are: E0 = 125 MPa, E1 = 2000

MPa, σtw0 = 1 MPa, k = 16 MPa, ε0 = 0.055, Ku = 2.2E5 J/m3, Ms = 700 kA/m,

N = 0.2. Magnetoelastic coefficient B1 is the maximum stress produced with 5.5%

blocked strain, which is 1 MPa.

Our mechanically-blocked measurements and thermodynamic model for constant

volume fraction describe the stress and magnetization dependence on field, and pro-

vide a measure of the work capacity of Ni-Mn-Ga. The work capacity, defined as the

106

−200 0 200 400 600 800−200

0

200

400

600

800

Applied Field (kA/m)

Mag

netiz

atio

n (k

A/m

)

Bias strain (%)

1 %

2 %

3 %

4 %

5 %

Figure 3.18: Magnetization vs field at varied blocked strains. Dashed line: experi-ment; solid line: model.

1 2 3 4 50

1

2

3

4

5

6

Applied Field (kA/m)

Initi

al s

usce

ptib

ility

Model resultExperimental values

Figure 3.19: Variation of initial susceptibility with biased blocked strain.

107

area under the σbl − σ0 curve, is 72.4 kJ/m3 for this material. This value compares

favorably with that of Terfenol-D and PZT (18-73 kJ/m3 [40]). However, the work

capacity of Ni-Mn-Ga is strongly biased towards high deformations at the expense of

low generated forces, which severely limits the actuation authority of the material.

Terfenol-D exhibits a measured stress of 8.05 MPa at a field of 25 kA/m and prestress

of −6.9 MPa [21]. The lower blocking stress of 1.47 MPa produced by Ni-Mn-Ga is

attributed to a low magnetoelastic coupling.

The maximum available blocking stress is observed at a bias strain of 3%, though

the maximum blocking stress occurs, as expected, when the sample is completely

prevented from deforming. Due to the competing effect of the stress-preferred and

field-preferred variants, the stress is highest when the volume fractions are approxi-

mately equal (ξ = 0.5) as seen in Figure 3.20.

The magnetoelastic energy in Ni-Mn-Ga is considerably smaller than the Zeeman,

magnetostatic, and anisotropy energies. The magnetostrictive strains in Ni-Mn-Ga

are of the order of 50-300 ppm [44, 123], which are negligible when compared to the

typical 6% deformation that occurs by twin-variant reorientation. The contribution

of magnetoelastic coupling can thus be ignored when describing the sensing and actu-

ation behaviors in which the material deforms by several percent strain. In the special

case of field application in mechanically-blocked condition, twin-variant reorientation

is completely suppressed and the magnetoelastic coupling becomes significant as it

remains the only source of stress generation. This is validated from the experimental

stress data as there is no coercive field associated with the twin-variant rearrange-

ment. In summary, the magnetoelastic coupling in Ni-Mn-Ga is relatively low but

becomes significant when the material is prevented from deforming.

108

−101234560

1

2

3

4

5

Bias Strain (%)

Str

ess

(M

Pa

)

σbl

σ0

σbl

−σ0

Figure 3.20: Experimental blocking stress σbl, minimum stress σ0, and availableblocking stress σbl − σ0 vs. bias strain.

3.9 Discussion

A unified magnetomechanical model based on the continuum thermodynamics

approach is presented to describe the sensing [101], actuation [103] and blocked-

force [108] behaviors of ferromagnetic shape memory Ni-Mn-Ga. The model requires

only seven parameters which are identified from two simple experiments: stress-strain

plot at zero magnetic field, and easy-axis and hard-axis magnetization curves. The

model parameter B1 is incorporated to describe the blocked-force behavior. The

model is low-order, with up to quadratic terms, which makes it convenient from the

viewpoint of FEA implementation, and incorporation in the structural dynamics of

109

a system. The model spans three magneto-mechanical characterization spaces, de-

scribing the interdependence of strain, stress, field, and magnetization. The model

accurately quantifies the dependent variables over large ranges of the bias indepen-

dent variable, which is rarely seen in literature. The magnetic Gibbs energy is the

thermodynamic potential for sensing and blocked force models, whereas the Gibbs

energy is the thermodynamic potential for actuation effect.

Several important characteristics are investigated in concert with magnetomechan-

ical characterization of single crystal Ni-Mn-Ga, along with the model predictions.

The flux density sensitivity with strain

(∂B

∂ε

)varies from 0 to a maximum value of

4.19 T/%ε at bias field of 173 kA/m, and has maximum reversible value of 2.38 T/%ε

at bias field of 368 kA/m (Figure 3.8). The stress induced due to magnetic field has

a theoretical maximum value of 2.84 MPa (Figure 3.6). The maximum field in-

duced strain has maximum reversible value of 5.8% at bias stresses of 0.89 MPa and

1.16 MPa, which are optimum for actuation (Figure 3.14). The initial susceptibility(∂M

∂H|H=0

)changes by 59% over a range of 4% strain (Figure 3.19) when mechani-

cally blocked. The maximum stress generation capacity is 1.47% at 3% strain, which

is 37% higher than that at the end values of blocked strain (Figure 3.20). These

parameters provide key insight into the magnetomechanical coupling of Ni-Mn-Ga.

Although the emphasis of the work is on a specific material-single crystal Ni-

Mn-Ga, the developed model can be applicable to any class of ferromagnetic shape

memory materials. With recent advances in increased blocking stress [61], FSMAs

are a promising new class of multi-functional smart materials. Modeling polycrys-

talline behavior is one of the future opportunities which could be explored based on

the results of this research. Possible future work could also involve extending the

110

model framework for 3-D case which will enable design of structures that incorpo-

rate FSMAs. Constitutive 3-D models will also facilitate implementation of finite

element analysis of structures to solve various magnetomechanical boundary value

problems. Several aspects of this model are also applicable to the dynamic behavior

of Ni-Mn-Ga, some of which is discussed in subsequent chapters.

111

CHAPTER 4

DYNAMIC ACTUATOR MODEL FOR FREQUENCYDEPENDENT STRAIN-FIELD HYSTERESIS

In this chapter, a model is developed to describe the relationship between mag-

netic field and strain in dynamic Ni-Mn-Ga actuators. Due to magnetic field diffusion

and structural actuator dynamics, the strain-field relationship changes significantly

relative to the quasistatic response as the magnetic field frequency is increased. The

magnitude and phase of the magnetic field inside the sample are modeled as a 1-

D magnetic diffusion problem with applied dynamic fields known on the surface of

the sample, from where an averaged or effective field is calculated. The continuum

thermodynamics constitutive model described in Chapter 3 is used to quantify the

hysteretic response of the martensite volume fraction due to this effective magnetic

field. It is postulated that the evolution of volume fractions with effective field ex-

hibits a zero-order response. To quantify the dynamic strain output, the actuator

is represented as a lumped-parameter, single-degree-of-freedom resonator with force

input dictated by the twin-variant volume fraction. This results in a second order,

linear ODE whose periodic force input is expressed as a summation of Fourier series

terms. The total dynamic strain output is obtained by superposition of strain solu-

tions due to each harmonic force input. The model accurately describes experimental

112

measurements at frequencies of up to 250 Hz. The application of this new approach is

also demonstrated for a dynamic magnetostrictive actuator to show the wider impact

of the presented work on the area of smart materials.

4.1 Introduction

As seen in the literature review (Chapter 1), most of the prior experimental and

modeling work on Ni-Mn-Ga is focused on the quasistatic actuation, i.e., dependence

of strain on magnetic field at low frequencies [65, 113]. Achieving the high saturation

fields of Ni-Mn-Ga (around 400 kA/m) requires large electromagnet coils with high

electrical inductance, which limits the effective spectral bandwidth of the material.

For this reason, perhaps, the dynamic characterization and modeling of Ni-Mn-Ga

has received limited attention.

Henry [48] presented measurements of magnetic field induced strains for drive

frequencies of up to 250 Hz and a linear model which describes the phase lag between

strain and field and system resonance frequencies. Peterson [97] presented dynamic

actuation measurements on piezoelectrically assisted twin boundary motion in Ni-Mn-

Ga. The acoustic stress waves produced by a piezoelectric actuator complement the

externally applied fields and allow for reduced field strengths. Scoby and Chen [111]

presented a preliminary magnetic diffusion model for cylindrical Ni-Mn-Ga material

with the field applied along the long axis, but they did not quantify the dynamic

strain response.

The modeling of dynamic piezoelectric or magnetostrictive transducers usually

requires the structural dynamics of the device to be coupled with the externally ap-

plied electric or magnetic fields through the active element’s strain. This is often

113

done by considering a spring-mass-damper resonator subjected to a forcing function

given by the product of the elastic modulus of the material, its cross-sectional area,

and the active strain due to electric or magnetic fields. The active strain is related

to the field by constitutive relations which can be linearized, without significant loss

of accuracy, when a suitable bias field is present [26]. The actuation response of Ni-

Mn-Ga is dictated by the rearrangement of martensite twin variants, which are either

field-preferred or stress-preferred depending on whether the magnetically easy crystal

axis is aligned with the field or the stress. The rearrangement and evolution of twin

variants with a.c. magnetic fields always exhibit large hysteresis, hence the consti-

tutive strain-field relation of Ni-Mn-Ga cannot be accurately quantified by linearized

models.

This chapter presents a new approach to quantifying the hysteretic relationship

between magnetic fields and strains in dynamic actuators consisting of a Ni-Mn-Ga

element, return spring, and external mechanical load. The key contribution of this

work is the modeling of coupled structural and magnetic dynamics in Ni-Mn-Ga ac-

tuators by means of a simple (yet accurate) framework. The framework constitutes a

useful tool for the design of actuators with straightforward geometries and provides

a set of core equations for finite element solvers applicable to more complex geome-

tries. Further, it offers the possibility of obtaining input field profiles that produce a

prescribed strain profile, which can be a useful tool in actuator control.

The model is focused on describing properties of measured Ni-Mn-Ga data [48]

observed as the frequency of the applied magnetic field is increased, as follows: (1) For

a given a.c. voltage magnitude, the maximum current and associated maximum ap-

plied field decrease due to an increase in the impedance of the coils; (2) The field at

114

zero strain (i.e., field required to change the sign of the deformation rate) increases

over a defined frequency range, indicating an increasing phase lag of the strain rela-

tive to the applied field; and (3) For a given applied field magnitude, the maximum

strain magnitude decreases and the shape of the hysteresis loop changes significantly.

It is proposed that overdamped second-order structural dynamics and magnetic field

diffusion due to eddy currents are the primary causes for the observed behaviors. The

two effects are coupled: eddy currents reduce the magnitude and delays the phase of

the magnetic field towards the center of the material, which in turn affects the corre-

sponding strain response through the structural dynamics. Magnetization dynamics

and twin boundary motion response times are considered relatively insignificant.

The model is constructed as illustrated in Figure 4.1. First, the magnitude and

phase of the magnetic field inside a prismatic Ni-Mn-Ga sample are modeled as a

1-D magnetic diffusion problem with applied a.c. fields known on the surface of the

sample. In order to calculate the bulk magnetic field-induced deformation, an effec-

tive or average magnetic field acting on the material is calculated. With this effective

field, a previous continuum thermodynamics constitutive model described in Chapter

3 [99, 101, 103], is used to quantify the hysteretic response of the martensite volume

fraction. The evolution of the volume fraction defines an equivalent forcing function

dependent on the elastic modulus of the Ni-Mn-Ga sample, its cross-sectional area,

and the maximum reorientation strain. Assuming steady-state excitation, this forcing

function is periodic and can be expressed as a Fourier series. This Fourier series pro-

vides the force excitation to a lumped-parameter, single-degree-of-freedom resonator

representing the Ni-Mn-Ga actuator. The dynamic strain response is obtained by

superposition of the strain response to forces of different frequencies.

115

Input field Diffusion

(Eddy currents)

Constitutive

model

Fourier series expansion

Structural

dynamics Dynamic strain

Figure 4.1: Flow chart for modeling of dynamic Ni-Mn-Ga actuators.

For model validation, dynamic measurements presented by Henry [48] are utilized.

A 10×10×20 mm3 single crystal Ni-Mn-Ga sample was placed between the poles

of an E-shaped electromagnet with the 10×20 mm2 sides facing the magnet poles.

The magnetic field was applied perpendicular to the longitudinal axis of the sample,

which tends to elongate it. A spring of stiffness 36 kN/m provided a compressive bias

stress of 1.7 MPa along the longitudinal axis of the sample to achieve reversible field-

induced actuation in response to cyclic fields. Figure 4.2 shows dynamic actuation

measurements. The strain response of Ni-Mn-Ga depends on the magnitude of the

applied field but not on its direction, thus giving two strain cycles per field cycle. The

frequencies shown in Figure 4.2 are the inverse of the time period of one strain cycle.

Thus, the frequency of applied field ranges from 1-250 Hz. It is also noted that the

applied field amplitude decays with increasing frequency, likely due to a combination

of high electromagnet inductance and the measurements having been conducted at

constant voltage rather than at constant current.

116

(a) (b)

Figure 4.2: Dynamic actuation data by Henry [48] for (a) 2−100 Hz (fa = 1−50 Hz)and (b) 100− 500 Hz (fa = 50− 250 Hz).

Since the experimental magnetic field waveform is not described in [48], sinusoidal

and triangular waveforms are studied. It is proposed that the experimental field

waveform deviates from an exact waveform (sinusoidal or triangular) as the applied

field frequency increases. Nonetheless, study of these two ideal waveforms provides

insight on the physical experiments.

4.2 Magnetic Field Diffusion

The application of an alternating magnetic field to a conducting material results in

the generation of eddy currents and an internal magnetic field which partially offsets

the applied field. The relationship between the eddy currents and applied fields is

117

described by Maxwell’s electromagnetic equations,

∇×H = j +∂D

∂t,

∇× E = −∂B

∂t,

∇ ·B = 0,

∇ ·D = ρe,

(4.1)

with H the magnetic field strength (A/m), j the free current density (A/m2), D the

electric flux density (C/m2), E the electric field strength (V/m), B the magnetic flux

density (T), and ρe the volume density of free charge (C/m3). The corresponding

constitutive equations are given by

j = σE,

B = µH,

D = εE,

(4.2)

where σ is the conductivity, µ is the magnetic permeability, and ε is the dielectric

constant. In the case of a stationary conductor exposed to alternating magnetic fields,

combination of (4.1)a, (4.2)a, and (4.2)c gives an expression for the Ampere-Maxwell

circuital law,

∇×∇×H = ∇× (σE) +∂

∂t[∇× (εE)]. (4.3)

After mathematical manipulation, (4.3) yields a magnetic field diffusion equation

which describes the penetration of dynamic magnetic field in a conducting medium [69].

For one-dimensional geometries, assuming that the magnetization is uniform and does

not saturate, the diffusion equation has the form,

∇2H − µσ∂H

∂t= 0, (4.4)

118

where σ is the conductivity, µ is the magnetic permeability, and ε is the dielectric

constant. The assumption of uniform magnetization is not necessarily met experi-

mentally due to nonuniform twin boundary motion [91, 85] and saturation effects.

However, comparison of model results and measurements (Section 4.4) suggests that

the simplified diffusion model is able to describe the problem qualitatively. This is

attributed to the susceptibilities of field-preferred and stress-preferred variants being

relatively close (4.7 and 1.1, respectively [99]) and not differing too much from zero

as twin boundary motion and magnetization rotation processes take place. It is also

speculated that the variants are sufficiently fine in the tested material.

The solution to (4.4) gives the magnetic field values H(x, t) at position x (inside

a material of thickness 2d) and time t. The boundary condition at the two ends is

the externally applied magnetic field. In the case of harmonic fields, the boundary

condition is given by

H(±d, t) = H0eiωt, (4.5)

where H0 is the amplitude and ω = 2πfa is the circular frequency (rad/s) of the

magnetic field on the surface of the Ni-Mn-Ga sample. Assuming no leakage flux in

the gap between the electromagnet and sample, this field is the same as the applied

field. The solution for magnetic fields inside the material has the form [69]

H(x, t) = H0 h(X) eiωt. (4.6)

119

In this expression, the complex magnitude scale factor is

h(X) = A(B + iC),

A =1

cosh2 Xd cos2 Xd + sinh2 Xd sin2 Xd

,

B = cosh X cos X cosh Xd cos Xd + sinh X sin X sinh Xd sin Xd,

C = sinh X sin X cosh Xd cos Xd − cosh X cos X sinh Xd sin Xd,

(4.7)

with

X =x

δ, Xd =

d

δ, δ =

√2

ωµσ, (4.8)

where δ is the skin depth, or the distance inside the material at which the diffused

field is 1/e times the external field. If the external field is an arbitrary periodic

function, the corresponding boundary condition is represented as a Fourier series

expansion. The diffused internal field is then obtained by superposition of individual

solutions (4.6) to each harmonic component of the applied field. Figure 4.3 shows the

variation of the internal magnetic field at different depths inside the sample. As the

depth increases, the amplitude of the magnetic field decays, accompanied by a phase

delay. For the case of triangular input fields, the amplitude decay and phase change

is accompanied by a shape change in the waveform.

4.2.1 Diffused Average Field

In order to model the bulk material behavior, an effective field acting on the mate-

rial needs to be obtained. This effective field can be used along with the constitutive

model to get the corresponding volume fraction response. To estimate the effective

magnetic field, an average of the field waveforms at various positions is calculated,

Havg(t) =1

Nx

Xd∑

X=−Xd

H(x, t). (4.9)

120

0 0.2 0.4 0.6 0.8 1−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

Nondimensional time (t*fa)

Nor

mal

ized

Fie

ld (

H/H

0)

d3d/4d/2d/40

(a)

0 0.2 0.4 0.6 0.8 1−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

Non−dimensional time (t*fa)

Nor

mal

ized

fiel

d (H

/H0)

d3d/4d/2d/40

(b)

Figure 4.3: Magnetic field variation inside the sample at varied depths for (a) sinu-soidal input and (b) triangular input. x = d represents the edge of the sample, x = 0represents the center.

121

Here, Nx represents the number of uniformly spaced points inside the material where

the field waveforms are calculated.

Figure 4.4 shows averaged field waveforms at several applied field frequencies for

sinusoidal and triangular inputs. In these simulations the resistivity has a value of

ρ = 1/σ = 6e-8 Ohm-m and the relative permeability is µr = 3. At 1 Hz, the magnetic

field intensity is uniform throughout the material and equal to the applied field H0,

and there is no phase lag. With increasing actuation frequency, the magnetic field

diffusion results in a decrease in the amplitude and an increase in the phase lag of the

averaged field relative to the field on the surface of the material. Figure 4.5 shows

the decay of the magnetic field amplitude with position inside the material at several

applied field frequencies.

When the applied field is sinusoidal, the diffused average field is also sinusoidal

regardless of frequency (Figure 4.4a). When the applied field is triangular, the shape

of the diffused average field increasingly differs from the input field as the frequency is

increased (Figure 4.4b). The corresponding strain waveforms are modified accordingly

as they are dictated by the material response to the effective averaged field. Thus, the

shape of the input field waveform can alter the final strain profile. This is discussed

in Section 4.4.

4.3 Quasistatic Strain-Field Hysteresis Model

To quantify the constitutive material response, the constitutive magnetomechani-

cal model for twin variant rearrangement is used, which is detailed in Chapter 3. The

model incorporates thermodynamic potentials to define reversible processes in combi-

nation with evolution equations for internal state variables associated with dissipative

122

0 0.2 0.4 0.6 0.8 1−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

Non−dimensional time (t*fa)

Nor

mal

ized

fiel

d (H

avg/H

0)

1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz

(a)

0 0.2 0.4 0.6 0.8 1−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

Non−dimensional time (t*fa)

Nor

mal

ized

fiel

d (H

avg/H

0)

1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz

(b)

Figure 4.4: Average field waveforms with increasing actuation frequency for (a) sinu-soidal input and (b) triangular input.

123

−5 −4 −3 −2 −1 0 1 2 3 4 50.7

0.75

0.8

0.85

0.9

0.95

1

Position (mm)

Max

imum

Nor

mal

ized

Fie

ld

1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz

(a)

−5 −4 −3 −2 −1 0 1 2 3 4 50.5

0.6

0.7

0.8

0.9

1

Position (mm)

Max

imum

Nor

mal

ized

Fie

ld

1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz

(b)

Figure 4.5: Dependence of normalized field amplitude on position with increasingactuation frequency for (a) sinusoidal input and (b) triangular input.

124

effects. The model naturally quantifies the actuation or sensing effects depending on

which variable pairs among stress, strain, magnetic field, and magnetization, are se-

lected as independent and dependent variables. For the actuation problem under

consideration, the average or effective field Havg (for simplicity denoted H from now

on) and bias compressive stress σb are the independent variables, and the strain ε

and magnetization M are the dependent variables. The constitutive actuation model

described in Section 3.7 gives the variation of the volume fraction ξ and total strain ε

on field H.

Overall model procedure remains the same as detailed earlier. A few minor changes

are made to the model to account for different initial conditions. Experimental data

collected by Henry [48] is used to validate the model results. In these measurements,

the sample is not converted to a complete stress-preferred state before the application

of field. The sample is first converted to a complete field-preferred state and is

then subjected to the given bias stress. The configuration of the sample before the

application of the field thus consists of a twin-variant structure dictated by the bias

stress. This situation is modeled by introducing a new model variable, the initial

volume fraction ξs, which represents the fraction of field preferred variants before

the application of field and after the application of the bias stress. Therefore, the

definition of the twinning strain with respect to the initial configuration and the

expression for mechanical Gibbs energy is modified. The expression for mechanical

energy is different during the forward ( ˙|H| ≥ 0) and reverse ( ˙|H| ≤ 0) application of

field.

ρφmech = − 1

2Eσ2

b +1

2aε2

0(ξ − ξs) ( ˙|H| > 0),

ρφmech = − 1

2Eσ2

b +1

2aε2

0(ξ − ξf + ξs) ( ˙|H| < 0),

(4.10)

125

The volume fraction obtained using the procedure detailed in Section 3.7. It is given

by,

ξ =πξ

mag + σbε0 + aε20ξs − πcr

aε20

( ˙|H| ≥ 0),

ξ =πξ

mag + σbε0 + aε20ξf − aε2

0ξs + πcr

aε20

( ˙|H| ≤ 0),

(4.11)

All the variables in equations (4.10) and (4.11) are defined in Chapter 3, with

the exception of ξs which is the initial volume fraction. Total strain is given by the

summation of the elastic and the twinning component as,

ε = εe + εtw = εe + ε0ξ. (4.12)

Figure 4.6 shows a comparison of model results with actuation data for a 1 Hz

applied field. The model parameters used are: ε0 = 0.04, k = 70 MPa, Ms = 0.8 T, Ku

= 1.7 J/m3, and σtw0 = 0.5 MPa. The hysteresis loop in Figure 4.6 is dominated by the

twinning strain ε0ξ (proportional to volume fraction), which represents around 99%

of the total strain. The variation of volume fraction with effective field is proposed to

exhibit a zero-order response, without any dynamics of its own, and thus independent

of the frequency of actuation. The second order structural dynamics associated with

the transducer vibrations modify the constitutive behavior shown in Figure 4.6 in the

manner detailed in Section 4.4.

4.4 Dynamic Actuator Model

The average field Havg (denoted H for simplicity) acting on the Ni-Mn-Ga sample

is calculated by applying expression (5.4) to a given input field waveform. Using

this effective field, the actuator model discussed in Section 4.3 is used to calculate

the field-preferred martensite volume fraction ξ. By ignoring the dynamics of twin

126

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

Applied Field (kA/m)

Str

ain

(%)

Figure 4.6: Model result for quasistatic strain vs. magnetic field. The circles denoteexperimental data points (1 Hz line in Figure 4.2) while the solid and dashed lines

denote model simulations for ˙|H| > 0 and ˙|H| < 0, respectively.

127

boundary motion, the dependence of volume fraction on applied field given by rela-

tions (4.11) is that of a zero-order system (ξ = f [H(t)]). Marioni et. al. [86] studied

the actuation of Ni-Mn-Ga single crystal using magnetic field pulses lasting 620 µs.

It was observed that the full 6% magnetic field induced strain was obtained in less

than 250 µs implying that the studied Ni-Mn-Ga sample has a bandwidth of around

2000 Hz. As the frequencies encountered in the present work are below 250 Hz, one

can accurately assume that twin boundary motion, and hence the evolution of vol-

ume fractions, occurs in concert with the applied field according to the dynamics of

a zero-order system.

4.4.1 Discrete Actuator Model

The mechanical properties of a dynamic Ni-Mn-Ga actuator are illustrated in Fig-

ure 4.7. Although the position of twin boundaries in the crystal affects the inertial re-

sponse of the material [84], this effect is ignored with the assumption of a lumped mass

system. The actuator is modeled as a lumped-parameter, single-degree-of-freedom,

lumped-parameter resonator in which the Ni-Mn-Ga rod acts as an equivalent spring

of stiffness EA/L, with E the modulus, A the area, and L the length of the Ni-Mn-Ga

sample. This equivalent spring is in parallel with the load spring of stiffness ke, which

is also used to pre-compress the sample. The overall system damping is represented

by ce and the combined mass of the Ni-Mn-Ga sample and output pushrod are mod-

eled as a lumped mass me. When an external field Ha(t) is applied to the Ni-Mn-Ga

sample, an equivalent force F (t) is generated which drives the motion of mass me.

A similar approach to that used for the modeling of dynamic magnetostrictive

actuators is employed. The motion of mass m is represented by a second order

128

ce

E, A, L

F(t)

me

ke

H (t) = H0ejwt

x (t)

Mechanical load

Figure 4.7: Dynamic Ni-Mn-Ga actuator consisting of an active sample (spring) con-nected in mechanical parallel with an external spring and damper. The mass includesthe dynamic mass of the sample and the actuator’s output pushrod.

differential equation,

mex + cex + kex = F (t) = −σ(t)A, (4.13)

with x the displacement of mass m. An expression for the normal stress is obtained

from constitutive relation (4.12) as,

ε = εe + εtw =σ

E+ ε0ξ, (4.14)

σ = E(ε− ε0ξ) = E(x

L− ε0ξ). (4.15)

The bias strain resulting from initial and final volume fractions (ξs, ξf ) is compensated

for when plotting the total strain. Substitution of (4.15) into (4.13) gives

mex + cex + (ke +AE

L)x = AEε0ξ. (4.16)

129

Equation (4.16) represents a second-order dynamic system driven by the volume frac-

tion. The dependence of volume fraction on applied field given by relations (4.11) is

nonlinear and hysteretic, and follows the dynamics of a zero-order system, i.e., the

volume fraction does not depend on the frequency of the applied magnetic field. This

is in contrast to biased magnetostrictive actuators, in which the drive force can be

approximated by a linear function of the magnetic field since the amount of hysteresis

in minor magnetostriction loops often is significantly less than in Ni-Mn-Ga.

4.4.2 Fourier Series Expansion of Volume Fraction

For periodic applied fields, the volume fraction also follows a periodic waveform

and hence the properties of Fourier series are utilized to calculate model solutions.

Figure 4.8 shows the calculated variation of volume fraction with time for the cases

of sinusoidal and triangular external fields. The reconstructed waveforms shown in

the figure are discussed later.

Using a Fourier series expansion, the periodic volume fraction is represented as a

sum of sinusoidal functions with coefficients

Zk =1

Ta

∫ Ta

0

ξ(t)e−iωktdt, k = 0,±1,±2, ..., (4.17)

where Ta = 1/fa, with fa the fundamental frequency. The frequency spectrum of

the volume fraction thus consists of discrete components at the frequencies ±ωk, k =

0, 1, 2...; Zk is the complex Fourier coefficient corresponding to the kth harmonic.

Equation (4.17) yields a double sided discrete frequency spectrum consisting of fre-

quencies −fs/2...fs/2, where fs = 1/dt represents the sampling frequency which

depends on the time domain resolution dt of the signal. The double sided frequency

130

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2

2.5

3

Time (sec)

ξ ε

0 (

%)

Sin: orig

Sin: recon

Tri: orig

Tri: recon

Figure 4.8: Volume fraction profile vs. time (fa = 1 Hz).

spectrum is converted to a single sided spectrum through the relations

|Z0| = |Z0| (k = 0),

|Zk| = |Zk|+ |Z−k| = 2|Zk| (k > 0).(4.18)

The phase angles remain unchanged,

∠Zk = ∠Zk (k ≥ 0). (4.19)

The reconstructed volume fraction ξr(t) is

ξr(t) = ξr(t± Ta) =K∑

k=−K

Zkeiωkt, (4.20)

in which K represents the number of terms in the series. The single sided frequency

spectrum of the volume fraction is shown in Figure 4.9 for sinusoidal and triangular

applied field waveforms. This spectrum consists of frequencies 0...fs/2. It is noted

131

0 2 4 6 8 100

0.5

1

1.5

Frequency (Hz)|

ξ |ε

0 (%

)

0 2 4 6 8 10−200

0

200

Frequency (Hz)

Ang

( ξ

) (d

eg

)

Sinusoidal

Triangular

Figure 4.9: Single sided frequency spectrum of volume fraction (fa = 1 Hz).

that the plotted spectrum has a resolution df = fa/4, as four cycles of the applied field

are included. The actuation frequency in the presented case is fa = 1 Hz. For an input

field frequency of fa Hz, the volume fraction spectrum consists of non-zero components

at frequencies 2fa, 4fa, 6fa,... Hz. Mathematically, the phase angles appear to be

leading; the physically correct phase angle values are obtained by subtracting π from

the mathematical values.

Finally, if the applied field has the form

Ha(t) = H0 sin(2πfat), (4.21)

132

with H0 constant, then the reconstructed volume fraction ξr(t) is represented in terms

of the single sided Fourier coefficients by

ξr(t) = ξr(t± Ta)K∑

k=0

|Zk| cos(2πkfat + ∠Zk). (4.22)

The reconstructed volume fraction signal overlapped over the original is shown in

Figure 4.8, for both the sinusoidal and triangular input fields. The number of terms

used is K = 20. Substitution of (4.22) into (4.16) gives,

mex + cex + (ke +AE

L)x = AEε0

K∑

k=0

|Zk| cos(2πkfat + ∠Zk), (4.23)

which represents a second-order dynamic system subjected to simultaneous harmonic

forces at the frequencies kfa, k = 0, . . . , K. The steady state solution for the net

displacement x(t) is given by the superposition of steady state solutions to each

forcing function. Thus, the steady state solution for the dynamic strain εd has the

form

εd(t) =x(t)

L=

EAε0

EA + keL

K∑

k=0

|Zk||Xk| cos(2πkfat + ∠Zk − ∠Xk). (4.24)

In (4.45), Xk represents the non-dimensional transfer function relating the force at

the kth harmonic and the corresponding displacement,

Xk =1

[1− (kfa/fn)2] + j(2ζkfa/fn)= |Xk|e−i∠Xk , (4.25)

where

|Xk| = 1√[1− (kfa/fn)2]2 + (2ζkfa/fn)2

, (4.26)

∠Xk = tan−1

(2ζkfa/fn

1− (kfa/fn)2

). (4.27)

133

The natural frequency and damping ratio in these expressions have the form

fn =1

2π

√ke + AE/L

me

, (4.28)

ζ =ce

2√

(ke + AE/L)me

. (4.29)

4.4.3 Results of Dynamic Actuation Model

Figure 4.10 shows experimental and calculated strain versus field curves for sinu-

soidal and triangular waveforms at varied frequencies. The model parameters used

are fn = 700 Hz, ζ = 0.95, ρ = 62×10−8 Ohm-m, and µr = 3. The natural frequency

is obtained by using a modulus E=166 MPa, which is estimated from the stress-

strain plots in [48]. The dynamic mass of the Ni-Mn-Ga sample and pushrods is

me=0.027 kg. It is seen that the assumption of triangular input field waveform tends

to model the higher frequency data well. This implies that the shape of the applied

field waveform may not remain exactly sinusoidal at higher frequencies. For example,

the experimental data at 250 Hz shows a slight discontinuity when the applied field

changes direction, thus verifying the proposed claim of triangular shape.

The model results match the experimental data well with the assumption of tri-

angular input field waveform, except for the case of 200 Hz. Otherwise, the model

accurately describes the increase of coercive field, the magnitude of maximum strain,

and the overall shape change of the hysteresis loop with increasing actuation fre-

quency. The lack of overshoot in the experimental data for any of the frequencies

justifies the assumption of overdamped system. The average error between the ex-

perimental data and the model results is 2.37%, which increases to 4.24% in the case

of fa = 200 Hz. The relationship between strain and field is strongly nonlinear and

134

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

Applied Field (kA/m)

Str

ain

(%)

250 Hz

1 Hz

100 Hz

50 Hz

150 Hz

200 Hz

175 Hz

(a)

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

Applied Field (kA/m)

Str

ain

(%)

250 Hz

1 Hz

100 Hz

50 Hz

150 Hz

200 Hz

175 Hz

(b)

Figure 4.10: Model results for strain vs. applied field at different frequencies for(a) sinusoidal, (b) triangular input waveforms. Dotted line: experimental, solid line:model.

135

hysteretic due to factors such as magnetic field diffusion, constitutive coupling, and

structural dynamics.

Maximum strain and hysteresis loop area

The maximum strain generated at a given frequency is of interest to understand

the dynamic properties of the system. It is observed that the applied field magnitude

decreases with increasing frequency, because the electromagnet inductance increases

with increasing frequency. As the applied field decreases, the field induced strain de-

creases too. The decay in the strain is therefore caused by the dynamics of the system

as well as the decreasing field magnitudes. Therefore, the comparison of maximum

strain at various frequencies is not useful for the available experimental data. Never-

theless, an attempt is made to understand the system behavior and gauge the model

performance by dividing the maximum strain at a given frequency by the applied

field amplitude at that frequency. Figure 4.11(b) shows variation of the normalized

maximum strain with frequency and its comparison with model calculations. The

normalized maximum strain reaches a peak at 175 Hz. However, this behavior should

not be confused with resonance, because the system is hysteretic. At 175 Hz, due to

the inductive losses, the applied field amplitude is reduced. However, this amplitude

is just sufficient to saturate the sample. Further increase in the applied field results

in negligible increase in the strain, as seen at frequencies lower than 175 Hz. There-

fore, the ratio of maximum strain over the field amplitude is maximum at 175 Hz. A

similar trend is observed in the hysteresis loop area enclosed by the strain-field curve

in half-cycle (H ≥ 0) as shown in Figure 4.11(a).

Assumption of the triangular input field waveform matches the experimental val-

ues better than assumption of the sinusoidal field. This indicates that the applied

136

0 50 100 150 200 2502

3

4

5

6

7

8x 10

−6

Actuation frequency (Hz)

Max

imum

Str

ain

/ H0

ExperimentalModel: SineModel:Triangular

(a)

0 50 100 150 200 2500

20

40

60

80

100

120

140

Actuation frequency (Hz)

Enc

lose

d A

rea×

10−

2 (kA

/m)

ExperimentalModel: SineModel: Triangular

(b)

Figure 4.11: (a) Normalized maximum strain vs. Frequency (b) Hysteresis loop areavs. Frequency

137

field waveform was either close to the triangular, or the sinusoidal waveform was dis-

torted due to the eddy current losses in the electromagnet cores. Further discussion

on the experimental data of maximum strain is given in Section 4.4.4.

4.4.4 Frequency Domain Analysis

Figure 4.12 shows a comparison of model calculations and experimental data in the

frequency domain. Only the results for triangular input field waveform are shown,

as the actual input field is proposed to be close to the triangular function from

the simulations. The frequency spectrum of the experimental strain data shows a

monotonous decay of strain magnitudes with increasing even harmonics up to an

actuation frequency of 100 Hz. For actuation frequencies from 150 Hz onwards, the

decay is not monotonous, for example, the strain magnitudes corresponding to the

4th and 6th harmonic are almost equal, with the magnitude corresponding to the 2nd

harmonic being comparatively high. This behavior is reflected in the strain-field plots

as the hysteresis loop shows increasing rounding-off for frequencies higher than 150

Hz. The model accurately describes these responses as the magnitudes match the

experimental values well for most cases. The phase angles for the experimental and

model spectra also show a good match. In some cases, the angles show a discrepancy

of about 180, though they are physically equivalent.

Figure 4.13(a) shows ‘order domain’, or ‘non-dimensional frequency domain’ spec-

trum of Fourier magnitudes of the experimental strain signal. The magnitudes cor-

responding to the zero frequency represent the average strain value in a cycle. The

variation of higher orders with actuation frequency is of interest. Though the spec-

trum under study is discrete, continuous curves are shown in Figure 4.13(a) to better

138

0 2 4 6 8 100

0.51

1.52

Non−dimensional frequency (f/fa)

|ε| (

%)

0 2 4 6 8 100

25

100

150175

Non−dimensional frequency (f/fa)

Ang

le(ε

) (d

eg)

(a)

0 2 4 6 8 100

1

Non−dimensional frequency (f/fa)

|ε| (

%)

0 2 4 6 8 100

50100150200

Non−dimensional frequency (f/fa)

Ang

le(ε

) (d

eg)

(b)

0 2 4 6 8 100

1

Non−dimensional frequency (f/fa)

|ε| (

%)

0 2 4 6 8 100

255075

100125

Non−dimensional frequency (f/fa)

Ang

le(ε

) (d

eg)

(c)

0 2 4 6 8 100

1

Non−dimensional frequency (f/fa)

|ε| (

%)

0 2 4 6 8 100

255075

100125

Non−dimensional frequency (f/fa)

Ang

le(ε

) (d

eg)

(d)

0 2 4 6 8 100

0.5

1

Non−dimensional frequency (f/fa)

|ε| (

%)

0 2 4 6 8 100

255075

100125

Non−dimensional frequency (f/fa)

Ang

le(ε

) (d

eg)

(e)

0 2 4 6 8 100

0.5

Non−dimensional frequency (f/fa)

|ε| (

%)

0 2 4 6 8 100

255075

100125

Non−dimensional frequency (f/fa)

Ang

le(ε

) (d

eg)

(f)

Figure 4.12: Model results for strain vs. applied field in frequency domain for trian-gular input waveform for (a) fa = 50 Hz, (b) fa = 100 Hz, (c) fa = 150 Hz, (d) fa

= 175 Hz, (e) fa = 200 Hz, (e) fa = 250 Hz. Dotted line: experimental, solid line:model.

139

visualize the trends of strain magnitudes. Figure 4.13(b) shows the variation of the

corresponding phase angles with harmonic order. The phase angle spectrum does

not differentiate the trends at various actuation frequencies as clearly as the mag-

nitude spectrum. Nevertheless, a correlation exists between Figure 4.13(a) and Fig-

ure 4.13(b). There is a trend of monotonic decrease at 1, 50, and 100 Hz. There

is a dip in the phase angle at 6th order for frequencies higher than 150 Hz, which is

associated with a rise in magnitude at 6th order in Figure 4.13(a).

The strain magnitudes decay almost linearly, in a monotonic fashion for actuation

frequencies up to 100 Hz. These characteristics indicate a blocky dependence in time

domain, similar to a rectified square wave signal (Figure 4.10). However, at higher

actuation frequencies, the magnitudes corresponding to the 6th order show a distinct

increase. This behavior can be attributed to the ’shape change’ of the strain-field

plots observed in Figure 4.10 at frequencies higher than 150 Hz. It is concluded that

the dynamic properties of the system show a distinct change at frequencies higher

than 150 Hz. A ’rounding off’ effect occurs in the strain-field relationship at the

higher drive frequencies.

The Fourier series magnitudes are plotted as a function of actuation frequency in

Figure 4.14(a). The variation of 2nd order with actuation frequency shows a distinct

peak at 175 Hz. For a linear system, it would have meant that the natural frequency

is near 350 Hz. However, no such conclusion can be reached for the hysteretic system

under consideration. Also, the decay of field with increasing frequencies complicates

a comparative study in the order domain.

The 6th order variation shows a peak at 150 Hz. The changes hysteresis loop shape,

and 6th order peaks associated with frequencies higher than 150 Hz may also be a

140

2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

Non−dimensional frequency (f/fa)

Str

ain

Mag

nitu

des

(%)

1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz

(a)

2 4 6 8 10−250

−200

−150

−100

−50

0

Non−dimensional frequency (f/fa)

Pha

se A

ngle

(de

g)

1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz

(b)

Figure 4.13: (a) Strain magnitude vs. harmonic order, (b) Phase angle vs. harmonicorder at varied actuation frequencies.

141

result of the decrease in the maximum applied field magnitude as seen in Figure 4.15.

If this reduced magnitude of field is applied for frequencies of 2, 50, and 100 Hz, then

the order domain spectrum at these frequencies may look similar to those for the

higher frequencies. The strain response is only dependent on the maximum applied

field, and the inertial effect of the system. However, the maximum applied field itself

depends on the inductive eddy current losses. Thus, strain response or strain order

spectrum depends on a number of different factors, which need to be analyzed in a

careful manner. The variation of the phase angles at a given order show a correlation

with Figure 4.14(b). The phase angle associated with 2nd order shows a dip at 175 Hz,

which is related the resonance of magnitude at the same frequency.

The maximum strain, maximum applied field, and their ratio is shown in Fig-

ure 4.15. The maximum applied field reduces after frequencies higher than 120 Hz.

The reason for the decay of maximum applied field is the increasing inductance of

the electromagnetic coil, and the eddy currents losses in the core. The maximum

strain also decreases with increasing frequency since its magnitude is directly related

to the maximum applied field. However, this relation is non-linear and hysteretic.

The strain to field ratio shows a clear jump at 175 Hz, which is strongly correlated

to the peak shown by the 2nd order harmonic in Figure 4.14(a). However, too much

importance should not be placed on the maximum strain to maximum field ratio as

the relationship is not linear, and this ratio can not be defined on the similar lines

as a transfer function. It is just a tool of measure for the particular case under

consideration.

142

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2

Actuation frequency (Hz)

Str

ain

Mag

nitu

des

(%)

2fa4fa6fa8fa10fa

(a)

0 50 100 150 200 250−200

−150

−100

−50

0

Actuation frequency (Hz)

Pha

se A

ngle

(de

g)

2fa4fa6fa8fa10fa

(b)

Figure 4.14: (a) Strain magnitude vs. actuation frequency, (b) Phase angle vs. actu-ation frequency at varied harmonic orders.

143

0 50 100 150 200 2500.4

0.5

0.6

0.7

0.8

0.9

1

Actuation frequency (Hz)

Nor

mal

ized

Str

ain

and

Fie

ld

εmax

Hmax

εmax

/Hmax

Figure 4.15: Variation of maximum strain and field with actuation frequency.

4.5 Conclusion

A model is presented to describe the dependence of strain on applied field at

varied frequencies in ferromagnetic shape memory Ni-Mn-Ga [107, 106]. The essential

components of the model include magnetomechanical constitutive responses, magnetic

field diffusion, and structural dynamics. The presented method can be extended to

arrive at the input field profiles which will result in the desired strain profile at a

given frequency. If the direction of flow in Figure 4.1 is reversed, the input field

profile can be designed from a desired strain profile. It is comparatively easy to

obtain the inverse Fourier transform, whereas calculation of the average field from a

144

desired strain profile through the constitutive model, and estimation of the external

field from the averaged diffused field inside the sample, can be complex.

The frequency spectra of the field-preferred volume fraction and the resulting

dynamic strain include even harmonics. The corresponding magnitudes at the 2nd

harmonic are comparatively high indicating frequency doubling similar to that asso-

ciated with magnetostrictive actuators. However, additional components at higher

harmonics are present due to the large hysteresis in FSMAs compared to biased

magnetostrictive materials. If the overall system including the active material is un-

derdamped, then it is possible to achieve system resonance at a frequency which is

1/4th or 1/6th of the system natural frequency. In magnetically-active material ac-

tuators, the application of magnetic fields at high frequencies becomes increasingly

difficult as the coil inductances tend to increase rapidly. If the actuator can be made

to resonate at a fraction of the system natural frequency, then this problem can be

simplified. However, the strain magnitudes corresponding to the higher harmonics

tend to diminish rapidly as well, which creates a compensating effect. Further, in

some cases the system natural frequency and damping may be beyond the control of

the designer. Nevertheless, our approach suggests a way to drive a magnetic actuator

at a fraction of the natural frequency to achieve resonance. A case study on a mag-

netostrictive actuator is presented in Section 4.6 to demonstrate the wide application

of this presented approach.

4.6 Dynamic Actuation Model for Magnetostrictive Materi-als

Magnetostrictive materials deform when exposed to magnetic fields and change

their magnetization state when stressed. These behaviors are nonlinear, hysteretic,

145

and frequency-dependent. Several models exist for describing the dependence of strain

on field at quasi-static frequencies. The strain-field behavior changes significantly rel-

ative to the quasi-static case as the frequency of applied field is increased. Modeling

the dynamic strain-field hysteresis has been a challenging problem because of the

inherent nonlinear and hysteretic behavior of the magnetostrictive material along

with the complexity of dynamic magnetic losses and structural vibrations of the

transducer device. Prior attempts use mathematical techniques such as the Preisach

model [121, 22, 4] and genetic algorithms [9]. A phenomenological approach including

eddy currents and structural dynamics was recently presented [54].

Chief intent of this section is to present a new approach for modeling the strain-

field hysteresis relationship of magnetostrictive materials driven with dynamic mag-

netic fields in actuator devices (Figure 4.1). The approach builds on the prior model

for dynamic hysteresis in ferromagnetic shape memory Ni-Mn-Ga [107] discussed ear-

lier in this chapter.

Magnetic Field Diffusion

As seen in Section 4.2, application of an alternating magnetic field to a conduct-

ing material such as magnetostrictive Terfenol-D results in the generation of eddy

currents and an internal magnetic field which partially offsets the applied field. The

relationship between the eddy currents and applied fields is described by Maxwell’s

electromagnetic equations. Assuming that the magnetization is uniform and does not

saturate, the diffusion equation describing the magnetic field inside a one-dimensional

conducting medium of cylindrical geometry has the form [69],

∂2H

∂r2+

1

r

∂H

∂r= µσ

∂H

∂t, (4.30)

146

where σ is electrical conductivity and µ is magnetic permeability. Cylindrical diffu-

sion equation is used because the typical geometry for magnetostrictive Terfenol-D

transducers is in the form of cylindrical rods.

For harmonic applied fields, the boundary condition at the edge (r = R) of the

cylindrical rod is given by,

H(R, t) = H0eiωt (4.31)

where H0 is the amplitude and ω = 2πfa is the circular frequency (rad/s) of the

magnetic field on the surface of the magnetostrictive material. The solution to (4.30)

gives the magnetic field values H(r, t) at radius r and time t. This solution is given

as,

H(x, t) = H0 h(R) eiωt. (4.32)

Therefore the diffusion equation (4.30) is transformed to,

d2h

dR2 +

1

R

dh

dR− h = 0, (4.33)

where the normalized complex and real radii are given as,

R =

√2ir

δ=

(1 + i)r

δ, Ra =

√2ia

δ

R =

√2r

δ, Ra =

√2a

δ, δ =

√2

ωµσ

(4.34)

This equation is solved by modified Bessel functions [69] of the first and second

kind and of order zero:

h(R) = CI0(R) + DK0(R) (4.35)

where the constants C and D are determined by the boundary conditions for the

specific problem.

147

For a solid cylindrical conductor, D = 0 since H remains finite for r = 0 and

K0(R) = 0. The constant C is determined by boundary condition (4.31):

h(R) =I0(R)

I0(Ra)(4.36)

Introducing the Kelvin functions [69],

I0(R) = ber(R) + ibei(R), (4.37)

and rearranging gives,

h(R) = heiα =berRberRa + beiRbeiRa + i(beiRberRa − berRbeiRa)

ber2Ra + bei2Ra

. (4.38)

The functions ber and bei are Kelvin functions, and their expressions are given

in Section B.2. To estimate the effective field, an average magnetic field is obtained

by integrating over the cross-section of the sample. For discrete points, this process

is similar to that of taking the average as given by equation (5.4). The difference for

the cylindrical geometry is that the number of points at a given radius are directly

proportional to that radius. This equation is given by,

Havg(t) =

(1∑Nr=Na

Nr=1 Nr

)r=a∑r=0

NrH(r, t). (4.39)

Figure 4.16 shows the average field waveforms at several applied field frequencies.

With increasing frequency, the magnetic field diffusion results in a decrease in the

amplitude and an increase in the phase lag of the averaged field relative to the field

on the surface of the material.

Actuator Structural Dynamics

It is proposed that the material response is dictated by this averaged field. The

constitutive material response is obtained from the Jiles-Atherton model [59] in com-

bination with a quadratic model for the magnetostriction. It is assumed that the

148

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Non−dimensional time (t*fa)

Nor

mal

ized

Fie

ld (

Hav

g/H

0)

10 Hz100 Hz500 Hz800 Hz1000 Hz1250 Hz1500 Hz2000 Hz

IncreasingFrequency

Figure 4.16: Normalized average field vs. non-dimensional time.

relationship between magnetostriction and field does not include additional dynamic

effects. The process to obtain the magnetostriction has been detailed before [59]. The

magnetostriction is assumed to be dependent on the square of magnetization as,

λ =3

2

(M

Ms

)2

, (4.40)

with λ magnetostriction, M magnetization, and Ms saturation magnetization. The

quadratic relationship is justified by the use of a sufficiently large bias stress in the

magnetostrictive material [21]. Under low magnetic fields, the total strain (ε) is given

by the superposition of the magnetostriction and elastic strain,

ε = λ + σ/E, (4.41)

in which E is the open-circuit elastic modulus. Note that equation (4.41) gives the

material response to a dynamic average field. However, the response of a dynamic

149

actuator including a magnetostrictive driver and external load must be obtained by

incorporating structural dynamics.

A dynamic magnetostrictive actuator is illustrated in Figure 4.17. The actuator

is modeled as a 1-DOF lumped-parameter resonator in which a magnetostrictive rod

acts as an equivalent spring of stiffness EA/L, with A the area and L the length. This

equivalent spring is in parallel with the load spring of stiffness ke, which is also used

to pre-compress the sample. The overall system damping is represented by lumped

damping coefficient ce; the combined mass of the magnetostrictive sample and output

pushrod are modeled as a lumped mass me. When an external field Ha(t) is applied

to the sample, an equivalent force F (t) is generated which drives the motion of the

mass.

ce

E, A, L

F(t)

me

ke

H (t) = H0ejwt

x (t)

Mechanical load

Magnetostric!ve Rod

Figure 4.17: Dynamic magnetostrictive actuator.

The dynamic system equation is written as

mx + cx + kx = F (t) = −σ(t)A, (4.42)

150

with x the displacement of mass m. Substitution of (4.41) into (4.42) combined with

ε = x/L gives

mex + cex + (ke +AE

L)x = AEλ(t). (4.43)

Equation (4.43) represents a second-order dynamic system driven by the mag-

netostriction. The dependence of magnetostriction on applied field is nonlinear and

hysteretic, and follows the dynamics of a zero-order system, i.e., the magnetostriction

does not depend on the frequency of the applied magnetic field. For periodic applied

fields, the magnetostriction also follows a periodic waveform and hence the properties

of Fourier series are utilized to express the magnetostriction as

λ(t) =N∑

n=0

|Λn| cos(2πnfat + ∠Λn), (4.44)

where |Λn| and ∠Λn respectively represent the magnitude and angle of the nth har-

monic of actuation frequency fa. The term AEλ(t) represents an equivalent force that

dictates the dynamic response of the actuator. Using the superposition principle, the

total dynamic strain (εd) is given by

εd(t) =x(t)

L=

EA

EA + keL

N∑n=0

|Λn|([1− (nfa/fn)2]2 + (2ζnfa/fn)2

)−1/2

cos

[2πnfat + ∠Λn − tan−1

(2ζnfa/fn

1− (nfa/fn)2

)],

(4.45)

with fn natural frequency and ζ damping ratio.

Model Results

Figure 4.18 shows a comparison of model results and experimental measurements

collected from a Terfenol-D transducer [8]. The model parameters, which remain

the same at all the frequencies, are: µ = 5µ0, 1/σ = 58e − 8Ωm, fn = 1150 Hz, and

151

ζ = 0.2. The model accurately describes the changing hysteresis loop shape and peak-

to-peak strain magnitude with increasing frequency. These results show improvement

over previous work using the same data [54]. With increasing frequency, the strain lags

behind the applied field due to the combined contributions of the system vibrations

and dynamic magnetic losses.

−10 −5 0 5 10−4

−2

0

2

4x 10

−4

Applied field (kA/m)

Str

ain

1500 Hz−10 −5 0 5 10−4

−2

0

2

4x 10

−4

Applied field (kA/m)

Str

ain

2000 Hz

−10 −5 0 5 10−4

−2

0

2

4x 10

−4

Applied field (kA/m)

Str

ain

800 Hz−10 −5 0 5 10−4

−2

0

2

4x 10

−4

Applied field (kA/m)

Str

ain

1000 Hz−10 −5 0 5 10−4

−2

0

2

4x 10

−4

Applied field (kA/m)

Str

ain

1250 Hz

−10 −5 0 5 10−4

−2

0

2

4x 10

−4

Applied field (kA/m)

Str

ain

10 Hz−10 −5 0 5 10−4

−2

0

2

4x 10

−4

Applied field (kA/m)

Str

ain

100 Hz−10 −5 0 5 10−4

−2

0

2

4x 10

−4

Applied field (kA/m)

Str

ain

500 Hz

Figure 4.18: Strain vs. applied field at varied actuation frequencies. Dashed line:experimental, solid line: model.

152

The maximum strain and largest hysteresis loop area are seen at a frequency

near resonance (1000 Hz) indicating a phase angle of about -90. As the frequency

increases beyond resonance, the strain magnitude diminishes rapidly accompanied by

further delay of the phase angle.

Model results and experimental data are shown in the non-dimensional frequency

domain or harmonic order domain in Figure 4.19. It is noted that the frequency

spectra contain the contribution of higher harmonics of the actuation frequency be-

cause of the hysteretic nature of the system. However, Terfenol-D exhibits relatively

small hysteresis compared to ferromagnetic shape memory alloys such as Ni-Mn-Ga.

Therefore, the contribution of higher harmonics of the actuation frequency is not as

significant as seen in Ni-Mn-Ga [107]. Figure 4.20 shows the variation of the mag-

nitude and phase of the first harmonic. Note that the frequency at which the peak

strain magnitude is observed (1000 Hz), occurs below the mechanical resonance fre-

quency (1150 Hz). This is because the contribution of actuator dynamics to the phase

angle is complemented by the phase angle due to the diffusion. Thus, the phase angle

of −90 deg and hence the corresponding maximum strain magnitude occur below

mechanical resonance.

Discussion

A model is presented to describe the dependence of strain on applied fields in

dynamic magnetostrictive actuators [105]. The essential components of the model

include the magnetomechanical constitutive response (obtained through the Jiles-

Atherton model), magnetic field diffusion, and actuator dynamics. Our intuitive and

physics-based approach has been successfully implemented for two classes of mag-

netically activated smart materials: Terfenol-D and Ni-Mn-Ga [107]. The presented

153

0 2 4 6 80

5

10

15

Harmonic Order (f/fa)

Str

ain

Ma

g. x

10

5

1500 Hz

Experimental

Model

0 2 4 6 80

5

10

15

20

25

Harmonic Order (f/fa)

Str

ain

Ma

g. x

10

5

800 Hz

Experimental

Model

0 2 4 6 80

1

2

3

4

Harmonic Order (f/fa)

Str

ain

Ma

g. x

10

5

2000 Hz

Experimental

Model

0 2 4 6 80

10

20

30

40

Harmonic Order (f/fa)

Str

ain

Ma

g. x

10

5

1000 Hz

Experimental

Model

0 2 4 6 80

5

10

15

20

25

Harmonic Order (f/fa)

Str

ain

Ma

g. x

10

5

1250 Hz

Experimental

Model

0 2 4 6 80

5

10

15

20

Harmonic Order (f/fa)

Str

ain

Ma

g. x

10

5

10 Hz

Experimental

Model

0 2 4 6 80

5

10

15

20

Harmonic Order (f/fa)

Str

ain

Ma

g. x

10

5

100 Hz

Experimental

Model

0 2 4 6 80

5

10

15

20

25

Harmonic Order (f/fa)

Str

ain

Ma

g. x

10

5

500 Hz

Experimental

Model

Figure 4.19: Frequency domain strain magnitudes at varied actuation frequencies.Dashed line: experimental, solid line: model.

method can be extended to arrive at the input field profiles which will result in the

desired strain profile at a given frequency. If the direction of flow in Figure 4.1 is

reversed, the input field profile can be designed from a desired strain profile. It is

comparatively easy to obtain the inverse Fourier transform, whereas calculation of

the average field from a desired strain profile through a constitutive model, and esti-

mation of the external field from the averaged diffused field inside the sample can be

complex.

154

The frequency spectra of the strain include even and odd harmonics. The con-

tribution of higher harmonics is very small because the Terfenol-D actuator under

consideration is biased with a field of 16 kA/m, which results in reduced hysteresis.

An unbiased actuator exhibits larger hysteresis and would consist of only even har-

monics, with increased contribution of the higher harmonics. The biased actuator

resonates when the applied field frequency is close to the natural frequency of the

actuator, whereas an unbiased actuator resonates when the applied field frequency

is half of the natural frequency. Our approach can successfully model the unbiased

actuator configuration also as seen for Ni-Mn-Ga in the earlier sections of this chapter.

155

0 500 1000 1500 20000

5

10

15

20

25

30

35

Actuation frequency (Hz)

Str

ain

Mag

. x 1

05

ExperimentalModel

0 500 1000 1500 2000−250

−200

−150

−100

−50

0

Actuation frequency (Hz)

Pha

se a

ngle

(de

g)

ExperimentalModel

Figure 4.20: Variation of (a) magnitude and (b) phase of the first harmonic.

156

CHAPTER 5

DYNAMIC SENSING BEHAVIOR: FREQUENCYDEPENDENT MAGNETIZATION-STRAIN HYSTERESIS

This chapter addresses the characterization and modeling of NiMnGa for use as

a dynamic deformation sensor. The flux density is experimentally determined as a

function of cyclic strain loading at frequencies from 0.2 Hz to 160 Hz. With in-

creasing frequency, the stress-strain response remains almost unchanged whereas the

flux density-strain response shows increasing hysteresis. This behavior indicates that

twin-variant reorientation occurs in concert with the mechanical loading, whereas

the rotation of magnetization vectors occurs with a delay as the loading frequency

increases. The increasing hysteresis in magnetization must be considered when utiliz-

ing the material in dynamic sensing applications. A modeling strategy is developed

which incorporates magnetic diffusion and a linear constitutive equation.

5.1 Experimental Characterization of Dynamic Sensing Be-havior

This section details the experimental characterization of the dependence of flux

density and stress on dynamic strain at a bias field of 368 kA/m for frequencies of up

to 160 Hz, with a view to determining the feasibility of using Ni-Mn-Ga as a dynamic

157

H

Hall probe

Load cell

ε

Ni-Mn-Ga sample

Electromagnet

Pole piece(s)

Pushrod(s)

Figure 5.1: Experimental setup for dynamic magnetization measurements.

deformation sensor. This bias field was determined as optimum for obtaining maxi-

mum reversible flux density change [99] as seen in Section 2.3.2. The measurements

also illustrate the dynamic behavior of twin boundary motion and magnetization

rotation in Ni-Mn-Ga. As shown in Fig. 5.1, the experimental setup consists of a

custom designed electromagnet and a uniaxial MTS 831 test frame. This frame is

designed for cyclic fatigue loading, with special servo valves which allow precise stroke

control up to 200 Hz. The setup is similar to that described in Section 2.2 for the

characterization of the quasi-static sensing behavior. The custom-built electromagnet

described in Section 2.1 is used along with the MTS frame.

158

A 6×6×10 mm3 single crystal NiMnGa sample (AdaptaMat Ltd.) is placed in

the center gap of the electromagnet. In the low-temperature martensite phase, the

sample exhibits a free magnetic field induced deformation of 5.8% under a transverse

field of 700 kA/m. The material is first converted to a single field-preferred variant

by applying a high field along the transverse (x) direction, and is subsequently com-

pressed slowly by a strain of 3.1% at a bias field of 368 kA/m. While being exposed to

the bias field, the sample is further subjected to a cyclic uniaxial strain loading of 3%

amplitude (peak to peak) along the longitudinal (y) direction at a desired frequency.

This process is repeated for frequencies ranging between 0.2 Hz and 160 Hz. The flux

density inside the material is measured by a Hall probe placed in the gap between a

magnet pole and a face of the sample. The Hall probe measures the net flux density

along the x-direction, from which the x-axis magnetization can be calculated. The

compressive force is measured by a load cell, and the displacement is measured by a

linear variable differential transducer. The data is recorded using a dynamic data ac-

quisition software at a sampling frequency of 4096 Hz. All the measuring instruments

have a bandwidth in the kHz range, well above the highest frequency employed in

the study.

Fig. 5.2(a) shows stress versus strain measurements for frequencies ranging from

4 Hz to 160 Hz. The strain axis is biased around the initial strain of 3.1%. These

plots show typical pseudoelastic minor loop behavior associated with single crystal

Ni-Mn-Ga at a high bias field. With increasing compressive strain, the stress increases

elastically, until a critical value is reached, after which twin boundary motion starts

and the stress-preferred variants grow at the expense of the field-preferred variants.

During unloading, the material exhibits pseudoelastic reversible behavior because

159

the bias field of 368 kA/m results in the generation of field-preferred variants at the

expense of stress-preferred variants.

The flux density dependence on strain shown in Fig. 5.2(b) is of interest for sensing

applications. The absolute value of flux density decreases with increasing compres-

sion. During compression, due to the high magnetocrystalline anisotropy of NiMnGa,

the nucleation and growth of stress-preferred variants is associated with rotation of

magnetization vectors into the longitudinal direction, which causes a reduction of

the permeability and flux density in the transverse direction. At low frequencies of

up to 4 Hz, the flux-density dependence on strain is almost linear with little hys-

teresis. This low-frequency behavior is consistent with some of the previous obser-

vations [45, 99, 73]. The net flux density change for a strain range of 3% is around

0.056 T (560 Gauss) for almost all frequencies, which shows that the magnetization

vectors rotate in the longitudinal direction by the same amount for all the frequen-

cies. The applied strain amplitude does not remain exactly at ±1.5% because the

MTS controller is working at very low displacements (≈±0.15 mm) and high frequen-

cies. Nevertheless, the strain amplitudes are maintained within a sufficiently narrow

range (±8%) so that a comparative study is possible on a consistent basis for different

frequencies.

With increasing frequency, the stress-strain behavior remains relatively unchanged (Fig. 5.2(a)).

This indicates that the twin-variant reorientation occurs in concert with the applied

loading for the frequency range under consideration. This behavior is consistent with

work by Marioni [86] showing that twin boundary motion occurs in concert with the

applied field for frequencies of up to 2000 Hz. On the other hand, the flux den-

sity dependence on strain shows a monotonic increase in hysteresis with increasing

160

−0.02 −0.01 0 0.01 0.020

1

2

3

4

5

Compressive Strain

Com

pres

sive

Str

ess

(MP

a)

4 Hz20 Hz50 Hz90 Hz120 Hz160 Hz

(a)

−0.02 −0.01 0 0.01 0.02−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

Compressive Strain

Rel

ativ

e F

lux

Den

sity

(T

)

4 Hz20 Hz 50 Hz 90 Hz120 Hz160 Hz

(b)

Figure 5.2: (a) Stress vs. strain and (b) flux-density vs. strain measurements forfrequencies of up to 160 Hz.

161

frequency. The hysteresis loss in the stress versus strain plots is equal to the area

enclosed by one cycle (∮

σdε), whereas the loss in the flux density versus strain plots

is obtained by multiplying the enclosed area (∮

Bdε) by a constant that has units of

magnetic field [125, 27]. Fig. 5.3 shows the hysteresis loss for the stress versus strain

and the flux density versus strain plots. The hysteresis in the stress plots is relatively

flat over the measured frequency range, whereas the hysteresis in the flux density

increases about 10 times at 160 Hz compared to the quasistatic case. The volumetric

energy loss, i.e., the area of the hysteresis loop is approximately linearly proportional

to the frequency. The bias field of 368 kA/m is strong enough to ensure that the

180-degree domains disappear within each twin variant, hence each variant consists

of a single magnetic domain throughout the cyclic loading process [101]. Therefore,

the only parameter affecting the magnetization hysteresis is the rotation angle of the

magnetization vectors with respect to the easy c-axis. This angle is independent of

the strain and variant volume fraction [101], and is therefore a constant for the given

bias field.

The process that leads to the observed magnetization dependence on strain is

postulated to occur in three steps: (i) As the sample is compressed, twin variant

rearrangement occurs and the number of crystals with easy c-axis in the longitudi-

nal (y) direction increases. The magnetization vectors remain attached to the c-axis,

therefore the magnetization in these crystals is oriented along the y-direction. (ii) Sub-

sequently, the magnetization vectors in these crystals rotate away from the c-axis to

settle at a certain equilibrium angle defined by the competition between the Zee-

man and magnetocrystalline anisotropy energies. This rotation process is proposed

162

0 25 50 75 100 125 1500

1000

2000

3000

4000

Flu

x de

nsity

hys

tere

sis

loss

(J/

m3 )

Frequency (Hz)0 50 100 150

0

2

4

6

8x 10

4

Str

ess

hyst

eres

is lo

ss (

J/m

3 )

Figure 5.3: Hysteresis loss with frequency for stress-strain and flux-density strainplots. The plots are normalized with respect to the strain amplitude at a givenfrequency.

163

to occur according to the dynamics of a first order system. Time constants for first-

order effects in Ni-Mn-Ga have been previously established for the time-dependent

long-time strain response [41, 81], and strain response to pulsed field [86]. The time

constant associated with pulse field response provides a measure of the dynamics of

twin-boundary motion, which is estimated to be around 157 µs [86]. In contrast,

the time constant associated with magnetization rotation in our measurements is

estimated to be around 1 ms. (iii) As the sample is unloaded, twin variant rearrange-

ment occurs due to the applied bias field. Crystals with the c-axis oriented along the

y-direction rotate into the x-direction, and an increase in the flux density along the

x-direction is observed. At low frequencies, magnetization rotation occurs in concert

with twin-variant reorientation. As the frequency increases, the delay associated with

the rotation of magnetization vectors into their equilibrium position increases, which

leads to the increase in hysteresis seen in Fig. 5.2(b). The counterclockwise direction

of the magnetization hysteresis loops implies that the dynamics of magnetization ro-

tation occur as described in steps (i)-(iii). If the magnetization vectors had directly

settled at the equilibrium angle without going through step (i), the direction of the

hysteresis excursions would have been clockwise.

5.2 Model for Frequency Dependent Magnetization-StrainHysteresis

A continuum thermodynamics constitutive model has been developed to describe

the quasi-static stress and flux density dependence on strain at varied bias fields [101].

The hysteretic stress versus strain curve is dictated by the evolution of the variant

volume fractions. We propose that the evolution of volume fraction is independent

of frequency for the given range, and therefore, no further modification is required

164

Dynamic

Strain

Linear

Constitutive

Equation

Field

Diffusion

Equation

avgM e Hε χ= +

2

0 0

H MH

t tµσ µσ

∂ ∂∇ − =

∂ ∂

( , )H H x t=

( )M M t=( )tε ε=

Dynamic

Magnetization

BC: ( , ) biasH d t H± =

( )avgH t

Figure 5.4: Scheme for modeling the frequency dependencies in magnetization-strainhysteresis.

to model the stress versus strain behavior at higher frequencies. However, the mag-

netization dependence on strain changes significantly with increasing frequencies due

to the losses associated with the dynamic magnetization rotation resulting from me-

chanical loading. The modeling strategy is summarized in Figure 5.4.

The constitutive model (Section 3.6.2) shows that at high bias fields, the depen-

dence of flux density on strain is almost linear and non-hysteretic. Therefore, a linear

constitutive equation for magnetization is assumed as an adequate approximation at

quasi-static frequencies and modified to address dynamic effects. If the strain is ap-

plied at a sufficiently slow rate, the magnetization response can be approximated as

follows,

M = eε + χHavg (5.1)

165

where e and χ are constants dependent on the given bias field. For the given data,

these constants are estimated as, e = −4.58 × 106 A/m, and χ = 2.32. The average

field Havg acting on the material is not necessarily equal to the bias field Hbias.

Equation (5.1) works well at low frequencies. However, as the frequency increases,

consideration of dynamic effects becomes necessary. The dynamic losses are modeled

using a 1-D diffusion equation that describes the interaction between the dynamic

magnetization and the magnetic field inside the material,

∇2H − µ0σ∂H

∂t= µ0σ

∂M

∂t, (5.2)

This treatment is similar to that in Ref. [107] for dynamic actuation, although the

final form of the diffusion equation and the boundary conditions are different. The

boundary condition on the two faces of the sample is the applied bias field,

H(±d, t) = Hbias. (5.3)

Although the field on the edges of the sample is constant, the field inside the

material varies as dictated by the diffusion equation. The diffusion equation is nu-

merically solved using the backward difference method to obtain the magnetic field

at a given position and time H(x, t) inside the material.

For sinusoidal applied strain, the magnetization given by equation (5.1) varies in a

sinusoidal fashion. This magnetization change dictates the variation of the magnetic

field inside the material given by (5.2). The internal magnetic field thus varies in a

sinusoidal fashion as seen in Figure 5.5(a). The magnitude of variation increases with

166

increasing depth inside the material. In order to capture the bulk material behavior,

the average of the internal field is calculated by,

Havg(t) =1

Nx

Xd∑

X=−Xd

H(x, t), (5.4)

where Nx represents the number of uniformly spaced points inside the material where

the field waveforms are calculated.

Figure 5.5 shows the results of various stages in the model. The parameters used

are, µr=3.0, and ρ = 1/σ = 62 × 10−8 Ohm-m, Nx=40. Figure 5.5(a) shows the

magnetic field at various depths inside the sample for a loading frequency of 140 Hz.

It is seen that as the depth inside the sample increases, the variation of the magnetic

field increases. At the edges of the sample (x = ±d), the magnetic field is constant,

with a value equal to the applied bias field.

Figure 5.5(b) shows the variation of the average field at varied frequencies. The

variation of the average field is directly proportional to the frequency of applied

loading: as the frequency increases, the amplitude of the average field increases.

Finally, the magnetization is recalculated by using the updated value of the average

field as shown by the block diagram in Figure 5.4. The flux-density is obtained from

the magnetization (see Figure 5.5(c)) by accounting for the demagnetization factor.

It is seen that the model adequately captures the increasing hysteresis in flux density

with increasing frequency. Further refinements in the model are possible, such as

including a 2-D diffusion equation, and updating the permeability of the material

while numerically solving the diffusion equation.

167

0 0.2 0.4 0.6 0.8 1320

340

360

380

400

420

Non−dimensional Time (t*fa)M

agne

tic F

ield

(kA

/m)

d0.8d0.6d0.4d0.2d0

IncreasingDepth

(a)

0 0.2 0.4 0.6 0.8 1340

350

360

370

380

390

400

Non−dimensional Time (t*fa)

Ave

rage

Fie

ld (

kA/m

)

4 Hz20 Hz60 Hz100 Hz140 Hz

Increasingfrequency

(b)

0.01 0.02 0.03 0.04 0.050

0.01

0.02

0.03

0.04

0.05

0.06

Strain

Rel

. Flu

x D

ensi

ty (

Tes

la)

4 Hz20 Hz60 Hz100 Hz140 Hz

(c)

Figure 5.5: Model results: (a) Internal magnetic field vs. time at varying depth forthe case of 140 Hz strain loading (sample dim:±d), (b) Average magnetic field vs.time at varying frequencies, and (c) Flux-density vs. strain at varying frequencies.

168

5.3 Discussion

The magnetization and stress response of single-crystal Ni-Mn-Ga subjected to

dynamic strain loading for frequencies from 0.2 Hz to 160 Hz is presented [109, 104].

This frequency range is significantly higher than previous characterizations of Ni-Mn-

Ga which investigated frequencies from d.c. to only 10 Hz. The rate of twin-variant

reorientation remains unaffected by frequency; however, the rate of rotation of mag-

netization vectors away from the easy c-axis is lower than the rate of loading and

of twin-variant reorientation. This behavior can be qualitatively explained by the

dynamics of a first-order system associated with the rotation of magnetization vec-

tors. The increasing hysteresis in the flux density could complicate the use of this

material for dynamic sensing. However, the “sensitivity” of the material, i.e., net

change in flux-density per percentage strain input remains relatively unchanged (≈

190 G per % strain) with increasing frequency. Thus the material retains the advan-

tage of being a large-deformation, high-compliance sensor as compared to materials

such as Terfenol-D [99] at relatively high frequencies. The significant magnetization

change at structural frequencies also illustrates the feasibility of using Ni-Mn-Ga for

energy harvesting applications. To employ the material as a dynamic sensor or in

energy harvesting applications, permanent magnets can be used instead of an elec-

tromagnet. The electromagnet provides the flexibility of turning the field on and

off at a desired magnitude, but the permanent magnets provide an energy efficiency

advantage. The dynamic magnetization process in the material is modeled using a

linear constitutive equation, along with a 1-D diffusion equation similar to that used

a previous dynamic actuation model. The model adequately captures the frequency

169

dependent magnetization versus strain hysteresis and describes the dynamic sensing

behavior of Ni-Mn-Ga.

170

CHAPTER 6

STIFFNESS AND RESONANCE TUNING WITH BIASMAGNETIC FIELDS

This chapter presents the dynamic characterization of mechanical stiffness changes

under varied bias magnetic fields in single-crystal ferromagnetic shape memory Ni-

Mn-Ga. The material is first converted to a single variant through the application and

subsequent removal of a bias magnetic field. Mechanical base excitation is then used to

measure the acceleration transmissibility across the sample, from where the resonance

frequency is directly identified. The tests are repeated for various longitudinal and

transverse bias magnetic fields ranging from 0 to 575 kA/m. A single degree of

freedom (DOF) model for the Ni-Mn-Ga sample is used to calculate the mechanical

stiffness and damping from the transmissibility measurements. An abrupt resonance

frequency increase of 21% and a stiffness increase of 51% are obtained with increasing

longitudinal fields. A gradual resonance frequency change of −35% and a stiffness

change of −61% are obtained with increasing transverse fields. A constitutive model

is used to describe the dependence of material stiffness on transverse bias magnetic

fields. The damping exhibited by the system is low in all cases (≈ 0.03). The

measured dynamic behaviors make Ni-Mn-Ga well suited for vibration absorbers with

electrically-tunable stiffness.

171

6.1 Introduction

FSMA applications other than actuation have received limited attention. Stud-

ies have shown the viability of Ni-Mn-Ga in sensing and energy harvesting applica-

tions [119, 62, 101]. As a sensor material, Ni-Mn-Ga has been shown to exhibit a

reversible magnetization change of 0.15 T when compressed by 5.8% strain at a bias

field of 368 kA/m [101]. In addition, the stiffness of Ni-Mn-Ga varies with externally

applied fields and stresses. In the low temperature martensitic phase, application of

a sufficiently large transverse magnetic field (> 700 kA/m) produces a Ni-Mn-Ga mi-

crostructure with a single “field preferred” variant configuration (Figure 6.1, center);

application of a sufficiently large longitudinal field (> 350 kA/m) or sufficiently large

compressive stress (> 3 MPa) creates a single “stress preferred” variant configuration

(Figure 6.1, right). The quasistatic stress-strain curve for Ni-Mn-Ga [101] shows that

the two configurations have significantly different stiffness. At intermediate fields and

stresses, both variants coexist and the material exhibits a bulk stiffness between the

two extreme values (Figure 6.1, left). This microstructure offers the opportunity to

control the bulk material stiffness through the control of variant volume fractions

with magnetic fields or stresses. Magnetic fields are the preferred method for stiff-

ness control as they can be applied remotely and can be adjusted precisely. Faidley

et al. [28] investigated stiffness changes in research grade, single crystal Ni-Mn-Ga

driven with magnetic fields applied along the [001] (longitudinal) direction. The ma-

terial they used exhibits reversible field induced strain when the longitudinal field is

removed, which is attributed to internal bias stresses associated with pinning sites.

The fields were applied with permanent magnets bonded onto the material, which

172

makes it difficult to separate resonance frequency changes due to magnetic fields or

mass increase. Analytical models were developed to address this limitation.

In this study we isolate the effect of magnetic field on the stiffness of Ni-Mn-Ga

by applying the magnetic fields in a non-contact manner, and investigate the stiffness

characteristics under both longitudinal and transverse magnetic fields. Base excita-

tion is used to measure the acceleration transmissibility across a prismatic Ni-Mn-Ga

sample, from where its resonance frequency is directly identified. Prior to the trans-

missibility measurements, a stress-preferred or field-preferred variant configuration

is established through the application and subsequent removal of a bias field using

a solenoid coil or an electromagnet, respectively. We show that longitudinal and

transverse bias magnetic fields have drastically different effects on the stiffness of Ni-

Mn-Ga: varying the former produces two distinct stiffness states whereas varying the

latter produces a continuous range of stiffnesses. We present a constitutive model

that describes the continuous stiffness variation.

6.2 Experimental Setup and Procedure

The measurements are conducted on commercial single crystal Ni-Mn-Ga manu-

factured by AdaptaMat, Inc. A sample with dimensions 6×6×10 mm3 is tested in

its low-temperature martensite phase. The sample exhibits 5.8% free strain in the

presence of transverse fields of about 400 kA/m. The broadband mechanical excita-

tion is provided by a Labworks ET126-B shaker table which has a frequency range

of dc to 8500 Hz and a 25 lb peak sine force capability. The shaker is driven by an

MB Dynamics SL500VCF power amplifier which has a power rating of 1000 VA and

173

a

c

H

Transverse Field

H

Longitudinal Field Field Preferred

Stress Preferred

Figure 6.1: Left: simplified 2-D twin variant microstructure of Ni-Mn-Ga. Center:microstructure after application of a sufficiently high transverse magnetic field. Right:after application of a sufficiently high longitudinal field.

maximum voltage gain of 48 with 40 V peak and 16 A rms. The shaker is controlled

by a Data Physics SignalCalc 550 vibration controller.

A schematic of the test setup for longitudinal field measurements is shown in

Figure 6.2. The sample is mounted on an aluminum pushrod fixed on the shaker

table, and a dead weight is mounted on top of the sample. Two PCB accelerometers

measure the base and top accelerations. The longitudinal field is applied by a custom-

made water cooled solenoid transducer which is made from AWG 15 insulated copper

wire with 28 layers and 48 turns per layer [83]. The solenoid is driven by two Technol

7790 amplifiers connected in series which produce an overall voltage gain of 60 and

a maximum output current of 56 A into the 3.7 Ω coil. The solenoid has a magnetic

field rating of 11.26 (kA/m)/A.

174

The transverse field experiment is illustrated in Figure 6.3. The magnetic fields are

applied by a custom-made electromagnet made from laminated E-cores with 2 coils

of about 550 turns each made from AWG 16 magnet wire. The coils are connected in

parallel. The electromagnet has a magnetic field rating of 63.21 (kA/m)/A and can

produce fields of up to 750 kA/m.

For the longitudinal field tests, the sample is initially configured as a single field-

preferred variant. The sample microstructure can be changed with increasing longi-

tudinal fields by favoring the growth of stress-preferred variants, which results in a

stiffening with increasing magnetic field. The sample in zero-field condition is first

subjected to band-limited white noise base excitation with a frequency range from 0

to 4000 Hz and reference RMS acceleration of 0.2 g. After completion of the zero-

field test, a DC voltage is applied across the solenoid to produce a DC longitudinal

magnetic field on the sample. Due to the fast response of Ni-Mn-Ga [86], application

of the field for a small time period is enough to change the variant configuration. In

this study we apply the fields for about 1 to 2 seconds. If the field is strong enough

to initiate twin boundary motion, stress-preferred variants are generated from the

original field-preferred variants. The sample is again subjected to band-limited white

noise base excitation to record the top and base acceleration response, from which

the transfer function between the top and base acceleration is obtained. This process

is continued until the sample reaches a complete stress-preferred variant state.

For the transverse field tests, the sample is initially configured as a single stress-

preferred variant. This configuration is obtained by applying a high longitudinal field

in excess of 400 kA/m. The sample is mounted on the shaker table between the pole

faces of the electromagnet using aluminum pushrods, and a dead weight is mounted

175

Water cooled

Solenoid

Ni-Mn-Ga

Aluminum rod

Dead

weight

Shaker table

Accelerometer(s)

x x

x x

x x

x x

x x

Figure 6.2: Schematic of the longitudinal field test setup.

Ni-Mn-Ga

Aluminum

pushrod(s)

Accelerometer(s)

Electromagnet

pole piece(s)

Dead weight

Shaker table

Figure 6.3: Schematic of the transverse field test setup.

176

on the top of the sample. The test procedure is the same as in the longitudinal field

test: the transverse bias field is incremented by a small amount and subsequently

removed before each run. When the field is sufficiently high, field-preferred variants

are generated at the expense of stress-preferred variants, resulting in a change in

stiffness and resonance frequency.

6.3 Theory

The system is represented by the DOF spring-mass-damper model shown in Fig-

ure 6.4, where Ks represents the stiffness of the Ni-Mn-Ga sample, Kr is the total

stiffness of the aluminum pushrods, M is the dead weight on the sample, and C is

the overall damping present in the system. The base motion is represented by x, and

the top motion is represented by y.

The system is subjected to band-limited white noise base excitation with reference

acceleration to the shaker controller having an RMS value of 0.2 g. The reference

acceleration has uniform autospectral density (PSD) over the range from 0 to 4000 Hz:

Grr(f) = G 0 ≤ f ≤ 4000

= 0 f > 4000,(6.1)

with f the frequency (Hz), Grr the reference acceleration PSD (g2/Hz), and G the

constant value of reference acceleration PSD (g2/Hz) over the given frequency band.

The measured base acceleration PSD, or actual input acceleration PSD differs from

the reference PSD, and is denoted by Gxx (g2/Hz). The top acceleration PSD is de-

noted by Gyy (g2/Hz). The RMS acceleration values are related to the corresponding

177

M

C

Ks

Kr

y&&

x&&

Shaker table

Figure 6.4: DOF spring-mass-damper model used for characterization of the Ni-Mn-Ga material.

178

acceleration PSDs by

ψ2r =

∫ fmax

fmin

Grr(f)df,

ψ2x =

∫ fmax

fmin

Gxx(f)df,

ψ2y =

∫ fmax

fmin

Gyy(f)df,

(6.2)

where ψr, ψx, ψy represent the reference, input, and output RMS acceleration (m/s2)

values, respectively. Frequencies fmin and fmax respectively represent the lower and

upper limits on the band limited signal. Figure 6.5 shows the experimentally obtained

PSDs for input, output, and reference acceleration signals in one of the test runs.

In this case, the RMS acceleration values obtained from (6.2) are ψr = 0.2 g2/Hz,

ψx = 0.2036 g2/Hz, and ψy = 0.7048 g2/Hz. It is noted that the measured input PSD

does not have an exactly uniform profile as the reference PSD does. However, the

RMS values for the input and reference PSDs differ by less than 2%.

Since the cross-PSD between the input and output signals (Gxy) cannot be mea-

sured by the shaker controller, only the magnitude (and not the phase) of the transfer

function between the top and base acceleration signals are obtained experimentally.

The transfer function magnitude calculated from the experimental data is given as

|Hxy(f)|2 =Gyy

Gxx

, (6.3)

where Hxy(f) represents the experimentally obtained transfer function between the

top and base accelerations. For the DOF system shown in Figure 6.4, the transfer

function between the top and base acceleration is given as

H(f) =1 + j(2ζf/fn)

1− (f/fn)2 + j(2ζf/fn), (6.4)

179

500 1000 1500 2000 2500 3000 35000

0.5

1

1.5

2

2.5

3

3.5x 10

−3

Frequency (Hz)

Acc

eler

atio

n P

SD

(g2 /H

z)

Gxx

× 100

Gyy

Grr × 100

Figure 6.5: Experimentally obtained acceleration PSDs.

where ζ is the overall damping ratio of the system and fn is the natural frequency of

the system (Hz). The natural frequency is experimentally obtained as the frequency

at which the output PSD is maximum,

fn = arg maxf

[Gyy(f)]. (6.5)

If the system in Figure 6.4 is subjected to band-limited input acceleration of uniform

PSD G, the RMS value of output acceleration is given as [2],

ψ2y =

Gπfn(1 + 4ζ2)

4ζ. (6.6)

Although the measured input acceleration PSD is not uniform, it can be assumed to be

uniform with sufficient accuracy for calculation of the damping ratio. An expression

180

500 1000 1500 2000 2500 3000 3500−10

−5

0

5

10

15

20

25

Frequency (Hz)

Tra

nsfe

r fu

nctio

n m

agni

tude

(dB

)

ExperimentalCalculated

Figure 6.6: Transfer function between top and base accelerations.

for the damping ratio is given as

ζ =ψ2

y −√

(ψ2y)

2 − (Gπfn)2

2Gπfn

, (6.7)

where the RMS value of output acceleration (ψy) is obtained from the measured out-

put PSD (Gyy) from equation (6.2), fn is obtained from (6.5), and G is the uniform

reference PSD. The experimental and calculated transfer function for the case under

consideration are shown in Figure 6.6. It is noted that the assumption of a linear,

DOF spring-mass-damper system, and the approximation of using the reference ac-

celeration PSD to calculate the effective damping ratio work well for describing the

experimentally obtained transfer function.

181

Further, the analytical expression for the natural frequency of the system is given

as,

fn =1

2π

√KsKr

(Ks + Kr)M, (6.8)

from where the mechanical stiffness of the Ni-Mn-Ga sample is obtained as,

Ks =M(2πfn)2Kr

Kr −M(2πfn)2. (6.9)

Further, the viscous damping coefficient is has the form

C = 2ζ√

KM. (6.10)

The stiffness change and resonance frequency change are calculated with respect to

the initial material stiffness, which depends on whether the test involves longitudinal

or transverse fields. The stiffness change is given by

∆Ks =Ks −Ks0

Ks0

× 100, (6.11)

where ∆Ks is the overall stiffness change (%), and Ks0 is the initial, zero-field stiffness.

6.4 Results and Discussion

6.4.1 Longitudinal Field Tests

The transmissibility ratio transfer function relating the acceleration of the top

to the acceleration of the base provides information on the resonance frequency and

damping present in the system. The measurements obtained in the longitudinal field

configuration are shown in Figure 6.7 for one of the test runs. The sample exhibits

only two distinct resonances after subjecting it to several fields ranging from 0 to

430 kA/m. At fields below 330 kA/m, the sample exhibits a resonance frequency of

approximately 1913 Hz; at fields of more than 330 kA/m, the resonance is observed at

182

2299 Hz. These results point to an ON/OFF effect with a threshold field of 330 kA/m.

This result was validated through repeated runs under the same conditions, as shown

in Figure 6.8, which shows the two distinct resonances for three different tests as well

as calculations.

The stiffness of the aluminum pushrod used in these tests is 1.36e8 N/m, and

the mass of the dead weight is 60.97 g. The two average stiffnesses calculated with

expression (6.9) are Ks1 = 9.33e6 N/m and Ks2 = 1.41e7 N/m. The average damping

ratios calculated with expression (6.7) are ζ1 = 0.0334 and ζ2 = 0.0422. The average

field at which the resonance shift takes place is 285 kA/m. The variation in the field

at which the threshold occurs may be due to small variations in the position of the

sample with respect to the solenoid. If the sample is not exactly aligned along the

length of the solenoid, the effective field in the sample might change. This can give

rise to varied magnitudes of field even when the current in the solenoid is the same.

A field of 330 kA/m can be considered optimum for achieving the second resonance

frequency as compared to the resonance at lower fields. The results of the longitudinal

tests are summarized in Table 6.1. It is seen that there is an average resonance shift

of 20.9% and an average stiffness shift of around 51.0%, both relative to the zero-field

value. The stiffness increases with increasing field since the sample is initially in its

field-preferred, mechanically softest state. Although the damping ratios also show

a large average shift of about 42.0%, the damping values are small at all magnetic

fields. This is beneficial for the implementation of Ni-Mn-Ga in tunable vibration

absorbers with a targeted absorption frequency.

183

500 1000 1500 2000 2500 3000 3500−10

−5

0

5

10

15

20

25

Tra

nsfe

r fu

nctio

n m

agni

tude

(dB

)

Frequency (Hz)

066132198264330396462

H (kA/m)

Figure 6.7: Acceleration transmissibility with longitudinal field.

500 1000 1500 2000 2500 3000 3500−10

−5

0

5

10

15

20

25

30

Frequency (Hz)

Tra

nsfe

r fu

nctio

n m

agni

tude

(dB

)

Test 1Test 2Test 3Calculated

Figure 6.8: Longitudinal field test model results and repeated measurements underthe same field inputs.

184

Run# fn1 fn2 ∆fn(%) Ks1 Ks2 ∆Ks(%) ζ1 ζ2 ∆ζ(%)

1 1902 2299 20.9 9.31e6 1.40e7 50.8 0.035 0.046 32.0

2 1963 2401 22.3 9.96e6 1.54e7 55.1 0.030 0.038 28.1

3 1846 2207 19.6 8.73e6 1.28e7 51.0 0.036 0.043 20.0

Ave. 20.9 51.0 26.7

Table 6.1: Summary of longitudinal field test results. Units: fn: (Hz), Ks: (N/m)

6.4.2 Transverse field Tests

The measurements conducted in the transverse field case are shown in Figure 6.9

for one of the test runs. Two differences with respect to the longitudinal tests are

observed. First, since the sample is initially configured in its stiffest state as a single

stress-preferred variant, an increase in transverse magnetic field produces a decrease

in the mechanical stiffness and associated resonance frequency. Secondly, the material

exhibits a gradual change in resonance frequency with changing field, in this case from

values of around 2300 Hz for zero applied field to around 1430 Hz for a dc magnetic

field of 439 kA/m. Further, the effective resonance frequency change from 2300 Hz

to 1430 Hz occurs for a relatively narrow field range from 245 kA/m to 439 kA/m.

Similar behavior was identified after conducting several test runs, three of which are

considered here for estimating the relevant model parameters. The frequency shifts for

the three cases are −36.4%, −33.0%, and −35.9%, giving an average shift of −35.1%

between the extreme values. Figure 6.10 shows these additional measurements, which

reflect the same trend.

185

500 1000 1500 2000 2500 3000 3500−15

−10

−5

0

5

10

15

20

25

Tra

nsfe

r fu

nctio

n m

agni

tude

(dB

)

Frequency (Hz)

0245255260273301315341356380412439

H (kA/m)

Figure 6.9: Transmissibility ratio measurements with transverse field configuration.

The overall resonance frequency shift and sample stiffness shift in the transverse

field tests are higher than in the longitudinal field test. For the longitudinal measure-

ments, the average sample stiffness in the stress-preferred configuration is 1.38e7 N/m,

whereas for the transverse tests it is 1.34e7 N/m. However, the average stiffness when

the sample is supposed to be in the completely field-preferred state is 9.1e6 N/m in

the case of longitudinal field tests and 5.27e6 N/m in the case of transverse field

tests. This state occurs at the start for the longitudinal field test and at the end in

the transverse field test. The possible reason behind this difference is that the manual

pressure applied while mounting the sample for longitudinal field tests results in ini-

tiation of twin boundary motion and a certain fraction of the sample is transformed

into the stress-preferred variant. This results in an increased stiffness as compared to

186

500 1000 1500 2000 2500 3000 3500−15

−10

−5

0

5

10

15

20

25

Tra

nsfe

r fu

nctio

n m

agni

tude

(dB

)

Frequency (Hz)

0236249262274284297310323423

H (kA/m)

500 1000 1500 2000 2500 3000 3500−15

−10

−5

0

5

10

15

20

25

Tra

nsfe

r fu

nctio

n m

agni

tude

(dB

)

Frequency (Hz)

101235248262286300314407508522

H (kA/m)

Figure 6.10: Additional measurements of transmissibility ratio with transverse fieldconfiguration.

187

Run# fn1 fn2 ∆fn(%) Ks1 Ks2 ∆Ks(%) ζ1 ζ2 ∆ζ(%)

1 2294 1460 −36.4 1.41e7 0.53e7 −62.5 0.030 0.037 24.5

2 2147 1439 −33.0 1.21e7 0.51e7 −57.7 0.032 0.037 15.5

3 2294 1470 −35.9 1.41e7 0.54e7 −61.9 0.030 0.046 55.6

Ave. −35.1 −60.7 31.8

Table 6.2: Summary of transverse field test results. Units: fn: (Hz), Ks: (N/m)

a completely field-preferred configuration. However, this behavior is consistent in the

three runs indicating that the structure assumed by the sample had been nearly the

same during the tests. The sample stiffness and the overall system damping ratio are

calculated as detailed in Section 6.3. The aluminum pushrods used in these tests have

different dimensions than those used in the longitudinal measurements, and hence the

stiffness has a different value of 1.068e7 N/m. The dead weight has the same mass

of 60.87 g. Table 6.2 shows the resonance frequency, stiffness, and damping ratio

variation in extreme values for the transverse field tests.

The damping ratio, viscous damping coefficient and resonance frequency variations

with initial bias field for the test run in Figure 6.9 are shown in Figures 6.11, 6.12,

and 6.13, respectively. The damping ratios show small overall magnitudes, hence the

material is suitable for tunable vibration absorption applications. It is also noted that

the damping ratio values are relatively flat over the bias fields when compared with

the resonance frequency. The slight rise and drop in the damping can be attributed

to the presence of twin boundaries [37]. With increasing bias field, twin boundaries

are created which leads to increased damping. At high bias fields, the sample is

converted to a complete field-preferred state. In this condition, the number of twin

188

200 250 300 350 400 4500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Bias Field (kA/m)

Dam

ping

Rat

io

Figure 6.11: Variation of damping ratio with initial transverse bias field.

boundaries decreases again, resulting in relatively lower damping. Figure 6.11 shows

this trend: the damping coefficient is maximum at intermediate fields, and attains

relatively lower values at the lowest and highest fields.

The variation of stiffness with changing bias field is modeled with an existing

continuum thermodynamics model developed by Sarawate et al. [101, 103]. With

increasing field, the Ni-Mn-Ga sample starts deforming because its twin variant con-

figuration changes. The variation of the field-preferred (ξ) and stress-preferred (1−ξ)

martensite volume fractions with field is described by the magnetomechanical con-

stitutive model. The model is formulated by writing a thermodynamic Gibbs energy

potential consisting of magnetic and mechanical components. The magnetic energy

has Zeeman, anisotropy and magnetostatic contributions; the mechanical energy has

189

200 250 300 350 400 4500

500

1000

1500

Field (kA/m)

Vis

cous

Dam

ping

Con

stan

t (N

m/s

)

Figure 6.12: Variation of viscous damping coefficient with initial transverse bias field.

200 250 300 350 400 4501000

1500

2000

2500

Bias Field (kA/m)

Res

onan

ce F

requ

ency

(H

z)

Figure 6.13: Variation of resonance frequency with initial transverse bias field.

190

elastic and twinning energy contributions. Mechanical dissipation and the microstruc-

ture of Ni-Mn-Ga are incorporated in the continuum thermodynamics framework by

considering the internal state variables volume fraction, domain fraction, and magne-

tization rotation angle. The constitutive strain response of the material is obtained by

restricting the process through the second law of thermodynamics, as detailed in [103].

The net compliance of the Ni-Mn-Ga sample is given by a linear combination of the

field-preferred and stress-preferred volume fractions. Thus, the net material modulus

is given as,

E(ξ) =1

S(ξ)=

1

S0 + (1− ξ)(S1 − S0)(6.12)

where E is the net material modulus, S is the net compliance, S0 is the compliance

of the material in complete field-preferred state, and S1 is the compliance of the

material in complete stress-preferred state. The twin variants are separated by a

twin boundary, and each side of the twin boundary contains a specific variant. If

the bulk material is subjected to a force, the stiffnesses associated with the stress-

preferred and field-preferred variants will be under equal forces, i.e., the two stiffnesses

will be in series. Therefore, the net compliance of the system is assumed to be a linear

combination of the compliances of the field-preferred and stress-preferred variants.

Further, the net stiffness is related to the modulus by

Ks =AE

L, (6.13)

with A the cross-sectional area, and L the length of the Ni-Mn-Ga element. Using

the constitutive model for volume fraction, and equations (6.12), (6.13), the stiffness

change of the material with initial bias field can be calculated. Model calculations

191

200 250 300 350 400 4504

6

8

10

12

14

16

Bias Field (kA/m)

Stif

fnes

s (N

/m)

× 10

6

ExperimentModel

Figure 6.14: Variation of stiffness with initial bias field.

are shown in Figure 6.14 along with the experimental values. The model accurately

predicts the stiffness variation with initial bias field.

Because of the relatively high demagnetization factor (0.385) in the transverse

direction, it takes higher external fields to fully elongate the sample. Thus, a contin-

uous change of resonance frequency and hence stiffness is observed with increasing

bias fields. In the case of the longitudinal field tests, the demagnetization factor is

0.229. Thus, once the twin boundary motion starts, it takes a very small range of

fields to transform the sample fully into the stress-preferred state. Thus, an abrupt

change in the resonance frequency and hence stiffness is seen in the longitudinal field

tests.

192

6.5 Concluding Remarks

The single-crystal Ni-Mn-Ga sample characterized in this study exhibits varied

dynamic stiffness with changing bias fields [102, 110]. The non-contact method of

applying the magnetic fields ensures consistent testing conditions. This is an im-

provement over the prior work by Faidley et al. [28], in which permanent magnets

were used to apply magnetic fields along the longitudinal direction. Unlike that

study, the characterization presented here was conducted on commercial Ni-Mn-Ga

material, under both longitudinal and transverse drive configurations. The field is

not applied throughout the duration of a given test, but only initially in order to

transform the sample into a given twin variant configuration. This is an advantage

of Ni-Mn-Ga over magnetostrictive materials like Terfenol-D in which a continuous

supply of magnetic field, and hence current in the electromagnetic coil, is required in

order to maintain the required resonance frequency. A study on a 0.63-cm-diameter,

5.08-cm-long Terfenol-D rod driven within a dynamic resonator has shown that this

material exhibits continuously variable resonance frequency tuning from 1375 Hz to

2010 Hz [34].

If a bi-directional resonance change was required, the system involving Ni-Mn-

Ga would need a restoring mechanism. A magnetic field source perpendicular to

the original field source could be used to maintain the advantage of low electrical

energy consumption. Another option is to use a restoring spring; but the presence

of the restoring spring results in reversible behavior of Ni-Mn-Ga, thus requiring a

continuous source of current to maintain the field. Nevertheless, this work shows

the suitability of using Ni-Mn-Ga in tunable vibration absorbers as it provides a

broad resonance frequency bandwidth comparable to Terfenol-D, with the option of

193

utilizing magnetic field pulse activation with very low energy consumption. Twin

boundary motion occurs almost instantaneously with the application of the field,

and the material configuration remains unchanged unless a restoring field or stress is

applied.

The overall resonance frequency and stiffness change in the transverse field tests

are −35.1% and −60.7% respectively. The equivalent values for the longitudinal field

tests are 21.3% and 51.5%, respectively. The damping values observed in the tests

are small (≈ 0.03) and are conducive to the use of Ni-Mn-Ga in active vibration

absorbers. An ON/OFF behavior is observed in the longitudinal field tests, whereas

a continuously changing resonance frequency is observed in the transverse field tests.

Thus, depending on the application and the frequency range under consideration,

the sample can be operated either in transverse or longitudinal field configuration.

The transverse field configuration offers more options regarding the ability to select

a particular resonance frequency. The longitudinal field configuration only offers two

discrete resonance frequencies but can be implemented in a more compact manner.

The evolution of volume fraction with increasing transverse field is described by the

existing continuum thermodynamics model, which is used to model the dependence

of material stiffness on the initial bias field assuming a linear variation of compliance

with volume fraction. Therefore, the stiffness exhibits a hyperbolic dependence on

the bias transverse field, which is also validated by experiments. The acceleration

transmissibility transfer function is accurately quantified by assuming a discretized

SDOF linear system. The development of a continuous dynamic model is desirable

for handling different sample geometries and higher modes. Although in this study

the magnetic field was switched off during the dynamic tests, development of a model

194

with the sample immersed in an external magnetic field during testing might be useful

for creating a more complete characterization of the dynamic behavior exhibited by

Ni-Mn-Ga.

195

CHAPTER 7

CONCLUSION

This dissertation was written to advance the understanding of the complex rela-

tionships under various static and dynamic conditions in ferromagnetic shape memory

alloys, specifically single crystal Ni-Mn-Ga. The key tasks were to characterize the

sensing behavior, to develop a coupled magnetomechanical model, and to investigate

the dynamic behavior. Key observations and conclusions are detailed at the end of

each of the prior chapters, and this chapter presents an overall summary of the entire

work.

7.1 Summary

7.1.1 Quasi-static Behavior

Sensing Characterization

One focus of the dissertation was to investigate whether Ni-Mn-Ga can be utilized

in sensing applications. For this purpose, an experimental setup was built to apply

uniaxial mechanical compression in presence of suitable bias magnetic fields. The

measurements revealed that the magnetization or flux density of Ni-Mn-Ga can be

altered by means of mechanical compression, thereby validating its ability to sense.

Furthermore, it was observed that the stress-strain behavior exhibits a transition

196

from irreversible behavior at low fields to the reversible behavior at high fields. This

phenomenon is similar to that in thermal shape memory alloys, except that the role of

temperature is replaced by the magnetic field. There is a strong correlation between

the stress and flux density behavior regarding the reversibility.

The presented characterization demonstrates that Ni-Mn-Ga can be useful as a

sensor. Its advantages with respect to other smart materials are the large deforma-

tion range, high-compliance, and high sensitivity at lower forces. Majority of the prior

focus on Ni-Mn-Ga applications has been on actuation. However, the low blocking

stress and requirement of large magnetic fields limit the use of the material as an

actuator. Large magnetic fields necessitate the construction of a bulky electromag-

net. However, in a sensor configuration, the required bias field can be applied using

small permanent magnets. Therefore, a sensor made using Ni-Mn-Ga can exhibit

significantly higher energy density than an actuator made using the same Ni-Mn-Ga

sample. This research opens up the possibilities for future research in this area.

Blocked-Force Characterization

The force generation capacity of single crystal Ni-Mn-Ga is also characterized.

When the material is subjected to a magnetic field and is mechanically blocked, it

tries to push against the loading arms, thus generating a force. The blocked force

characterization is one of the key properties of smart materials, and it gives an indica-

tion of the actuation performance and the work capacity of the material. Though it is

observed that Ni-Mn-Ga provides higher work capacity than materials such as piezo-

electrics and magnetostrictives, the actuation authority of the material is severely

restricted due to the low blocking stress of around 3.5 MPa.

197

Magnetomechanical Constitutive Model

A continuum thermodynamics based model is presented which describes the cou-

pled magnetomechanical behavior of the material in variety of operating conditions.

The model describes the sensing, actuation, and blocked force behavior of single

crystal Ni-Mn-Ga ferromagnetic shape memory alloy. The nonlinearities and path

dependencies leading to hysteresis are well captured by the model. The classical

continuum mechanics framework is used with addition of magnetic terms; and the

internal state variables are used to incorporate the material microstructure and dis-

sipation. The model is physics based, which makes it flexible for additional of other

complex effects such as the exchange energy, magnetomechanical coupling energy, etc.

The model uses only seven non-adjustable parameters which are identified from two

simple experiments. The model is low-order, which makes it suitable for incorpora-

tion into custom finite element codes. The constitutive model is rate-independent,

and the material behavior at higher frequencies needs to be described by including

additional physics.

Chief utility of the model will be in designing and predicting the performance of

Ni-Mn-Ga sensors and actuators by describing the macroscopic relationships between

various magnetomechanical variables. In addition to modeling these primary vari-

ables (stress, strain, magnetization, field), closed form solutions are derived to obtain

certain key variables such as the maximum strain, coercive field, twinning stress,

residual field, sensitivity, etc. The optimum bias field for a sensor and an optimum

bias stress for an actuator can be obtained from the model. These calculations pro-

vide a powerful tool as the model can be used to readily obtain an optimum actuator

or sensor design for a given Ni-Mn-Ga sample. The model can be easily modified to

198

describe the minor loops, which are critical for cyclic operation of the material around

a bias stress or bias field.

7.1.2 Dynamic Behavior

Dynamic Actuator Model

A new model is developed to describe the frequency dependent strain-field hys-

teresis in dynamic Ni-Mn-Ga actuators. This model is successfully implemented on

a dynamic magnetostrictive actuator to show its possible impact on the community

of hysteretic smart materials. The model uses the constitutive actuation model to

obtain a key variable such as the volume fraction or magnetostriction which is directly

related to the material’s strain. In addition to the constitutive model, the dynamic

magnetic losses due to eddy current are modeled using magnetic field diffusion and

the structural dynamics of the actuator is included by modeling the system as a

single-degree-of-freedom system. The applied magnetic field generates a force on the

actuator which makes the material vibrate. This force is expressed in terms of the

volume fraction which couples the dynamic strain to the magnetic field. The Fourier

series expansion of the volume fraction gives the net force acting on the actuator, and

the dynamic strain is obtained by superposition of the displacement response to each

harmonic component of the force. Analysis of strain in frequency domain at different

actuation frequencies reveals an interconnection with the shape of the macroscopic

hysteresis loop. This new approach can enable calculation of the input field profile

from the desired output strain profile by reversing the model flow.

199

Dynamic Sensing Characterization and Modeling

Characterization of the dynamic sensing properties of Ni-Mn-Ga was not addressed

in the literature. This research presents the first evidence that the stress induced mag-

netization change in Ni-Mn-Ga can also occur at higher frequencies (up to 160 Hz). It

is observed that the twin-variant reorientation remains unaffected for this frequency

range, which means that the stress-strain plots remain unaffected by the frequency.

On the contrary, the magnetization-strain plots show increasing hysteresis with fre-

quency, which indicates that the magnetization rotation process occurs with a delay.

This behavior can be explained by magnetic diffusion equation in a similar fashion

to that for the dynamic actuator model. The peak-to-peak magnetization values do

not decay significantly for the given range, indicating that the material can be used

as a sensor at higher frequencies. Ni-Mn-Ga sensors can thus give an advantage over

piezoelectric sensors, because they can be operated in quasi-static as well as dynamic

conditions.

Stiffness Tuning

Several smart materials can be used as tunable stiffness devices, because their

stiffness can be altered by application of electric or magnetic fields. This research

demonstrates the suitability of Ni-Mn-Ga as a tunable vibration absorber by char-

acterizing the resonance and stiffness with bias fields. The stiffness variation under

different collinear and transverse bias fields is characterized. Suitable drive configu-

ration can be chosen depending on the application.

200

Quasi-static Dynamic

Behavior Sensing Actuation Blocked-

force Actuation Sensing

Stiffness

Tuning

Input

Variable(s) Strain Field Field Field Strain

Base

Acceleration

Output

Variable(s)

Magnetization,

Stress

Strain,

Magnetization

Stress,

Magnetization Strain

Magnetization,

Stress

Top

Acceleration

Bias

Variable(s) Field Stress Strain

Stress,

Frequency

Field,

Frequency Field

Energy

Potential Magnetic Gibbs Gibbs

Magnetic

Gibbs - - -

Experiment In house Outside data In house Outside data In house In house

Modeling Continuum

Thermodynamics

Derived from

sensing work

Derived from

sensing work

Diffusion +

Constitutive

+Dynamics

Diffusion+

Lin.Constitutive

Second

order sys.

Figure 7.1: Characterization map of Ni-Mn-Ga. Plain blocks in “Experiment” and“Modeling” rows show the new contribution of the work; Light gray blocks show thata limited prior work existed, which was completely addressed in this research; Darkgray blocks indicate that prior work was available, and no new contribution was made.

7.1.3 Characterization Map

The presented research addresses the properties of Ni-Mn-Ga in a variety of static

and dynamic conditions. Figure 7.1 shows the contribution made by this research

regarding both experimental and modeling work pertaining to ferromagnetic shape

memory alloys. The presented work covers a significant realm of the possible charac-

terizations. Few additions to this work could be possible, such as modeling magne-

tization in dynamic actuation, or using the flux-density as a bias variable. However,

majority of the real world applications using smart materials are covered by the pre-

sented characterization map.

201

7.2 Contributions

• Hardware and test setups are developed for conducting characterization of the

sensing behavior of single Ni-Mn-Ga to measure stress, magnetization response

to strain input under bias fields.

• Increasing bias field marks the transition from irreversible (pseudoelastic) to

reversible (quasi-plastic) behavior.

• A bias field of 368 kA/m is identified as the optimum bias field which results

in reversible flux density change of 145 mT for strain of 5.8% and stress of

4.4 MPa.

• Flux density vs. strain behavior is linear and almost non-hysteretic whereas the

flux density vs. stress behavior is highly hysteretic, indicating that the material

will be more useful as a deformation sensor than a force sensor.

• A continuum thermodynamics based magnetomechanical constitutive model is

developed to quantify the non-linear and hysteretic behavior of Ni-Mn-Ga for

sensing, actuation and blocked-force cases.

• The microstructure and dissipation is included in the continuum framework via

internal state variables, the evolution of which dictates the material response.

• The work capacity of Ni-Mn-Ga is around 72.4 kJ/m3, which is higher than

that of piezoelectric and magnetostrictive, however, the actuation authority of

the material is limited as the maximum blocking force is only around 4 MPa.

202

• Quasi-static characterization chows a flux density sensitivity with strain

(∂B

∂ε

)

as 4.19T/%ε at 173 kA/m, and 2.38T/%ε at 368 kA/m; maximum field induced

twinning stress as 2.84 MPa; variation of initial susceptibility

(∂M

∂H|H=0

)of

59%; and maximum stress generation of 1.47% at 3% strain.

• Dynamic actuation model to was developed by including eddy currents and

structural dynamics along with constitutive volume fraction model to describe

the frequency dependent strain-field hysteresis.

• The dynamic actuator model was applied for magnetostrictive materials to

demonstrate its wider application.

• The dynamic sensing behavior of Ni-Mn-Ga was characterized by subjecting Ni-

Mn-Ga to compressive strain loading of 3% at frequencies from 0.2 to 160 Hz

in presence of bias field of 368 kA/m.

• The dynamic stress vs. strain plots show negligible change with increasing

frequency, whereas the flux-density vs. strain plots show an increasing hysteresis

that is linearly proportional to the frequency.

• The net flux-density change per unit strain remains almost constant (≈ 159 G)

with increasing strain, which can offer applications in broadband sensing and

energy harvesting.

• Stiffness of Ni-Mn-Ga was characterized by conducting broadband white-noise

base excitation tests under collinear and transverse bias magnetic fields.

203

• Measured stiffness changes of 51% and 61% for the collinear and transverse con-

figurations respectively indicate that Ni-Mn-Ga is suitable for tunable vibration

absorption applications.

• Ni-Mn-Ga is therefore demonstrated as a new multi-functional smart material

with applications in sensing, actuation and vibration absorption.

7.3 Future Work

This research has led to a thorough understanding about several aspects of Ni-

Mn-Ga FSMAs which were previously not investigated. Following list enumerates the

possible improvements in this work, as well as the future research opportunities that

have been opened up as a result of this research:

7.3.1 Possible Improvements

• The thermodynamic energy potentials in the constitutive model can be revisited

to add more complex effects such as the exchange energy, and magnetoelastic

coupling energy.

• A more accurate expression for magnetostatic energy can be used as that in

Ref. [82]. However, the usefulness of the additional accuracy against the in-

creased complexity and computational time needs to be evaluated.

• The blocked-force model can be improved to add the hysteretic effects in the

stress response.

• 2-D magnetic diffusion equations can be used in the dynamic actuation and

sensing models, and the current averaging technique can be reconsidered.

204

7.3.2 Future Research Opportunities

• The sensor device using permanent magnets as that shown in Appendix B (see

Section B.3) could be refined to make it more compact and robust. Such a

device would lead to realistic evaluation of the energy density of Ni-Mn-Ga

sensors and the effect of the system dynamics on the sensor performance. The

effect of prestress on the system properties could be of interest.

• The constitutive model could be extended to address the 3-D behavior, which

would enable the implementation of the model in finite element analysis codes.

• A continuous structural model of the Ni-Mn-Ga rod could be used for dynamic

actuator. This will enable further development towards predicting the dynamic

performance of structures made using Ni-Mn-Ga, or structures with patches of

Ni-Mn-Ga, encompassing various shapes such as rods, beams and plates.

• The dynamic actuator model can be augmented to add the electromagnet

impedances so that the voltage and currents can be used as input variables

instead of magnetic field.

205

APPENDIX A

MISCELLANEOUS ISSUES WITH QUASI-STATICCHARACTERIZATION AND MODELING

A.1 Electromagnet Design and Calibration

A.1.1 Effect of Dimensions on Field

To design the electromagnet, influence of various parameters on the final field

must be studied to maximize its efficiency. Figure A.1 shows a 2-D view of the

laminates. Once the overall dimensions of the E-shaped laminates are chosen, certain

dimensions are fixed, such as the width of the central legs (D). But, there are two

major dimensions that affect the magnetic field generated per given current density

in the coils (J). They are the length of the E-shaped legs (L) and the width of the

central leg at the end of the taper (d). The angle of the taper (Φ) on the central leg

is,

Φ = tan−1

(D − d

2W

)(A.1)

The objective of designing the electromagnet is to generate maximum magnetic

field in the central air gap for a given current density in the coils. A finite element

software for electromagnetics such as FEMM or COMSOL provide a quick way to

investigate the effect of these dimensions on the generated magnetic field. Using

206

Laminated core

Air Gap

Coils

L

D d

(Current Density J) W

Figure A.1: Schematic of the Electromagnet.

FEMM, various simulations are conducted to find the effect of the ratio (d/D) on the

magnetic field at a given length (L). A snapshot of one of the simulations is shown in

Figure2.3. The results of these simulations are summarized in Figures A.2 and A.3.

It is observed that the length of 5 inches gives maximum field ratios in the range

of around 0.3-0.6. However, this results in a steep taper angle of around 20-30 deg.

Such a steep taper angle is usually not recommended because it can result in excessive

leakage which may not be accurately simulated by the FEMM software. Furthermore,

a steep angle or very small width (d) may not provide a uniform field over the entire

length of the sample. Considering these issues, the length of the legs is chosen as

6 inches, and the width of the legs is chosen as 1.4 inch. These dimensions correspond

to a ratio of 0.62, and taper angle of 10.04 deg.

207

0.2 0.4 0.6 0.8 1750

800

850

900

950

1000

Ratio of Small width / Large width

Mag

netic

Fie

ld (

kA/m

)

5 in6 in7 in8.3 in

Increasing L

Figure A.2: Effect of ratio (d/D) on field.

0 5 10 15 20 25 30 35750

800

850

900

950

1000

Taper Angle (deg)

Mag

netic

Fie

ld (

kA/m

)

5 in6 in7 in8.3 in

Increasing L

Figure A.3: Effect of angle (Φ) on field.

208

1 1.5 2 2.5 3 3.5 4 4.5300

400

500

600

700

800

900

Current Density (MA/m2)

Mag

netic

Fie

ld (

kA/m

)

Figure A.4: Variation of current density with field.

For these final dimensions, the variation of field with current density (J) in the

coils is plotted in Figure A.4. It is observed that the field increases linearly with cur-

rent density values of up to J ≈ 2.25 M/A2. Further increase in current density does

not increase the field by a significant amount because the electromagnet core starts to

saturate. Therefore, the coils are designed to carry maximum current corresponding

to the current density of around 2.5 MA/m2.

Wire Selection

The wire is selected based upon the available area, maximum current carrying

capacity, resistance of the wire, and most importantly, the magnetomotive force (NI)

it can produce within the given constraints. The available area (Aw) for winding a

209

coil is fixed, which corresponds to a rectangle (lw × ww) of around 2 in× 1.075 in. If

the wire has a diameter of dw, the maximum possible turns per layer (n) are,

n =lwdw

, (A.2)

and maximum number possible number of turns (Nm) are,

Nm =lwww

d2w

, (A.3)

The area occupied by one turn is assumed to be equal to the square of the wire

diameter. The packing efficiency is assumed to be around (ηp = 80%), which gives

the actual number of turns as,

N = ηpNm. (A.4)

If the maximum current carrying capacity of the wire is Im, the maximum MMF

produced by the wire is,

MMFmax = NIm. (A.5)

For the given purpose, the objective of the coil design is to maximize this MMF for

a given wire. Additional considerations include the total resistance of the wire (Rw),

which dictates the power requirements and the Joule heating (IR2w), which places

restrictions on the resistance and current. A wire of small diameter would pack a

very large number of turns, however, its current carrying capacity would be low, and

the resistance would be high, leading to increased heating. On the other hand, a wire

with large diameter would carry a high amount of current, but its size could place

restrictions on the maximum possible turns. A detailed study of various wire sizes

from AWG 12 to AWG 20 is conducted to arrive at the optimum wire size. These

results are summarized in Figure A.5.

210

AW

G W

ire S

ize

12

13

14

15

16

17

18

19

20

Len

gth

(in

)2

22

22

22

22

Heig

ht

dif

f (i

n)

1.075

1.075

1.075

1.075

1.075

1.075

1.075

1.075

1.075

Are

a (

in2)

2.15

2.15

2.15

2.15

2.15

2.15

2.15

2.15

2.15

Wir

e d

iam

ete

r (i

n)

0.0808

0.072

0.0641

0.0571

0.0508

0.0453

0.0403

0.0359

0.032

Tu

rns p

er

layer

24.7524752527.77777778

31.20124835.026269739.370078744.150110449.627791655.7103064

62.5

Max. P

ossib

le t

urn

s329.3

182041

414.7

376543

523.2

6586

659.4

26268

833.1

26666

1047.7

1233

1323.8

1826

1668.2

0555

2099.6

0938

Rate

d C

urr

en

t (A

mp

)11.5

10

8.5

7.5

6.5

5.75

54.375

3.75

Th

eo

reti

cal M

MF

(N

*I)

3787.1

59347

4147.3

76543

4447.7

5981

4945.6

9701

5415.3

2333

6024.3

4591

6619.0

9131

7298.3

993

7873.5

3516

Cu

rren

t d

en

sit

y (

A/in

2)

1761.4694641929.0123462068.725492300.324192518.755042802.021353078.647123394.604323662.10938

N_I/A

2.7302831292.9899751163.206530933.565509633.904078124.343141784.771912585.261647235.67628088

Mean

Peri

mete

r12.34

12.34

12.34

12.34

12.34

12.34

12.34

12.34

12.34

R_p

er

1000 f

t1.63

2.06

2.525

3.184

4.016

5.064

6.385

8.051

10.15

R_co

il0.5519976850.8785664221.358681612.15910228

3.44063545.455941028.6920693513.811260121.9148478

Ind

ucta

nce*p

hi*

e-6

0.108450480.1720073220.27380716

0.4348430.694100041.097701131.752494792.782909774.40835953

Jo

ule

Heat

= I^

2*R

73.0

0169385

87.8

5664223

98.1

647463

121.4

49503

145.3

66846

180.3

8705

217.3

01734

264.3

5615

308.1

77547

Packin

g e

ffic

ien

cy

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.8

Actu

al tu

rns (

Na)

263.4

545633

331.7

901235

418.6

12688

527.5

41015

666.5

01333

713.0

43478

820

937.1

42857

1093.3

3333

Actu

al M

MF

(N

a*I

)3029.7

27478

3317.9

01235

3558.2

0785

3956.5

5761

4332.2

5866

4100

4100

4100

4100

R_co

il (

Oh

m)

0.4

41598148

0.7

02853138

1.0

8694529

1.7

2728182

2.7

5250832

3.7

1315965

5.3

8404483

7.7

5871036

11.4

117578

Ind

ucta

nce*p

hi*

10^

(-6)

0.0

69408307

0.1

10084686

0.1

7523658

0.2

7829952

0.4

4422403

0.5

08431

0.6

724

0.8

7823673

1.1

9537778

Jo

ule

Heat

= I^

2*R

58.4

0135508

70.2

8531379

78.5

31797

97.1

596026

116.2

93476

122.7

66341

134.6

01121

148.5

06566

160.4

77844

Po

ssib

le m

ore

tu

rns

--

--

334.668854503.818261731.0626971006.27604

Fig

ure

A.5

:C

ompar

ison

ofva

riou

sw

ire

size

s.

211

It is seen from this comparison that AWG 16 wire gives the maximum MMF among

the chosen sizes. The comparison of the various wires regarding their maximum

current capacity, maximum possible turns and MMF is given in Figure A.6. The wire

size of AWG 16 clearly turns out to be the optimum size as it provides a balance

between the maximum current carrying capacity and maximum allowed turns, which

leads to maximum possible MMF. This wire has a diameter of 0.0508 in.

The coil is wound on a rectangular shaped bobbin using a stepper motor and a

custom-made fixture. A thin layer of epoxy is applied after each layer to hold the

wires together, and to provide extra insulation. Two such coils are placed on the

central legs of the electromagnet, and are connected in parallel. Figure A.7 shows a

picture of the assembled electromagnet. Drawings of the electromagnet and relevant

parts are given in D.1.

212

12 13 14 15 16 17 18 19 201000

1500

2000

2500

3000

3500

4000

4500

5000

AWG Wire Number

Turns (N×4)Current (I

max×400)

MMF (N×Imax

)

Figure A.6: Comparison of current carrying capacity, possible turns and MMF pro-duced by various wires (The current and turns are multiplied by scaling factors) Wiresize AWG 16 is seen as an optimum size.

213

Air

gap

Laminated

core

coil

Figure A.7: Picture of the assembled electromagnet.

A.1.2 Electromagnet Calibration with Sample

The magnetic bias field for the sensing characterization presented in Section 2.2

is assumed to be that given by the calibration curve in Figure 2.4. The applied bias

magnetic field is not measured during the tests, and the calibration curve is used to

obtain the field from the measured current in the electromagnet coils.

Naturally, one of the issues during theses tests is whether the observed change in

flux density is only due to the change of sample variant configuration or also due to

the change in the reluctance in the electromagnet gap. When the sample is placed

in the electromagnet gap, the permeability of the air gap decreases as the reluctance

due to the sample is higher than that due to the air. Furthermore, as the sample is

compressed, its permeability changes which again changes the reluctance of the air

gap.

214

If the change in reluctance is significant, it can introduce errors in the results

obtained in sensing as well as blocked force characterizations, because the applied

magnetic field can no longer be accurately predicted by the electromagnet calibra-

tion curve in Figure 2.4. Therefore, it is necessary to check the effect of sample

configuration on the applied field.

Electromagnet calibration tests similar to that in Section 2.1.2 are conducted with

the sample in the air-gap. First, the sample with complete field-preferred variant con-

figuration (easy axis configuration) is placed in the air gap, and the electromagnet

is calibrated. The easy axis configuration implies that the sample has highest per-

meability and thus the reluctance in the electromagnet gap is lowest. Therefore,

this configuration can have maximum impact on the applied field, of all other con-

figurations of the sample. This process is also repeated with sample in hard-axis

configuration placed in the electromagnet air-gap. Finally, the sample is removed

from the air gap, and the applied field in the air gap is measured.

The test results are shown in Figure A.8. It is seen that there is almost no change

in the measured field for the same values of current with or without the presence

of sample. The maximum variation is obtained as 3%, which is sufficiently small to

allow the approximation that the presence of sample does not affect the applied field.

Possible reasons for negligible variation in the field magnitude can be:

• The permeability of the sample in both easy-axis and hard-axis case is too low

compared to that of the iron core, and thus does not affect the total reluctance

much.

215

−8 −6 −4 −2 0 2 4 6 8−600

−400

−200

0

200

400

600

Current (Amp)

Mag

netic

fiel

d (k

A/m

)

Sample easy axisSample hard axisNo sample

6 6.5 7 7.5 8300

350

400

450

500

550

Current (Amp)

Mag

netic

fiel

d (k

A/m

)

Sample easy axisSample hard axisNo sample

Figure A.8: Electromagnet calibration curve in presence of sample, the easy axiscurve shows maximum variation.

216

• There is a significant reluctance and flux leakage in the electromagnet core itself,

therefore a small change in the reluctance of the sample does not change the

overall behavior of the magnetic circuit.

These tests thus confirm that the issue of electromagnet reluctance change can be

neglected, and the applied fields in all the cases can be assumed to be equal to the

applied fields measured in air.

217

A.2 Verification of Demagnetization Factor

As seen in Section 3.6.2, the relationship between the measured flux-density (Bm)

and magnetization of the sample in x-direction (M) is given as,

Bm = µ0(H + NxM), (A.6)

with Nx the demagnetization factor, H is the applied field, and M is the magnetization

inside the sample. The schematic of the demagnetization process is illustrated in

Figure A.9. The demagnetization factor is obtained from the geometry of the sample.

Therefore by measuring the flux-density outside the sample, the magnetization inside

the sample is calculated, and is used for comparison with the model results. Validation

of equation (A.6) is therefore critical from the viewpoint of both the characterization

and modeling of the sensing behavior.

To simulate this situation, a finite element software, COMSOL is used. As seen

in Section A.1.2, the sample does not affect the applied magnetic field of the elec-

tromagnet. Hence, the source of magnetic field can be represented by electromagnet

as well as permanent magnets, and the latter is used for simplicity. Moreover, the

use of permanent magnets as a constant magnetic field source is a better choice for

these simulations because COMSOL can not realistically model the reluctance in the

electromagnet cores.

The problem under consideration is modeled by with two permanent magnets

and the Ni-Mn-Ga sample in the gap between them (Figure A.10). This is a 3-

D magneto-static problem with no currents. Two Nd-Fe-B permanent magnets are

considered to be applying a bias field. The magnets are modeled by using a rem-

nant flux density value from the manufactures’ catalogue (Br = 1.32 T in this case).

218

+

+

+

+

+

-

-

-

-

-

M

Hd

H

Figure A.9: Schematic of the demagnetization field inside the sample. The appliedfield (H) creates a magnetization (M) inside the sample, which results in north andsouth poles on its surface. H and M are shown by solid arrows. The demagnetizationfield (Hd = NxM) is directed from north to south poles as shown by dashed arrows.Although inside the sample, the demagnetization field opposes the applied field, itadds to the applied field outside the sample. Therefore, the net field inside the sampleis given as H −NxM , whereas the net field outside the sample is given as H + NxM .

219

magnet Ni-Mn-Ga

Figure A.10: A snapshot from COMSOL simulation.

The sample is located in the central gap. The medium of the sample is varied

as (i) Air (µr = 1), (ii) Ni-Mn-Ga with complete field preferred (easy-axis) with

(µr = 3.06), and (iii) stress preferred (hard-axis) with (µr = 1.46). The field, flux

density and magnetization are plotted as a function of the air gap in the middle of the

two magnets. The horizontal line is shown in red color over which the three quantities

are plotted.

In the experimental setup, the Hall probe is placed outside the sample, in the

gap between the sample and magnet. Naturally, the flux density measured by the

220

Hall probe is not the same as that inside the sample. Consequently, the magnetiza-

tion inside the sample also can not be obtained directly. Therefore to calculate the

magnetization, expression A.6 is used.

When there is no sample present in the gap of the electromagnet, the simulated

magnetic field corresponds to H. However when a sample is present in the air gap,

this simulated field increases because the demagnetization field adds to the applied

field. Therefore the effect of the demagnetization field on simulated field is obtained

by,

Hre = H + NxM, (A.7)

where Hre is the recalculated magnetic field.

The magnetization inside the material is obtained as,

M =

(Bm

µ0

−H

)1

Nd

, (A.8)

The flux density outside the sample is then reiterated by using the calculated

magnetization as,

Bre = µ0Hr, (A.9)

Figure A.11 shows the magnetic field variation in the gap, figure A.12 shows the

flux density variation, and figure A.13 shows the magnetization variation. The solid

lines show the quantities obtained from COMSOL directly, whereas the dashed lines

show quantities calculated from equations (A.6) to (A.7).

In Figure A.11, it is seen that with increasing permeability of the media in the

gap (µair < µhard < µeasy), the applied field increases. This behavior may not be

obvious, since the magnetic field from the permanent magnets is expected to constant.

However, the demagnetization field from the sample adds to the applied field from the

221

0.025 0.03 0.035 0.04 0.045150

200

250

300

350

400

450

500

550

600

650

Distance (inch)

Mag

netic

fiel

d (k

A/m

)

EasyHardNo sampleEasy reiteratedHard reiterated

Figure A.11: Magnetic field vs distance. Solid: COMSOL, Dashed: recalculated.

0.025 0.03 0.035 0.04 0.0450.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Distance (inch)

Flu

x de

nsity

(T

esla

)

EasyHardNo sampleEasy reiteratedHard reiterated

Figure A.12: Flux density vs distance. Solid: COMSOL, Dashed: recalculated.

222

0.025 0.03 0.035 0.04 0.0450

50

100

150

200

250

300

350

400

450

500

Distance (inch)

Mag

netiz

atio

n (k

A/m

)

EasyHardEasy reiteratedHard reiterated

Figure A.13: Magnetization. Solid: COMSOL, Dashed: recalculated.

magnets, and thus results in an apparent increase in the applied field. The addition of

the demagnetization field to the field produced by magnets is given by equation (A.7).

Referring to Figure A.12, in case of easy and hard axis, the Hall probe measures the

flux density value that is given by COMSOL simulation. The aim of this measurement

of flux density outside the sample is to obtain the flux density and magnetization

inside the sample. In Figures A.12 and A.13, it is seen that the flux density and

magnetization inside the sample for easy axis case is higher than that for the hard axis

case, which is expected because of the higher permeability. However, the magnetic

field inside the sample varies in opposite manner to that of the flux density and

magnetization.

223

From the measured flux density (Bm), the magnetization inside the sample (M)

can be calculated from equation (A.8), and from this magnetization, the field and

flux density can be recalculated from equations (A.7) and (A.9). These recalculated

values will be comparable to values simulated by COMSOL only if the method to

calculate the magnetization is valid. Equations (A.6) through (A.9) are accurate

when the demagnetization field can be added algebraically to the applied field to

obtain the recalculated field. The best chance for these relations to hold true is

when the calculations are performed at points that are very close to the edges of the

sample (just to the left or right of the sample). As seen by the two thick circles in

all the figures, the values of the simulated and calculated fields match well in this

vicinity. Therefore, the Hall probe is placed on the edge of the sample, where the

measured and recalculated values match with good accuracy.

224

A.3 Damping Properties of Ni-Mn-Ga

As seen in Chapter 2, the stress-strain behavior of Ni-Mn-Ga is highly hysteretic,

indicating the potential of the material for damping applications. The damping ca-

pacity is measured as a function of energy absorbed by the material relative to the

mechanical energy input to the system. Furthermore, this damping capacity of Ni-

Mn-Ga can be altered by the bias magnetic field because the stress-strain behavior is

highly dependent on the bias field. Traditional high modulus damping materials find

limited applications since their damping capacities (tan delta≈0.01) are significantly

lower than that seen in polymers. The structure of Ni-Mn-Ga is inherently stiffer

than that of the viscoelastic materials. Also, the twin-variant rearrangement can be

initiated at relatively low stresses of around 1 MPa. The combination of high mod-

ulus and high damping capacity can give Ni-Mn-Ga advantages over the currently

available systems.

Damping capacity (Ψ) is a unit-less quantity is given as the ratio of the energy

dissipated per cycle of oscillation (∆W ) to the energy input to the system per cycle

of oscillation (W ) in the form [36]:

Ψ =∆W

W(A.10)

In the context of the hysteretic stress-strain loops, ∆W represents the area en-

closed within one cycle whereas the net energy input (W =∫

σdε) is the area within

the loading curve. Typically, the damping properties of viscoelastic materials are

calculated from the phase difference (δ) between the stress and strain response in

time-domain. If the phase lag is constant, the damping capacity (Ψ) can be directly

225

related to (tan δ). Recently, a relationship has been developed to relate Ψ to δ [71].

Ψ =∆W

W=

π tan δ(1 + (π

2+ δ) tan δ

) (A.11)

For small δ (i.e. δ << 1), this equation becomes [36],

Ψ =∆W

W≈ π tan δ (A.12)

Equations (A.10)-(A.12) provide relationships between mechanical hysteresis loops,

damping capacity of the material, and tan δ. From the stress-strain curves showed in

Chapter 2, various properties such as energy absorbed, energy input, damping capac-

ity (Ψ) and tan δ can be calculated. Figure A.14 shows the variation of the energy

absorbed and mechanical energy input with the bias field. It is observed that the

mechanical energy input increases almost linearly with the magnetic field. This is

because the twinning stress increases with field, requiring more energy to compress

the sample completely. The energy absorbed by the material in one cycle increases

monotonically for fields of up to around 360 kA/m, after which it remains almost

constant.

The damping capacity of Ni-Mn-Ga is shown in Figure A.15, which is obtained

directly from the plots in Figure A.14. The damping capacity is almost constant up

to magnetic fields of around 251 kA/m, after which it decreases in a linear fashion.

As the bias field is increased, more mechanical energy input is required to compress

the material. But, this additional energy input does not result in the energy that is

absorbed by the material.

The variation of tan δ is shown in Figure A.16. Ni-Mn-Ga shows significantly

higher values of tan δ than materials such as aluminum (tan δ ≈ 0.01). Although the

phase lag δ varies at different locations in the stress-strain curve, the values shown

226

0 100 200 300 400 50050

100

150

200

250

Bias Field (kA/m)

Ene

rgy

(kJ/

m3 )

Total EnergyEnergy absorbed

Figure A.14: Energy absorbed in the stress-strain curves of Ni-Mn-Ga.

0 100 200 300 400 5000.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Bias Field (kA/m)

Dam

ping

cap

city

(Ψ

)

Figure A.15: Damping capacity as a function of bias field.

227

0 100 200 300 400 5000.2

0.22

0.24

0.26

0.28

0.3

0.32

Bias Field (kA/m)

Tan

δ

Figure A.16: Variation of tan δ with magnetic bias field.

in Figure A.16 represent an average estimate. These values are of a similar order as

those for viscoelastic materials, however, Ni-Mn-Ga provides an advantage of higher

stiffness.

228

A.4 Magnetization Angles

Figure A.17 shows an assumption of the microstructure with four angles. Consider

the case of constant volume fraction, and an assumption of reversible evolution of the

four angles and the domain fraction (to be discussed later).

q3

Ms

Ms

x

1 - x

1 - a

a

x

y

e

H

a 1 - a

q4

q2

q1

Figure A.17: Schematic of Ni-Mn-Ga microstructure assuming four different anglesin the four regions.

ρφze =− ξµ0HMs[α cos(θ1)− (1− α) cos(θ2)]

− (1− ξ)µ0HMs[α sin(θ3) + (1− α) sin(θ4)](A.13)

ρφms =1

2ξµ0NM2

s [α cos(θ1)− (1− α) cos(θ2)]2

+1

2(1− ξ)µ0NM2

s [α sin(θ3) + (1− α) sin(θ4)]2

(A.14)

229

ρφan =ξ[Kuα sin(θ1)2 + Ku(1− α) sin(θ2)

2]

+ (1− ξ)[Kuα sin(θ3)2 + Ku(1− α) sin(θ4)

2](A.15)

Assuming reversible rotation of the magnetization vectors, we propose that the

derivatives of the energy expression with the two angles is zero,

π3 = −∂(ρφ)

∂θ3

= 0, π4 = −∂(ρφ)

∂θ4

= 0 (A.16)

The above equation leads to a result,

θ1 = 0, θ2 = 0. (A.17)

Therefore, it is concluded that the magnetization vectors in the field-preferred vari-

ant always remain attached to the c-axis of the crystals. This result is physically

consistent as both the Zeeman and anisotropy energies favor the attachment of the

magnetization vectors to the c-axis of the crystals.

Following a similar treatment for the angles θ3 and θ4, we get,

π3 = −∂(ρφ)

∂θ3

= 0, π4 = −∂(ρφ)

∂θ4

= 0 (A.18)

Further, using the above equation and ignoring extreme cases of α = 0, 1, ξ = 1 and

θ3 = π/2, θ4 = π/2, we get,

1

(1− ξ)α cos(θ3)π3 +

1

(1− ξ)(1− α) cos(θ4)π4 = 0 (A.19)

The above equation leads to a result,

θ4 = −θ3. (A.20)

Thus, directions of these two angles will always maintain a relation as given by equa-

tion (A.20).

230

APPENDIX B

MISCELLANEOUS ISSUES WITH DYNAMICCHARACTERIZATION AND MODELING

B.1 Jiles-Atherton Model

This section briefly addresses the Jiles-Atherton model used in Section 4.6, which

is utilized for modeling the frequency dependent strain-field hysteresis in dynamic

magnetostrictive actuators. The Jiles-Atherton model is for quasi-static behavior,

however, it is discussed in this section because it is augmented to model the dynamic

behavior by including magnetic diffusion and actuator dynamics. The detailed model

development can be found in [21, 59]. Key equations used to model the behavior are

summarized here.

The effective applied field (He) is different from the actual applied field (H) since

the term corresponding to Weiss interaction field (αM) also contributes towards the

effective field. This term in turn depends on the net magnetization M .

He = H + αM (B.1)

The anhysteric magnetization (Man) is calculated from the Langevin function (L(x) =

coth(x) − 1/x). It corresponds to the magnetization due to applied magnetic field

231

without any losses due to domain wall motion and external stress. This magnetization

is required to be calculated iteratively since Jiles had proposed that anhysteric mag-

netization should be calculated by considering effective field and not just the applied

field.

Man = MsL(He/a) (B.2)

where a is a model parameter, effective domain density.

In reality, the actual magnetization curve varies from anhysteric as there are en-

ergy losses due to domain wall pinning. Hence, only a part of energy is utilized

for magnetizing the material as the remaining energy is lost in overcoming domain

wall motion. The differential equation relating irreversible magnetization compo-

nent (Mirr) to effective field is given as,

Mirr = Man − kδ∂Mirr

∂He

, (B.3)

which is modified by chain rule as,

∂Mirr

∂H=

Man −Mirr

δk

∂He

∂H, (B.4)

where k is the energy to break a pinning site, and δ is a binary factor, with value 1

when (dH/dt > 0) and -1 when (dH/dt < 0). Furthermore,

∂He

∂H= 1 + α

∂Mirr

∂H, (B.5)

which is further modified as,

∂Mirr

∂H= ζ

Man −Mirr

δk − α(Man −Mirr), (B.6)

232

where

ζ =

1, (dH/dt > 0andM < Mirr), or1 (dH/dt < 0andM > Mirr),0, otherwise

(B.7)

The three magnetization components are related as follows because the reversible

component (Mrev) attempts to reduce the difference between irreversible and anhys-

tertic components.

Mrev = c(Man −Mirr), (B.8)

where c quantifies the amount of reversible domain wall bulging. Finally the net

magnetization component is given as,

M = Mrev + Mirr. (B.9)

The magnetostriction is related to the magnetization as,

λ =3

2

(M

Ms

)2

, (B.10)

with Ms the saturation magnetization.

The MATLAB code for Jiles-Atherton model is given in Section C.2.3. The model

results for magnetization and magnetostriction are shown in Figures B.1 and B.2

respectively. The magnetization results are consistent with the expected behavior.

The reversible component Mrev has slightly higher values at low applied fields, because

the domain walls bend reversibly at low field values. At high fields, they have sufficient

energy to break the pinning sites and therefore the reversible component reduces. The

effect of ζ is also evident in case of minor loop as the reversible component remains

constant for the corresponding short period of time because the susceptibility would

have had negative value according to equation without ζ.

233

−40 −20 0 20 40−800

−600

−400

−200

0

200

400

600

800

Applied Field (kA/m)

Mag

netiz

atio

n (k

A/m

)

ManMMirrMrev

Figure B.1: Magnetization vs. field using Jiles model.

−40 −20 0 20 400

2

4

6

8x 10

−4

Appplied Field (kA/m)

Mag

neto

stric

tion

Figure B.2: Magnetostriction vs. field using Jiles model.

234

B.2 Kelvin Functions

The functions berv(x) and beiv(x) are termed as Kelvin functions. They are real

and imaginary parts of vth order Bessel function of the first kind. For the special

case of v = 0, the functions are simply termed as ber(x) and bei(x), which are used

in Section 4.6. They are given as,

ber(x) =1

2

(J0(xe3iπ/4) + J0(xe−3iπ/4)

), (B.11)

bei(x) =1

2i

(J0(xe3iπ/4)− J0(xe−3iπ/4)

), (B.12)

where J0(x) is the zeroth order Bessel function of the first kind. Figure B.3 shows

these functions for x = 0 to 10.

0 2 4 6 8 10−20

0

20

40

60

80

100

120

140

x

ber

(x)

(a)

0 2 4 6 8 10−40

−20

0

20

40

60

x

bei (

x)

(b)

Figure B.3: Kelvin functions (a) ber(x) and bei(x).

235

B.3 Prototype Device for Ni-Mn-Ga Sensor

A prototype Ni-Mn-Ga sensor device is built as shown in Figure B.4. This de-

vice consists of aluminum plates to form the body. The construction of the device is

inspired from that used for piezoelectric accelerometers [25], which employ a seismic

mass based device. Two Nd-Fe-B permanent magnets of dimensions 1 × 1× 1 inch3

are used to apply a bias magnetic field of around 368 kA/m, which is the optimum

bias field as seen in Section 2.3.2. These permanent magnets are very strong, with

remnant magnetization of around 1.3 Tesla on the surface. The single crystal Ni-Mn-

Ga sample with dimensions 6 × 6 × 10 mm3 is placed in the gap between the two

permanent magnets. A seismic mass of around 80 grams is placed on top of the sam-

ple, followed by a PCB force sensor for measurement of dynamic forces. The seismic

mass is supposed to generate dynamic stresses in the Ni-Mn-Ga sample, which could

induce a change in its magnetization by twin variant reorientation. A preload spring

of OD 1.00 in, and ID 0.73 in is used for applying preload of around 3.5 MPa. The

stiffness of the spring is around 53 lb/in. This bias stress results in a material con-

figuration with approximately half field-preferred and half stress-preferred variants.

An adjust plate can be rotated up and down to vary the preload in presence of the

magnetic field. The load is varied according to the stress-strain curve corresponding

to 368 kA/m in Figure 2.6. Drawings of the device are given in Section D.2.

There are two configurations or boundary conditions in which the device could

be operated. In first case, the device is mounted on a shaker, with accelerometers

on the base plate and seismic mass to measure their motion. A Hall probe placed in

the gap between the magnets and Ni-Mn-Ga sample is used to measure the change in

flux-density. When subjected to a base excitation through the shaker, the sample will

236

Magnet(s)

Ni-Mn-Ga

Seismic mass

PCB Force sensor

Preload spring

Base plate

Adjust plate

Threaded rod

Figure B.4: Prototype device for Ni-Mn-Ga sensor.

be subjected to a dynamic stress because of the seismic mass. This stress can lead to

a change in magnetization of the material, which can be measured by the Hall probe.

In second case, the base plate is fixed to the ground, and a pushrod is attached to

the plate above the PCB force sensor. This rod extends through the hollow threaded

rod. This rod provides an input excitation to the material, which can be operated

via a vibration shaker or MTS machine. In this configuration, the seismic mass could

be removed to make the device more compact.

237

APPENDIX C

MODEL CODES

C.1 Quasi-static Model

C.1.1 Model Flowchart

Figure C.1 shows the model flowchart for loading case.

Figure C.1: Flowchart of the sensing model for loading case (ξ < 0).

238

C.1.2 Sensing Model Code

clear allclcclose all

warning off MATLAB:divideByZero

mu0=4*pi*10^(-7); % Permeability of vacuum (Tm/A)H =445; % Bias field magnitude (kA/m)sig_tw_0 = 0.6*10^6; % Twinning stress at zero field (N/m2)Ms=625*1000; % Saturation magnetization (A/m)e0 = 0.058; % Reorientation strainE0= 400e6; E1 = 2400e6; % Extreme values of modulli (N/m2)S0=1/E0; S1 = 1/E1; % Compliance (m2/N)Ku = 1.67e5; % Anisotropy constant (J/m3)Nd=0.434; % Demagnetization factorN= Nd - (1 - 2*Nd) % Factor to calculate magnetostatic energyk = 24*10^6; % Stiffness of twinning region (N/m2)pi_cr_0 = e0*sig_tw_0; % Threshold driving force

%% Code for loading

zs=[]; zs(1,1)=0; % Stress-preferred volume fractionz=[]; z(1,1)=1-zs(1,1); % Field-preferred volume fractiondt_started = 0; % Variable for identifying twin-onsete_max = 0.07;e = (0:e_max/1000:e_max); % Loading strain

for i=1:length(e)e_tw(i) = zs(i)*e0; % Twinning straine_e(i)=e(i) - e_tw(i); % Elastic strain

% ComplianceS(i) = S0 + (1-z(i))*(S1-S0);% Modulus associated with elastic strainE(i) = 1/S(i);% Modulus associated with twinning straina(i) = E(i)*k/(E(i)-k);

% Driving force due to stressF_zs(i) = 1/2*(e(i)-e0*(1-z(i)))^2*(-S1+S0)/(S0+(1-z(i))*(S1-S0))^2 ...

-(e(i)-e0*(1-z(i)))*e0/(S0+(1-z(i))*(S1-S0)) ...+1/2*k*e0^2*(1-z(i))^2*(-S1+S0)/((S0+(1-z(i))*(S1-S0))^2 ...*(1/(S0+(1-z(i))*(S1-S0))-k))-1/2*k*e0^2*(1-z(i))^2*(-S1+S0) .../((S0+(1-z(i))*(S1-S0))^3*(1/(S0+(1-z(i))*(S1-S0))-k)^2) ...

239

+k*e0^2*(1-z(i))/((S0+(1-z(i))*(S1-S0))*(1/(S0+(1-z(i))*(S1-S0))-k));

% Domain fractionalpha(i) = 1/2*(H+N*Ms)/(Ms*N);

% Constraints on alphaif(alpha(i)>=1)

alpha(i)=1;endif(alpha(i)<=0)

alpha(i)=0;end

% Rotation anglesin_theta(i) = mu0*H*Ms/(2*Ku+mu0*N*Ms^2);

% Constraints on thetaif(sin_theta(i) > 1)

sin_theta(i) = 1;endif(sin_theta(i) <-1)

sin_theta(i) = -1;end

theta(i) = asin(sin_theta(i));

% MagnetizationM(i) = (1-z(i))*(Ms*sin(theta(i))) + z(i)*Ms*(alpha(i) - (1-alpha(i)));% Flux-density (inside sample)B(i) = mu0*(M(i)+H); %Expression for induction inside sample% Flux-density (measured)Bm(i)=mu0*(H + M(i)*Nd);

% Constraint on magnetizationif (M(i)>=Ms)

M(i)=Ms;end

% Driving force due to fieldF_z_H(i) = - (mu0*H*Ms*sin(theta(i))-2*mu0*H*Ms*alpha(i) ...

+mu0*H*Ms+2*mu0*N*Ms^2*alpha(i)^2-2*mu0*N*Ms^2*alpha(i) ...-Ku+1/2*cos(theta(i))^2*mu0*N*Ms^2+cos(theta(i))^2*Ku );

% Net thermodynamic driving forceF(i) = +F_zs(i) + F_z_H(i);

240

% Code to check if the threshold is metif(i > 1 & F(i) <= -pi_cr & (z(i)-z(i-1)) <=0)

% Function to calculate volume fractionz(i+1) = fzero(@(xx) loading_newmag(xx,alpha(i),theta(i),...a(i),e0,S1,S0,pi_cr,e(i),N,Ms,H,mu0,Ku,k),0.5);

elsez(i+1)=z(i);

end

% Constraint on volume fractionif(z(i+1) >= 1)

z(i+1)=1;dt_started =0;

else if (z(i+1) <= 0)z(i+1)= 0;

endend

zs(i+1)=1-z(i+1);

% Stresssig(i)=E(i)*(e(i) - e0*zs(i));

end

%% Code for unloading

clear F_zs F_z_H z zszs=[];zs(1,1)=1;z(1,1)=1-zs(1,1);dt_started = 0;

e=(e_max:-e_max/1000:0); % Unloading strainfor i=1:length(e)

e_tw(i) = z(i)*e0; % Twinning straine_e(i)=e(i) + e_tw(i)-e0; % Elastic strain

S(i) = S0 + (1-z(i))*(S1-S0);E(i) = 1/S(i);a(i) = E(i)*k/(E(i)-k);

F_zs(i) = 1/2*(e(i)-(1-z(i))*e0)^2*(-S1+S0)/(S0+(1-z(i))*(S1-S0))^2 ...-(e(i)-(1-z(i))*e0)*e0/(S0+(1-z(i))*(S1-S0))+1/2*k*z(i)^2*e0^2 ...

241

*(-S1+S0)/((S0+(1-z(i))*(S1-S0))^2*(1/(S0+(1-z(i))*(S1-S0))-k)) ...-1/2*k*z(i)^2*e0^2*(-S1+S0)/((S0+(1-z(i))*(S1-S0))^3 ...*(1/(S0+(1-z(i))*(S1-S0))-k)^2)-k*z(i)*e0^2 .../((S0+(1-z(i))*(S1-S0))*(1/(S0+(1-z(i))*(S1-S0))-k));

alpha(i) = 1/2*(H+N*Ms)/(Ms*N);

if(alpha(i)>=1)alpha(i)=1;

endif(alpha(i)<=0)

alpha(i)=0;endsin_theta(i) = mu0*H*Ms/(2*Ku+mu0*N*Ms^2);

if(sin_theta(i) > 1)sin_theta(i) = 1;

endif(sin_theta(i) <-1)

sin_theta(i) = -1;end

theta(i) = asin(sin_theta(i));

M(i) = (1-z(i))*(Ms*sin(theta(i))) + z(i)*Ms*(alpha(i) - (1-alpha(i)));

if (M(i)>=Ms)M(i)=Ms;

endB(i) = mu0*(M(i)+H);Bm(i)=mu0*(H + M(i)*Nd);F_z_H(i) = - (mu0*H*Ms*sin(theta(i))-2*mu0*H*Ms*alpha(i) ...

+mu0*H*Ms+2*mu0*N*Ms^2*alpha(i)^2-2*mu0*N*Ms^2*alpha(i)-Ku ...+1/2*cos(theta(i))^2*mu0*N*Ms^2+cos(theta(i))^2*Ku );

F(i) = +F_zs(i) + F_z_H(i);

% Code to check if the threshold is metif( (i > 1 & (F(i)) >= pi_cr ) & z(i)>=z(i-1))

z(i+1)= fzero(@(xx) unloading_newmag(xx,alpha(i),theta(i), ...a(i),e0,S1,S0,pi_cr,e(i),N,Ms,H,mu0,Ku,k),0.5);

elsez(i+1)=z(i);

end

242

if(z(i+1) >= 1)z(i+1)=1;

else if (z(i+1) <= 0)z(i+1)= 0;

endend

zs(i+1)=1-z(i+1);sig(i) = E(i)*(e(i) - e0*(1-z(i)));

% Constraint to ensure that stress is always compressiveif(sig(i) <0)

sig(i) = 0;z(i+1)=z(i);zs(i+1)=zs(i);dt_started =0;

endend

%% Function file: loading_newmag.m

function y = loading_newmag(xx,alpha,theta,a,e0, ...S1,S0,pi_cr,e,N,Ms,H,mu0,Ku,k)

y = -mu0*H*Ms*sin(theta)+2*mu0*H*Ms*alpha-mu0*H*Ms-2*mu0*N*Ms^2*alpha^2...+2*mu0*N*Ms^2*alpha+Ku-1/2*cos(theta)^2*mu0*N*Ms^2-cos(theta)^2*Ku...

+1/2*(e-e0*(1-xx))^2*(-S1+S0)/(S0+(1-xx)*(S1-S0))^2...-(e-e0*(1-xx))*e0/(S0+(1-xx)*(S1-S0))+1/2*k*e0^2*(1-xx)^2*(-S1+S0)...

/((S0+(1-xx)*(S1-S0))^2*(1/(S0+(1-xx)*(S1-S0))-k))...-1/2*k*e0^2*(1-xx)^2*(-S1+S0)/((S0+(1-xx)*(S1-S0))^3...

*(1/(S0+(1-xx)*(S1-S0))-k)^2)+k*e0^2*(1-xx)/((S0+(1-xx)*(S1-S0))...*(1/(S0+(1-xx)*(S1-S0))-k)) + pi_cr;

%% Function file: unloading_newmag.m

function y = unloading_newmag(xx,alpha,theta,a,e0, ...S1,S0,pi_cr,e,N,Ms,H,mu0,Ku,k)

y =-mu0*H*Ms*sin(theta)+2*mu0*H*Ms*alpha-mu0*H*Ms-2*mu0*Ms^2*N*alpha^2...+2*mu0*Ms^2*N*alpha+Ku-1/2*cos(theta)^2*mu0*Ms^2*N-cos(theta)^2*Ku ...

+ 1/2*(e-(1-xx)*e0)^2*(-S1+S0)/(S0+(1-xx)*(S1-S0))^2 ...-(e-(1-xx)*e0)*e0/(S0+(1-xx)*(S1-S0))+1/2*k*xx^2*e0^2*(-S1+S0).../((S0+(1-xx)*(S1-S0))^2*(1/(S0+(1-xx)*(S1-S0))-k))...-1/2*k*xx^2*e0^2*(-S1+S0)/((S0+(1-xx)*(S1-S0))^3 ...

*(1/(S0+(1-xx)*(S1-S0))-k)^2)-k*xx*e0^2/((S0+(1-xx)*(S1-S0))...

243

*(1/(S0+(1-xx)*(S1-S0))-k))-pi_cr;

C.1.3 Actuation Model Code

clear all;close all;clc;

warning off MATLAB:divideByZero

mu0=4*pi*10^(-7); % Permeability of vacuum (Tm/A)sig_b = -1.43*10^6; % Bias stresssig_tw_0 = 0.8*10^6; % Twinning stress at zero field (N/m2)Ms=0.65/mu0; % Saturation magnetization (A/m)Ku=1.68*10^5; % Anisotropy constant (J/m3)k =13*10^6; % Stiffness of twinning region (N/m2)Nd=0.42; % Demagnetization factorN = 3*Nd - 1; % Factor for Magnetostatic energye0 = 0.058; % Reorientation strainE = 800*10^6; % average modulus (N/m2)S = 1/E; % Compliance (m2/N)

H_max = 800;H=0:0.5:H_max;H=H*1000; % Field during forward application

pi_cr = e0*sig_tw_0 ; % Threshold driving forcexi(1)=0;

zs=[]; % Stress-preferred volume fractionz=[]; % Field-preferred volume fraction

%% Code for forward field applicationz_start = 0.0; % Initial configurationz(1) = z_start;dt_started =0; % Variable for twin-onset

for i=1:length(H)

zs(i)=1-z(i);a = k*E/(E-k); % Stiffness associated with twinning strain

% Domain fractionalpha(i) = 1/2*(H(i)+N*Ms)/(Ms*N);

% Constraints on alpha

244

if(alpha(i)>=1)alpha(i)=1;

endif(alpha(i)<=0)

alpha(i)=0;end

% Rotation anglesin_theta(i) = mu0*H(i)*Ms/(2*Ku+mu0*N*Ms^2);

% Constraints on thetaif(sin_theta(i) > 1)

sin_theta(i) = 1;endif(sin_theta(i) <-1)

sin_theta(i) = -1;end

theta(i) = asin(sin_theta(i));

% MagnetizationM(i) = (1-z(i))*(Ms*sin(theta(i))) + z(i)*Ms*(alpha(i) - (1-alpha(i)));% Flux density (inside sample)B(i) = mu0*(M(i)+H(i));% Flux density (measured)Bm(i)=mu0*(H(i) + M(i)*Nd);

% Constraint on magnetizationif (M(i)>=Ms)

M(i)=Ms;end

% Driving force due to magnetic fieldF_z_H(i) = - (mu0*H(i)*Ms*sin(theta(i))-2*mu0*H(i)*Ms*alpha(i)...

+mu0*H(i)*Ms+2*mu0*N*Ms^2*alpha(i)^2-2*mu0*N*Ms^2*alpha(i)...-Ku+1/2*cos(theta(i))^2*mu0*N*Ms^2+cos(theta(i))^2*Ku );

% Driving force due to stressF_zs(i) = -a*e0^2*(z(i) ) + sig_b*e0;

% Net thermodynamic driving forceF_z(i) = F_z_H(i) + F_zs(i);

% Code to check if the threshold is metif ( i>1 & (F_z(i)) >= pi_cr & z(i)>=z(i-1) )

245

if (dt_started ==0)H(i)/1000

enddt_started =1;% Calculation of volume fractionz(i+1) = (F_z_H(i) + sig_b*e0 - pi_cr)/(a*e0^2);% Constraint on volume fractionif(z(i+1)<0)

z(i+1)=0;endif(z(i+1)>1)

z(i+1)=1;end

elsez(i+1)=z(i);

end

e_tw(i) = z(i)*e0; % Twinning straine_e(i) = sig_b/E; % Elastic strain

e(i) = e_e(i) + e_tw(i); % Total strainend

%% Code for reverse field applicationclear H F_z F_z_H e e_tw z zs F_zs M Bm theta alphaz=[];zs=[];z(1)=z_end;H = H_max:-0.5:00;H =H*1000; % Field during reverse applicationfor i=1:length(H)

zs(i)=1-z(i);a=k*E/(E-k);alpha(i) = 1/2*(H(i)+N*Ms)/(Ms*N);

if(alpha(i)>=1)alpha(i)=1;

endif(alpha(i)<=0)

alpha(i)=0;end

sin_theta(i) = mu0*H(i)*Ms/(2*Ku+mu0*N*Ms^2);

if(sin_theta(i) > 1)sin_theta(i) = 1;

end

246

if(sin_theta(i) <-1)sin_theta(i) = -1;

end

theta(i) = asin(sin_theta(i));

M(i) = (1-z(i))*(Ms*sin(theta(i))) + z(i)*Ms*(alpha(i) - (1-alpha(i)));B(i) = mu0*(M(i)+H(i));Bm(i)=mu0*(H(i) + M(i)*Nd);

if (M(i)>=Ms)M(i)=Ms;

end

F_z_H(i) = - (mu0*H(i)*Ms*sin(theta(i))-2*mu0*H(i)*Ms*alpha(i) ...+mu0*H(i)*Ms+2*mu0*N*Ms^2*alpha(i)^2-2*mu0*N*Ms^2*alpha(i) ...-Ku+1/2*cos(theta(i))^2*mu0*N*Ms^2+cos(theta(i))^2*Ku );

F_zs(i) = -a*e0^2*( z(i) - z_end*0 ) + sig_b*e0 ;F_z(i) = F_z_H(i) + F_zs(i);

if ( i>1 & (F_z(i)) <=-pi_cr & z(i)<=z(i-1) )dt_started =1;z(i+1)= (F_z_H(i) + sig_b*e0 + pi_cr + a*e0^2*z_end*0)/(a*e0^2);if(z(i+1)>1)

z(i+1)=1;endif(z(i+1)<z_start)

z(i+1)=z_start;end

elsez(i+1)=z(i);

ende_tw(i) = z(i)*e0;e_e(i) = sig_b/E; %% Constitutive equatione(i) = e_e(i) + e0*z(i);

end

C.2 Dynamic Model

C.2.1 Dynamic Actuator Model

247

clear all;close all;clc;

warning off MATLAB:dividebyzero

m = 4; % Number of cycles

% Parameters for constitutive modelf0=1; % FrequencyT0=1/f0; % Time period of one cycleT = m*T0; % Total time perioddf=1/T; % Frequency resolutionNp = 2^12; % Total pointsN0=Np/m; % Points in one cycleh=T/Np; % Time resolutionfs=1/h; % Sampling frequencyt=0:h:T-h; % Time vectort2=-h*N0:h:T-h;

% Matrix to select various frequenciesFF =[1 2 3 4 5 6 7

1 50 100 150 175 200 2506.25 6.25 6.25 5.5 4.5 3.875 3];

COL = [ ’g’ ’c’ ’m’ ’b’ ’y’ ’r’ ’k’ ];

ff = 7 % Index to select a frequencyxi = .95; % Damping ratiofn = 700; % Natural frequencyfa = FF(2,ff); % Actuation (applied field) frequencycol=COL(1,ff);H00 = FF(3,ff)*79.577*10^3; % Magnitude of applied fieldH0 = H00*1; % Magnitude of applied fieldmu0=4*pi*10^(-7); % Permeability of vacuummur = 3; % Relative permeabilityrho =62*10^(-8); % Resistivitye0=0.04; % Reorientation strainmu=mu0*mur; % Permeability of samplesigma = 1/rho; % Conductivityomega = 2*pi*fa; % Circular frequencydelta=sqrt(2/(mu*sigma*omega)); % Skin depth

d=5e-3; % Sample widthx = -d:d/100:d; % Distance vector

248

X= x/delta;Xd = d/delta;

% Parameters to triangular field profileNN = 2^10; TT = 1/fa; dtt = TT/NN;tt = 0:dtt:TT-dtt;mm=1;Npp = mm*NN; TTp = mm*TT;dff = 1/TTp; fss = 1/dtt; fhh= fss/2;

% Code to generate triangular field profilefor kk = 1:N0

if (kk <= N0/4+1)hh1(kk) = tt(kk)/(TT/4)*H0;

else if (kk<=3*N0/4+1)hh1(kk) = -H0/(TT/4)*(tt(kk)-TT/4)+H0;

elsehh1(kk) = H0/(TT/4)*(tt(kk)- 3*TT/4) - H0;

endend

endH_ext = [hh1 hh1 hh1 hh1 hh1];H_ext = [ hh1 hh1 hh1 hh1];H_ext = hh1; % Applied field vector

fft_H_ext = fft(H_ext); % Fourier transform of applied field

freqq = 0:dff:fss-dff;freqq1 = 0:dff:fhh-dff;ttt = 0:dtt:mm*TT-dtt;

% Code to generate single sided magnitude and phase spectrumfor kk = 1:length(freqq1)

% Magnitudeif (kk==1)

M_H_ext(kk) = abs(fft_H_ext(kk))/Npp;else

M_H_ext(kk) = abs(fft_H_ext(kk))*2/Npp;end% PhaseP_H_ext(kk) = unwrap(angle(fft_H_ext(kk)));

end

figure(33); subplot(2,1,1); stem(freqq1, M_H_ext/10^3);figure(33); subplot(2,1,2); stem(freqq1, P_H_ext);

249

% Code to regenerate input field profilefor ii = 1:length(ttt)

sum = 0;for kk = 1:512

sum=sum + M_H_ext(kk)*sin(2*pi*(kk-1)*dff*ttt(ii)+ (P_H_ext(kk)) + pi/2);

end% Regenerated input field profileH_ext_calc(ii)=sum;

end

figure(34); plot(ttt, H_ext/10^3,’b’);hold on;figure(34); plot(ttt, H_ext_calc/10^3,’r’);hold on;

% Code for calculation of Diffused Internal Fieldfor ii = 1:length(x)

X(ii) = x(ii)/delta;

% Complex solution to Diffusion Equationh_ans(ii) = 1/(cosh(Xd)^2*cos(Xd)^2+sinh(Xd)^2*sin(Xd)^2)...

*( cosh(X(ii))*cos(X(ii))*cosh(Xd)*cos(Xd)+sinh(X(ii)) ...*sin(X(ii))*sinh(Xd)*sin(Xd)+ j*(sinh(X(ii))*sin(X(ii))...*cosh(Xd)*cos(Xd)-cosh(X(ii))*cos(X(ii))*sinh(Xd)*sin(Xd)) );

hhh(ii)= abs(h_ans(ii)); % Magnitude

alpha(ii) = angle(h_ans(ii)); % Phase

% Calculation of field H(x,t) inside the sample using superpositionfor jj = 1:length(tt)

H(ii,jj)=0;for kk = 1:101

H(ii,jj) = H(ii,jj)+ M_H_ext(kk)*hhh(ii).*sin(2*pi*(kk-1)...*dff*tt(jj) + alpha(ii) + P_H_ext(kk) + pi/2);

endend

end

figure(35); plot(tt,H(1,:)/10^3,tt,H(25,:)/10^3,tt,...H(50,:)/10^3,tt,H(100,:)/10^3);hold on;

% Maximum field at a given distance inside samplefor kk = 1:length(x)

H_maxx(kk) = max(H(kk,:));

250

end

figure(36); plot(x*10^3,H_maxx/10^3,[col],’linewidth’,2);hold on;xlabel(’Position (mm)’,’Fontsize’,16);ylabel(’Maximum Field (kA/m)’,’Fontsize’,16);set(gca,’Fontsize’,14);

% Average field at a given timefor jj = 1:length(tt)

H_avg(jj) = mean(H(:,jj));endfigure(35); plot(tt,H_avg/10^3,’k’,’linewidth’,2);hold on;figure(37); plot(tt*fa,H_avg/10^3,[col],’linewidth’,2);hold on;

xlabel(’Nondimensional time (t*fa)’,’Fontsize’,16);ylabel(’Average Field (kA/m)’,’Fontsize’,16);set(gca,’Fontsize’,14);

% Cyclic average field that is used as an input to constitutive modelhh = [H_avg H_avg H_avg H_avg H_avg];

figure(80); plot(t2 ,hh ,’g--’); hold on;figure(80); plot(t(1:length(hh1)) ,hh1 ,’r ’); hold on;

% ---------------------------------------------------------

H_e=hh(N0+1:end);

z_end = 0;z_net = []; % Volume fractionH_net = [];z_start = 0.35; % Initial volume fractionz_end = 1; % Maximum volume fraction

count_f = 0; count_r = 0;loss =0;z=[];

% Function is used to obtain volume fraction from field[z_start z_end z_net H_net] = act_comb_mod_3(hh,z_start,z_end,fa,loss);z_net = z_net - z_start;

H_net = H_net(N0+1:end); z_net = z_net(N0+1:end);

figure(81); plot(H_net,z_net,’r’);figure(1);plot(t,H_net/10^3,’b’,’linewidth’,2);hold on;

251

xlabel(’Time (sec)’,’Fontsize’,16);ylabel(’Field (kA/m)’,’Fontsize’,16);set(gca,’Fontsize’,14);

figure(2);plot(t,z_net*e0*100,’r’,’linewidth’,2);hold on;xlabel(’Time (sec)’,’Fontsize’,16);ylabel(’| \xi |e_0 (%)’,’Fontsize’,16);set(gca,’Fontsize’,14); xlim([0 1]);

figure(3);plot(H_net/10^3,z_net*e0*100,[col,’--’],’linewidth’,1);hold on;xlabel(’Field (kA/m)’,’Fontsize’,16);ylabel(’Volume fraction’,’Fontsize’,16);set(gca,’Fontsize’,14);

figure(3);plot(H_e/10^3,z_net*e0*100,[col,’.’],’linewidth’,1);hold on;xlabel(’Field (kA/m)’,’Fontsize’,16);ylabel(’Volume fraction’,’Fontsize’,16);set(gca,’Fontsize’,14);

freq = 0:df:fs-df;w = hanning(Np);

fh=fs/2;

freq1 = 0:df:fh-df;fft_H = fft(H_net);fft_z = fft(z_net);

% Calculation of magnitudes and angles for Volume Fraction and net field% to create a single sided spectrumfor kk = 1:length(freq)/2

if (kk==1)M_H(kk) = abs(fft_H(kk))/Np ;M_z(kk) = abs(fft_z(kk))/Np ;

elseM_H(kk) = (abs(fft_H(kk)) + abs(fft_H(Np - kk+2)) )/Np;M_z(kk) = (abs(fft_z(kk)) + abs(fft_z(Np - kk+2)) )/Np;

endif(M_H(kk) > max(abs(fft_H))*1e-5/Np)

P_H(kk) = unwrap(angle(fft_H(kk)));else

P_H(kk) = 0;endif(M_z(kk) > max(abs(fft_z))*1e-5/Np)

P_z(kk) = unwrap(angle(fft_z(kk)));kk;

elseP_z(kk) = 0;

252

endend

figure(54); subplot(2,1,1); stem(freq1,M_z*e0*100,’r’,’linewidth’,2);hold on;xlabel(’Frequency (Hz)’,’Fontsize’,16);ylabel(’| \xi |e_0 (%)’,’Fontsize’,16);set(gca,’Fontsize’,14); xlim([ -0.1 10]); ylim([ 0 max(M_z)*e0*100]);

figure(54); subplot(2,1,2); stem(freq1,P_z*180/pi,’r’,’linewidth’,2);hold on;xlabel(’Frequency (Hz)’,’Fontsize’,16);ylabel(’Ang( \xi ) (deg)’,’Fontsize’,16);set(gca,’Fontsize’,14); xlim([ -0.1 10]);

% Creation a vector of freqs 0,2,4,.. for storing the magnitudes and phases% of the FFT of volume fraction mag and phase of FFTkkk=0;for kk = 1:length(freq1)

if ( mod(freq1(kk),2)==0)kkk = kkk+1;F(kkk)=freq1(kk);Mag_z(kkk) = M_z(kk);Ph_z(kkk) = P_z(kk);

endend

% Regeneration of the original signal of volume fractionfor ii = 1:length(t)

sum = 0;for kk = 1:20

sum=sum + Mag_z(kk)*cos(2*pi*F(kk)*t(ii) + (Ph_z(kk)) );endz_calc(ii)=sum;

endfigure(2);plot(t,z_calc*e0*100,’y’);xlim([0 1]);

legend(’Sin: orig’,’Sin: recon’,’Tri: orig’,’Tri: recon’);figure(3);plot(H_net/10^3,z_calc*e0*100,’r’);

% ---------------------------------------------------------

k = 12*10^3 % Spring constantf = 0:0.1:400; % Frequency vector

wa = 2*pi*fa; % Actuation frequency (rad/s)wn = 2*pi*fn; % Natural frequency (rad/s)% Sampling parameters

253

T = 1/fa; TT = 1/fa/2;N = 2^8; freq = fa*2;dt = T/N;t_t = 0:dt:T-dt;t = 0:dt:TT-dt;j=sqrt(-1);

% Creation of triangular input field waveform at actuation frequencyfor kk = 1:N/2

if (kk <= N/4+1)HH(kk) = t(kk)/(T/4)*H0;

else if (kk<=3*N/4+1)HH(kk) = -H0/(T/4)*(t(kk)-T/4)+H0;

elseHH(kk) = H0/(T/4)*(t(kk)- 3*T/4) - H0;

endend

end

% Calculation of dynamic strain using actuator dynamicsfor ii = 1:length(t)

sum2 = 0;for kk = 1:100

r = F(kk)*fa/fn; % Frequency RatioXX =1./((1-r.^2)+j*(2*xi.*r)); % MagnitudeX0 = abs(XX); phi = -angle(XX); % Phase% Superposition of individual displacement responsessum2=sum2 + e0*Mag_z(kk)*X0*cos(2*pi*F(kk)*fa*t(ii) ...

+ Ph_z(kk) - phi ) ;endx_calc(ii)=sum2; % Final dynamic strain

end% ---------------------------------------------------------

C.2.2 Dynamic Sensing Model

f0 = 12; % Frequency of applied strainH_mag = 368*1000; % Bias field magnitude (kA/m)fa=f0;

% Generating strain vectors from the experimental datapp = textread(’cyclic_data_12Hz.txt’);t_av = pp(:,1);

254

e_av = pp(:,2);sig_av = pp(:,3);B_av = (pp(:,4));B_av1 = -(max(B_av) - B_av);e_b = 0.03% figure(1); plot(e_av + e_b,sig_av);% figure(9); plot(e_av + e_b, B_av); %ylim([ 0 1]);t = t_av;ee = e_av+0.03;

mm = 1:length(t);Np = mm;[val indd] = max(ee);% Loading strain vectoree_load = ee(1:indd); t_load = t(1:indd);% Unloading strain vectoree_unload = ee(indd:end); t_unload = t(indd:end);

% Sampling parametersnnnn=14Np = 2^8;T = 1/f0;TT = nnnn*T;dt = T/Np;df = 1/TT;

mu0 =4*pi*10^(-7); % Permeability of vacuummur = 4.5; % Relative permeabilitymu = mu0*mur; % Permeability of samplerho = .6*10^(-8); % Resistivitysigma = 1/rho; % Conductivitykd = 1/(mu0*sigma);d = 6e-3; % Sample widthNx = 21; % Number of pointsdx = d/(Nx-1);x = 0:dx:d;scaling = kd*dt/dx^2r = scaling

% Parameters for backward difference methodt = 0:dt:TT-dt;fs = 1/dt;fh = fs/2;freq = 0:df:fs-df;freq1 = 0:df:fh-df;

255

H0 = H_magNt = length(t)

%% Code for Backward Difference Method% Code to generate matricesH = H0*ones(Nx,nnnn*Nt);H = H0*zeros(Nx,Nt);A = zeros(Nx,Nx);A(1,1) = (1+2*r) ; A(1,2) = -r;A(Nx,Nx-1) = -r; A(Nx,Nx) = (1+2*r) ;

A(1,1) = 1; A(1,2) = 0;A(Nx,Nx-1)=0; A(Nx,Nx)= 1;

for ii = 2:Nx-1A(ii,ii-1) = -r;A(ii,ii) = 1+2*r;A(ii,ii+1) = -r;

end

e_mean = (min(ee_load)+max(ee_load))/2;e_ampl = (max(ee_load)-min(ee_load))/2;e_net = e_mean + e_ampl*sin(2*pi*f0*t - pi/2);

ee_load = e_net(1:Np/2); t_load = t(1:Np/2);ee_unload = e_net(Np/2+1:Np); t_unload = t(Np/2+1:Np);

% Parameters for linear constitutive modelmax_z = .7793min_z = .3069ee_net = e_net;max_zs = 1 - min_zmin_zs = 1 - max_z

H_mag = H_mag*ones(1, length(ee_net));

% Code to obtain magnetic field inside the sample% and dynamic magnetization using backward difference

for i=1:length(ee_net)if (i==1)

M_d(i) = -3.7333e6*ee_net(1) + 1.76*H_mag(1);H_d(i) = H_mag(1);

end

256

% Linear constitutive equationM_d(i+1) = -3.7333e6*ee_net(i) + 1.76*H_mag(1);

b = H(:,i) - (M_d(i+1)-M_d(i));b(1) = H0 ;b(end) = H0;H(:,i+1) = A\b;

% Internal magnetic fieldH_d(i+1) = mean(H(:,i+1));% Dynamic magnetizationM_lin(i) = -3.7333e6*ee_net(i) + 1.76*H_d(i);% Dynamic flux densityBm_lin(i) = mu0*(H0 + Nd*M_lin(i));

end

C.2.3 Jiles-Atherton Model

clear allclcclose all

M=0; % Net magnetizationMrev=0; % Reversible magnetizationMirr=0; % Irreversible magnetizationMan=0; % Anhysteretic magnetizationMs=7.65*10^5; % Saturation magnetization

%Model parametersa = 5000;k=4000;c=0.18;alpha=0.0033;

T=10; % Final timedt=0.005; % Time resolutionHmax = 40000; % Applied field amplitudet=0;i=0;count=0;col = ’b’

while ( t <= 4*10)t;if ((t > 2*T - 0.001) & (t < 2*T +0.001) & count==0)

t=2*T+T/2count=1;

257

T=T;Hmax=Hmax/2;

endt=t+dt;i=i+1;tt=t-T*floor(t/T);zeta=1;zeta=0;% Construction of magnetic field vector for minor loops% and to identify parameter zetaif(tt < T/4)

H(i)=Hmax*tt/(T/4);dHdt=Hmax/(T/4);delta=1;if (i~=1 & M(i-1)<Man(i-1) & t>T/4)

zeta=1;end

else if (tt< 3*T/4)H(i)=-Hmax*(tt-T/2)/(T/4);dHdt=-Hmax/(T/4);delta=-1;if (i~=0 & M(i-1)>Man(i-1) & t>T/4)

zeta=1;end

else if (tt<=T)H(i)=Hmax*(tt-T)/(T/4);dHdt=Hmax/(T/4);delta=1;if (i~=0 & M(i-1)<Man(i-1) & t>T/4)

zeta=1;end

endend

endif (t<=T/4)

zeta=1;endif(i>1)

dH(i)=H(i)-H(i-1);else

dH(i)=H(i);endif (i==1)

M(i)=0;Mrev(i)=0;

258

Mirr(i)=0;Man(i)=0;dM=0;dMirr=0;slope1=0;

else% Langevin function [Lv(x)=coth(x)-1/x]Man(i)=Ms*Lv((H(i)+alpha*Man(i-1))/a);

if (i==2)slope1=0;

else% Slope dMirr/dMslope1=(Mirr(i-1)-Mirr(i-2))/(M(i-1)-M(i-2));

endslope=zeta*dHdt*(Man(i)-Mirr(i-1)) ...

/(delta*k-alpha*(Man(i)-Mirr(i-1))*slope1);Mirr(i)=Mirr(i-1)+slope*dt;Mrev(i)=c*(Man(i)-Mirr(i));M(i)=Mrev(i)+Mirr(i);lambda(i) = M(i)^2/Ms^2;hold on;grid on;

endzetta(i)=zeta;tttt(i) = t;

end

259

APPENDIX D

TEST SETUP DRAWINGS

D.1 Electromagnet Drawings (Figures D.1-D.6)

D.2 Dynamic Sensing Device Drawings (Figures D.7-D.15)

260

Fig

ure

D.1

:E

-shap

edla

min

ates

for

elec

trom

agnet

.

261

Figure D.2: Plate for mounting electromagnet.

262

Figure D.3: Holding plates for electromagnet.

263

Figure D.4: Base channels for mounting electromagnet.

264

Figure D.5: Bottom pushrod for applying compression using MTS machine.

265

Figure D.6: Top pushrod for applying compression using MTS machine.

266

Fig

ure

D.7

:2-

Dvie

wof

the

asse

mble

ddev

ice.

267

Fig

ure

D.8

:B

otto

mpla

te.

268

Fig

ure

D.9

:Top

pla

te.

269

Figure D.10: Side plate.

270

Figure D.11: Support disc.

Figure D.12: Disc to adjust the compression of spring.

271

Figure D.13: Seismic mass (material: brass).

Figure D.14: Plate to secure magnets (2 nos).

272

Figure D.15: Grip to hold the sample (2 nos).

273

BIBLIOGRAPHY

[1] Adaptamat inc. www.adaptamat.com.

[2] J.S. Bendat and A.G. Piersol. Random Data: Analysis and Measurement Pro-cedures. John Wiley and Sons, Inc., New York, NY, 2000.

[3] Z. Bo and D.C. Lagoudas. “Thermomechanical modeling of polycrystallineSMAs under cyclic loading, Part I: theoretical derivations”. International Jour-nal of Engineering Science, 37(9):10891140, 1999.

[4] O. Bottauscio, M. Chiampi, A. Lovisolo, P.E. Roccato, and M. Zucca. “Dynamicmodeling and experimental analysis of Terfenol-D rods for magnetostrictiveactuators”. Journal of Applied Physics, 103(7):07F121–07F121–3, 2008.

[5] J.G. Boyd and D.C. Lagoudas. A thermodynamical constitutive model for shapememory materials. part I. The monolithic shape memory alloy. InternationalJournal of Plasticity, 12(6):805–842, 1996.

[6] L.C. Brinson. “One Dimensional Constitutive Behavior of Shape MemoryAlloys: thermo-mechanical derivation with non-constant material functions”.Journal of Intelligent Material Systems and Structures, 4(2):229–242, 1993.

[7] L.C. Brinson and M.S. Huang. “Simplifications and Comparisons of ShapeMemory Alloy Constitutive Models”. Journal of Intelligent Material Systemsand Structures, 7(1):108–114, 1995.

[8] F.T. Calkins. Design, analysis, and modeling of giant magnetostrictive trans-ducers. PhD thesis, Iowa State University, 1997.

[9] S. Cao, B. Wang, J. Zheng, W. Huang, Y. Sun, and Q. Xiang. “Modelingdynamic hysteresis for giant magnetostrictive actuator using hybrid genetic al-gorithm”. IEEE Transactions on Magnetics, 42(4):911–914, 2006.

[10] S. Chakraborty and G.R. Tomlinson. “An initial experimental investigation intothe change in magnetic induction of a Terfenol-D rod due to external stress”.Smart Materials and Structures, 12:763–768, 2003.

274

[11] V. A. Chernenko, V. A. L’vov, P. Mullner, G. Kostorz, and T. Takagi.“Magnetic-field-induced superelasticity of ferromagnetic thermoelastic marten-sites: Experiment and modeling”. Phys. Rev. B, 69:134410.1–10, 2004.

[12] V.A. Chernenko, V.A. Lvov, E. Cesari, J. Pons, A.A. Rudenko, H. Date,M. Matsumoto, and T. Kanomatad. “Stressstrain behaviour of NiMnGa alloys:experiment and modelling”. Material Science and Engineering A, 378:349–352,2004.

[13] A.E. Clark, J.B. Restorff, M. Wun-Fogle, T.A. Lograsso, and D.L. Schlagel.“Magnetostrictive properties of body-centered cubic Fe-Ga and Fe-Ga-Al al-loys”. IEEE Transactions on Magnetics, 36(5):3238–3240, 2000.

[14] B.D. Coleman and M.E. Gurtin. Thermodynamics with internal state variables.Journal of Chemical Physics, 47(2):597–613, 1967.

[15] R. N. Couch and I. Chopra. “Quasi-static modeling of NiMnGa magnetic shapememory alloy”. In Proceedings of SPIE Smart Structures and Materials, volume5764, pages 1–12, San Diego, CA, March 2005.

[16] R.N. Couch and I. Chopra. “A quasi-static model for NiMnGa magnetic shapememory alloy”. Smart Materials and Structures, 16:S11–S21, 2007.

[17] R.N. Couch and I.Chopra. “Testing and modeling of NiMnGa ferromagneticshape memory alloy for static and dynamic loading conditions”. In Proceedingsof SPIE Smart Structures and Materials, volume 6173, pages 501–516, SanDiego, CA, March 2006.

[18] R.N. Couch, J. Sirohi, and I. Chopra. “Development of a Quasi-static Modelof NiMnGa Magnetic Shape Memory Alloy”. Journal of Intelligent MaterialSystems and Structures, 18:611–620, 2007.

[19] N. Creton and L. Hirsinger. “Rearrangement surfaces under magnetic fieldand/or stress in NiMnGa”. Journal Magnetism and Magnetic Materials, 290-291:832835, 2005.

[20] M.J. Dapino. Encyclopedia of Smart Materials. John Wiley and Sons, Inc.,New York, NY, 2000.

[21] M.J. Dapino, R.C. Smith, L.E. Faidley, , and A.B. Flatau. “A CoupledStructural-Magnetic Strain and Stress Model for Magnetostrictive Transduc-ers”. Journal of Intelligent Material Systems and Structures, 11(2):135–152,2000.

275

[22] D. Davino, C. Natale, S. Pirzzori, and C. Visone. “Phenomenological dynamicmodel of a magnetostrictive actuator”. Physica B, Condensed Matter, 343(1-4):112–116, 2004.

[23] A. DeSimone and R.D. James. “A constrained theory of magnetoelasticity”.Journal of the Mechanics and Physics of Solids, 50(2):283–320, 1997.

[24] A. DeSimone and R.D. James. “A theory of magnetostriction oriented towardsapplications”. Journal of Applied Physics, 81(8):5706–5708, 1997.

[25] E.O. Doebelin. Measurement Systems: Application and Design. McGraw-HillPublishing Co., 2003.

[26] P.R. Downey and M. J. Dapino. “Extended Frequency Bandwidth and ElectricalResonance Tuning in Hybrid Terfenol-D/PMN-PT Transducers in MechanicalSeries Configuration ”. Journal of Intelligent Material Systems and Structures,16(9):757–772, 2005.

[27] A. Ercuta and I. Mihalca. Magnetomechanical damping and magnetoelastichysteresis in permalloy. Journal of Physics D: Applied Physics, 35:2902–2908,2002.

[28] L. E. Faidley, M. J. Dapino, G. N. Washington, and T. A. Lograsso. “Modu-lus Increase with Magnetic Field in Ferromagnetic Shape Memory NiMnGa”.Journal of Intelligent Material Systems and Structures, 17(2):123–131, 2006.

[29] L.E. Faidley. Systematic Analysis of the crystal structure, chemical ordering,and microstructure of Ni-Mn-Ga ferromagnetic shape memory alloys. PhD the-sis, The Ohio State University, 2002.

[30] L.E. Faidley, M.J. Dapino, and G.N. Washington. “Dynamic behavior andstiffness tuning in solenoid-based Ni-Mn-Ga transducers”. In Proceedings ofSPIE Smart Structures and Materials, volume 5387, pages 519–527, San Diego,CA, March 2004.

[31] L.E. Faidley, M.J. Dapino, and G.N. Washington. “Homogenized Strain Modelfor NiMnGa Driven with Collinear Magnetic Field and Stress”. Journal ofIntelligent Material Systems and Structures, 19:681–694, 2008.

[32] L.E. Faidley, M.J. Dapino, G.N. Washington, and T.A. Lograsso. “ReversibleStrain in Ni-Mn-Ga with Colinear Field and stress”. In Proceedings of SPIESmart Structures and Materials, volume 5761, pages 501–512, San Diego, CA,March 2005.

276

[33] L.E. Faidley, M.J. Dapino, G.N. Washington, and T.A. Lograsso. “ReversibleStrain in Ni-Mn-Ga with Colinear Field and stress”. In Proceedings of SPIESmart Structures and Materials, volume 5761, pages 501–512, San Diego, CA,March 2005.

[34] A.B. Flatau, M.J. Dapino, and F.T. Calkins. “High Bandwidth Tunabilityin a Smart Vibration Absorber”. Journal of Intelligent Material Systems andStructures, 11(12), 2000.

[35] A.B. Flatau, M.J. Dapino, and F.T. Calkins. “Wide Band Tunable Mechani-cal Resonator Employing the ∆E Effect of Terfenol-D”. Journal of IntelligentMaterial Systems and Structures, 15:355–368, 2004.

[36] E. Gans, C. Henry, and G.P. Carman. “High Energy Absorption in Bulk Fer-romagnetic Shape Memory Alloys (Ni50Mn29Ga21)”. In Proceedings of SPIESmart Structures and Materials, volume 5387, pages 177–185, San Diego, CA,March 2004.

[37] E. Gans, C. Henry, and G.P. Carman. “Reduction in required magnetic field toinduce twin-boundary motion in ferromagnetic shape memory alloys”. Journalof Magnetism and Magnetic Materials, 95(11):6965–6967, 2004.

[38] J.Y. Gauthier, C. Lexcellent, A. Hubert, J. Abadie, and N. Chaillet. “ModelingRearrangement Process of Martensite Platelets in a Magnetic Shape MemoryAlloy Ni2MnGa Single Crystal under Magnetic Field and (or) Stress Action”.Journal of Intelligent Material Systems and Structures, 18:289299, 2007.

[39] Y. Ge, O. Heczko, O. Soderberg, and V. K. Lindroos. “Various magnetic domainstructures in a NiMnGa martensite exhibiting magnetic shape memory effect”.Journal of Applied Physics, 96(4):2159–2163, 2004.

[40] V. Giurgiutiu, C.A. Rogers, and Z. Chaudhry. “Energy-Based Comparison ofSolid-State Induced-Strain Actuators”. Journal of Intelligent Material Systemsand Structures, 7(1):4–14, 1996.

[41] N.I. Glavatska and A.A. Rudenko V.A. L’vov. “Time-dependent magnetostraineffect and stress relaxation in the martensitic phase of NiMnGa”. Journal ofMagnetism and Magnetic Materials, 241(2):287–291, 2002.

[42] N.I. Glavatska, A.A. Rudenko, I.N. Glavatskiy, and V.A. L’vov. “Statisticalmodel of magnetostrain effect in martensite”. Journal of Magnetism and Mag-netic Materials, 265:142–151, 2003.

[43] D.L. Hall and A.B. Flatau. “Broadband performance of a magnetostrictiveshaker”. Journal of Intelligent Material Systems and Structures, 6(1):109–116,1995.

277

[44] O. Heczko. “Determination of ordinary magnetostriction in NiMnGa magneticshape memory alloy”. Journal of Magnetism and Magnetic Materials, 290-291:846–849, 2005.

[45] O. Heczko. “Magnetic shape memory effect and magnetization reversal”. Jour-nal of Magnetism and Magnetic Materials, 290-291(2):787–794, 2005.

[46] O. Heczko, V.A. L’vov, L. Straka, and S.P. Hannula. “Magnetic indication ofthe stress-induced martensitic transformation in ferromagnetic NiMnGa alloy”.Journal of Magnetism and Magnetic Materials, 302:387390, 2006.

[47] O. Heczko, A. Sozinov, and K. Ullakko. “Giant Field-Induced Reversible Strainin Magnetic Shape Memory NiMnGa Alloy”. IEEE Transactions on Magnetics,36(5):3266–3268, 2000.

[48] C.P. Henry. Dynamic actuation response of Ni-Mn-Ga ferromagnetic shapememory alloys. PhD thesis, Massachusetts Institute of Technology, 2002.

[49] C.P. Henry, P.G. Te11o, D. Bono, J. Hong, R. Wager, J. Dai, S.M. Allen,and R.C. O’Handley. Frequency response of single crystal ni-mn-ga fsmas. InProceedings of SPIE Smart Structures and Materials, volume 5053, pages 207–211, San Diego, CA, March 2003.

[50] L. Hirsinger. “Modelling of magneto-mechanical hysteresis loops in Ni-Mn-Gashape memory alloys”. Physica Status Solidi (C), 1(12):3458–3462, 2004.

[51] L. Hirsinger. “Ni-Mn-Ga Shape Memory Alloys: Modelling of Magneto-Mechanical Behaviour”. International Journal of Applied Electromagnetics andMechanics, 19:473–477, 2004.

[52] L. Hirsinger and C. Lexcellent. “Internal variable model for magneto-mechanicalbehavior of ferromagnetic shape memory alloys Ni-Mn-Ga”. Journal dePhysique IV, 112(2):977–980, 2003.

[53] L. Hirsinger and C. Lexcellent. “Modelling detwinning of martensite plateletsunder magnetic and (or) stress actions on NiMnGa alloys”. Journal of Mag-netism and Magnetic Materials, 254-255:275–277, 2003.

[54] W. Huang, B. Wang, S. Cao, Y. Sun, L. Weng, and H. Chen. “DynamicStrain Model With Eddy Current Effects for Giant Magnetostrictive Trans-ducer”. IEEE Transactions on Magnetics, 43(4):1381–1384, 2007.

[55] A. Jaaskelainen, K. Ullakko, and V. K. Lindroos. “Magnetic field-induced strainand stress in a Ni-Mn-Ga alloy”. Journal De Physique IV, 112(2):1005–1008,2003.

278

[56] R. James and M. Wuttig. “Magnetostriction of martensite”. PhilosophicalMagazine, 77(5):1273–1299, 2004.

[57] R.D. James, R. Tickle, and M. Wuttig. “Large field-induced strains in ferromag-netic shape memory materials”. Journal of Magnetism and Magnetic Materials,A273275:320–325, 1999.

[58] D.C. Jiles. Introduction to Magnetism and Magnetic Materials. Chapman andHall, London, 1991.

[59] D.C. Jiles. “Theory of the magnetomechanical effect”. Journal of AppliedPhysics, 28(8):1537–1546, 1995.

[60] X. Jin, M. Marioni, D. Bono, S.M. Allen, R.C. O’Handley, and T.Y.Hsu. “Em-pirical mapping of NiMnGa properties with composition and valence electronconcentration”. Journal of Applied Physics, 91(10):8222–8224, 2002.

[61] H.E. Karaca, I. Karaman, B. Basaran, Y.I. Chumlyakov, and H.J. Maier. “Mag-netic field and stress induced martensite reorientation in NiMnGa ferromagneticshape memory alloy single crystals”. Acta Materialia, 54(1):233–245, 2006.

[62] I. Karaman, B. Basaran, H.E. Karaca, A.I. Karsilayan, and Y.I. Chumlyakov.“Energy harvesting using martensite variant reorientation mechanism in a NiM-nGa magnetic shape memory alloy”. Applied Physics Letters, 90:172505.1–3,2007.

[63] R.A. Kellogg. The Delta-E effect in terfenol-D and its applications in a tunablemechanical resonator. Master’s thesis.

[64] R.A. Kellogg. Development and modeling of iron gallium alloys. PhD thesis,Iowa State University, 2003.

[65] J. Kiang and L. Tong. “Modelling of magneto-mechanical behaviour of NiMnGasingle crystals”. Journal of Magnetism and Magnetic Materials, 292:394–412,2005.

[66] B. Kiefer, H.E. Karaca, D.C. Lagoudas, and I. Karaman. “Characterization andModeling of the Magnetic Field-Induced Strain and Work Output in Ni2MnGaMagnetic Shape Memory Alloys”. Journal of Magnetism and Magnetic Mate-rials, 312(1):164–175, 2007.

[67] B. Kiefer and D.C. Lagoudas. “Magnetic field-induced martensitic variant re-orientation in magnetic shape memory alloys”. Philosophical Magazine, 85(33-35):4289–4329, 2005.

279

[68] B. Kiefer and D.C. Lagoudas. “Modeling of the Magnetic Field-Induced Marten-sitic Variant Reorientation and the Associated Magnetic Shape Memory Effectin MSMAs”. In Proceedings of SPIE Smart Structures and Materials, volume5761, pages 454–465, San Diego, CA, March 2005.

[69] H.E. Knoepfel. Magnetic Fields: A Comprehensive Theoretical Treatise forPractical Use. John Wiley and Sons, Inc., New York, NY, 2000.

[70] K. Kumada. “A piezoelectric ultrasonic motor”. Japanese Journal of AppliedPhysics, 24(2):739–741, 1986.

[71] G.F. Lee and B. Hartman. “Specific Damping Capacity for Arbitrary LossAngle”. Journal of Sound and Vibrations, 211(2):265–272.

[72] G. Li and Y. Liu. “Stress-induced change of magnetization in a NiMnGasingle crystal under magnetomechanical training”. Applied Physics Letters,88:232504.1–3, 2006.

[73] G. Li, Y. Liu, and B.K.A. Ngoi. “Some aspects of strain-induced change ofmagnetization in a NiMnGa single crystal”. Scripta Materialia, 53(7):829–834,2005.

[74] A.A. Likhachev, A. Sozinov, and K. Ullakko. “Optimizing work output in Ni-Mn-Ga and other ferromagnetic shape memory alloys”. In Proceedings of SPIESmart Structures and Materials, volume 4699, pages 553–563, 2002.

[75] A.A. Likhachev, A. Sozinov, and K. Ullakko. “Magnetic shape memory - Mech-anism, modeling principles and their application to Ni-Mn-Ga”. Journal dePhysique IV, 112:981–984, 2003.

[76] A.A. Likhachev, A. Sozinov, and K. Ullakko. “Different modeling concepts ofmagnetic shape memory and their comparison with some experimental resultsobtained in NiMnGa”. Materials Science and Engineering A, 378(1-2):513–518,2004.

[77] A.A. Likhachev, A. Sozinov, and K. Ullakko. “Modeling the strain response,magneto-mechanical cycling under the external stress, work output and energylosses in NiMnGa”. Mechanics of Materials, 38:551563, 2006.

[78] A.A. Likhachev and K. Ullakko. “Magnetic-field-controlled twin boundariesmotion and giant magneto-mechanical effects in NiMnGa shape memory alloy”.Physics Letters A, 275(1-2):142–151, 2000.

[79] A.A. Likhachev and K. Ullakko. “Quantitative model of large magnetostraineffect in ferromagnetic shape memory alloys”. European Physical Journal B,14:263–268, 2000.

280

[80] A.A. Likhachev and K. Ullakko. “The model development and experimentalinvestigation of giant magneto-mechanical effects in Ni-Mn-Ga”. Journal ofMagnetism and Magnetic Materials, 226-230:1541–1543, 2001.

[81] V. L’vov, O. Rudenko, and N. Glavatska. “Fluctuating stress as the origin ofthe time-dependent magnetostrain effect in Ni-Mn-Ga martensites”. PhysicalReview B, 71(2):024421.1–6, 2005.

[82] M. C. Sanchez C. Aroca M. Maicas, E. Lopez and P. Sanchez. “MagnetostaticEnergy Calculations in Two- and Three-Dimensional Arrays of FerromagneticPrisms”. IEEE Transactions on Magnetics, 34(4):601–607, 1998.

[83] A. Malla, M. Dapino, T.A. Lograsso, and D.L. Schlagel. “Large magnetically in-duced strains in Ni50Mn28.7Ga21.3 driven with collinear field and stress”. Journalof Applied Physics, 99(6):063903.1–9, 2006.

[84] M. Marioni. Pulsed magnetic field-induced twin boundary motion on Ni-Mn-Ga.PhD thesis, Massachusetts Institute of Technology, 2003.

[85] M.A. Marioni, S.M. Allen, and R.C. O’Handley. “Nonuniform twin-boundarymotion in NiMnGa single crystals”. Applied Physics Letters, 84(20):4071–4073,2004.

[86] M.A. Marioni, R.C. O’Handley, and S.M. Allen. “Pulsed magnetic field-inducedactuation of NiMnGa single crystals”. Applied Physics Letters, 83(19):3966–3968, 2003.

[87] P. Mullner, V. A. Chernenko, and G. Kostorz. “Stress-induced twin rearrange-ment resulting in change of magnetization in a NiMnGa ferromagnetic marten-site”. Scripta Materialia, 49(2):129–133, 2003.

[88] S. Murray, S.M. Allen, R.C. O’Handley, and T. Lograsso. “Magneto-mechanicalperformance and mechanical properties of Ni-Mn-Ga ferromagnetic shape mem-ory alloys”. In Proceedings of SPIE Smart Structures and Materials, volume3992, pages 387–395, San Diego, CA, March 2000.

[89] S.J. Murray, M. Marioni, S.M. Allen, R.C. O’Handley, and T.A. Lograsso. “6percent magnetic-field-induced strain by twin-boundary motion in ferromag-netic NiMnGa”. Applied Physics Letters, 77(6):886–888, 2000.

[90] S.J. Murray, M. Marioni, P.G. Tello, S.M. Allen, and R.C. O’Handley. “Giantmagnetic-field-induced strain in Ni-Mn-Ga crystals: experimental results andmodeling”. Applied Physics Letters, 77(6):886–888, 2000.

281

[91] S.J. Murray, M.A. Marioni, A.M. Kukla, J. Robinson, R.C. OHandley, and S.M.Allen. “Large field induced strain in single crystalline NiMnGa ferromagneticshape memory alloy”. Journal of Applied Physics, 87:5774–5776, 2000.

[92] R.C. O’Handley. “Model for strain and magnetization in magnetic shape-memory alloys”. Journal of Applied Physics, 83(6):3263–3270, 1998.

[93] R.C. O’Handley. Modern Magnetic Materials: Principles and Applications.John Wiley and Sons, Inc., New York, NY, 2000.

[94] R.C. O’Handley, S.J. Murray, M. Marioni, H. Nembach, and S.M. Allen.“Phenomenology of giant magnetic-field-induced strain in ferromagnetic shape-memory materials (invited)”. Journal of Applied Physics, 87(9):4712–4717,2000.

[95] R.C. O’Handley, D.I. Paul, M. Marioni, C.P. Henry, M. Richard, P.G. Tello,and S.M. Allen. “Micromagnetics and micromechanics of Ni-Mn-Ga actuation”.Journal de Physique IV, 112:972–976, 2003.

[96] Y.H. Pao and K. Hutter. “Electrodynamics for Moving Elastic Solids and Vis-cous Fluids”. Proceedings of IEEE, 63(7):1011–1021, 1975.

[97] B. Peterson. Acoustic assisted actuation of Ni-Mn-Ga ferromagnetic shapememory alloys. PhD thesis, Massachusetts Institute of Technology, 2006.

[98] M.L. Richard. Systematic Analysis of the crystal structure, chemical order-ing, and microstructure of Ni-Mn-Ga ferromagnetic shape memory alloys. PhDthesis, Massachusetts Institute of Technology, 2005.

[99] N. Sarawate and M. Dapino. “Experimental characterization of the sensoreffect in ferromagnetic shape memory NiMnGa”. Applied Physics Letters,88:121923.1–3, 2006.

[100] N. Sarawate and M. Dapino. “Sensing behavior of ferromagnetic shape memoryNi-Mn-Ga”. In Proceedings of SPIE Smart Structures and Materials, volume6170, pages 376–384, San Diego, CA, March 2006.

[101] N. Sarawate and M. Dapino. “A continuum thermodynamics model for thesensing effect in ferromagnetic shape memory NiMnGa”. Journal of AppliedPhysics, 101(12):123522.1–11, 2007.

[102] N. Sarawate and M. Dapino. “Electrical stiffness tuning in ferromagnetic shapememory Ni-Mn-Ga”. In Proceedings of SPIE Smart Structures and Materials,volume 6529, pages 652916.1–11, San Diego, CA, March 2007.

282

[103] N. Sarawate and M. Dapino. “Magnetomechanical characterization and unifiedactuator/sensor modeling of ferromagnetic shape memory alloy Ni-Mn-Ga”.In Proceedings of SPIE Smart Structures and Materials, volume 6526, pages652629.1–11, San Diego, CA, March 2007.

[104] N. Sarawate and M. Dapino. “Characterization and Modeling of Dynamic Sens-ing Behavior in Ferromagnetic Shape Memory Alloys”. ASME Conference onSmart Materials, Adaptive Structures and Intelligent Systems (in review), Oc-tober 2008.

[105] N. Sarawate and M. Dapino. “Dynamic actuation model for magnetostrictivematerials”. Smart Materials and Structures (in review), 2008.

[106] N. Sarawate and M. Dapino. “Dynamic strain-field hysteresis model for ferro-magnetic shape memory Ni-Mn-Ga”. In Proceedings of SPIE Smart Structuresand Materials (in press), volume 6929, pages xxx–xxx, San Diego, CA, March2008.

[107] N. Sarawate and M. Dapino. “Frequency Dependent Strain-Field HysteresisModel for Ferromagnetic Shape Memory NiMnGa”. IEEE Transactions onMagnetics, 44(5):566–575, 2008.

[108] N. Sarawate and M. Dapino. “Magnetic-field-induced stress and magnetizationin mechanically blocked NiMnGa”. Journal of Applied Physics, 103:083902.1–4,2008.

[109] N. Sarawate and M. Dapino. “Magnetization dependence on dynamic strain inferromagnetic shape memory Ni-Mn-Ga”. Applied Physics Letters (in review),2008.

[110] N. Sarawate and M. Dapino. “Stiffness tuning with bias magnetic fields in fer-romagnetic shape memory Ni-Mn-Ga”. Journal of Intelligent Material Systemsand Structures (in review), 2008.

[111] S. Scoby and Y.C. Chen. “Dynamic behavior of ferromagnetic shape mem-ory alloys”. In Proceedings of 47th AIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics, and Materials Conference, volume 1630, pages 1768–1771,Newport, CA, May 2006.

[112] R.C. Smith. Smart Material Systems: Model Development. Society of Industrialand Applied Mathematics, Philadelphia, PA, 2005.

[113] O. Soderberg, Y. Ge, A. Sozniov, S. Hannula, and V. K. Lindroos. “Recentbreakthrough development of the magnetic shape memory effect in NiMnGaalloys”. Smart Materials and Structures, 14(5):S223–S235, 2005.

283

[114] A. Sozniov, A.A. Likhachev, N. Lanska, and K. Ullakko. Appl. Phys. Lett.,80:1746, 2002.

[115] L. Straka and O. Heczko. “Superelastic Response of NiMnGa Martensite inMagnetic Fields and a Simple Model”. IEEE Transactions on Magnetics,39(5):3402–3404, 2003.

[116] L. Straka and O. Heczko. “Magnetization changes in NiMnGa magnetic shapememory single crystal during compressive stress reorientation”. Scripta Mate-rialia, 54(9):1549–1552, 2006.

[117] I. Suorsa. Performance And Modeling Of Magnetic Shape Memory ActuatorsAnd Sensors. PhD thesis, Helsinki University of Technology, 2005.

[118] I. Suorsa, E. Pagounis, and K. Ullakko. “Magnetization dependence on strainin the NiMnGa magnetic shape memory material”. Applied Physics Letters,84(23):4658–4660, 2004.

[119] I. Suorsa, E. Pagounis, and K. Ullakko. “Voltage generation induced by me-chanical straining in magnetic shape memory materials”. Journal of AppliedPhysics, 95(12):8054–8058, 2004.

[120] I. Suorsa, E. Pagounis, and K. Ullakko. “Position dependent inductance basedon magnetic shape memory materials”. Sensors and Actuators A, 121:136–141,2005.

[121] X. Tan and J.S. Baras. “Modeling and Control of Hysteresis in MagnetostrictiveActuators”. Automatica, 40(9):1469–1480, 2004.

[122] R. Tickle. Ferromagnetic Shape Memory Materials. PhD thesis, University ofMinnesota, 2000.

[123] R. Tickle and R.D. James. “Magnetic and magnetomechanical properties ofNi2MnGa”. Journal of Magnetism and Magnetic Materials, 195(3):627–638,1999.

[124] R. Tickle, R.D. James, T. Shield, M. Wuttig, and V.V. Kokorin. “FerromagneticShape Memory in the Ni-Mn-Ga System”. IEEE Transactions on Magnetics,35(5):4301–4310, 1999.

[125] K. Uchino and S. Hirose. Loss Mechanisms in Piezoelectrics: How to MeasureDifferent Losses Separately. IEEE Transactions on Ultrasonics, Ferroelectrics,and Frequency Control, 48(1):307–321, 2001.

284

[126] K. Ullakko. “Magnetically controlled shape memory alloys: A new class of actu-ator materials.”. Journal of Materials Engineering and Performance, 5(3):405–409, 1996.

[127] K. Ullakko. “Magnetically controlled shape memory alloys: A new class of actu-ator materials”. Journal of Materials Engineering and Performance, 5(3):405–409, 1996.

[128] K. Ullakko, J. K. Huang, C. Kantner, V. V. Kokorin, and R. C. O’Handley.“Large magnetic-field-induced strains in Ni2MnGa single crystals”. AppliedPhysics Letters, 69(13):1966–1968, 1996.

[129] K. Ullakko, J.K. Huang, V.V. Kokorin, and R.C. O’Handley. “MagneticallyControlled Shape Memory Effect in Ni2MnGa Intermetallics”. Scripta Materi-alia, 36(10):1133–1138, 1997.

[130] G. Washington. “Shape Memory Alloys: Smart Materials Course Notes”. TheOhio State University, 2007.

285