CHARACTERIZATION and MODELING of the FERROMAGNETIC SHAPE MEMORY ALLOY Ni-Mn-Ga for SENSING and ACTUATION

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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

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

*****

The Ohio State University

2008

Dissertation Committee: Approved by

Marcelo Dapino, Adviser Rajendra Singh Adviser Stephen Bechtel Graduate Program in Rebecca Dupaix Mechanical 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 actuation 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 blocking 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 suggestions. 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 Engineer Hodek Vibration Technologies, India 2004 ...... M.S. Mechanical Engineering University of Missouri-Rolla, Rolla MO 2004-present ...... Graduate Research Associate, The Ohio State University Columbus OH

PUBLICATIONS

Journal Publications

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

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

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

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

v N. Sarawate and M. Dapino, “Magnetization dependence on dynamic strain in ferromagnetic 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 in mechanically 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 for ferromagnetic 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 effect 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 ferromagnetic 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 sensing behavior of ferromagnetic shape memory alloys,” Proceedings of ASME Conference on Smart Materials, Adaptive Structures and Intelligent Systems, Paper #656, Ellicott City, MD, October 2008.

N. Sarawate, and M. Dapino, “Dynamic strain-ﬁeld hysteresis model for ferromagnetic shape 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 memory 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 modeling 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 ferromagnetic 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 memory Ni-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. Dapino Applied Mechanics Prof. Dapino, Prof. Bechtel, Prof. Dupaix System 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 ...... 1 1.2 Overview of Smart Materials ...... 5 1.2.1 Ferroelectrics ...... 6 1.2.2 Magnetostrictives ...... 8 1.2.3 Shape Memory Alloys ...... 9 1.3 Ferromagnetic Shape Memory Alloys ...... 14 1.3.1 Early Work ...... 15 1.3.2 Properties and Crystal Structure ...... 18 1.3.3 Magnetocrystalline Anisotropy ...... 19 1.3.4 Strain Mechanism ...... 20 1.4 Literature Review on Ni-Mn-Ga ...... 22 1.4.1 Sensing Behavior ...... 23 1.4.2 Modeling ...... 26 1.4.3 Dynamic Behavior ...... 30 1.5 Research Objectives ...... 33 1.6 Outline of Dissertation ...... 33

viii 1.6.1 Quasi-static Behavior ...... 34 1.6.2 Dynamic Behavior ...... 35

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

2.1 Electromagnet Design and Construction ...... 38 2.1.1 Magnetic Circuit ...... 39 2.1.2 Electromagnet Construction and Calibration ...... 42 2.2 Experimental Characterization ...... 45 2.2.1 Stress-Strain Behavior ...... 46 2.2.2 Flux Density Behavior ...... 50 2.3 Discussion ...... 51 2.3.1 Magnetic Field Induced Stress and Flux Density Recovery . 53 2.3.2 Optimum Bias Field for Sensing ...... 57

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

3.1 Thermodynamic Framework ...... 62 3.2 Incorporation of the Ni-Mn-Ga Microstructure in the Thermody- namic Framework ...... 66 3.3 Energy Formulation ...... 70 3.3.1 Magnetic Energy ...... 70 3.3.2 Mechanical Energy ...... 73 3.4 Evolution of Domain Fraction and Magnetization Rotation Angle . 75 3.5 Evolution of Volume Fraction ...... 78 3.6 Sensing Model Results ...... 80 3.6.1 Stress-Strain Results ...... 80 3.6.2 Flux Density Results ...... 82 3.6.3 Thermodynamic Driving Force and Volume Fraction .... 87 3.7 Extension to Actuation Model ...... 89 3.7.1 Actuation Model ...... 92 3.7.2 Actuation Model Results ...... 95 3.8 Blocked Force Model ...... 101 3.8.1 Results of Blocked-Force Behavior ...... 105 3.9 Discussion ...... 109

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

4.1 Introduction ...... 113 4.2 Magnetic Field Diﬀusion ...... 117 4.2.1 Diﬀused Average Field ...... 120

ix 4.3 Quasistatic Strain-Field Hysteresis Model ...... 122 4.4 Dynamic Actuator Model ...... 126 4.4.1 Discrete Actuator Model ...... 128 4.4.2 Fourier Series Expansion of Volume Fraction ...... 130 4.4.3 Results of Dynamic Actuation Model ...... 134 4.4.4 Frequency Domain Analysis ...... 138 4.5 Conclusion ...... 144 4.6 Dynamic Actuation Model for Magnetostrictive Materials ..... 145

5. Dynamic Sensing Behavior: Frequency Dependent Magnetization-Strain Hysteresis ...... 157

5.1 Experimental Characterization of Dynamic Sensing Behavior .... 157 5.2 Model for Frequency Dependent Magnetization-Strain Hysteresis . 164 5.3 Discussion ...... 169

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

6.1 Introduction ...... 172 6.2 Experimental Setup and Procedure ...... 173 6.3 Theory ...... 177 6.4 Results and Discussion ...... 182 6.4.1 Longitudinal Field Tests ...... 182 6.4.2 Transverse ﬁeld Tests ...... 185 6.5 Concluding Remarks ...... 193

7. Conclusion ...... 196

7.1 Summary ...... 196 7.1.1 Quasi-static Behavior ...... 196 7.1.2 Dynamic Behavior ...... 199 7.1.3 Characterization Map ...... 201 7.2 Contributions ...... 202 7.3 Future Work ...... 204 7.3.1 Possible Improvements ...... 204 7.3.2 Future Research Opportunities ...... 205

Appendices:

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

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

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

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

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

C. Model Codes ...... 238

C.1 Quasi-static Model ...... 238 C.1.1 Model Flowchart ...... 238 C.1.2 Sensing Model Code ...... 239 C.1.3 Actuation Model Code ...... 244 C.2 Dynamic Model ...... 247 C.2.1 Dynamic Actuator Model ...... 247 C.2.2 Dynamic Sensing Model ...... 254 C.2.3 Jiles-Atherton Model ...... 257

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

D.1 Electromagnet Drawings (Figures D.1-D.6) ...... 260 D.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 approximately proportional to the current i that passes through the solenoid when a voltage is applied to it, (b) the rotation of magnetic dipoles changes the length of the sample, (c) and (d) curves M vs. H and ∆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) above Af ...... 14

1.6 (a) Relative orientation of sample, strain gauge, and applied ﬁeld for measurements 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

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 transverse ﬁeld and longitudinal stress...... 21

2.1 Schematic of the electromagnet. Two E-shaped legs form the ﬂux path indicated 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 electron microscope [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 applied ﬁeld...... 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-stress curves. 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: experiment; 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 available blocking 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) sinusoidal input and (b) triangular input. x = d represents the edge of the 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 increasing actuation frequency for (a) sinusoidal input and (b) triangular input. 124

4.6 Model result for quasistatic strain vs. magnetic ﬁeld. The circles denote experimental 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

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’s output 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 area vs. Frequency ...... 137

4.12 Model results for strain vs. applied ﬁeld in frequency domain for triangular 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

4.13 (a) Strain magnitude vs. harmonic order, (b) Phase angle vs. harmonic order 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 for frequencies of up to 160 Hz...... 161

5.3 Hysteresis loss with frequency for stress-strain and ﬂux-density strain plots. The plots are normalized with respect to the strain amplitude at 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 depth for the 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

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...... 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 under the same ﬁeld inputs...... 184

6.9 Transmissibility ratio measurements with transverse ﬁeld conﬁguration. 186

6.10 Additional measurements of transmissibility ratio with transverse ﬁeld conﬁguration...... 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 gray blocks show that a limited prior work existed, which was completely addressed in this research; Dark gray blocks indicate that prior work was 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 produced by various wires (The current and turns are multiplied by scaling factors) 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 axis curve shows maximum variation...... 216

A.9 Schematic of the demagnetization field inside the sample. The applied field (H) creates a magnetization (M) inside the sample, which results in north and south poles on its surface. H and M are shown by solid arrows. The demagnetization field (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, it adds to the applied field outside the sample. Therefore, the net field inside the sample is given as H − NxM, whereas the net field outside the sample is 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 angles in 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 thermally 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 sensing, 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 aforementioned 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 deformation (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 thoroughly 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 electromagnet, which results in the generation of displacement that can be measured by a suitable 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 magnetization change, a comprehensive characterization under a wide range of inputs and bias variables is lacking. Development of models that can describe the macroscopic behavior 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 relationships 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 magnetization 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 energy 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 ferromagnetic shape memory. In general, all of these smart materials are transducers, and they convert energy from one form to another. The smart materials have potential 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 actuators. 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 sensitivity to harsh environmental conditions.

5 Output/ Charge, Magnetic Strain Temperature Light Input Current field Electric Permittivity, Electro- Converse Electro Electro Op- field Conductivity magnetism Piezo Effect Caloric tic Effect Effect Magnetic Mag-elect Permeability Magneto- Magneto Magneto Op- field Effect striction Caloric effect tic Effect Stress Piezo- Piezo- Compliance - Photo Elas- electric magnetic tic Effect Effect Effect Heat Pyroelectric - Thermal Ex- Specific Heat - Effect pansion Light Photovoltaic - Photostriction - Refractive Effect Index

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 commonly 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 having 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 characterized 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 produce 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 moments 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 proportional 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 magnetization, 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 vectors turning away from their preferred direction, it can be understood that magnetostriction 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 disadvantages 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)

Figure 1.2: Joule magnetostriction produced by a magnetic ﬁeld H. (a) H is approximately proportional to the current i that passes through the solenoid when a voltage is applied to it, (b) the rotation of magnetic dipoles changes the length of the sample, (c) and (d) curves M vs. H and ∆L/L vs. H obtained by varying the ﬁeld sinusoidally [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 temperature 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

10-50◦C.

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 initial microstructure consists of randomly oriented austenite. The elastic region (o → a) is followed by the transformation region (a → b), where the stress-induced martensite 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 unloading, 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) above Af .

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 frequency 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 ferromagnetic 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 initial years of research, the magnetic field induced strain was assumed because of the

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

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

Ga that demonstrated larger strains at higher temperatures. Tickle and James presented 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 for measurements 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 -15 C 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 effect 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 concentration. 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 temperature 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 publications 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 1000◦C for 24 hours and ordered at

800◦C 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, F m3m) 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 martensite 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) Tetragonal structure, 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 material 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],

2 4 Ua = Ku0 + Ku1 sin θ + Ku2 sin 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, represented 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

c c

a a H H

a a c c

H = 0

(a) (b) (c)

c c a a

a H a c

Saturation H = 0

(d) (e) (f)

Figure 1.8: Schematic of strain mechanism in Ni-Mn-Ga FSMA under transverse ﬁeld 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 constant 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 exhibits magnetic field induced pseudoelastic behavior, similar to that in SMAs which is temperature induced. The stress-strain response was hysteretic, whereas the magnetization 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 superelastic 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 interconnection 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 transformation 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 stoichiometric 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 measurements 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 behavior. 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 inductance 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 sensing behavior of Ni-Mn-Ga under different conditions, a comprehensive investigation of the simultaneous measurement of stress and magnetization at wide range of magnetic 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 magnetization.

James and Wuttig [56] presented a model based on a constrained theory of micromagnetics (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 microstructural 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 domain 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 dependence on field by energy minimization. The Zeeman energy difference (∆M · H) across the twin boundary is determined as the driving force responsible for strain generation. 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 between 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 ferromagnetic 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 systematic 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 demagnetization 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 behavior. 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 phenomena. 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 tuning: 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 conducted 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 sinusoidal 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 measurements 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 system 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 experimental 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 problem 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 conducted 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 density 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 boundary 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 vibration 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 difficult 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 deformation 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 crystal 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 responds 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 constitutive 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 under 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 measured 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 characterized 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 under 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 compression. 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 presence 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 providing 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 circuit 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

Laminated core

Flux path

Air Gap

Coil(s)

Figure 2.1: Schematic of the electromagnet. Two E-shaped legs form the ﬂux path indicated 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. I −→ −→ 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. l R = (2.2) m µA

The magnetic circuit of the electromagnet is shown in Figure 2.2. The net magnetomotive 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.

2Vm Φ = BAair = (2.3) Rair + 2Rsteel

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 product 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.

Rsteel

MMF MMF Rair f f

Rsteel

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 accommodating 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

43 800

600

400

200

−200 Field (kA/m) −400

−600

−800 −20 −15 −10 −5 0 5 10 15 20 Current (Amp)

Figure 2.4: Electromagnet calibration curve.

the electromagnet with coils on them are bolted together to complete the construction. 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 electromagnet 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 sample 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 mechanical 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 magnitude 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 under 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.

Hall probe Load cell ε

Electromagnet Pole piece(s)

Ni-Mn-Ga sample Pushrod(s)

Figure 2.5: Experimental setup for quasi-static sensing characterization.

47 6 6 6

5 H=0 kA/m 5 H=55 kA/m 5 H=94 kA/m

4 4 4

3 3 3

2 2 2

1 1 1

0 0 0 Compressive Stress (MPa) Compressive Stress (MPa) Compressive Stress (MPa) −1 −1 −1 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 Compressive Strain Compressive Strain Compressive Strain

6 6 6

5 H=133 kA/m 5 H=173 kA/m 5 H=211 kA/m

4 4 4

3 3 3

2 2 2

1 1 1

6 6 6

5 H=251 kA/m 5 H=291 kA/m 5 H=330 kA/m

4 4 4

3 3 3

2 2 2

1 1 1

6 6 6

5 H=368 kA/m 5 H=407 kA/m 5 H=445 kA/m

4 4 4

3 3 3

2 2 2

1 1 1

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 ‘twinning 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 feasibility 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 0.9

0.8 445 0.7 407 368 0.6 330 0.5 291 0.4 251 211 0.3 173 0.2 133 Flux Density (Tesla) 0.1 94 55 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Compressive Strain

Figure 2.7: Flux density vs. strain at varied bias ﬁelds.

0.9

0.8 445 0.7 407 368 0.6 330 0.5 291 0.4 251 211 0.3 173 0.2 133 Flux Density (Tesla) 0.1 94 55 0 0 1 2 3 4 5 Compressive Stress (MPa)

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 directly 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 redistribution 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 magnetomechanical 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 magnetization 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 transverse 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

top pushrod

bottom pushrod

ε, σ εmax, σmax 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.

top pushrod

bottom pushrod

ε, σ εmax, σmax 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 consists 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 magnetization 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 direction, is subsequently exposed to a 0.5 Hz sinusoidal field while being prevented

57 0.3

0.25 Reversible/ 0.2 Pseudoelastic Behavior

0.15 Partially Reversible 0.1 Behavior Irreversible/ 0.05 Quasiplastic

Flux Density Change (Tesla) Behavior 0 0 100 200 300 400 500 Applied Field (kA/m)

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 rotate 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 1.5

1 Start points during compression 0.5 End points 0 during compression

Density (Tesla) Density Hard Axis −0.5

Flux Easy Axis −1

−1.5 −1000 −500 0 500 1000 Applied Field (kA/m)

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 maximum 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 reversible 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 characterization is modeled using a continuum thermodynamics approach. It is discussed in the next section.

60 CHAPTER 3

CONSTITUTIVE MODEL FOR COUPLED MAGNETOMECHANICAL 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 presented 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 thermodynamics. 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 observations. 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,

˙ −→ −→˙ ρ²˙ = P · F + µ0 H · M + ρr − Divq, (3.1) where ² 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 inequality. 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,

r q ρη˙ ≥ ρ − Div( ), (3.2) Θ Θ 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

−→˙ 1 ρΘη ˙ − ρ²˙ + P · F˙ + µ H · M − q · GradΘ ≥ 0. (3.3) 0 Θ

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,

1 ρΘη ˙ − ρ²˙ + σε˙ + µ HM˙ − q · GradΘ ≥ 0. (3.4) 0 Θ

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, ², 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 ² with ψ through the Legendre transformation,

ψ = ² − 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, 1 −ρψ˙ − ρηΘ˙ + σε˙ + µ HM˙ − q · GradΘ ≥ 0. (3.9) 0 Θ We now impose the assumption of isothermal condition. This is because the coupled magnetomechanical behavior of interest in ferromagnetic shape memory Ni-Mn-

Ga occurs in the low-temperature martensite phase. The eﬀect of changing temperature 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) (3.12)

ψ =ψ(ε, M)

For majority of the applications involving magneto-mechanical materials, such as sensing and actuation, the magnetic ﬁeld is chosen as an independent variable because 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 the Thermodynamic Framework

The framework discussed in Section 3.1 pertains to thermo-magneto-mechanical materials which have a perfect memory of their reference conﬁguration and temperature. 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.

θ θ e Ms α H 1 - ξ Ms ξ 1 - α y

1 - α α x

Figure 3.1: Simpliﬁed two-variant microstructure of Ni-Mn-Ga.

Figure 3.2: Image of twin-variant Ni-Mn-Ga microstructure by Scanning electron microscope [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 relative 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) (3.15)

M =M(ε, H, α, θ, ξ). The rate of magnetic Gibbs energy can be expressed using the chain rule as,

∂(ρϕ) ∂(ρϕ) ∂(ρϕ) ∂(ρϕ) ∂(ρϕ) ρϕ˙ = ε˙ + H˙ + α˙ + θ˙ + ξ˙ (3.16) ∂ε ∂H ∂α ∂θ ∂ξ

Using (3.16) along with (3.14), we get, · ¸ ∂(ρϕ) ∂(ρϕ) ∂(ρϕ) ∂(ρϕ) ∂(ρϕ) − ε˙ + H˙ + α˙ + θ˙ + ξ˙ + σε˙ − µ MH˙ ≥ 0 (3.17) ∂ε ∂H ∂α ∂θ ∂ξ 0

This expression can be expanded as, · ¸ · ¸ ∂(ρϕ) ∂(ρϕ) σ − ε˙ + −µ M − H˙ + παα˙ + πθθ˙ + πξξ˙ ≥ 0 (3.18) ∂ε 0 ∂H 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, ∂(ρϕ) πα : = − , ∂α ∂(ρϕ) πθ : = − , (3.19) ∂θ ∂(ρϕ) πξ : = − . ∂ξ In inequality (3.18), the termsε ˙ and H˙ are independent of each other, and of other rates. Therefore, for an arbitrary process, the coeﬃcients ofε ˙ and H˙ must vanish in order for the inequality to hold. This leads to the constitutive equations,

∂(ρϕ) σ = , (3.20) ∂ε 69 1 ∂(ρϕ) M = − . (3.21) µ0 ∂H The Clausius-Duhem inequality is reduced to,

παα˙ + πθθ˙ + πξξ˙ ≥ 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 magnetic and mechanical components. The energy associated with the conventional magnetoelastic 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 sample 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,

1 1 ρϕ = ξ[ µ N(M α − M (1 − α))2] + (1 − ξ)[ µ NM 2 sin2 θ], (3.24) ms 2 0 s s 2 0 s 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,

2 ρϕan = (1 − ξ)[Ku sin 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 component 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,

1 2 ρϕmag =ξ[−µ0HMsα + µ0HMs(1 − α) + µ0N(Msα − Ms(1 − α)) ] 2 (3.27) 1 + (1 − ξ)[−µ HM sin θ + µ NM 2 sin2 θ + K sin2 θ]. 0 s 2 0 s u 3.3.2 Mechanical Energy

The mechanical energy typically represents the elastic strain energy contribution 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 ≤) (3.28) ˙ Unloading: εtw = ε0ξ(ξ ≥) 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 maximum 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, 1 1 ρϕ = E(ξ)ε2 + a(ξ)ε2 (3.30) mech 2 e 2 tw

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 1 1 = − . (3.31) a(ξ) E(ξ) k

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,

1 1 E(ξ) = = . (3.32) S(ξ) S0 + (1 − ξ)(S1 − S0)

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)

stw0 E1 E0

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 magnetization 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 magnetization 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,

∂(ρϕ) πα = − = 0, (3.36) ∂α ∂(ρϕ) πθ = − = 0 (3.37) ∂θ

The closed form solutions for domain fraction and magnetization rotation angle are obtained from (3.27), (3.33), (3.36), and (3.37) as,

H 1 α = + , (3.38) 2MsN 2

µ ¶ −1 µ0HMs θ = sin 2 (3.39) µ0NMs + 2Ku 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 1 ) α 0.9

0.8

0.7

0.6 Domain Fraction ( 0.5

0 100 200 300 400 500 600 700 Applied Field (kA/m)

100 )

0 80 θ

Rotation Angle ( 20

0 0 100 200 300 400 500 600 700 Applied Field (kA/m)

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

πξξ˙ ≥ 0. (3.40)

The total thermodynamic driving force associated with the evolution of volume fraction consists of magnetic and mechanical contributions.

ξ ξ ξ π = πmag + πmech, (3.41) with the magnetic and mechanical driving forces given by

∂(ρϕ ) 1 πξ = − mag =µ HM α − µ HM (1 − α) − µ N(M α − M (1 − α))2 mag ∂ξ 0 s 0 s 2 0 s s 1 − µ HM sin(θ) + µ NM 2 sin(θ)2 + K sin(θ)2, 0 s 2 0 s u (3.42)

∂(ρϕ ) 1 ∂E(ξ) Loading : πξ = − mech = − E(ξ)[ε − ε (1 − ξ)]ε − [ε − ε (1 − ξ)]2 mech ∂ξ 0 0 2 ∂ξ 0 1 ∂a(ξ) + a(ξ)(1 − ξ)ε2 − ε2(1 − ξ)2. 0 2 ∂ξ 0 (3.43)

∂(ρϕ ) 1 ∂E(ξ) Unloading : πξ = − mech = −E(ξ)[ε − ε (1 − ξ)]ε − [ε − ε (1 − ξ)]2 mech ∂ξ 0 0 2 ∂ξ 0 1 ∂a(ξ) − a(ξ)ε2ξ − ε2ξ2. 0 2 ∂ξ 0 (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 associated energy or critical driving force (πcr) required for twin variant rearrangement to start is estimated from the twinning stress at zero field (σtw0) as,

cr π = 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 deformation (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

3 = 1.67E5 J/m , 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 6 6 Experiment Experiment 94 kA/m 133 kA/m 5 Model:loading 5 Model:loading Model:unloading Model:unloading 4 4

3 3

2 2

1 1

0 0 Compressive Stress (MPa) Compressive Stress (MPa) −1 −1 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 Compressive Strain Compressive Strain

6 6 Experiment Experiment 211 kA/m 251 kA/m 5 Model:loading 5 Model:loading Model:unloading Model:unloading 4 4

3 3

2 2

1 1

0 0 Compressive Stress (MPa) Compressive Stress (MPa) −1 −1 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 Compressive Strain Compressive Strain

6 6 Experiment Experiment 291 kA/m 368 kA/m 5 Model:loading 5 Model:loading Model:unloading Model:unloading 4 4

3 3

2 2

1 1

0 0 Compressive Stress (MPa)

−1 Compressive Stress (MPa) −1 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 Compressive Strain Compressive Strain

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

81 As the bias field is increased, more energy is required for twin variant rearrangement 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 complete field-preferred state (ξ = 1). Therefore, an expression for the twinning stress can be obtained as detailed below:

ξ ξ cr πmag + πmech = −π

At start of twinning (ξ=1), (3.48) ξ πmag(H) − σtw(H)ε0 = −σtw0ε0 ξ πmag(H) σtw(H) = + σtw0 ε0 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 3.5

2.5

1.5

Twinning Stress (MPa) ξ Model prediction: π /ε +σ 1 mag 0 tw0 Experimental values (σ (H)) tw 0.5 0 100 200 300 400 500 600 700 Applied Field (kA/m)

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, ∂B H or a factor similar to piezoelectric coupling coefficient at constant field, . This ∂ε 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 1 Bias H (kA/m) 445 0.8 368

0.6 291

211 0.4

173

Flux Density (Tesla) 0.2 94 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Compressive Strain (a)

1 Bias H (kA/m) 445 0.8 368

0.6 291

0.4 211

173

Flux Density (Tesla) 0.2 94 0 0 1 2 3 4 5 6 7 Compressive Stress (MPa) (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 5 Model result Experimental values 4

3 (Tesla) H )

∂ε 2 B/ ∂ ( 1 Irreversible Partially Reversible reversible

0 0 100 200 300 400 500 Applied Field (kA/m)

Figure 3.8: Variation of sensitivity factor with applied bias ﬁeld.

(4.15). These expressions are given as, H M = M = easy (ξ=1) N 2 µ0HMs Mhard = M(ξ=0) = 2 µ0NMs + 2Ku (3.50)

If M > Ms,M = Ms

If M < −Ms,M = −Ms. 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 thermodynamic 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 1.4

1.2

0.8

0.6

0.4

0.2 Experimental (Easy Axis)

Flux Density (Tesla) Experimental (Hard Axis) 0 Model (Easy Axis) Model (Hard Axis) −0.2

−200 0 200 400 600 800 1000 Applied Field (kA/m) (a)

700

600

500

400

300

200

100 Experimental (Easy Axis) 0

Magnetization (kA/m) Experimental (Hard Axis) −100 Model (Easy Axis) Model (Hard Axis) −200 0 200 400 600 800 Applied Field (kA/m) (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 magnetization on varying ﬁeld under bias compressive stress. The actuator model utilizes the exact same parameters as the sensing model. Further, the actuation model framework 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 application (|H˙ | < 0), if the prestress is sufficiently large, the original length of the sample is restored to complete one strain cycle.

89 5 5 x 10 x 10 ) )

3 2 3 2 ±πcr ±πcr 1.5 π 1.5 π mech mech 1 π 1 π mag mag π π 0.5 0.5

0 0

−0.5 −0.5

−1 −1

−1.5 H=94 kA/m −1.5 H=133 kA/m

Thermodynamic Driving Force (J/m −2 Thermodynamic Driving Force (J/m −2 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 Compressive Strain Compressive Strain 5 5 x 10 x 10 ) cr ) 3 2 ±π 3 2 ±πcr π 1.5 1.5 π mech mech π π 1 mag 1 mag π π 0.5 0.5

0 0

−0.5 −0.5

−1 −1

−1.5 H=211 kA/m −1.5 H=251 kA/m

Thermodynamic Driving Force (J/m −2 Thermodynamic Driving Force (J/m −2 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 Compressive Strain Compressive Strain 5 5 x 10 x 10 ) ) cr 3 2 ±πcr 3 2 ±π 1.5 π 1.5 π mech mech π π 1 mag 1 mag π π 0.5 0.5

0 0

−0.5 −0.5

−1 −1 H=368 kA/m −1.5 −1.5 H=291 kA/m

Thermodynamic Driving Force (J/m −2 Thermodynamic Driving Force (J/m −2 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 Compressive Strain Compressive Strain

Figure 3.10: Evolution of thermodynamic driving forces.

90 1 1

0.8 Stress 0.8 Stress preferred (1−ξ) preferred (1−ξ) 0.6 0.6

0.4 0.4 Field Field preferred (ξ) preferred (ξ) Volume Fraction 0.2 Volume Fraction 0.2 H=94 kA/m H=133 kA/m 0 0 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 Compressive Strain Compressive Strain

1 1

0.8 Stress 0.8 Stress preferred (1−ξ) preferred (1−ξ) 0.6 0.6

0.4 0.4 Field Field preferred (ξ) preferred (ξ) Volume Fraction 0.2 Volume Fraction 0.2 H=211 kA/m H=251 kA/m 0 0 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 Compressive Strain Compressive Strain

1 1

0.8 Stress 0.8 Stress preferred (1−ξ) preferred (1−ξ) 0.6 0.6

0.4 0.4 Field Field preferred (ξ) preferred (ξ) Volume Fraction 0.2 Volume Fraction 0.2 H=291 kA/m H=368 kA/m 0 0 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 Compressive Strain Compressive Strain

Figure 3.11: Evolution of volume fraction.

91 0.06 Model result Experimental values 0.05

0.04

0.03

0.02 Residual Strain

0.01

0 0 100 200 300 400 500 Applied Field (kA/m)

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 original (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 energy. From (3.11) and (3.51), we get the Clausius-Duhem inequality of the form,

˙ ˙ −ρφ − σε˙ e − µ0MH + σεtw˙ ≥ 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) (3.54)

M =M(σ, H, ξ, α, θ).

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,

∂(ρφ) ε = − , (3.55) e ∂σ 1 ∂(ρφ) M = − . (3.56) µ0 ∂H

The Clausius-Duhem inequality reduces to the form, µ ¶ ∂(ρφ) − + σε ξ˙ ≥ 0 (3.57) ∂ξ 0

πξ∗ ξ˙ ≥ 0 (3.58)

93 where the total thermodynamic driving force πξ∗ is deﬁned as,

∗ ∂(ρφ) πξ = − + σε = πξ + σε = πξ + πξ + σε . (3.59) ∂ξ 0 0 mag mech 0

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,

1 1 ρφ = − Sσ2 + aε2ξ2. (3.60) mech 2 2 0

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 mechanical energy equation for actuation remains the same during application of both increasing and decreasing ﬁeld. The parameters associated with the mechanical energy 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

ξ∗ 2 πmech = −aε0ξ + 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 3 E = 800 MPa, σtw0 = 0.8 MPa, k = 14 MPa, ε0 = 0.058, Ku = 1.7E5 J/m , 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 0.06

0.05 0.25 0.89 0.04 Bias Stress 1.16 (MPa) 0.03

Strain 1.43 0.02 1.63 0.01

2.11 0

0 200 400 600 800 Applied Field (kA/m)

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.

ξ ξ cr πmag(Hsat) + πmech = π cr Ku + σbε0 − π ξmax = 2 (3.65) aε0

ε(Hsat) = Sσb + ε0ξmax

97 0.06

0.05

0.04 Irreversible Reversible 0.03

0.02

Maximum strain 0.01

0 Model result Experimental values −0.01 0 0.5 1 1.5 2 2.5 Bias compressive stress (MPa)

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 rearrangement occurs with relatively smaller increase in the magnetic field. Therefore, accurate evaluation of the coercive field gives an estimate of the magnetic field requirements for the electromagnet design. The coercive field determines the resistance to the twin boundary motion due to the added contributions of the internal material 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 properties, the coercive field (Hc) is obtained by solving equation (3.66), which is obtained from (3.42), (3.45), and (3.62).

2 2 2 − µ0HcMs sin θ + 2µ0HcMsα − µ0HcMs − 2µ0NMs α + 2µ0NMs α 1 (3.66) + σ ε + K − µ NM 2 cos2 θ − K cos2 θ = σ ε b 0 u 2 0 s u tw0 0 The expression for the coercive ﬁeld is therefore given as, p 2 2 2 µ0NMs + 2Ku − 2µ0NMs [Ku + ε0(σb − σtw0)] + 4Kuε0(σb − σtw0) + 4Ku Hc = 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,

99 700 Model Experiment 600

500

400

300

Coercive Field (kA/m) 200

100 0 0.5 1 1.5 2 2.5 Bias Stress (MPa)

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

100 600

0.25 500 0.89 1.16 Bias Stress 400 (MPa) 1.43

300 1.63

200 2.11 Magnetization (kA/m) 100

0 0 200 400 600 800 Applied Field (kA/m)

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 during 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 blocking 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 magnetization 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 subsequently 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 electromagnet. 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 desired 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 direction (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 configuration. 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

2 2 ρϕme = B1εy(1 − ξ)(− sin θ) + σ0εyξ(− sin 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

2 σme = [B1(1 − ξ) + σ0ξ](− sin 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 magnetoelastic 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 3.5 Bias strain (%) 5 % 3 4 % 3 % 2.5

2 % 2 1 %

Stress (MPa) 1.5

0.5 −800 −600 −400 −200 0 200 400 600 800 Applied field

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

3 MPa, σtw0 = 1 MPa, k = 16 MPa, ε0 = 0.055, Ku = 2.2E5 J/m , 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 provide a measure of the work capacity of Ni-Mn-Ga. The work capacity, deﬁned as the

106 800

600 1 % 2 % 400 3 % 4 % 200 5 %

Magnetization (kA/m) 0 Bias strain (%)

−200 −200 0 200 400 600 800 Applied Field (kA/m)

Figure 3.18: Magnetization vs ﬁeld at varied blocked strains. Dashed line: experiment; solid line: model.

6 Model result Experimental values 5

2 Initial susceptibility 1

0 1 2 3 4 5 Applied Field (kA/m)

Figure 3.19: Variation of initial susceptibility with biased blocked strain.

107 3 area under the σbl − σ0 curve, is 72.4 kJ/m 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 approximately 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 actuation 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 rearrangement. In summary, the magnetoelastic coupling in Ni-Mn-Ga is relatively low but becomes significant when the material is prevented from deforming.

108 5 σ bl σ 4 0 σ −σ bl 0

) 3 MPa

2 Stress (

0 6 5 4 3 2 1 0 −1 Bias Strain (%)

Figure 3.20: Experimental blocking stress σbl, minimum stress σ0, and available blocking 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, describing the interdependence of strain, stress, field, and magnetization. The model accurately quantifies the dependent variables over large ranges of the bias independent 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 magnetomechanical characterization of single crystal Ni-Mn-Ga, along with the model predictions. µ ¶ ∂B The flux density sensitivity with strain varies from 0 to a maximum value of ∂ε 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 induced 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 µ ¶ ∂M | changes by 59% over a range of 4% strain (Figure 3.19) when mechani- ∂H H=0 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 polycrystalline 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 incorporate 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 FREQUENCY DEPENDENT STRAIN-FIELD HYSTERESIS

In this chapter, a model is developed to describe the relationship between magnetic 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 exhibits 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 solutions 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 applied 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 constitutive 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 actuators 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 geometries. 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 applied 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 relative 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 corresponding 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 effective 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 provides 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 Constitutive (Eddy currents) model

Structural Dynamic strain Fourier series dynamics expansion

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, ∂D ∇ × H = j + , ∂t ∂B ∇ × E = − , ∂t (4.1) ∇ · B = 0,

∇ · D = ρe, 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

3 density (T), and ρe the volume density of free charge (C/m ). The corresponding constitutive equations are given by

j = σE,

B = µH, (4.2)

D = εE, 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 Amp`ere-Maxwell circuital law, ∂ ∇ × ∇ × H = ∇ × (σE) + [∇ × (εE)]. (4.3) ∂t

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,

∂H ∇2H − µσ = 0, (4.4) ∂t

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 experimentally 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

iωt H(±d, t) = H0e , (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]

iωt H(x, t) = H0 h(X) e . (4.6)

119 In this expression, the complex magnitude scale factor is

h(X) = A(B + iC), 1 A = , 2 2 2 2 cosh Xd cos Xd + sinh Xd sin Xd (4.7) 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, with r x d 2 X = , X = , δ = , (4.8) δ d δ ωµσ 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 material 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,

X 1 Xd Havg(t) = H(x, t). (4.9) Nx X=−Xd

120 1 d 0.75 3d/4 )

0 d/2 0.5 d/4 0 0.25

−0.25

−0.5 Normalized Field (H/H −0.75

−1 0 0.2 0.4 0.6 0.8 1 Nondimensional time (t*fa) (a)

1 d 0.75 3d/4

) d/2 0 0.5 d/4 0 0.25

−0.25

−0.5 Normalized field (H/H −0.75

−1 0 0.2 0.4 0.6 0.8 1 Non−dimensional time (t*fa) (b)

Figure 4.3: Magnetic ﬁeld variation inside the sample at varied depths for (a) sinusoidal input and (b) triangular input. x = d represents the edge of the sample, x = 0 represents 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 magnetomechanical model for twin variant rearrangement is used, which is detailed in Chapter 3. The model incorporates thermodynamic potentials to deﬁne reversible processes in combination with evolution equations for internal state variables associated with dissipative

122 1 1 Hz 0.75 50 Hz ) 0 100 Hz /H 0.5 150 Hz

avg 175 Hz 0.25 200 Hz 250 Hz 0

−0.25

−0.5

Normalized field (H −0.75

−1 0 0.2 0.4 0.6 0.8 1 Non−dimensional time (t*fa) (a)

1 1 Hz 0.75 50 Hz )

0 100 Hz

/H 0.5 150 Hz 175 Hz avg 200 Hz 0.25 250 Hz 0

−0.25

−0.5

Normalized field (H −0.75

−1 0 0.2 0.4 0.6 0.8 1 Non−dimensional time (t*fa) (b)

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

123 1 1 Hz 50 Hz 0.95 100 Hz 150 Hz 0.9 175 Hz 200 Hz 250 Hz 0.85

0.8

0.75 Maximum Normalized Field

0.7 −5 −4 −3 −2 −1 0 1 2 3 4 5 Position (mm) (a)

1 1 Hz 50 Hz 100 Hz 0.9 150 Hz 175 Hz 200 Hz 0.8 250 Hz

0.7

0.6 Maximum Normalized Field

0.5 −5 −4 −3 −2 −1 0 1 2 3 4 5 Position (mm) (b)

Figure 4.5: Dependence of normalized ﬁeld amplitude on position with increasing actuation 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 selected 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.

1 2 1 2 ˙ ρφmech = − σb + aε0(ξ − ξs)(|H| > 0), 2E 2 (4.10) 1 1 ρφ = − σ2 + aε2(ξ − ξ + ξ )(|H˙ | < 0), mech 2E b 2 0 f s

125 The volume fraction obtained using the procedure detailed in Section 3.7. It is given by,

ξ 2 cr πmag + σbε0 + aε0ξs − π ˙ ξ = 2 (|H| ≥ 0), aε0 ξ 2 2 cr (4.11) πmag + σbε0 + aε0ξf − aε0ξs + π ˙ ξ = 2 (|H| ≤ 0), aε0 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

3 = 1.7 J/m , 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 3

2.5

1.5 Strain (%) 1

0.5

0 0 100 200 300 400 500 600 Applied Field (kA/m)

Figure 4.6: Model result for quasistatic strain vs. magnetic ﬁeld. The circles denote experimental 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 relations (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 volume 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 response 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 modeled 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

F(t)

k e E, A, L me

ce x (t) H (t) = H ejwt 0 Mechanical load

Figure 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’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,

σ ε = ε + ε = + ε ξ, (4.14) e tw E 0

x σ = E(ε − ε ξ) = E( − ε ξ). (4.15) 0 L 0

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

AE m x¨ + c x˙ + (k + )x = AEε ξ. (4.16) e e e L 0

129 Equation (4.16) represents a second-order dynamic system driven by the volume fraction. 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 Z 1 Ta −iωkt Zk = ξ(t)e dt, k = 0, ±1, ±2, ..., (4.17) Ta 0 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 =

th 0, 1, 2...; Zk is the complex Fourier coeﬃcient corresponding to the k harmonic.

Equation (4.17) yields a double sided discrete frequency spectrum consisting of frequencies −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 3

2.5

1.5 %) ( 0 ε

ξ 1

0.5 Sin: orig Sin: recon 0 Tri: orig Tri: recon

−0.5 0 0.2 0.4 0.6 0.8 1 Time (sec)

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), (4.18) |Zk| = |Zk| + |Z−k| = 2|Zk| (k > 0). The phase angles remain unchanged,

∠Zk = ∠Zk (k ≥ 0). (4.19)

The reconstructed volume fraction ξr(t) is XK iωkt ξr(t) = ξr(t ± Ta) = Zke , (4.20) k=−K 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 1.5 Sinusoidal %)

( Triangular

0 1 |ε

ξ 0.5 | 0 0 2 4 6 8 10 Frequency (Hz) 200 ) g e d (

) 0 ξ ( g n

A −200 0 2 4 6 8 10 Frequency (Hz)

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

XK ξr(t) = ξr(t ± Ta) |Zk| cos(2πkfat + ∠Zk). (4.22) k=0

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,

AE XK m x¨ + c x˙ + (k + )x = AEε |Z | cos(2πkf t + ∠Z ), (4.23) e e e L 0 k a k k=0 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

XK x(t) EAε0 εd(t) = = |Zk||Xk| cos(2πkfat + ∠Zk − ∠Xk). (4.24) L EA + keL k=0

In (4.45), Xk represents the non-dimensional transfer function relating the force at the kth harmonic and the corresponding displacement,

1 −i∠Xk Xk = 2 = |Xk|e , (4.25) [1 − (kfa/fn) ] + j(2ζkfa/fn) where 1 |Xk| = p , (4.26) 2 2 2 [1 − (kfa/fn) ] + (2ζkfa/fn)

µ ¶ −1 2ζkfa/fn ∠Xk = tan 2 . (4.27) 1 − (kfa/fn)

133 The natural frequency and damping ratio in these expressions have the form s 1 ke + AE/L fn = , (4.28) 2π me

c ζ = p e . (4.29) 2 (ke + AE/L)me

4.4.3 Results of Dynamic Actuation Model

Figure 4.10 shows experimental and calculated strain versus ﬁeld curves for sinusoidal and triangular waveforms at varied frequencies. The model parameters used

−8 are fn = 700 Hz, ζ = 0.95, ρ = 62 × 10 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 triangular 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 frequency. The lack of overshoot in the experimental data for any of the frequencies justifies the assumption of overdamped system. The average error between the experimental 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 3

2.5

1.5 1 Hz 50 Hz Strain (%) 1 100 Hz 150 Hz 175 Hz 0.5 200 Hz 250 Hz 0 0 100 200 300 400 500 600 Applied Field (kA/m) (a)

2.5

1.5 1 Hz 50 Hz

Strain (%) 100 Hz 1 150 Hz 175 Hz 0.5 200 Hz 250 Hz 0 0 100 200 300 400 500 600 Applied Field (kA/m) (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 decreases 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 values better than assumption of the sinusoidal ﬁeld. This indicates that the applied

136 −6 x 10 8

0 7

4 Experimental

Maximum Strain / H 3 Model: Sine Model:Triangular 2 0 50 100 150 200 250 Actuation frequency (Hz) (a)

140

120

(kA/m) 100 −2

10 80 ×

40 Experimental 20 Model: Sine Model: Triangular Enclosed Area 0 0 50 100 150 200 250 Actuation frequency (Hz) (b)

Figure 4.11: (a) Normalized maximum strain vs. Frequency (b) Hysteresis loop area vs. 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’ spectrum of Fourier magnitudes of the experimental strain signal. The magnitudes corresponding 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 spectrum under study is discrete, continuous curves are shown in Figure 4.13(a) to better

138 2 1.5 1 1 | (%) | (%) ε ε

| 0.5 | 0 0 0 2 4 6 8 10 0 2 4 6 8 10 Non−dimensional frequency (f/fa) Non−dimensional frequency (f/fa) 175 200 150 150 ) (deg) ) (deg)

ε 100 ε 100 25 50 0 0

Angle( 0 2 4 6 8 10 Angle( 0 2 4 6 8 10 Non−dimensional frequency (f/fa) Non−dimensional frequency (f/fa) (a) (b)

1 1 | (%) | (%) ε ε | | 0 0 0 2 4 6 8 10 0 2 4 6 8 10 Non−dimensional frequency (f/fa) Non−dimensional frequency (f/fa) 125 125 100 100

) (deg) 75 ) (deg) 75 ε 50 ε 50 25 25 0 0

Angle( 0 2 4 6 8 10 Angle( 0 2 4 6 8 10 Non−dimensional frequency (f/fa) Non−dimensional frequency (f/fa) (c) (d)

1 0.5

| (%) 0.5 | (%) ε ε | | 0 0 0 2 4 6 8 10 0 2 4 6 8 10 Non−dimensional frequency (f/fa) Non−dimensional frequency (f/fa) 125 125 100 100

) (deg) 75 ) (deg) 75 ε 50 ε 50 25 25 0 0

Angle( 0 2 4 6 8 10 Angle( 0 2 4 6 8 10 Non−dimensional frequency (f/fa) Non−dimensional frequency (f/fa) (e) (f)

Figure 4.12: Model results for strain vs. applied ﬁeld in frequency domain for triangular 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 magnitude 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 1.2 1 Hz 50 Hz 1 100 Hz 150 Hz 0.8 175 Hz 200 Hz 0.6 250 Hz

0.4

0.2 Strain Magnitudes (%)

0 2 4 6 8 10 Non−dimensional frequency (f/f ) a (a)

−50

−100

1 Hz −150 50 Hz 100 Hz 150 Hz −200

Phase Angle (deg) 175 Hz 200 Hz 250 Hz −250 2 4 6 8 10 Non−dimensional frequency (f/f ) a (b)

Figure 4.13: (a) Strain magnitude vs. harmonic order, (b) Phase angle vs. harmonic order 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 1.2 2fa 4fa 1 6fa 8fa 0.8 10fa

0.6

0.4

0.2 Strain Magnitudes (%)

0 0 50 100 150 200 250 Actuation frequency (Hz) (a)

0 2fa 4fa 6fa −50 8fa 10fa

−100

−150 Phase Angle (deg)

−200 0 50 100 150 200 250 Actuation frequency (Hz) (b)

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

143 1

0.9

0.8

0.7

0.6 ε max H 0.5 max ε

Normalized Strain and Field /H max max 0.4 0 50 100 150 200 250 Actuation frequency (Hz)

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 associated 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 actuators, 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 magnetostrictive 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 relative 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 magnetic 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 earlier in this chapter.

Magnetic Field Diﬀusion

As seen in Section 4.2, application of an alternating magnetic field to a conducting 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 1 ∂H ∂H + = µσ , (4.30) ∂r2 r ∂r ∂t

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,

iωt H(R, t) = H0e (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,

iωt H(x, t) = H0 h(R) e . (4.32)

Therefore the diﬀusion equation (4.30) is transformed to,

d2h 1 dh 2 + − h = 0, (4.33) dR R dR where the normalized complex and real radii are given as, √ √ 2ir (1 + i)r 2ia R = = , R = δ δ a δ √ √ r (4.34) 2r 2a 2 R = ,R = , δ = δ a δ ωµσ

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): I (R) h(R) = 0 (4.36) I0(Ra) Introducing the Kelvin functions [69],

I0(R) = ber(R) + ibei(R), (4.37) and rearranging gives,

iα berRberRa + beiRbeiRa + i(beiRberRa − berRbeiRa) h(R) = he = 2 2 . (4.38) ber Ra + bei Ra 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, Ã ! 1 Xr=a Havg(t) = P NrH(r, t). (4.39) Nr=Na N Nr=1 r r=0 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 combination with a quadratic model for the magnetostriction. It is assumed that the

148 1 10 Hz )

0 0.8 100 Hz /H 0.6 500 Hz

avg 800 Hz 0.4 1000 Hz 0.2 1250 Hz 0 Increasing 1500 Hz 2000 Hz −0.2 Frequency −0.4 −0.6 −0.8 Normalized Field (H −1 0 0.2 0.4 0.6 0.8 1 Non−dimensional time (t*fa)

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, µ ¶ 3 M 2 λ = , (4.40) 2 Ms 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.

F(t)

k e E, A, L me Magnetostricve Rod ce jwt x (t) H (t) = H0e Mechanical load

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 AE m x¨ + c x˙ + (k + )x = AEλ(t). (4.43) e e e L

Equation (4.43) represents a second-order dynamic system driven by the magnetostriction. 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

XN λ(t) = |Λn| cos(2πnfat + ∠Λn), (4.44) n=0 where |Λn| and ∠Λn respectively represent the magnitude and angle of the nth harmonic 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

x(t) EA XN ¡ ¢ ε (t) = = |Λ | [1 − (nf /f )2]2 + (2ζnf /f )2 −1/2 d L EA + k L n a n a n · e n=0 µ ¶¸ (4.45) −1 2ζnfa/fn cos 2πnfat + ∠Λn − tan 2 , 1 − (nfa/fn) 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.

−4 −4 −4 x 10 x 10 x 10 4 4 4

2 2 2

in 0 in 0 in 0 Stra Stra Stra −2 −2 −2 10 Hz 100 Hz 500 Hz −4 −4 −4 −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 Applied field (kA/m) Applied field (kA/m) Applied field (kA/m) −4 −4 −4 x 10 x 10 x 10 4 4 4

2 2 2

in 0 in 0 in 0 Stra Stra Stra −2 −2 −2 800 Hz 1000 Hz 1250 Hz −4 −4 −4 −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 Applied field (kA/m) Applied field (kA/m) Applied field (kA/m) −4 −4 x 10 x 10 4 4

2 2

0 0 Strain Strain −2 −2 1500 Hz 2000 Hz −4 −4 −10 −5 0 5 10 −10 −5 0 5 10 Applied field (kA/m) Applied field (kA/m)

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 because 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 magnitude 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 frequency (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 magnetically activated smart materials: Terfenol-D and Ni-Mn-Ga [107]. The presented

153 20 20 25 Experimental Experimental Experimental 5 5 5 Model Model 20 Model 15 15 10 10 10 15 g. x 10 10 Hz g. x 10 g. x 100 Hz 10 5 5 in Ma in Ma in Ma 5 500 Hz

Stra 0 Stra 0 Stra 0 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 Harmonic Order (f/fa) Harmonic Order (f/fa) Harmonic Order (f/fa) 25 40 25 Experimental Experimental Experimental 5 5 5 20 Model Model 20 Model 30

x 10 15 x 10 x 10 15 20 10 800 Hz 1000 Hz 10 1250 Hz 10 5 5

Strain Mag. 0 Strain Mag. 0 Strain Mag. 0 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 Harmonic Order (f/fa) Harmonic Order (f/fa) Harmonic Order (f/fa)

15 4 Experimental Experimental 5 Model 5 Model 3 10 x 10 x 10 1500 Hz 2 2000 Hz 5 1

Strain Mag. 0 Strain Mag. 0 0 2 4 6 8 0 2 4 6 8 Harmonic Order (f/fa) Harmonic Order (f/fa)

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 estimation 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 contribution 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 harmonics, 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 35 Experimental 30 Model 5 25

10 Strain Mag. x 10 5

0 0 500 1000 1500 2000 Actuation frequency (Hz)

0 Experimental Model −50

−100

−150

−200 Phase angle (deg)

−250 0 500 1000 1500 2000 Actuation frequency (Hz)

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

156 CHAPTER 5

DYNAMIC SENSING BEHAVIOR: FREQUENCY DEPENDENT 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 increasing 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 utilizing 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

Hall probe Load cell ε

Electromagnet Pole piece(s)

Ni-Mn-Ga sample Pushrod(s)

Figure 5.1: Experimental setup for dynamic magnetization measurements.

deformation sensor. This bias ﬁeld was determined as optimum for obtaining maximum 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 compressed 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 acquisition 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 compression. 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 hysteresis. This low-frequency behavior is consistent with some of the previous observations [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 frequencies. 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 frequencies. 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 density dependence on strain shows a monotonic increase in hysteresis with increasing

160 5 4 Hz 20 Hz 4 50 Hz 90 Hz 120 Hz 3 160 Hz

1 Compressive Stress (MPa) 0 −0.02 −0.01 0 0.01 0.02 Compressive Strain (a)

0 4 Hz 20 Hz −0.01 50 Hz 90 Hz −0.02 120 Hz 160 Hz −0.03

−0.04

−0.05

−0.06 Relative Flux Density (T)

−0.07 −0.02 −0.01 0 0.01 0.02 Compressive Strain (b)

Figure 5.2: (a) Stress vs. strain and (b) ﬂux-density vs. strain measurements for frequencies of up to 160 Hz.

161 frequency. The hysteresis loss in the stress versus strain plots is equal to the area H enclosed by one cycle ( σdε), whereas the loss in the flux density versus strain plots H 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 longitudinal (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 4 x 10 4000 8 ) 3 ) 3

3000 6

2000 4

1000 2 Stress hysteresis loss (J/m

Flux density hysteresis loss (J/m 0 0 0 25 50 75 100 125 150 Frequency (Hz)

Figure 5.3: Hysteresis loss with frequency for stress-strain and ﬂux-density strain plots. The plots are normalized with respect to the strain amplitude at a given frequency.

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 rearrangement 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 rotation 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-Strain Hysteresis

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

M= eε + χ H ε= ε ()t avg M= M() t Linear Dynamic Dynamic Constitutive Strain Magnetization Equation

Havg () t Diffusion Field Equation = H H(,) x t ∂HM ∂ ∇2 H −µσ = µσ 0∂t 0 ∂ t ± = BC: H(,) d t Hbias

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

to model the stress versus strain behavior at higher frequencies. However, the magnetization dependence on strain changes signiﬁcantly with increasing frequencies due to the losses associated with the dynamic magnetization rotation resulting from mechanical loading. The modeling strategy is summarized in Figure 5.4.

The constitutive model (Section 3.6.2) shows that at high bias fields, the dependence 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 applied 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,

∂H ∂M ∇2H − µ σ = µ σ , (5.2) 0 ∂t 0 ∂t

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 numerically 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,

X 1 Xd Havg(t) = H(x, t), (5.4) Nx X=−Xd 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

−8 are, µr=3.0, and ρ = 1/σ = 62 × 10 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.

167 420 Increasing d Depth 0.8d 400 0.6d 0.4d 380 0.2d 0

360

340 Magnetic Field (kA/m)

320 0 0.2 0.4 0.6 0.8 1 Non−dimensional Time (t*fa) (a)

400 Increasing 4 Hz frequency 20 Hz 390 60 Hz 100 Hz 380 140 Hz 370

360

350 Average Field (kA/m)

340 0 0.2 0.4 0.6 0.8 1 Non−dimensional Time (t*fa) (b)

0.06 4 Hz 0.05 20 Hz 60 Hz 100 Hz 0.04 140 Hz

0.03

0.02

0.01 Rel. Flux Density (Tesla) 0 0.01 0.02 0.03 0.04 0.05 Strain (c)

Figure 5.5: Model results: (a) Internal magnetic ﬁeld vs. time at varying depth for the 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 magnetization 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 vectors. 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 advantage 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 electromagnet. 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 BIAS MAGNETIC 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 applications [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 microstructure 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 stiffness 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 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

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 excitation 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 transmissibility 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 excitation 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

H a c

Field Preferred Transverse Field Longitudinal Field

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 longitudinal 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 Dead x x weight x x

x x Ni-Mn-Ga x x

x x Aluminum rod

Accelerometer(s)

Shaker table

Figure 6.2: Schematic of the longitudinal ﬁeld test setup.

Dead weight

Ni-Mn-Ga

Electromagnet pole piece(s) Aluminum pushrod(s)

Accelerometer(s)

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 (6.1) = 0 f > 4000,

2 with f the frequency (Hz), Grr the reference acceleration PSD (g /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

2 the reference PSD, and is denoted by Gxx (g /Hz). The top acceleration PSD is de-

2 noted by Gyy (g /Hz). The RMS acceleration values are related to the corresponding

177

&&y M

Kr &&x

Shaker table

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

178 acceleration PSDs by

Z fmax 2 ψr = Grr(f)df, fmin Z fmax 2 ψx = Gxx(f)df, (6.2) fmin Z fmax 2 ψy = Gyy(f)df, fmin

2 where ψr, ψx, ψy represent the reference, input, and output RMS acceleration (m/s ) 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.

2 In this case, the RMS acceleration values obtained from (6.2) are ψr = 0.2 g /Hz,

2 2 ψx = 0.2036 g /Hz, and ψy = 0.7048 g /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 measured 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

2 Gyy |Hxy(f)| = , (6.3) Gxx 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

1 + j(2ζf/fn) H(f) = 2 , (6.4) 1 − (f/fn) + j(2ζf/fn)

179 −3 x 10 3.5 G × 100 xx 3 G yy

/Hz) G × 100 2 2.5 rr

1.5

Acceleration PSD (g 0.5

0 500 1000 1500 2000 2500 3000 3500 Frequency (Hz)

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 max[Gyy(f)]. (6.5) f

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],

Gπf (1 + 4ζ2) ψ2 = n . (6.6) y 4ζ

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 25 Experimental 20 Calculated

−5 Transfer function magnitude (dB) −10 500 1000 1500 2000 2500 3000 3500 Frequency (Hz)

Figure 6.6: Transfer function between top and base accelerations.

for the damping ratio is given as q 2 2 2 2 ψy − (ψy) − (Gπfn) ζ = , (6.7) 2Gπfn where the RMS value of output acceleration (ψy) is obtained from the measured output 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 acceleration 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, s 1 KsKr fn = , (6.8) 2π (Ks + Kr)M from where the mechanical stiﬀness of the Ni-Mn-Ga sample is obtained as,

2 M(2πfn) Kr Ks = 2 . (6.9) Kr − M(2πfn)

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 − Ks0 ∆Ks = × 100, (6.11) Ks0 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 25 0 H (kA/m) 20 66 132 198 15 264 330 10 396 462 5

−5 Transfer function magnitude (dB)

−10 500 1000 1500 2000 2500 3000 3500 Frequency (Hz)

Figure 6.7: Acceleration transmissibility with longitudinal ﬁeld.

30 Test 1 25 Test 2 Test 3 20 Calculated

−5 Transfer function magnitude (dB) −10 500 1000 1500 2000 2500 3000 3500 Frequency (Hz)

Figure 6.8: Longitudinal ﬁeld test model results and repeated measurements under the 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 25 0 20 245 255 15 260 273 301 10 315 341 5 356 380 0 412 439 −5 H (kA/m) −10 Transfer function magnitude (dB) −15 500 1000 1500 2000 2500 3000 3500 Frequency (Hz)

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 measurements, 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 25 0 20 236 249 15 262 274 284 10 297 310 5 323 423 0

−5 H (kA/m) −10 Transfer function magnitude (dB) −15 500 1000 1500 2000 2500 3000 3500 Frequency (Hz)

25 101 20 235 248 15 262 286 300 10 314 407 5 508 522 0

−5 H (kA/m)

Transfer function magnitude (dB) −10

−15 500 1000 1500 2000 2500 3000 3500 Frequency (Hz)

Figure 6.10: Additional measurements of transmissibility ratio with transverse ﬁeld conﬁguration.

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 0.08

0.07

0.06

0.05

0.04

0.03 Damping Ratio 0.02

0.01

0 200 250 300 350 400 450 Bias Field (kA/m)

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 constitutive 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 1500

1000

500

Viscous Damping Constant (Nm/s) 0 200 250 300 350 400 450 Field (kA/m)

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

2500

2000

1500 Resonance Frequency (Hz)

1000 200 250 300 350 400 450 Bias Field (kA/m)

Figure 6.13: Variation of resonance frequency with initial transverse bias ﬁeld.

190 elastic and twinning energy contributions. Mechanical dissipation and the microstructure of Ni-Mn-Ga are incorporated in the continuum thermodynamics framework by considering the internal state variables volume fraction, domain fraction, and magnetization 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, 1 1 E(ξ) = = (6.12) S(ξ) S0 + (1 − ξ)(S1 − S0) 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

AE K = , (6.13) s L 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 16 Experiment Model 14 6 10

× 12

8 Stiffness (N/m) 6

4 200 250 300 350 400 450 Bias Field (kA/m)

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 continuous 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 improvement 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 relationships 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 deformation 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 electromagnet. 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 indication 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 piezoelectrics 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 coupled 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 dissipation. 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 incorporation 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 variables (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 provide 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 hysteresis 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 magnetization 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 frequency, 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 characterizing the resonance and stiffness with bias fields. The stiffness variation under different collinear and transverse bias fields is characterized. Suitable drive configuration can be chosen depending on the application.

200 Quasi-static Dynamic Blocked- Stiffness Behavior Sensing Actuation Actuation Sensing force Tuning Input Base Strain Field Field Field Strain Variable(s) Acceleration Output Magnetization, Strain, Stress, Magnetization, Top Strain Variable(s) Stress Magnetization Magnetization Stress Acceleration Bias Stress, Field, Field Stress Strain Field Variable(s) Frequency Frequency Energy Magnetic Magnetic Gibbs Gibbs - - - Potential Gibbs

Experiment In house Outside data In house Outside data In house In house

Diffusion + Continuum Derived from Derived from Diffusion+ Second Modeling Constitutive Thermodynamics sensing work sensing work Lin.Constitutive order sys. +Dynamics

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 that a limited prior work existed, which was completely addressed in this research; Dark gray 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 characterizations. Few additions to this work could be possible, such as modeling magnetization 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 presented 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 µ ¶ ∂B • Quasi-static characterization chows a ﬂux density sensitivity with strain ∂ε as 4.19T/%ε at 173 kA/m, and 2.38T/%ε at 368 kA/m; maximum ﬁeld induced µ ¶ ∂M twinning stress as 2.84 MPa; variation of initial susceptibility | of ∂H H=0 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-STATIC CHARACTERIZATION 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, µ ¶ D − d Φ = tan−1 (A.1) 2W

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

D d

W Coils (Current Density J)

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 1000

950

900

Increasing L 850

5 in Magnetic Field (kA/m) 800 6 in 7 in 8.3 in 750 0.2 0.4 0.6 0.8 1 Ratio of Small width / Large width

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

1000

950

900

850 Increasing L 5 in Magnetic Field (kA/m) 800 6 in 7 in 8.3 in 750 0 5 10 15 20 25 30 35 Taper Angle (deg)

Figure A.3: Eﬀect of angle (Φ) on ﬁeld.

208 900

800

700

600

500 Magnetic Field (kA/m) 400

300 1 1.5 2 2.5 3 3.5 4 4.5 Current Density (MA/m2)

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 current 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,

l n = w , (A.2) dw and maximum number possible number of turns (Nm) are,

lwww Nm = 2 , (A.3) dw

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),

2 which dictates the power requirements and the Joule heating (IRw), 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 AWG Wire Size 12 13 14 15 16 17 18 19 20 Length (in) 2 2 2 2 2 2 2 2 2 Height diff (in) 1.075 1.075 1.075 1.075 1.075 1.075 1.075 1.075 1.075 Area (in2) 2.15 2.15 2.15 2.15 2.15 2.15 2.15 2.15 2.15 Wire diameter (in) 0.0808 0.072 0.0641 0.0571 0.0508 0.0453 0.0403 0.0359 0.032 Turns per layer 24.75247525 27.77777778 31.201248 35.0262697 39.3700787 44.1501104 49.6277916 55.7103064 62.5 Max. Possible turns 329.3182041 414.7376543 523.26586 659.426268 833.126666 1047.71233 1323.81826 1668.20555 2099.60938 Rated Current (Amp) 11.5 10 8.5 7.5 6.5 5.75 5 4.375 3.75 Theoretical MMF (N*I) 3787.159347 4147.376543 4447.75981 4945.69701 5415.32333 6024.34591 6619.09131 7298.3993 7873.53516 Current density (A/in2) 1761.469464 1929.012346 2068.72549 2300.32419 2518.75504 2802.02135 3078.64712 3394.60432 3662.10938 N_I/A 2.730283129 2.989975116 3.20653093 3.56550963 3.90407812 4.34314178 4.77191258 5.26164723 5.67628088 Mean Perimeter 12.34 12.34 12.34 12.34 12.34 12.34 12.34 12.34 12.34 R_per 1000 ft 1.63 2.06 2.525 3.184 4.016 5.064 6.385 8.051 10.15

211 R_coil 0.551997685 0.878566422 1.35868161 2.15910228 3.4406354 5.45594102 8.69206935 13.8112601 21.9148478 Inductance*phi*e-6 0.10845048 0.172007322 0.27380716 0.434843 0.69410004 1.09770113 1.75249479 2.78290977 4.40835953 Joule Heat = I^2*R 73.00169385 87.85664223 98.1647463 121.449503 145.366846 180.38705 217.301734 264.35615 308.177547 Packing efficiency 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 Actual turns (Na) 263.4545633 331.7901235 418.612688 527.541015 666.501333 713.043478 820 937.142857 1093.33333 Actual MMF (Na*I) 3029.727478 3317.901235 3558.20785 3956.55761 4332.25866 4100 4100 4100 4100 R_coil (Ohm) 0.441598148 0.702853138 1.08694529 1.72728182 2.75250832 3.71315965 5.38404483 7.75871036 11.4117578 Inductance*phi*10^(-6) 0.069408307 0.110084686 0.17523658 0.27829952 0.44422403 0.508431 0.6724 0.87823673 1.19537778

Joule Heat = I^2*R 58.40135508 70.28531379 78.531797 97.1596026 116.293476 122.766341 134.601121 148.506566 160.477844 Possible more turns - - - - 334.668854 503.818261 731.062697 1006.27604

Figure A.5: Comparison of various wire sizes. 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 Turns (N×4) 5000 Current (I ×400) max 4500 MMF (N×I ) max 4000

3500

3000

2500

2000

1500

1000 12 13 14 15 16 17 18 19 20 AWG Wire Number

Figure A.6: Comparison of current carrying capacity, possible turns and MMF produced by various wires (The current and turns are multiplied by scaling factors) Wire size AWG 16 is seen as an optimum size.

213 coil

Laminated Air core gap

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 calibration 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 permeability 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.

215 600 Sample easy axis Sample hard axis 400 No sample

200

−200 Magnetic field (kA/m) −400

−600 −8 −6 −4 −2 0 2 4 6 8 Current (Amp)

550 Sample easy axis Sample hard axis No sample 500

450

400

Magnetic field (kA/m) 350

300 6 6.5 7 7.5 8 Current (Amp)

Figure A.8: Electromagnet calibration curve in presence of sample, the easy axis curve 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 electromagnet. 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 remnant ﬂux density value from the manufactures’ catalogue (Br = 1.32 T in this case).

218

- + - + H M - + Hd - +

- +

Figure A.9: Schematic of the demagnetization field inside the sample. The applied field (H) creates a magnetization (M) inside the sample, which results in north and south poles on its surface. H and M are shown by solid arrows. The demagnetization field (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, it adds to the applied field outside the sample. Therefore, the net field inside the sample is 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 magnetization 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, µ ¶ B 1 M = m − H , (A.8) µ0 Nd

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

221 650 Easy 600 Hard No sample 550 Easy reiterated 500 Hard reiterated

450

400

350

300

Magnetic field (kA/m) 250

200

150 0.025 0.03 0.035 0.04 0.045 Distance (inch)

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

0.9 Easy 0.85 Hard No sample 0.8 Easy reiterated 0.75 Hard reiterated

0.7

0.65

0.6

0.55 Flux density (Tesla) 0.5

0.45

0.4 0.025 0.03 0.035 0.04 0.045 Distance (inch)

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

222 500 Easy 450 Hard Easy reiterated 400 Hard reiterated 350

300

250

200

150

Magnetization (kA/m) 100

0 0.025 0.03 0.035 0.04 0.045 Distance (inch)

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 capacity 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 modulus 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]:

∆W Ψ = (A.10) W

In the context of the hysteretic stress-strain loops, ∆W represents the area en- R 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].

∆W π tan δ ¡ ¢ Ψ = = π (A.11) W 1 + ( 2 + δ) tan δ For small δ (i.e. δ << 1), this equation becomes [36],

∆W Ψ = ≈ π tan δ (A.12) W

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 capacity (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

226 250 Total Energy Energy absorbed

200 ) 3

150 Energy (kJ/m 100

50 0 100 200 300 400 500 Bias Field (kA/m)

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

0.95 )

Ψ 0.9

0.85

0.8

0.75 Damping capcity ( 0.7

0.65 0 100 200 300 400 500 Bias Field (kA/m)

Figure A.15: Damping capacity as a function of bias ﬁeld.

227 0.32

0.3

0.28 δ 0.26 Tan

0.24

0.22

0.2 0 100 200 300 400 500 Bias Field (kA/m)

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).

q4 q3 e Ms a q1 H 1 - x Ms x 1 - q2 a y

1 - a a x

Figure A.17: Schematic of Ni-Mn-Ga microstructure assuming four diﬀerent angles in the four regions.

ρφze = − ξµ0HMs[α cos(θ1) − (1 − α) cos(θ2)] (A.13) − (1 − ξ)µ0HMs[α sin(θ3) + (1 − α) sin(θ4)]

1 2 2 ρφms = ξµ0NMs [α cos(θ1) − (1 − α) cos(θ2)] 2 (A.14) 1 + (1 − ξ)µ NM 2[α sin(θ ) + (1 − α) sin(θ )]2 2 0 s 3 4

229 2 2 ρφan =ξ[Kuα sin(θ1) + Ku(1 − α) sin(θ2) ] (A.15) 2 2 + (1 − ξ)[Kuα sin(θ3) + Ku(1 − α) sin(θ4) ] Assuming reversible rotation of the magnetization vectors, we propose that the derivatives of the energy expression with the two angles is zero,

∂(ρφ) ∂(ρφ) π3 = − = 0, π4 = − = 0 (A.16) ∂θ3 ∂θ4

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 variant 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,

∂(ρφ) ∂(ρφ) π3 = − = 0, π4 = − = 0 (A.18) ∂θ3 ∂θ4

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 π3 + π4 = 0 (A.19) (1 − ξ)α cos(θ3) (1 − ξ)(1 − α) cos(θ4)

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 equation (A.20).

230 APPENDIX B

MISCELLANEOUS ISSUES WITH DYNAMIC CHARACTERIZATION 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 magnetization 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 energy 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 component (Mirr) to effective field is given as,

∂Mirr Mirr = Man − kδ , (B.3) ∂He which is modiﬁed by chain rule as,

∂M M − M ∂H irr = an irr e , (B.4) ∂H δk ∂H 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,

∂H ∂M e = 1 + α irr , (B.5) ∂H ∂H which is further modiﬁed as,

∂M M − M irr = ζ an irr , (B.6) ∂H δk − α(Man − Mirr)

232 where   1, (dH/dt > 0andM < Mirr), or ζ = 1 (dH/dt < 0andM > M ), (B.7)  irr 0, otherwise 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, µ ¶ 3 M 2 λ = , (B.10) 2 Ms 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.

233 800 Man 600 M Mirr 400 Mrev

200

−200

−400 Magnetization (kA/m)

−600

−800 −40 −20 0 20 40 Applied Field (kA/m)

Figure B.1: Magnetization vs. ﬁeld using Jiles model. −4 x 10 8

Magnetostriction 2

0 −40 −20 0 20 40 Appplied Field (kA/m)

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,

1 ¡ ¢ ber(x) = J (xe3iπ/4) + J (xe−3iπ/4) , (B.11) 2 0 0

1 ¡ ¢ bei(x) = J (xe3iπ/4) − J (xe−3iπ/4) , (B.12) 2i 0 0 where J0(x) is the zeroth order Bessel function of the ﬁrst kind. Figure B.3 shows these functions for x = 0 to 10.

140 60

120 40 100

80 20 60 bei (x) ber (x) 40 0

20 −20 0

−20 −40 0 2 4 6 8 10 0 2 4 6 8 10 x x (a) (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 device 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 sample, 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 Threaded rod

Adjust plate Preload spring PCB Force sensor Seismic mass Magnet(s) Ni-Mn-Ga Base plate

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 all clc close 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 strain E0= 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 factor N= Nd - (1 - 2*Nd) % Factor to calculate magnetostatic energy k = 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 fraction z=[]; z(1,1)=1-zs(1,1); % Field-preferred volume fraction dt_started = 0; % Variable for identifying twin-onset e_max = 0.07; e = (0:e_max/1000:e_max); % Loading strain for i=1:length(e) e_tw(i) = zs(i)*e0; % Twinning strain e_e(i)=e(i) - e_tw(i); % Elastic strain

% Compliance S(i) = S0 + (1-z(i))*(S1-S0); % Modulus associated with elastic strain E(i) = 1/S(i); % Modulus associated with twinning strain a(i) = E(i)*k/(E(i)-k);

% Driving force due to stress F_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 fraction alpha(i) = 1/2*(H+N*Ms)/(Ms*N);

% Constraints on alpha if(alpha(i)>=1) alpha(i)=1; end if(alpha(i)<=0) alpha(i)=0; end

% Rotation angle sin_theta(i) = mu0*H*Ms/(2*Ku+mu0*N*Ms^2);

% Constraints on theta if(sin_theta(i) > 1) sin_theta(i) = 1; end if(sin_theta(i) <-1) sin_theta(i) = -1; end theta(i) = asin(sin_theta(i));

% Magnetization M(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 magnetization if (M(i)>=Ms) M(i)=Ms; end

% Driving force due to field 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 );

% Net thermodynamic driving force F(i) = +F_zs(i) + F_z_H(i);

240 % Code to check if the threshold is met if(i > 1 & F(i) <= -pi_cr & (z(i)-z(i-1)) <=0) % Function to calculate volume fraction z(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); else z(i+1)=z(i); end

% Constraint on volume fraction if(z(i+1) >= 1) z(i+1)=1; dt_started =0; else if (z(i+1) <= 0) z(i+1)= 0; end end

zs(i+1)=1-z(i+1);

% Stress sig(i)=E(i)*(e(i) - e0*zs(i)); end

%% Code for unloading clear F_zs F_z_H z zs zs=[]; zs(1,1)=1; z(1,1)=1-zs(1,1); dt_started = 0; e=(e_max:-e_max/1000:0); % Unloading strain for i=1:length(e)

e_tw(i) = z(i)*e0; % Twinning strain e_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; end if(alpha(i)<=0) alpha(i)=0; end sin_theta(i) = mu0*H*Ms/(2*Ku+mu0*N*Ms^2); if(sin_theta(i) > 1) sin_theta(i) = 1; end 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))); if (M(i)>=Ms) M(i)=Ms; end B(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 met if( (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); else z(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; end end

zs(i+1)=1-z(i+1); sig(i) = E(i)*(e(i) - e0*(1-z(i)));

% Constraint to ensure that stress is always compressive if(sig(i) <0) sig(i) = 0; z(i+1)=z(i); zs(i+1)=zs(i); dt_started =0; end end

%% 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 stress sig_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 factor N = 3*Nd - 1; % Factor for Magnetostatic energy e0 = 0.058; % Reorientation strain E = 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 force xi(1)=0; zs=[]; % Stress-preferred volume fraction z=[]; % Field-preferred volume fraction

%% Code for forward field application z_start = 0.0; % Initial configuration z(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 fraction alpha(i) = 1/2*(H(i)+N*Ms)/(Ms*N);

% Constraints on alpha

244 if(alpha(i)>=1) alpha(i)=1; end if(alpha(i)<=0) alpha(i)=0; end

% Rotation angle sin_theta(i) = mu0*H(i)*Ms/(2*Ku+mu0*N*Ms^2);

% Constraints on theta if(sin_theta(i) > 1) sin_theta(i) = 1; end if(sin_theta(i) <-1) sin_theta(i) = -1; end theta(i) = asin(sin_theta(i));

% Magnetization M(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 magnetization if (M(i)>=Ms) M(i)=Ms; end

% Driving force due to magnetic field 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 );

% Driving force due to stress F_zs(i) = -a*e0^2*(z(i) ) + sig_b*e0;

% Net thermodynamic driving force F_z(i) = F_z_H(i) + F_zs(i);

% Code to check if the threshold is met if ( i>1 & (F_z(i)) >= pi_cr & z(i)>=z(i-1) )

245 if (dt_started ==0) H(i)/1000 end dt_started =1; % Calculation of volume fraction z(i+1) = (F_z_H(i) + sig_b*e0 - pi_cr)/(a*e0^2); % Constraint on volume fraction if(z(i+1)<0) z(i+1)=0; end if(z(i+1)>1) z(i+1)=1; end else z(i+1)=z(i); end

e_tw(i) = z(i)*e0; % Twinning strain e_e(i) = sig_b/E; % Elastic strain

e(i) = e_e(i) + e_tw(i); % Total strain end

%% Code for reverse field application clear H F_z F_z_H e e_tw z zs F_zs M Bm theta alpha z=[];zs=[]; z(1)=z_end; H = H_max:-0.5:00;H =H*1000; % Field during reverse application for 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; end if(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; end if(z(i+1)

else z(i+1)=z(i); end e_tw(i) = z(i)*e0; e_e(i) = sig_b/E; %% Constitutive equation e(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 model f0=1; % Frequency T0=1/f0; % Time period of one cycle T = m*T0; % Total time period df=1/T; % Frequency resolution Np = 2^12; % Total points N0=Np/m; % Points in one cycle h=T/Np; % Time resolution fs=1/h; % Sampling frequency t=0:h:T-h; % Time vector t2=-h*N0:h:T-h;

% Matrix to select various frequencies FF =[1 2 3 4 5 6 7 1 50 100 150 175 200 250 6.25 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 frequency xi = .95; % Damping ratio fn = 700; % Natural frequency fa = FF(2,ff); % Actuation (applied field) frequency col=COL(1,ff); H00 = FF(3,ff)*79.577*10^3; % Magnitude of applied field H0 = H00*1; % Magnitude of applied field mu0=4*pi*10^(-7); % Permeability of vacuum mur = 3; % Relative permeability rho =62*10^(-8); % Resistivity e0=0.04; % Reorientation strain mu=mu0*mur; % Permeability of sample sigma = 1/rho; % Conductivity omega = 2*pi*fa; % Circular frequency delta=sqrt(2/(mu*sigma*omega)); % Skin depth d=5e-3; % Sample width x = -d:d/100:d; % Distance vector

248 X= x/delta; Xd = d/delta;

% Parameters to triangular field profile NN = 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 profile for 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; else hh1(kk) = H0/(TT/4)*(tt(kk)- 3*TT/4) - H0; end end end H_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 spectrum for kk = 1:length(freqq1) % Magnitude if (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 % Phase P_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 profile for 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 profile H_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 Field for ii = 1:length(x) X(ii) = x(ii)/delta;

% Complex solution to Diffusion Equation h_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 superposition for 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); end end 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 sample for 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 time for jj = 1:length(tt) H_avg(jj) = mean(H(:,jj)); end figure(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 model hh = [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 fraction H_net = []; z_start = 0.35; % Initial volume fraction z_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 spectrum for kk = 1:length(freq)/2

if (kk==1) M_H(kk) = abs(fft_H(kk))/Np ; M_z(kk) = abs(fft_z(kk))/Np ; else M_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; end if(M_H(kk) > max(abs(fft_H))*1e-5/Np) P_H(kk) = unwrap(angle(fft_H(kk))); else P_H(kk) = 0; end if(M_z(kk) > max(abs(fft_z))*1e-5/Np) P_z(kk) = unwrap(angle(fft_z(kk))); kk; else P_z(kk) = 0;

252 end end 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 FFT kkk=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); end end

% Regeneration of the original signal of volume fraction for 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)) ); end z_calc(ii)=sum; end figure(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 constant f = 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 frequency for 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; else HH(kk) = H0/(T/4)*(t(kk)- 3*T/4) - H0; end end end

% Calculation of dynamic strain using actuator dynamics for ii = 1:length(t) sum2 = 0; for kk = 1:100 r = F(kk)*fa/fn; % Frequency Ratio XX =1./((1-r.^2)+j*(2*xi.*r)); % Magnitude X0 = abs(XX); phi = -angle(XX); % Phase % Superposition of individual displacement responses sum2=sum2 + e0*Mag_z(kk)*X0*cos(2*pi*F(kk)*fa*t(ii) ... + Ph_z(kk) - phi ) ; end x_calc(ii)=sum2; % Final dynamic strain end % ------

C.2.2 Dynamic Sensing Model f0 = 12; % Frequency of applied strain H_mag = 368*1000; % Bias field magnitude (kA/m) fa=f0;

% Generating strain vectors from the experimental data pp = 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 vector ee_load = ee(1:indd); t_load = t(1:indd); % Unloading strain vector ee_unload = ee(indd:end); t_unload = t(indd:end);

% Sampling parameters nnnn=14 Np = 2^8; T = 1/f0; TT = nnnn*T; dt = T/Np; df = 1/TT; mu0 =4*pi*10^(-7); % Permeability of vacuum mur = 4.5; % Relative permeability mu = mu0*mur; % Permeability of sample rho = .6*10^(-8); % Resistivity sigma = 1/rho; % Conductivity kd = 1/(mu0*sigma); d = 6e-3; % Sample width Nx = 21; % Number of points dx = d/(Nx-1); x = 0:dx:d; scaling = kd*dt/dx^2 r = scaling

% Parameters for backward difference method t = 0:dt:TT-dt; fs = 1/dt; fh = fs/2; freq = 0:df:fs-df; freq1 = 0:df:fh-df;

255 H0 = H_mag Nt = length(t)

%% Code for Backward Difference Method % Code to generate matrices H = 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-1 A(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 model max_z = .7793 min_z = .3069 ee_net = e_net; max_zs = 1 - min_z min_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 equation M_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 field H_d(i+1) = mean(H(:,i+1)); % Dynamic magnetization M_lin(i) = -3.7333e6*ee_net(i) + 1.76*H_d(i); % Dynamic flux density Bm_lin(i) = mu0*(H0 + Nd*M_lin(i)); end

C.2.3 Jiles-Atherton Model clear all clc close all

M=0; % Net magnetization Mrev=0; % Reversible magnetization Mirr=0; % Irreversible magnetization Man=0; % Anhysteretic magnetization Ms=7.65*10^5; % Saturation magnetization

%Model parameters a = 5000; k=4000; c=0.18; alpha=0.0033;

T=10; % Final time dt=0.005; % Time resolution Hmax = 40000; % Applied field amplitude t=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/2 count=1;

257 T=T; Hmax=Hmax/2; end t=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 zeta if(tt < T/4) H(i)=Hmax*tt/(T/4); dHdt=Hmax/(T/4); delta=1; if (i~=1 & M(i-1)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)T/4) zeta=1; end end end end if (t<=T/4) zeta=1; end if(i>1) dH(i)=H(i)-H(i-1); else dH(i)=H(i); end if (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/dM slope1=(Mirr(i-1)-Mirr(i-2))/(M(i-1)-M(i-2)); end slope=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; end zetta(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)

260 261

Figure D.1: E-shaped laminates for electromagnet. 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.

266 267

Figure D.7: 2-D view of the assembled device. 268

Figure D.8: Bottom plate. 269

Figure D.9: Top plate. 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).

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