CHARACTERIZATION AND MODELING OF THE FERROMAGNETIC SHAPE MEMORY ALLOY Ni-Mn-Ga FOR SENSING AND ACTUATION DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Neelesh Nandkumar Sarawate, B.E., M.S. ***** The Ohio State University 2008 Dissertation Committee: Marcelo Dapino, Adviser Rajendra Singh Stephen Bechtel Rebecca Dupaix Approved by Adviser Graduate Program in Mechanical Engineering
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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
N. Sarawate and M. Dapino, “Characterization and modeling of the dynamic sensingbehavior of Ni-Mn-Ga”, Smart Materials and Structures, Draft in preparation.
N. Sarawate and M. Dapino, “Magneto-mechanical energy model for nonlinear andhysteretic quasi-static behavior of Ni-Mn-Ga”, Journal of Intelligent Material Systemsand Structures, in review.
N. Sarawate and M. Dapino, “Dynamic actuation model for magnetostrictive mate-rials,” Smart Materials and Structures, in review.
N. Sarawate and M. Dapino, “Stiffness tuning using bias fields in ferromagnetic shapememory alloys,” Journal of Intelligent Material Systems and Structures, in review.
v
N. Sarawate and M. Dapino, “Magnetization dependence on dynamic strain in ferro-magnetic shape memory Ni-Mn-Ga,” Applied Physics Letters, Vol. 93(6), p. 062501,2008.
N. Sarawate and M. Dapino, “Magnetic field induced stress and magnetization inmechanically blocked Ni-Mn-Ga,” Journal of Applied Physics. Vol. 103(1), p. 083902,2008.
N. Sarawate and M. Dapino, “Frequency dependent strain-field hysteresis model forferromagnetic shape memory Ni-Mn-Ga,” IEEE Transactions on Magnetics, Vol. 44(5),pp. 566-575, 2008.
N. Sarawate and M. Dapino, “Continuum thermodynamics model for the sensing ef-fect in ferromagnetic shape memory Ni-Mn-Ga,” Journal of Applied Physics, Vol. 101(12), p. 123522, 2007.
N. Sarawate and M. Dapino, “Experimental characterization of the sensor effect in fer-romagnetic shape memory Ni-Mn-Ga,” Applied Physics Letters, Vol. 88(1), p. 121923,2006.
Conference Publications
N. Sarawate, and M. Dapino, “Characterization and modeling of dynamic sensingbehavior of ferromagnetic shape memory alloys,” Proceedings of ASME Conference onSmart Materials, Adaptive Structures and Intelligent Systems, Paper #656, EllicottCity, MD, October 2008.
N. Sarawate, and M. Dapino, “Dynamic strain-field hysteresis model for ferromagneticshape memory Ni-Mn-Ga,” Proceedings of SPIE Smart Structures and Materials,Vol. 6929, p. 69291R, San Diego, CA, March 2008.
N. Sarawate, and M. Dapino, “Electrical stiffness tuning in ferromagnetic shape mem-ory Ni-Mn-Ga,” Proceedings of SPIE Smart Structures and Materials, Vol. 6529,p. 652916, San Diego, CA, March 2007.
N. Sarawate, and M. Dapino, “Magnetomechanical characterization and unified mod-eling of Ni-Mn-Ga,” Proceedings of SPIE Smart Structures and Materials, Vol. 6526,p. 652629, San Diego, CA, March 2007.
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N. Sarawate, and M. Dapino, “A thermodynamic model for the sensing behavior of fer-romagnetic shape memory Ni-Mn-Ga,” Proceedings of ASME IMECE, Paper #14555,Chicago, IL, November 2006.
N. Sarawate, and M. Dapino, “Sensing behavior of ferromagnetic shape memoryNi-Mn-Ga,” Proceedings of SPIE Smart Structures and Materials,” Vol. 6170, pp.61701B, San Diego, CA, February 2006.
FIELDS OF STUDY
Major Field: Mechanical Engineering
Studies in:
Smart Materials and Structures Prof. DapinoApplied Mechanics Prof. Dapino, Prof. Bechtel, Prof. DupaixSystem Dynamics and Vibrations Prof. Dapino, Prof. Singh
1.1 Overview of transduction principles in smart materials. . . . . . . . . 6
6.1 Summary of longitudinal field test results. Units: fn: (Hz), Ks: (N/m) 185
6.2 Summary of transverse field test results. Units: fn: (Hz), Ks: (N/m) . 188
xii
LIST OF FIGURES
Figure Page
1.1 Comparison of FSMAs with other classes of smart materials. . . . . . 2
1.2 Joule magnetostriction produced by a magnetic field H. (a) H is ap-proximately proportional to the current i that passes through thesolenoid when a voltage is applied to it, (b) the rotation of magneticdipoles changes the length of the sample, (c) and (d) curves M vs. Hand ∆L/L vs. H obtained by varying the field sinusoidally [20]. . . . 10
1.3 SMA transformation between high and low temperature phases. . . . 11
1.6 (a) Relative orientation of sample, strain gauge, and applied field formeasurements shown in (b) and (c). (b) Strain vs applied field in theL21 (austenite) phase at 283 K. (c) Same as (b) but data taken at265 K in the martensitic phase [128]. . . . . . . . . . . . . . . . . . . 16
4.1 Flow chart for modeling of dynamic Ni-Mn-Ga actuators. . . . . . . 116
4.2 Dynamic actuation data by Henry [48] for (a) 2 − 100 Hz (fa = 1 −50 Hz) and (b) 100− 500 Hz (fa = 50− 250 Hz). . . . . . . . . . . . 117
4.3 Magnetic field variation inside the sample at varied depths for (a) si-nusoidal input and (b) triangular input. x = d represents the edge ofthe sample, x = 0 represents the center. . . . . . . . . . . . . . . . . 121
xv
4.4 Average field waveforms with increasing actuation frequency for (a)sinusoidal input and (b) triangular input. . . . . . . . . . . . . . . . 123
4.5 Dependence of normalized field amplitude on position with increasingactuation frequency for (a) sinusoidal input and (b) triangular input. 124
4.6 Model result for quasistatic strain vs. magnetic field. The circlesdenote experimental data points (1 Hz line in Figure 4.2) while the solidand dashed lines denote model simulations for ˙|H| > 0 and ˙|H| < 0,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.7 Dynamic Ni-Mn-Ga actuator consisting of an active sample (spring)connected in mechanical parallel with an external spring and damper.The mass includes the dynamic mass of the sample and the actuator’soutput pushrod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.12 Model results for strain vs. applied field in frequency domain for tri-angular input waveform for (a) fa = 50 Hz, (b) fa = 100 Hz, (c) fa =150 Hz, (d) fa = 175 Hz, (e) fa = 200 Hz, (e) fa = 250 Hz. Dottedline: experimental, solid line: model. . . . . . . . . . . . . . . . . . . 139
5.2 (a) Stress vs. strain and (b) flux-density vs. strain measurements forfrequencies of up to 160 Hz. . . . . . . . . . . . . . . . . . . . . . . . 161
5.3 Hysteresis loss with frequency for stress-strain and flux-density strainplots. The plots are normalized with respect to the strain amplitudeat a given frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.4 Scheme for modeling the frequency dependencies in magnetization-strain hysteresis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.5 Model results: (a) Internal magnetic field vs. time at varying depthfor the case of 140 Hz strain loading (sample dim:±d), (b) Averagemagnetic field vs. time at varying frequencies, and (c) Flux-density vs.strain at varying frequencies. . . . . . . . . . . . . . . . . . . . . . . . 168
6.1 Left: simplified 2-D twin variant microstructure of Ni-Mn-Ga. Center:microstructure after application of a sufficiently high transverse mag-netic field. Right: after application of a sufficiently high longitudinalfield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.2 Schematic of the longitudinal field test setup. . . . . . . . . . . . . . 176
6.3 Schematic of the transverse field test setup. . . . . . . . . . . . . . . 176
6.4 DOF spring-mass-damper model used for characterization of the Ni-Mn-Ga material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.1 Characterization map of Ni-Mn-Ga. Plain blocks in “Experiment” and“Modeling” rows show the new contribution of the work; Light grayblocks show that a limited prior work existed, which was completelyaddressed in this research; Dark gray blocks indicate that prior workwas available, and no new contribution was made. . . . . . . . . . . . 201
A.6 Comparison of current carrying capacity, possible turns and MMF pro-duced by various wires (The current and turns are multiplied by scalingfactors) Wire size AWG 16 is seen as an optimum size. . . . . . . . . 213
A.8 Electromagnet calibration curve in presence of sample, the easy axiscurve shows maximum variation. . . . . . . . . . . . . . . . . . . . . 216
A.9 Schematic of the demagnetization field inside the sample. The appliedfield (H) creates a magnetization (M) inside the sample, which resultsin north and south poles on its surface. H and M are shown by solidarrows. The demagnetization field (Hd = NxM) is directed from northto south poles as shown by dashed arrows. Although inside the sam-ple, the demagnetization field opposes the applied field, it adds to theapplied field outside the sample. Therefore, the net field inside thesample is given as H −NxM , whereas the net field outside the sampleis given as H + NxM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
D.15 Grip to hold the sample (2 nos). . . . . . . . . . . . . . . . . . . . . . 273
xx
CHAPTER 1
INTRODUCTION AND LITERATURE REVIEW
1.1 Introduction and Motivation
Ferromagnetic Shape Memory Alloys (FSMAs) in the nickel-manganese-gallium
(Ni-Mn-Ga) system are a recent class of smart materials that have generated great
research interest because of their ability to produce large strains of up to 10% in the
presence of magnetic fields. This strain magnitude is around 100 times larger than
that exhibited by other smart materials such as piezoelectrics and magnetostrictives.
Due to the magnetic field activation, FSMAs exhibit faster response than the ther-
mally activated Shape Memory Alloys (SMAs). The combination of large strains and
fast response gives FSMAs a unique advantage over other smart materials. As seen in
Figure 1.1, FSMAs bridge the gap between various existing classes of smart materials.
Ni-Mn-Ga FSMAs therefore open up opportunities for various possible applications
such as sonar transducers, structural morphing, energy harvesting, motion/force sens-
ing, vibration control, etc. However, Ni-Mn-Ga FSMAs are still relatively new, and
their behavior and mechanics are not fully understood. Also, most of the aforemen-
tioned applications can be classified into two fundamental behaviors: sensing and
1
Figure 1.1: Comparison of FSMAs with other classes of smart materials.
actuation. Actuation refers to the application of magnetic field to generate deforma-
tion (strain), whereas sensing refers to the application of mechanical input (stress or
strain) to alter the magnetization of the material. If these two behaviors are thor-
oughly studied in various static and dynamic conditions, it will lead to a significant
advancement in the state of the art of this technology.
Because of the ability of Ni-Mn-Ga to generate large strains under magnetic fields,
most of the prior work has been focused on the experimental characterization and
modeling of the actuation effect. The characterization of the actuator effect is usually
conducted by subjecting the material to magnetic fields created with an electromag-
net, which results in the generation of displacement that can be measured by a suit-
able sensor. Early challenges in conducting the actuation characterization involved
2
construction of electromagnets that can apply the large magnetic field required to
saturate the material to generate maximum strain. Typically the magnetic field is
applied in the presence of a constant compressive (bias) stress. This bias stress is
used to restore the original configuration of the material when the magnetic field is
removed. Ni-Mn-Ga exhibits a low-blocking stress (≈ 3 MPa), which can limit its
actuation authority. Investigation of applications other than actuation is necessary
to fully understand the capabilities of Ni-Mn-Ga FSMAs.
The sensing effect has received limited attention. Although a few prior studies
have shown the ability of Ni-Mn-Ga to respond to mechanical inputs by magnetiza-
tion change, a comprehensive characterization under a wide range of inputs and bias
variables is lacking. Development of models that can describe the macroscopic behav-
ior of the material in sensing mode is also required. The presented work will provide
a physics-based model that describes the coupled magnetomechanical behavior of the
material in sensing mode.
A major advantage of FSMAs over thermal SMAs is their fast response, or high
operating frequency. Even so, most of the prior work on Ni-Mn-Ga is focused on quasi-
static behavior. While experimental work on dynamic actuation does exist, there are
no models to describe the frequency-dependent behavior of a dynamic Ni-Mn-Ga
actuator. The applications of Ni-Mn-Ga as a dynamic sensor and as a vibration
absorber have not been fully explored. Understanding the dynamic behavior of Ni-
Mn-Ga is required to realize its potential as a dynamic actuator, sensor or a vibration
absorber.
The presented research addresses various unresolved aspects with the modeling
and characterization of commercial quality single crystal Ni-Mn-Ga in quasi-static
3
and dynamic conditions. In the quasi-static part, experimental characterization of
the sensing effect of Ni-Mn-Ga is conducted. A magnetomechanical test setup is
developed to conduct the characterization. Further, a continuum thermodynamics
based energy model is developed to describe the sensing behavior of Ni-Mn-Ga. The
thermodynamic framework is extended to also describe the actuation and blocked-
force behaviors, thus fully describing the non-linear and hysteretic constitutive re-
lationships in Ni-Mn-Ga. In the dynamic part, study of Ni-Mn-Ga under dynamic
mechanical and magnetic excitation is conducted. To model the strain dependence on
dynamic fields (magnetic excitation), the constitutive actuation model is augmented
with magnetic field diffusion and system-level structural dynamics. The dynamic
mechanical excitation includes two characterizations: dynamic sensing and tunable
stiffness. Dynamic sensing characterization is conducted by altering the magneti-
zation of Ni-Mn-Ga by subjecting it to cyclic strain loading at frequencies of up
to 160 Hz. The stiffness of Ni-Mn-Ga is characterized under varied collinear and
transverse magnetic field drive configurations, to illustrate its viability for tunable
vibrations absorption applications.
This chapter reviews existing state of the art on Ni-Mn-Ga. An overview of
various smart materials is presented. Properties of Ni-Mn-Ga FSMAs are discussed
and the active strain mechanism is introduced. The details of prior experimental work
on the sensing behavior of Ni-Mn-Ga are reviewed, followed by a review of various
approaches to model the coupled magnetomechanical quasi-static behavior. Finally,
the characterization and modeling of the dynamic behavior of Ni-Mn-Ga is reviewed,
which includes dynamic actuation (frequency dependent strain-field hysteresis) and
stiffness tuning under varied bias fields.
4
1.2 Overview of Smart Materials
A smart material is an engineered substance that converts one form of input en-
ergy into different form of output energy. These “active” or “smart” materials can
react with a change in dimensional, electrical, elastic, magnetic, thermal or rheological
properties to external stimuli such as heat, electric or magnetic field, stress and light.
In most operating regimes, smart materials have the ability to recover the original
shape and properties when the external driving input is removed which makes them
suitable candidates for use in actuator and sensor applications. Smart materials can
be broadly categorized into several classes based on the type of driving input and
the phenomenon by which the response is produced: piezoelectric, electrostrictive,
magnetostrictive, electrorheological and magnetorheological, shape memory, and fer-
romagnetic shape memory. In general, all of these smart materials are transducers,
and they convert energy from one form to another. The smart materials have poten-
tial to replace conventional hydraulic and pneumatic actuators. Table 1.1 shows the
transduction principles or the effects that couple one domain to another.
Smart materials have been widely utilized in various commercial sensors and actu-
ators. Major advantages of smart material actuators and sensors include high energy
density, fast response, compact size, and less-moving parts. The disadvantages of
these materials are limited strain outputs, limited blocking forces, high cost and sen-
sitivity to harsh environmental conditions.
5
Output/Input
Charge,Current
Magneticfield
Strain Temperature Light
Electricfield
Permittivity,Conductivity
Electro-magnetism
ConversePiezo Effect
ElectroCaloricEffect
Electro Op-tic Effect
Magneticfield
Mag-electEffect
Permeability Magneto-striction
MagnetoCaloric effect
Magneto Op-tic Effect
Stress Piezo-electricEffect
Piezo-magneticEffect
Compliance - Photo Elas-tic Effect
Heat PyroelectricEffect
- Thermal Ex-pansion
Specific Heat -
Light PhotovoltaicEffect
- Photostriction - RefractiveIndex
Table 1.1: Overview of transduction principles in smart materials.
1.2.1 Ferroelectrics
Ferroelectric materials constitute a class of smart materials that exhibit coupling
between the mechanical and electrical domains. Piezoelectrics are the most com-
monly known examples of ferroelectric class. Piezoelectric materials produce strains
of up to 0.1% (PZT) and 0.07% (PVDF) when exposed to an electric field [112]
and also produce a voltage when subjected to an applied stress. They have found
numerous applications as both actuators and sensors. Piezoelectric devices are also
known for their high frequency capability; this technology is often used in ultrasonic
applications [70]. Microscopically, piezoelectric materials are characterized by hav-
ing an off-center charged ion in a tetragonal unit cell which can be moved from one
axis to another through the application of an electric field or stress [112]. As the
ion changes position, it causes strain in the material due to the electromechanical
coupling. In order for bulk strain to occur, these materials are generally polarized.
Typical piezoelectric materials, PZT and PVDF, are generally employed in stacks,
6
where the strain amplitude is amplified by placing many devices in series and in bi-
morphs and THUNDER actuators where the strain is amplified through the elastic
structure to which the active material is attached. In general, piezoelectrics are char-
acterized as a moderate force, low stroke, solid state device. For actuation, excitation
voltages required to energize these materials can be as high as 1-2 kV, although 100 V
is typical. Because piezoelectrics have high energy density, operate over wide band-
widths, and are easy to incorporate into structures, they are a good candidate for
smart actuation. Piezoelectric materials also find wide applications as sensors, for
example in accelerometers.
Electrostrictive materials are similar to piezoelectrics in terms of the operating
principle, but they typically generate larger strains (0.1%), and are highly nonlinear
and hysteretic. They require higher fields to generate the saturating strain, and
have stringent temperature requirements. Furthermore, only a unidirectional strain
is possible as the strain depends on the magnitude of the electric field, and not
the polarity. All ferroelectric materials typically exhibit a domain structure and
a spontaneous polarization, when cooled below the Curie temperature. When an
electric field is applied to the material, the domains tend to align along the direction
of applied field, resulting in the strain generation. Single crystal materials exhibit
higher energy density and large strain, whereas polycrystalline materials exhibit lesser
strain and higher hysteresis. But, polycrystalline materials are significantly cheaper
and easy to manufacture than the single crystal materials.
7
1.2.2 Magnetostrictives
Magnetostrictive materials are similar to Ferromagnetic Shape Memory Alloys
(FSMAs) in that they both strain when exposed to a magnetic field and both pro-
duce a change in magnetization when a stress is applied. However, the mechanism
responsible for these phenomena is distinctly different for the two materials. Giant-
magnetostrictive materials such as Terfenol-D and Galfenol have strong spin-orbit
coupling. Thus, when an applied magnetic field rotates the spins, the orbital mo-
ments rotate and considerable distortion of the crystal lattice occurs resulting in
large macroscopic strains [20]. A diagram of this strain mechanism is shown in Fig-
ure 1.2. The magnetostrictive material is usually pre-compressed in order to orient
the magnetic moments perpendicular to its longitudinal axis. When a longitudinal
magnetic field is applied to the material, the magnetic moments tend to align along
the direction of the field. This results in orientation of the domains in longitudinal
direction, which results in the strain generation. The strain is approximately pro-
portional to the square of magnetization, which results in butterfly curves that give
two strain cycles per magnetization and field cycle. The magnetostrictive materials
respond to the applied mechanical stress by producing a change in their magnetiza-
tion, which can be detected by measuring the induced voltage in a pickup coil or a
suitable magnetic sensor. This ‘inverse’ phenomenon is termed as Villari effect.
Since the magnetostriction of Terfenol-D is dependent on the magnetization vec-
tors turning away from their preferred direction, it can be understood that mag-
netostriction depends on a relatively low value of magnetic anisotropy whereas the
opposite is a requirement for FSMAs. Terfenol-D achieves maximum strains of around
0.12% and can be operated for frequencies of up to 10 kHz [43] including a Delta-E
8
effect [63] similar to that discussed for Ni-Mn-Ga in Chapter 6. Some of the disadvan-
tages of Terfenol-D are that it is relatively expensive to produce and is highly brittle.
A similar material, Galfenol, which is easier to produce and has higher strength is
gaining in popularity. Galfenol can produce 0.03% strain [64] and is machinable with
common techniques [13]. Both of these materials are commonly employed in solenoid
based actuators as opposed to Ni-Mn-Ga actuators that consist of an electromagnet.
Magnetostrictives have found applications as actuators and sensors in a broad range
of fields including industry, bio-medicine, and defense [20].
1.2.3 Shape Memory Alloys
Shape Memory Alloys (SMAs) are alloys that undergo significant deformation
at low temperatures and retain this deformation until they are heated [130]. In
comparison to piezoelectric and magnetostrictive materials, SMAs have the advantage
of generating significantly large strains of around 10%. SMAs produce strain by a
similar mechanism as that in the FSMAs. Thus, an in-depth review of these materials
is useful from the viewpoint of understanding the behavior of Ni-Mn-Ga FSMAs.
At high temperatures, SMAs such as Nickel-Titanium (Ni-Ti) alloy exhibits a body
centered cubic austenite phase. At low temperatures, the material exhibits martensite
phase, which has a monoclinic crystal structure. The transformation between the
low and high temperature phases is shown in Figure 1.3. When the material is
cooled from the high temperature austenite phase, a “twinned” martensite structure is
formed. This twinned structure consists of alternating rows of atoms tilted in opposite
direction. The atoms form twins of themselves with respect to a plane of symmetry
called as a twinning plane, or twin boundary. When a stress is applied to the material,
9
(a) (b)
(c) (d)
Figure 1.2: Joule magnetostriction produced by a magnetic field H. (a) H is ap-proximately proportional to the current i that passes through the solenoid when avoltage is applied to it, (b) the rotation of magnetic dipoles changes the length of thesample, (c) and (d) curves M vs. H and ∆L/L vs. H obtained by varying the fieldsinusoidally [20].
10
Figure 1.3: SMA transformation between high and low temperature phases.
the twins are reoriented so that they all lie in the same direction. This process is called
as “detwinning”. When the material is heated, the deformed martensite reverts to the
cubic austenite form, and the original shape of the component is restored. Therefore
this behavior is called as “shape memory effect” as the material remembers its original
shape. This entire process is shown in Figure 1.3.
This process is highly hysteretic. The hysteresis associated with temperature is
shown in Figure 1.4. The amount of martensite in the material is quantified by the
martensite volume fraction (ξ). Naturally the austenite volume fraction is (1-ξ).
Referring to Figure 1.4, at a temperature below Mf , the material is 100% martensite.
When heated, the material does not transform to the austenite phase until a temper-
ature As is reached, after which the material starts transforming to austenite. The
11
Figure 1.4: Schematic of phase transformation.
material consists of 100% austenite when a temperature Af is exceeded. When the
material is cooled below Af , it does not start transforming to the martensite phase
until a temperature Ms is reached. The martensite transformation is completed when
the temperature reaches Mf . The values of the four critical temperatures (Mf ,Ms,Af ,
and As) depend on the composition of alloy, with typical width of hysteresis loop being
10-50C.
The temperature and associated phase transformations also significantly affect the
stress-strain behavior of the material. Figure 1.5 shows the stress-strain behavior of
SMAs at two constant temperatures, namely below Mf and above Af . At temperature
below Mf (Figure 1.5(a)), the material consists of a complete martensite phase, and in
absence of load, the material consists of a twinned structure. The elastic region (o →
12
a) corresponds to the elastic compression of the material until the stress level is
sufficient to start detwinning. In the detwinning region (a → b), the twins reorient
themselves until they all lie in the same crystallographic region. The amount of
stress needed is relatively small (beyond the elastic region) to cause detwinning, which
corresponds to a low slope region. The material again gets compressed elastically (b →
c) after the detwinning is completed. In the plastic region (beyond c), the subsequent
shape memory effect is destroyed. In the unloading region (c → d), the material
does not come back to its original shape because the material is deformed when it is
detwinned. Only the elastic deformation is recovered. The residual strains can only
be recovered if the material is heated to Af .
The second configuration in the Figure 1.5(b) is at temperature above Af . The ini-
tial microstructure consists of randomly oriented austenite. The elastic region (o → a)
is followed by the transformation region (a → b), where the stress-induced marten-
site is formed upon loading, which is again followed by an elastic region (b → c).
Upon unloading (c → e), the stress induced martensite goes through elastic unload-
ing, which is followed by the transformation back to the austenite phase. Thus the
shape memory behavior is seen in the stress-strain curves also, where the stress and
temperature are both responsible for the phase change. Later (in Chapter 2) it will
be seen that in case of FSMAs, the magnetic field acts in an analogous manner to
the temperature: at high magnetic fields, the stress-strain plots of FSMAs exhibit
pseudoelastic or reversible behavior, whereas at low magnetic fields, the stress-strain
behavior is irreversible.
The major advantage of SMAs is that they generate large strain of around 10%.
Also, their Young’s modulus changes by about 3-5 times when constrained. The major
disadvantage of SMAs is their limited bandwidth due to the slow heating process,
which limits their use when fast actuation response is required. They have found
numerous applications in the aerospace, medical, safety devices, robotics, etc. Some
of their applications include couplers in fighter planes, tweezers, orthodontic wires,
eyeglass frames, fire-sprinklers, and micromanipulators to simulate human muscle
motion.
1.3 Ferromagnetic Shape Memory Alloys
Ferromagnetic Shape Memory Alloys (FSMAs), which are also called Magnetic
Shape Memory Alloys (MSM-Alloys), were first identified by Ullakko at MIT in
1996 [128]. This new class of materials, which generates strain when subjected to
a magnetic field, showed promise of relatively high strain and high operating fre-
quency of several hundred Hz [126]. Therefore, they have been the subject of much
14
research over the past 10 years. This section provides an overview of the work done
by key contributors to the field and motivates the importance of the investigations
performed for this dissertation.
1.3.1 Early Work
Ferromagnetic shape memory effect occurs in various alloys such as nickel-manganese-
gallium (Ni-Mn-Ga), iron-palladium (Fe-Pd), and cobalt-nickel-aluminum (Co-Ni-
Al). The problem of slow thermally-induced phase transformation response exhibited
by the nickel-titanium (Ni-Ti) alloys has been addressed with the discovery of ferro-
magnetic shape memory alloys. Of these, Ni-Mn-Ga is the most commonly studied
FSMA, which is also commercially available [1].
The first report of the significant magnetic field induced strain in Heusler type
non-stoichiometric Ni2MnGa alloys was presented in 1996 by Ullakko et al. [128]. This
phenomenon was further validated through a series of publications by Ullakko [126,
127, 129]. The experimental results for unstressed crystals of Ni2MnGa at 77 K
showed strains of 0.2% under a 8 kOe magnetic field. This original data is repro-
duced in Figure 1.6. The tests are conducted with two directions of applied field,
namely along [001] and [110] direction with respect to the bcc parent phase. The
strain is measured in the direction along the field and perpendicular to it. In the ini-
tial years of research, the magnetic field induced strain was assumed because of the
magnetostriction, and was reported as λs = 133× 10−6, with e|| − e⊥ = 0.20× 10−3.
Experimental advancement continued with testing of off-stoichiometric Ni-Mn-
Ga that demonstrated larger strains at higher temperatures. Tickle and James pre-
sented several results in their publications [123, 124, 57]. The measurements on
15
Figure 1.6: (a) Relative orientation of sample, strain gauge, and applied field formeasurements shown in (b) and (c). (b) Strain vs applied field in the L21 (austenite)phase at 283 K. (c) Same as (b) but data taken at 265 K in the martensitic phase [128].
16
Ni51.3Mn24.0Ga24.7 at -15C exposed to fields of less than 10 kOe were presented, which
showed strains of up to 0.2% due to cyclic application of an axial magnetic field and
strains of 1.3% when fields were applied transverse to the sample that started from
a stress biased state. This finding shifted the focus of Ni-Mn-Ga research towards
the orthogonal stress-field orientation. The transverse field tends to oppose the ef-
fect of the collinear compressive stress, and therefore this configuration provides the
opportunity to obtain maximum possible strain.
Further work was focused on compositional dependence on the strain generation
ability. Murray et al. [88] reported compositional and temperature dependence on the
performance of polycrystalline Ni-Mn-Ga alloys. Jin et al. [60] studied the empirical
mapping of Ni-Mn-Ga properties with composition and valence electron concentra-
tion. A range between Ni52.5Mn24.0Ga23.5 and Ni49.4Mn29.2Ga21.4 was identified, in
which the martensitic transformation temperature, Tm, is higher than room temper-
ature and lower than the Curie temperature, Tc, and the saturation magnetization
is larger than 60 emu/g. These conditions are suggested as optimum for creating
samples with the best capability for large, room temperature strains.
Large strains of 6% in Ni-Mn-Ga single crystals were reported in numerous pub-
lications by Murray et al. [89, 90, 88], Heczko et al. [47], and Likhachev [79]. The
alloys used in these measurements consisted of tetragonal martensite structure with a
five-layer (5M) shuffle type modulation. Strains of 9.5% were reported by Sozinov et
al. [114] having seven-layer (7M) modulation, which is the most promising result in
Heusler type of ferromagnetic shape memory alloys of the family Ni2+x+yMn1−xGa1−y.
17
1.3.2 Properties and Crystal Structure
Currently, the ferromagnetic shape memory alloys are grown by conventional single
crystal growth techniques such as Bridgman [113]. After producing the single crystal
bars, the materials are homogenized at about 1000C for 24 hours and ordered at
800C for another 20 hours. The material is then oriented using X-ray techniques
to produce the desired crystallographic structure for the MSM effect. Following the
crystal orientation, the material is cut and thermomechanically treated. The key to
obtaining high strains is to cut the samples so that the twin boundaries are aligned
at 45 to the sample axis (when magnetic field is applied transverse to the bar).
Ni2MnGa is an intermetallic compound that exhibits Heusler Structure. At high
temperatures, it exhibits cubic austenite (L21, Fm3m) structure as shown in Fig-
ure 1.7(a) [98, 29]. Ni-Mn-Ga exhibits a paramagnetic/ferromagnetic transition with
a Curie temperature of about 373 K. When cooled below the Curie temperature, the
material undergoes a phase change to a martensite, tetragonal (l4/mmm) structure
as shown in Figure 1.7(b). The unique c-axis of the tetragonal unit cell is shorter
than the a-axis, c/a < 1 [98]. Therefore, the theoretical maximum strain can be given
as,
εmax = 1− c/a (1.1)
Most commonly observed value of the c/a ratio is 0.94, and therefore a strain of
around 6% is typically observed.
The self accommodating twin-variant martensite structure is similar to the marten-
site structure in SMAs. Because of the tetragonal nature of the martensitic phase,
three twin orientations are possible of which two are identical relative to the axis of
the sample. The variants with their c-axis aligned with the sample axis are referred
Here, B1 represents the magnetoelastic coupling coefficient [93] obtained by measuring
the maximum stress generated when the sample is biased by 5.5% (when ξ = 0), and εy
represents the magnetostrictive strain in the y direction. The first term represents the
magnetoelastic energy contribution due to magnetic fields, which contributes only in
the stress preferred variant (1-ξ). The second term represents the energy contribution
due to the initial compressive stress σ0. The applied field leads to increase of energy
in stress preferred variants, whereas the stress leads to increase of energy in field
preferred variants. The stress generated due to magnetoelastic coupling thus has the
104
form
σme = [B1(1− ξ) + σ0ξ](− sin2 θ). (3.69)
The magnetoelastic energy is not considered while evaluating the domain fraction
and magnetization rotation angle because it is around 1000 times smaller than the
Zeeman, magnetostatic, and anisotropy energies. On the other hand, the magnetoe-
lastic energy becomes significant as it is the sole source of stress generation when
field-induced deformations are prevented.
3.8.1 Results of Blocked-Force Behavior
Figure 3.17 shows experimental and calculated stress vs. applied field curves
at varied bias strains. Hysteresis is not included in the model. The significance of
magnetoelastic coupling is evident as the stress starts increasing as soon as the field is
applied, with the rotation of magnetization vectors. The increase in stress is directly
related to the angle of rotation (θ) predicted by the magnetization model. On the
contrary, the variant reorientation process is typically associated with a high amount
of coercive field that increases with increasing bias stress [67, 101]. The absence of
a coercive field, and of discontinuity in stress profiles, confirms the magnetoelastic
coupling rather than twin reorientation as origin of the stress.
Figure 3.18 shows the magnetization dependence on applied field at varied blocked
strains. The negligible hysteresis is typical of single crystal Ni-Mn-Ga when the
volume fraction is approximately constant. Thus, the model assumption of reversible
evolution of α and θ is validated along with the assumption of constant volume
fraction. This is in contrast to Figure 3.16,where the hysteresis occurs in concert with
twin variant rearrangement. The initial susceptibility of Ni-Mn-Ga varies significantly
105
−800 −600 −400 −200 0 200 400 600 8000.5
1
1.5
2
2.5
3
3.5
Applied field
Str
ess
(MP
a)
1 %
2 %
3 %
4 %
5 %Bias strain (%)
Figure 3.17: Stress vs field at varied blocked strains. Dotted: experiment; solid line:model.
with bias strains, as the M −H curve shifts between the two extreme cases of easy
axis and hard axis curves. A 59% change in susceptibility is observed over a range of
4% change in strain experimentally. Figure 3.19 shows the variation of susceptibility
with varied blocked strains. The model parameters are: E0 = 125 MPa, E1 = 2000
MPa, σtw0 = 1 MPa, k = 16 MPa, ε0 = 0.055, Ku = 2.2E5 J/m3, Ms = 700 kA/m,
N = 0.2. Magnetoelastic coefficient B1 is the maximum stress produced with 5.5%
blocked strain, which is 1 MPa.
Our mechanically-blocked measurements and thermodynamic model for constant
volume fraction describe the stress and magnetization dependence on field, and pro-
vide a measure of the work capacity of Ni-Mn-Ga. The work capacity, defined as the
106
−200 0 200 400 600 800−200
0
200
400
600
800
Applied Field (kA/m)
Mag
netiz
atio
n (k
A/m
)
Bias strain (%)
1 %
2 %
3 %
4 %
5 %
Figure 3.18: Magnetization vs field at varied blocked strains. Dashed line: experi-ment; solid line: model.
1 2 3 4 50
1
2
3
4
5
6
Applied Field (kA/m)
Initi
al s
usce
ptib
ility
Model resultExperimental values
Figure 3.19: Variation of initial susceptibility with biased blocked strain.
107
area under the σbl − σ0 curve, is 72.4 kJ/m3 for this material. This value compares
favorably with that of Terfenol-D and PZT (18-73 kJ/m3 [40]). However, the work
capacity of Ni-Mn-Ga is strongly biased towards high deformations at the expense of
low generated forces, which severely limits the actuation authority of the material.
Terfenol-D exhibits a measured stress of 8.05 MPa at a field of 25 kA/m and prestress
of −6.9 MPa [21]. The lower blocking stress of 1.47 MPa produced by Ni-Mn-Ga is
attributed to a low magnetoelastic coupling.
The maximum available blocking stress is observed at a bias strain of 3%, though
the maximum blocking stress occurs, as expected, when the sample is completely
prevented from deforming. Due to the competing effect of the stress-preferred and
field-preferred variants, the stress is highest when the volume fractions are approxi-
mately equal (ξ = 0.5) as seen in Figure 3.20.
The magnetoelastic energy in Ni-Mn-Ga is considerably smaller than the Zeeman,
magnetostatic, and anisotropy energies. The magnetostrictive strains in Ni-Mn-Ga
are of the order of 50-300 ppm [44, 123], which are negligible when compared to the
typical 6% deformation that occurs by twin-variant reorientation. The contribution
of magnetoelastic coupling can thus be ignored when describing the sensing and actu-
ation behaviors in which the material deforms by several percent strain. In the special
case of field application in mechanically-blocked condition, twin-variant reorientation
is completely suppressed and the magnetoelastic coupling becomes significant as it
remains the only source of stress generation. This is validated from the experimental
stress data as there is no coercive field associated with the twin-variant rearrange-
ment. In summary, the magnetoelastic coupling in Ni-Mn-Ga is relatively low but
becomes significant when the material is prevented from deforming.
108
−101234560
1
2
3
4
5
Bias Strain (%)
Str
ess
(M
Pa
)
σbl
σ0
σbl
−σ0
Figure 3.20: Experimental blocking stress σbl, minimum stress σ0, and availableblocking stress σbl − σ0 vs. bias strain.
3.9 Discussion
A unified magnetomechanical model based on the continuum thermodynamics
approach is presented to describe the sensing [101], actuation [103] and blocked-
force [108] behaviors of ferromagnetic shape memory Ni-Mn-Ga. The model requires
only seven parameters which are identified from two simple experiments: stress-strain
plot at zero magnetic field, and easy-axis and hard-axis magnetization curves. The
model parameter B1 is incorporated to describe the blocked-force behavior. The
model is low-order, with up to quadratic terms, which makes it convenient from the
viewpoint of FEA implementation, and incorporation in the structural dynamics of
109
a system. The model spans three magneto-mechanical characterization spaces, de-
scribing the interdependence of strain, stress, field, and magnetization. The model
accurately quantifies the dependent variables over large ranges of the bias indepen-
dent variable, which is rarely seen in literature. The magnetic Gibbs energy is the
thermodynamic potential for sensing and blocked force models, whereas the Gibbs
energy is the thermodynamic potential for actuation effect.
Several important characteristics are investigated in concert with magnetomechan-
ical characterization of single crystal Ni-Mn-Ga, along with the model predictions.
The flux density sensitivity with strain
(∂B
∂ε
)varies from 0 to a maximum value of
4.19 T/%ε at bias field of 173 kA/m, and has maximum reversible value of 2.38 T/%ε
at bias field of 368 kA/m (Figure 3.8). The stress induced due to magnetic field has
a theoretical maximum value of 2.84 MPa (Figure 3.6). The maximum field in-
duced strain has maximum reversible value of 5.8% at bias stresses of 0.89 MPa and
1.16 MPa, which are optimum for actuation (Figure 3.14). The initial susceptibility(∂M
∂H|H=0
)changes by 59% over a range of 4% strain (Figure 3.19) when mechani-
cally blocked. The maximum stress generation capacity is 1.47% at 3% strain, which
is 37% higher than that at the end values of blocked strain (Figure 3.20). These
parameters provide key insight into the magnetomechanical coupling of Ni-Mn-Ga.
Although the emphasis of the work is on a specific material-single crystal Ni-
Mn-Ga, the developed model can be applicable to any class of ferromagnetic shape
memory materials. With recent advances in increased blocking stress [61], FSMAs
are a promising new class of multi-functional smart materials. Modeling polycrys-
talline behavior is one of the future opportunities which could be explored based on
the results of this research. Possible future work could also involve extending the
110
model framework for 3-D case which will enable design of structures that incorpo-
rate FSMAs. Constitutive 3-D models will also facilitate implementation of finite
element analysis of structures to solve various magnetomechanical boundary value
problems. Several aspects of this model are also applicable to the dynamic behavior
of Ni-Mn-Ga, some of which is discussed in subsequent chapters.
111
CHAPTER 4
DYNAMIC ACTUATOR MODEL FOR FREQUENCYDEPENDENT STRAIN-FIELD HYSTERESIS
In this chapter, a model is developed to describe the relationship between mag-
netic field and strain in dynamic Ni-Mn-Ga actuators. Due to magnetic field diffusion
and structural actuator dynamics, the strain-field relationship changes significantly
relative to the quasistatic response as the magnetic field frequency is increased. The
magnitude and phase of the magnetic field inside the sample are modeled as a 1-
D magnetic diffusion problem with applied dynamic fields known on the surface of
the sample, from where an averaged or effective field is calculated. The continuum
thermodynamics constitutive model described in Chapter 3 is used to quantify the
hysteretic response of the martensite volume fraction due to this effective magnetic
field. It is postulated that the evolution of volume fractions with effective field ex-
hibits a zero-order response. To quantify the dynamic strain output, the actuator
is represented as a lumped-parameter, single-degree-of-freedom resonator with force
input dictated by the twin-variant volume fraction. This results in a second order,
linear ODE whose periodic force input is expressed as a summation of Fourier series
terms. The total dynamic strain output is obtained by superposition of strain solu-
tions due to each harmonic force input. The model accurately describes experimental
112
measurements at frequencies of up to 250 Hz. The application of this new approach is
also demonstrated for a dynamic magnetostrictive actuator to show the wider impact
of the presented work on the area of smart materials.
4.1 Introduction
As seen in the literature review (Chapter 1), most of the prior experimental and
modeling work on Ni-Mn-Ga is focused on the quasistatic actuation, i.e., dependence
of strain on magnetic field at low frequencies [65, 113]. Achieving the high saturation
fields of Ni-Mn-Ga (around 400 kA/m) requires large electromagnet coils with high
electrical inductance, which limits the effective spectral bandwidth of the material.
For this reason, perhaps, the dynamic characterization and modeling of Ni-Mn-Ga
has received limited attention.
Henry [48] presented measurements of magnetic field induced strains for drive
frequencies of up to 250 Hz and a linear model which describes the phase lag between
strain and field and system resonance frequencies. Peterson [97] presented dynamic
actuation measurements on piezoelectrically assisted twin boundary motion in Ni-Mn-
Ga. The acoustic stress waves produced by a piezoelectric actuator complement the
externally applied fields and allow for reduced field strengths. Scoby and Chen [111]
presented a preliminary magnetic diffusion model for cylindrical Ni-Mn-Ga material
with the field applied along the long axis, but they did not quantify the dynamic
strain response.
The modeling of dynamic piezoelectric or magnetostrictive transducers usually
requires the structural dynamics of the device to be coupled with the externally ap-
plied electric or magnetic fields through the active element’s strain. This is often
113
done by considering a spring-mass-damper resonator subjected to a forcing function
given by the product of the elastic modulus of the material, its cross-sectional area,
and the active strain due to electric or magnetic fields. The active strain is related
to the field by constitutive relations which can be linearized, without significant loss
of accuracy, when a suitable bias field is present [26]. The actuation response of Ni-
Mn-Ga is dictated by the rearrangement of martensite twin variants, which are either
field-preferred or stress-preferred depending on whether the magnetically easy crystal
axis is aligned with the field or the stress. The rearrangement and evolution of twin
variants with a.c. magnetic fields always exhibit large hysteresis, hence the consti-
tutive strain-field relation of Ni-Mn-Ga cannot be accurately quantified by linearized
models.
This chapter presents a new approach to quantifying the hysteretic relationship
between magnetic fields and strains in dynamic actuators consisting of a Ni-Mn-Ga
element, return spring, and external mechanical load. The key contribution of this
work is the modeling of coupled structural and magnetic dynamics in Ni-Mn-Ga ac-
tuators by means of a simple (yet accurate) framework. The framework constitutes a
useful tool for the design of actuators with straightforward geometries and provides
a set of core equations for finite element solvers applicable to more complex geome-
tries. Further, it offers the possibility of obtaining input field profiles that produce a
prescribed strain profile, which can be a useful tool in actuator control.
The model is focused on describing properties of measured Ni-Mn-Ga data [48]
observed as the frequency of the applied magnetic field is increased, as follows: (1) For
a given a.c. voltage magnitude, the maximum current and associated maximum ap-
plied field decrease due to an increase in the impedance of the coils; (2) The field at
114
zero strain (i.e., field required to change the sign of the deformation rate) increases
over a defined frequency range, indicating an increasing phase lag of the strain rela-
tive to the applied field; and (3) For a given applied field magnitude, the maximum
strain magnitude decreases and the shape of the hysteresis loop changes significantly.
It is proposed that overdamped second-order structural dynamics and magnetic field
diffusion due to eddy currents are the primary causes for the observed behaviors. The
two effects are coupled: eddy currents reduce the magnitude and delays the phase of
the magnetic field towards the center of the material, which in turn affects the corre-
sponding strain response through the structural dynamics. Magnetization dynamics
and twin boundary motion response times are considered relatively insignificant.
The model is constructed as illustrated in Figure 4.1. First, the magnitude and
phase of the magnetic field inside a prismatic Ni-Mn-Ga sample are modeled as a
1-D magnetic diffusion problem with applied a.c. fields known on the surface of the
sample. In order to calculate the bulk magnetic field-induced deformation, an effec-
tive or average magnetic field acting on the material is calculated. With this effective
field, a previous continuum thermodynamics constitutive model described in Chapter
3 [99, 101, 103], is used to quantify the hysteretic response of the martensite volume
fraction. The evolution of the volume fraction defines an equivalent forcing function
dependent on the elastic modulus of the Ni-Mn-Ga sample, its cross-sectional area,
and the maximum reorientation strain. Assuming steady-state excitation, this forcing
function is periodic and can be expressed as a Fourier series. This Fourier series pro-
vides the force excitation to a lumped-parameter, single-degree-of-freedom resonator
representing the Ni-Mn-Ga actuator. The dynamic strain response is obtained by
superposition of the strain response to forces of different frequencies.
115
Input field Diffusion
(Eddy currents)
Constitutive
model
Fourier series expansion
Structural
dynamics Dynamic strain
Figure 4.1: Flow chart for modeling of dynamic Ni-Mn-Ga actuators.
For model validation, dynamic measurements presented by Henry [48] are utilized.
A 10×10×20 mm3 single crystal Ni-Mn-Ga sample was placed between the poles
of an E-shaped electromagnet with the 10×20 mm2 sides facing the magnet poles.
The magnetic field was applied perpendicular to the longitudinal axis of the sample,
which tends to elongate it. A spring of stiffness 36 kN/m provided a compressive bias
stress of 1.7 MPa along the longitudinal axis of the sample to achieve reversible field-
induced actuation in response to cyclic fields. Figure 4.2 shows dynamic actuation
measurements. The strain response of Ni-Mn-Ga depends on the magnitude of the
applied field but not on its direction, thus giving two strain cycles per field cycle. The
frequencies shown in Figure 4.2 are the inverse of the time period of one strain cycle.
Thus, the frequency of applied field ranges from 1-250 Hz. It is also noted that the
applied field amplitude decays with increasing frequency, likely due to a combination
of high electromagnet inductance and the measurements having been conducted at
constant voltage rather than at constant current.
116
(a) (b)
Figure 4.2: Dynamic actuation data by Henry [48] for (a) 2−100 Hz (fa = 1−50 Hz)and (b) 100− 500 Hz (fa = 50− 250 Hz).
Since the experimental magnetic field waveform is not described in [48], sinusoidal
and triangular waveforms are studied. It is proposed that the experimental field
waveform deviates from an exact waveform (sinusoidal or triangular) as the applied
field frequency increases. Nonetheless, study of these two ideal waveforms provides
insight on the physical experiments.
4.2 Magnetic Field Diffusion
The application of an alternating magnetic field to a conducting material results in
the generation of eddy currents and an internal magnetic field which partially offsets
the applied field. The relationship between the eddy currents and applied fields is
117
described by Maxwell’s electromagnetic equations,
∇×H = j +∂D
∂t,
∇× E = −∂B
∂t,
∇ ·B = 0,
∇ ·D = ρe,
(4.1)
with H the magnetic field strength (A/m), j the free current density (A/m2), D the
electric flux density (C/m2), E the electric field strength (V/m), B the magnetic flux
density (T), and ρe the volume density of free charge (C/m3). The corresponding
constitutive equations are given by
j = σE,
B = µH,
D = εE,
(4.2)
where σ is the conductivity, µ is the magnetic permeability, and ε is the dielectric
constant. In the case of a stationary conductor exposed to alternating magnetic fields,
combination of (4.1)a, (4.2)a, and (4.2)c gives an expression for the Ampere-Maxwell
circuital law,
∇×∇×H = ∇× (σE) +∂
∂t[∇× (εE)]. (4.3)
After mathematical manipulation, (4.3) yields a magnetic field diffusion equation
which describes the penetration of dynamic magnetic field in a conducting medium [69].
For one-dimensional geometries, assuming that the magnetization is uniform and does
not saturate, the diffusion equation has the form,
∇2H − µσ∂H
∂t= 0, (4.4)
118
where σ is the conductivity, µ is the magnetic permeability, and ε is the dielectric
constant. The assumption of uniform magnetization is not necessarily met experi-
mentally due to nonuniform twin boundary motion [91, 85] and saturation effects.
However, comparison of model results and measurements (Section 4.4) suggests that
the simplified diffusion model is able to describe the problem qualitatively. This is
attributed to the susceptibilities of field-preferred and stress-preferred variants being
relatively close (4.7 and 1.1, respectively [99]) and not differing too much from zero
as twin boundary motion and magnetization rotation processes take place. It is also
speculated that the variants are sufficiently fine in the tested material.
The solution to (4.4) gives the magnetic field values H(x, t) at position x (inside
a material of thickness 2d) and time t. The boundary condition at the two ends is
the externally applied magnetic field. In the case of harmonic fields, the boundary
condition is given by
H(±d, t) = H0eiωt, (4.5)
where H0 is the amplitude and ω = 2πfa is the circular frequency (rad/s) of the
magnetic field on the surface of the Ni-Mn-Ga sample. Assuming no leakage flux in
the gap between the electromagnet and sample, this field is the same as the applied
field. The solution for magnetic fields inside the material has the form [69]
H(x, t) = H0 h(X) eiωt. (4.6)
119
In this expression, the complex magnitude scale factor is
h(X) = A(B + iC),
A =1
cosh2 Xd cos2 Xd + sinh2 Xd sin2 Xd
,
B = cosh X cos X cosh Xd cos Xd + sinh X sin X sinh Xd sin Xd,
C = sinh X sin X cosh Xd cos Xd − cosh X cos X sinh Xd sin Xd,
(4.7)
with
X =x
δ, Xd =
d
δ, δ =
√2
ωµσ, (4.8)
where δ is the skin depth, or the distance inside the material at which the diffused
field is 1/e times the external field. If the external field is an arbitrary periodic
function, the corresponding boundary condition is represented as a Fourier series
expansion. The diffused internal field is then obtained by superposition of individual
solutions (4.6) to each harmonic component of the applied field. Figure 4.3 shows the
variation of the internal magnetic field at different depths inside the sample. As the
depth increases, the amplitude of the magnetic field decays, accompanied by a phase
delay. For the case of triangular input fields, the amplitude decay and phase change
is accompanied by a shape change in the waveform.
4.2.1 Diffused Average Field
In order to model the bulk material behavior, an effective field acting on the mate-
rial needs to be obtained. This effective field can be used along with the constitutive
model to get the corresponding volume fraction response. To estimate the effective
magnetic field, an average of the field waveforms at various positions is calculated,
Havg(t) =1
Nx
Xd∑
X=−Xd
H(x, t). (4.9)
120
0 0.2 0.4 0.6 0.8 1−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Nondimensional time (t*fa)
Nor
mal
ized
Fie
ld (
H/H
0)
d3d/4d/2d/40
(a)
0 0.2 0.4 0.6 0.8 1−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Non−dimensional time (t*fa)
Nor
mal
ized
fiel
d (H
/H0)
d3d/4d/2d/40
(b)
Figure 4.3: Magnetic field variation inside the sample at varied depths for (a) sinu-soidal input and (b) triangular input. x = d represents the edge of the sample, x = 0represents the center.
121
Here, Nx represents the number of uniformly spaced points inside the material where
the field waveforms are calculated.
Figure 4.4 shows averaged field waveforms at several applied field frequencies for
sinusoidal and triangular inputs. In these simulations the resistivity has a value of
ρ = 1/σ = 6e-8 Ohm-m and the relative permeability is µr = 3. At 1 Hz, the magnetic
field intensity is uniform throughout the material and equal to the applied field H0,
and there is no phase lag. With increasing actuation frequency, the magnetic field
diffusion results in a decrease in the amplitude and an increase in the phase lag of the
averaged field relative to the field on the surface of the material. Figure 4.5 shows
the decay of the magnetic field amplitude with position inside the material at several
applied field frequencies.
When the applied field is sinusoidal, the diffused average field is also sinusoidal
regardless of frequency (Figure 4.4a). When the applied field is triangular, the shape
of the diffused average field increasingly differs from the input field as the frequency is
increased (Figure 4.4b). The corresponding strain waveforms are modified accordingly
as they are dictated by the material response to the effective averaged field. Thus, the
shape of the input field waveform can alter the final strain profile. This is discussed
in Section 4.4.
4.3 Quasistatic Strain-Field Hysteresis Model
To quantify the constitutive material response, the constitutive magnetomechani-
cal model for twin variant rearrangement is used, which is detailed in Chapter 3. The
model incorporates thermodynamic potentials to define reversible processes in combi-
nation with evolution equations for internal state variables associated with dissipative
122
0 0.2 0.4 0.6 0.8 1−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Non−dimensional time (t*fa)
Nor
mal
ized
fiel
d (H
avg/H
0)
1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz
(a)
0 0.2 0.4 0.6 0.8 1−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Non−dimensional time (t*fa)
Nor
mal
ized
fiel
d (H
avg/H
0)
1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz
(b)
Figure 4.4: Average field waveforms with increasing actuation frequency for (a) sinu-soidal input and (b) triangular input.
123
−5 −4 −3 −2 −1 0 1 2 3 4 50.7
0.75
0.8
0.85
0.9
0.95
1
Position (mm)
Max
imum
Nor
mal
ized
Fie
ld
1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz
(a)
−5 −4 −3 −2 −1 0 1 2 3 4 50.5
0.6
0.7
0.8
0.9
1
Position (mm)
Max
imum
Nor
mal
ized
Fie
ld
1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz
(b)
Figure 4.5: Dependence of normalized field amplitude on position with increasingactuation frequency for (a) sinusoidal input and (b) triangular input.
124
effects. The model naturally quantifies the actuation or sensing effects depending on
which variable pairs among stress, strain, magnetic field, and magnetization, are se-
lected as independent and dependent variables. For the actuation problem under
consideration, the average or effective field Havg (for simplicity denoted H from now
on) and bias compressive stress σb are the independent variables, and the strain ε
and magnetization M are the dependent variables. The constitutive actuation model
described in Section 3.7 gives the variation of the volume fraction ξ and total strain ε
on field H.
Overall model procedure remains the same as detailed earlier. A few minor changes
are made to the model to account for different initial conditions. Experimental data
collected by Henry [48] is used to validate the model results. In these measurements,
the sample is not converted to a complete stress-preferred state before the application
of field. The sample is first converted to a complete field-preferred state and is
then subjected to the given bias stress. The configuration of the sample before the
application of the field thus consists of a twin-variant structure dictated by the bias
stress. This situation is modeled by introducing a new model variable, the initial
volume fraction ξs, which represents the fraction of field preferred variants before
the application of field and after the application of the bias stress. Therefore, the
definition of the twinning strain with respect to the initial configuration and the
expression for mechanical Gibbs energy is modified. The expression for mechanical
energy is different during the forward ( ˙|H| ≥ 0) and reverse ( ˙|H| ≤ 0) application of
field.
ρφmech = − 1
2Eσ2
b +1
2aε2
0(ξ − ξs) ( ˙|H| > 0),
ρφmech = − 1
2Eσ2
b +1
2aε2
0(ξ − ξf + ξs) ( ˙|H| < 0),
(4.10)
125
The volume fraction obtained using the procedure detailed in Section 3.7. It is given
by,
ξ =πξ
mag + σbε0 + aε20ξs − πcr
aε20
( ˙|H| ≥ 0),
ξ =πξ
mag + σbε0 + aε20ξf − aε2
0ξs + πcr
aε20
( ˙|H| ≤ 0),
(4.11)
All the variables in equations (4.10) and (4.11) are defined in Chapter 3, with
the exception of ξs which is the initial volume fraction. Total strain is given by the
summation of the elastic and the twinning component as,
ε = εe + εtw = εe + ε0ξ. (4.12)
Figure 4.6 shows a comparison of model results with actuation data for a 1 Hz
applied field. The model parameters used are: ε0 = 0.04, k = 70 MPa, Ms = 0.8 T, Ku
= 1.7 J/m3, and σtw0 = 0.5 MPa. The hysteresis loop in Figure 4.6 is dominated by the
twinning strain ε0ξ (proportional to volume fraction), which represents around 99%
of the total strain. The variation of volume fraction with effective field is proposed to
exhibit a zero-order response, without any dynamics of its own, and thus independent
of the frequency of actuation. The second order structural dynamics associated with
the transducer vibrations modify the constitutive behavior shown in Figure 4.6 in the
manner detailed in Section 4.4.
4.4 Dynamic Actuator Model
The average field Havg (denoted H for simplicity) acting on the Ni-Mn-Ga sample
is calculated by applying expression (5.4) to a given input field waveform. Using
this effective field, the actuator model discussed in Section 4.3 is used to calculate
the field-preferred martensite volume fraction ξ. By ignoring the dynamics of twin
126
0 100 200 300 400 500 6000
0.5
1
1.5
2
2.5
3
Applied Field (kA/m)
Str
ain
(%)
Figure 4.6: Model result for quasistatic strain vs. magnetic field. The circles denoteexperimental data points (1 Hz line in Figure 4.2) while the solid and dashed lines
denote model simulations for ˙|H| > 0 and ˙|H| < 0, respectively.
127
boundary motion, the dependence of volume fraction on applied field given by rela-
tions (4.11) is that of a zero-order system (ξ = f [H(t)]). Marioni et. al. [86] studied
the actuation of Ni-Mn-Ga single crystal using magnetic field pulses lasting 620 µs.
It was observed that the full 6% magnetic field induced strain was obtained in less
than 250 µs implying that the studied Ni-Mn-Ga sample has a bandwidth of around
2000 Hz. As the frequencies encountered in the present work are below 250 Hz, one
can accurately assume that twin boundary motion, and hence the evolution of vol-
ume fractions, occurs in concert with the applied field according to the dynamics of
a zero-order system.
4.4.1 Discrete Actuator Model
The mechanical properties of a dynamic Ni-Mn-Ga actuator are illustrated in Fig-
ure 4.7. Although the position of twin boundaries in the crystal affects the inertial re-
sponse of the material [84], this effect is ignored with the assumption of a lumped mass
system. The actuator is modeled as a lumped-parameter, single-degree-of-freedom,
lumped-parameter resonator in which the Ni-Mn-Ga rod acts as an equivalent spring
of stiffness EA/L, with E the modulus, A the area, and L the length of the Ni-Mn-Ga
sample. This equivalent spring is in parallel with the load spring of stiffness ke, which
is also used to pre-compress the sample. The overall system damping is represented
by ce and the combined mass of the Ni-Mn-Ga sample and output pushrod are mod-
eled as a lumped mass me. When an external field Ha(t) is applied to the Ni-Mn-Ga
sample, an equivalent force F (t) is generated which drives the motion of mass me.
A similar approach to that used for the modeling of dynamic magnetostrictive
actuators is employed. The motion of mass m is represented by a second order
128
ce
E, A, L
F(t)
me
ke
H (t) = H0ejwt
x (t)
Mechanical load
Figure 4.7: Dynamic Ni-Mn-Ga actuator consisting of an active sample (spring) con-nected in mechanical parallel with an external spring and damper. The mass includesthe dynamic mass of the sample and the actuator’s output pushrod.
differential equation,
mex + cex + kex = F (t) = −σ(t)A, (4.13)
with x the displacement of mass m. An expression for the normal stress is obtained
from constitutive relation (4.12) as,
ε = εe + εtw =σ
E+ ε0ξ, (4.14)
σ = E(ε− ε0ξ) = E(x
L− ε0ξ). (4.15)
The bias strain resulting from initial and final volume fractions (ξs, ξf ) is compensated
for when plotting the total strain. Substitution of (4.15) into (4.13) gives
mex + cex + (ke +AE
L)x = AEε0ξ. (4.16)
129
Equation (4.16) represents a second-order dynamic system driven by the volume frac-
tion. The dependence of volume fraction on applied field given by relations (4.11) is
nonlinear and hysteretic, and follows the dynamics of a zero-order system, i.e., the
volume fraction does not depend on the frequency of the applied magnetic field. This
is in contrast to biased magnetostrictive actuators, in which the drive force can be
approximated by a linear function of the magnetic field since the amount of hysteresis
in minor magnetostriction loops often is significantly less than in Ni-Mn-Ga.
4.4.2 Fourier Series Expansion of Volume Fraction
For periodic applied fields, the volume fraction also follows a periodic waveform
and hence the properties of Fourier series are utilized to calculate model solutions.
Figure 4.8 shows the calculated variation of volume fraction with time for the cases
of sinusoidal and triangular external fields. The reconstructed waveforms shown in
the figure are discussed later.
Using a Fourier series expansion, the periodic volume fraction is represented as a
sum of sinusoidal functions with coefficients
Zk =1
Ta
∫ Ta
0
ξ(t)e−iωktdt, k = 0,±1,±2, ..., (4.17)
where Ta = 1/fa, with fa the fundamental frequency. The frequency spectrum of
the volume fraction thus consists of discrete components at the frequencies ±ωk, k =
0, 1, 2...; Zk is the complex Fourier coefficient corresponding to the kth harmonic.
Equation (4.17) yields a double sided discrete frequency spectrum consisting of fre-
quencies −fs/2...fs/2, where fs = 1/dt represents the sampling frequency which
depends on the time domain resolution dt of the signal. The double sided frequency
130
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2
2.5
3
Time (sec)
ξ ε
0 (
%)
Sin: orig
Sin: recon
Tri: orig
Tri: recon
Figure 4.8: Volume fraction profile vs. time (fa = 1 Hz).
spectrum is converted to a single sided spectrum through the relations
|Z0| = |Z0| (k = 0),
|Zk| = |Zk|+ |Z−k| = 2|Zk| (k > 0).(4.18)
The phase angles remain unchanged,
∠Zk = ∠Zk (k ≥ 0). (4.19)
The reconstructed volume fraction ξr(t) is
ξr(t) = ξr(t± Ta) =K∑
k=−K
Zkeiωkt, (4.20)
in which K represents the number of terms in the series. The single sided frequency
spectrum of the volume fraction is shown in Figure 4.9 for sinusoidal and triangular
applied field waveforms. This spectrum consists of frequencies 0...fs/2. It is noted
131
0 2 4 6 8 100
0.5
1
1.5
Frequency (Hz)|
ξ |ε
0 (%
)
0 2 4 6 8 10−200
0
200
Frequency (Hz)
Ang
( ξ
) (d
eg
)
Sinusoidal
Triangular
Figure 4.9: Single sided frequency spectrum of volume fraction (fa = 1 Hz).
that the plotted spectrum has a resolution df = fa/4, as four cycles of the applied field
are included. The actuation frequency in the presented case is fa = 1 Hz. For an input
field frequency of fa Hz, the volume fraction spectrum consists of non-zero components
at frequencies 2fa, 4fa, 6fa,... Hz. Mathematically, the phase angles appear to be
leading; the physically correct phase angle values are obtained by subtracting π from
the mathematical values.
Finally, if the applied field has the form
Ha(t) = H0 sin(2πfat), (4.21)
132
with H0 constant, then the reconstructed volume fraction ξr(t) is represented in terms
of the single sided Fourier coefficients by
ξr(t) = ξr(t± Ta)K∑
k=0
|Zk| cos(2πkfat + ∠Zk). (4.22)
The reconstructed volume fraction signal overlapped over the original is shown in
Figure 4.8, for both the sinusoidal and triangular input fields. The number of terms
used is K = 20. Substitution of (4.22) into (4.16) gives,
mex + cex + (ke +AE
L)x = AEε0
K∑
k=0
|Zk| cos(2πkfat + ∠Zk), (4.23)
which represents a second-order dynamic system subjected to simultaneous harmonic
forces at the frequencies kfa, k = 0, . . . , K. The steady state solution for the net
displacement x(t) is given by the superposition of steady state solutions to each
forcing function. Thus, the steady state solution for the dynamic strain εd has the
form
εd(t) =x(t)
L=
EAε0
EA + keL
K∑
k=0
|Zk||Xk| cos(2πkfat + ∠Zk − ∠Xk). (4.24)
In (4.45), Xk represents the non-dimensional transfer function relating the force at
the kth harmonic and the corresponding displacement,
Xk =1
[1− (kfa/fn)2] + j(2ζkfa/fn)= |Xk|e−i∠Xk , (4.25)
where
|Xk| = 1√[1− (kfa/fn)2]2 + (2ζkfa/fn)2
, (4.26)
∠Xk = tan−1
(2ζkfa/fn
1− (kfa/fn)2
). (4.27)
133
The natural frequency and damping ratio in these expressions have the form
fn =1
2π
√ke + AE/L
me
, (4.28)
ζ =ce
2√
(ke + AE/L)me
. (4.29)
4.4.3 Results of Dynamic Actuation Model
Figure 4.10 shows experimental and calculated strain versus field curves for sinu-
soidal and triangular waveforms at varied frequencies. The model parameters used
are fn = 700 Hz, ζ = 0.95, ρ = 62×10−8 Ohm-m, and µr = 3. The natural frequency
is obtained by using a modulus E=166 MPa, which is estimated from the stress-
strain plots in [48]. The dynamic mass of the Ni-Mn-Ga sample and pushrods is
me=0.027 kg. It is seen that the assumption of triangular input field waveform tends
to model the higher frequency data well. This implies that the shape of the applied
field waveform may not remain exactly sinusoidal at higher frequencies. For example,
the experimental data at 250 Hz shows a slight discontinuity when the applied field
changes direction, thus verifying the proposed claim of triangular shape.
The model results match the experimental data well with the assumption of tri-
angular input field waveform, except for the case of 200 Hz. Otherwise, the model
accurately describes the increase of coercive field, the magnitude of maximum strain,
and the overall shape change of the hysteresis loop with increasing actuation fre-
quency. The lack of overshoot in the experimental data for any of the frequencies
justifies the assumption of overdamped system. The average error between the ex-
perimental data and the model results is 2.37%, which increases to 4.24% in the case
of fa = 200 Hz. The relationship between strain and field is strongly nonlinear and
134
0 100 200 300 400 500 6000
0.5
1
1.5
2
2.5
3
Applied Field (kA/m)
Str
ain
(%)
250 Hz
1 Hz
100 Hz
50 Hz
150 Hz
200 Hz
175 Hz
(a)
0 100 200 300 400 500 6000
0.5
1
1.5
2
2.5
3
Applied Field (kA/m)
Str
ain
(%)
250 Hz
1 Hz
100 Hz
50 Hz
150 Hz
200 Hz
175 Hz
(b)
Figure 4.10: Model results for strain vs. applied field at different frequencies for(a) sinusoidal, (b) triangular input waveforms. Dotted line: experimental, solid line:model.
135
hysteretic due to factors such as magnetic field diffusion, constitutive coupling, and
structural dynamics.
Maximum strain and hysteresis loop area
The maximum strain generated at a given frequency is of interest to understand
the dynamic properties of the system. It is observed that the applied field magnitude
decreases with increasing frequency, because the electromagnet inductance increases
with increasing frequency. As the applied field decreases, the field induced strain de-
creases too. The decay in the strain is therefore caused by the dynamics of the system
as well as the decreasing field magnitudes. Therefore, the comparison of maximum
strain at various frequencies is not useful for the available experimental data. Never-
theless, an attempt is made to understand the system behavior and gauge the model
performance by dividing the maximum strain at a given frequency by the applied
field amplitude at that frequency. Figure 4.11(b) shows variation of the normalized
maximum strain with frequency and its comparison with model calculations. The
normalized maximum strain reaches a peak at 175 Hz. However, this behavior should
not be confused with resonance, because the system is hysteretic. At 175 Hz, due to
the inductive losses, the applied field amplitude is reduced. However, this amplitude
is just sufficient to saturate the sample. Further increase in the applied field results
in negligible increase in the strain, as seen at frequencies lower than 175 Hz. There-
fore, the ratio of maximum strain over the field amplitude is maximum at 175 Hz. A
similar trend is observed in the hysteresis loop area enclosed by the strain-field curve
in half-cycle (H ≥ 0) as shown in Figure 4.11(a).
Assumption of the triangular input field waveform matches the experimental val-
ues better than assumption of the sinusoidal field. This indicates that the applied
136
0 50 100 150 200 2502
3
4
5
6
7
8x 10
−6
Actuation frequency (Hz)
Max
imum
Str
ain
/ H0
ExperimentalModel: SineModel:Triangular
(a)
0 50 100 150 200 2500
20
40
60
80
100
120
140
Actuation frequency (Hz)
Enc
lose
d A
rea×
10−
2 (kA
/m)
ExperimentalModel: SineModel: Triangular
(b)
Figure 4.11: (a) Normalized maximum strain vs. Frequency (b) Hysteresis loop areavs. Frequency
137
field waveform was either close to the triangular, or the sinusoidal waveform was dis-
torted due to the eddy current losses in the electromagnet cores. Further discussion
on the experimental data of maximum strain is given in Section 4.4.4.
4.4.4 Frequency Domain Analysis
Figure 4.12 shows a comparison of model calculations and experimental data in the
frequency domain. Only the results for triangular input field waveform are shown,
as the actual input field is proposed to be close to the triangular function from
the simulations. The frequency spectrum of the experimental strain data shows a
monotonous decay of strain magnitudes with increasing even harmonics up to an
actuation frequency of 100 Hz. For actuation frequencies from 150 Hz onwards, the
decay is not monotonous, for example, the strain magnitudes corresponding to the
4th and 6th harmonic are almost equal, with the magnitude corresponding to the 2nd
harmonic being comparatively high. This behavior is reflected in the strain-field plots
as the hysteresis loop shows increasing rounding-off for frequencies higher than 150
Hz. The model accurately describes these responses as the magnitudes match the
experimental values well for most cases. The phase angles for the experimental and
model spectra also show a good match. In some cases, the angles show a discrepancy
of about 180, though they are physically equivalent.
Figure 4.13(a) shows ‘order domain’, or ‘non-dimensional frequency domain’ spec-
trum of Fourier magnitudes of the experimental strain signal. The magnitudes cor-
responding to the zero frequency represent the average strain value in a cycle. The
variation of higher orders with actuation frequency is of interest. Though the spec-
trum under study is discrete, continuous curves are shown in Figure 4.13(a) to better
138
0 2 4 6 8 100
0.51
1.52
Non−dimensional frequency (f/fa)
|ε| (
%)
0 2 4 6 8 100
25
100
150175
Non−dimensional frequency (f/fa)
Ang
le(ε
) (d
eg)
(a)
0 2 4 6 8 100
1
Non−dimensional frequency (f/fa)
|ε| (
%)
0 2 4 6 8 100
50100150200
Non−dimensional frequency (f/fa)
Ang
le(ε
) (d
eg)
(b)
0 2 4 6 8 100
1
Non−dimensional frequency (f/fa)
|ε| (
%)
0 2 4 6 8 100
255075
100125
Non−dimensional frequency (f/fa)
Ang
le(ε
) (d
eg)
(c)
0 2 4 6 8 100
1
Non−dimensional frequency (f/fa)
|ε| (
%)
0 2 4 6 8 100
255075
100125
Non−dimensional frequency (f/fa)
Ang
le(ε
) (d
eg)
(d)
0 2 4 6 8 100
0.5
1
Non−dimensional frequency (f/fa)
|ε| (
%)
0 2 4 6 8 100
255075
100125
Non−dimensional frequency (f/fa)
Ang
le(ε
) (d
eg)
(e)
0 2 4 6 8 100
0.5
Non−dimensional frequency (f/fa)
|ε| (
%)
0 2 4 6 8 100
255075
100125
Non−dimensional frequency (f/fa)
Ang
le(ε
) (d
eg)
(f)
Figure 4.12: Model results for strain vs. applied field in frequency domain for trian-gular input waveform for (a) fa = 50 Hz, (b) fa = 100 Hz, (c) fa = 150 Hz, (d) fa
= 175 Hz, (e) fa = 200 Hz, (e) fa = 250 Hz. Dotted line: experimental, solid line:model.
139
visualize the trends of strain magnitudes. Figure 4.13(b) shows the variation of the
corresponding phase angles with harmonic order. The phase angle spectrum does
not differentiate the trends at various actuation frequencies as clearly as the mag-
nitude spectrum. Nevertheless, a correlation exists between Figure 4.13(a) and Fig-
ure 4.13(b). There is a trend of monotonic decrease at 1, 50, and 100 Hz. There
is a dip in the phase angle at 6th order for frequencies higher than 150 Hz, which is
associated with a rise in magnitude at 6th order in Figure 4.13(a).
The strain magnitudes decay almost linearly, in a monotonic fashion for actuation
frequencies up to 100 Hz. These characteristics indicate a blocky dependence in time
domain, similar to a rectified square wave signal (Figure 4.10). However, at higher
actuation frequencies, the magnitudes corresponding to the 6th order show a distinct
increase. This behavior can be attributed to the ’shape change’ of the strain-field
plots observed in Figure 4.10 at frequencies higher than 150 Hz. It is concluded that
the dynamic properties of the system show a distinct change at frequencies higher
than 150 Hz. A ’rounding off’ effect occurs in the strain-field relationship at the
higher drive frequencies.
The Fourier series magnitudes are plotted as a function of actuation frequency in
Figure 4.14(a). The variation of 2nd order with actuation frequency shows a distinct
peak at 175 Hz. For a linear system, it would have meant that the natural frequency
is near 350 Hz. However, no such conclusion can be reached for the hysteretic system
under consideration. Also, the decay of field with increasing frequencies complicates
a comparative study in the order domain.
The 6th order variation shows a peak at 150 Hz. The changes hysteresis loop shape,
and 6th order peaks associated with frequencies higher than 150 Hz may also be a
140
2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
Non−dimensional frequency (f/fa)
Str
ain
Mag
nitu
des
(%)
1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz
(a)
2 4 6 8 10−250
−200
−150
−100
−50
0
Non−dimensional frequency (f/fa)
Pha
se A
ngle
(de
g)
1 Hz50 Hz100 Hz150 Hz175 Hz200 Hz250 Hz
(b)
Figure 4.13: (a) Strain magnitude vs. harmonic order, (b) Phase angle vs. harmonicorder at varied actuation frequencies.
141
result of the decrease in the maximum applied field magnitude as seen in Figure 4.15.
If this reduced magnitude of field is applied for frequencies of 2, 50, and 100 Hz, then
the order domain spectrum at these frequencies may look similar to those for the
higher frequencies. The strain response is only dependent on the maximum applied
field, and the inertial effect of the system. However, the maximum applied field itself
depends on the inductive eddy current losses. Thus, strain response or strain order
spectrum depends on a number of different factors, which need to be analyzed in a
careful manner. The variation of the phase angles at a given order show a correlation
with Figure 4.14(b). The phase angle associated with 2nd order shows a dip at 175 Hz,
which is related the resonance of magnitude at the same frequency.
The maximum strain, maximum applied field, and their ratio is shown in Fig-
ure 4.15. The maximum applied field reduces after frequencies higher than 120 Hz.
The reason for the decay of maximum applied field is the increasing inductance of
the electromagnetic coil, and the eddy currents losses in the core. The maximum
strain also decreases with increasing frequency since its magnitude is directly related
to the maximum applied field. However, this relation is non-linear and hysteretic.
The strain to field ratio shows a clear jump at 175 Hz, which is strongly correlated
to the peak shown by the 2nd order harmonic in Figure 4.14(a). However, too much
importance should not be placed on the maximum strain to maximum field ratio as
the relationship is not linear, and this ratio can not be defined on the similar lines
as a transfer function. It is just a tool of measure for the particular case under
consideration.
142
0 50 100 150 200 2500
0.2
0.4
0.6
0.8
1
1.2
Actuation frequency (Hz)
Str
ain
Mag
nitu
des
(%)
2fa4fa6fa8fa10fa
(a)
0 50 100 150 200 250−200
−150
−100
−50
0
Actuation frequency (Hz)
Pha
se A
ngle
(de
g)
2fa4fa6fa8fa10fa
(b)
Figure 4.14: (a) Strain magnitude vs. actuation frequency, (b) Phase angle vs. actu-ation frequency at varied harmonic orders.
143
0 50 100 150 200 2500.4
0.5
0.6
0.7
0.8
0.9
1
Actuation frequency (Hz)
Nor
mal
ized
Str
ain
and
Fie
ld
εmax
Hmax
εmax
/Hmax
Figure 4.15: Variation of maximum strain and field with actuation frequency.
4.5 Conclusion
A model is presented to describe the dependence of strain on applied field at
varied frequencies in ferromagnetic shape memory Ni-Mn-Ga [107, 106]. The essential
components of the model include magnetomechanical constitutive responses, magnetic
field diffusion, and structural dynamics. The presented method can be extended to
arrive at the input field profiles which will result in the desired strain profile at a
given frequency. If the direction of flow in Figure 4.1 is reversed, the input field
profile can be designed from a desired strain profile. It is comparatively easy to
obtain the inverse Fourier transform, whereas calculation of the average field from a
144
desired strain profile through the constitutive model, and estimation of the external
field from the averaged diffused field inside the sample, can be complex.
The frequency spectra of the field-preferred volume fraction and the resulting
dynamic strain include even harmonics. The corresponding magnitudes at the 2nd
harmonic are comparatively high indicating frequency doubling similar to that asso-
ciated with magnetostrictive actuators. However, additional components at higher
harmonics are present due to the large hysteresis in FSMAs compared to biased
magnetostrictive materials. If the overall system including the active material is un-
derdamped, then it is possible to achieve system resonance at a frequency which is
1/4th or 1/6th of the system natural frequency. In magnetically-active material ac-
tuators, the application of magnetic fields at high frequencies becomes increasingly
difficult as the coil inductances tend to increase rapidly. If the actuator can be made
to resonate at a fraction of the system natural frequency, then this problem can be
simplified. However, the strain magnitudes corresponding to the higher harmonics
tend to diminish rapidly as well, which creates a compensating effect. Further, in
some cases the system natural frequency and damping may be beyond the control of
the designer. Nevertheless, our approach suggests a way to drive a magnetic actuator
at a fraction of the natural frequency to achieve resonance. A case study on a mag-
netostrictive actuator is presented in Section 4.6 to demonstrate the wide application
of this presented approach.
4.6 Dynamic Actuation Model for Magnetostrictive Materi-als
Magnetostrictive materials deform when exposed to magnetic fields and change
their magnetization state when stressed. These behaviors are nonlinear, hysteretic,
145
and frequency-dependent. Several models exist for describing the dependence of strain
on field at quasi-static frequencies. The strain-field behavior changes significantly rel-
ative to the quasi-static case as the frequency of applied field is increased. Modeling
the dynamic strain-field hysteresis has been a challenging problem because of the
inherent nonlinear and hysteretic behavior of the magnetostrictive material along
with the complexity of dynamic magnetic losses and structural vibrations of the
transducer device. Prior attempts use mathematical techniques such as the Preisach
model [121, 22, 4] and genetic algorithms [9]. A phenomenological approach including
eddy currents and structural dynamics was recently presented [54].
Chief intent of this section is to present a new approach for modeling the strain-
field hysteresis relationship of magnetostrictive materials driven with dynamic mag-
netic fields in actuator devices (Figure 4.1). The approach builds on the prior model
for dynamic hysteresis in ferromagnetic shape memory Ni-Mn-Ga [107] discussed ear-
lier in this chapter.
Magnetic Field Diffusion
As seen in Section 4.2, application of an alternating magnetic field to a conduct-
ing material such as magnetostrictive Terfenol-D results in the generation of eddy
currents and an internal magnetic field which partially offsets the applied field. The
relationship between the eddy currents and applied fields is described by Maxwell’s
electromagnetic equations. Assuming that the magnetization is uniform and does not
saturate, the diffusion equation describing the magnetic field inside a one-dimensional
conducting medium of cylindrical geometry has the form [69],
∂2H
∂r2+
1
r
∂H
∂r= µσ
∂H
∂t, (4.30)
146
where σ is electrical conductivity and µ is magnetic permeability. Cylindrical diffu-
sion equation is used because the typical geometry for magnetostrictive Terfenol-D
transducers is in the form of cylindrical rods.
For harmonic applied fields, the boundary condition at the edge (r = R) of the
cylindrical rod is given by,
H(R, t) = H0eiωt (4.31)
where H0 is the amplitude and ω = 2πfa is the circular frequency (rad/s) of the
magnetic field on the surface of the magnetostrictive material. The solution to (4.30)
gives the magnetic field values H(r, t) at radius r and time t. This solution is given
as,
H(x, t) = H0 h(R) eiωt. (4.32)
Therefore the diffusion equation (4.30) is transformed to,
d2h
dR2 +
1
R
dh
dR− h = 0, (4.33)
where the normalized complex and real radii are given as,
R =
√2ir
δ=
(1 + i)r
δ, Ra =
√2ia
δ
R =
√2r
δ, Ra =
√2a
δ, δ =
√2
ωµσ
(4.34)
This equation is solved by modified Bessel functions [69] of the first and second
kind and of order zero:
h(R) = CI0(R) + DK0(R) (4.35)
where the constants C and D are determined by the boundary conditions for the
specific problem.
147
For a solid cylindrical conductor, D = 0 since H remains finite for r = 0 and
K0(R) = 0. The constant C is determined by boundary condition (4.31):
This chapter addresses the characterization and modeling of NiMnGa for use as
a dynamic deformation sensor. The flux density is experimentally determined as a
function of cyclic strain loading at frequencies from 0.2 Hz to 160 Hz. With in-
creasing frequency, the stress-strain response remains almost unchanged whereas the
flux density-strain response shows increasing hysteresis. This behavior indicates that
twin-variant reorientation occurs in concert with the mechanical loading, whereas
the rotation of magnetization vectors occurs with a delay as the loading frequency
increases. The increasing hysteresis in magnetization must be considered when utiliz-
ing the material in dynamic sensing applications. A modeling strategy is developed
which incorporates magnetic diffusion and a linear constitutive equation.
5.1 Experimental Characterization of Dynamic Sensing Be-havior
This section details the experimental characterization of the dependence of flux
density and stress on dynamic strain at a bias field of 368 kA/m for frequencies of up
to 160 Hz, with a view to determining the feasibility of using Ni-Mn-Ga as a dynamic
157
H
Hall probe
Load cell
ε
Ni-Mn-Ga sample
Electromagnet
Pole piece(s)
Pushrod(s)
Figure 5.1: Experimental setup for dynamic magnetization measurements.
deformation sensor. This bias field was determined as optimum for obtaining maxi-
mum reversible flux density change [99] as seen in Section 2.3.2. The measurements
also illustrate the dynamic behavior of twin boundary motion and magnetization
rotation in Ni-Mn-Ga. As shown in Fig. 5.1, the experimental setup consists of a
custom designed electromagnet and a uniaxial MTS 831 test frame. This frame is
designed for cyclic fatigue loading, with special servo valves which allow precise stroke
control up to 200 Hz. The setup is similar to that described in Section 2.2 for the
characterization of the quasi-static sensing behavior. The custom-built electromagnet
described in Section 2.1 is used along with the MTS frame.
158
A 6×6×10 mm3 single crystal NiMnGa sample (AdaptaMat Ltd.) is placed in
the center gap of the electromagnet. In the low-temperature martensite phase, the
sample exhibits a free magnetic field induced deformation of 5.8% under a transverse
field of 700 kA/m. The material is first converted to a single field-preferred variant
by applying a high field along the transverse (x) direction, and is subsequently com-
pressed slowly by a strain of 3.1% at a bias field of 368 kA/m. While being exposed to
the bias field, the sample is further subjected to a cyclic uniaxial strain loading of 3%
amplitude (peak to peak) along the longitudinal (y) direction at a desired frequency.
This process is repeated for frequencies ranging between 0.2 Hz and 160 Hz. The flux
density inside the material is measured by a Hall probe placed in the gap between a
magnet pole and a face of the sample. The Hall probe measures the net flux density
along the x-direction, from which the x-axis magnetization can be calculated. The
compressive force is measured by a load cell, and the displacement is measured by a
linear variable differential transducer. The data is recorded using a dynamic data ac-
quisition software at a sampling frequency of 4096 Hz. All the measuring instruments
have a bandwidth in the kHz range, well above the highest frequency employed in
the study.
Fig. 5.2(a) shows stress versus strain measurements for frequencies ranging from
4 Hz to 160 Hz. The strain axis is biased around the initial strain of 3.1%. These
plots show typical pseudoelastic minor loop behavior associated with single crystal
Ni-Mn-Ga at a high bias field. With increasing compressive strain, the stress increases
elastically, until a critical value is reached, after which twin boundary motion starts
and the stress-preferred variants grow at the expense of the field-preferred variants.
During unloading, the material exhibits pseudoelastic reversible behavior because
159
the bias field of 368 kA/m results in the generation of field-preferred variants at the
expense of stress-preferred variants.
The flux density dependence on strain shown in Fig. 5.2(b) is of interest for sensing
applications. The absolute value of flux density decreases with increasing compres-
sion. During compression, due to the high magnetocrystalline anisotropy of NiMnGa,
the nucleation and growth of stress-preferred variants is associated with rotation of
magnetization vectors into the longitudinal direction, which causes a reduction of
the permeability and flux density in the transverse direction. At low frequencies of
up to 4 Hz, the flux-density dependence on strain is almost linear with little hys-
teresis. This low-frequency behavior is consistent with some of the previous obser-
vations [45, 99, 73]. The net flux density change for a strain range of 3% is around
0.056 T (560 Gauss) for almost all frequencies, which shows that the magnetization
vectors rotate in the longitudinal direction by the same amount for all the frequen-
cies. The applied strain amplitude does not remain exactly at ±1.5% because the
MTS controller is working at very low displacements (≈±0.15 mm) and high frequen-
cies. Nevertheless, the strain amplitudes are maintained within a sufficiently narrow
range (±8%) so that a comparative study is possible on a consistent basis for different
frequencies.
With increasing frequency, the stress-strain behavior remains relatively unchanged (Fig. 5.2(a)).
This indicates that the twin-variant reorientation occurs in concert with the applied
loading for the frequency range under consideration. This behavior is consistent with
work by Marioni [86] showing that twin boundary motion occurs in concert with the
applied field for frequencies of up to 2000 Hz. On the other hand, the flux den-
sity dependence on strain shows a monotonic increase in hysteresis with increasing
160
−0.02 −0.01 0 0.01 0.020
1
2
3
4
5
Compressive Strain
Com
pres
sive
Str
ess
(MP
a)
4 Hz20 Hz50 Hz90 Hz120 Hz160 Hz
(a)
−0.02 −0.01 0 0.01 0.02−0.07
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
Compressive Strain
Rel
ativ
e F
lux
Den
sity
(T
)
4 Hz20 Hz 50 Hz 90 Hz120 Hz160 Hz
(b)
Figure 5.2: (a) Stress vs. strain and (b) flux-density vs. strain measurements forfrequencies of up to 160 Hz.
161
frequency. The hysteresis loss in the stress versus strain plots is equal to the area
enclosed by one cycle (∮
σdε), whereas the loss in the flux density versus strain plots
is obtained by multiplying the enclosed area (∮
Bdε) by a constant that has units of
magnetic field [125, 27]. Fig. 5.3 shows the hysteresis loss for the stress versus strain
and the flux density versus strain plots. The hysteresis in the stress plots is relatively
flat over the measured frequency range, whereas the hysteresis in the flux density
increases about 10 times at 160 Hz compared to the quasistatic case. The volumetric
energy loss, i.e., the area of the hysteresis loop is approximately linearly proportional
to the frequency. The bias field of 368 kA/m is strong enough to ensure that the
180-degree domains disappear within each twin variant, hence each variant consists
of a single magnetic domain throughout the cyclic loading process [101]. Therefore,
the only parameter affecting the magnetization hysteresis is the rotation angle of the
magnetization vectors with respect to the easy c-axis. This angle is independent of
the strain and variant volume fraction [101], and is therefore a constant for the given
bias field.
The process that leads to the observed magnetization dependence on strain is
postulated to occur in three steps: (i) As the sample is compressed, twin variant
rearrangement occurs and the number of crystals with easy c-axis in the longitudi-
nal (y) direction increases. The magnetization vectors remain attached to the c-axis,
therefore the magnetization in these crystals is oriented along the y-direction. (ii) Sub-
sequently, the magnetization vectors in these crystals rotate away from the c-axis to
settle at a certain equilibrium angle defined by the competition between the Zee-
man and magnetocrystalline anisotropy energies. This rotation process is proposed
162
0 25 50 75 100 125 1500
1000
2000
3000
4000
Flu
x de
nsity
hys
tere
sis
loss
(J/
m3 )
Frequency (Hz)0 50 100 150
0
2
4
6
8x 10
4
Str
ess
hyst
eres
is lo
ss (
J/m
3 )
Figure 5.3: Hysteresis loss with frequency for stress-strain and flux-density strainplots. The plots are normalized with respect to the strain amplitude at a givenfrequency.
163
to occur according to the dynamics of a first order system. Time constants for first-
order effects in Ni-Mn-Ga have been previously established for the time-dependent
long-time strain response [41, 81], and strain response to pulsed field [86]. The time
constant associated with pulse field response provides a measure of the dynamics of
twin-boundary motion, which is estimated to be around 157 µs [86]. In contrast,
the time constant associated with magnetization rotation in our measurements is
estimated to be around 1 ms. (iii) As the sample is unloaded, twin variant rearrange-
ment occurs due to the applied bias field. Crystals with the c-axis oriented along the
y-direction rotate into the x-direction, and an increase in the flux density along the
x-direction is observed. At low frequencies, magnetization rotation occurs in concert
with twin-variant reorientation. As the frequency increases, the delay associated with
the rotation of magnetization vectors into their equilibrium position increases, which
leads to the increase in hysteresis seen in Fig. 5.2(b). The counterclockwise direction
of the magnetization hysteresis loops implies that the dynamics of magnetization ro-
tation occur as described in steps (i)-(iii). If the magnetization vectors had directly
settled at the equilibrium angle without going through step (i), the direction of the
hysteresis excursions would have been clockwise.
5.2 Model for Frequency Dependent Magnetization-StrainHysteresis
A continuum thermodynamics constitutive model has been developed to describe
the quasi-static stress and flux density dependence on strain at varied bias fields [101].
The hysteretic stress versus strain curve is dictated by the evolution of the variant
volume fractions. We propose that the evolution of volume fraction is independent
of frequency for the given range, and therefore, no further modification is required
164
Dynamic
Strain
Linear
Constitutive
Equation
Field
Diffusion
Equation
avgM e Hε χ= +
2
0 0
H MH
t tµσ µσ
∂ ∂∇ − =
∂ ∂
( , )H H x t=
( )M M t=( )tε ε=
Dynamic
Magnetization
BC: ( , ) biasH d t H± =
( )avgH t
Figure 5.4: Scheme for modeling the frequency dependencies in magnetization-strainhysteresis.
to model the stress versus strain behavior at higher frequencies. However, the mag-
netization dependence on strain changes significantly with increasing frequencies due
to the losses associated with the dynamic magnetization rotation resulting from me-
chanical loading. The modeling strategy is summarized in Figure 5.4.
The constitutive model (Section 3.6.2) shows that at high bias fields, the depen-
dence of flux density on strain is almost linear and non-hysteretic. Therefore, a linear
constitutive equation for magnetization is assumed as an adequate approximation at
quasi-static frequencies and modified to address dynamic effects. If the strain is ap-
plied at a sufficiently slow rate, the magnetization response can be approximated as
follows,
M = eε + χHavg (5.1)
165
where e and χ are constants dependent on the given bias field. For the given data,
these constants are estimated as, e = −4.58 × 106 A/m, and χ = 2.32. The average
field Havg acting on the material is not necessarily equal to the bias field Hbias.
Equation (5.1) works well at low frequencies. However, as the frequency increases,
consideration of dynamic effects becomes necessary. The dynamic losses are modeled
using a 1-D diffusion equation that describes the interaction between the dynamic
magnetization and the magnetic field inside the material,
∇2H − µ0σ∂H
∂t= µ0σ
∂M
∂t, (5.2)
This treatment is similar to that in Ref. [107] for dynamic actuation, although the
final form of the diffusion equation and the boundary conditions are different. The
boundary condition on the two faces of the sample is the applied bias field,
H(±d, t) = Hbias. (5.3)
Although the field on the edges of the sample is constant, the field inside the
material varies as dictated by the diffusion equation. The diffusion equation is nu-
merically solved using the backward difference method to obtain the magnetic field
at a given position and time H(x, t) inside the material.
For sinusoidal applied strain, the magnetization given by equation (5.1) varies in a
sinusoidal fashion. This magnetization change dictates the variation of the magnetic
field inside the material given by (5.2). The internal magnetic field thus varies in a
sinusoidal fashion as seen in Figure 5.5(a). The magnitude of variation increases with
166
increasing depth inside the material. In order to capture the bulk material behavior,
the average of the internal field is calculated by,
Havg(t) =1
Nx
Xd∑
X=−Xd
H(x, t), (5.4)
where Nx represents the number of uniformly spaced points inside the material where
the field waveforms are calculated.
Figure 5.5 shows the results of various stages in the model. The parameters used
are, µr=3.0, and ρ = 1/σ = 62 × 10−8 Ohm-m, Nx=40. Figure 5.5(a) shows the
magnetic field at various depths inside the sample for a loading frequency of 140 Hz.
It is seen that as the depth inside the sample increases, the variation of the magnetic
field increases. At the edges of the sample (x = ±d), the magnetic field is constant,
with a value equal to the applied bias field.
Figure 5.5(b) shows the variation of the average field at varied frequencies. The
variation of the average field is directly proportional to the frequency of applied
loading: as the frequency increases, the amplitude of the average field increases.
Finally, the magnetization is recalculated by using the updated value of the average
field as shown by the block diagram in Figure 5.4. The flux-density is obtained from
the magnetization (see Figure 5.5(c)) by accounting for the demagnetization factor.
It is seen that the model adequately captures the increasing hysteresis in flux density
with increasing frequency. Further refinements in the model are possible, such as
including a 2-D diffusion equation, and updating the permeability of the material
while numerically solving the diffusion equation.
167
0 0.2 0.4 0.6 0.8 1320
340
360
380
400
420
Non−dimensional Time (t*fa)M
agne
tic F
ield
(kA
/m)
d0.8d0.6d0.4d0.2d0
IncreasingDepth
(a)
0 0.2 0.4 0.6 0.8 1340
350
360
370
380
390
400
Non−dimensional Time (t*fa)
Ave
rage
Fie
ld (
kA/m
)
4 Hz20 Hz60 Hz100 Hz140 Hz
Increasingfrequency
(b)
0.01 0.02 0.03 0.04 0.050
0.01
0.02
0.03
0.04
0.05
0.06
Strain
Rel
. Flu
x D
ensi
ty (
Tes
la)
4 Hz20 Hz60 Hz100 Hz140 Hz
(c)
Figure 5.5: Model results: (a) Internal magnetic field vs. time at varying depth forthe case of 140 Hz strain loading (sample dim:±d), (b) Average magnetic field vs.time at varying frequencies, and (c) Flux-density vs. strain at varying frequencies.
168
5.3 Discussion
The magnetization and stress response of single-crystal Ni-Mn-Ga subjected to
dynamic strain loading for frequencies from 0.2 Hz to 160 Hz is presented [109, 104].
This frequency range is significantly higher than previous characterizations of Ni-Mn-
Ga which investigated frequencies from d.c. to only 10 Hz. The rate of twin-variant
reorientation remains unaffected by frequency; however, the rate of rotation of mag-
netization vectors away from the easy c-axis is lower than the rate of loading and
of twin-variant reorientation. This behavior can be qualitatively explained by the
dynamics of a first-order system associated with the rotation of magnetization vec-
tors. The increasing hysteresis in the flux density could complicate the use of this
material for dynamic sensing. However, the “sensitivity” of the material, i.e., net
change in flux-density per percentage strain input remains relatively unchanged (≈
190 G per % strain) with increasing frequency. Thus the material retains the advan-
tage of being a large-deformation, high-compliance sensor as compared to materials
such as Terfenol-D [99] at relatively high frequencies. The significant magnetization
change at structural frequencies also illustrates the feasibility of using Ni-Mn-Ga for
energy harvesting applications. To employ the material as a dynamic sensor or in
energy harvesting applications, permanent magnets can be used instead of an elec-
tromagnet. The electromagnet provides the flexibility of turning the field on and
off at a desired magnitude, but the permanent magnets provide an energy efficiency
advantage. The dynamic magnetization process in the material is modeled using a
linear constitutive equation, along with a 1-D diffusion equation similar to that used
a previous dynamic actuation model. The model adequately captures the frequency
169
dependent magnetization versus strain hysteresis and describes the dynamic sensing
behavior of Ni-Mn-Ga.
170
CHAPTER 6
STIFFNESS AND RESONANCE TUNING WITH BIASMAGNETIC FIELDS
This chapter presents the dynamic characterization of mechanical stiffness changes
under varied bias magnetic fields in single-crystal ferromagnetic shape memory Ni-
Mn-Ga. The material is first converted to a single variant through the application and
subsequent removal of a bias magnetic field. Mechanical base excitation is then used to
measure the acceleration transmissibility across the sample, from where the resonance
frequency is directly identified. The tests are repeated for various longitudinal and
transverse bias magnetic fields ranging from 0 to 575 kA/m. A single degree of
freedom (DOF) model for the Ni-Mn-Ga sample is used to calculate the mechanical
stiffness and damping from the transmissibility measurements. An abrupt resonance
frequency increase of 21% and a stiffness increase of 51% are obtained with increasing
longitudinal fields. A gradual resonance frequency change of −35% and a stiffness
change of −61% are obtained with increasing transverse fields. A constitutive model
is used to describe the dependence of material stiffness on transverse bias magnetic
fields. The damping exhibited by the system is low in all cases (≈ 0.03). The
measured dynamic behaviors make Ni-Mn-Ga well suited for vibration absorbers with
electrically-tunable stiffness.
171
6.1 Introduction
FSMA applications other than actuation have received limited attention. Stud-
ies have shown the viability of Ni-Mn-Ga in sensing and energy harvesting applica-
tions [119, 62, 101]. As a sensor material, Ni-Mn-Ga has been shown to exhibit a
reversible magnetization change of 0.15 T when compressed by 5.8% strain at a bias
field of 368 kA/m [101]. In addition, the stiffness of Ni-Mn-Ga varies with externally
applied fields and stresses. In the low temperature martensitic phase, application of
a sufficiently large transverse magnetic field (> 700 kA/m) produces a Ni-Mn-Ga mi-
crostructure with a single “field preferred” variant configuration (Figure 6.1, center);
application of a sufficiently large longitudinal field (> 350 kA/m) or sufficiently large
compressive stress (> 3 MPa) creates a single “stress preferred” variant configuration
(Figure 6.1, right). The quasistatic stress-strain curve for Ni-Mn-Ga [101] shows that
the two configurations have significantly different stiffness. At intermediate fields and
stresses, both variants coexist and the material exhibits a bulk stiffness between the
two extreme values (Figure 6.1, left). This microstructure offers the opportunity to
control the bulk material stiffness through the control of variant volume fractions
with magnetic fields or stresses. Magnetic fields are the preferred method for stiff-
ness control as they can be applied remotely and can be adjusted precisely. Faidley
et al. [28] investigated stiffness changes in research grade, single crystal Ni-Mn-Ga
driven with magnetic fields applied along the [001] (longitudinal) direction. The ma-
terial they used exhibits reversible field induced strain when the longitudinal field is
removed, which is attributed to internal bias stresses associated with pinning sites.
The fields were applied with permanent magnets bonded onto the material, which
172
makes it difficult to separate resonance frequency changes due to magnetic fields or
mass increase. Analytical models were developed to address this limitation.
In this study we isolate the effect of magnetic field on the stiffness of Ni-Mn-Ga
by applying the magnetic fields in a non-contact manner, and investigate the stiffness
characteristics under both longitudinal and transverse magnetic fields. Base excita-
tion is used to measure the acceleration transmissibility across a prismatic Ni-Mn-Ga
sample, from where its resonance frequency is directly identified. Prior to the trans-
missibility measurements, a stress-preferred or field-preferred variant configuration
is established through the application and subsequent removal of a bias field using
a solenoid coil or an electromagnet, respectively. We show that longitudinal and
transverse bias magnetic fields have drastically different effects on the stiffness of Ni-
Mn-Ga: varying the former produces two distinct stiffness states whereas varying the
latter produces a continuous range of stiffnesses. We present a constitutive model
that describes the continuous stiffness variation.
6.2 Experimental Setup and Procedure
The measurements are conducted on commercial single crystal Ni-Mn-Ga manu-
factured by AdaptaMat, Inc. A sample with dimensions 6×6×10 mm3 is tested in
its low-temperature martensite phase. The sample exhibits 5.8% free strain in the
presence of transverse fields of about 400 kA/m. The broadband mechanical excita-
tion is provided by a Labworks ET126-B shaker table which has a frequency range
of dc to 8500 Hz and a 25 lb peak sine force capability. The shaker is driven by an
MB Dynamics SL500VCF power amplifier which has a power rating of 1000 VA and
173
a
c
H
Transverse Field
H
Longitudinal Field Field Preferred
Stress Preferred
Figure 6.1: Left: simplified 2-D twin variant microstructure of Ni-Mn-Ga. Center:microstructure after application of a sufficiently high transverse magnetic field. Right:after application of a sufficiently high longitudinal field.
maximum voltage gain of 48 with 40 V peak and 16 A rms. The shaker is controlled
by a Data Physics SignalCalc 550 vibration controller.
A schematic of the test setup for longitudinal field measurements is shown in
Figure 6.2. The sample is mounted on an aluminum pushrod fixed on the shaker
table, and a dead weight is mounted on top of the sample. Two PCB accelerometers
measure the base and top accelerations. The longitudinal field is applied by a custom-
made water cooled solenoid transducer which is made from AWG 15 insulated copper
wire with 28 layers and 48 turns per layer [83]. The solenoid is driven by two Technol
7790 amplifiers connected in series which produce an overall voltage gain of 60 and
a maximum output current of 56 A into the 3.7 Ω coil. The solenoid has a magnetic
field rating of 11.26 (kA/m)/A.
174
The transverse field experiment is illustrated in Figure 6.3. The magnetic fields are
applied by a custom-made electromagnet made from laminated E-cores with 2 coils
of about 550 turns each made from AWG 16 magnet wire. The coils are connected in
parallel. The electromagnet has a magnetic field rating of 63.21 (kA/m)/A and can
produce fields of up to 750 kA/m.
For the longitudinal field tests, the sample is initially configured as a single field-
preferred variant. The sample microstructure can be changed with increasing longi-
tudinal fields by favoring the growth of stress-preferred variants, which results in a
stiffening with increasing magnetic field. The sample in zero-field condition is first
subjected to band-limited white noise base excitation with a frequency range from 0
to 4000 Hz and reference RMS acceleration of 0.2 g. After completion of the zero-
field test, a DC voltage is applied across the solenoid to produce a DC longitudinal
magnetic field on the sample. Due to the fast response of Ni-Mn-Ga [86], application
of the field for a small time period is enough to change the variant configuration. In
this study we apply the fields for about 1 to 2 seconds. If the field is strong enough
to initiate twin boundary motion, stress-preferred variants are generated from the
original field-preferred variants. The sample is again subjected to band-limited white
noise base excitation to record the top and base acceleration response, from which
the transfer function between the top and base acceleration is obtained. This process
is continued until the sample reaches a complete stress-preferred variant state.
For the transverse field tests, the sample is initially configured as a single stress-
preferred variant. This configuration is obtained by applying a high longitudinal field
in excess of 400 kA/m. The sample is mounted on the shaker table between the pole
faces of the electromagnet using aluminum pushrods, and a dead weight is mounted
175
Water cooled
Solenoid
Ni-Mn-Ga
Aluminum rod
Dead
weight
Shaker table
Accelerometer(s)
x x
x x
x x
x x
x x
Figure 6.2: Schematic of the longitudinal field test setup.
Ni-Mn-Ga
Aluminum
pushrod(s)
Accelerometer(s)
Electromagnet
pole piece(s)
Dead weight
Shaker table
Figure 6.3: Schematic of the transverse field test setup.
176
on the top of the sample. The test procedure is the same as in the longitudinal field
test: the transverse bias field is incremented by a small amount and subsequently
removed before each run. When the field is sufficiently high, field-preferred variants
are generated at the expense of stress-preferred variants, resulting in a change in
stiffness and resonance frequency.
6.3 Theory
The system is represented by the DOF spring-mass-damper model shown in Fig-
ure 6.4, where Ks represents the stiffness of the Ni-Mn-Ga sample, Kr is the total
stiffness of the aluminum pushrods, M is the dead weight on the sample, and C is
the overall damping present in the system. The base motion is represented by x, and
the top motion is represented by y.
The system is subjected to band-limited white noise base excitation with reference
acceleration to the shaker controller having an RMS value of 0.2 g. The reference
acceleration has uniform autospectral density (PSD) over the range from 0 to 4000 Hz:
Grr(f) = G 0 ≤ f ≤ 4000
= 0 f > 4000,(6.1)
with f the frequency (Hz), Grr the reference acceleration PSD (g2/Hz), and G the
constant value of reference acceleration PSD (g2/Hz) over the given frequency band.
The measured base acceleration PSD, or actual input acceleration PSD differs from
the reference PSD, and is denoted by Gxx (g2/Hz). The top acceleration PSD is de-
noted by Gyy (g2/Hz). The RMS acceleration values are related to the corresponding
177
M
C
Ks
Kr
y&&
x&&
Shaker table
Figure 6.4: DOF spring-mass-damper model used for characterization of the Ni-Mn-Ga material.
178
acceleration PSDs by
ψ2r =
∫ fmax
fmin
Grr(f)df,
ψ2x =
∫ fmax
fmin
Gxx(f)df,
ψ2y =
∫ fmax
fmin
Gyy(f)df,
(6.2)
where ψr, ψx, ψy represent the reference, input, and output RMS acceleration (m/s2)
values, respectively. Frequencies fmin and fmax respectively represent the lower and
upper limits on the band limited signal. Figure 6.5 shows the experimentally obtained
PSDs for input, output, and reference acceleration signals in one of the test runs.
In this case, the RMS acceleration values obtained from (6.2) are ψr = 0.2 g2/Hz,
ψx = 0.2036 g2/Hz, and ψy = 0.7048 g2/Hz. It is noted that the measured input PSD
does not have an exactly uniform profile as the reference PSD does. However, the
RMS values for the input and reference PSDs differ by less than 2%.
Since the cross-PSD between the input and output signals (Gxy) cannot be mea-
sured by the shaker controller, only the magnitude (and not the phase) of the transfer
function between the top and base acceleration signals are obtained experimentally.
The transfer function magnitude calculated from the experimental data is given as
|Hxy(f)|2 =Gyy
Gxx
, (6.3)
where Hxy(f) represents the experimentally obtained transfer function between the
top and base accelerations. For the DOF system shown in Figure 6.4, the transfer
function between the top and base acceleration is given as
this trend: the damping coefficient is maximum at intermediate fields, and attains
relatively lower values at the lowest and highest fields.
The variation of stiffness with changing bias field is modeled with an existing
continuum thermodynamics model developed by Sarawate et al. [101, 103]. With
increasing field, the Ni-Mn-Ga sample starts deforming because its twin variant con-
figuration changes. The variation of the field-preferred (ξ) and stress-preferred (1−ξ)
martensite volume fractions with field is described by the magnetomechanical con-
stitutive model. The model is formulated by writing a thermodynamic Gibbs energy
potential consisting of magnetic and mechanical components. The magnetic energy
has Zeeman, anisotropy and magnetostatic contributions; the mechanical energy has
189
200 250 300 350 400 4500
500
1000
1500
Field (kA/m)
Vis
cous
Dam
ping
Con
stan
t (N
m/s
)
Figure 6.12: Variation of viscous damping coefficient with initial transverse bias field.
200 250 300 350 400 4501000
1500
2000
2500
Bias Field (kA/m)
Res
onan
ce F
requ
ency
(H
z)
Figure 6.13: Variation of resonance frequency with initial transverse bias field.
190
elastic and twinning energy contributions. Mechanical dissipation and the microstruc-
ture of Ni-Mn-Ga are incorporated in the continuum thermodynamics framework by
considering the internal state variables volume fraction, domain fraction, and magne-
tization rotation angle. The constitutive strain response of the material is obtained by
restricting the process through the second law of thermodynamics, as detailed in [103].
The net compliance of the Ni-Mn-Ga sample is given by a linear combination of the
field-preferred and stress-preferred volume fractions. Thus, the net material modulus
is given as,
E(ξ) =1
S(ξ)=
1
S0 + (1− ξ)(S1 − S0)(6.12)
where E is the net material modulus, S is the net compliance, S0 is the compliance
of the material in complete field-preferred state, and S1 is the compliance of the
material in complete stress-preferred state. The twin variants are separated by a
twin boundary, and each side of the twin boundary contains a specific variant. If
the bulk material is subjected to a force, the stiffnesses associated with the stress-
preferred and field-preferred variants will be under equal forces, i.e., the two stiffnesses
will be in series. Therefore, the net compliance of the system is assumed to be a linear
combination of the compliances of the field-preferred and stress-preferred variants.
Further, the net stiffness is related to the modulus by
Ks =AE
L, (6.13)
with A the cross-sectional area, and L the length of the Ni-Mn-Ga element. Using
the constitutive model for volume fraction, and equations (6.12), (6.13), the stiffness
change of the material with initial bias field can be calculated. Model calculations
191
200 250 300 350 400 4504
6
8
10
12
14
16
Bias Field (kA/m)
Stif
fnes
s (N
/m)
× 10
6
ExperimentModel
Figure 6.14: Variation of stiffness with initial bias field.
are shown in Figure 6.14 along with the experimental values. The model accurately
predicts the stiffness variation with initial bias field.
Because of the relatively high demagnetization factor (0.385) in the transverse
direction, it takes higher external fields to fully elongate the sample. Thus, a contin-
uous change of resonance frequency and hence stiffness is observed with increasing
bias fields. In the case of the longitudinal field tests, the demagnetization factor is
0.229. Thus, once the twin boundary motion starts, it takes a very small range of
fields to transform the sample fully into the stress-preferred state. Thus, an abrupt
change in the resonance frequency and hence stiffness is seen in the longitudinal field
tests.
192
6.5 Concluding Remarks
The single-crystal Ni-Mn-Ga sample characterized in this study exhibits varied
dynamic stiffness with changing bias fields [102, 110]. The non-contact method of
applying the magnetic fields ensures consistent testing conditions. This is an im-
provement over the prior work by Faidley et al. [28], in which permanent magnets
were used to apply magnetic fields along the longitudinal direction. Unlike that
study, the characterization presented here was conducted on commercial Ni-Mn-Ga
material, under both longitudinal and transverse drive configurations. The field is
not applied throughout the duration of a given test, but only initially in order to
transform the sample into a given twin variant configuration. This is an advantage
of Ni-Mn-Ga over magnetostrictive materials like Terfenol-D in which a continuous
supply of magnetic field, and hence current in the electromagnetic coil, is required in
order to maintain the required resonance frequency. A study on a 0.63-cm-diameter,
5.08-cm-long Terfenol-D rod driven within a dynamic resonator has shown that this
material exhibits continuously variable resonance frequency tuning from 1375 Hz to
2010 Hz [34].
If a bi-directional resonance change was required, the system involving Ni-Mn-
Ga would need a restoring mechanism. A magnetic field source perpendicular to
the original field source could be used to maintain the advantage of low electrical
energy consumption. Another option is to use a restoring spring; but the presence
of the restoring spring results in reversible behavior of Ni-Mn-Ga, thus requiring a
continuous source of current to maintain the field. Nevertheless, this work shows
the suitability of using Ni-Mn-Ga in tunable vibration absorbers as it provides a
broad resonance frequency bandwidth comparable to Terfenol-D, with the option of
193
utilizing magnetic field pulse activation with very low energy consumption. Twin
boundary motion occurs almost instantaneously with the application of the field,
and the material configuration remains unchanged unless a restoring field or stress is
applied.
The overall resonance frequency and stiffness change in the transverse field tests
are −35.1% and −60.7% respectively. The equivalent values for the longitudinal field
tests are 21.3% and 51.5%, respectively. The damping values observed in the tests
are small (≈ 0.03) and are conducive to the use of Ni-Mn-Ga in active vibration
absorbers. An ON/OFF behavior is observed in the longitudinal field tests, whereas
a continuously changing resonance frequency is observed in the transverse field tests.
Thus, depending on the application and the frequency range under consideration,
the sample can be operated either in transverse or longitudinal field configuration.
The transverse field configuration offers more options regarding the ability to select
a particular resonance frequency. The longitudinal field configuration only offers two
discrete resonance frequencies but can be implemented in a more compact manner.
The evolution of volume fraction with increasing transverse field is described by the
existing continuum thermodynamics model, which is used to model the dependence
of material stiffness on the initial bias field assuming a linear variation of compliance
with volume fraction. Therefore, the stiffness exhibits a hyperbolic dependence on
the bias transverse field, which is also validated by experiments. The acceleration
transmissibility transfer function is accurately quantified by assuming a discretized
SDOF linear system. The development of a continuous dynamic model is desirable
for handling different sample geometries and higher modes. Although in this study
the magnetic field was switched off during the dynamic tests, development of a model
194
with the sample immersed in an external magnetic field during testing might be useful
for creating a more complete characterization of the dynamic behavior exhibited by
Ni-Mn-Ga.
195
CHAPTER 7
CONCLUSION
This dissertation was written to advance the understanding of the complex rela-
tionships under various static and dynamic conditions in ferromagnetic shape memory
alloys, specifically single crystal Ni-Mn-Ga. The key tasks were to characterize the
sensing behavior, to develop a coupled magnetomechanical model, and to investigate
the dynamic behavior. Key observations and conclusions are detailed at the end of
each of the prior chapters, and this chapter presents an overall summary of the entire
work.
7.1 Summary
7.1.1 Quasi-static Behavior
Sensing Characterization
One focus of the dissertation was to investigate whether Ni-Mn-Ga can be utilized
in sensing applications. For this purpose, an experimental setup was built to apply
uniaxial mechanical compression in presence of suitable bias magnetic fields. The
measurements revealed that the magnetization or flux density of Ni-Mn-Ga can be
altered by means of mechanical compression, thereby validating its ability to sense.
Furthermore, it was observed that the stress-strain behavior exhibits a transition
196
from irreversible behavior at low fields to the reversible behavior at high fields. This
phenomenon is similar to that in thermal shape memory alloys, except that the role of
temperature is replaced by the magnetic field. There is a strong correlation between
the stress and flux density behavior regarding the reversibility.
The presented characterization demonstrates that Ni-Mn-Ga can be useful as a
sensor. Its advantages with respect to other smart materials are the large deforma-
tion range, high-compliance, and high sensitivity at lower forces. Majority of the prior
focus on Ni-Mn-Ga applications has been on actuation. However, the low blocking
stress and requirement of large magnetic fields limit the use of the material as an
actuator. Large magnetic fields necessitate the construction of a bulky electromag-
net. However, in a sensor configuration, the required bias field can be applied using
small permanent magnets. Therefore, a sensor made using Ni-Mn-Ga can exhibit
significantly higher energy density than an actuator made using the same Ni-Mn-Ga
sample. This research opens up the possibilities for future research in this area.
Blocked-Force Characterization
The force generation capacity of single crystal Ni-Mn-Ga is also characterized.
When the material is subjected to a magnetic field and is mechanically blocked, it
tries to push against the loading arms, thus generating a force. The blocked force
characterization is one of the key properties of smart materials, and it gives an indica-
tion of the actuation performance and the work capacity of the material. Though it is
observed that Ni-Mn-Ga provides higher work capacity than materials such as piezo-
electrics and magnetostrictives, the actuation authority of the material is severely
restricted due to the low blocking stress of around 3.5 MPa.
197
Magnetomechanical Constitutive Model
A continuum thermodynamics based model is presented which describes the cou-
pled magnetomechanical behavior of the material in variety of operating conditions.
The model describes the sensing, actuation, and blocked force behavior of single
crystal Ni-Mn-Ga ferromagnetic shape memory alloy. The nonlinearities and path
dependencies leading to hysteresis are well captured by the model. The classical
continuum mechanics framework is used with addition of magnetic terms; and the
internal state variables are used to incorporate the material microstructure and dis-
sipation. The model is physics based, which makes it flexible for additional of other
complex effects such as the exchange energy, magnetomechanical coupling energy, etc.
The model uses only seven non-adjustable parameters which are identified from two
simple experiments. The model is low-order, which makes it suitable for incorpora-
tion into custom finite element codes. The constitutive model is rate-independent,
and the material behavior at higher frequencies needs to be described by including
additional physics.
Chief utility of the model will be in designing and predicting the performance of
Ni-Mn-Ga sensors and actuators by describing the macroscopic relationships between
various magnetomechanical variables. In addition to modeling these primary vari-
ables (stress, strain, magnetization, field), closed form solutions are derived to obtain
certain key variables such as the maximum strain, coercive field, twinning stress,
residual field, sensitivity, etc. The optimum bias field for a sensor and an optimum
bias stress for an actuator can be obtained from the model. These calculations pro-
vide a powerful tool as the model can be used to readily obtain an optimum actuator
or sensor design for a given Ni-Mn-Ga sample. The model can be easily modified to
198
describe the minor loops, which are critical for cyclic operation of the material around
a bias stress or bias field.
7.1.2 Dynamic Behavior
Dynamic Actuator Model
A new model is developed to describe the frequency dependent strain-field hys-
teresis in dynamic Ni-Mn-Ga actuators. This model is successfully implemented on
a dynamic magnetostrictive actuator to show its possible impact on the community
of hysteretic smart materials. The model uses the constitutive actuation model to
obtain a key variable such as the volume fraction or magnetostriction which is directly
related to the material’s strain. In addition to the constitutive model, the dynamic
magnetic losses due to eddy current are modeled using magnetic field diffusion and
the structural dynamics of the actuator is included by modeling the system as a
single-degree-of-freedom system. The applied magnetic field generates a force on the
actuator which makes the material vibrate. This force is expressed in terms of the
volume fraction which couples the dynamic strain to the magnetic field. The Fourier
series expansion of the volume fraction gives the net force acting on the actuator, and
the dynamic strain is obtained by superposition of the displacement response to each
harmonic component of the force. Analysis of strain in frequency domain at different
actuation frequencies reveals an interconnection with the shape of the macroscopic
hysteresis loop. This new approach can enable calculation of the input field profile
from the desired output strain profile by reversing the model flow.
199
Dynamic Sensing Characterization and Modeling
Characterization of the dynamic sensing properties of Ni-Mn-Ga was not addressed
in the literature. This research presents the first evidence that the stress induced mag-
netization change in Ni-Mn-Ga can also occur at higher frequencies (up to 160 Hz). It
is observed that the twin-variant reorientation remains unaffected for this frequency
range, which means that the stress-strain plots remain unaffected by the frequency.
On the contrary, the magnetization-strain plots show increasing hysteresis with fre-
quency, which indicates that the magnetization rotation process occurs with a delay.
This behavior can be explained by magnetic diffusion equation in a similar fashion
to that for the dynamic actuator model. The peak-to-peak magnetization values do
not decay significantly for the given range, indicating that the material can be used
as a sensor at higher frequencies. Ni-Mn-Ga sensors can thus give an advantage over
piezoelectric sensors, because they can be operated in quasi-static as well as dynamic
conditions.
Stiffness Tuning
Several smart materials can be used as tunable stiffness devices, because their
stiffness can be altered by application of electric or magnetic fields. This research
demonstrates the suitability of Ni-Mn-Ga as a tunable vibration absorber by char-
acterizing the resonance and stiffness with bias fields. The stiffness variation under
different collinear and transverse bias fields is characterized. Suitable drive configu-
ration can be chosen depending on the application.
200
Quasi-static Dynamic
Behavior Sensing Actuation Blocked-
force Actuation Sensing
Stiffness
Tuning
Input
Variable(s) Strain Field Field Field Strain
Base
Acceleration
Output
Variable(s)
Magnetization,
Stress
Strain,
Magnetization
Stress,
Magnetization Strain
Magnetization,
Stress
Top
Acceleration
Bias
Variable(s) Field Stress Strain
Stress,
Frequency
Field,
Frequency Field
Energy
Potential Magnetic Gibbs Gibbs
Magnetic
Gibbs - - -
Experiment In house Outside data In house Outside data In house In house
Modeling Continuum
Thermodynamics
Derived from
sensing work
Derived from
sensing work
Diffusion +
Constitutive
+Dynamics
Diffusion+
Lin.Constitutive
Second
order sys.
Figure 7.1: Characterization map of Ni-Mn-Ga. Plain blocks in “Experiment” and“Modeling” rows show the new contribution of the work; Light gray blocks show thata limited prior work existed, which was completely addressed in this research; Darkgray blocks indicate that prior work was available, and no new contribution was made.
7.1.3 Characterization Map
The presented research addresses the properties of Ni-Mn-Ga in a variety of static
and dynamic conditions. Figure 7.1 shows the contribution made by this research
regarding both experimental and modeling work pertaining to ferromagnetic shape
memory alloys. The presented work covers a significant realm of the possible charac-
terizations. Few additions to this work could be possible, such as modeling magne-
tization in dynamic actuation, or using the flux-density as a bias variable. However,
majority of the real world applications using smart materials are covered by the pre-
sented characterization map.
201
7.2 Contributions
• Hardware and test setups are developed for conducting characterization of the
sensing behavior of single Ni-Mn-Ga to measure stress, magnetization response
to strain input under bias fields.
• Increasing bias field marks the transition from irreversible (pseudoelastic) to
reversible (quasi-plastic) behavior.
• A bias field of 368 kA/m is identified as the optimum bias field which results
in reversible flux density change of 145 mT for strain of 5.8% and stress of
4.4 MPa.
• Flux density vs. strain behavior is linear and almost non-hysteretic whereas the
flux density vs. stress behavior is highly hysteretic, indicating that the material
will be more useful as a deformation sensor than a force sensor.
• A continuum thermodynamics based magnetomechanical constitutive model is
developed to quantify the non-linear and hysteretic behavior of Ni-Mn-Ga for
sensing, actuation and blocked-force cases.
• The microstructure and dissipation is included in the continuum framework via
internal state variables, the evolution of which dictates the material response.
• The work capacity of Ni-Mn-Ga is around 72.4 kJ/m3, which is higher than
that of piezoelectric and magnetostrictive, however, the actuation authority of
the material is limited as the maximum blocking force is only around 4 MPa.
202
• Quasi-static characterization chows a flux density sensitivity with strain
(∂B
∂ε
)
as 4.19T/%ε at 173 kA/m, and 2.38T/%ε at 368 kA/m; maximum field induced
twinning stress as 2.84 MPa; variation of initial susceptibility
(∂M
∂H|H=0
)of
59%; and maximum stress generation of 1.47% at 3% strain.
• Dynamic actuation model to was developed by including eddy currents and
structural dynamics along with constitutive volume fraction model to describe
the frequency dependent strain-field hysteresis.
• The dynamic actuator model was applied for magnetostrictive materials to
demonstrate its wider application.
• The dynamic sensing behavior of Ni-Mn-Ga was characterized by subjecting Ni-
Mn-Ga to compressive strain loading of 3% at frequencies from 0.2 to 160 Hz
in presence of bias field of 368 kA/m.
• The dynamic stress vs. strain plots show negligible change with increasing
frequency, whereas the flux-density vs. strain plots show an increasing hysteresis
that is linearly proportional to the frequency.
• The net flux-density change per unit strain remains almost constant (≈ 159 G)
with increasing strain, which can offer applications in broadband sensing and
energy harvesting.
• Stiffness of Ni-Mn-Ga was characterized by conducting broadband white-noise
base excitation tests under collinear and transverse bias magnetic fields.
203
• Measured stiffness changes of 51% and 61% for the collinear and transverse con-
figurations respectively indicate that Ni-Mn-Ga is suitable for tunable vibration
absorption applications.
• Ni-Mn-Ga is therefore demonstrated as a new multi-functional smart material
with applications in sensing, actuation and vibration absorption.
7.3 Future Work
This research has led to a thorough understanding about several aspects of Ni-
Mn-Ga FSMAs which were previously not investigated. Following list enumerates the
possible improvements in this work, as well as the future research opportunities that
have been opened up as a result of this research:
7.3.1 Possible Improvements
• The thermodynamic energy potentials in the constitutive model can be revisited
to add more complex effects such as the exchange energy, and magnetoelastic
coupling energy.
• A more accurate expression for magnetostatic energy can be used as that in
Ref. [82]. However, the usefulness of the additional accuracy against the in-
creased complexity and computational time needs to be evaluated.
• The blocked-force model can be improved to add the hysteretic effects in the
stress response.
• 2-D magnetic diffusion equations can be used in the dynamic actuation and
sensing models, and the current averaging technique can be reconsidered.
204
7.3.2 Future Research Opportunities
• The sensor device using permanent magnets as that shown in Appendix B (see
Section B.3) could be refined to make it more compact and robust. Such a
device would lead to realistic evaluation of the energy density of Ni-Mn-Ga
sensors and the effect of the system dynamics on the sensor performance. The
effect of prestress on the system properties could be of interest.
• The constitutive model could be extended to address the 3-D behavior, which
would enable the implementation of the model in finite element analysis codes.
• A continuous structural model of the Ni-Mn-Ga rod could be used for dynamic
actuator. This will enable further development towards predicting the dynamic
performance of structures made using Ni-Mn-Ga, or structures with patches of
Ni-Mn-Ga, encompassing various shapes such as rods, beams and plates.
• The dynamic actuator model can be augmented to add the electromagnet
impedances so that the voltage and currents can be used as input variables
instead of magnetic field.
205
APPENDIX A
MISCELLANEOUS ISSUES WITH QUASI-STATICCHARACTERIZATION AND MODELING
A.1 Electromagnet Design and Calibration
A.1.1 Effect of Dimensions on Field
To design the electromagnet, influence of various parameters on the final field
must be studied to maximize its efficiency. Figure A.1 shows a 2-D view of the
laminates. Once the overall dimensions of the E-shaped laminates are chosen, certain
dimensions are fixed, such as the width of the central legs (D). But, there are two
major dimensions that affect the magnetic field generated per given current density
in the coils (J). They are the length of the E-shaped legs (L) and the width of the
central leg at the end of the taper (d). The angle of the taper (Φ) on the central leg
is,
Φ = tan−1
(D − d
2W
)(A.1)
The objective of designing the electromagnet is to generate maximum magnetic
field in the central air gap for a given current density in the coils. A finite element
software for electromagnetics such as FEMM or COMSOL provide a quick way to
investigate the effect of these dimensions on the generated magnetic field. Using
206
Laminated core
Air Gap
Coils
L
D d
(Current Density J) W
Figure A.1: Schematic of the Electromagnet.
FEMM, various simulations are conducted to find the effect of the ratio (d/D) on the
magnetic field at a given length (L). A snapshot of one of the simulations is shown in
Figure2.3. The results of these simulations are summarized in Figures A.2 and A.3.
It is observed that the length of 5 inches gives maximum field ratios in the range
of around 0.3-0.6. However, this results in a steep taper angle of around 20-30 deg.
Such a steep taper angle is usually not recommended because it can result in excessive
leakage which may not be accurately simulated by the FEMM software. Furthermore,
a steep angle or very small width (d) may not provide a uniform field over the entire
length of the sample. Considering these issues, the length of the legs is chosen as
6 inches, and the width of the legs is chosen as 1.4 inch. These dimensions correspond
to a ratio of 0.62, and taper angle of 10.04 deg.
207
0.2 0.4 0.6 0.8 1750
800
850
900
950
1000
Ratio of Small width / Large width
Mag
netic
Fie
ld (
kA/m
)
5 in6 in7 in8.3 in
Increasing L
Figure A.2: Effect of ratio (d/D) on field.
0 5 10 15 20 25 30 35750
800
850
900
950
1000
Taper Angle (deg)
Mag
netic
Fie
ld (
kA/m
)
5 in6 in7 in8.3 in
Increasing L
Figure A.3: Effect of angle (Φ) on field.
208
1 1.5 2 2.5 3 3.5 4 4.5300
400
500
600
700
800
900
Current Density (MA/m2)
Mag
netic
Fie
ld (
kA/m
)
Figure A.4: Variation of current density with field.
For these final dimensions, the variation of field with current density (J) in the
coils is plotted in Figure A.4. It is observed that the field increases linearly with cur-
rent density values of up to J ≈ 2.25 M/A2. Further increase in current density does
not increase the field by a significant amount because the electromagnet core starts to
saturate. Therefore, the coils are designed to carry maximum current corresponding
to the current density of around 2.5 MA/m2.
Wire Selection
The wire is selected based upon the available area, maximum current carrying
capacity, resistance of the wire, and most importantly, the magnetomotive force (NI)
it can produce within the given constraints. The available area (Aw) for winding a
209
coil is fixed, which corresponds to a rectangle (lw × ww) of around 2 in× 1.075 in. If
the wire has a diameter of dw, the maximum possible turns per layer (n) are,
n =lwdw
, (A.2)
and maximum number possible number of turns (Nm) are,
Nm =lwww
d2w
, (A.3)
The area occupied by one turn is assumed to be equal to the square of the wire
diameter. The packing efficiency is assumed to be around (ηp = 80%), which gives
the actual number of turns as,
N = ηpNm. (A.4)
If the maximum current carrying capacity of the wire is Im, the maximum MMF
produced by the wire is,
MMFmax = NIm. (A.5)
For the given purpose, the objective of the coil design is to maximize this MMF for
a given wire. Additional considerations include the total resistance of the wire (Rw),
which dictates the power requirements and the Joule heating (IR2w), which places
restrictions on the resistance and current. A wire of small diameter would pack a
very large number of turns, however, its current carrying capacity would be low, and
the resistance would be high, leading to increased heating. On the other hand, a wire
with large diameter would carry a high amount of current, but its size could place
restrictions on the maximum possible turns. A detailed study of various wire sizes
from AWG 12 to AWG 20 is conducted to arrive at the optimum wire size. These
It is seen from this comparison that AWG 16 wire gives the maximum MMF among
the chosen sizes. The comparison of the various wires regarding their maximum
current capacity, maximum possible turns and MMF is given in Figure A.6. The wire
size of AWG 16 clearly turns out to be the optimum size as it provides a balance
between the maximum current carrying capacity and maximum allowed turns, which
leads to maximum possible MMF. This wire has a diameter of 0.0508 in.
The coil is wound on a rectangular shaped bobbin using a stepper motor and a
custom-made fixture. A thin layer of epoxy is applied after each layer to hold the
wires together, and to provide extra insulation. Two such coils are placed on the
central legs of the electromagnet, and are connected in parallel. Figure A.7 shows a
picture of the assembled electromagnet. Drawings of the electromagnet and relevant
parts are given in D.1.
212
12 13 14 15 16 17 18 19 201000
1500
2000
2500
3000
3500
4000
4500
5000
AWG Wire Number
Turns (N×4)Current (I
max×400)
MMF (N×Imax
)
Figure A.6: Comparison of current carrying capacity, possible turns and MMF pro-duced by various wires (The current and turns are multiplied by scaling factors) Wiresize AWG 16 is seen as an optimum size.
213
Air
gap
Laminated
core
coil
Figure A.7: Picture of the assembled electromagnet.
A.1.2 Electromagnet Calibration with Sample
The magnetic bias field for the sensing characterization presented in Section 2.2
is assumed to be that given by the calibration curve in Figure 2.4. The applied bias
magnetic field is not measured during the tests, and the calibration curve is used to
obtain the field from the measured current in the electromagnet coils.
Naturally, one of the issues during theses tests is whether the observed change in
flux density is only due to the change of sample variant configuration or also due to
the change in the reluctance in the electromagnet gap. When the sample is placed
in the electromagnet gap, the permeability of the air gap decreases as the reluctance
due to the sample is higher than that due to the air. Furthermore, as the sample is
compressed, its permeability changes which again changes the reluctance of the air
gap.
214
If the change in reluctance is significant, it can introduce errors in the results
obtained in sensing as well as blocked force characterizations, because the applied
magnetic field can no longer be accurately predicted by the electromagnet calibra-
tion curve in Figure 2.4. Therefore, it is necessary to check the effect of sample
configuration on the applied field.
Electromagnet calibration tests similar to that in Section 2.1.2 are conducted with
the sample in the air-gap. First, the sample with complete field-preferred variant con-
figuration (easy axis configuration) is placed in the air gap, and the electromagnet
is calibrated. The easy axis configuration implies that the sample has highest per-
meability and thus the reluctance in the electromagnet gap is lowest. Therefore,
this configuration can have maximum impact on the applied field, of all other con-
figurations of the sample. This process is also repeated with sample in hard-axis
configuration placed in the electromagnet air-gap. Finally, the sample is removed
from the air gap, and the applied field in the air gap is measured.
The test results are shown in Figure A.8. It is seen that there is almost no change
in the measured field for the same values of current with or without the presence
of sample. The maximum variation is obtained as 3%, which is sufficiently small to
allow the approximation that the presence of sample does not affect the applied field.
Possible reasons for negligible variation in the field magnitude can be:
• The permeability of the sample in both easy-axis and hard-axis case is too low
compared to that of the iron core, and thus does not affect the total reluctance
much.
215
−8 −6 −4 −2 0 2 4 6 8−600
−400
−200
0
200
400
600
Current (Amp)
Mag
netic
fiel
d (k
A/m
)
Sample easy axisSample hard axisNo sample
6 6.5 7 7.5 8300
350
400
450
500
550
Current (Amp)
Mag
netic
fiel
d (k
A/m
)
Sample easy axisSample hard axisNo sample
Figure A.8: Electromagnet calibration curve in presence of sample, the easy axiscurve shows maximum variation.
216
• There is a significant reluctance and flux leakage in the electromagnet core itself,
therefore a small change in the reluctance of the sample does not change the
overall behavior of the magnetic circuit.
These tests thus confirm that the issue of electromagnet reluctance change can be
neglected, and the applied fields in all the cases can be assumed to be equal to the
applied fields measured in air.
217
A.2 Verification of Demagnetization Factor
As seen in Section 3.6.2, the relationship between the measured flux-density (Bm)
and magnetization of the sample in x-direction (M) is given as,
Bm = µ0(H + NxM), (A.6)
with Nx the demagnetization factor, H is the applied field, and M is the magnetization
inside the sample. The schematic of the demagnetization process is illustrated in
Figure A.9. The demagnetization factor is obtained from the geometry of the sample.
Therefore by measuring the flux-density outside the sample, the magnetization inside
the sample is calculated, and is used for comparison with the model results. Validation
of equation (A.6) is therefore critical from the viewpoint of both the characterization
and modeling of the sensing behavior.
To simulate this situation, a finite element software, COMSOL is used. As seen
in Section A.1.2, the sample does not affect the applied magnetic field of the elec-
tromagnet. Hence, the source of magnetic field can be represented by electromagnet
as well as permanent magnets, and the latter is used for simplicity. Moreover, the
use of permanent magnets as a constant magnetic field source is a better choice for
these simulations because COMSOL can not realistically model the reluctance in the
electromagnet cores.
The problem under consideration is modeled by with two permanent magnets
and the Ni-Mn-Ga sample in the gap between them (Figure A.10). This is a 3-
D magneto-static problem with no currents. Two Nd-Fe-B permanent magnets are
considered to be applying a bias field. The magnets are modeled by using a rem-
nant flux density value from the manufactures’ catalogue (Br = 1.32 T in this case).
218
+
+
+
+
+
-
-
-
-
-
M
Hd
H
Figure A.9: Schematic of the demagnetization field inside the sample. The appliedfield (H) creates a magnetization (M) inside the sample, which results in north andsouth poles on its surface. H and M are shown by solid arrows. The demagnetizationfield (Hd = NxM) is directed from north to south poles as shown by dashed arrows.Although inside the sample, the demagnetization field opposes the applied field, itadds to the applied field outside the sample. Therefore, the net field inside the sampleis given as H −NxM , whereas the net field outside the sample is given as H + NxM .
219
magnet Ni-Mn-Ga
Figure A.10: A snapshot from COMSOL simulation.
The sample is located in the central gap. The medium of the sample is varied
as (i) Air (µr = 1), (ii) Ni-Mn-Ga with complete field preferred (easy-axis) with
(µr = 3.06), and (iii) stress preferred (hard-axis) with (µr = 1.46). The field, flux
density and magnetization are plotted as a function of the air gap in the middle of the
two magnets. The horizontal line is shown in red color over which the three quantities
are plotted.
In the experimental setup, the Hall probe is placed outside the sample, in the
gap between the sample and magnet. Naturally, the flux density measured by the
220
Hall probe is not the same as that inside the sample. Consequently, the magnetiza-
tion inside the sample also can not be obtained directly. Therefore to calculate the
magnetization, expression A.6 is used.
When there is no sample present in the gap of the electromagnet, the simulated
magnetic field corresponds to H. However when a sample is present in the air gap,
this simulated field increases because the demagnetization field adds to the applied
field. Therefore the effect of the demagnetization field on simulated field is obtained
by,
Hre = H + NxM, (A.7)
where Hre is the recalculated magnetic field.
The magnetization inside the material is obtained as,
M =
(Bm
µ0
−H
)1
Nd
, (A.8)
The flux density outside the sample is then reiterated by using the calculated
magnetization as,
Bre = µ0Hr, (A.9)
Figure A.11 shows the magnetic field variation in the gap, figure A.12 shows the
flux density variation, and figure A.13 shows the magnetization variation. The solid
lines show the quantities obtained from COMSOL directly, whereas the dashed lines
show quantities calculated from equations (A.6) to (A.7).
In Figure A.11, it is seen that with increasing permeability of the media in the
gap (µair < µhard < µeasy), the applied field increases. This behavior may not be
obvious, since the magnetic field from the permanent magnets is expected to constant.
However, the demagnetization field from the sample adds to the applied field from the
221
0.025 0.03 0.035 0.04 0.045150
200
250
300
350
400
450
500
550
600
650
Distance (inch)
Mag
netic
fiel
d (k
A/m
)
EasyHardNo sampleEasy reiteratedHard reiterated
Figure A.11: Magnetic field vs distance. Solid: COMSOL, Dashed: recalculated.
0.025 0.03 0.035 0.04 0.0450.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
Distance (inch)
Flu
x de
nsity
(T
esla
)
EasyHardNo sampleEasy reiteratedHard reiterated
Figure A.12: Flux density vs distance. Solid: COMSOL, Dashed: recalculated.
%% Code for reverse field applicationclear H F_z F_z_H e e_tw z zs F_zs M Bm theta alphaz=[];zs=[];z(1)=z_end;H = H_max:-0.5:00;H =H*1000; % Field during reverse applicationfor i=1:length(H)
% Parameters for constitutive modelf0=1; % FrequencyT0=1/f0; % Time period of one cycleT = m*T0; % Total time perioddf=1/T; % Frequency resolutionNp = 2^12; % Total pointsN0=Np/m; % Points in one cycleh=T/Np; % Time resolutionfs=1/h; % Sampling frequencyt=0:h:T-h; % Time vectort2=-h*N0:h:T-h;
% Matrix to select various frequenciesFF =[1 2 3 4 5 6 7
% Creation a vector of freqs 0,2,4,.. for storing the magnitudes and phases% of the FFT of volume fraction mag and phase of FFTkkk=0;for kk = 1:length(freq1)
if ( mod(freq1(kk),2)==0)kkk = kkk+1;F(kkk)=freq1(kk);Mag_z(kkk) = M_z(kk);Ph_z(kkk) = P_z(kk);
endend
% Regeneration of the original signal of volume fractionfor ii = 1:length(t)
Figure D.4: Base channels for mounting electromagnet.
264
Figure D.5: Bottom pushrod for applying compression using MTS machine.
265
Figure D.6: Top pushrod for applying compression using MTS machine.
266
Fig
ure
D.7
:2-
Dvie
wof
the
asse
mble
ddev
ice.
267
Fig
ure
D.8
:B
otto
mpla
te.
268
Fig
ure
D.9
:Top
pla
te.
269
Figure D.10: Side plate.
270
Figure D.11: Support disc.
Figure D.12: Disc to adjust the compression of spring.
271
Figure D.13: Seismic mass (material: brass).
Figure D.14: Plate to secure magnets (2 nos).
272
Figure D.15: Grip to hold the sample (2 nos).
273
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