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Magnetic and Magnetostrictive Characterization and
Modeling of High Strength Steel
by Christopher Donald Burgy
B.S. in Physics, May 2008, The Pennsylvania State University
M.S. in Applied Physics, May 2010, Johns Hopkins University
A Dissertation submitted to
The Faculty of
The School of Engineering and Applied Science
of The George Washington University
in partial satisfaction of the requirements
for the degree of Doctor of Philosophy
May 18th
, 2014
Dissertation directed by
Edward Della Torre
Professor of Engineering and Applied Science
ii
The School of Engineering and Applied Science of The George Washington University
certifies that Christopher Donald Burgy has passed the Final Examination for the degree
of Doctor of Philosophy as of March 20th
, 2014. This is the final and approved form of
the dissertation.
Magnetic and Magnetostrictive Characterization and Modeling of
High Strength Steel
Christopher Donald Burgy
Dissertation Research Committee:
Edward Della Torre, Professor of Engineering and Applied Science,
Dissertation Director
Can E. Korman, Professor of Engineering and Applied Science,
Committee Director
Roger H. Lang, L. Stanley Crane Professor of Engineering and Applied
Science, Committee Member
iii
Dedication
I dedicate this dissertation to my family and friends. I am forever indebted to my
parents, Donald and Evelyn, for their constant love and support. My sister Katherine and
my brother David have always lent an ear when needed, and have been a great blessing in
my life. Here’s to helping me keep it all in perspective.
I also dedicate this dissertation to my friends, who have provided much needed
distraction throughout this whole process. I will always appreciate what they have and
continue to do for me. Thank you for helping me to balance heavy workloads with the
better things in life.
Last but not least, I dedicate this dissertation to my remarkable wife Maureen. You
and Pez have kept me going through thick and thin; and you have been immensely patient
while I drone on and on about research. I could never have completed this work without
you, and I promise not to talk about magnetism for at least a month.
iv
Acknowledgements
The author wishes to acknowledge all those whose help and guidance made this
dissertation possible. I wish to thank my committee members who were very generous in
their time and expertise. This immense undertaking would never have gotten off the
ground if it weren’t for the dedication, patience, and knowledge of my advisor, Dr.
Edward Della Torre. A special thanks to my colleagues Marilyn Wun-Fogle and Dr.
James Restorff, who have spent many hours toiling alongside me to make this research
possible. I am sure that you have both forgotten more about magnetism than I could ever
hope to learn. Thank you to Dr. Lawrence Bennett for the many helpful discussions from
countless weekly meetings. And finally, thank you Dr. Can Korman and Dr. Roger Lang
for agreeing to serve on my committee, and providing invaluable guidance throughout
this process.
I would like to acknowledge and thank Dr. George Stimak from the Office of Naval
Research, and Dr. Jack Price from the Naval Surface Warfare Center, Carderock Division
(NSWCCD) for providing the funding which paid for most of this effort. Special thanks
go to the United States Navy and its personnel, specifically my fellow NSWCCD
colleagues for their ongoing support and flexibility in this endeavor. I would specifically
like to thank: Dr. Brian Glover, Dr. John Holmes, Dr. Bruce Hood, Dr. Stephen
Potashnik, and Mr. John Scarzello for their insightful discussions and expertise.
Finally, I would like to thank my fellow Institute for Magnetics Research colleagues;
specifically Hatem Elbidweihy, Mohammedreza Ghahremani, Maryam Ovichi, Dr. Yi
Jin, Dr. Shou Gu, and Dr. Virgil Provenzano for the many collaborations, conversations,
and pieces of advice throughout the years.
v
Abstract
Magnetic and Magnetostrictive Characterization and Modeling of High Strength
Steel
High strength steels exhibit small amounts of magnetostriction, which is a useful
property for non-destructive testing amongst other things. This property cannot currently
be fully utilized due to a lack of adequate measurements and models. This thesis reports
measurements of these material parameters, and derives a model using these parameters
to predict magnetization changes due to the application of compressive stresses and
magnetic fields. The resulting Preisach model, coupled with COMSOL Multiphysics®
finite element modeling, accurately predicts the magnetization change seen in a separate
high strength steel sample previously measured by the National Institute of Standards and
Technology.
Three sets of measurements on low-carbon, low-alloy high strength steel are
introduced in this research. The first experiment measured magnetostriction in steel rods
under uniaxial compressive stresses and magnetic fields. The second experiment
consisted of magnetostriction and magnetization measurements of the same steel rods
under the influence of bi-axially applied magnetic fields. The final experiment quantified
the small effect that temperature has on magnetization of steels. The experiments
demonstrated that the widely used approximation of stress as an “effective field” is
inadequate, and that temperatures between -50 and 100 °C cause minimal changes in
magnetization.
Preisach model parameters for the prediction of the magnetomechanical effect were
derived from the experiments. The resulting model accurately predicts experimentally
vi
derived major and minor loops for a high strength steel sample, including the bulging and
coincident points attributed to compressive stresses. A framework is presented which
couples the uniaxial magnetomechanical model with a finite element package, and was
used successfully to predict experimentally measured magnetization changes on a
complex sample. These results show that a 1-D magnetomechanical model can be
applied to predict 3-D magnetization changes due to stress, if adequately coupled.
vii
Table of Contents
Dedication ......................................................................................................................... iii
Acknowledgements .......................................................................................................... iv
Abstract .............................................................................................................................. v
Table of Contents ............................................................................................................ vii
List of Figures .................................................................................................................... x
List of Tables .................................................................................................................. xiii
Chapter 1 — Introduction ............................................................................................... 1
Chapter 2 — Literature Review ...................................................................................... 4
2.1 Hysteresis and Magnetization ................................................................................. 5
2.1.1 Anisotropies ................................................................................................... 5
2.1.2 Temperature ................................................................................................... 6
2.1.3 Stress .............................................................................................................. 6
2.1.4 Magnetic Relaxation and Viscosity ............................................................... 7
2.2 Magnetostriction ..................................................................................................... 7
2.3 Magnetostriction and the Magnetomechanical Effect Specific to Steels................ 9
2.3.1 General Properties of Steels ......................................................................... 10
2.3.2 Crystalline Structure .................................................................................... 12
2.3.3 Anisotropies ................................................................................................. 13
2.3.4 Magnetomechanical Effect in Steels ............................................................ 14
2.3.5 Magnetostriction in Steels............................................................................ 15
2.4 Hysteresis Models ................................................................................................. 15
2.4.1 Jiles/Atherton Model .................................................................................... 16
2.4.2 Preisach Models ........................................................................................... 17
2.4.2.1 Classical Preisach................................................................................ 18
2.4.2.2 Della Torre/Pinzaglia/Cardelli (DPC) Model ..................................... 21
2.4.2.3 Della Torre/Oti/Kádár (DOK) Model ................................................. 23
2.4.2.4 Complete Moving Hysteresis (CMH) Model ..................................... 24
2.4.2.5 Stress-dependent Preisach ................................................................... 26
2.5 Magnetostriction Modeling ................................................................................... 28
2.5.1 Schneider/Cannell/Watts Model .................................................................. 29
viii
2.5.2 Jiles/Sablik Model ........................................................................................ 30
2.5.3 Della Torre/Reimers Model ......................................................................... 31
Chapter 3 — Experiments and Characterization ........................................................ 33
3.1 Experiment #1 - Magnetostriction with Respect to Rolling Direction ................. 33
3.1.1 Experiment #1 Abstract ............................................................................... 33
3.1.2 Experiment #1 Introduction ......................................................................... 34
3.1.3 Experiment #1 Methods ............................................................................... 35
3.1.4 Experiment #1 Results ................................................................................. 38
3.1.5 Experiment #1 Discussion ........................................................................... 42
3.1.6 Experiment #1 Conclusions ......................................................................... 44
3.2 Experiment #2 - Magnetostriction with Bi-Axial Applied Magnetic Fields ........ 45
3.2.1 Experiment #2 Abstract ............................................................................... 45
3.2.2 Experiment #2 Introduction ......................................................................... 45
3.2.3 Experiment #2 Methods ............................................................................... 47
3.2.4 Experiment #2 Model .................................................................................. 50
3.2.5 Experiment #2 Results ................................................................................. 51
3.2.6 Experiment #2 Discussion ........................................................................... 56
3.2.7 Experiment #2 Conclusions ......................................................................... 57
3.3 Experiment #3 – Temperature Dependence of Hysteresis .................................... 57
3.3.1 Experiment #3 Abstract ............................................................................... 58
3.3.2 Experiment #3 Introduction ......................................................................... 58
3.3.3 Experiment #3 Methods ............................................................................... 59
3.3.4 Experiment #3 Results ................................................................................. 61
3.3.5 Experiment #3 Discussion ........................................................................... 65
3.3.6 Experiment #3 Conclusions ......................................................................... 65
Chapter 4 — Numerical Model Development .............................................................. 67
4.1 Modeling Abstract ................................................................................................ 67
4.2 Modeling Introduction .......................................................................................... 67
4.3 DOK Stress-Dependent Preisach Model ............................................................... 68
4.4 Comparison Experiment ....................................................................................... 70
4.5 Coupling Framework and Finite Element Model ................................................. 71
ix
4.6 Model Results and Discussion .............................................................................. 73
4.7 Model Conclusions ............................................................................................... 76
Chapter 5 — Conclusions and Future Work ............................................................... 77
5.1 Summary of Findings ............................................................................................ 77
5.2 Future Work .......................................................................................................... 79
List of Published and Pending Papers .......................................................................... 81
References ........................................................................................................................ 82
Appendix A — Modeling Framework in Detail ........................................................... 89
x
List of Figures
Figure 2-1: Effect of an applied field on an acicular particle ........................................................... 8
Figure 2-2: Effect of field on magnetostrictive materials ................................................................. 9
Figure 2-3: Phase diagram for steels ............................................................................................ 12
Figure 2-4: Elementary hysteresis operator .................................................................................. 19
Figure 2-5: The α-β half plane of the Preisach model ................................................................... 20
Figure 2-6: First order reversal curves for determining Preisach parameters ............................... 21
Figure 2-7: Details of a critical surface and related energy landscape ......................................... 23
Figure 2-8: Diagram of a CMH hysteresis loop ............................................................................. 25
Figure 2-9: Modified definitions of the staircase function L(T) in the α-β half plane ..................... 27
Figure 3-1: Magnetostriction of a high strength steel sample ....................................................... 34
Figure 3-2: Detailed sample diagrams........................................................................................... 36
Figure 3-3: Simplified experimental setup schematic .................................................................... 37
Figure 3-4: Magnetization vs. applied field under -175 MPa compression for cylinders cut
parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate ................. 38
Figure 3-5: Magnetization vs. applied field under increasing compression................................... 39
Figure 3-6: Magnetostriction vs. magnetization for “zero” applied stress for cylinders cut
parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate ................. 39
Figure 3-7: Magnetostriction vs. magnetization for -125 MPa applied stress for cylinders cut
parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate ................. 40
Figure 3-8: Susceptibility (differential) vs. applied field for “zero” applied stress for cylinders
cut parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate ........... 40
Figure 3-9: Susceptibility (differential) vs. applied field for -175 MPa applied stress for
cylinders cut parallel, perpendicular, and 45 degrees to the rolling direction from a
steel plate ................................................................................................................................ 41
Figure 3-10: Major loops (intrinsic magnetic flux density vs. applied stress) for cylinders cut
parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate ................. 41
xi
Figure 3-11: Major loops (intrinsic magnetic flux density vs. applied stress) for a cylinder cut
parallel to the rolling direction from a steel plate .................................................................... 42
Figure 3-12: Difference between the transverse field measurements and the uni-axially
applied compressive stress and field measurements ............................................................. 47
Figure 3-13: Sample holder for transverse field test ..................................................................... 49
Figure 3-14: Magnetization vs. longitudinal applied field .............................................................. 52
Figure 3-15: Magnetization vs. longitudinal applied field (zoomed in) .......................................... 52
Figure 3-16: Normalized magnetization versus longitudinal applied field for one of the
samples cut 45 degrees to the rolling direction of the steel plate ........................................... 53
Figure 3-17: Normalized magnetization versus field ratio for cylindrical samples cut parallel,
perpendicular, and 45 degrees to the rolling direction from a steel plate ............................... 53
Figure 3-18: J-H loops for one of the samples cut 45 degrees to the rolling direction of the
original steel plate ................................................................................................................... 54
Figure 3-19: J-H loops for the same 45 degree samples shown in Figure 3-18 ........................... 54
Figure 3-20: Single Domain Model output for a sample cut parallel to the rolling direction of
the steel plate .......................................................................................................................... 55
Figure 3-21: Magnetostriction versus magnetization for transverse fields applied to samples
cut perpendicular to the rolling direction of the steel plate...................................................... 55
Figure 3-22: NIST HSS sample for temperature measurements .................................................. 60
Figure 3-23: Magnetization versus applied field (major loops) for the toroidal-like NIST steel
sample ..................................................................................................................................... 62
Figure 3-24: Magnetization versus applied field zoomed in (major loops) for the toroidal-like
NIST steel sample ................................................................................................................... 62
Figure 3-25: Magnetization versus applied field (minor loops) for the toroidal-like NIST steel
sample ..................................................................................................................................... 63
Figure 3-26: Magnetization versus applied field zoomed in (minor loops) for the toroidal-like
NIST steel sample ................................................................................................................... 64
xii
Figure 3-27: Magnetization versus applied field (major and minor loops) for the toroidal-like
NIST steel sample ................................................................................................................... 64
Figure 4-1: Comparison of DOK minor loop predictions versus measured data for parallel
cylindrical rod steel sample ..................................................................................................... 70
Figure 4-2: Example DOK stress-dependent model output .......................................................... 70
Figure 4-3: NIST high strength steel sample for model comparison ............................................. 71
Figure 4-4: COMSOL®
model of the NIST toroidal-shaped high strength steel sample with
simulated drive coils ................................................................................................................ 72
Figure 4-5: B-H loop for σz = 0 MPa .............................................................................................. 73
Figure 4-6: B-H loop for σz = 160 MPa (tension) ........................................................................... 74
Figure 4-7: B-H loop for σz = 400 MPa (tension) ........................................................................... 74
xiii
List of Tables
Table 3-1: Sample compositions ................................................................................................... 35
Table 3-2: Sample compositions for each experiment .................................................................. 59
Table 4-1: DOK model parameters for each of the stress values ................................................. 69
Table 4-2: NIST tensile stresses and the equivalent compressive stress data sets used ............ 72
Table 4-3: RMS error percentages for each leg of the measured versus predicted data ............. 75
Table 4-4: RMS error percentages for the measured versus predicted data (by stress) .............. 75
1
Chapter 1 — Introduction
High strength steels are incredibly useful materials. With their high strength-to-
weight ratios, they are often used as the base material for drive shafts, structural steel
building supplies, and the hulls of naval vessels. It is unfortunate, therefore, that the
magnetic properties of high strength steels are routinely overlooked in both the
engineering and scientific communities. While there has been some effort expended on
the characterization of iron and electrical steels over the last century, the focus of most
research institutions has drifted more towards the study of “giant” magnetostrictive
materials like Terfenol-D. On the surface, this is a logical diversion of attention, as the
applications for steels in which detailed knowledge of their magnetic and
magnetostrictive properties are pivotal to their use seem quite limited. However, there
are a number of areas, such as non-destructive measurements of torque, in which a robust
model predicting the magnetic and magnetostrictive properties of steel would be prudent
to have.
While usually known for its relative strength, a lesser known property of steels is that
their ferromagnetic content exhibits extremely useful side effects such as the magnetic
response of the steel to changes in applied and internal strains. Because the measured
permeability of steel is affected by stress, voids, cracks, impurities, etc., it is possible for
engineers to use magnetic sensors in order to perform non-destructive testing on things
like pipes, buildings, and machinery. Additionally, since steel exhibits magnetostrictive
properties, a wise engineer can harness the interaction between magnetic fields and
changes in physical length to actively monitor torque and shear stresses without any
additional sensors. For example, MagCanica, Inc. markets a torque sensor that uses the
2
existing drive shaft steel as the active material. Finally, it is interesting to note that while
steel exhibits a relatively minute magnetomechanical effect, its application is usually
done in such a large scale that these properties tend to produce large effects. This is an
important characteristic in areas in which these magnetic properties are viewed as
negative side effects, as in the magnetic signature control of naval vessels.
In order to take advantage of these secondary material qualities, a rigorous set of
experiments must be undertaken to fully characterize high strength steel and a model
must be designed and validated. Additionally, due to the complex geometry of most steel
applications, any model created should be capable of being implemented into a 3-D finite
element modeling (FEM) package. Although there have been attempts to create a model
capable of characterizing these effects in high strength steels, a reliable and robust model
like the one needed does not currently exist in the literature.
In order to make accurate numerical predictions, a new semi-empirical model should
be constructed based on the physics of magnetostriction, and parameters determined
empirically from laboratory measurements. This characterization would require accurate
measurements of the magnetic properties of the steel with respect to temperature, stress,
and applied fields. The data can then be used to condition a semi-empirical physics based
model of stress-induced changes of the magnetic properties. Combined with commercial
FEM solvers, a validated magnetostriction model such as this one could be used to
predict the performance of transducers, linear actuators, and torque sensors for load-
bearing shafts manufactured from high strength steels. This dissertation details a set of
experiments which were undertaken on high strength steels, and the resulting material
characterization model as implemented in a numerical modeling software package.
3
This dissertation includes 5 chapters organized in the following manner. Chapter 1
gives a brief overview of the background for this effort and the necessary experiments for
the creation of a semi-empirical model.
Chapter 2 gives a full literature review for the dissertation, highlighting the
information which pertains to the design of the experiments and development of the
numerical model framework. These topics of interest include: hysteresis,
magnetostriction, the magnetic properties of steel, existing models of hysteresis and
magnetostriction, and the existing research pertaining to measurement and modeling of
magnetostriction in steels.
Chapter 3 gives an overview of the experiments which have been completed in order
to characterize the magnetic properties of the high strength steel. This chapter includes
the details of each experiment and highlights some of the results for each.
Chapter 4 details the creation of the Preisach material characterization model, the
details on its implementation in a numerical modeling framework, and the resulting
comparison to an independently taken data set.
Chapter 5 details a summary and conclusion for each of the experimental and
modeling efforts, and outlines areas of future study which are beyond the scope of this
dissertation.
4
Chapter 2 — Literature Review
A comprehensive study on the magnetic properties of high strength steels will
necessarily span a number of fields in science and engineering. Magnetostriction of high
strength steels has been used for the non-destructive testing of mechanical structures such
as oil pipelines, as well as the magnetic sensing of stresses due to torques. Additionally,
magnetic properties of steels have been used to determine the electrical losses incurred in
laminated electrical steel transformers by accurately predicting the hysteresis and eddy
currents within these components. Furthermore, the various applications of these
magnetostrictive and magnetic properties are inextricably influenced by the
micromagnetic crystal structure of iron based materials, as well as the environment in
which they are used. In order to address these various components competently, a
comprehensive literature review was undertaken in order to capture an adequate survey of
the state of the art for the field while avoiding the limitless dialogue possible for each of
these vast subjects. This chapter will outline the findings of a literature review for the
basis of this research effort.
Section 2.1 Hysteresis and Magnetization discusses the effects and causes of
hysteresis in materials, and how those effects determine the magnetization. Section 2.2
Magnetostriction defines the concept of magnetostriction and the predominant
explanations for its occurrence. Section 2.3 Magnetostriction and the
Magnetomechanical Effect Specific to Steels identifies the crystalline structures and
unique magnetic attributes of steel with respect to other magnetostrictive materials.
Section 2.4 Hysteresis Modeling details the past and current modeling efforts to
characterize and predict hysteresis in materials. Finally, Section 2.5 Magnetostriction
5
Modeling explores the extension of the hysteresis modeling capabilities to predict
magnetostriction, and the capabilities and limitations of the current state of the art steel
magnetostriction models.
2.1 Hysteresis and Magnetization
The study of hysteresis and magnetization in ferromagnets can take place on many
scales, ranging from the quantum interactions of atoms, to the micromagnetic viewpoint
of distributed magnetic domains formed by regions of magnetically saturated material.
The magnetic domain theory, first presented by Weiss in 1906, combined the competing
components of ferromagnetism theory at the time into a single elegant explanation
[CUL09, p. 116], [WEI06]. The key tenets to this theory were that: (a) magnetic
moments were always magnetically saturated, (b) those moments were grouped together
in magnetic domains, and (c) the spacing, rotation, and interaction of those domains was
the process which gave ferromagnets their bulk magnetic properties [JIL91, p. 109]. The
interaction of these magnetic domains with their neighbors in the presence of an applied
magnetic field gives rise to the hysteresis which we associate with ferromagnetic
materials. By definition, hysteresis is irreversible, and the area of the B-H loop
corresponds to the hysteretic energy loss (in the form of heat) throughout the cycle
[CUL09, p. 224]. Some of the most influential effects on the hysteresis of materials are
listed in the following sections.
2.1.1 Anisotropies
Anisotropies within magnetic materials can be attributed to the crystal lattice structure
or to previous material handling, i.e. heat treatments, rolling, etc. These anisotropies can
be exploited in order to alter the magnetostriction of a material [MEL11]. Electrical
6
transformers are similarly made of laminated electrical steels arranged to take advantage
of rolling induced anisotropies, which produce higher permeabilities along the rolling
direction [SHI11].
Ferromagnetic anisotropy constants may be determined from single crystal specimens
by fitting measurements to mathematically predicted curves, but only if great care is
taken not to impart additional strains to the material during the measurements [WIL37].
If the material being tested is not a single crystal, principal stress axes can be determined
for exploitation with a field rotation method or some derivative thereof [FAN12].
However, while the literature describes a number of anisotropies regarding the handling
of iron and nickel, it tends to lack a comprehensive study of the effects of built-in
anisotropies from the handling of steels (i.e. from cold working materials).
2.1.2 Temperature
Raising the temperature of ferromagnetic materials above the Curie temperature TC
will randomize the directions of magnetic domains and reduce the total magnetization to
zero. It is not surprising then that the application of temperatures below but near TC have
been shown to have a measureable effect on the hysteresis and magnetization of materials
[HIR65]. However, at temperatures which are much less than TC, the changes in
magnetization are much smaller. These temperature variations have been shown to
follow a Brillouin function [CUL09, p. 124].
2.1.3 Stress
The effect of stress on magnetization is defined as the magnetomechanical effect,
which is the inverse of magnetostriction. Through the magnetomechanical effect,
magnetization is generated from the application of stress. The application of stress in this
7
instance must be done in the presence of a magnetic field, or else the domains will all
rotate in random directions and leave the sample with a net magnetization of zero. A
compressive stress will serve to align the domains with their longest dimension
perpendicular to the direction of the applied stress. If this compression is done under the
influence of external magnetic fields, the magnetization of the sample will be altered due
to the presence of many (small) aligned magnetic dipoles. Taken as a whole over the
sample, these tiny alignments can add up to significant changes in the magnetization.
Bozorth collected many measurements of iron, nickel, and other alloys under
compressive and tensile stresses [BOZ93]. His measurements identified interesting
coincident points within the B-H loops for iron under various compressive stresses, as
well as for nickel when it was placed under tension [BOZ93, p. 606]. Additionally, the
application of stress has even been shown to increase or decrease the Curie temperature
in these materials [KOU61].
2.1.4 Magnetic Relaxation and Viscosity
Although most of the change in magnetization of a sample under the influence of an
applied field happens instantaneously, a small percentage of the magnetization shows
time dependence. These time-dependent characteristics are referred to as magnetic
relaxation and magnetic viscosity. The slow (on the order of minutes to months)
magnetic relaxation of most ferromagnetic materials refers to the eventual loss of
magnetization over time, and can be greatly affected by temperature [KIT46]. Magnetic
viscosity, also referred to as “creeping”, can be measured when an applied field is held
constant and the magnetization can slowly change by up to 1% [BUL01],[BUL02a].
2.2 Magnetostriction
8
Magnetostriction is a mechanical strain caused by an applied magnetic field. The
phenomenon of magnetostriction has been discussed thoroughly in the literature. Brown
detailed a theoretical explanation of magnetostriction based on the movement of domain
walls; where he averaged their behavior and created a model which was capable of
qualitative predictions with only magnetic data and crystal coefficients as inputs
[BRO49]. In Figure 2-1, magnetic domains are idealized as ellipses in order to illustrate
the effect of an external magnetic field applied to a magnetostrictive material. The
ellipsoidal shape is a consequence of non spherically-symmetric electron orbits. The
orbit’s spatial direction can be altered by an applied magnetic field via the spin-orbit
interaction.
Figure 2-1: Effect of an applied field on an acicular particle
[DEL97]
Each magnetic domain, which essentially contains an internal dipole, will tend to turn
and align with an applied magnetic field. This alignment serves to physically elongate
the structure in the direction of the magnetic field. A transverse applied field would have
the opposite effect, causing the structure to shrink in size along the longitudinal
dimension. This change in length, divided by the overall length of the sample, is what we
define as magnetostriction. This effect can be seen in Figure 2-2.
9
Saturation magnetostriction λS, which is defined at the maximum magnetostriction, is
the most commonly used coefficient to describe the magnetostriction of a material. This
can tend to be more complicated in iron based materials such as steel that exhibit “Villari
reversal”, where increasing the magnetic field will result in an initial peak of
magnetostriction, but increasing it further will lead to a gradually decreasing value for λ
(see section 2.3).
Figure 2-2: Effect of field on magnetostrictive materials
[CUL09, p. 257]
Most of the recent literature that may be found on magnetostriction focuses on
Terfenol-D, which is a highly magnetostrictive material. Once placed under the
appropriate mechanical and magnetic conditions, Terfenol-D has proved to be an
excellent material to use in actuators and transducers [MOF91]. However, there has
recently been an increase in interest on the magnetostriction of steels, as their magnetic
properties have been shown useful for the non-destructive measurement of strains due to
torques.
2.3 Magnetostriction and the Magnetomechanical Effect Specific to Steels
Steel shows inherently different material characteristics than giant magnetostrictive
materials like Terfenol-D. A striking difference is shown in the “Villari reversal” seen in
10
some high strength steels. This effect is characterized by the magnetostriction reaching a
maximum value and then decreasing with larger applied fields instead of reaching a
saturation value. Currently, the literature does not fully explain this phenomenon, but it
is believed to be caused by the polycrystalline nature of steel. However, this is just one
of the many unique differences which need to be explored if the magnetic behaviors of
high strength steels are to be fully exploited.
Nondestructive testing is a perfect example of how one would use this knowledge.
Flaws and defects within steels distort magnetic field lines and leave telltale signs of flux
leakage exterior to the surface of the material [JIL90]. These leakage paths allow
engineers to identify and treat the problem areas in steel structures. Similar methods
have been deployed to determine localized surface characteristics which are independent
from the averages volume magnetic properties [VAN07]. Finally, the determination of
inhomogeneities within steels has been accomplished in a preliminary experimental form
by the “drag force method” [GAR08] which can non-destructively indicate changes in
permeability and hysteresis loss in electrical steels. Techniques such as these are
currently being applied to investigate flaws in steel structures before they lead to
catastrophic failures within the materials. However, to fully utilize the useful properties
of ferromagnetic materials such as steels, one must first investigate the causes of these
phenomena.
2.3.1 General Properties of Steels
Steels are mainly comprised of carbon and iron, with additional elements added
throughout the process to affect different material properties. Effects of the normal
chemical element additions on magnetization of steels are well documented. Examples
11
of these effects are the hardening capabilities provided by manganese, and the added
ductility and shock-resistance provided by small additions of silicon [BET72]. Elements
commonly added besides those listed above include: phosphorus, sulfur, nickel,
chromium, molybdenum, vanadium, copper, boron, lead, nitrogen, and aluminum.
High strength steels are created by the careful selection of alloy materials and a
specific set of thermal treatments. Generally, these treatments require a cycle of steps
involving heating the steel ore to high temperatures, and then rapid cooling via
immersion in water or oil (known as “quenching”). Heating the steel above 750 °C leads
to the formation of austenite, which is a stable phase of steel with a face centered
crystalline structure [MCR46]. Without quenching, the steel’s carbon atoms would
redistribute via diffusion as the temperature slowly dropped, and the steel would attain a
stable body-centered cubic structure (ferrite). However for high strength steels, the
quenching process leads to the formation of martensite, which is a much harder form of
ferrite and has a tetragonal structure. This structure differs from body-centered-cubic due
to the inclusion of large amounts of carbon atoms which could not diffuse out of the
crystal structure quickly enough with the quenching process [MCR46]. For most steels,
the process of heating and quenching is repeated until an 80% martensite to 20%
austenite balance is achieved [BET72]. This mixture leads to the most desirable
combination of ductility and strength. Figure 2-3 shows a general phase diagram for
steels with respect to carbon content.
12
Figure 2-3: Phase diagram for steels [COM13] This diagram shows what temperatures are needed to enter the austenitic phase for steels, given a certain carbon percentage. Martensite is found within the Ferrite phase. High strength steels usually contain less than 0.2% carbon. Cementite is the Fe3C compound.
2.3.2 Crystalline Structure
Steels show inherently different magnetomechanical and magnetostrictive effects in
part due to their crystalline structure. This structure is cubic in nature and varies
depending on the additional elements and the mechanical treatment of the material.
Heaps made some of the first comprehensive magnetic measurements of steel, iron, and
magnetite in which he identified most of the characteristics contributed to these materials
[HEA23]. Properties such as the Villari reversal and magnetostriction are attributed to
crystal arrangements and interactions, including those imparted from mechanical
treatments of the materials such as rolling. Heaps also postulated that the ordinary
magnetostriction curve could be the sum of two or more curves which are determined by
13
the crystalline structure of the material; a point which was taken advantage of by
ElBidweihy et al. [ELB12].
More recent investigations have determined that crystalline grain structures at the ~10
µm level can greatly influence the magnetic properties of steels, and are determined
mainly by heat treatments [JIL88a]. The application of different heat treatments on
samples taken from the same bulk material creates drastically different microstructural
patterns. Of these samples, those with a martensitic structure had the highest coercivities
and hysteresis losses compared to those with pearlite or bainite microstructures which
consists of layers of ferrite and cementite, [JIL88a]. High strength steel, which is
comprised of martensite with very small amounts of austenite, mimics these properties.
2.3.3 Anisotropies
If steels are rolled or cold worked in a specific direction, they can exhibit
magnetostrictive and magnetic anisotropies in those directions [BOZ93, p. 638].
Attempts to quantify the effect of these rolling-induced anisotropies have been met with
limited success. Del Veccio was one of the first to come up with a model which utilized
a statistical averaging to approximate the distribution of domain angles from the rolling
direction in electrical steels [DEL84]. Additional characterization efforts have been
undertaken, which were generally able to quantify the energy losses and hysteresis loops
in strip samples cut at angles parallel and transverse to the sheet rolling direction
[FIO02].
One of the only efforts to investigate the effect of anisotropies in non-electrical steels
was just recently carried out on pipeline steels [GRO08]. In this paper, Grössinger shows
that mild steel samples with larger grain sizes, which were elongated in the direction of
14
rolling, show larger magnetostrictions and coercivities even with ferrite/pearlite
microstructures. This texture induced anisotropy should be larger in martensitic
structures, as they will retain more internal strains. However, a similar experiment
measuring the magnetostriction in low carbon steels with parallel and perpendicular
applied fields and with parallel applied compressive stresses showed little change with
respect to sample orientation from rolling direction [YAM96]. It is difficult to rectify
these seemingly contrary results, as they are on separate materials and were completed
with varying degrees of accuracy.
2.3.4 Magnetomechanical Effect in Steels
The earliest comprehensive experiments to quantify the magnetomechanical effects in
steels were undertaken by Langman [LAN85], [LAN90]. These measurements detail the
effects of stress on the magnetization characteristics of a mild steel, but stop short of
explaining the phenomenon behind the results. Bulte continued the research on steel
samples where Langman had left off, and produced a theory as to the origins of the
magnetomechanical effect; highlighting interesting coincident points which arise from
the plotting of B-H loops at different compressive stresses [BUL02b].
Recently, Perevertov has published a number of papers which outline the influences
of residual and compressive stresses on the hysteresis observed in mild and electrical
steels [PER08], [PER12]. His papers also indicate a clear example of the coincident
points, as well as bulging in the B-H loops of the sample due to the application of stress.
It is interesting to note that these measurements were taken with respect to tension being
applied to the sample, whereas previous measurements of high strength steels have
shown these characteristics when under compressive forces [LAN85], [WUN09]. This is
15
indicative that the mild steels measured by Perevertov are negative magnetostrictive
materials [PER08], [PER12], while the high strength and mild steels measured by Wun-
Fogle, Langman, and Sablik et al. are positive magnetostrictive materials [WUN09],
[LAN85], [LAN90], [SAB87].
2.3.5 Magnetostriction in Steels
Although closely related to the magnetomechanical effect, the study of
magnetostriction in steels is a subject unto itself. Once again, most of the research in this
field is related to the application of electrical steels, where designers seek to minimize
power losses from hysteresis, vibration, and noise. In these applications, however,
electrical steels are usually manufactured into thin sheets which are then placed together
in laminated structures to act as magnetic cores for transformers. Accordingly, some of
the theories presented for the magnetostrictive deformation [HIL05] are useful but not
entirely applicable to those which would be needed for high strength steel applications.
However, the magnetostriction of both types of steels is similarly influenced by the
cutting techniques used to create the samples, in that there is an inherent shift in the
magnetostriction curves due to the presence of built in stress domain patterns from
machining [KLI12]. Moreover, while these materials are similar in many ways, a new
model must be created for the magnetostriction of high strength steels in order to capture
the unique characteristics and circumstances in which they are used.
2.4 Hysteresis Models
While extensive studies of hysteresis have been completed over the years, the theories
presented to explain the nature of this phenomenon are mostly incomplete. However,
although these studies have faced mixed success, there have still been a number of
16
advances in the predictive modeling of these characteristics. Several hysteresis modeling
techniques have been developed, from energy based predictions, to more novel concepts
relating hysteresis to predator-prey pursuit curves [BUL09]. The most promising model
presented is the Preisach model of hysteresis, which utilizes a phenomenological
approach to quantify the effects of hysteresis. This section will explore each of the
leading hysteretic models, and then give a full background as to why the Preisach model
is the most promising.
2.4.1 Jiles/Atherton Model
One of the most comprehensive hysteresis models to date was created by Jiles and
Atherton. This model is based around the use of an “effective field” Be, which is based on
the Weiss mean field, and expresses the field which magnetic moments in a specific
domain will experience [JIL84]. Previous theories for hysteresis simply addressed wall
motion and were insufficient for describing materials with imperfections, which act as
domain “pinning” sites. Jiles and Atherton used a combination of Maxwell-Boltzmann
statistics and Langevin functions to accurately describe the domain rotations and wall
movements in the presence of material imperfections (pinning sites). This model works
very well for predicting the anhysteretic curve, and can predict major and minor loops for
certain materials (although the latter loops are forced closed in order to satisfy the
conditions of the model). This is to be expected, as one of the core tenets of the model is
that a ferromagnet’s magnetization will approach the anhysteretic curve at equilibrium
[JIL84].
The Jiles/Atherton model was altered a number of times, and eventually expanded to
incorporate temperature and stress dependencies in the form of calculated energy
17
densities [HAU09], [NAU11]. Sablik et al. have used these altered theories to predict
biaxial stress effects on hysteresis in steels [SAB99], utilizing the “stress demagnetization
factors” described by Schneider et al. [SCH92].
Viana et al. have also used these modified Jiles magnetization laws in order to predict
the magnetomechanical effects in a ferromagnetic cylinder under (internally applied)
hydrostatic pressure in order to validate methods for magnetic signature reduction for
naval vessels [VIA10], [VIA11a]. In their efforts, they extended the theories of Jiles in
order to incorporate shape dependent demagnetizing fields. They applied the resulting
magnetomechanical models to a non-trivial shape via mathematically dividing their
geometry into a number of discretized volume elements (much like a FEM package
would “mesh” a given geometry before solving), and solving over each element. Viana
et al. claim to have matched the measured and modeled data to within 5%. However, it
should be noted that their use of the model incorporates a number of fitting parameters
which are determined through the use of a least squares algorithm [VIA10].
The Jiles/Atherton model captures a number of the magnetic properties of a given
material. However, the main issue with this modeling approach is that it has a number of
fitting parameters, which for a given data set will make the model fit extremely well, but
will not apply directly to a different data set. Since a general model of hysteresis for high
strength steel is the desired output of this Ph.D., this fact eliminates the Jiles/Atherton
modeling approach. In contrast, the Preisach model has very few fitting parameters, and
in theory should be able to accurately capture the effects for all geometries given an
adequate series of measurements for parameter identification [PHI95].
2.4.2 Preisach Models
18
Ferenc Preisach developed his now famous model for hysteresis 78 years ago
[PRE35]. Since then, it has been successfully applied to a number of applications,
including the predictions for magnetic recording device behavior and for the control of
non-linear actuators. Over the last 30 years, this model has been modified a number of
times to address different phenomena. These different branches of the Preisach model
have been used to characterize: magnetic aftereffect, reversible and irreversible
magnetization, the influence of stress on hysteresis, and many more nonlinear effects.
This section will include an overview of the classical Preisach model as well as some of
the most prevalent offshoots from the literature.
2.4.2.1 Classical Preisach
The classical Preisach model is a scalar formation built upon the idea of an elemental
hysteresis operator, the hysteron. A Russian mathematician named Krasnoselskii added a
formulation to the Preisach model, which expressed hysterons in a purely mathematical
form, as described by Mayergoyz [MAY85]. In this formulation, a hysteron is the basic
switching node of the Preisach model, and can only have two possible values, 1 and -1.
In this hysteron, or γαβ operator, the points at which a single hysteron will switch values
are α and β, which are referred to as the “up” and “down” switching fields, and usually
defined with α > β. As such, a hysteron can be visualized as a rectangular loop as in
Figure 2-4.
19
Figure 2-4: Elementary hysteresis operator [MAY85]
Starting from the lower left branch, if the applied field H is increased from some H <
β, the hysteron’s output will remain -1 until H equals α, at which point the output of the
γαβ operator will be 1. Decreasing the magnetic field at that point will not change the
output of the operator until reaching the field β, at which the output will once again be -1.
It is in this way that the hysteron simulates the basic principle of hysteresis, in that it is
irreversible. The hysteron also requires the tracking of the history of the extremum field
values, as the field will only change when passing these points. When a set of hysterons
are combined with a weighting function µ(α,β), a formulation may be created, which
when integrated over all field values, can accurately model hysteresis without any
physical knowledge of the sources of it [MAY86]. Thus, the Preisach model can be
written in the form
( ) ∬ ( ) ( ) (2.1)
over the range of -HsβαHs. A useful visual tool for the Preisach model is the α-β half
plane, as shown in Figure 2-5.
20
Figure 2-5: The α-β half plane of the Preisach model
[MAY86]
In Figure 2-5, the area marked S-(t) refers to the hysterons which are outputting -1,
and similarly, S+(t) is the area corresponding to the hysterons outputting +1. The
interface between the two, L(t), is created through its attachment with the line designated
by α = β and is a series of links which are dependent on the extremum of the applied
fields (or other input). The last link of this line moves from bottom to top when
increasing field, and from right to left when decreasing [MAY86]. The summation of
each of the S(t) areas gives the output for the Preisach function.
The weighting function µ(α,β) is derived from a series of first order reversal curves
(FORCs). FORCs are the resulting curves from increasing the magnetic field from
negative saturation until some point (α,fα), and then decreasing that field to the point
(β,fαβ). The function F(α,β) is thus defined to be F(α,β)=(fα - fαβ)/2. The weighting
function is then the second order partial derivative of F(α,β). This relationship is
discussed in detail and applied with success in the literature [RES90] and shown in
Figure 2-6. In this way, FORCs can be used to fully characterize the irreversible
21
magnetic behavior of a material without knowledge of the contributing phenomena.
Figure 2-6: First order reversal curves for determining Preisach parameters [RES90]
In order to capture the reversible magnetization of a material, the Preisach model
must be expanded [DEL90], [DEL92]. These expansions can be seen in the moving
model and product model shown below, in which the reversible term of the magnetization
is computed from the bulk magnetic curves [DEL90]. The Extended Preisach model is
another variant of the classical approach, which uses a continuous distribution of
hysteron outputs from [-1, 1] instead of having two discrete values [OPP10]. This
extended version has shown nominal increases in accuracy over the classical model and
was created specifically for Terfenol-D actuators. Finally, there have been a number of
vector Preisach models created which extend the classical formulation to 2- and 3-
dimensions [CAR05].
2.4.2.2 Della Torre/Pinzaglia/Cardelli (DPC) Model
Della Torre et al. have altered the Preisach model for a myriad of reasons in the last
30 years. The Della Torre/Pinzaglia/Cardelli (DPC) model is one such variation,
whereby a time constant is added to the operative field. This addition enables the
22
Preisach model to accurately predict aftereffect, which is important for magnetic
recording applications [DEL98]. This model was then extended to a vector formulation
via the definition of a critical surface (CS) for each hysteron. The CS is defined as a
closed convex surface, which will be in the form of an ellipse in two dimensions or an
ellipsoid in three dimensions. Each CS serves the same switching-field purpose as the
scalar hysteron from the classical Preisach formulation, and will be asymmetrical about
the origin depending on how it interacts with neighboring particles. The CS for an
isotropic single domain in a zero applied magnetic field would be a circle or sphere
depending on the dimension modeled [DEL11].
The rules for computing the magnetization using the CS and the DPC model are
outlined by Della Torre et al [DEL10], [DEL11]. These rules generally dictate the
situations in which the magnetization will stay constant or follow a “conservative
function” of magnetization depending on whether the vector sum of the fields lies
internally or externally to the CS. A diagram of this relationship is shown in Figure 2-
7(a). Plotting the ΔH field with respect to the angle from the x axis θ will yield an energy
landscape of the hysteron as in Figure 2-7(b). Hysterons which enter the CS at various
angles with respect to the x axis will rotate until they fall in the closest local energy well.
For example, in Figure 2-7(b), a hysteron entering the CS at an angle of 220° will rotate
until it lies in the ~190° energy well despite there being a global minimum at ~350°.
23
Figure 2-7: Details of a critical surface and related energy landscape (a) Critical surface and the applied field (where the axes reflect arbitrarily-applied magnetic fields) (b) Corresponding energy landscape of the CS [DEL11]
Through the application of the CS, Della Torre et al. were able to model aftereffect
and accommodation while simultaneously preserving the saturation and loss properties,
which were shortcomings in the previous models [DEL07]. These properties can be
addressed using models based on the Stoner-Wohlfarth (SW) particles [STO91], but are
then limited by a lack of field interaction between the particles [EVA10]. Some
combinations of the SW and Preisach models have been recently attempted, and seem to
overcome these shortfalls [KOH00], [DEL06].
2.4.2.3 Della Torre/Oti/Kádár (DOK) Model
The Della Torre/Oti/Kádár (DOK) model combines pieces of a number of different
models in order to resolve differences between the Classical Preisach predictions and real
world measurements. This is a necessary combination, as the Classical Preisach model
can only represent irreversible magnetization changes and congruent loops. The DOK
model, therefore, utilizes a combination of the Classical Preisach model for the
irreversible components, and a moving or product model in order to incorporate the
reversible components.
24
The moving model is designed to compute the reversible component of magnetization
separately and then add it to the irreversible component [DEL90]. This separation allows
the accurate modeling of incongruent loops [CAR00]. This is accomplished by adding a
term of αM to the Preisach variables. In this addition, α is a parameter derived for each
material. In this model, the reversible magnetization is defined as
( )
( )
( )
( ), (2.2)
for . This addition to the Classical Preisach model has been used successfully
to model the hysteresis of magnetic recording media [REI98], [KAH03].
Another form of the DOK model utilizes the Product model in order to incorporate
incongruent loops and reversible magnetization. In the Product model, the reversible and
irreversible magnetizations are not treated as separate components [DEL90]. A
noncongruency function R(m) is utilized to keep the magnetization from exceeding
saturation during the modeling process, while a factor of β (normalized initial
susceptibility) produces reversible magnetization [DEL90]. However, due to the
intractability of the Product model, it tends to be more computationally expensive.
2.4.2.4 Complete Moving Hysteresis (CMH) Model
The Complete Moving Hysteresis Model, or CMH, is a moving-type Preisach model
which computes the irreversible and locally reversible components of magnetization
[VAJ93]. As such, the CMH still uses the moving parameter α, and then adds additional
complexity over the previous models. The main difference between this model and the
others is that it is state-dependent, in which the remanence of each hysteron is different
for the +1 and -1 states of the particle. The remanence states are given by
( ) ( )
25
( ) ( ) . (2.3)
Moreover, the CMH model uses a more realistic, non-rectangular hysteron loop. This
loop is comprised of a regular rectangular irreversible magnetization loop, as well as a
locally reversible magnetization component [DEL94]. The diagram of a CMH hysteresis
loop on the Preisach plane can be seen in Figure 2-8.
Figure 2-8: Diagram of a CMH hysteresis loop
Note the interaction field Hi and the critical field Hc [DEL94].
As seen in the figure above, the field which the magnetic particle is subjected to is
factorized into two parts: one part from the applied field and one from the interaction
field within the medium. The locally reversible magnetization is computed from a
Preisach integral which is dependent on the interacting fields.
A CMH model has already been developed for HTS (high tensile steel) steels in
which the seven Preisach parameters needed were obtained from the major hysteresis
loops and virgin curve only [KAH94]. Kahler defines the seven Preisach parameters as:
Saturation magnetization, Ms, squareness, S, zero field susceptibility, χ0, moving
26
parameter, α, critical field, hci, and the standard deviations for the interaction and critical
fields, σi and σc, respectively [KAH94].
The CMH shows a marked improvement over the previous iterations of the Preisach
model and the identification of the required parameters is well defined [DEL94]. As
such, the CMH model (or some variation thereof) would be a good candidate to base a
magnetostrictive and magnetomechanical model for high strength steel on. Additionally,
a CMH model can be successfully applied to FEM solutions through the use of lookup
tables without prohibitively increasing the complexity of the computations, which should
make the implementation of such a model straightforward [VAJ93]. However, the main
drawback of this model versus the DOK model is the increase in complexity and the
computational resources required.
2.4.2.5 Stress-dependent Preisach
When incorporating stress into the Preisach model, the approach taken in the
literature is usually a derivative of the “effective field” championed by Jiles/Atherton
[JIL84]. These stress-dependent Preisach models incorporate an effective field, He,
which depends on applied field, H, stress, σ, and the switching fields α and β [BER91].
By allowing each of the µ(α,β) operators to have their own effective field functions,
Bergqvist altered the classical Preisach model into the general form
( ) ∬ ( ) ( ( ) ( ) ) , (2.4)
which includes the same limits of integration -HsβαHs as in the classical Preisach
function [BER91]. It is interesting to note that this form could even be used to
incorporate temperature eventually. Special treatment of the staircase function L(t) from
the classical model is needed in this formulation, as it will take on a different shape with
27
the new definitions as seen in Figure 2-9.
Figure 2-9: Modified definitions of the staircase function L(T) in the α-β half plane [BER91]
Due to this added complexity, a number of rules are derived and defined in [BER91]
in order to simplify the resulting integrals for numerical implementation. In this
formulation, ( ) is defined as the maximum value of β at the point (β+ν,β) included
in the area. Similarly,
( ) is defined as the minimum value of B at the point
(β+ν,β) included in the area. Accordingly, a new definition of the staircase function
is made which negates the need for determining He explicitly, in which
( ) [ ( ) ( ) ( ) ( )] ( ) ( )
( ) ; (2.5)
and thus the magnetization can be computed directly from
( ) ∫ ( ( ) ( )) ( )
, (2.6)
as per Bergqvist, where ( ) ∫ ( )
. This model formulation was shown to
be approximately as accurate as the classical Preisach model [BER91].
Another study found in the literature in which the Preisach function was altered to
28
incorporate stress dependence was that of Ktena and Hristoforou [KTE12]. Their model
incorporates a vector Preisach model which utilized SW particles and a weighted set of
normal distributions in order to account for variations due to magnetoelastic coupling. In
addition to the Gaussian distributions used to capture the interaction effects, a function
was added to take into account the angular dispersion of easy axes. The measurements
they compared their model to were made on low carbon electrical steel under tension.
While the accuracy of the model is not stated explicitly, the model seems to only match
the measured data qualitatively; identifying the major characteristics of the material
under stress. Additionally, the steel which they are measuring and fitting to the model
tends to show the opposite magnetic properties as high strength steels, in that tensile
stress decreases their sample’s magnetic induction and coercivity [KTE12], whereas high
strength steels show the opposite effect [WUN09].
Jiles has previous stated that a Preisach model would be inappropriate for the
modeling of magnetostriction for non-destructive testing purposes [JIL88b] due to the
number of parameters that need to be tracked; however, with the increases in computing
power in the last 25 years, this should be less of an issue. The Bergqvist stress Preisach
model was applied successfully for the prediction of magnetostriction and
magnetomechanical effect in Terfenol-D despite the lack of computing power 20 years
ago [KVA92]; although, it is important to note that this was for a simple rod structure.
Nonetheless, the formulation of the Preisach model is in general a good fit for the FEM
process, and it would be feasible to apply the resulting vector Preisach models in an
appropriate software package with today’s technology.
2.5 Magnetostriction Modeling
29
The literature contains three main attempts to develop all-encompassing models for
magnetoelastic effects. They are discussed here in the order of appearance, and their
relative strengths and weaknesses. In general, the magnetostriction models listed below
follow derivations which branch from their associated hysteresis models listed in the
previous section. As such, the first two models discussed below are energy based
models, and the last one is a phenomenological vector Preisach based model.
2.5.1 Schneider/Cannell/Watts Model
The Schneider/Cannell/Watts (SCW) model utilizes a stress effective field which is
first defined by [BRO49] and is a product of only two functions: one of magnetization
and one of stress [PER12]. By this formulation, the internal field is defined as
Hi = H + Hσ - DσM, (2.7)
where Dσ=3λsσ/MsBs, and Hσ=-3λsσcos(θσ)/Bs. Accordingly, the change in magnetization
is found from the differential susceptibility by
( )
( )
, (2.8)
and ∑ ∫ ( ) , (2.9)
where the summation over i refers to all domain wall types, and fi is an appropriate
weight factor for each domain wall type [SAB94b]. Since the increases in coercive field
are not predicted by this model, the authors applied the “Kondorsky’s Rule”, in which
( ) ( ) [SCH92]. Despite this addition, and after the use of a number of
unsupported fitting parameters, the model does not fit the data well.
Sablik et al. extended this model to incorporate biaxial stresses by defining an
effective stress [SAB94b]. In this case the model behaved well qualitatively, capturing
30
most of the effects from the biaxial stress when the remanence was normalized
[SAB94a].
2.5.2 Jiles/Sablik Model
The Jiles/Sablik model is a magnetostriction model based on additions to the original
Jiles/Atherton hysteresis model. In as such, a stress-dependent term is added to the
effective field He to account for the effect of the stress on the hysteresis of the sample
[SAB87]. As per the original model, the magnetization of a ferromagnet is assumed to
approach the anhysteretic curve in the equilibrium state [JIL84]. The effective field is
thus altered to be
( ), (2.10)
where α is a mean field parameter for coupling between magnetic domains [SAB87], and
Hσ(σ,M) is a component added to incorporate the stress applied to the medium defined as
( )
(
) . (2.11)
Sablik gives a number of different possible values for the magnetostriction λ term,
and reports the various performance of each outcome [SAB87]. The change in
irreversible magnetization is thus defined as a derivation of the original Jiles/Atherton
result,
( )⁄ [ ( ) ]( ) , (2.12)
and the reversible magnetization is then found from Mrev = c(Man-Mirr), where c is the
ratio of the susceptibilities between the normal and anhysteretic magnetizations [SAB87].
Like the Jiles/Atherton model discussed earlier, this derivation suffers from the same
inconsistencies brought about by the use of fitting parameters and uncertainty over the
31
form of the magnetostriction function λ. However, this new model does qualitatively
capture the magnetoelastic effects shown in the mild steel samples tested, including the
decrease in susceptibility and the overall stationary nature of the coercive field.
2.5.3 Della Torre/Reimers Model
The Della Torre/Reimer’s model was created by the extension of the existing DOK
Preisach hysteresis model. The DOK model, as described in the previous sections,
utilizes a magnetization-dependent reversible magnetization, and has the general form of
[ ( ) ( )] , (2.13)
where S is the squareness, Ms is the saturization magnetization, and f(H) is the normalized
reversible component of the magnetization, equaling ( ( )) [DEL97]. The
symbol ξ is experimentally derived as the normalized zero field susceptibility. The a’s
listed are defined as
and
, (2.14)
where Mi is the irreversible magnetization [DEL97]. Della Torre and Reimers changed
this original formulation in order to incorporate magnetostrictive susceptibility, so that
the stress is thus defined as
( ) [ ( ) ( )] . (2.15)
The term ν is the ratio of magnetostrictive to magnetic susceptibility, and the constant K
is determined by the shape of the hysterons [DEL97]. Finally, the irreversible
magnetization is calculated in the same manner as before for the DOK model
(
), (2.16)
where the σ value is the standard deviation of the (Gaussian) switching field, and Hci is
the remanent coercivity. This early version of the Della Torre/Reimers magnetostriction
32
model was successful in replicating the magnetostrictive effects seen in Terfenol-D
[DEL97], but did not fully implement the moving parameter of the DOK model.
Further implementations involved a Fast DOK model version for bimodal materials
which did include the operative field (h = H + αM) [REI99a]. This Fast DOK method
was extended to a Simplified Vector Preisach Model (SVPM) to compute the irreversible
and reversible susceptibilities in a vector form [REI01b]. Finally, the SVPM was
coupled with a commercial FEM package for high anisotropic materials in [REI01a].
The robust mathematical derivations of the Della Torre/Reimers Preisach approach,
combined with the phenomenological nature of the model, suggested that it was the most
ideal magnetostrictive model type to build upon for this Ph.D. research. A model based
on this approach would be computationally fast and would yield improvements upon the
current state of the art for the characterization of magnetic and magnetomechanical
properties of high strength steels.
33
Chapter 3 — Experiments and Characterization
In order to characterize the high strength steel properties, a number of experiments
were made on high strength steel rod samples while varying key parameters. These
parameters include temperature, stress, and applied magnetic fields. In section 3.1,
Experiment #1 showcases measurements on high strength steel rods under different
applied magnetic fields and stresses. In section 3.2, Experiment #2 showcases
measurements of the rods with bi-axial magnetic fields applied. In section 3.3,
Experiment #3 showcases measurements on the rods under different applied magnetic
fields and temperatures. A magnetomechanical model developed from these experiments
and its application for predicting magnetization changes via a numerical modeling
framework is highlighted in Chapter 4.
3.1 Experiment #1 - Magnetostriction with Respect to Rolling Direction
Note: The results of this experiment was presented as an oral presentation at the 2012
INTERMAG conference in Vancouver, Canada. The paper was subsequently published
in IEEE Trans. Magn. [BUR12]
3.1.1 Experiment #1 Abstract
Previous studies on the magnetostriction in high strength steels have ignored the
internal anisotropies due to previous material handling. This experiment presents data
taken on rods of a high strength steel that have been machined parallel, perpendicular and
45° to the rolling direction. Magnetization, magnetostriction, susceptibility, and stress-
strain curves have been measured under various stresses and fields. In general, the
parallel cylinders showed altered B-H, susceptibility, and magnetostriction curves
compared to the other two orientations. These measurements were incorporated into a
34
Preisach model allowing detailed predictions of the magnetic state after stress and field
changes [ELB14a].
3.1.2 Experiment #1 Introduction
Steel shows inherently different material characteristics than giant magnetostrictive
materials like Terfenol-D. A striking difference is shown in the Villari reversal seen in
some high strength steels. Figure 3-1 illustrates an example this characteristic, shown by
the magnetostriction versus applied magnetic field for one of the parallel samples.
Figure 3-1: Magnetostriction of a high strength steel sample
This sample was oriented parallel to the rolling direction. These measurements were taken under a 25 MPa compressive stress. The data were broken into increasing and decreasing magnetic field legs and low pass filtered.
While previous work [WUN09] took measurements principally in a single direction,
we have taken into account directional anisotropies. Cold-rolling an iron alloy stretches
and distorts the magnetic domains in the direction of rolling [BOZ93, p. 638]. These
altered domain shapes impact the magnetic characteristics of the alloy; adding an
additional preferred direction of magnetization to the easy or hard axes within the
crystalline structure. Some previous measurements have mentioned directional
anisotropies within rolled steels but did not characterize the differences [GRO08]. Our
goal is to incorporate anisotropic stress-induced differences into a Preisach model.
35
Measurements reported here will be used to obtain insight into the vector nature of stress
induced magnetization.
In this section we report magnetic properties of high strength steels oriented parallel,
perpendicular and 45° to the rolling direction. These measurements include the
differential susceptibility, magnetization, major/minor hysteresis loops, major/minor
stress-strain loops, and magnetostriction under compressive stresses between -1 and -175
MPa.
3.1.3 Experiment #1 Methods
Solid cylinders with their longitudinal axis oriented parallel, perpendicular and 45°
with respect to the rolling direction were machined from each of three locations on the
original rolled plate of high strength steel, a total of nine samples. The sample
compositions are shown in Table 3-1 below. All of the samples were taken from the
same sheet of rolled steel, and compositional differences between locations are assumed
to be negligible. Consideration was given to the process of machining the samples from
the rolled sheet, as the effect of adding additional stress to steel samples has been well
documented [KLI12]. Sample orientations and a picture of one of the parallel cylinders
are shown in Figure 3-2.
Table 3-1: Sample compositions Principal additions to iron (in percent) for steel samples used in these measurements.
C Cr Ni Mo Mn Si Cu
0.16 1.37 2.68 0.25 0.27 0.3 0.11
36
Figure 3-2: Detailed sample diagrams
Diagram of the how samples were cut from the sheet of steel (top), as well as one of the parallel cylinders outfitted with a strain gauge (bottom).
Each cylinder was ~5.71 mm in diameter and ~55.8 mm in length. An MTS-858
hydraulic load frame compressed each sample under predetermined stresses while a
longitudinal applied field HL was varied. HL was actively controlled using a Hall probe
and a PI controller to reduce the effect of the stress and field dependence of the
magnetostrictive sample’s permeability. Strain was measured by two
MicroMeasurements WK-06-500GB-350 strain gages mounted on opposite sides of the
rod with AE-15 resin. Other details of the experimental setup are given in [WUN09].
The testing setup is shown in Figure 3-3.
Rolling direction
Pe
rpen
dic
ula
r
45o
Parallel
Gauged sample
37
Figure 3-3: Simplified experimental setup schematic
The closed-flux path provided by the load frame negates the demagnetization factor of the steel sample.
The experimental procedure started with a decreasing AC-field demagnetization of
each sample in the longitudinal direction. The demagnetization was completed at -1
MPa, which we have taken as zero applied stress due to the need for minor compression
to hold the sample in place.
For the major loop measurements with fixed stress, once demagnetized, the sample
was placed under the desired load, and the strain gauges zeroed. Each cylinder was
tested for 8 different fixed stresses between -1 and -175 MPa. Then HL was cycled over
± 90 kA/m. Each measurement set was preceded by this same demagnetization and
compression cycle.
For the major loop experiments with fixed HL, once demagnetized, the sample was
placed under the desired HL of the measurement and the strain gauges zeroed. Then the
applied longitudinal load was cycled between -1 and -175 MPa. Each measurement set
was preceded by this same demagnetization and applied field cycle for a total of ten fixed
HL values.
38
For the minor loop experiments with fixed HL, the procedure was nearly identical to
the major loop measurements with fixed HL except for two differences. First, the minor
loops were performed under four smaller-ranges of stress; between -1 and: -40, -80, -150,
and -175 MPa. Second, each measurement set was preceded by this same
demagnetization and applied field cycle, albeit for a total of four fixed HL values. Each
of the measurements listed above were repeated for all nine cylinders.
3.1.4 Experiment #1 Results
Figure 3-4 shows an example of the magnetization versus applied field loops,
comparing one set of the machined cylinders. As expected, the magnetization curves do
not start from zero. Due to the experimental procedure, the application of stress after the
AC demagnetization increases the starting magnetic induction before the application of
external magnetic field. The compressive stress (-175 MPa) can be seen in the tilting and
deformation of the loops. Figure 3-4 shows a typical B-H loop under compression.
Figure 3-5 shows a series of B-H loops under varying compression from -1MPa to -125
MPa for one of the parallel cylinders.
Figure 3-4: Magnetization vs. applied field under -175 MPa compression for cylinders cut parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate
39
Figure 3-5: Magnetization vs. applied field under increasing compression Data shown are for a cylinder cut parallel to the rolling direction of a steel plate.
The magnetostriction exhibits a similar difference between the orientations of the axis
of each sample with the rolling direction. The reversal of the magnetostriction at high
field values has been shown in previous measurements [WUN09]. The double “U” shape
seen in the magnetostriction vs. magnetization curves tends to align into a single “U”
shape at -100 MPa. The difference in magnetostriction during increasing and decreasing
fields shows mechanical hysteresis. Figure 3-6 shows magnetostriction vs. magnetization
for “zero” applied stress, and Figure 3-7 shows the same for -125 MPa applied stress.
Figure 3-6: Magnetostriction vs. magnetization for “zero” applied stress for cylinders cut parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate
Each curve has two parts: ascending and descending applied field. The curves with the minimum on the left are for the descending applied field, and the curves with the minimum on the right are for the ascending curves.
40
Figure 3-7: Magnetostriction vs. magnetization for -125 MPa applied stress for cylinders cut parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate
Note that the curves have collapsed into a single “U” shape.
Figures 3-8 and 3-9 show the difference in susceptibility as the applied stress is
increased. In Figure 3-8, the samples are under the “zero” stress situation, and show a
large distinction between the sample taken parallel to the rolling direction with the other
two directions. Figure 3-9 shows the same plot for the -175 MPa applied stress, yielding
a smaller and more aligned differential susceptibility. In each plot there are two parts,
corresponding to the increasing and decreasing field values.
Figure 3-8: Susceptibility (differential) vs. applied field for “zero” applied stress for cylinders cut parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate
Each curve has two parts: ascending and descending applied field.
41
Figure 3-9: Susceptibility (differential) vs. applied field for -175 MPa applied stress for cylinders cut parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate
Each curve has two parts: ascending and descending applied field.
Figures 3-10 and 3-11 illustrate the effect of an externally applied field on the
intrinsic magnetic flux density with respect to stress cycles. In Figure 3-10 we show an
example of stress cycles for steel cylinders machined with respect to rolling direction.
Figure 3-11 shows the effect of varying the applied field for one of the samples (a parallel
cylinder).
Figure 3-10: Major loops (intrinsic magnetic flux density vs. applied stress) for cylinders cut parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate
Data taken for ~2300 A/m applied field.
42
Figure 3-11: Major loops (intrinsic magnetic flux density vs. applied stress) for a cylinder cut parallel to the rolling direction from a steel plate
Data taken for ~4000 A/m and 0 A/m applied field. The applied field is held constant and stress is cycled from 0 to -175 MPa.
The magnetostriction and susceptibility data have been low passed with a 5 Hz
Butterworth filter to smooth out high frequency noise present in the testing apparatus.
This noise is due to a combination of building power noise (60 Hz), and the noise
associated with the piston load-frame apparatus used (the hydraulic piston requires a
constant position dither in order to overcome static friction). All other plots show
unfiltered data.
3.1.5 Experiment #1 Discussion
In all but one set of cylinders, the B-H curves for each sample show distinct
differences between the rods taken parallel to the rolling direction versus the
perpendicular and 45 degree rods. As increasing compressive loads are applied, the B-H
curves follow the same tilting pattern shown in Figure 3-5; showing that increasing
magnetic fields will need to be applied for the same relative changes in magnetization.
The peak susceptibility of the cylinders varied with respect to angle from the rolling
direction, and occurred near the field where the slope of the coefficient of
magnetostriction changes sign. However, there was some difficulty distinguishing
43
between minor loop measurements and the noise of the experimental apparatus at lower
field values due to the noise floor of the strain gauges.
The susceptibility plots tend to make the differences between rolling directions easier
to visualize. When the rods are placed under the “zero” applied strain condition, the
differences between the three rolling directions are the largest. As expected, applying
increasing loads to the samples tend to diminish the magnitude of the susceptibility, with
the added effect of reducing the relative differences between the rolling directions as the
stress anisotropy begins to dominate the rolling induced anisotropies.
The magnetostriction data show that at low applied stresses, the built in forces from
the rolling and machining of the cylinders tend to give more distinct characteristics to the
shape of the curves. A combination of physical and magnetic hysteresis is observable in
the difference in magnetostriction minima depending on increasing or decreasing field
legs. At larger applied stress levels, the magnetostriction curves increase in magnitude,
and align to have a more uniform response to external applied field.
The stress-strain curves for this material were linear within the applied stress values
chosen for the experiments (-1 to -175 MPa). This is to be expected, as high strength
steels are designed to have linear responses to stress well past the range of stress
parameters tested.
The intrinsic magnetic flux density vs. applied stress major loops for constant applied
field HL did not show as distinct or as repeatable variations with regard to the rolling
direction as the magnetostriction plots. While the stress applied varied the same amount
as before (-1 to -175 MPa), the measurements were carried out using much smaller
44
applied fields. This could have been the source of errors in the test setup with regards to
drift and noise in the magnetization measurements. The HL-field controller was unable to
maintain tight control at low applied field values. Accordingly, the smaller field range
could introduce errors due to the diminished differences between sample rolling
directions found at lower applied fields.
Figure 3-11 illustrates a typical reaction of the material to an applied external field
while cycling stress. At larger field values the loops shown tend to widen and dip
downwards, reflecting hysteresis in the sample. It is possible that a small dither used by
the load-frame to reduce static friction produced some of the sinusoidal-like-noise seen in
the measurements.
In all cases, the application of compressive load served to align the material
characteristics for each set of cylinders. The amount of compressive load which aligned
these parameters is related to the internal stresses caused by the cold rolling and
machining of the cylinders, with the point at which they align corresponding to the point
in which the external applied load negates the “built-in” directional anisotropies in each
sample.
3.1.6 Experiment #1 Conclusions
The results show interesting differences between the cylinders depending on their
orientation with respect to the rolling direction. In general, the parallel cylinders showed
altered B-H curves with larger peak susceptibility and correspondingly different
magnetostriction curves compared to the other samples. In related work, such
differences, including anisotropies, in material characteristics were incorporated into a
45
Preisach model to characterize the magnetomechanical responses of high strength steel
[ELB14a].
3.2 Experiment #2 - Magnetostriction with Bi-Axial Applied Magnetic Fields
Note: The results of this experiment was presented as an oral presentation at the 2013
HMM conference in Taormina, Italy. The paper is in print for Physica B [BUR14a].
3.2.1 Experiment #2 Abstract
A detailed knowledge of a material’s microscopic texture is required in order to
produce a realistic model of the magnetization process under applied fields. Previous
studies on the magnetostriction in high strength steels have ignored the internal
anisotropies due to prior material handling. To this end, a measurement utilizing two
perpendicular fields was designed to interrogate the magnetic texture and microstructure
of high-strength steel rods. These magnetization and magnetostriction measurements
were then fitted to an energy-based domain rotation model which had been altered to
address vector fields and uniaxial anisotropies. Given the simplicity of the model it is
surprising to see that it captures a number of the general trends in the Data, however the
fit is generally poor. Improving upon this data set will allow us to determine general
magnetic characteristics of microstructure in the steels. These measurements were
incorporated into a Preisach model allowing detailed predictions of the magnetic state
after magnetic field changes in multiple directions [ELB14b].
3.2.2 Experiment #2 Introduction
The magneto-mechanical effect in steels has been documented in the literature for
well over a century. Although steels exhibit very small magnetostrictions, their prolific
uses in industrial applications lead to situations in which those attributes may either be
46
harnessed or deemed undesirable. The magnetostriction of high strength steels is useful
in particular for noncontact torque sensing on high strength-to-weight ratio drive shafts
[WUN09].
Accordingly, many measurements and theories have been developed to characterize
the magnetostriction of steels. Most studies of magnetostriction in high strength steels
have ignored the internal anisotropies due to previous material handling. Cold-rolling
steels leaves magnetic domains stretched in the direction of rolling, allowing for an
additional easy axis to exist along with the six crystalline axes found in cubic lattices
[BOZ93]. Our previous work has shown differences in magnetic properties of high
strength steels with regard to their orientation relative to the rolling direction in sheet
steel [BUR12].
While we have been able to show magnetic differences between sample directions
due to the original forming of the material, it is not possible to fully characterize the
shapes of the magnetic domains within the bulk of the material. Preparing the surface of
a sample for microscopic measurements of domain shapes and sizes can alter the
characteristics which one is aiming to measure. Even if the anisotropies on the surface
are not destroyed during polishing, the visible surface domains which remain may not be
an accurate description of the domains within the bulk of the sample [CUL09]. Due to
this fact, we have devised a way to interrogate those domains via the application of
orthogonal magnetic fields and the rough fitting of the data to a magnetic rotation model.
This experiment is part of a cumulative effort which will eventually allow the
characterization of domain sizes, shapes, and distributions for use as future Preisach
modeling parameters.
47
In this section we report magnetic properties of a high strength steel oriented parallel,
perpendicular, and 45° to the rolling direction while under the influence of longitudinal
and transverse fields. These measurements include the differential susceptibility,
magnetization, and magnetostriction without additional applied stress. While our
previous work [BUR12] relied on applied compressive stress to deform the shape of the
J-H loop, these new measurements utilize a constant transverse field in order to induce
this domain-pinning. Figure 3-12 shows the difference between these measurement
types. The black lines (shown in a vertical top-down view of the cylinder samples)
indicate the direction of magnetic domain “pinning” due to the compressive stressed
and/or transverse fields applied. While the transverse field applied to the sample has a
similar overall effect as a compressive longitudinal stress, there are more options for
lowest energy states in the latter.
Figure 3-12: Difference between the transverse field measurements and the uni-axially applied compressive stress and field measurements
(a) The axially applied compressive stress experiment applied longitudinal field and stress, pushing the domains into any preferred direction along the central plane (b) the transverse field experiments apply a longitudinal field as well, but replace the stress with a transverse field, pushing domains towards a preferred direction close to the transverse field direction.
3.2.3 Experiment #2 Methods
Solid cylinders with their longitudinal axis oriented parallel, perpendicular, and 45°
with respect to the rolling direction were machined from each of three locations on the
original rolled plate of high strength steel, a total of nine samples. The sample
48
compositions and details are listed in our previous publication [BUR12]. All samples
were taken from the same sheet of steel, and differences in composition and heat
treatments are assumed to be negligible.
Each cylinder was ~5.71 mm in diameter and ~55.8 mm in length. A sample holder
was constructed which would position a steel rod, which was wrapped in longitudinal
field induction and pickup coils, between the iron pole faces of a large transverse field
magnet. This transverse magnet was used to apply a constant transverse field HT to the
samples while a longitudinal applied field HL was varied. Strain was measured by two
MicroMeasurements WK-06-500GB-350 strain gauges mounted on opposite sides of the
rod with AE-15 resin. The measurements of these strain gauges were averaged to give
the magnetostriction shown in the results section.
The water-cooled transverse magnet was capable of applying uniform fields in excess
of 1400 kA/m between the 15.24 cm diameter pole faces. The air gap between the pole
faces was measured to be approximately 7.5 cm. The sample holder was placed with the
longitudinal coil centered within this gap. A brass turnbuckle was utilized to hold the
sample tightly within the sample holder. Figure 3-13 shows a picture of the disassembled
and assembled sample holder.
49
Figure 3-13: Sample holder for transverse field test
Sample holder disassembled, showing longitudinal HL coil and sample between Al end-pieces (left). Sample holder fully assembled showing placement of coil and brass turn-buckle (center). Sample holder assembled and mounted between the poles of the transverse field magnet (right).
One drawback of this test setup was the lack of a closed flux path for the field lines.
While iron end-pieces were originally used to try and contain the flux within the sample,
they were found to disproportionally distort the field, creating a large dipole. This dipole,
acted upon by the transverse field, put a significant torque on the sample and distorted the
magnetostriction measurements. A finite element model was created in COMSOL
Multiphysics® in order to determine a suitable testing arrangement to alleviate the torque
imparted on the sample. As the strain measurements and pickup coils are located in the
center of the sample, it was determined that aluminum end-pieces would be sufficient for
holding the sample in place as they maintained a fairly constant field within the center
region of the steel rod. This did not negate the longitudinal demagnetization factor
within the sample, but the quadra-pole created did not feel the same realigning force from
the transverse field.
50
The experimental procedure started with setting HT = 0 A/m, and the taking of
calibration points at each longitudinally applied field extrema for later data correction.
After the calibration points were obtained, a decreasing AC-field demagnetization was
performed on each sample in the longitudinal direction. The demagnetization was
completed at HT = 0 A/m. The temperature within the longitudinal coil was monitored
and attempts were made to keep it fairly constant. However, the design of the sample
holder made it impossible to cool the sample directly, so a correction was made to the
data for the temperature-dependent drift of the strain gauges throughout the course of the
measurements.
For the major loop measurements, once demagnetized, the sample was set to the
desired HT and the strain gauges zeroed. Then HL was cycled over ±80 kA/m. Each
cylinder was tested for 10 different fixed transverse fields between 0 and 1400 kA/m.
Each measurement set was preceded by this same demagnetization and application of HT
cycle. Each of the measurements listed above were repeated for all nine cylinders.
3.2.4 Experiment #2 Model
The model used to fit the data was developed from a previous Energy-based domain
rotation approach [RES06]. This single-domain rotation model utilizes the same αi’s as
direction cosines of the magnetization with respect to the cubic crystal axes, but applies
them in a different manner. In our notation below, z refers to the longitudinal direction,
and x refers to the transverse field direction. The energy equation used is given by two
terms, and is different for each individual rolling direction j (parallel, perpendicular, and
45 degree)
∑ , (3.1)
51
where Ms is the saturation magnetization, Hi is the field applied in each direction, and
Kuniaxial is the anisotropy found in the sample due to the rolling of the original steel. The
value changes based on the direction of the sample being measured as follows
, (3.2)
This serves as a useful approximation for the effects of the transverse fields on the
sample magnetizations. As they were calculated in [RES06], the strain S and
magnetization M are calculated from energy weighted averages incorporating a
smoothing factor Ω:
[∑ ( ⁄ )] [∑ ( ⁄ )] and
[∑ ( ⁄ )] [∑ ( ⁄ )] , (3.3)
where and .
3.2.5 Experiment #2 Results
Figure 3-14 shows an example of the magnetization versus longitudinally applied
field loops for one of the parallel cylinders. As expected, the magnetization curves do
not start from zero. Due to the experimental procedure, the application of the transverse
field after the AC demagnetization increases the starting magnetic induction before the
application of longitudinal magnetic field. The increasing transverse fields can be seen in
the tilting and deformation of the loops. Figure 3-15 shows the same J-H loop with the
center portion enlarged. It is interesting to note that the center of the J-H loops tend to
skew upwards and towards the negative longitudinally applied field direction when a
certain transverse field is applied.
52
Figure 3-14: Magnetization vs. longitudinal applied field
Shown for various transverse fields for a cylindrical sample cut parallel to the rolling direction of a steel plate.
Figure 3-15: Magnetization vs. longitudinal applied field (zoomed in)
Shown for various transverse fields for a cylindrical sample cut parallel to the rolling direction of a steel plate.
The effect of the transverse field is difficult to quantify in this test setup. A useful
visualization tool is the plotting of a number of J-H loops, and then calculating the
corresponding decrease in “saturation magnetization” Js (which we are truncating to be at
HL = 80 kA/m) with each increase in the applied transverse field. An example of this
analysis is shown in Figures 3-16 and 3-17. In Figure 3-16, the Data shows a series of J-
H loops with 10 transverse fields between 0 and 1400 kA/m for one of the parallel
samples. While harder to see in this graph, the center of the high-transverse field J-H
loops show the same shift to the upper left quadrant as seen in Figure 3-15. Figure 3-17
shows the normalized values of Js versus the calculated ratio of the perpendicular fields
53
inside of the steel cylinders. In this calculation, we have used a demagnetization factor of
0.5 for the transverse field direction, and 0.04 for the parallel field direction. The “zero”
field ratio corresponds to a maximum longitudinal applied field of 80 kA/m, and a
transverse field of zero. The field ratio of 5 indicates an internal transverse field five
times larger than the longitudinal field applied.
Figure 3-16: Normalized magnetization versus longitudinal applied field for one of the samples cut 45 degrees to the rolling direction of the steel plate
These curves were taken for increasing transverse field values; note that the crossover of the highest transverse field J-H loops have shifted to the left of the origin.
Figure 3-17: Normalized magnetization versus field ratio for cylindrical samples cut parallel, perpendicular, and 45 degrees to the rolling direction from a steel plate
The magnetization is normalized to the saturation for 0 HT (the other magnetization values are taken at 80 kA/m) versus a calculated ratio of the HT and HL in the sample. The horizontal axis corresponds to the ratio of HT (with a Demag factor of 0.5) to the maximum HL.
54
Figure 3-18: J-H loops for one of the samples cut 45 degrees to the rolling direction of the original steel plate
These curves lie congruent to each other at zero transverse applied field.
Figure 3-18 shows the J-H loops for the three 45 degree samples that were measured.
At zero transverse field, these loops are congruent. Increasing the transverse applied
field to a certain point yielded a separation of these loops. This effect is shown in Figure
3-19.
Although it was generally inappropriate for fitting the data, the single domain rotation
model did predict the general shapes and features of the measured data. An example of
the model output is shown in Figure 3-20.
Figure 3-19: J-H loops for the same 45 degree samples shown in Figure 3-18
These curves taken with 637 kA/m HT. Note the separation of the J-H loops, showing inhomogeneity in the samples.
55
Figure 3-20: Single Domain Model output for a sample cut parallel to the rolling direction of the steel plate
The model under-predicts the effect of HT, but roughly captures inflection points and general trends.
Figure 3-21: Magnetostriction versus magnetization for transverse fields applied to samples cut perpendicular to the rolling direction of the steel plate
Each curve shows the magnetostriction versus magnetization for a different constantly applied transverse field for one of the perpendicular samples. Note: The red curves correspond to increasing longitudinal applied field, while the black curves correspond to decreasing applied longitudinal field. H// here refers to HT.
Accurate measurements for the magnetostrictions were obtained for the lower
transverse field values. The peak magnetostriction decreases from 8.16 ppm to 4.41 ppm
when HT is increased from 159 kA/m to 637 kA/m, as seen in Figure 3-21. The
difference in magnetostriction during increasing and decreasing fields shows mechanical
hysteresis. The magnetostriction data have been low passed with a 0.5 Hz Butterworth
filter, and then averaged between increasing and decreasing legs to smooth out some of
the noise present in the testing apparatus. This noise is due to a combination of building
56
power noise (60 Hz), and thermal noise in the strain gauges. All other plots show
unfiltered data.
3.2.6 Experiment #2 Discussion
As expected, for every cylinder measured, the “saturation magnetization” (which we
truncate at HL = ±80 kA/m) decreased with increasing HT. Increasing HT also led to a
dramatic decrease in the susceptibility and coercive field for each loop.
The domain rotation model is useful for visualizing the effects of the transverse field
on the magnetic domains in the sample. As the field in the transverse direction increases,
the percentage of domains aligned with the longitudinal direction are decreased, and the
domains elongate in the transverse plane. Once the transverse field is large enough, it
irreversibly pins these domains in the transverse direction and reduces their contribution
to the longitudinal magnetization, tilting the M-H curve. Usually, the model is assumed
to have a constant distribution of direction cosines, but this version has been altered to
include anisotropies due to the rolling direction.
While our measurements show distinct differences between the rolling directions at
lower transverse fields, the application of larger transverse fields tended to blur those
differences until they were indistinguishable. This is in agreement with our previous
measurements. However, this experiment is unique in that samples which initially had
the same characteristics (Figure 3-18) were shown to have different responses in larger
transverse fields (Figure 3-19), which was not observed before. These discrepancies are
assumed to stem from the experimental differences highlighted in Figure 3-12.
When the rods are placed under the zero applied field condition, the differences
between the three rolling directions are the largest, as the built-in forces from the rolling
57
and machining of the cylinders tend to give more distinct characteristics to the shape of
the curves. Applying increasing transverse fields to the samples tend to diminish the
magnitude of the magnetostrictions, with the added effect of reducing the relative
differences between the rolling directions, as inhomogeneities in the samples begin to
overcome the uniaxial anisotropy from the rolling. As in our previous measurements
[BUR12], a combination of physical and magnetic hysteresis is observable in the
difference in magnetostriction minima depending on increasing or decreasing field legs.
At larger transverse field levels, the magnetostriction curves decrease in magnitude, and
align to have a more uniform response to external applied field.
In all cases, the application of larger transverse applied fields tended to align material
characteristics for each set of cylinders until the inhomogeneities of the samples blurred
these differences.
3.2.7 Experiment #2 Conclusions
The results show interesting differences in magnetization versus applied field
between the cylinders depending on their orientation with respect to the rolling direction,
but it was found that these differences were not distinguishable at higher transverse field
values. In general, the model used captured the general trends and shapes of the M-H
curves, but failed to accurately predict the magnitude of the effect of the transverse fields.
This is expected as the model was originally designed to characterize single domains
instead of multi-domain materials. These measurements proved to be very useful in
visualizing the interaction of magnetic domains within steel samples, and they were used
in related work to develop parameters for a new Preisach model [ELB14b].
3.3 Experiment #3 – Temperature Dependence of Hysteresis
58
3.3.1 Experiment #3 Abstract
The models derived from the results of Experiments 1 and 2 were made under the
assumption that magnetization is independent of temperature. However, since high
strength steels are utilized in a number of situations where there can be large variations in
temperature, it is necessary to quantify these effects. To test for temperature dependence
of the magnetization of high strength steel, an experiment was designed in which major
and minor hysteresis loops were measured on a non-trivial geometry while subjected to
constant temperatures. The sample used was measured previously by the National
Institute of Standards and Technology for the simultaneous application of longitudinal
stresses and transverse magnetic fields. In this new experiment, we have shown that the
variation with respect to temperature of magnetization in high strength steels is much
greater than predicted for minor magnetization loops, and confirms the effects which
were predicted in the literature for major magnetization loops for similar materials.
3.3.2 Experiment #3 Introduction
The temperature dependence of magnetization in iron based materials has been well
documented in the literature. Raising the temperature above the Curie temperature, TC,
has been shown to randomize the orientation of magnetic domains, leading to a zero net
magnetization within ferromagnetic materials [BOZ49, p. 713]. Due to this effect, there
has been a great deal of research performed at high temperatures and at temperatures near
phase transition regions; however, the behavior of magnetism in steel at temperatures
well below TC has not been as well documented. Moreover, while there have been a
number of efforts to develop models and theories which explain the temperature
59
dependence of magnetization [RAG09], [RAG10], these tend to address materials which
are unrelated to steels.
Iron, which is a major component of high strength steels, has a TC of ~726 °C
[CUL09, p. 120]. Higher temperatures (close to the phase transitions highlighted in
Figure 2-3) can significantly alter the crystalline state of the steels, and thus their
magnetic properties. While steels usually only reach these temperatures during the initial
forging period, they are regularly operated within the temperature range of -50 and 100
°C, which was subsequently chosen for these measurements. In this temperature range
well below TC, the temperature dependence of magnetization in steels is greatly reduced.
While there have been predictions which assume that steel will behave according to the
laws outlined for iron, the accuracy of our model requires us to check this assumption. In
this section, we report on magnetization measurements of high strength steel with respect
to constant temperatures and varying applied magnetic fields.
3.3.3 Experiment #3 Methods
The sample used for this measurement was previously measured by the National
Institute of Standards and Technology (NIST) [PET91]. In the previous experiments,
NIST measured the B-H, susceptibility, and normal induction curves for the steel under
tensile and compressive stresses which were transverse to the applied magnetic field.
The high strength steel used for this sample was the same type characterized in our
previous experiments, [BUR12], [BUR14a], but with slightly different chemical
composition as listed in Table 3-2.
Table 3-2: Sample compositions for each experiment Sample principal additions to iron (in percent).
Data C Cr Ni Mo Mn Si Cu
Solid cylinders 0.16 1.37 2.68 0.25 0.27 0.3 0.11
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NIST 0.16 1.57 2.78 0.42 0.26 0.19 0.00
A diagram of the sample can be seen in Figure 3-22.
Figure 3-22: NIST HSS sample for temperature measurements [PET91]
From left to right: (a) three plane cutout view of the sample with windings, (b) The dark shaded area is considered to be the active area due to the wrapping of the coils.
The NIST sample is a hollow cylinder made from high strength steel, with the
necked-down center region of smallest outer diameter designated as the “active region”.
A drive coil consisting of 64 turns of copper wire wound between slots machined into the
sample was used to apply circumferential magnetic fields Hφ, and a similarly wound
pick-up coil was used to measure the magnetic flux density Bφ.
For this experiment, the NIST sample was placed within a large non-magnetic
Tenney series 942 environmental chamber, which is capable of sustaining temperatures
between -50 and 100 °C. The sample was centered in the environmental chamber, and
had its own independent drive and sense coils. Each measurement started with setting the
61
desired temperature, measuring the resistance of the drive coil for calibration, and then
performing an AC demagnetization cycle on the sample. After allowing the system to
cool back to the set temperature, the flux meter was zeroed and then the measurements
were made. Measurements were taken at a three constant temperatures (-50, 25, and 100°
C) while the sample was subjected to major Hφ loops between ± 11 kA/m, and minor Hφ
loops between ± (2.39, 1.59, 1.19, 0.80, 0.66, 0.53, and 0.3978) kA/m. The minor loop
measurements were preceded by the application of a large negative and then large
positive Hφ to facilitate centering of the B and H values during post-processing. The
application of large fields before the minor loops increased repeatability of the
measurements, as the sample was more likely to start from the same magnetic state for
each measurement. Each measurement set was preceded by this same calibration and
application of Hφ cycle.
The temperature within the environmental chamber was monitored and attempts were
made to keep it fairly constant. However, the presence of the drive coils and their heating
during magnetic field applications made it impossible to prevent the temperature from
increasing ~2 to 4° C over the course of a measurement. The measurements were made
slowly to compensate for this inductive heating.
3.3.4 Experiment #3 Results
Figures 3-23 shows an example of the magnetization versus applied field loops for
the NIST sample under the three temperatures tested. Figure 3-24 shows the same J-H
loop with the center portion enlarged. As expected, the saturation magnetization, MS,
decreases with increasing temperature. The loops begin at slightly different
62
magnetization values due to order of demagnetization and the zeroing of the fluxmeter
before the measurement at each desired temperature.
Figure 3-23: Magnetization versus applied field (major loops) for the toroidal-like NIST steel sample Data shown for various temperatures (in °C). The data starts at negative saturation,
proceeds to positive saturation, and then returns to positive saturation.
Figure 3-24: Magnetization versus applied field zoomed in (major loops) for the toroidal-like NIST steel sample Data shown for various temperatures (in °C). The data starts at negative saturation,
proceeds to positive saturation, and then returns to positive saturation.
63
The increasing temperature can be seen in the tilting and reduction in magnitude of
the magnetization loops. Interestingly, the temperature difference does not have as large
of an effect on the coercivity or coercive field for the steel within the center portion of the
J-H loop. Figure 3-25 shows the effect of temperature on minor J-H loops, and Figure 3-
26 shows the same loop with the center portion enlarged. It is interesting to note that the
minor loops show much larger differences in magnetization than the major ones wrt
temperature. Since the slope is so steep within the center region of the J-H loop, small
differences in the approach history have a much larger effect.
Figure 3-25: Magnetization versus applied field (minor loops) for the toroidal-like NIST steel sample Data shown for various temperatures (in °C). The data starts at negative saturation,
proceeds to positive saturation, and then performs a minor loop between ±1.19 kA/m.
64
Figure 3-26: Magnetization versus applied field zoomed in (minor loops) for the toroidal-like NIST steel sample Data shown for various temperatures (in °C). The data starts at negative saturation,
proceeds to positive saturation, and then performs a minor loop between ±1.19 kA/m.
Figure 3-27 shows an example of a set of major and minor J-H loops for high strength
steel in 100°C.
Figure 3-27: Magnetization versus applied field (major and minor loops) for the toroidal-like NIST steel sample Data shown for various major and minor loop field values for 100°C. The data starts at
negative saturation, proceeds to positive saturation, and then performs each loop.
65
Differences between repeated minor loops are assumed to be from heating of the
drive coil during the measurement process and magnetic viscosity within the material.
3.3.5 Experiment #3 Discussion
As expected, the “saturation magnetization” (which we truncate at Hφ = ± 11.93
kA/m) decreased with increasing T. For iron, Bozorth predicted that the saturation
magnetization, MS, would decrease by ~0.8% between -50 and 100° C [BOZ49, p. 717]
as predicted by the equation
⁄
⁄ (3.4)
where IS is the saturization magnetization, T is the absolute temperature in Kelvin, I0 is
the saturation magnetization at T = 0, and θ is the Curie temperature, TC, in Kelvin. For
this calculation we have assumed TC = 770 °C and I0 = 1.95 T. For the high strength steel
measured here, the actual differences in magnetization varied from ~2% for the major
loops and up to ~8% for the minor loops measured.
Some of these differences are to be expected, as steel has a number of impurities
which would not have been expected in Bozorth’s calculation for iron. However, it is
interesting to see the rather large difference seen in the minor magnetization loops, given
the fact that increasing temperature did not lead to a measurable change in the
susceptibility and coercive field for each loop. Future efforts to characterize the
magnetization in steels should take into account the temperature fluctuations within
experimental setups, as they can lead to appreciable variations in magnitude.
3.3.6 Experiment #3 Conclusions
The results of this experiment show interesting differences in magnetization with
respect to temperature, but it was found that these differences vary depending on the
66
magnitude of the applied fields. Major loop magnetization measurements of high
strength steels between -50 and 100°C were identical to within ~2% error. Moreover,
while it was shown that variances in magnetization due to temperature can be as much at
8% when high strength steels are subjected to lower magnetic fields, it is generally an
accurate assumption to assume that temperature has very little effect. In future modeling
efforts it will be important to be mindful of the effect of temperature on the magnetic
properties if accuracy above this threshold is required.
67
Chapter 4 — Numerical Model Development
Note: This model was presented as a poster presentation at the 2013Magnetism and
Magnetic Materials Conference in Denver, Colorado. The paper, which is titled
“Application of a DOK Stress-dependent Preisach Model through a Numerical Model,” is
in print for Journal of Applied Physics [BUR13b].
4.1 Modeling Abstract
Many magnetomechanical models fit the data sets they were originally developed
from very well, but are not transportable to different data sets or geometries. In order to
test the portability of a previously developed Della Torre-Oti-Kadar (DOK) stress-
dependent Preisach model for high strength steels, a numerical model was implemented
to replicate data taken on a non-trivial geometry made of the same material. The data
used for comparison were measured previously by the National Institute of Standards and
Technology (NIST) on a toroidal-like sample which allowed for the simultaneous
application of longitudinal stresses and transverse magnetic fields. The geometry was
modeled in a finite element modeling package and coupled with a DOK model via
material parameters. A coupling framework was developed and B-H loops were modeled
and compared to the NIST data available with some agreement. In the future this
modeling approach should be extended to incorporate more complex field and stress
interactions and applied to additional data sets as available.
4.2 Modeling Introduction
Previous studies of the magnetostriction in steels have led to models which fit well
for a given physical experiment, but are not transportable to secondary data sets without
additional curve fitting [PHI95]. To address these shortfalls, we have extended a
previously developed Della Torre-Oti-Kadar (DOK) model, and then applied this model
68
to predict magnetization changes in a complex geometry through the use of numerical
modeling.
Our previous studies on the magnetostriction in high strength steels have yielded a
strong set of uniaxial stress and field data [BUR12], as well as biaxial field data
[BUR14a]. These measurements were made on simple solid cylinders, which negated the
need to consider complex geometrical relations. A DOK model was developed on the
former measurements which accurately predicted the effects of stress on major loop
magnetization in high strength steels [ELB14a]. This new effort has extended that DOK
model to predict minor magnetization loops as in [DEL99], [DEL90], and applied the
resulting model numerically to a separate existing data set.
The data used for verification of the coupled method were measured by the National
Institute of Standards and Technology (NIST) on a sample which was made of the same
high strength steel as our previous measurements [BUR12], [BUR14a]. Details of the
measurements are shown in [SCH92], [PET91]. Accordingly, a finite element modeling
(FEM) method was coupled with the existing DOK model in order to predict the
magnetization changes within a non-trivial geometry.
4.3 DOK Stress-Dependent Preisach Model
For simplicity, the Preisach differential equation method is used in this section to
compute the magnetization for a given field sequence. A useful approximation is to
assume that the Preisach function is Gaussian. More details about the method and the
approximation can be found in [DEL99]. We have implemented this method using a
MATLAB® program. The program’s input is the desired sequence of the applied field
values, and the resolution. Its output is the normalized magnetization. A simple
69
reversible magnetization component using the DOK model was also included in the
algorithm [DEL90]. The model was calibrated to the solid cylinder data from our
previous paper [BUR12] for 8 different fixed compressive stresses between -1 and -175
MPa, and varying applied magnetic fields between ± 90 kA/m, and details of the model
can be found in [ELB14a]. The parameters for used for each stress value are shown in
Table 4-1, where σi is the standard deviation of the interaction field, σk is the standard
deviation of the critical field, hk is the mean switching field, S is the squareness, and χ0 is
the zero-field susceptibility.
Table 4-1: DOK model parameters for each of the stress values Negative stress values refer to compression.
Stress (MPa) σi σk hk S χ0
-1 0.1 0.05 0.15 0.76 0.9
-25 0.12 0.05 0.15 0.76 0.95
-50 0.15 0.05 0.15 0.76 0.98
-75 0.20 0.05 0.15 0.76 1.01
-100 0.25 0.05 0.15 0.76 1.05
-125 0.28 0.05 0.15 0.76 1.1
-150 0.3 0.05 0.15 0.76 1.15
-175 0.32 0.05 0.15 0.76 1.2
Figure 4-1 shows a comparison of the DOK minor loop model predictions compared
to measured data derived from our previous measurements [BUR12]. An example of
more complex output of the model for a decreasing-cycling field starting at positive
saturation can be seen in Figure 4-2.
70
Figure 4-1: Comparison of DOK minor loop predictions versus measured data for parallel cylindrical rod steel sample This data was taken from our previous simple rod measurements for -1 MPa
(compressive) stress [BUR12]. The data starts at positive saturation, proceeds to negative saturation, and then performs a minor loop between 950 and -850 A/m.
Figure 4-2: Example DOK stress-dependent model output This model output is for -125 MPa compressive stress and varying the applied field, H, according to the input sequence (9.3, -8.5, 7.6, -5.1, 3.4, -1.7, and 0.8) kA/m
4.4 Comparison Experiment
The data used to test our coupled model results against were previously gathered
during a joint experiment between the Naval Surface Warfare Center, Carderock Division
and NIST [SCH92], [PET91]. This experiment utilized a hollow toroidal-shaped sample
71
of high strength steel which had been machined to allow for the application of stress
transverse to the application of magnetic fields. A picture of the sample can be seen in
Figure 4-3. The high strength steel used for this sample was the same type characterized
in our previous experiments, [BUR12], [BUR14a], but with slightly different chemical
composition as listed in Table 3-2.
Figure 4-3: NIST high strength steel sample for model comparison NIST sample showing half of the drive coils used to apply circumferential magnetic fields, and the threaded ends used to apply transverse stresses
The NIST sample is a hollow cylinder, with the necked-down center region of
smallest outer diameter designated as the “active region”. The forces applied in the
longitudinal direction of the sample were calibrated to apply a desired stress σz within
this region. A drive coil consisting of 64 turns of copper wire wound between slots
machined into the sample was used to apply circumferential magnetic fields Hφ, and a
similarly wound pick-up coil was used to measure the magnetic flux density Bφ. This
sample was placed under constant σz ranging between ± 400 MPa, and then subjected to
Hφ between ± 8 kA/m.
4.5 Coupling Framework and Finite Element Model
Prediction of the B-H loops for the NIST sample was completed in a number of steps.
First, a structural-analysis model was built in COMSOL Multiphysics® to calculate the
longitudinal σz, radial σr, and circumferential stresses σφ of the NIST sample. The
application of a tensile (positive) σz creates corresponding compressive (negative) σφ and
72
σr stresses. Since the magnetic fields are applied in the φ direction, we assumed that the
σφ stresses would have the greatest effect and should be used for the selection of material
parameters. The structural model revealed that the NIST sample had two regions of
roughly constant σφ values for each σz measured: one σφ for the active-region and one σφ
for the rest of the sample (non-active region). Equivalent stress values were determined
via rounding to the closest stresses measured in the solid cylinder data sets [BUR12] and
used to generate the parameters for the Preisach model. The parameter sets selected are
listed in Table 4-2.
Table 4-2: NIST tensile stresses and the equivalent compressive stress data sets used All stresses are shown in MPa, with negative values indicating compression and positive values indicating tension [PET91]
NIST σz Active region equivalent σφ
Non-active region equivalent σφ
0 -1 -1
160 -50 -1
400 -125 -25
The parameter sets were used to generate B-H curves for each of the sample regions
via the DOK Preisach model implemented in MATLAB®. These curves were then
imported into COMSOL® as material properties in a stationary magnetic fields
simulation. Figure 4-4 shows the FEM model with the magnetic field drive coils in
place.
Figure 4-4: COMSOL®
model of the NIST toroidal-shaped high strength steel sample with simulated drive coils
73
From left to right: (a) lengthwise view, and (b) end on view showing hollow center. The “active region” is shown within the dashed red line.
NIST assumed that these drive coils acted as a uniform solenoid over the active
region, and calculated [PET91] that the current within the coil would generate an average
magnetic field of:
( ⁄ )
( ), (4.1)
where Np is the number of coil turns, I is the current, and ro and ri the inner and outer radii
of the active region in the sample. A maximum I for the COMSOL® model coil of 8.094
A was derived from this equation and the reported maximum field of ± 8 kA/m.
4.6 Model Results and Discussion
Figures 4-5 through 4-7 show the results of the COMSOL® simulation for 0, 160, and
400 MPa, respectively. For figure clarity, only the top half of each B-H loop is plotted.
As expected, the application of tension in a transverse direction to the magnetic field
causes the B-H curves to tilt downwards and increase in coercive field.
Figure 4-5: B-H loop for σz = 0 MPa The NIST data [PET91] is the solid line, and the COMSOL
® prediction is the dashed line.
74
Figure 4-6: B-H loop for σz = 160 MPa (tension) The NIST data [PET91] is the solid line, and the COMSOL
® prediction is the dashed line.
Figure 4-7: B-H loop for σz = 400 MPa (tension) The NIST data [PET91] is the solid line, and the COMSOL
® prediction is the dashed line.
These are the same characteristics one would expect to see from the application of
compressive stress and magnetic field along the same axis. The decrease in saturation
magnetization and bulging of the loops near the origin were previously shown [BUR12],
[BUL02b], [PER12].
It is interesting that the 0 MPa B-H loop shows the worst fit out of the three. In
Figure 4-5, the reported saturation magnetization for the NIST sample was ~13% higher
than for the solid cylinders we have measured [BUR12]. This is to be expected to some
extent, as the equivalent σφ data available from the solid cylinder measurements were
actually taken under -1 MPa compressive stress; however, the discrepancy is too large to
75
be solely contributed to this fact. To compare the measured and modeled data further, the
data were interpolated to a common grid via MATLAB®, and a root mean square (RMS)
error percentage was calculated using the following formula:
√∑ (( )
)
√∑ (( ) )
(4.2)
where N is the number of points in the data sets, MEASn is the data measured by NIST for
the nth
data point, and NUMn are the solutions predicted by the numerical model
framework for the nth
data point. The results of the RMS calculations for each increasing
or decreasing magnetic field leg are shown in Table 4-3 below.
Table 4-3: RMS error percentages for each leg of the measured versus predicted data All errors are shown in percentages. Data were split into individual increasing- and decreasing-magnetic field legs.
Data Set RMS Error Percentages
σz = 0 MPa increasing-field leg 23.21
σz = 0 MPa decreasing-field leg 16.15
σz = 160 MPa (tension) increasing-field leg 9.58
σz = 160 MPa (tension) decreasing-field leg 9.07
σz = 400 MPa (tension) increasing-field leg 6.48
σz = 400 MPa (tension) decreasing-field leg 9.59
The total RMS error results per stress value (which each include one increasing and
one deceasing magnetic field leg) are shown in Table 4-4.
Table 4-4: RMS error percentages for the measured versus predicted data (by stress) All errors are shown in percentages. Data from the up and down legs for each stress were combined for the total RMS errors.
Data Set RMS Error Percentages
σz = 0 MPa total 19.53
σz = 160 MPa (tension) total 9.29
σz = 400 MPa (tension) total 8.45
76
The differences between the data and predictions could stem from a number of
factors. As shown in Table 3-2, the materials are not identical even though they are the
same type of steel. Also, the NIST measurement applied stress after demagnetizing the
sample, whereas the solid cylinder measurements applied stress before demagnetization.
Another likely source of error could be how Hφ was estimated, as a solenoid
approximation was assumed to determine I. Any errors in that approximation would be
compounded in COMSOL®
, especially at higher fields, as we have used that same
calculation to determine what I to use numerically. NIST also reported that they had
difficulty calculating the flux density, as they were not able to install the pickup coils flat
on the sample surface. This meant that they could not measure the area of the sensing
coil as accurately, and could lead to overestimation of Bφ. However, despite the errors
present in the measurements and modeling, the fit for the B-H loops under stress are
generally good and estimate the coercive fields well.
4.7 Model Conclusions
The implementation of a previously developed DOK stress-dependent Preisach model
in a numerical model framework is studied and compared to data taken on a non-trivial
geometry. The model predictions show many of the characteristics seen in the measured
data, and the fit of the B-H curves are generally good given the information available.
These results show that a 1-D magnetomechanical model can be applied to predict 3-D
magnetization changes due to stress, if adequately coupled. In the future this modeling
approach could be extended to incorporate more complex field and stress interactions.
77
Chapter 5 — Conclusions and Future Work
This is an area of research which has barely been explored despite over a century of
sporadic effort. The literature review, and measurements taken throughout this effort,
indicate that these high strength steels show interesting and useful capabilities which
have yet to be exploited by the scientific community. The work accomplished by this
effort is summarized in Section 5.1 Summary of Findings. The final products of this
research include three data sets and three models for the characterization of the
magnetomechanical effect in high strength steels, as well as a numerical modeling
framework which has been used to match a previous data set. Future research topics
which could stem from this effort are highlighted in Section 5.2 Future Work.
5.1 Summary of Findings
The main objective of the research presented here was to develop a robust set of
measurements from which model parameters could be derived, and then used, to create a
model capable of predicting magnetization changes within high strength steels due to the
application of stresses and magnetic fields. As a result of this research, valuable
modeling insights were obtained, a number of models have been created, and one was
implemented in a new numerical framework.
The first major finding of this research was that the widely used approximation of
stress as an “effective field” is inadequate. This conclusion is apparent through the
comparisons between Experiment #1, which involved the uniaxial application of stress
and magnetic fields, with Experiment #2, which involved the application of bi-axial
magnetic fields. The results of these two experiments proved that the application of bi-
axial magnetic fields does not yield the same magnetization loop bulging, tilting, or
78
coincident points apparent during the uniaxial application of stress and magnetic fields.
This is an important discovery, as it highlights erroneous assumptions found in previous
modeling attempts [JIL84], [SAB87].
The second major finding of this research effort was that temperatures between -50
and 100°C were proven to have a minimal impact on magnetization and magnetostriction
in high strength steels. Although the temperature dependence of these variables was
assumed to be similar to that seen in iron, these measurements are the first time that this
small effect has been quantified in low-carbon, low-alloy high strength steels.
Temperature was shown to have small but noticeable effect when looking at major and
minor loops, and on the same order of magnitude as predicted for iron.
The final major finding of this research is that a 1-D magnetomechanical model can
be applied to predict 3-D magnetization changes due to stress, if adequately coupled.
This is a fundamentally different approach than the one used to develop the
Schneider/Cannel/Watts and Jiles/Sablik models, which were made under the assumption
that they characterized bi-directional stress and magnetic field applications. However,
since a compressive stress in one direction will create corresponding tensile stresses in
the perpendicular plane (as determined by the Poisson’s Ratio for the material), these
models incorrectly characterize magnetization changes with respect to complex out-of-
plane stresses. While these differences would most likely cause minor issues in simple
measurement setups, the fact that stress is a tensor would make it nearly impossible to use
the aforementioned models to predict magnetization changes due to stress in a complex
structure. This finding would explain the difficulty in transporting these models to
separate-but-related data sets [PHI95], and the need for fitting parameters such as “stress
79
demagnetizing fields” [SCH92]. In contrast, it was reasoned that a truly 1-D model could
overcome these pitfalls if applied numerically with respect to stress and magnetic fields.
The 1-D Preisach model defined here was created from material parameters derived
from the experiments listed above. Once characterized, this stress and field model was
coupled with COMSOL Multiphysics® and used to accurately predict the magnetization
change seen in a separate high strength steel sample previously measured by the National
Institute of Standards and Technology. The 1-D model, used as a material parameter for
the numerical study, was generally able to predict the effects seen in a complex, 3-D
object to within ~9% RMS error, with a peak RMS error of 19.5% for the 0 MPA case.
The results indicate that this Preisach model and framework approach can be considered
a general magnetomechanical model and applied to related high strength steel data sets.
5.2 Future Work
There are many future experiments recommended after the findings of this research
effort. First, additional rotating field experiments should be made in order to more
accurately characterize the sizes, shapes, and distributions of magnetic domains within
high strength steels. These measurements would aid in the modeling of more complex
crystalline structures, and more accurate hysteretic predictions.
Similarly, more effort should be expended to characterize how the
magnetomechanical effect changes with respect to varying angles between applied field
and magnetization. While we have characterized some of the effect for three directions, a
more general approach would be more fitting.
Secondly, the study of the magnetomechanical effect on high strength steels should
be expanded to include tensile forces. While the modeling efforts shown in this research
80
support the theory that a tensile stress can be modeled as a corresponding compressive
stress, these assumptions should be validated on a number of material data sets.
Furthermore, these tensile and compressive stress measurements should be expanded
to include different angles (besides 90 and 180°) between applied fields and applied
stresses; the measurements from Experiment #1 were uniaxial, and the NIST
measurements had orthogonally applied stresses and fields.
Finally, more research should be undertaken to more accurately characterize and
model the magnetomechanical effects of stress on minor hysteretic loops. These are
notoriously hard to measure, and would provide an expanded knowledge of how the
materials behave magnetically.
81
List of Published and Pending Papers
The following is a list of published and pending papers related to this dissertation
proposal in chronological order.
1. C. D. Burgy, E. Della Torre, M. Wun-Fogle, J. B. Restorff, “Magnetostrictive study
of high strength steels with respect to angle from rolling direction”, IEEE Trans.
Magn., 48, 3088-3091 (2012).
2. H. ElBidweihy, C. D. Burgy, E. Della Torre, M. Wun-Fogle, “Modeling and
Experimental Analysis of Magnetostriction of High Strength Steels,” European
Physical Journal Web of Conferences, 40, 13005, (2013).
3. H. ElBidweihy, C. D. Burgy, and E. Della Torre, “Stress-associated changes in the
magnetic properties of high strength steels”, Physica B, 435, 16-20 (2014).
4. C. D. Burgy, M. Wun-fogle, J.B. Restorff, E. Della Torre, H. ElBidweihy,
“Magnetostriction measurements of high strength steels under the influence of bi-
axial magnetic fields,” Physica B, 435, 129-133 (2014).
5. C. D. Burgy, H. ElBidweihy, E. Della Torre, “Application of a Della Torre-Oti-Kadar
stress-Preisach Model through a Numerical Model,” J. Appl. Phys., 115, 17D112
(2014).
6. H. ElBidweihy, C. D. Burgy, E. Della Torre, L. H. Bennett, “Biaxial Preisach-type
Model for Sequential Application of Orthogonal Fields,” J. Appl. Phys., 115, 17D106
(2014).
82
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Appendix A — Modeling Framework in Detail
This appendix will further outline the numerical modeling framework discussed in
Chapter 4. The numerical modeling framework as it currently stands is a very hands-on
process, which could be much more automated in the future via closer integration with
the COMSOL® modeling software. The next pages will highlight each one of the steps
in the framework, and provide sufficient detail to allow this work to be duplicated and
expanded elsewhere.
The numerical modeling framework consisted of the following steps:
1. Generate DOK Preisach model parameters for each of the 8 uniaxial stresses
measured in Experiment #1
2. Model the NIST sample in COMSOL Multiphysics® version 4.3b
3. Use the structural-analysis solver to solve the numerical model and determine the
circumferential compressive stress, σφ, for a given longitudinal tensile stress, σz,
(listed in the NIST report [PET91]) as an input
4. Determine equivalent stress values via rounding to the closest stresses measured
in the solid cylinder data sets from Experiment #1
5. Use the equivalent stress values to choose the correct parameters for each DOK
Preisach model
6. Generate a B-H curve (for each stress) in MATLAB® over the range of
circumferential magnetic field values specified by the NIST report [PET91]
7. Export the B-H curves from MATLAB® to COMSOL®, and apply them as
material parameters
8. Solve the COMSOL® model again using the static-magnetic field analysis solver
9. Compare the predicted magnetization to the published NIST results
For the first step, a DOK Preisach model was developed in MATLAB® from the
uniaxial stress and magnetic field experiments. The parameters for this model were
derived from the data according to the identification schemes outlines in [DEL99]: σi is
the standard deviation of the interaction field, σk is the standard deviation of the critical
field, hk is the mean switching field, S is the squareness, and χ0 is the zero-field
susceptibility. Of these parameters, hk, S and χ0 are derived from the major loop. The
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mean switching field, hk, is the value of the applied field which reduces the magnetization
to zero. Similarly, χ0 is determined from the susceptibility in the major loop when the
applied field is zero. The squareness, S, is the measured ration of the maximum
remanence to the saturation magnetization. The final two parameters, σk and σi, are
obtained from two separate steps. First, the sum of their squares, σ2, is determined from
fitting the major loop to a Gaussian. Then the ratio of the two is determined by the
measurement of a first order reversal curve as defined in [DEL99, p. 48]. The parameters
for each of the 8 stresses measured in Experiment #1 are shown in Table 4-1.
Once these parameters were determined for each of the 8 stress values measured in
Experiment #1, the DOK model was formulated in a MATLAB® program. The inputs of
this program were: the saturation magnetization, the stress applied, and the series of
magnetic fields applied to the sample since saturation. For Experiment #1, the samples
were first brought to positive magnetic saturation via applying large magnetic fields.
Starting from saturation has the effect of deleting the previous magnetic history of the
sample and makes accurate modeling much easier and more repeatable. This is
especially evident in the modeling of minor loops, which are greatly dependent on the
previous magnetization history of the material. The output of the MATLAB® program is
a B-H curve for the specific series of applies fields which were used as inputs.
In the second step, the NIST sample geometry was recreated with the COMSOL®
finite element software. COMSOL® is a commercial finite element software which
allows for the solving of multiple-physics simulations with a single model. This is
accomplished by: modeling the sample geometry, applying appropriate boundary
conditions, incorporating material data, choosing an appropriate solver, and then
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calculating the results. This NIST numerical model was slightly simplified from the
exact geometry, in that the threads of the sample were not included (see Figure 4-4).
However, care was taken to ensure that the model was accurate to the dimensions listed
in the NIST report [PET91], including the unique arrangement of the magnetic field
windings.
Each of the solvers used within COMSOL® require unique boundary conditions in
order to converge properly. To calculate the stresses within the NIST geometry under
different applied loads, HSS material constants and the COMSOL® structural solver
were used. The HSS material parameters chosen for the structural model included: a
Poisson’s ratio of 0.3, a Young’s Modulus of 205*109 MPa, and a density of 7850 kg/m
3.
For the structural solver, a fixed-position boundary was applied to one of the end faces of
the sample, and then a structural load was applied the opposite end-face to simulate the
longitudinal tension applied during the experiment. As the NIST report [PET91] defined
the tension applied to the sample as being measured in the active region, the structural
load was chosen based on that assumption as well. The stress within an elastic structure
is equal to the stress applied times the area. As the cross sectional area of the active
region is smaller than the end-faces of the NIST sample, a ratio was determined to apply
the correct boundary load. It was found that a stress of 1 MPa applied to the end-face
surface yielded a stress of 3.93 MPa in the active region. Accordingly, this ratio was
utilized in the third step of the framework, when the structural-analysis solver was used
to solve the numerical model and determine the circumferential compressive stress, σφ,
for a given longitudinal tensile stress, σz, (listed in the NIST report [PET91]) as an input.
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Having solved for the circumferential compressive stresses within the NIST sample,
the fourth step of the numerical modeling framework entailed determining equivalent
stress values via rounding to the closest stresses measured in the solid cylinder data sets
from Experiment #1. Experiment #1 involved the measurement of 8 compressive
stresses: 1, 25, 50, 75, 100, 125, 150, and 175 MPa. Accordingly, each compressive
stress determined from the structural analysis solver was matched to an existing data set
from Experiment #1.
In the fifth step of the numerical modeling framework, the equivalent stress values
determined from the structural analysis and comparison steps were used to choose the
correct sets of DOK parameters. For each of the three tensile stresses measured by NIST
(0, 160, and 400 MPa), two sets of parameters were chosen: one for the active region, and
one for the non-active region. The equivalent stress data sets are shown in Table 4-2.
In the sixth step of the framework, B-H curves were generated in the MATLAB®
program for each stress listed in Table 4-2 over the range of circumferential magnetic
field values specified by the NIST report [PET91]. These curves were exported from
MATLAB® corresponding to increasing and decreasing applied magnetic field legs.
They were exported in this manner so that they would be treated as single-valued
functions for COMSOL®. Furthermore, it can be seen that interaction with MATLAB®
was a necessary component of this framework, as the COMSOL® software does not have
built in capability for handling hysteretic (non-single-valued) curves as of version 4.3b.
In the seventh step of the framework, these curves were exported from MATLAB® to
COMSOL®, and were applied within the NIST numerical model as non-linear material
parameters. In the static magnetic field solver, COMSOL® uses the boundary conditions
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and the magnetic fields applied to predict the magnetic fields within the modeled
structure. These linear fields are then combined with the B-H curves imported in from
the DOK MATLAB® program in order to predict the resulting magnetization. As
COMSOL® allows for the definition of a B-H curve, this was easily accomplished.
In the second to last step of the framework, the NIST numerical model was re-solved
using the static-magnetic fields solver within COMSOL®. The magnetic fields were
applied within the numerical model by specifying a current, I, within the modeled
magnetic field windings. This was accomplished by a current to magnetic field ratio of
8.094 A for every 8 kA/m. As mentioned above, the B-H curves imported into COMSOL
were applied as material parameters. As the fields are linear within the material, the
COMSOL® solver simply iterates between solving for the field within the sample and
then calculating the corresponding magnetization. This is accomplished for every
element within the numerical model, and then the resulting magnetization is computed by
summing those elements via superposition.
Finally, in the last step of the framework, the resulting magnetization predicted by the
COMSOL® model was compared to the measured NIST results. This was accomplished
via fitting the predicted and measured data to common grids of measurement points in
MATLAB®, and then calculating the RMS percentage error as outlined in Equation 4.2.
In the future, this framework can be expanded and/or altered at the desire of the
researcher involved. There are opportunities afforded within the COMSOL® finite
element modeling software in which some parts of this framework could be streamlined
and automated. For more information on areas of improvement and suggestions for
future work, see section 5.2.