magnetic and magnetostrictive characterization and

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




3.2.4 Experiment #2 Model .................................................................................. 50




3.3 Experiment #3 – Temperature Dependence of Hysteresis .................................... 57







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


Figure 3-7: Magnetostriction vs. magnetization for -125 MPa applied stress for cylinders cut


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


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


xii

Figure 3-27: Magnetization versus applied field (major and minor loops) for the toroidal-like


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].


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].


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.


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 .


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.


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.


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


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.


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.


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

60

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.


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.


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.


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


Figure 4-7: B-H loop for σz = 400 MPa (tension) The NIST data [PET91] is the solid line, and the COMSOL


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.

magnetic and magnetostrictive characterization and

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