University of Southern California Dissertations and Theses

Long range internal stresses in cyclically deformed copper single crystals; and, Understanding large-strain softening of aluminum in shear at elevated temperatures

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Long range internal stresses in cyclically deformed copper single crystals; and, Understanding large-strain softening of aluminum in shear at elevated temperatures

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Content Long Range Internal Stresses in Cyclically Deformed Copper Single Crystals
and
Understanding Large-Strain Softening of Aluminum in Shear at Elevated Temperatures

by

Roya Ermagan

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATERIALS SCIENCE)

August 2021

2021 Roya Ermagan

2

DEDICATION

Whole-heartedly, and gratefully so, I dedicate this work to my husband, Dr. Roozbeh Sheikhi,
my caring parents, Soudabeh and Ghandy, and my sister, Vina, who have all offered me
unconditional love and support and made it possible for me to complete this work.

In Loving Memory of my dear friend, Paniz Soltani,
A Materials Science PhD Candidate at Max Planck Institute,
who we sorrowfully lost in downed flight PS752.

ii

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ACKNOWLEDGEMENTS

I have had the honor to learn and work among some of the most knowledgeable and caring
individuals over the past few years. Without the continuous support of these truly amazing mentors
and friends, finishing my PhD studies would have not been possible.
I would like to express my deepest gratitude to Professor Michael Kassner. He gave me the
opportunity to work on a variety of interesting projects and provided immense insights on how to
perform a well-organized research project. Professor Kassner provided me with knowledge
necessary to conclude my PhD studies and to grow in my future career.
I would also like to express my appreciation to Professor Andrea Hodge, who has been really
generous in extending her laboratory to me. Without the access, much of the sample preparation
and chemical analysis would not have been possible.
I would also like to thank my PhD. committee members, Professor Nicholas Graham, Professor
Ivan Bermejo Moreno, and Dr. Lessa Grunenfelder for their insightful comments on my research.
I am truly grateful for the time they set aside to hear my defense.
In addition, the only ones who truly understand the intensity and the challenges of this program
are those who experience it with you. Thus, I would also like to thank my very good friends and
colleagues, Dr. Roozbeh Sheikhi, Dr. Sepehr Maktabi, Dr. Mahshad Samnejad, Dr. Marjan
Sherafati, Dr. Saeed Abrishami, Reza Pejman, and Kiana Askari for their continuous support.
Lastly, I am exceptionally grateful to my husband, Roozbeh, my caring parents, Soudabeh and
Ghandy, my sister, Vina, and my chosen family, Tannaz Sattari and Reza Khoie for their
continuous mental and emotional support. They have always supported me and provided me
invaluable advice. Their limitless support and encouragement kept me motivated throughout my
studies and has enabled me to conclude my PhD journey. I am forever grateful to you all.
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TABLE OF CONTENTS

Dedication....................................................................................................................................... ii
Acknowledgements ....................................................................................................................... iii
List of Figures................................................................................................................................ vi
List of Tables................................................................................................................................ xii
Abstract........................................................................................................................................ xiii
1 Chapter 1: Long Range Internal Stresses in Cyclically Deformed Copper Single Crystals
………………………………………………………………………………………………..1
1.1 Motivation ...................................................................................................................... 1
1.2 Background ................................................................................................................... 2
1.2.1 Heterogeneous Dislocation Microstructure ................................................................ 2
1.2.2 The Labyrinth Structure .............................................................................................. 4
1.2.3 The Bauschinger Effect............................................................................................. 10
1.2.4 Previous LRIS Determination Studies ...................................................................... 17
1.3 Experimental Techniques ........................................................................................... 29
1.3.1 Cyclic Deformation ................................................................................................... 29
1.3.2 Sample Preparation Techniques ................................................................................ 29
1.3.3 Microstructural and LRIS Characterization Methods ............................................... 34
1.4 Work to Date ............................................................................................................... 48
1.4.1 Experimental Methods .............................................................................................. 48
1.5 Conclusions .................................................................................................................. 84
1.6 Future Suggested Work .............................................................................................. 85
2 Chapter 2: Understanding Large-Strain Softening Of Aluminum In Shear At Elevated
Temperatures ................................................................................................................................ 86
2.1 Motivation .................................................................................................................... 86
2.2 Background ................................................................................................................. 87
2.2.1 Aluminum Softening ................................................................................................. 87
2.2.2 Microstructural Changes during Thermomechanical Processing ............................. 89
2.2.3 Textures and Taylor Factor ....................................................................................... 93
2.2.4 Suggested Reasonings for Aluminum’s Softening at Elevated Temperatures ......... 97
2.3 Results and Discussion .............................................................................................. 103
2.4 Conclusions ................................................................................................................ 111
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5
3 Chapter 3: The Creep and Fracture Behavior of Additively Manufactured Inconel 625
and 718 ....................................................................................................................................... 112
3.1 Motivation .................................................................................................................. 112
3.2 Background ............................................................................................................... 113
3.2.1 Description of Creep Deformation.......................................................................... 113
3.2.2 Inconel 718 Superalloy ........................................................................................... 117
3.2.3 Metal Additive Manufacturing ............................................................................... 122
3.3 Material and Experimental Details ......................................................................... 123
3.3.1 Inconel Superalloys ................................................................................................. 123
3.3.2 Creep Instrument ..................................................................................................... 126
3.3.3 Creep Tests and Curves .......................................................................................... 128
3.4 Results and Discussion .............................................................................................. 132
3.5 Conclusion ................................................................................................................. 141
4 Chapter 4: Power Law Breakdown in the Creep in Single-Phase Metals ...................... 142
4.1 Motivation .................................................................................................................. 142
4.2 Background ............................................................................................................... 142
4.3 Work to date .............................................................................................................. 145
4.3.1 Results and Discussion ........................................................................................... 145
4.4 Conclusions ................................................................................................................ 152

References .................................................................................................................................. 153

Appendices ................................................................................................................................. 175
Appendix A: Bethe Parameter Codes .................................................................................. 176
Appendix B: EMSoft Codes ................................................................................................ 177

v

6
LIST OF FIGURES
Figure 1-1. Fracture surface with fatigue indications on a fan blade of a Southwest aircraft [1]. .. 2
Figure 1-2. (a) The dislocation patterns in cyclically deformed copper single crystals having the
labyrinth structure when deformed in [0 0 1], wall structure when deformed in [0 1 1] and cell
structure when deformed in [1 1 1] directions [6], (b) In the (0 1 0) longitudinal section, regularly
spaced thin walls parallel to (0 0 1) and (1 0 0) were found by Jin and Winter in a cyclically
deformed copper single crystal for the first time [8]. ..................................................................... 5
Figure 1-3. Slip system of an FCC crystal. ..................................................................................... 6
Figure 1-4. Illustration of the labyrinth structure formation [6]. .................................................... 8
Figure 1-5. Dependence of the scaled saturation shear stress, τ/μ on b/dc, for single crystals
cyclically loaded along a <0 0 1> axis at 300 K. Labyrinth structures are observed in all the
specimens [5]. ................................................................................................................................. 9
Figure 1-6. A generalized illustration of the Bauschinger effect. The material yields at a stress that
is much lower than expected for isotropic hardening with reversal of the direction of straining [14].
....................................................................................................................................................... 11
Figure 1-7. The composite model indicating ‘‘hard’’ regions (high dislocation density) and ‘‘soft’’
regions (low dislocation density) and the corresponding LRIS present [14]................................ 14
Figure 1-8. A profile of long range internal stresses as they would relate to the microstructure
shown [14]. ................................................................................................................................... 15
Figure 1-9. An illustration of an idealized dislocation pile-up model [14]. ................................. 16
Figure 1-10. Dislocation dipoles in γ-TiAl. (a) A dislocation dipole across its two edge dislocation
arms on a high resolution image. (b) The simulated atomic configuration of the same dislocation
dipole [24]. .................................................................................................................................... 18
Figure 1-11. (a) TEM micrograph of cyclically deformed Cu single crystal in single slip. A channel
bounded by dipole bundles and the locations of where the CBED was used to determine the lattice
parameter. (b) The variation in the lattice parameter and stress was measured by CBED, versus
position within the channel (typically 3.8 mm wide). No substantial LRIS was evident [28]. .... 21
Figure 1-12. The variation in calculated long range internal stresses from dislocation bowing
versus position within the channel in a cyclically deformed copper single crystal to saturation
(PSBs present). The dashed lines are the modified data by Mughrabi considering interactions with
dipoles and multipoles of the wall [2], [34]. The dislocation loops assessed were within PSBs. 23
Figure 1-13. (a) X-ray diffraction peak in monotonically deformed copper. The asymmetric
broadening can be observed with comparison to the undeformed specimen. (b) A decomposition
can be performed that leads to two separate diffraction sub peaks coming from the two regions
within the microstructure (high and low dislocation density regions) with different lattice
vi

7
parameters and hence, different stresses [35]. (c) Schematic illustration on how a dislocation can
broaden the X-ray diffraction peak. .............................................................................................. 25
Figure 1-14. The X-ray beam hits the sample and is diffracted onto the CCD. A platinum wire
scans across the sample in sub-micron steps, and provides depth resolution to the measurements
[39]. ............................................................................................................................................... 27
Figure 1-15. The cyclically deformed single crystal copper cylinder. ......................................... 30
Figure 1-16. (a) Schematic illustration of a jet electropolisher. (b) An SEM image showing the thin
regions around the perforation on a copper TEM sample made by jet electropolishing. (c) Ideal
sample to jet distance. (d) The current versus voltage diagram of electropolishing. Increasing the
voltage until a plateau in current is reached results in electrolytic polishing [40]. ...................... 32
Figure 1-17. Schematic illustration of a dual-beam FIB–SEM instrument. Expanded view shows
the electron and ion beam sample interaction [41]. ...................................................................... 33
Figure 1-18. Schematic arrangement of EBSD system [43]. ........................................................ 35
Figure 1-19. A schematic configuration of a TEM [45]. .............................................................. 36
Figure 1-20. Schematic illustrations of the transmission electron microscope for (a) bright field
imaging and (b) selected area diffraction mode and (c) dark field imaging [45]. ........................ 37
Figure 1-21. The Ewald sphere. When the sphere cuts the reciprocal lattice points, the Bragg
condition is satisfied. Note that the radius of the Ewald sphere is very large compared to the
reciprocal lattice so that the sphere projection is almost flat [44]. .............................................. 39
Figure 1-22. Rotation of the FCC sample to the zone axis of [100], [111], and [110] produces the
diffraction patterns shown............................................................................................................. 40
Figure 1-23. The two beam condition with g <202>. ................................................................... 41
Figure 1-24. Formation of the Kikuchi lines [44]. ........................................................................ 43
Figure 1-25. Schematic illustration of the collection of a parallel and convergent beam electron
diffraction pattern [45]. ................................................................................................................. 44
Figure 1-26. Reciprocal space schematic showing the diffraction conditions for the zero order Laue
zone (ZOLZ), first order Laue zone (FOLZ), and second order Laue zone (SOLZ) by intersection
of the Ewald sphere with the reciprocal-lattice rod (relrod) [46]. ................................................ 46
Figure 1-27. (a) The cyclic deformation of [001]-oriented copper single crystal at ambient
temperature. Strains are plastic and elastic. (b) The evolution of the maximum and minimum peak
stress with respect to the number of cycles. (c) The logarithmic plot of the evolution of the
maximum peak stress with respect to the number of cycles confirms saturation is reached. ....... 50
vii

8
Figure 1-28. Diffraction patterns of the two orthogonal directions in the single crystal specimen.
Note that for each disk two different diffraction patterns were found due to bends created by jet
electropolishing. ............................................................................................................................ 51
Figure 1-29. EBSD orientation map of the side disks showing the crystal orientation as <110>. 54
Figure 1-30. (a) EBSD map of the TEM sample prepared using a FIB showing the (0 1 0) single
crystal and the damaged crystal at the top. (b) TEM micrograph of the copper sample prepared
with FIB. Note that there is no labyrinth structure in the single crystal section at the bottom right.
....................................................................................................................................................... 55
Figure 1-31. Transmission electron microscope “cube” based on images taken from the (100),
(010), and (001) planes of a specimen cyclically deformed to saturation as illustrated in Figure 1-
27.a. The stress axis is parallel to the vertical direction [001]. ..................................................... 56
Figure 1-32. TEM micrographs of the labyrinth structure from the (010) planes of the cyclically
deformed copper. The <202> two-beam conditions were used to image dislocations in this study.
....................................................................................................................................................... 58
Figure 1-33. The EELS analysis showing the thickness measurements. ...................................... 60
Figure 1-34. Dependence of the scaled shear stress τa/G on b/d ×10
-3
, for single crystals cyclically
loaded at 300 K. Data from [8–10], [54–56]. ............................................................................... 61
Figure 1-35. A three-dimensional (3-D) schematic representation of the dislocation structure
illustrating the dipole height “h”. .................................................................................................. 62
Figure 1-36. A part of the labyrinth structure showing dislocation dipoles. ................................ 64
Figure 1-37. (a) simulated Kikuchi pattern generated from a perfect FCC Ni crystal model, (b)
standard textbook FCC Kikuchi pattern and (c) experimentally collected and stitched TEM
diffraction pattern [72]. ................................................................................................................. 71
Figure 1-38. CBED patterns were recorded across channels to assess the internal stresses in the
direction of applied stress [0 0 1]. (a) A channel illustrating the locations where the CBED patterns
were acquired. (b) The <4 1 1> CBED pattern that was recorded closer to the wall and (c) in the
middle of the channel. ................................................................................................................... 72
Figure 1-39. The cross correlation between <4 1 1> CBED patterns of (b) and (c) in figure 1-38
shows minimal difference in the position of the lines in the patterns. .......................................... 73
Figure 1-40. Simulated CBED pattern of [0 0 1] zone axis using EMSoft [76] with a thickness of
(a) 120 nm and (b) 300 nm. .......................................................................................................... 75
Figure 1-41. Dynamically simulated <411> HOLZ line patterns illustrating the HOLZ line
intersections used for strain assessment. ....................................................................................... 76
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9
Figure 1-42. The obtained lattice parameters and corresponding axial stress values in different
locations within a dislocation channel. The lattice parameter of an unloaded copper was found to
be 0.3592 nm. ................................................................................................................................ 78
Figure 1-43. Change in the [4 1 1] CBED pattern HOLZ lines of cyclically deformed copper in a
channel and undeformed copper relative to simulated pattern with lattice parameter of 0.3592 nm.
The data is shown as a χ2 fit between the HOLZ lines of the aforementioned CBED patterns. .. 79
Figure 2-0-1. Elevated-temperature equivalent-uniaxial stress versus equivalent-uniaxial strain of
high-purity Al at strain rates of (a) strain rate of 5.8 × 10
-4
s
-1
and (b) 1.3 × 10
-2
, s
-1
from [82]. . 88
Figure 2-0-2. Recovery, recrystallization, and grain growth relate to grain size, hardness, ductility,
and residual stress in the material [101]. ...................................................................................... 90
Figure 2-0-3. DRX can occur in various forms: discontinuous dynamic recrystallization (DDRX),
continuous dynamic recrystallization (CDRX), and geometric dynamic recrystallization (GDRX)
[102]. ............................................................................................................................................. 91
Figure 2-0-4. GDRX progress diagram: a) grain boundaries flatten with well-defined substructures
in the matrix when deformation is comparatively small. (b) as deformation continues, the serrated
HABs become closer while the subgrain size remains roughly the same. (c) As a consequence, the
HABs eventually impinge, giving rise to a microstructure of 1/3 to 1/2 HABs [102]. ................ 93
Figure 2-0-5. Orientation factors for (a) single slip "Schmid" and for (b) multiple slip "Taylor"
calculated for uniaxial (tensile or compression) deformation [107]. ............................................ 94
Figure 2-0-6. Calculations of the evolution of the average Taylor factor with strain. (a) Tension
and compression. (b) Shear [106]. ................................................................................................ 96
Figure 2-0-7. The aluminum deformed in torsion at 371
o
C with the peak stress,
ss p,

. The flow
stress subsequently decreases to a flow stress,
ss

, which is nearly constant, and a steady-state
condition is reached. The peak stress,
ss p,

, seems equivalent to the steady-state creep stress
observed in tension. ...................................................................................................................... 98
Figure 2-0-8. The dislocation density (a) and the subgrain size (b) are approximately constant
throughout deformation. ............................................................................................................... 99
Figure 2-0-9. Schematic of the torsion specimen with the indicated shear stress and the principal
stresses for a pure shear stress state. ........................................................................................... 105
Figure 2-0-10. The 371°C equivalent uniaxial stress versus equivalent uniaxial strain of hollow
torsion specimens and the corresponding compressive yield stress (0.10 strain offset) at the same
strain rate and elevated temperature subsequent to various (pre)strains in torsion. ................... 109
Figure 2-0-11. The equivalent uniaxial stress versus equivalent uniaxial strain of hollow torsion
specimens and the corresponding compressive yield stress. Specimens were tested under torsion
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10
at elevated-temperature followed by a quench, and ambient-temperature compression along the
torsion axis. The ambient temperature compression hardening is consistent with elevated
temperature torsion texture development followed by ambient temperature glide control with
a compression axis identical with the torsion axis. .................................................................... 110
Figure 3-1. The creep behavior at temperatures higher than 0.5 Tm. (a) at constant stress condition.
(b) the stress-strain curve of the same material at constant strain rate (creep rate) condition. The
three regimes of I, II and III are the primary, secondary (steady-state), and tertiary (fracture)
regimes [123]. ............................................................................................................................. 114
Figure 3-2. Transformation-time-temperature diagram of IN718 [131, 133]. ............................ 121
Figure 3-3. (a) The Arcweld Manufacturing Company creep machine used for elevated
temperature creep rupture tests. (b) The creep machine schematic [140]. ................................. 127
Figure 3-4. The one cycle of the cyclic heating profile of the longtime heat treatment cycles for 6
months and 1 year in this study [141]. ........................................................................................ 129
Figure 3-5. The stress versus strain behavior of wrought IN718 at ambient and 650˚C.
Embrittlement is evident at 650˚C. LPBF corresponds to laser powder bed fusion (described in
section 3.2.3). .............................................................................................................................. 130
Figure 3-6. Representative creep curves of the high temperature creep tests at 650 ˚C on AM and
wrought IN718 ............................................................................................................................ 132
Figure 3-7. EBSD images of solution annealed wrought and HIPed AM 718. Both specimens
underwent the standard double age heat treatment. The average grain size of AM HIPed IN718
measured about 68 μm while the wrought was about 11 μm. Although the grain size is different
by a factor of six, grain size strengthening is not particularly substantial [146] at elevated
temperatures within the five power-law regime. ........................................................................ 133
Figure 3-8. (a) and (b). The creep rate and ductility of wrought and AM 718 at 650oC. Literature
values were taken from Special Metals [143]. The creep ductility values of wrought and AM 718
at 650oC and higher stresses of 730 and 819 MPa indicate significant embrittlement of the AM
IN718. The tests with arrows indicate that failure was not observed within one day. The higher
stress tests to failure occurred over a period of about one week. ............................................... 137
Figure 3-9. Wrought and AM Inconel 625 creep test results (a) minimum (steady-state) creep rate
versus the modulus compensated (normalized) stress. (b) creep ductility at 650 ˚C (c) creep
ductility at 800 ˚C. The ductility at the two test temperatures is indicated by solid dots for 24 hour
(short creep tests or SCT) specimens, X marks for 6 months at 650 ˚C samples and + marks for 1
year at 650 ˚C specimens [119]. ................................................................................................. 138
Figure 3-10. The element distribution as a function of distance from a grain boundary and a 𝛾 ’’
interface from atom probe topography (APT) for AM IN625 specimens: (a) creep deformed at 294
MPa and 650˚C within 24 hours (b) creep deformed after 6 months at 650˚C at 658 MPa (c)
x

11
concentration profile near a 𝛾 ’’ particle. The shaded area are grain boundary regions or 𝛾 ’’
precipitates [119]. ....................................................................................................................... 139
Figure 3-11. The fatigue behavior of wrought and additively manufactured IN625 and 718 at
650oC [152]. ............................................................................................................................... 139
Figure 3-12. Strong oxygen, carbon and sulfur signals were observed at the Al2O3 particles. No
evidence of grain boundary segregation of impurities was observed in the nanoSIMS results for
AM as-HIPed 625 alloy. ............................................................................................................. 140
Figure 4-1. The steady-state creep behavior of high purity aluminum from temperatures ranging
from sub-ambient to near the melting point [160, 163-164].This figure is adapted from [164]. 144
Figure 4-2. The activation energy for steady-state creep of silver as a function of temperature
[169,170]. .................................................................................................................................... 146
Figure 4-3. (a) the steady-state (n) and constant structure (N) stress exponents for annealed (broken
line) and steady-state (solid line) structures of AISI 304 stainless steel as a function of lattice
diffusion coefficient compensated strain-rate. (b) comparison of the constant structure stress
exponents (N) of annealed aluminum with the steady-state stress exponent (n) based on earlier
work by the author [184] and others [166,185,186]. Deff is the effective diffusion coefficient that
is the combined diffusivity considering dislocation pipes as described in [166]. ...................... 149

xi

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LIST OF TABLES

Table 1-1. The ratios of N=h2+k2+l2 of different planes (hkl). Note that considering the extinction
Equation (1-14) planes that can diffract in an FCC crystal are highlighted in green [49]. ........... 53
Table 1-2. Dipole heights and stress calculations (Equation (1-10) for [26], [28], and [30] and
Equation (1-15) for this study) based on observed maximum “h” values. ................................... 69
Table 1-3. Convergent Beam Electron Diffraction studies for long range internal stress
assessment on creep and cyclically deformed materials. .............................................................. 83
Table 2-1. Textures and Taylor Factors (M) observed in aluminum deformed to large strains at
elevated temperatures.................................................................................................................. 102
Table 2-2. Normalized climb stress for the observed textures.................................................... 106
Table 2-3. Changes in flow stress with torsion to compression at high temperature in terms of
climb stress analysis. ................................................................................................................... 107
Table 3-1. The compositions of the wrought and AM Inconel alloys. ....................................... 125

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ABSTRACT

Four projects were performed and reported in the present dissertation with concentration on
mechanical behavior of metals.
The first project focuses on understanding Long Range Internal Stresses (LRIS), which is critical
for explaining the basis of the Bauschinger effect, spring-back in metal forming, and plastic
deformation in cyclically deformed metals. Few studies have assessed LRIS in cyclically deformed
single crystals in single-slip while there are no such studies in multiple-slip. Here, we report on
LRIS in a cyclically deformed copper single crystal in multiple-slip via two methods: 1- Lattice
parameter determination using Convergent Beam Electron Diffraction (CBED) and 2- Measuring
the maximum dipole heights. TEM micrographs show a labyrinth dislocation microstructure with
high dislocation density walls and low dislocation density channels. Lattice parameters and dipole
heights were assessed in the channels and walls of the labyrinth structure. Lattice parameters
obtained were almost identical near the walls and in the channels. The maximum dipole heights
were also approximately independent of location. Thus, a homogenous stress state within the
heterogeneous dislocation structure is suggested.
The second project describes large strains of aluminum in pure shear at elevated temperature. This
pronounced softening has been attributed to a variety of phenomena. The most widely accepted
early explanations involve the development of a texture leading to a decrease in the average Taylor
factor. That is, there is a decrease in Schmid factors in the deformed grains. Our work suggests
that the texture leads to softening through an increase in the dislocation climb stress. This appears
to be particularly reasonable as dislocation climb is widely regarded the rate-controlling
mechanism for high temperature plasticity.
xiii

14
Additive manufacturing by selective laser powder bed fusion of alloys followed by hot isostatic
pressing (HIP) has gained attention by industry as it provides economical production of
complicated-configuration engine parts with fewer joining steps and greater geometric freedom.
The creep behavior of additively manufactured (AM) Inconel 625 and 718 at 650 and 800
o
C was
compared with that of the wrought alloy after 24 hour and long-term tests up to about one year.
Inconel 718 is precipitation strengthened while 625 is solid-solution strengthened. It was
discovered that the creep strength of the 625 and 718 produced by additively manufacturing is
essentially identical than that of the wrought alloy. Complications occur, however, with a marked
loss of ductility in the AM alloys. Fatigue and impact toughness were adversely affected as well.
The basis for these degradations was particularly investigated in the third project.
The last project provides insight into the basis of power-law breakdown (PLB) in the steady-state
creep of metals and alloys. A variety of theories has been presented in the past but this new
examination suggests that there is evidence that a dramatic supersaturation of vacancies leading to
very high diffusion rates and enhanced dislocation climb is associated with the rate-controlling
process for creep in PLB. The effect of vacancy supersaturation may be enhanced by dislocation
short circuit diffusion paths at lower temperatures due to the dramatic increase in dislocation
density.

xiv

1

1 Chapter 1: Long Range Internal Stresses in Cyclically Deformed
Copper Single Crystals
1.1 Motivation
Fatigue failures are associated with components experiencing cyclic stresses or strains resulting
in permanent damage. Fatigue is one of the most common catastrophic-failure mechanisms in
structural materials. For instance, a Southwest aircraft had to make an emergency landing in 2018
due to an engine failure caused by metal fatigue in one of the fan blades (see figure 1-1). Thus, it
goes without saying there is much more to be comprehended in regards to the mechanisms enabling
fatigue. A better understanding of fatigue would lead to enhanced performance of structural
materials. The research performed and discussed here aims to provide experimental evidence for
some of the mechanisms and behaviors that take place during cyclic deformation of metals. When
a crystalline metallic material is cyclically deformed, a dislocation microstructure develops within
the material. Along with this a heterogeneous stress state might be present which correlates directly
with this microstructure. These stresses are known as long range internal stresses (LRIS) or
backstress and it is this phenomenon which will be discussed in great depth. Understanding
backstresses in metal forming operations could lead to enormous savings in the metal-forming
manufacturing sector. Along with the larger goals, there are also more tangible and immediate
questions for which this research will provide insight. The Bauschinger Effect, which is the lower
yield stress present when reversing the deformation direction, may have some origins in LRIS.
Perhaps most importantly, the research presented here provides important insight into the
deformation mechanics of cyclically deformed single crystalline material in general.

2

Figure 1-1. Fracture surface with fatigue indications on a fan blade of a Southwest aircraft [1].

1.2 Background
In this section, we begin by presenting the origins of dislocations microstructure development
during deformation of metallic materials and the corresponding stresses (LRIS). Then a detailed
explanation of the Bauschinger Effect is presented followed by a background on specific models
developed to explain these stresses. Finally, there is a detailed explanation of the methods we used
to quantify the internal stresses in the heterogeneous dislocation microstructure. By performing an
in depth analysis of the heterogeneous dislocations microstructure perhaps some light can be shed
on the nature of long range internal stresses.
1.2.1 Heterogeneous Dislocation Microstructure
During deformation, dislocations glide through the material and interact with each other. If
obstacles are present in the crystal lattice it becomes more difficult for dislocations to glide through
the material. Dislocations act as obstacles to other dislocations and as plastic deformation
continues they may begin to tangle together. Additionally, as plastic deformation occurs, the
number of dislocations increases dramatically (e.g. Frank Reed source). This in turn increases the
difficulty with which they can move freely, having the effect of strengthening the material (strain
hardening). As these processes continue with ever increasing numbers of dislocations and

3
interactions, dislocation microstructures emerge within the material, consisting of dislocation
dense regions, and regions with little or no dislocations present. It is these dislocation
microstructures which are integral to the understanding of LRIS and will be discussed in depth in
the following sections.
Two broad categories for deformation are cyclic and monotonic. Each class produces different
kinds of substructure (e.g. in terms of the dislocation dipole content of the dislocation
heterogeneities and the crystallographic misorientation across these structures). Each of these
categories can be further subdivided. Cyclic can be separated by the number of cycles into lower
total-cycles or higher total-cycles and pre-saturation or saturated stresses (in terms of the applied
stress). Monotonic is subdivided into low and high (creep) temperature deformation. Both
monotonic and cyclic can be divided into deformations where multiple slips are activated versus
only a single slip is activated. In monotonic deformation, when multiple slip systems are active,
the microstructure takes on a particular form. Regions of dislocation free (or relatively free) cell
interiors are surrounded by dislocation dense cell walls. In single slip deformation, a
microstructure develops consisting of elongated clusters of dislocations. At higher temperatures
cell walls may take the form of subgrain boundaries. Persistent slip bands and dense dipole bundles
(or veins) and channels with relatively low dislocation densities form if cyclically deformed in
single slip [2]. Dislocation walls and channels (regions with high and low dislocation densities,
respectively) develop if cyclically deformed in multiple slip. The driving force for the formation
of dislocation walls and cell structures is explained by the reduction of the strain energy of the
dislocations which form the cell walls by energetically favorable dislocation reactions [3].
Similarly, Holt [4] has treated the problem of dislocation clustering in analogy to spinodal

4
decomposition as the clustering of atoms during spinodal decomposition and the clustering of
dislocations are very similar: both result in a modulated structure.
There have been a number of studies on LRIS measurements with different methods in cyclically
deformed FCC single crystals in single slip while there are no such studies in multiple slip. Sauzay
and Kubin [5], in a review article on the dislocation microstructures in cyclically deformed FCC
metals, note that there are no publications in which dislocation densities were measured in fatigued
crystals oriented for multiple slip. All experiments performed for this research are of this type of
microstructure (i.e. copper single crystal cyclically deformed with multiple slip systems active).
1.2.2 The Labyrinth Structure
In comparison with the single-slip oriented copper single crystals, the multiple-slip oriented ones
show very different dislocation patterns. Figure 1-2 depicts this by showing the dislocation patterns
in cyclically deformed copper single crystals having the labyrinth structure when deformed with
the stress axis in [0 0 1], wall structure when deformed in [0 1 1] and cell structure when deformed
in [1 1 1] directions [6]. Here we only focus on the labyrinth structure as it is a unique dislocation
microstructure of cyclically deformed FCC metals. The labyrinth microstructure was first observed
in a polycrystalline Cu-Ni alloy fatigued in multiple glide by Charsley and Kuhlmann-Wilsdorf
[7] and then observed by Jin and Winter [8] for the first time in single crystals in cyclically
deformed [001] oriented copper (multiple slip) as shown in Figure 1-2.b.

5

Figure 1-2. (a) The dislocation patterns in cyclically deformed copper single crystals having the
labyrinth structure when deformed in [0 0 1], wall structure when deformed in [0 1 1] and cell
structure when deformed in [1 1 1] directions [6], (b) In the (0 1 0) longitudinal section, regularly
spaced thin walls parallel to (0 0 1) and (1 0 0) were found by Jin and Winter in a cyclically
deformed copper single crystal for the first time [8].

The Labyrinth structure has very distinctive features: two sets of mutual perpendicular walls, but
how do these walls form? When FCC metals are cyclically deformed along a [0 0 1] direction
(tension and compression with R = −1), a labyrinth microstructure consisting of almost
(a)
(b)

6
perpendicular dislocation walls and channels (regions with high and low dislocation densities,
respectively) develops. An FCC metal stressed in [0 0 1] orientation has the following eight sets
of equivalent slip systems with the same Schmid factor of 0.408, which can be activated
simultaneously: four slip planes of {111} type and two slip directions of [ī 0 1] and [0 ī 1]. Figure
1-3 illustrates one set of slip system as an example.

Figure 1-3. Slip system of an FCC crystal.

Li et al. [6] found out that the formation of the complex dislocation patterns like the labyrinth
structure depends on the activating slip system and is related to the dislocation reactions. Figure
1-4.a illustrates that on the (0 1 0) plane, two sets of persistent slip bands belonging to the activated
slip systems have formed, which agrees with the movement and development of dislocations in

7
their respective regions. These PSBs will disappear quickly when cyclically deformed at higher
strain amplitudes. With continuing the cyclic deformation, the reaction between dislocations along
[ī 0 1] and [1 0 1] will occur as shown in Figure 1-4.b. The result of the reactions can be expressed
as:
½ [ī 0 1] + ½ [1 0 1] = [0 0 1]
½ [1 0 ī] + ½ [1 0 1] = [1 0 0]
Their intersections with the (0 1 0) plane form the eventual Labyrinth structure. It should be noted
that among all orientations, the (0 1 0) is the best plane to observe the Labyrinth structure (since
the (0 1 0) plane is the only plane where it’s normal direction is perpendicular to the two slip
directions of [ī 0 1] and [1 0 1] simultaneously).

8

Figure 1-4. Illustration of the labyrinth structure formation [6].

1.2.2.1 The Labyrinth Structure Characteristics
The details of these dislocation structures strongly depend on the strain amplitude, crystal
orientation, temperature, and number of cycles [9–12]. For instance Fuji et al. [9] showed that the
channel width decreases with an increase in the plastic shear strain amplitude. This relationship is
known as the Similitude Law and is shown in Figure 1-5. The similitude relation is derived from
the dislocation strengthening relation provided that an assumption is made to enforce similitude.
The quantity 1/√𝜌 is proportional to the average length “l ” of a uniform dislocation microstructure
with average dislocation density “𝜌 ”. The physical interpretation of the dislocation strengthening
relation implies that l is related to dislocation intersection processes. As smaller average lengths
imply stronger interactions and higher values of the coefficient 𝛼 , one has 𝑙 ̅
=
1
𝛼 √𝜌 = 𝜇𝑏 /𝜏 .
Similitude is obtained by assuming that the characteristic wavelength d of dislocation patterns is
proportional to l, hence:
𝜏 = 𝐾𝜇𝑏 /𝑑 (1-1)

9
For cell structures, the cell size includes the average diameter of the cell interiors plus an average
wall thickness. By combining the similitude relation and the strengthening relation one can find:
𝐾 𝛼 = 𝑑 √𝜌 = 𝑐𝑜𝑛𝑠𝑡 . (1-2)
Similitude implies that
𝐾 𝛼 and d√𝜌 achieve the same values in all FCC metals (linear relationship
between 𝜏 /𝜇 and b/d [5], [9]. The validity of the Similitude law will be assessed in this research.

Figure 1-5. Dependence of the scaled saturation shear stress, τ/μ on b/dc, for single crystals
cyclically loaded along a <0 0 1> axis at 300 K. Labyrinth structures are observed in all the
specimens [5].

10
1.2.3 The Bauschinger Effect
Understanding LRIS is essential for a variety of reasons including understanding the basis of the
Bauschinger effect, spring back in metal forming, and, of course, understanding the plastic
deformation of cyclically deformed metals (fatigue) [13], [14]. The concept of long range internal
stresses in metals has been first discussed in connection with the Bauschinger effect [15]. When a
crystalline metal is deformed monotonically, the material hardens considerably. If the applied
stress is reversed, the material plastically deforms at a lower stress than if it continued to be
monotonically deformed. This is in opposition to what is expected based on the isotropic
hardening. The Bauschinger effect is illustrated in Figure 1-6 and Equation (1-3) quantifies the
deformation in relation to backstresses [16].
𝜎 𝑏 =
(𝜎 𝑓 + 𝜎 𝑟 )
2
(1 − 3)
In this equation, σb is the backstress, σr is the yield stress on reversal of the direction of straining,
and σf is the stress in the forward sense just prior to the reversal. Backstresses have sometimes
been explained by kinematic hardening (or a translation of the yield surface). If σb = 0, there is no
backstress and all hardening is isotropic. Not only is the flow stress lower on reversal but the
hardening features are different as well. The Bauschinger effect is important as it appears to be the
basis for low hardening rates and low saturation stresses (and failure stresses) in cyclic deformation
(fatigue) [14].

11

Figure 1-6. A generalized illustration of the Bauschinger effect. The material yields at a stress that
is much lower than expected for isotropic hardening with reversal of the direction of straining [14].
1.2.3.1 Composite Model - Mughrabi
This section will present the prominent composite model by Mughrabi [2] and Pedersen et al. [17]
which may explain the origins of long range internal stresses considering both the nature of
dislocations and the microstructure present. An explanation on the Mughrabi’s composite model
will provide a guide as to what to expect when analyzing long range internal stress experimental
results. The composite model proposed the internal stresses were of relatively high values.
Mughrabi presented the simple case where hard (high dislocation density walls) and soft (low
dislocation density channels or cell interiors) sections of the microstructure are compatibly
sheared. The different regions each yield plastically at different stresses, depending on the
dislocation density and arrangements. Due to the higher yield stress of the walls, the soft sections
deform plastically through dislocation movement, while the hard wall sections deform elastically

12
This deformation is allowed (compatible) due to a buildup of dislocations at the interface. Of
course, at higher deformation stress both sections begin to deform plastically. It is suggested that
the composite structure is under a heterogeneous stress-state. The high-dislocation density regions
experience a larger magnitude stress than the average or applied stress, and the channel interiors
experience a smaller magnitude stress than the applied stress.
Consider the applied flow stress τa which deforms the sample. Due to the large difference in
dislocation densities, different flow stresses are required to deform the cell wall (τw) and cell
interior (τC ). Once the applied stress reaches that of the cell interior flow stress τC , the cell interior
regions begin to deform plastically while the walls are still in the elastic regime. As the applied
flow stress continues to increase and reaches that of the wall flow stress τw, the bulk material begins
to flow plastically with cell interior and wall deforming in parallel. The relationship between the
cell walls, cell interiors, and applied stress can be expressed as,
τa = fw × τw + fC × τC (1-4)
where τa is the applied stress, fw is the area fraction of the wall, and fC is the area fraction of the
channel interior. Consequently, the stresses present in the microstructure are dependent on the
fraction of wall and channel interior. It’s also important to note that when the applied stress is
removed a back-stress is created in the cell interiors as the walls elastically relax. This is due to
the elastic stress being larger for the walls than the cell interiors. Along with the back-stress is a
balancing forward stress in the walls. With no applied stress, the sum of all LRIS must be zero.
Essentially, the LRIS can be considered a deviation from the applied stress. The following
Equations demonstrate this principle.
τw = τa +Δτw (1-5)

13
τC = τa +ΔτC (1-6)
where τw is the stress present in the walls under load and τC is the stress present in the cell interiors
under load. Once the applied stress has been removed, the bulk material must be in equilibrium
and have a net internal stress of zero. Combining Equation (1-4) with (1-5) and (1-6) and assuming
no applied stress, we have the following,
fw ×Δτw + fC × ΔτC= 0 (1-7)
To explain qualitatively, if a crystalline material is deformed in compression, the walls would
experience an internal stress greater (in magnitude) than the applied stress due to the higher yield
stress, while the cell interiors would experience a lower (in magnitude) stress than the applied.
Once the applied stress is removed, these LRIS are still present in the microstructure. While walls
are still experiencing a positive LRIS (when unloading from the tensile applied stress), to maintain
the equilibrium considering Equation 1-7, the cell interiors will experience a compressive LRIS,
also known as back stress [2].
Figure 1-7 demonstrates a two-dimensional dislocation wall and cell structure. Dislocations
accumulating at the walls act as obstacles to dislocation glide. Some dislocations can pass through
the wall, while others will remain on the walls. This type of dislocation arrangement allows for
deformation compatibility between the walls and cell interior (one can deform elastically, while
the other plastically) [14].

14

Figure 1-7. The composite model indicating ‘‘hard’’ regions (high dislocation density) and ‘‘soft’’
regions (low dislocation density) and the corresponding LRIS present [14].

The composite model has been proposed as an underlying cause of the Bauschinger effect. As soft
and hard regions are unloaded in parallel from tension, the hard region places the soft region in
compression while the stress in the hard region is still positive. When the average stress is zero,
the stress in the hard region is positive while negative in the soft region. Thus, it can be argued
that the backstress creates a Bauschinger effect due to reverse plasticity in the soft region. The
composite model for backstress is illustrated in Figure 1-7. Note that elastic perfectly plastic
behavior is assumed. The definition Kassner et al. used for LRIS is:
τl = τa + Δτi (1-8)

15
where τl is the local stress, τa is the applied stress and Δτi is the LRIS. Thus, LRIS is the local
deviation from the applied stress in a loaded material. This has been demonstrated in Figure 1-8
[14].

Figure 1-8. A profile of long range internal stresses as they would relate to the microstructure
shown [14].

1.2.3.2 Pile Up Model
Dislocation pile-ups leading to long range internal stresses was one of the early explanations as
the underlying reason for the Bauschinger effect [18]. The stress, τp, ahead of the pile-ups is
estimated by the following equation.
τp = nτa (1-9)
where n is the number of dislocations in the pile-up and τa is the applied stress [19]. Thus, high
local stresses develop when pile-ups are present. Local equilibrium is achieved when the high
stress against a barrier, τp, is balanced by a stress, −τp . Seeger et al. [18] and Mott [20] suggested
that on reversal of the direction of straining, this internal stress, −τp, assists the plasticity on

16
reversal direction and decreases the applied stress for reverse plasticity, which leads to a
Bauschinger effect. This is illustrated in figure 1-9.

Figure 1-9. An illustration of an idealized dislocation pile-up model [14].

1.2.3.3 Orowan–Sleeswyk Type Mechanism
This model is a non-LRIS explanation for the Bauschinger effect. Sleeswyk et al. [21] analyzed
the Bauschinger effect in an AISI 310 stainless steel and several single-phase metals at room
temperature and found that the hardening behavior on reversal can be modeled by the monotonic
behavior provided a small reversible strain that is subtracted from the early plastic strain associated
with each reversal. This is known as the Orowan-Sleeswyk type mechanism which as stated earlier
does not involve long range internal stresses [22] with dislocations easily reversing their motion
(e.g across cells or from dislocation walls or reversing from pile-ups). Sleeswyk et al. [21]
suggested when dislocations are gliding they encounter increasingly effective obstacles and the
stress necessary to activate further dislocation motion or plasticity continually increases (work

17
hardening occurs). On reversal straining, under σ, the dislocations will easily move past those
obstacles that have already been surmounted. Therefore, the flow stress on reversal is initially
relatively low. In-situ reverse deformation studies on high voltage transmission electron
microscope by Kassner et al. [23] on pure Al (with a cellular dislocation microstructure) showed
some unraveling in the cell-wall formation in monotonic deformation on reversal, consistent with
the Orowan-Sleeswyk explanation.
1.2.4 Previous LRIS Determination Studies
In this section we are going to review the methods available on how to determine the long range
internal stresses in cyclically deformed single crystalline metals at ambient temperature.
1.2.4.1 Dipole Height Measurements
One method to investigate the existence of long-range internal stresses is to measure the
dislocation dipole heights across the heterogeneous dislocation microstructure. Dislocation dipoles
are pairs of parallel dislocations with opposite Burgers vectors that are on different slip planes.
Figure 1-10 illustrates a dislocation dipole across its two edge dislocation arms on a high resolution
image along with its simulated atomic configuration [24]. Dipole height “h” is defined as the
distance between the <111> slip planes of the two dislocations of a dipole.

18

Figure 1-10. Dislocation dipoles in γ-TiAl. (a) A dislocation dipole across its two edge dislocation
arms on a high resolution image. (b) The simulated atomic configuration of the same dislocation
dipole [24].

Maximum dipole heights represent the upper limit of stable dipoles under the imposed stress. Thus,
local stresses in cyclically deformed materials may be predicted. In this study, the maximum dipole
heights of both walls as well as those of the channels were measured. The stress to separate (or
“break”) an infinite edge dipole, assuming isotropic elasticity, can be evaluated by the formula:
𝜏 =
𝐺𝑏
8𝜋 (1 − 𝜈 ) ℎ

(1-10)
where G is the shear modulus, 𝜈 is Poisson’s ratio, b is the Burgers vector, and h is the dipole
height. This equation will be discussed in greater detail in the work to date section.
There are no known studies of measurement of dipole heights in polyslip. Sauzay and Kubin [5],
in a review article on the dislocation microstructures in cyclically deformed FCC metals, note that
there are no publications in which dislocation densities were measured in fatigued crystals oriented
for multiple slip. Previous works on FCC single crystals oriented for single slip include Kassner
et al. [25–28] on Al and Cu, Antonopoulos et al. [29] on Cu, and Tippelt et al. [30] on Ni. All of
these studies discovered that the dipole heights in the walls and the channels were nearly equal.
(a) (b)

19
One interpretation is that the stress-state may be homogeneous since the stress required to separate
the widest dipole is a measure of the local stress (wider separation dipoles are unstable), and these
are the same in the walls and the channels. (It should be noted that according to Essmann and
Mughrabi [31], edge dipoles eliminate spontaneously in the walls if separated by <1.6 nm in
copper.)
Kassner et al. [12] suggested a homogenous but elevated stress about 2.7 times larger than the
applied stress in a fatigued copper single crystal oriented for single slip at ambient temperature.
Their interpretation is that differences between the applied stress and local stress suggested by
maximum dipole height may be rationalized by small dislocation pile-ups. In another study by
Kassner et al. [26], single crystal Al was cyclically deformed in single slip at 77 K. They
demonstrated that the strength is consistently about 0.8 of the applied stress based on the maximum
dipole height; this is nearly a perfect match between the dipole separation stress and the applied
shear stress, strongly suggesting an absence of LRIS using Equation (1-10), which has limitations
since it is relevant for infinite-length dipoles. As will be discussed in the discussion section, dipoles
are finite and this equation is limited. Maximum dipole separations in fatigued Ni crystals
cyclically deformed in single slip at ambient temperature also predict a homogenous but elevated
stress about four times the flow stress [30]. The present work intends to investigate the LRIS trends
through dipole separation measurements for multiple slip in single crystal copper. Most
investigators [2], [3], [30], as reviewed in [14], suggest that, using in-situ TEM dislocation loop
radii measurements and asymmetry measurements in X-ray peaks, the LRIS in the high dislocation
density heterogeneities varies from a factor of 1.0 (no LRIS) to 3, or more larger than the applied
stress.

20
1.2.4.2 Convergent Beam Electron Diffraction Measurements
An alternate technique for measuring lattice strains in dislocation heterogeneities is CBED
(convergent-beam electron diffraction). This method involves using a small convergent probe to
generate a diffraction pattern where the central disk contains the higher order Laue zone (HOLZ)
lines that are very sensitive to small elastic distortions in the lattice. This will be discussed in more
detail in the experimental procedure section. CBED studies have been done by Kassner et al. [28],
[32] on an unloaded monotonically (creep-deformed) aluminum and copper and cyclically
deformed copper oriented for single slip at ambient temperature to assess the internal stresses.
Figure 1-11 illustrates the lattice parameter and stress measurements of Kassner et al. studies on
cyclically deformed copper oriented for single slip. Lattice parameter measurements in the
channels and close to the vein bundles (within 80 nm) showed some minimal scatter in both studies
by Kassner et al, but no evidence emerged of a residual stress. Legros et al. [33] also assessed the
internal stresses in cyclically deformed single-slip oriented silicon single crystals between 1073
and 1173 K using the CBED technique. Their measurements discovered that almost no significant
internal stresses arising from the walls are felt by the channels, and that the stress profile is flat.
All the CBED studies are performed on unloaded material, and of course, the foil thickness is small
in the regions examined. Thus, LRIS relaxation is possible.

21

Figure 1-11. (a) TEM micrograph of cyclically deformed Cu single crystal in single slip. A channel
bounded by dipole bundles and the locations of where the CBED was used to determine the lattice
parameter. (b) The variation in the lattice parameter and stress was measured by CBED, versus
position within the channel (typically 3.8 mm wide). No substantial LRIS was evident [28].

22
1.2.4.3 Dislocation Loop Radii Measurements

Mughrabi [2], [34] in an influential experiment assessed long range internal stresses by measuring
dislocation loop radii as a function of position within the heterogeneous dislocation microstructure
of a cyclically deformed copper single crystal to saturation (PSBs). In this study dislocations in
PSBs were pinned by neutron irradiation of the deformed single crystal copper specimens before
thinning for TEM studies. After dislocation loop radii measurements, the standard Orowan bowing
equation was used to calculate the internal stresses. The Orowan bowing equation is:
𝜏 𝑙𝑜𝑐 =
𝑇 𝑏𝑟
(1-11)
where 𝜏 𝑙𝑜𝑐
is the local shear stress, T is the dislocation line tension, b is the Burgers vector, and r
is the radius of curvature. Figure 1-12 shows the variation in calculated long range internal stresses
from dislocation bowing, versus position within the channel. It can be observed that the
dislocations were bowed more drastically closer to the dislocation walls than in the center of the
channels. By implementing the Orowan bowing equation, stress values next to the walls were
calculated to triple the applied stress, and in the channels about 0.63 times the applied stress.
Interactions with dipoles and multipoles of the wall are considered in the final calculations as the
Orowan bowing equation does not account for these effects. It should be mentioned that this study
has long been cited as a proof of long-range internal stresses existence in cyclically deformed
metals and follows the Mughrabi’s composite model as well [3], [34].

23

Figure 1-12. The variation in calculated long range internal stresses from dislocation bowing
versus position within the channel in a cyclically deformed copper single crystal to saturation
(PSBs present). The dashed lines are the modified data by Mughrabi considering interactions with
dipoles and multipoles of the wall [2], [34]. The dislocation loops assessed were within PSBs.

1.2.4.4 Asymmetry in X-ray Peak

The details of X-ray peaks characteristics have been used to assess LRIS. When a material is
plastically deformed, the diffraction line profiles broaden and develop a characteristic asymmetric
shape. There is also a change in the location of the peak, as it shifts away from the perfect
unstrained crystal. The asymmetry of the line profile is strain dependent; if strain is increased, the
asymmetry and peak broadening increases. Considering the aforementioned composite model by
Mughrabi where the deformed material included two regions of dislocation dense walls and low

24
density cell interiors, each with different stresses, the asymmetric peak can be decomposed to two
separate diffraction sub peaks coming from the two regions within the microstructure. Each of
these two regions with different dislocation density diffracts in a characteristic way, such that the
sum of these two line profiles results in the final measured X-ray line profile. This can be seen in
figure 1-13 in a study by Schafler et al. [35]. The areas under each subpeak reflect the
corresponding relative volume. Thus, in figure 1-13, the smaller symmetric peak should be
associated with the high dislocation density regions such as walls, dipole bundles or PSBs [36,37].

25

Figure 1-13. (a) X-ray diffraction peak in monotonically deformed copper. The asymmetric
broadening can be observed with comparison to the undeformed specimen. (b) A decomposition
can be performed that leads to two separate diffraction sub peaks coming from the two regions
within the microstructure (high and low dislocation density regions) with different lattice

26
parameters and hence, different stresses [35]. (c) Schematic illustration on how a dislocation can
broaden the X-ray diffraction peak.

In another study by Ungar et al. [38], on a compression deformed single-crystal copper, internal
stresses of 0.1σa in the cell interiors and 0.4 σa in the cell walls were reported. It should be noted
that although these experiments are indirect, they have been used to infer local stresses based on
the irradiation of millions of cells [14].
1.2.4.5 Synchrotron X-ray Microbeam Measurement

The first synchrotron X-ray microbeam measurements of cell interiors were published by Levine
et al. [39] both on a single crystal copper that was monotonically deformed in compression and
tension and a cyclically deformed copper single crystal oriented for single slip. The synchrotron
X-ray microbeam at the Advanced Photon Source at Argonne National Laboratory was used to
scan the interior of the samples up to 50μm in depth.

27

Figure 1-14. The X-ray beam hits the sample and is diffracted onto the CCD. A platinum wire
scans across the sample in sub-micron steps, and provides depth resolution to the measurements
[39].

In the monotonically deformed specimen the sign of the cell interior stresses with relation to the
applied flow stress was interestingly opposite. This confirms the existence of back stresses in the
dislocation cell interiors with low dislocation density, exactly as the composite model predicts.
Although no measurements were done in the walls, for maintaining the net stress of the unloaded
bulk material to zero, the walls must contain a forward stress. They observed large strain
fluctuations within the cell interiors [39].

28
In the cyclically deformed specimen the microbeam was used to scan the dipole bundles and no
strains were measured in the cyclically deformed copper within the margin of error 1 × 10
−4
,
which is roughly equal to 7 MPa [39].

29

1.3 Experimental Techniques

The following section includes an overview on sample preparation methods used in this study
followed by the microstructural characterization techniques to observe the labyrinth structure.
Characterization techniques to assess the long range internal stresses will then be discussed.
1.3.1 Cyclic Deformation

[001]-oriented copper single crystals of 99.999% purity were cyclically deformed in tension and
compression with R = −1 using MAYES ESM100 servo-mechanical machine (at 298 K to 157
cycles at a strain amplitude of 4.0 × 10
−3
and a strain rate of 2 × 10
−3
s
−1
provided by our colleague,
Professor Maxime Sauzay, at CEA University of Paris Saclay.
1.3.2 Sample Preparation Techniques

1.3.2.1 Sample Sectioning and Polishing
Figure 1-15 depicts the cyclically deformed single crystal copper cylinder. A wire saw machine
was used to prepare 3 mm disks of 600 μm thickness to have the minimum stress applied to the
sample that could cause damage.

30

Figure 1-15. The cyclically deformed single crystal copper cylinder.

Microstructural observations like TEM and EBSD require a very smooth and scratch free surface
that can be acquired with high quality polishing. In this study a Buehler polishing wheel was used
to grind copper disks with SiC sandpapers starting from 600 grit, and followed by 800 and 1200
grit down to 250 m thickness. The thickness of 250μm was found to adequately preserve the
dislocation microstructure. For the EBSD samples, a subsequent polycrystalline 1 µm diamond
paste polishing was conducted with a polishing cloth after a set of sand paper polishing processes.
The rotating speed of the polishing wheel was set at 150 rpm for all hand polishing processes.
Finally, a Buehler Vibromet 2 with the MasterMet 2 colloidal silica suspension and power gauge
setting of 50 was used as an automatic vibratory polishing.

31
1.3.2.2 Jet Electro-Polishing
Transmission Electron Microscopy foils need to be thin enough for electrons to be transmitted
through the sample. There are different techniques to make such thin foils. In this study a TEM
sample preparation method that applies the minimum stress to the foil is desired. Thus, the
Fischione twin jet was used for the thinning process of the copper foils down to 200 nm.
A jet electropolisher uses two jets to direct electrolyte flow onto the specimen, which
simultaneously thins and polishes both sides. Cathode coils in the jet assemblies and the anodic
platinum contact in the specimen holder allow current to flow through the electrolyte (Figure 1-
16.a). For a pure copper a solution of 10% nitric acid and 90% methanol at −20 °C was used as
an electrolyte. Jet electropolishing is accomplished by pumping a stream of negatively charged
electrolyte against the surface of a positively charged sample. This process is used to create a
dished or dimpled area and is continued until perforation of the specimen occurs. The regions close
to the perforation are thin enough for the electron beam to transmit. This can be seen in the
Scanning Electron Microscopy (SEM) image (Figure 1-16.b). The ideal sample to jet distance is
illustrated in figure 1-16.c [40]. The polishing voltage and current can be adjusted at low levels to
selectively dissolve metal ions from a specimen, that is, to chemically etch the specimen.
Increasing the voltage until a plateau in current is reached results in electrolytic polishing (Figure
1-16.d [40]) .

32

Figure 1-16. (a) Schematic illustration of a jet electropolisher. (b) An SEM image showing the thin
regions around the perforation on a copper TEM sample made by jet electropolishing. (c) Ideal
sample to jet distance. (d) The current versus voltage diagram of electropolishing. Increasing the
voltage until a plateau in current is reached results in electrolytic polishing [40].

(a) (b)
(c) (d)

33
1.3.2.3 Focused Ion Beam

Another method to prepare specimens for TEM studies is focused ion beam (FIB). In FIB, ions
(such as Ga) are generated within the ion gun and are directed towards the specimen surface. This
interaction between ions and the specimen generates the emission of backscattered ions, radiation,
and ion induced secondary electrons. The interaction of Ga with the surface generates sputtered
ions that can be utilized to mill the specimen [41]. When combined with a micromanipulator and
gas injection system, the FIB can produce samples with dimensions suitable for TEM. To prepare
TEM samples, the sample is milled from a region of interest and placed on lift-out grids, where
milling continues to further reduce the thickness of the specimen. In this study a 200 nm thick
sample was prepared using a Tescan GAIA-3 GMH FIB-SEM FIB.

Figure 1-17. Schematic illustration of a dual-beam FIB–SEM instrument. Expanded view shows
the electron and ion beam sample interaction [41].

34
1.3.3 Microstructural and LRIS Characterization Methods
1.3.3.1 Electron backscatter diffraction (EBSD)
Electron backscatter diffraction (EBSD) is a method to obtain the crystallographic data including
crystal orientations from samples in the Scanning Electron Microscope (SEM). EBSD operates by
arranging a highly polished sample at a low angle, usually 20˚, to the incident electron beam (since
the SEM stage is often used to tilt the plane of the sample to this low angle, the value of stage tilt
is usually referred to 70˚). In SEM, an electron beam generated from a filament or field emission
gun interacts both elastically and inelastically with a crystalline sample after passing through
multiple electromagnetic lenses and apertures. In elastic interactions the colliding electrons are
reflected by high angles, which are known as back-scattered electrons. On the other hand, during
inelastic interactions, the incoming electrons interact with the surface of the sample generating
secondary electrons and x-rays. Images in SEM are produced from secondary electrons as elevated
regions of the sample will appear brighter, as it reflects a higher number of electrons. Electron
diffraction also occurs from the incident beam point on the sample surface as certain electrons
satisfy the Bragg’s law of a crystallographic plane. The diffracted electrons form the Kikuchi
pattern that can be detected on a fluorescent screen. This diffraction pattern can be utilized to
determine the crystal orientation, crystallographically different phases, and grain boundaries [42].
In this work the JSM-7001F-LV SEM equipped with EBSD was used to determine the desired
[010] direction on the copper cylinder for labyrinth dislocation microstructure observation.

35

Figure 1-18. Schematic arrangement of EBSD system [43].

1.3.3.2 Transmission Electron Microscope (TEM)

In transmission electron microscopy (TEM), an accelerated beam of electrons generated from a
field emission gun (FEG) or a filament (LaB6 or tungsten) is condensed using condenser
electromagnetic lenses and apertures down a column where it transmits a very thin specimen. All
illumination system electromagnetic lenses that are located before the specimen demagnify the
electron beam to acquire a parallel beam. Then the transmitted electrons are projected onto a
fluorescence screen (ZnS) or a charged couple device (CCD) camera to acquire the resulting image
after passing through a series of electromagnetic lenses and apertures. These electromagnetic
lenses (the imaging system) are all magnifying the electron beam to increase the resolution of the
image. The resulting interaction of the electrons with the specimen can be utilized to image and
characterize many aspects of the specimen including mass contrast, phase contrast, defects,

36
interfaces, density variations, atomic composition, crystal structure and orientation, and lattice
parameters [44]. Figure 1-19 illustrates a schematic configuration of a TEM [45].

Figure 1-19. A schematic configuration of a TEM [45].

Depending on the position of the objective aperture and the collected electron beam, different
TEM modes can be acquired. A schematic of the bright field, dark field, and selected area
diffraction imaging in the TEM is shown in figure 1-20 [45].

37

Figure 1-20. Schematic illustrations of the transmission electron microscope for (a) bright field
imaging and (b) selected area diffraction mode and (c) dark field imaging [45].

Diffraction in the TEM occurs via the same wavelength interference mechanism as in x-ray
diffraction. However, since the wavelength of electrons are orders of magnitude smaller than that
of the x-rays the angles of diffraction are much smaller.

38
In order to better understand diffraction within the framework of TEM, a geometrical
representation of the incident and scattered electrons or the Ewald’s sphere is useful. The Ewald
sphere is basically a geometrical representation of the incident and scattered electron beams,
reciprocal lattice of the crystal, and the angles of diffraction. As the name implies, this is a 3-
dimensional geometry. Figure 1-21 shows the two-dimensional equivalent, which is fundamentally
a section of the 3-dimensional geometry. It should be noted that each point in a reciprocal lattice
represents a set of crystallographic planes in real space. The Ewald sphere intersecting each of
these points indicates that diffraction has occurred in that specific point which is equivalent to a
plane in real space. In real space, the TEM specimen is very thin, that results in the points in
reciprocal space to be elongated causing a number of (extra) reciprocal lattice points to intersect
the Ewald sphere. This is quite different from x-ray diffraction. For one, the Ewald sphere for x-
ray diffraction is substantially smaller than for an electron beam as the radius of the sphere is
inversely proportional to the wavelength of the incoming beam, and secondly, the crystals are also
substantially thicker in the case of x-ray studies [44].

39

Figure 1-21. The Ewald sphere. When the sphere cuts the reciprocal lattice points, the Bragg
condition is satisfied. Note that the radius of the Ewald sphere is very large compared to the
reciprocal lattice so that the sphere projection is almost flat [44].

If the incident beam hits the sample at specific points of symmetry or the so-called zone axes,
constructive interference of the diffracted beams would result in recognizable diffraction patterns.
Figure 1-22 shows a number of diffraction patterns from different zone axes for an FCC crystal.
These zone axis diffraction patterns can be used to specify orientation of the crystal and to create
specific diffraction conditions. Most importantly, diffraction patterns can be utilized to find
specific diffraction conditions (beams/spots for g
.
b) that can result in observation and
characterization of dislocations within a crystal using TEM. This topic will be further discussed
in the following sections [44].

40

Figure 1-22. Rotation of the FCC sample to the zone axis of [100], [111], and [110] produces the
diffraction patterns shown.

In figure 1-22, note that some diffraction spots are missing. For example, the [001] diffraction
pattern does not show diffraction spots corresponding to [100] or [010]. The absence of these
diffraction spots is due to an extinction rule for diffracting beams which is dependent on the
symmetries within the crystal structure. For FCC crystals, the following extinction equation
applies,
Γ(hkl) = 1+e
iπ(h+k)
+ e
iπ(h+l)
+ e
iπ(k+l)
(1-12)

The diffraction point does not exist if Γ (hkl) = 0. Using this formula, it is rather easy to predict if
a certain plane in the crystal would form a diffraction spot in the TEM micrograph. For example,
inserting the (100) plane indices into equation (1-12) results in:
Γ (100) = 1 − 1 – 1 + 1 = 0
Therefore, a diffraction spot for the plane would be absent.

41
1.3.3.2.1 The Two-Beam Condition

When a zone axis is found, a special technique can be used to look at only a single specific
diffracting beam. In this technique, the incident beam is tilted so that the Ewald sphere would
intersect both the origin and the desired diffraction spot. This is known as the two-beam condition
(Figure 1-23). An aperture can then be used to select only the incident beam and block the
diffracted beam (bright field), or to use a diffracting beam and block the incident beam (dark field).
In other words, when selecting one of these beams to pass through the aperture, the projection of
the sample onto the projection screen is composed of only the incident beam minus the diffraction
(bright field), or conversely only the diffracted beam (dark field) [44].

Figure 1-23. The two beam condition with g <202>.

42
1.3.3.2.2 Kikuchi Lines

When rotating the sample inside the TEM, the best method for determining the crystallographic
orientation is through the use of Kikuchi lines. Kikuchi lines are produced by diffuse electron
scattering as the electrons hit the sample. The electrons scatter and end up meeting every Bragg
condition, and diffract off of every crystallographic plane. This creates a complex pattern of lines
which correspond to the crystallographic planes. A diagram is shown in figure 1-24 [44].
Kikuchi lines are far more pronounced in thicker samples, and in regions of perfect crystal. It is
therefore fairly easy to focus the beam down into a single dislocation cell interior, to find the exact
crystallographic orientation. Using this orientation, two beam conditions can be set up. Often two
cell interiors have slightly different orientations, so when analyzing a dislocation wall, it’s essential
to use Kikuchi lines to constantly be aware of the diffraction conditions. Kikuchi lines were used
extensively in the study presented here to orient and set up all of the diffraction conditions.

43

Figure 1-24. Formation of the Kikuchi lines [44].

44
1.3.3.2.3 Convergent Beam Electron Diffraction (CBED)

If the electron beam is focused (rather than parallel to specimen) onto the surface of the sample in
TEM then resultant diffraction pattern would be convergent beam electron diffraction (CBED).
Diffraction images of a single crystal created using a parallel electron beam configuration result
in a diffraction spot pattern, whereas in a convergent beam arrangement, these spots are broadened
out into disks. Schematic drawings comparing the two diffraction configurations are depicted in
figure 1-25.

Figure 1-25. Schematic illustration of the collection of a parallel and convergent beam electron
diffraction pattern [45].

45
There are two major advantages in utilizing CBED over conventional parallel beam diffraction
conditions. First is that with CBED a smaller area can be probed and very localized information
can be obtained. Second, the pattern obtained is in fact three-dimensional information about the
structure. This three-dimensional information can be used to determine the thickness, crystal space
group, and lattice parameter (and therefore strain) of the specimen. The crystal structure can be
determined by finding the CBED patterns in several major zone axes. In a known crystal structure,
higher order Laue zone, or HOLZ lines can be used to determine deviations from the equilibrium
lattice parameter in a given region or relative deviations between regions. HOLZ lines arise due to
diffraction from planes not parallel to the incident electron beam. As shown in the Ewald sphere
construction (Figure 1-26), the first set of reciprocal lattice points above the zero order Laue zone
(ZOLZ) are called the first order Laue zone (FOLZ) and correspond to the condition hu+kv+lw=1
where h, k, and l are the indices for the diffracting plane and u, v, and w are the indices for the
incident beam direction. The next set are called the second order Laue zone (SOLZ), etc. All of
these planes above the ZOLZ are generically referred to as HOLZ. The HOLZ reciprocal lattice
points that are intersected by the Ewald sphere meet the diffraction conditions of Bragg’s law but
are not oriented parallel to the incident electron beam axis.

46

Figure 1-26. Reciprocal space schematic showing the diffraction conditions for the zero order Laue
zone (ZOLZ), first order Laue zone (FOLZ), and second order Laue zone (SOLZ) by intersection
of the Ewald sphere with the reciprocal-lattice rod (relrod) [46].

The diffracted intensity from these planes is scattered out to relatively high angles, resulting in a
series of concentric, bright rings around the ZOLZ pattern. Inside the central 000 disk of the ZOLZ
are fine lines. Due to their sensitivity to small lattice parameter changes, the potential for using
HOLZ lines for strain determination (LRIS) was recognized with their discovery and explanation
by Jones et al. [47]. This technique involves using a convergent probe to generate a diffraction
pattern where the central disk contains the HOLZ lines that are very sensitive to slight elastic
distortions to the lattice. This will be discussed in more detail in the 4th chapter.
1.3.3.2.4 Electron Energy Loss Spectroscopy (EELS)

Electron energy loss spectroscopy (EELS) is the use of the energy distribution of electrons that
pass through a thin sample to analyze the content of the sample and create images with unique

47
contrast effects. EELS instrumentation is typically incorporated into a transmission electron
microscope (TEM) or a scanning TEM (STEM). As the incident electron interacts inelastically
with the sample, it changes both its energy and momentum. The amount of energy loss can be
detected in the spectrometer and gives rise to the electron energy loss signal. EELS allows quick
and reliable measurement of local thickness in TEM with the spatial resolution about 1 nm. The
thickness t is calculated as:
t = mfp × ln (I/I0) (1-13)
Where mfp is the mean free path of electron inelastic scattering, which has been tabulated for most
elemental solids and oxides. (I0) is the integral under the zero-loss peak (ZLP) and (I) is the integral
under the whole spectrum.

48
1.4 Work to Date on LRIS
The following describes work to date, from a paper titled “Assessment of Internal Stresses Using
Dislocation Dipole Heights in Cyclically Deformed [001] Copper Single Crystals”, published in
Metals Journal (MDPI) and selected to be highlighted on the front page [48]. This study explores
how long-range internal stresses compare with the single slip studies through measuring the
dislocation of dipole heights.
Next, the studies on the determination of long range internal stresses using Convergent Beam
Electron Diffraction technique from the paper titled “Determination of Long-Range Internal
Stresses in Cyclically Deformed Copper Single Crystals Using Convergent Beam Electron
Diffraction”, published in Crystals Journal (MDPI) will be discussed.
1.4.1 Experimental Methods
1.4.1.1 Cyclic Deformation

[001]-oriented copper single crystals of 99.999% purity were cyclically deformed in
tension/compression at 298 K to 157 cycles at a strain amplitude of 4.0 × 10
−3
and a strain rate of
2 × 10
−3
s
−1
provided by our colleague, Professor Maxime Sauzay, at CEA University of Paris
Saclay. Schmid factors for this polyslip were 0.408 (with 8 potentially active slip systems). The
crystal was fatigued to saturation with an axial stress of 275 MPa and a resolved shear stress of
112 MPa in the <110> direction on a {111} plane. Figure 1-27.a illustrates the stress versus strain
behavior and figure 1-27.b demonstrates the maximum and minimum peak stresses with respect
to the number of cycles. A continuous hardening was observed and it occurred very quickly–that
is, during the first 20 cycles. Then, the cyclic hardening rate decelerated until a maximum peak
stress was observed at the 108th cycle. After that, a very slow softening was observed up until the

49
157th cycle. This is more discernible in the logarithmic plot of the evolution of the maximum peak
stress with respect to the number of cycles in figure 1-27.c.

(a)

(b)

50

(c)
Figure 1-27. (a) The cyclic deformation of [001]-oriented copper single crystal at ambient
temperature. Strains are plastic and elastic. (b) The evolution of the maximum and minimum peak
stress with respect to the number of cycles. (c) The logarithmic plot of the evolution of the
maximum peak stress with respect to the number of cycles confirms saturation is reached.

1.4.1.2 Sample Preparation

As stated in the first chapter, the labyrinth structure can only be seen in the (010) plane. The only
known direction on the cyclically deformed copper cylinder was the stress direction [001]. Thus,
for finding the [010] direction, two random orthogonal foils from the side of the cylinder were
prepared using the jet electropolisher for diffraction pattern studies under TEM (Figure 1-28). As
it is illustrated in figure 1-28, diffraction patterns were found for each disk.

51

Figure 1-28. Diffraction patterns of the two orthogonal directions in the single crystal specimen.
Note that for each disk two different diffraction patterns were found due to bends created by jet
electropolishing.

1.4.1.2.1 Zone Axis Determination and EBSD Studies

For finding the zone axis from the diffraction pattern, first we need to determine the indices of at
least two spots on the diffraction pattern: g1 and g2. Then, the zone axis can be found from z = g1
× g2. For assigning specific (hkl) to an individual spot , the angles and distances between each
spot in the diffraction pattern were measured (R1 and R2) and the ratios between the distances were
found. Then, a set of possible zone axes were determined using the Table 1-1 and Equation (1-14).

(1-14)

52

Where (a) is the lattice parameter and L is the camera length. Thus, the 4 zone axes of [1 1 0], and
[1 -1 0] were found.

53
Table 1-1. The ratios of N=h2+k2+l2 of different planes (hkl). Note that considering the extinction
Equation (1-14) planes that can diffract in an FCC crystal are highlighted in green [49].

54
In order to confirm the orientation of the foils, EBSD studies were also carried out. Figure 1-29
shows that the bulk crystal orientation is [1 1 0].

Figure 1-29. EBSD orientation map of the side disks showing the crystal orientation as <110>.

Focused ion beam was used for preparing TEM samples. The TEM micrograph in figure 1-30
shows the labyrinth structure has been completely vanished as a result of the ion bombardment.
The impact of ions to the sample is sufficient to destroy the labyrinth dislocation structure. Thus,
TEM sample preparation using FIB technique was not pursued and all the TEM samples in this
study were prepared using the conventional jet electropolishing technique. EBSD map was
obtained from the sample prepared with FIB to determine possible damage induced in the
crystallinity of the specimen and to confirm that the crystallographic orientation of [0 1 0] was
achieved .

55

Figure 1-30. (a) EBSD map of the TEM sample prepared using a FIB showing the (0 1 0) single
crystal and the damaged crystal at the top. (b) TEM micrograph of the copper sample prepared
with FIB. Note that there is no labyrinth structure in the single crystal section at the bottom right.

Disks (3 mm) from the (001), (010), and (100) planes were electropolished with a Fischione twin
jet using 10% nitric acid and 90% methanol at −20 °C. After much trial and error the optimal
polishing condition was found to be 15 seconds at 40 volts and around 2 minutes at 15 volts. The
speed was set such that when the electrolyte is flowing and the jets are raised above the surface of
the solution, the solution softly curves downward and meets about 1 cm from straight across (for
a pure copper a solution of 10% nitric acid and 90% methanol at −20 °C was used as an
electrolyte). Each sample was then checked under a stereoscope and, of course, finally under the
TEM.
The specimens were examined in bright field using the JEOL JEM-2100F transmission electron
microscope at the University of Southern California at an accelerating voltage of 200 kV. The
specimens were stored in liquid nitrogen to prevent static recovery/recrystallization of the
dislocation substructure [50, 51].
(a) (b)

56
1.4.1.3 The Labyrinth Microstructure

Figure 1-31 depicts the 3-dimensional microstructure of the fatigued copper containing dislocation
walls and channels. The stress axis is parallel to the vertical [001] direction. This figure illustrates
that the microstructure of fatigued [001] copper single crystal consists of a labyrinth and cellular
structure. Both structures consist of relatively evenly spaced walls and channels with a much lower
dislocation density.

Figure 1-31. Transmission electron microscope “cube” based on images taken from the (100),
(010), and (001) planes of a specimen cyclically deformed to saturation as illustrated in Figure 1-
27.a. The stress axis is parallel to the vertical direction [001].

57
This heterogeneous structure has been referred to, by some, as consisting of ‘‘hard’’ regions with
high dislocation density and ‘‘soft’’ regions with low dislocation density [2, 3]. The labyrinth
structure was only uniquely identified on (010) planes where all the dipole height measurements
and CBED analysis were performed. Due to small misorientations by low angle dislocation tilt
boundaries in the crystal, the “perfect” labyrinth structure may not be observed. Figure 1-32
illustrates TEM micrographs of the labyrinth structure from the (010) planes of the cyclically
deformed copper. One can observe that only two wall planes of (001) and (100) are discovered in
the labyrinth microstructure where more walls are along [100] than along the perpendicular
direction.

58

Figure 1-32. TEM micrographs of the labyrinth structure from the (010) planes of the cyclically
deformed copper. The <202> two-beam conditions were used to image dislocations in this study.

The average channel width was approximately 0.36 µm at a shear stress of 112 MPa and the
volume fraction of the channels was about 70%. The average wall width was 0.12 µm. The channel
widths were smaller than what has been reported in earlier studies of the dislocation wall and
channel structures due to the higher amount of applied shear stress (e.g, a channel width of 0.52

59
µm at a shear stress of 44 MPa [8] and a channel width of 0.57 µm at a shear stress of 50 MPa [9]
both for [001] single crystal copper).
In the current study, the dislocation density in the walls was 8.6 × 10
14
m/m
3
and the density in the
channels was 1.55 × 10
13
m/m
3
using Equation (1-15) [52]. In this equation, N is the number of
dislocation intersections with a random line of length, L, and t is the thickness of the film.
𝜌 =
2𝑁 𝐿𝑡

(1-15)
The thickness of the film was determined by electron energy loss spectroscopy (EELS) (Figure 1-
33). The mean free path of electrons in copper is 103.6 nm at an accelerating voltage of 200 kV.
Thus, according to the plot in figure 1-33, the thickness is from 119.14 nm to 352.24 nm (up to 7
m from the hole). Of course, the thickness decreases gradually closer to the pinhole.

60

Figure 1-33. The EELS analysis showing the thickness measurements.

Kassner et al. [26] reported a dislocation density of 1.3 × 10
15
m
−2
and 8.3 × 10
12
m
−2
in the walls
and channels, respectively, for a cyclically deformed single crystal copper in single slip to pre-
saturation to a shear stress of 19 MPa where all of the dislocations were dipoles. Thus, perhaps
dipoles did in fact control the flow stress. In this study, where copper is cyclically deformed under
multiple slip, more non-dipole dislocations were available than dipoles. Therefore, perhaps the
total dislocation density controls the stress and the dipoles reflect what the value of the stress is.

61
1.4.1.3.1 The Labyrinth Structure Characteristics

At high plastic strain amplitudes, there is no more labyrinth structure forms and only cell structures
have been reported [53]. Therefore, in this study a low plastic strain amplitude was chosen to avoid
cell formation and study the similitude law in the labyrinth structure. This scaling law shows the
relationship between the applied stress and perhaps the inverse of channel width as stated in section
2.1.1.1. According to the review on this law by Sauzay et al. [5] the relationship between the
applied stress and the inverse of channel width is linear. However, in the present work an
exponential correlation between stress and inverse of channel width is observed at higher applied
stresses. In this study, the higher values of applied stress (112 MPa) compared to that of previous
studies (all below 70 MPa) is considered to be responsible for such change in behavior. All data
are summarized in a plot in figure 1-34.

Figure 1-34. Dependence of the scaled shear stress τa/G on b/d ×10
-3
, for single crystals cyclically
loaded at 300 K. Data from [8–10], [54–56].

62
1.4.1.4 LRIS Assessment through Dipole Height Measurements
Figure 1-35 represents the schematic configuration of the dislocation structure in part of a labyrinth
structure in a copper disk. Dipole height “h” which is the distance between the two slip planes of
the dislocation dipole is also illustrated.

Figure 1-35. A three-dimensional (3-D) schematic representation of the dislocation structure
illustrating the dipole height “h”.

A total of 100 dipole heights were measured. The error in dipole height measurements was
estimated to be ±1.0 nm. Dipoles were identified as two parallel dislocations with similar lengths
in the <112> direction. As stated before, in order to obtain the dipole height, which is defined as
the distance between the <111> slip planes of the two dislocations of a dipole, the spacing between
the dislocation lines in foils with a normal of (010) needs to be projected onto a single {111} plane.
Although secondary dipoles were observed in cyclically deformed single slip Cu single crystals,

63
only primary dipole heights were measured; secondary dipoles were rarely observed (in the present
work) [28]. As mentioned earlier, maximum dipole heights represent the upper limit of stable
dipoles under the imposed stress. Therefore, they can predict the local stresses in cyclically
deformed materials. The maximum dipole height in the walls and channels was 4.3 nm and 4 nm,
respectively, and the top 10% of the maximum dipole heights in the walls was 3.3 nm and in the
channels was 3.5 nm. This is analogous to what earlier studies found in fatigued copper, aluminum,
and nickel with single slip [25–28], [30]. A homogenous stress state within the crystal can be
interpreted since the local stresses based on the maximum dipole heights are nearly identical in the
walls and channels. This suggests that the internal stresses are minimal. An Orowan–Sleeswyk-
type mechanism, which involves no internal stresses, could be the underlying basis for the
Bauschinger effect that was observed with each reversal [13]. The mechanism may be responsible
for the observed saturation at a very low stress compared to the true (monotonic) fracture stress
[13, 21, 22].

64

Figure 1-36. A part of the labyrinth structure showing dislocation dipoles.

Choosing h = 4.15 nm, G = 46.5 GPa, and 𝜈 = 0.355, Equation (1-10) predicts a dipole-breaking
value of 177 MPa, which is higher than the remote shear stress of 112 MPa. That is in line with
various estimates made by Kassner and co-workers showing that the dipole separation stress is
generally higher than the remote shear stress value, in both hard and soft regions [28]. At 77 K,
aluminum was the exception, with a pre-saturation vein and channel microstructure [26]. In this
case, based on Equation (1-10), which as mentioned earlier has its limitations, the dipole separation
stress and the applied shear stress are almost the same.
Existing literature suggests negligible differences in the stress between hard and soft phases but a
much higher dipole separation stress than expected based on dislocation dipole height

65
measurements and asymmetry in x-ray peaks [2, 3, 27, 30]. In the following paragraphs, three
considerations are noted, which the authors believe should be incorporated into equation (1-10).
1. Contrary to nickel and copper, aluminum crystals obey almost isotropic elasticity. In aluminum,
the Zener anisotropy ratio amounts to 1.1, very close to 1. The Zener ratio reaches 2.5 in nickel
and 3.3 in copper, respectively [5]. The introduction of cubic elasticity in equations (1-10) makes
a difference between almost isotropic aluminum and anisotropic nickel and copper. For this
purpose, the anisotropic values of the shear modulus and the Poisson’s ratio proposed by Bacon
and Scattergood were used [57]. For copper at 300 K, the numerical application provides
anisotropic values of G = 42.1 GPa and 𝜈 = 0.431, which differ from their isotropic elasticity
counterparts (G = 46.5 GPa and 𝜈 = 0.355, respectively). For nickel, the difference remains but is
lower due to a lower anisotropy than copper. For aluminum, both values are very close as expected
in this almost isotropic metal. The use of the Bacon and Scattergood effective parameter in
Equation (1-10) does not significantly change the dipole separation stress, which reaches 180 MPa.
In fact, a lower shear modulus and a higher Poisson ratio act in opposite ways in equation (1-10).
This small effect (here around 5 MPa) of the anisotropy of cubic elasticity was confirmed by the
dislocation dynamics simulation studies carried out by Veyssière and Chiu on copper [58].
2. Aluminum’s normalized stacking fault energy (SFE) is higher than that of nickel and copper
(Table 2). This implies that the distance between the head and queue partial dislocations is shorter
in aluminum than in nickel while copper has the greatest distance in comparison with the other
two. Differences in the forces between the two opposite edge dislocations are expected because
each of them is now split into two partial dislocations. This may explain the specific behavior of
aluminum (almost equal dipole separation stress and applied stress) with respect to nickel and
copper. It should also be noted that in aluminum, the extinction distance is 2 or 3 times larger than

66
it is for copper, depending upon the reflections chosen [59]. This means that it is more difficult to
obtain an estimate of the maximum dipole height in aluminum. We once more refer to the work of
Veyssière and Chiu [58], who not only assessed the effect of cubic elasticity but also considered
SFE. Based on the 2D dislocation dynamics simulations of dipole separation stress in copper,
partial dissociation only has a visible effect on the dipole separation stress in dipole heights lower
than 2.5 nm. Since in our study the maximum dipole heights are 4 nm, the dissociations would not
influence the dipole separation stress [58]. It can be concluded that for the metals considered in
Table 1-2, only small increases in the edge dipole separation stress are induced by dislocation
dissociation, even in copper. In this case, the “conflicting” results exhibited by Kassner and
coworkers, which were observed in copper but not in aluminum, cannot be the effect of the
stacking fault distance alone.
3. Finally, another characteristic of dipoles in a fatigued microstructure should be discussed. The
dipole length is finite and its aspect ratio, h/L, ranges between 1/15 and 1/10 [5], [30]. Careful
examination of the maximum dipole heights provides a mean value of the aspect ratio, h/L, of 1/11
in the wall and channel regions of the labyrinths here. The discrete dislocation dynamics
computations of Sauzay and Dupuy, where a nodal DDD code was used [60] to assess the influence
of d/L on the separation stress. Various prismatic loops were built that take large ranges of the L
and h parameters into consideration. The two long primary dislocations of length L are indeed
linked together by short collinear dislocation segments corresponding to the h height. For each
prismatic loop, a shear stress was applied to the instability of each prismatic loop. It was found
that a correction factor, C (h,h/L), allows for the rewriting of equation (1-10), which is only valid
for infinite edge dipoles. Equation (1-15) was obtained based on numerous DDD computation
results:

67
𝜏 = 𝐶 (ℎ, ℎ/𝐿 )
𝐺𝑏
8𝜋 (1 − 𝜈 ) ℎ

(1-15)
Equation (1-15) was used to evaluate the dipole separation stress in this study. Using h = 4.15 nm
and h/L = 1/11, a pre-factor C of 1.5 was found [60]. Then, introducing the Bacon and Scattergood
elasticity parameters mentioned above for Cu [57], a dipole separation stress as high as 270 MPa
was found. That is about 2.4 times higher than the applied shear stress.
Therefore, none of the three topics discussed here–-cubic elasticity, dissociation into two partial
dislocations, nor finite aspect ratio of edge dipoles–-allows for the evaluation of an edge dipole
separation stress lower than the one predicted by the classical formula concerning infinite edge
dipoles without dislocation dissociation and assuming isotropic elasticity. All of the results are
summarized in Table 1-2 using Equation (1-10) and the Bacon and Scattergood effective elasticity
coefficients. As stated previously, only in the pre-saturation vein-channel microstructure of single
slip aluminum at 77 K are the evaluated dipole separation stress and the applied one comparable.
Tripoles or small dislocation pile-ups (just two to three dislocations) blocked at dipoles may lead
to such a local increase in shear stress [61], [62].
Further nodal DDD computations accounting for the separation of the dislocations into two partial
dislocations and the finite aspect ratio of dipoles may be of interest to assess the synergy between
both effects. It has been noticed previously that the evolution of the separation stress changes
drastically around h = 2.5 nm in copper for infinite edge dipoles. Therefore, the synergy between
both may lead to slightly different results. Finally, based on both our evaluation of the effect of
the anisotropy of cubic elasticity in copper and nickel and the one carried out by Veyssiere and

68
Chiu [58] in various conditions, it can be concluded that it is not an order one parameter for the
edge dipole separation stress evaluation.
As stated, the stress required to separate the dipoles with the maximum height was calculated to
be 270 MPa from Equation (1-15). Wider dipole heights than hmax would be unstable under the
local stress. The dipole stress for the maximum height of 270 MPa suggests a stress about a factor
of 2.4 larger than the applied stress. Kassner et al. [27] have reported this ratio to be 2.7 for fatigued
copper at shear stress of 19 MPa in single slip, and Tippelt et al. [30] reported a value close to 4
for cyclically deformed nickel at shear stress of 50 MPa in single slip. It should again be noted that
in both of these studies dislocation dipoles are assumed to have an infinite length. All values are
presented in Table 1-2. It is suggested that stress raisers, such as small dislocation pile-ups, can
increase the local stress. Brown [63] suggests that an infinite dipole approached by a single
dislocation can be broken up by an applied stress that is half of that required if the dipole is isolated.
This is the “tripole” effect of Neumann [61], [62] and is also related to Veyssiere’s [58] notions of
athermal dipole refinement. Thus, one expects the ratio in line 7 of Table 1-2 to be 2; this is not
far from what was observed.

69
Table 1-2. Dipole heights and stress calculations (Equation (1-10) for [26], [28], and [30] and
Equation (1-15) for this study) based on observed maximum “h” values.

This Study (Cu) Cu [28] Al [26] Ni [30]
Labyrinth
Walls
Labyrinth
Channels
Dipole
bundles
(Veins)
Channels
Dipole
bundles
(Veins)
Channels
PSB
Walls
PSB
Channels
Maximum Dipole Height
(nm)
4.3 4 15 12 31 32 6 7.1
Dipole Stress for hmax
(MPa)
270 ( h= 4.15) 52 (h = 13.5) 16 (h = 31.5) 186.6 (h = 6.55)
Applied Resolved Shear
Stress (MPa)
112 19 20 50
saturation half of saturation stress pre-saturation saturation
𝜏 d /𝜏 a (for hmax) 2.4 2.7 0.8 3.7
Slip polyslip single slip single slip single slip
SFE mJ/m
2
[65]–[68] 60 60 200 90
T/Tm 0.22 0.22 0.12 0.17
Strain Amplitude 0.40% Plastic only = 0.125% Plastic only = 0.12% 0.40%
Strain Rate (s
-1
) 2 × 10
−3
2.5 × 10
−3
2 × 10
−4
10
−3

Number of Cycles 157 200 560
-

70
1.4.1.5 LRIS Assessment through CBED Analysis

HOLZ lines in the CBED pattern are an effective way to measure local strain or change in the
lattice parameter since the electron beam can be focused locally to a smaller spot size than parallel
beam diffraction. Since large g-vectors are involved, the HOLZ lines position within the CBED
pattern are extremely sensitive to small changes in the lattice parameter. The shifts in HOLZ lines
have been shown to have sensitivities down to around 10
-4
to changes in strain [69 ,70]. Strain
determination is based on the shifts of the HOLZ lines in the strained specimen relative to the
unstrained pattern. The accuracy of strain calculation is dependent on the HOLZ lines being used
for the measurement. In other words, specific orientations are more sensitive to the changes in
material’s strain state. The HOLZ lines of <4 1 1> zone axis have been shown to have high
sensitivities to changes in strain in FCC crystals [71]. For acquiring the <4 1 1> CBED pattern,
the foil needs to be rotated under the TEM from its normal (which in this study is close to [0 1 0]
) to the <4 1 1> zone axis. This can be done using a Kikuchi map of an FCC crystal (Figure 1-37
[72]). Considering simulated patterns based on dynamical theory, the influence of foil inclination
on HOLZ pattern symmetry was found to be negligible for the tilt angles used in this study (below
~ 19°) [79].

71

Figure 1-37. (a) simulated Kikuchi pattern generated from a perfect FCC Ni crystal model, (b)
standard textbook FCC Kikuchi pattern and (c) experimentally collected and stitched TEM
diffraction pattern [72].

CBED studies were done on both the cyclically deformed copper and a 99.999% pure unstrained
copper using the JEOL JEM-2100F TEM at an accelerating voltage of 200 kV and a beam diameter
of about 40 nm. The <4 1 1> CBED patterns were acquired in small volumes of the cyclically
deformed copper very close to and remote from a dislocation heterogeneity (dislocation walls). A
channel with five points where a CBED pattern was acquired is illustrated in Figure 1-38. The
closest a CBED pattern could be acquired from a dislocation heterogeneity was approximately 30
nm. Below 30 nm, the dislocation tangles within the walls are too close to the electron probe,
causing perturbations within the CBED pattern.

(a) (b) (c)

72

Figure 1-38. CBED patterns were recorded across channels to assess the internal stresses in the
direction of applied stress [0 0 1]. (a) A channel illustrating the locations where the CBED patterns
were acquired. (b) The <4 1 1> CBED pattern that was recorded closer to the wall and (c) in the
middle of the channel.

The cross correlations between <4 1 1> CBED patterns of (b) and (c) in figure 1-38 show minimal
difference in the position of the lines in the patterns (see figure 1-39). A homogenous stress state
within the crystal can be interpreted since the <4 1 1> CBED patterns look identical close to the
walls and in the channel interior. This suggests that the internal stresses are minimal. However, in
this study we aim to quantify the internal stresses or the local lattice parameter. A comparison
between the recorded <4 1 1> CBED patterns with the software generated CBED patterns were

73
made in order to achieve this goal. The internal stress can be quantified by comparing the lattice
parameter of the deformed copper with that of the unstrained specimen.

Figure 1-39. The cross correlation between <4 1 1> CBED patterns of (b) and (c) in figure 1-38
shows minimal difference in the position of the lines in the patterns.

1.4.1.5.1 Higher Order Laue Zone Line Pattern Simulations

In order to obtain lattice parameter from a CBED pattern one can compare the experimental CBED
pattern with a simulated one [73]. Software simulation programs that generate CBED patterns
including the HOLZ lines have been developed to assess the relative HOLZ line shifts due to strain
[74–76]. These simulations assume either kinematical or dynamical behavior of the electrons
within the specimen. A kinematical approach assumes that each electron has only one diffraction
within the specimen while a dynamical behavior assumes the true behavior of the electron beam
in the specimen with multiple diffractions. The difference between the dynamical and kinematical
simulation CBED patterns is that the dynamical simulation can accurately regenerate the line
intensities seen in the experimental patterns, whereas no variation in the line intensity can be
created by the kinematical simulations. In this study the EMSoft codes that consider a dynamical
behavior of the electrons have been used for CBED patterns simulations to achieve a higher

74
accuracy [76]. The EMSoft simulation code needs an Xtal file that contains the crystal structure
information including the crystal system, space group number, atomic number, Debye-Waller
factor (DWF) and, of course, the lattice parameter. The Bethe parameter, often referred to as the
stopping power of material in front of charged particles, will be considered in a different set of
codes before running the final simulations. It is the Bethe parameters that take the dynamical
behavior effect into account. The aforementioned characteristics for copper along with the EMSoft
codes are shown in the appendix. The output file of the EMSoft simulations is an HDF5 file that
can be read with ImageJ if individual hyperstacks with the data set layout of “tzyx” is selected.
The HDF5 file contains individual diffraction disks (one per family of reflections).
Thickness is another important factor that significantly affects the CBED pattern in terms of both
the HOLZ lines position and the brightness and contrast of the pattern. This has also been
considered in the simulation codes. Figure 1-40 shows an example of an EMsoft generated CBED
pattern with [0 0 1] zone axis with different thicknesses. Note that the line intensities are different
since the dynamical behavior of the electron beam in the crystal is considered.

75

Figure 1-40. Simulated CBED pattern of [0 0 1] zone axis using EMSoft [76] with a thickness of
(a) 120 nm and (b) 300 nm.

1.4.1.5.2 Stress Determination Method

Lattice parameter determination from the HOLZ line patterns were performed using the
normalized distance between different HOLZ line intersections. Normalization by using the ratios
of the distance between different intersections is used to adjust for differences in magnification
between the different experimental patterns and also between the experimental and simulated
patterns. The comparison of simulated and experimental pattern is achieved by using the chi-
squared, which is the typical refinement method for producing the best match between the
simulated and experimental patterns [73]. Chi-squared is defined as:

𝜒 2
= ∑
1
𝑑 𝑖𝑠 (𝑑 𝑖𝑠
− 𝑑 𝑖𝑥
)
2 𝑁 𝑖 (1-16)

(a) (b)

76
where N is the number of data points, ds is the normalized distance between two intersections in
the simulated pattern, and dx is the normalized distance between the same intersections of the
experimental pattern.
As stated earlier, the changes in spacing between several intersection points of HOLZ lines were
measured as a function of different lattice parameters. The labeled intersection points of the HOLZ
lines utilized for the strain analysis are illustrated in figure 1-41.

Figure 1-41. Dynamically simulated <411> HOLZ line patterns illustrating the HOLZ line
intersections used for strain assessment.

The CBED pattern of an unstrained copper single crystal has also been recorded and compared
with the simulated patterns to determine the lattice parameter of the undeformed copper.
Consequently, the strain can be evaluated by comparing the lattice parameter of the cyclically

77
deformed copper with the unstrained value. The strain was converted into stress by implementing
the elastic modulus along the [0 0 1] direction into the stress-strain relationship:
σ11 = E[0 0 1] ε11 (1-17)
where σ11 is the internal stress calculated from the measured strain ε11, multiplied by the elastic
modulus E[0 0 1]. Since copper is an anisotropic metal, the elastic modulus is different in different
crystallographic orientations. In the current study, CBED measurements were recorded along the
[0 0 1] direction which is the direction of the applied stress. Thus, the elastic modulus of E [0 0 1]
= 66.6 ± 0.5 GPa was used accounting for the cubic elasticity anisotropy of copper [39].
Figure 1-42 shows the lattice parameters measurements and stress calculations corresponding to
different positions within a channel. The horizontal axis shows the distance from the walls
normalized by the channel width (The average channel width is 0.36 μm and the wall width is
about 0.12 μm). Figure 1-42 represents the data obtained from four channels in two TEM foils.
Minor scattering exists in the lattice parameters of different channels that is ±2 × 10
−4
nm. Identical
values of lattice parameters in each channel show that the internal stresses are homogenous in the
channel and close to the walls of the labyrinth dislocation microstructure. The lattice parameter
obtained from an unstrained copper single crystal is 0.3592. Comparing the lattice parameter of
cyclically deformed copper single crystal with that of unstrained copper using the Equation (1-17)
indicates that the internal stresses are minimal (less than 6.5% of the applies stress). Of course, it
is possible that internal stresses exist and are less than the measurement error. The accuracy of
lattice parameter measurements is about ±1 × 10
−4
nm. The error in stress measurements is then
approximately ±18 MPa that is 6.5% of the applied axial stress of 275 MPa.

78

Figure 1-42. The obtained lattice parameters and corresponding axial stress values in different
locations within a dislocation channel. The lattice parameter of an unloaded copper was found to
be 0.3592 nm.

Following the Legros et al. [33] method of chi-squared analysis, an effort was made to visualize
the local changes of lattice parameter within a single channel. Since the lattice parameter of the
unstrained copper was observed to be 0.3592 nm, channel four with the lattice parameter of 0.3592
nm in the cyclically deformed copper was chosen for this analysis. HOLZ lines of simulated CBED
patterns corresponding to the lattice parameter of 0.35920 nm were considered as the reference
point for chi-squared analysis. The data illustrated in figure 1-43 is a chi-squared fit between the
aforementioned HOLZ lines intersection ratios for cyclically deformed (yellow) and undeformed
(red) copper. Chi-squared analysis can “refine” the strain measurement below 10
−4
and provide
increased resolution of the elastic strain, although precise values of the strain within in the
10
−5
range is not possible. The chi-squared results plotted in figure 1-43 are somewhat qualitative.
Although the changes in the lattice parameter in a channel are minimal and less than 1 × 10
−4
nm,
the data shown in figure 1-43 can show that the difference between the lattice parameter values of

79
the cyclically deformed copper and the unstrained copper are slightly higher in the proximity of
the walls in comparison with the values in the channel interior. This result is consistent with the
composite model but with much lower values of internal stresses (less than 6.5% of the applied
stress). It should also be noted that the aforementioned differences in the chi-squared values might
be due to the higher quality of HOLZ lines in the middle of the channels as opposed to the vicinity
of the walls.

Figure 1-43. Change in the [4 1 1] CBED pattern HOLZ lines of cyclically deformed copper in a
channel and undeformed copper relative to simulated pattern with lattice parameter of 0.3592 nm.
The data is shown as a χ2 fit between the HOLZ lines of the aforementioned CBED patterns.

Although the changes in the lattice parameter in a channel are very minimal (in the range of 10
-5

nm), figure 1-43 indicates that the lattice parameter values for the strained specimen are slightly
lower than that of the unstrained copper within the channel, while higher in the proximity of the
walls. This notion agrees well with the composite model suggested by Mughrabi where the high-

80
dislocation density regions experience a larger magnitude of stress than the average stress, and the
channel interiors experience a smaller magnitude of stress than the average stress (described in
section 2.2.1). Therefore, it can be concluded that long range internal stresses have a minor effect
on occurrence of Bauschinger. This notion is similar to the composite model suggested by
Mughrabi. Considering the two renowned theories that rationalize the Bauschinger effect
(Composite model and Orowan-Sleeswyk mechanism), it appears that the dominant features of the
Bauschinger effect may need to include the Orowan–Sleeswyk type of explanation since both the
maximum dipole height measurements and the lattice parameter assessment through CBED
analysis carried out in this study suggest a relatively homogenous stress state. As stated earlier, no
internal stresses are involved in the Orowan-Sleeswyk mechanism where the Bauschinger effect
is rationalized by the lower lineal density of obstacles in reverse direction of straining.
It should be noted that dislocations may eject out of the surface in the thin areas of the TEM foil.
This will result in stress relaxation and can subsequently alter the values of the internal stress. This
is rather challenging since thin areas of the specimen are to be used for acquiring high quality
HOLZ lines in a CBED pattern. It must be emphasized that these relaxations may be negligible as
the labyrinth pattern with similar characteristics such as dislocation density, channel, and wall
width were observed both in the thin regions as well as thicker areas. The dislocation density in
the thinner regions (approximately 130 ± 10 nm) where CBED patterns were recorded was 8.2 ×
10
14
m/m
3
in the walls and 1.8 × 10
13
m/m
3
in the channels. This is very close to the wall
dislocation density of 8.6 × 10
14
m/m
3
and channel dislocation density of 1.5 × 10
13
m/m
3
in
relatively thicker regions (approximately 0.23 μm) where dislocation densities were measured.
Also, for the case of epitaxial layers, Treacy et al. [77] demonstrated that strong relaxation effects
are expected when the foil thickness is lower than 10% of the strain modulation wavelength which

81
is the dislocation channel diameter in the current study. Since this ratio is about 30% in this work,
stress relaxations are expected to be very minimal. Although copper has a fairly low stacking fault
energy of 60 mJ/m
2
and the labyrinth microstructure characteristics including dislocation densities
are consistent in the thick and thin regions, relaxations caused by dislocations ejecting the thin
regions of the foil cannot be completely neglected.
There have been few studies on internal stress assessment using CBED technique in creep
deformed and fatigued polycrystals and single crystals in single slip, while such studies on
cyclically deformed single crystals oriented in multiple slip are missing in the literature. Straub et
al. [78] and Maier et al. [79] examined internal stresses using CBED analysis in polycrystalline
copper specimens experiencing either creep or cyclic deformation. They did not quantify the
internal stresses but suggested that internal stresses exist. It should be noted that both of these
studies used kinematical simulations for deriving the position of the HOLZ lines, but dynamical
effects may be important and should always be included, otherwise the match is not accurate.
Kassner et al. suggested a homogenous stress distribution with no internal stresses in creep
deformed polycrystalline copper using CBED analysis [80]. In another CBED study by Kassner et
al. an absence of internal stresses in creep deformed aluminum single crystal was reported [32].
Legros et al. observed negligible internal stresses closer to the dislocation wall (7 MPa that was
about 14% of their applied stress) and no internal stresses within the cell interior in a cyclically
deformed silicon single crystal oriented for single slip [33]. In the most recent study by Kassner et
al. on a cyclically deformed copper single crystal oriented for single slip no internal stresses were
detected near or remote from the dipole bundles [28]. All of the deformation conditions for these
studies are summarized in Table 1-3. The current study shows a homogenous stress state within
the crystal since the lattice parameters are almost identical near the dislocation walls and in the

82
channels. This is analogous to what earlier studies found in fatigued copper and silicon oriented
for single slip and creep deformed aluminum, and copper [28, 32, 33, 77–80].

83
Table 1-3. Convergent Beam Electron Diffraction studies for long range internal stress assessment
on creep and cyclically deformed materials.

CBED Studies This Study Kassner[28] Legros[33] Kassner[32] Kassner[80] Straub[78] Maier[79]
Material
Copper [001]
Single Crystal
Copper [123]
Single Crystal
Silicone [231]
Single Crystal
Aluminum
Single Crystal
Copper
Polycrystal
Copper
Polycrystal
Copper
Polycrystal
Deformation
Type
Cyclic Cyclic Cyclic
Creep
Steady State
Creep
Steady State
Creep Cyclic
Applied Stress
(MPa)
112
(Shear)
Saturation
19
(Shear)
Presaturation
49
(Shear)
Presaturation
162
(Normal)
20 40
7
(Normal)
144
(Normal)
(Normal)
Strain 0.40%
0.125%
Plastic only
6 × 10
-4

Plastic only
- 0.04 0.05 3.6
5 × 10
-3
Plastic only
Strain Rate 2 × 10
−3
2.5 × 10
−3
3 × 10
−4
- 2.5×10
-5
5.6×10
-7
1.9×10
−3
-
Slip Polyslip Single Slip Single Slip - - - -
Number of
Cycles
157 200 - - - - 2000
Temperature
(K)
293 293 1078 663 823 573 293
LRIS None None None None None Observed Observed

84
1.5 Conclusions

1. The maximum dislocation dipole heights were nearly equal in the channels and walls of the
labyrinth structure for cyclically deformed [001] copper single crystals oriented for multiple
slip to stress saturation. This is the first study that examined dipole heights in cyclically
deformed metals in multiple slip. Our observations lead to the conclusion of a uniform stress
distribution with low internal stresses as the stress to separate the widest dipoles is independent
of the location. However, the widest dipole strengths correspond to a shear stress of about 2.4
of the applied stress. Similar behaviors (homogenous dipole heights and higher dipole
separation stresses) have also been reported for cyclically deformed metal single crystals in
single slip in other studies. The fact that the calculated separation stress based on the dipole
height is larger than the applied stress based on Equation (1-15) may be explained by mild
dislocation pile-ups. Earlier synchrotron work on monotonically deformed copper single
crystals [39] found that long-range internal stresses were about 10% of the applied stress. Thus,
the results of the current cyclic deformation study are consistent with the earlier monotonic
deformation work.
2. The lattice parameter assessment through convergent beam electron diffraction patterns
recorded in the channels and close to the walls of the labyrinth dislocation structure suggest
very low long range internal stresses (LRIS). The results of the CBED analysis study is
consistent with our maximum dipole height measurement work. Minimal changes (less than
10
−4
nm) were observed in the lattice parameters recorded throughout a single channel. These
values are less than 6.5% of the applied stress. Hence, negligible internal stresses in the channel
interior and near the dislocation walls were observed. The Kassner et al. x-ray synchrotron
study on monotonically deformed (to 30% strain at ambient temperature) copper single crystals

85
[39] suggest that long range internal stresses were nearly 10% of the applied stress. Thus, the
outcome of the present cyclic deformation study is consistent with the earlier monotonic
deformation work.
3. Although the changes in the lattice parameter in a channel are minimal (less than 10
−4
nm),
chi-squared analysis suggest that the difference between the lattice parameter values of the
cyclically deformed copper and the unstrained copper are slightly higher in the proximity of
the walls in comparison with the channel interior. These internal stresses are less than 6.5% of
the applied stress. This is consistent with the composite model originally suggested by
Mughrabi but with perhaps lower values of internal stresses. Therefore, it appears that a low
proportion of the Bauschinger effect may be influenced by the existence of long range internal
stresses. The dominant feature of the Bauschinger effect may include the Orowan–Sleeswyk
[6] mechanism type of explanation, since both the maximum dipole height measurements and
the lattice parameter assessment through CBED analysis suggest a relatively homogenous
stress state within the heterogeneous dislocation microstructure.
1.6 Future Suggested Work
Synchrotron X-ray microbeam measurements for internal stress assessments will be carried out at
the Advanced Photon Source at Argonne National Laboratory on a cyclically deformed copper
single crystal oriented for single slip to obtain a more comprehensive understanding of the LRIS
specially in the Persistent Slip Bands with very high dislocation densities.

86
2 Chapter 2: Understanding Large-Strain Softening of Aluminum
In Shear At Elevated Temperatures

2.1 Motivation

Aluminum in pure shear to large strains at elevated temperature shows pronounced strain
softening of roughly 15-20% which has been attributed to a variety of phenomena. In this study
we aim to better understand the basis for this large-strain softening of aluminum under pure shear
at elevated temperatures. The most widely accepted early explanations involve the development
of a texture leading to a decrease in the average Taylor factor. That is, there is a decrease in Schmid
factors in the deformed grains. Earlier work by the materials community only considered changes
in the dislocation glide stress with the evolving texture as an explanation for the softening. In this
study we suggest that the developed texture leads to softening through an increase in the
dislocation climb stress that can explain the softening trends. This appears to be particularly
reasonable as dislocation climb is widely regarded as the rate-controlling mechanism for high
temperature plasticity.

87
2.2 Background

In this chapter as we aim to describe the basis for the large-strain softening of aluminum under
pure shear at elevated temperatures. The associated stress-strain behavior will be presented first.
In addition, there are some terminologies like different types of recovery and recrystallization,
textures, and Taylor factor which are worth explaining before going into the details of the basis
for this softening. Then a detailed explanation of the suggested reasonings for aluminum’s
softening at elevated temperatures by the researchers in the materials community is presented. The
possibility that changes in the dislocation climb stress, induced by the texture, can rationalize this
softening is presented.
2.2.1 Aluminum Softening

Figure 2-1 [82] illustrates the large strain softening in aluminum at elevated temperatures. As
background, the stress versus strain behavior and microstructural evolution of aluminum deformed
in torsion (pure shear) at elevated temperatures was studied by a variety of investigators such as
Myshlyaev and coworkers [83–86], Monthiellet and coworkers [87–89], McQueen and coworkers
[90–93], Pettersen and Nes [94], and Kassner and coworkers [82], [95–98]. It was consistently
reported that the ductility of high and commercial purity Al can exceed equivalent uniaxial strains
of 100. Figure 2-1 illustrates that Al hardens to a peak stress
ss p,

at strains less than 0.5. The flow
stress subsequently decreases to a flow stress,
ss

, which is nearly constant, and a steady-state
condition is reached.

88

Figure 2-0-1. Elevated-temperature equivalent-uniaxial stress versus equivalent-uniaxial strain of
high-purity Al at strain rates of (a) strain rate of 5.8 × 10
-4
s
-1
and (b) 1.3 × 10
-2
, s
-1
from [82].

89
The peak stress,
ss p,

, seems equivalent to the steady-state creep stress observed in tension (tensile
ductility is generally limited to relatively small strains). The flow stress decreases by about 17%
and this occurs over a fairly broad range of 1-2 strain, depending on the temperature and strain
rate, as illustrated in figures 2-1 (Equation (2-2) governs the plastic flow). Other cubic materials
such as α-Fe show similar behavior [99].
2.2.2 Microstructural Changes during Thermomechanical Processing

The energy that is stored and increased in the metals during deformation mainly takes the form of
dislocations which causes the material to become thermodynamically unstable [100]. When a
metal is deformed at elevated temperatures, these defects tend to be removed through three main
processes of recovery, recrystallization, and grain growth to reduce the free energy of the system.
Recovery is an annealing processes that occurs in deformed materials without a high angle grain
boundary migration. In general, recovery occurs by arranging dislocations in a manner that lowers
their energy, for example through the formation of low-angle subgrain boundaries or through the
dislocation annihilation. The high angle grain boundaries (HABs) typically do not migrate during
the recovery stage. Dynamic recovery (DRV) is recovery that occurs during deformation whereas
static recovery (SRV) occurs after deformation. Recrystallization is the process in which a
deformed material forms a new grain structure by forming and migrating high angle grain
boundaries (with misorientations of more than 10-15°) driven by the stored energy of deformation.
These new grains are less strained, causing a decrease in the hardness of a material. Grain growth
can then be defined as a reduction in grain boundary area owing to the growth of the mean grain
size [100].

90

Figure 2-0-2. Recovery, recrystallization, and grain growth relate to grain size, hardness, ductility,
and residual stress in the material [101].

2.2.2.1 Recrystallization

Recrystallization can take place under static and dynamic conditions. For both of these cases, the
recrystallization comments made in the previous section are equally valid. As opposed to static
recrystallization (SRX), in dynamic recrystallization (DRX) the nucleation and growth of new
grains takes place during deformation rather than afterward as part of a separate heat
treatment/annealing [100]. The most common static instance is heating cold-worked metal that
results in a recrystallized microstructure. A static recrystallization is dependent on both prestrain
and temperature, and only very slightly on the strain rate. During processing, most metallic

91
materials are often subjected to hot deformation, during which dynamic recrystallization occurs
which is associated with plasticity. On a stress–strain curve, one can detect the onset of dynamic
recrystallization at a distinct peak in the flow stress which occurs as a result of the softening effect
of recrystallization [102]. DRX can occur in various forms of: 1. Discontinuous dynamic
recrystallization (DDRX), 2. Continuous dynamic recrystallization (CDRX), and 3. Geometric
dynamic recrystallization which will be briefly explained in the following sections (GDRX).

Figure 2-0-3. DRX can occur in various forms: discontinuous dynamic recrystallization (DDRX),
continuous dynamic recrystallization (CDRX), and geometric dynamic recrystallization (GDRX)
[102].

It should be noted that in figure 2-3, for GDRX about a third of the boundaries are HABs and the
HABs form continuous HABs. For CDRX all boundaries can be HABs but generally only a

92
fraction of boundaries are HABs that are random unlike GDRX. Once DDRX is complete, all
boundaries are HABs.

2.2.2.1.1 Discontinuous Dynamic Recrystallization (DDRX)

Recrystallization is described as a discontinuous process if it occurs heterogeneously with distinct
nucleation and growth stages. For low stacking fault energy materials, discontinuous dynamic
recrystallization is frequently observed during hot deformation, during which nucleation of new
strain-free grains occurs at the expense of dislocation-filled regions [102].
2.2.2.1.2 Continuous Dynamic Recrystallization (CDRX)

In contrast to DDRX, recrystallization can also occur uniformly, resulting in microstructures that
evolve with no clear nucleation and growth phases, exhibiting a continuous character.
Deformation of materials with high stacking fault energies results in subgrain or cell structures
with low-angle grain boundaries as a consequence of the efficient dynamic recovery.
Progressively, they develop into high angle grain boundaries (HABs) as a consequence of larger
deformations, a process known as continuous dynamic recrystallization [102].
2.2.2.1.3 Geometric Dynamic Recrystallization (GDX)

In Geometric dynamic recrystallization, the grains will become increasingly flattened and
elongated until all the boundaries on either side can only be separated by a very small distance
(e.g. two subgrain diameters). In the process of deformation, the grain boundaries are serrated due
to surface tension effects where they are in contact with the low angle grain boundaries of sub-
grains (i.e. triple points with one subgrain boundary). The serrations eventually will come into
contact with each other. The developed serrations will be pinched off and equiaxed grains with

93
HABs will be formed [95, 102]. The GDRX can be particularly observed in metals with high
stacking fault energies that are deformed at elevated temperatures[103,104].

Figure 2-0-4. GDRX progress diagram: a) grain boundaries flatten with well-defined substructures
in the matrix when deformation is comparatively small. (b) as deformation continues, the serrated
HABs become closer while the subgrain size remains roughly the same. (c) As a consequence, the
HABs eventually impinge, giving rise to a microstructure of 1/3 to 1/2 HABs [102].

2.2.3 Textures and Taylor Factor

The crystallographic orientation of each grain in a polycrystalline aggregate differs from its
neighbors and the biased distribution of these crystallographic orientations is called a texture. A
perfect single crystal is considered to have anisotropic texture while a polycrystal with fully
random orientated grains is considered as an isotropic texture. A polycrystal may have a weak,
moderate or strong texture if the crystallographic orientations are not random and have a preferred
orientation [105, 106].
The texture and related changes in microstructure of a material can strongly influence the
properties such as strength, corrosion resistance, deformation behavior, magnetic susceptibility,
and others. Through the average Taylor factor M, one can relate the macroscopic yield stress for a
random oriented FCC polycrystal to the critically resolved shear stress of a reference single crystal:

94
𝜎 = M
̅
𝜏 𝑐 (2-1)
Where 𝜎 is the macroscopic yield stress, M
̅
is the average Taylor factor, and 𝜏 𝑐 is the critically
resolved shear stress. Figure 2-5 illustrates the orientation factors a) Schmid (m) and b) Taylor (M)
calculated for uniaxial deformations.

Figure 2-0-5. Orientation factors for (a) single slip "Schmid" and for (b) multiple slip "Taylor"
calculated for uniaxial (tensile or compression) deformation [107].

95
Note that in figure 2-5, the Taylor factor is greatest at the <111> and <110> and the lowest at the
<100>. In other words, a cubic material with a perfect <111> or <110> fiber texture will be 1.5
times stronger (in tension or compression) than the same cubic material with a <100> fiber texture.
Obviously this is not as dramatic of a strengthening mechanism as can be achieved in other ways
like precipitation or solid solution hardening, but it is still significant. Additionally, this can be
accomplished with no sacrifice to other properties.
Deformation modes affect the Taylor factor. According to Bishop and Hill, the Taylor factor for
tension is (M=) 3.06 and for torsion is (M=) 1.65 [108]. According to figure 2-6, the Taylor factor
clearly varies with strain. These variations are distinct for each of the three deformation modes of
tension, compression, and torsion.

96

Figure 2-0-6. Calculations of the evolution of the average Taylor factor with strain. (a) Tension
and compression. (b) Shear [106].

97
2.2.4 Suggested Reasonings for Aluminum’s Softening at Elevated
Temperatures

This softening has not been widely attributed to discontinuous dynamic recrystallization. Rather
only dynamic recovery (DRV) is widely accepted to occur. The softening of 17% to strains of 1-2
is often followed by a slight, gradual, increase in stress (about 4%) beyond strains of about 10 [82].
The early explanations for the softening vary from decreases in the average Taylor factor due to
texture evolution leading to a decrease in the stress for dislocation glide as well as changes in the
dislocation substructure (e.g. increase in the average subgrain size). It is now widely agreed that
large-strain deformation results in a dramatic increase in the high-angle boundary area (HAB).
That is, one-third to one-half of the initially low-misorientation subgrain facets become HABs (e.g.
θ > 8°). Two groups [82, 91, 92, 95-98] attribute this primarily to geometric dynamic
recrystallization (GDRX), where the original grains of the polycrystal elongate, leading to an
increased high-angle boundary (HAB) area. Perdrix et al. [87] considered the formation to be
primarily, due to continuous dynamic recrystallization where subgrains gradually transform to
HABs with dislocation accumulation. The number of HABs in the deformed single crystals (where
of course HABs were initially absent) was much less than for large strain deformation of
polycrystals (9% vs. 35%) in support of GDRX.

98

Figure 2-0-7. The aluminum deformed in torsion at 371
o
C with the peak stress,
ss p,

. The flow
stress subsequently decreases to a flow stress,
ss

, which is nearly constant, and a steady-state
condition is reached. The peak stress,
ss p,

, seems equivalent to the steady-state creep stress
observed in tension.

HABs can form in some cases from dislocation reaction. The boundaries in the deformed single
crystal may be geometric necessary boundaries (GNBs) which are HABs that form from
dislocation reaction and occur as a result of incompatible slip within a given grain/crystal [95],
[100]. Earlier work by the author [82] found that both the subgrain size and the dislocation density
(not associated with subgrain boundaries) are approximately constant throughout deformation.
This is illustrated in figure 2-8.

99

Figure 2-0-8. The dislocation density (a) and the subgrain size (b) are approximately constant
throughout deformation.

100

Flow is described by the classic equation:
𝜀 ̇ss= A exp [(-Qc / kT) (𝜎 /G)
n
] (2-2)
where A is a constant, Q is the activation energy for creep that is approximately equal to the
activation energy for lattice self-diffusion, 𝜎 is the equivalent uniaxial stress, n is the stress
exponent (approximately 4–5 above 0.6Tm) [109].
Pettersen and Nes [94] suggested that the softening in aluminum alloys (AA 6060 and 6082) is a
result of these new HABs being particularly effective sinks for dislocations, leading to softening.
Only about one-third of decrease in strength was suggested to be due to a decrease in the average
Taylor factor. Perdrix et al. [87] suggested that softening can originate from three sources: a
decrease in Taylor factor, increase in subgrain size, and a small increase in stress exponent. They
suggested that the Taylor factor decreases by about 18%, which is enough to explain all of the
softening. McQueen [90] and Kassner et. al [82], [97] appeared to independently rationalize the
softening exclusively by texture. One problem with this microstructural-based explanation is that
it appears that changes in the subgrain size in Al, by itself, do not seem to affect the flow stress, as
discussed extensively in [109] and [110]. Rather, the density of dislocations not associated with
subgrain boundaries may be more important. This appears to be unchanged with strain.
The softening has not been attributed to any new deformation mechanism that might arise from
the increase in HAB area, such as Coble creep or an increased contribution to strain from grain
boundary sliding (GBS). The activation energy for creep-plasticity is unchanged from that of close
to lattice self-diffusion [84, 87]. Also the stress exponent is unchanged from that of about 4-5 over
the softening regime [84, 87]. Either Coble or GBS would be associated with smaller stress

101
exponents (1-2) and activation energies about half that of lattice self-diffusion. Table 2-1 reports
the observed textures with the large strain deformation. The first index is the crystallographic plane
in the shear plane and the second is the shear direction. McQueen and coworkers [90], [93]
observed the A, B1, and C textures in commercial purity Al deformed in torsion to large strains at
400°C, with the B1 being strongest. Later, McQueen [91] only observed the strong B1 texture
based on x-ray diffraction and STEM on pure Al. Perdrix et al. [87] noted the softest (111
̅
) [11
̅
0]
or an A texture at a strain of 31 in commercial purity Al at 400°C. Barnett and Montheillet [88],
examined the textures by EBSD from strains of 0-2 (softening regime), and concluded that four
texture components were developing, A, A2, B, and C in Al 1050 at 450°C, of equal magnitude,
with about 30% of material within 10° of the texture components. Kocks, Tome, and Wenk [111]
discuss textures in pure metals torsionally deformed at ambient and elevated temperatures. They
emphasized that B and (weak) A fiber torsion textures are common. Shrivastava et. al [112]
calculated the torsional Taylor factors (M) for A, C, and B1 textures. These are listed in Table 2-
1. The average value of M in the textured Al was 2.39, or about 16% less than 2.86 for a random
array of grains.

102
Table 2-1. Textures and Taylor Factors (M) observed in aluminum deformed to large strains at
elevated temperatures.

Torsion Texture M for Traditional Slip
A (111
̅
) [11
̅
0] 1.73
C (001) [110] 2.99
B1 (1
̅
12) [110] 2.44
Average - 2.34
Isotropic - 2.86

A decrease in the average Taylor factor from 2.86 to 2.44 leads to an increase in the resolved shear
stress of 15%, if the strongest (B1) texture is used and glide mechanisms are assumed. Pettersen
and Nes [94] argued that the “harder” C component texture that was observed in addition to the
B1 texture in the aluminum alloy only mandated a decrease in flow stress of only 5-7%, with,
again, the remaining softening resulting from increased dynamic recovery in association with the
dramatic increase in HABs which were suggested to be sinks for gliding dislocations.

103
2.3 Results and Discussion

The following describes work to date, from a paper titled “Large-Strain Softening of Aluminum
in Shear at Elevated Temperature: Influence of Dislocation Climb”, published principally in
Metallurgical and Materials Transactions A [113–115].
Earlier work by Kassner et. al. [97] summarized some of the above work in greater detail, and
performed additional experiments which included high temperature torsion tests quickly followed
by high temperature compression tests at the same temperature and strain-rate. Basically, it was
concluded that the high temperature strain softening was, as suggested above, a consequence of
textural softening from a decrease in the average Taylor factor. However, there is a serious problem
with the textural softening conclusion by the authors and the other investigators referenced. The
suggestion that the decrease (and the magnitude of the decrease) in flow stress is a result of the
decrease of the average Taylor factor (through the development of textures) suggests that the
influence of the texture is due to a decrease in the glide stress for plasticity. However, for pure
and commercial purity aluminum it has been fairly well established that the rate-controlling
process for plasticity within the five power-law creep regime is dislocation climb [109] governed
by Eq. 2-2. The activation is very close to that of lattice self-diffusion, which controls the
dislocation climb-rate. Thus, in order for the softening to occur, the climb force on the dislocations
must increase as a result of the observed texture changes. This was not considered in any of the
above investigations.

104
The purpose of the present analysis is to assess through the following two tasks:
1. Whether the large-strain softening observed in torsion by all of the investigations is
accompanied by an appropriate increase in the climb stress on the dislocations of the
various (active) slip systems associated with the observed textures.
2. Whether elevated-temperature torsion tests in the softening regime followed by a
compression test at the same temperature and strain rate also show a flow-stress-change
behavior consistent with the changes in the climb stress associated with the observed
textures. Changing to compression from torsion changes the glide (and climb) stresses and
activates new slip systems. This can additionally check the viability of dislocation-
glide vs dislocation-climb to rationalize the softening.
Interestingly, if these behaviors are inconsistent with dislocation climb behavior, this could
suggest, contrary to general thinking, a glide-controlled mechanism for creep in pure metals and
class M alloys at elevated temperatures which some have actually proposed [116]. Settling whether
the softening observations are explainable by climb is also, then, a check on the viability of
dislocation climb as the rate-controlling mechanism for five-power-law creep.
An analysis of the climb-stress changes with texture evolution is performed in this work in terms
of the preceding two tasks. Figure 2-9 shows the stress-state in torsion (pure shear). Of course, the
normal principal stresses (𝜎 1 and 𝜎 2) that are rotated by 45 degrees from the torsion axis are
indicated. Normal stresses are responsible for dislocation climb.

105

Figure 2-0-9. Schematic of the torsion specimen with the indicated shear stress and the principal
stresses for a pure shear stress state.

The results of the geometric analysis are listed in the Tables 2-2 and 2-3. The climb stress 𝜎 c is
determined by the angles 𝜃 between 𝜎 1 and 𝜎 2, and the Burgers vector where {111} <110> slip is
assumed. Slip on other planes (e.g. {100} and {110}) was not considered. Table 2-3 lists the values
of the normalized climb stresses, 𝜎 c/ 𝜎 1 which can be compared to that of untextured aluminum
of (0.71). Higher values of 𝜎 c/ 𝜎 1 leads to lower stresses for constant strain-rate deformation.
The average of the three (perfect) textures listed are expected to produce an increase in the climb
stress of about 7% which is less than the observed softening of about 17%. The important
conclusion is that climb-control does, in fact, predict softening in torsion.

106
Table 2-2. Normalized climb stress for the observed textures.

Torsion Texture 𝜎 c / 𝜎 1 (ave.)
A (111
̅
) [11
̅
0] 0.64
C (001) [110] 0.83
B1 (1
̅
12) [110] 0.79
Average - 0.75
Isotropic (random grain
orientation)
- 0.71

107

Table 2-3. Changes in flow stress with torsion to compression at high temperature in terms of
climb stress analysis.

Torsion Texture
𝜎 c / 𝜎 1 in compression
following HT torsion
(ave.)

A (111
̅
) [11
̅
0] 0.82
C (001) [110] 0.58
B1 (1
̅
12) [110] 0.71
Average - 0.70
Isotropic (random grain
orientation)
- 0.71

If the primary textures are absent of the A texture and B1 is the primary texture, as suggested by
McQueen [92] and Kocks et. al [111], then a larger value of softening of 11% is predicted based
on climb. Of course, some softening due to microstructural changes cannot be excluded as
suggested by Pettersen and Nes [94]. Also, these textures may not be perfectly established as
assumed in the calculation.

108
Figure 2-10 illustrates the 16% softening of a hollow torsion specimen to a strain of about 1.3.
Elevated-temperature torsion tests were performed to various strains within the softening regime.
The tests were terminated at three strain levels, of 0.2, 0.46, and the nearly fully softened state at
a strain of 0.69. The specimens were quickly unloaded and compressed upon termination of
torsion. Calculation of the climb stress changes predict that the elevated temperature flow stress
in compression (immediately after torsion) should be approximately unchanged just as observed
in the experiments of [97] illustrated in figure 2-10. This is an important observation since a
dislocation glide assumption predicts an 8% increase.

109

Figure 2-0-10. The 371°C equivalent uniaxial stress versus equivalent uniaxial strain of hollow
torsion specimens and the corresponding compressive yield stress (0.10 strain offset) at the same
strain rate and elevated temperature subsequent to various (pre)strains in torsion.

Interestingly, if the “fully softened” regime specimens are compressed at ambient temperature
(where dislocation glide is expected to be the rate-controlling mechanism) along the torsion axis,
the average Taylor factor for the textures increases by 7-13% [117]. This is illustrated in figure 2-
11. If just the B1 texture is used, an 8% increase is expected. Therefore, our experimental results
appear basically consistent with the predictions of a dominating texture explanation for elevated-
temperature softening. The solid specimen results in which ambient temperature compression tests,
subsequent to elevated-temperature torsion tests, appear consistent with the texture explanation

110
and dislocation-glide explanations for softening. The high temperature torsion tests and torsion
followed by compression results are consistent with dislocation climb stress changes from the
texture evolution.

Figure 2-0-11. The equivalent uniaxial stress versus equivalent uniaxial strain of hollow torsion
specimens and the corresponding compressive yield stress. Specimens were tested under torsion
at elevated-temperature followed by a quench, and ambient-temperature compression along the
torsion axis. The ambient temperature compression hardening is consistent with elevated
temperature torsion texture development followed by ambient temperature glide control with
a compression axis identical with the torsion axis.

111
2.4 Conclusions

The elevated-temperature softening observed in pure shear can be rationalized by the development
of textures. Specifically, at elevated temperature there is an increase in the average dislocation-
climb-stress with the development of the textures and this leads to softening. The traditional
explanation of a change in the glide stress leading to softening appears less likely.

112
3 Chapter 3: The Creep and Fracture Behavior of Additively
Manufactured Inconel 625 and 718

3.1 Motivation

Additive manufacturing of alloys has gained attention by industry as it provides economical
production of complicated-configuration engine parts with fewer joining steps and greater
geometric freedom. The creep behavior of additively manufactured (AM) Inconel 718 at 650
o
C
was compared with that of the wrought alloy up to failure. Inconel 718 is precipitation strengthened
(while 625 is solid-solution strengthened). It was discovered that the creep strength of the 625 and
718 produced by additive manufacturing is essentially identical to that of the wrought alloy.
Complications occur, however, with a marked loss of ductility in the AM alloys. Fatigue and
impact toughness were adversely affected as well.
Here we report the new additively manufactured (AM) Inconel 625 and 718 (IN625 and IN718)
creep tests and microstructural characterizations. The results are compared to conventional
wrought alloys. Some of the IN625 results were reported very recently by the other Kassner’s
materials research group members [118,119]. The tests occurred over time scales from 1 day to
about one year at 650 and 800 ˚C. These tests uniquely characterized the elevated temperature
creep strength as well and the high temperature ductility of the alloys which, prior to our work,
was unknown. The loss of ductility would be particularly investigated as this could affect the
elevated temperature fracture toughness. The elevated-temperature fatigue behavior of the AM
alloys compared to the wrought alloys is investigated as well.

113
3.2 Background

3.2.1 Description of Creep Deformation

Creep is defined as a time-dependent plasticity phenomenon that is influenced by dislocation
movement or vacancy diffusion during a certain period of time under a constant applied stress
[120, 121]. Generally, the applied stress is less than the yield stress (σ < σy) [121, 122], and tests
are usually performed at temperatures higher than 0.5 Tm where Tm is the absolute melting
temperature. Although a material can creep at any temperature, the creep process is more
observable at higher temperatures. This time-dependent plasticity that occurs during creep is in
contrast with the general observation, such as when a material is deformed at ambient temperature
(for example at 0.1 - 0.3 Tm), where negligible plasticity under constant stress at or below the yield
stress, at conventional tensile testing strain-rates, is usually observed (e.g.,10
-4
- 10
-3
). It should be
noted that some rate-sensitive and relatively low strain-hardening alloys such as titanium [120]
and steel [121], exhibit some primary creep (e.g., a few percent strain) under stresses less than the
yield stress within a relatively short period of time. As thermal energy increases at elevated
temperatures, the vibration frequency of atoms (including dislocations) rises. This activates the
movement and concentration of vacancies, resulting in an enhanced dislocation climb. It also
lowers the thermal energy barrier for dislocations to climb under a constant applied stress,
allowing deformation. In Figure 3-1, the plasticity under these conditions is shown for (a) constant
stress and (b) constant strain-rate (creep rate) conditions. The creep rate is defined as a change in
strain as a function of time.

114

Figure 3-1. The creep behavior at temperatures higher than 0.5 Tm. (a) at constant stress condition.
(b) the stress-strain curve of the same material at constant strain rate (creep rate) condition. The

115
three regimes of I, II and III are the primary, secondary (steady-state), and tertiary (fracture)
regimes [123].

Figure 3-1 (a) illustrates creep behavior at temperatures above 0.5 Tm at constant stress conditions.
Figure 3-1 (b) shows the stress-strain curve for the same material at constant strain rate (creep rate)
at the same temperature (above 0.5 Tm). Note that the strain rate in figure 3-1 (a) remains constant,
whereas it varies in figure 3-1 (b). The following three regimes are clearly discernible: I, II and III
which are the primary, secondary (steady-state), and tertiary (fracture). Stage I or primary creep
occurs immediately after the initial stress is applied to the sample. As plastic strain or time
increases, the strain rate (𝜀 ̇) rapidly increases but at a decreasing rate (figure 3-1.a). Figure 3-1(b)
shows that under constant strain-rate conditions, dynamic hardening occurs, which results in
increased flow stresses. It is important to note that occasionally in primary creep, the strain rate
can increase with strain for some creep types, such as with solute drag, leading to an 'inverted'
primary.
Then the strain rate decreases or the stress increases to a constant level over a wide range of
strain (Stage II) . Once the strain-rate reaches a steady state, the material enters secondary creep.
During secondary creep, since there is a balance between dynamic hardening and dynamic
recovery, the strain-rate is constant. A secondary creep regime is of particular importance as it
indicates that the microstructure has reached a state of dynamic equilibrium, which entails constant
structure. It will be discussed in more detail in the following sections.
Eventually, there is a point when the metal cannot withstand the applied stress. The material is
now in the tertiary creep regime, also known as the fracture regime (Stage III). Tertiary creep has
its origins in cavitation and cracking which increases the strain rate at an increasing rate that leads
to rupture (in the case of constant strain rates, the flow stress drops in this stage).

116

3.2.1.1 Steady-State Creep

As stated in the previous section, in steady-state creep, dynamic hardening occurring as a result of
deformation is balanced by dynamic recovery. The majority of total strain may be accumulated
during steady-state. Therefore, it is especially important to interpret creep behaviors during this
stage. For instance, the Monkman-Grant relation often can predict the fracture time based on
steady-state strain rate. The minimum creep rate (often the steady-state creep-rate) is generally
expressed by the classic Norton-like equation for pure metals and class M alloys that exhibit a
similar creep behavior to pure metals:
𝜀̇ 𝑚𝑖𝑛 = 𝐴 exp[𝑄 𝑐 ∕ 𝑘𝑇 ](𝜎 m ∕ 𝐸 )
𝑛 (3-1)
where A is a constant, that includes stacking fault energy, 𝑄 𝑐 is the creep activation energy, k is
the Boltzman constant, T is temperature, 𝜎 m is the minimum stress (or 𝜎 ss, the steady-state stress),
E is the Young’s modulus and n is the stress exponent [122]. These variables and constants are of
course, related to the operating creep mechanism(s). The creep results of this study will be related
to the details of Eq. (3-1). Stress exponents, or n, values in class M (pure metal like creep behavior)
are typically 5 (4-7) above 0.6Tm. The values of the stress exponent are nearly constant above these
temperatures and sometimes referred to as 5 power-law creep. In some alloys (e.g. Al-Mg) with
the viscous glide of dislocations, the exponent may be 3 at higher temperatures. Power-law-
breakdown is observed in both cases at lower temperatures and the stress sensitivity exponent is
non-constant and increases with decreasing temperature and higher stress. Although the term five-
power-law creep is also known as dislocation climb controlled creep, it may be misleading because
climb control could occur under other regimes, such as Harper-Dorn, superplasticity, and power-
law-breakdown [123].

117
3.2.2 Inconel 718 Superalloy

Nickel-based superalloys have been around since the 1930s, and are mainly used in aerospace
applications where high mechanical strength, good resistance to creep, fatigue, and corrosion, and
ability to operate continuously at high temperatures is needed. Inconel 718, a superalloy consisting
of nickel-chromium-iron which is hardened by precipitation, is one of the most widely used
superalloys due to its high resistance to creep, and fatigue at temperatures greater than 650° C (it
can be used at temperatures ranging from -253° to 705°C). The precipitation heat treatment of
Inconel 718 shall be performed by heating to 1325 ± 25°F (718 ± 14°C), hold at temperature for 8
h, furnace cool to 1150 ± 25°F (621 ± 14°C), hold until the total precipitation heat treatment time
has reached 18 h, and then air cool [124]. The good machinability and weldability of Inconel 718
enables this superalloy to be utilized in a variety of components ranging from gas turbine disks to
screw and tooling for high-strength aerospace components. For instance, Inconel 718 is used in
the engine manifolds of SpaceX's Merlin rocket, which powers the Falcon 9 spacecraft.
3.2.2.1 Composition

In general, chromium is added to the matrix (nickel and iron) to provide oxidation resistance and
solid solution hardening. Chromium is rarely employed in amounts that exceed 19 wt% to keep it
hot workable.
Molybdenum adds high temperature strength via solid solution hardening. Molybdenum seems to
remain in solution at moderate temperatures, rather than precipitate out of the solution. It can have
an adverse effect on hot workability as well as chromium if added in large amounts; thus, it is kept
at less than 3 wt% to avoid this from happening.

118
Niobium in Inconel 718 can dramatically increase its strength through precipitation hardening.
Despite a significant increase in strength, ductility can be reduced dramatically too, especially
when niobium more than 5% is added. This results in 5 wt% niobium being the optimal content so
that good ductility with acceptable strength can be achieved.
Other than niobium, titanium is also a component involved with precipitation hardening in the
gamma matrix. Again, it can improve strength while reducing ductility. It has been demonstrated
by Eiselstein [125] that the highest stress rupture strength is attained when 1% titanium is
added. Thus, the 1% titanium level is established as an optimal composition for achieving both
good tensile strength and stress rupture properties.
Aluminum is also a component of the precipitation hardening phase. Carbon, titanium, and heat
treatment variables all affect how the aluminum content affects mechanical properties. Aluminum
adds no ductility or stress rupture resistance, and it has been reported that above the nominal 0.5%
level of aluminum, the strength is decreased.
Boron additions are useful for improving the mechanical properties like stress rupture. The
optimum based on regression analysis by Eiselstein is about 0.005% [125].
The carbon reacts sharply with niobium to produce niobium carbide (NbC). Consequently,
niobium is deprived from the precipitation-hardening phase and strength is reduced. Thus, the
carbon level is kept low, at approximately 0.05%. The same phenomenon happens with nitrogen
and titanium. Nitrogen combines with titanium and prevents the hardening effect of titanium.
Therefore, the nitrogen content should be minimized as well. The silicon concentration must also
be kept low to avoid impairing the hot workability and the rupture strength of the superalloy [125].
The solid solution matrix is thought to become embrittled when silicon is present [126].

119
3.2.2.2 Microstructures

In general, Inconel 718 contains gamma (𝛾 ) as matrix material, gamma prime (𝛾 ′), gamma double
prime (𝛾 "), as well as NbC and TiN in trace amounts. Occasionally, other phases such as laves
phases ((Ni,Cr,Fe)2(Nb,Mo,Ti)), orthorhombic needle like Ni3Nb δ–phase at the grain boundaries,
and high-temperature carbides may appear due to improper heat treatment or excessive service
temperature [127].
The gamma phase (𝛾 ) is the matrix material and is made up of the elements: nickel, chromium,
iron, columbium, molybdenum, titanium, and aluminum in solid solution. Gamma has an FCC
crystal structure and has comparatively low yield strength when solution-annealed.
Gamma prime phase (𝛾 ′) is an A3B compound where nickel is generally at the A positions, and the
B atoms are mostly niobium with some trace amount of titanium and aluminum (γ′-Ni3(Al,Ti,Nb))
[125]. The gamma prime chemical composition in atomic percent is about 75% nickel, 15%
niobium, 5% titanium, and 5% aluminum according to Eiselstein [125]. Gama prime has an
ordered FCC crystal structure with nickel atoms at the center of the face and niobium, aluminum,
and titanium atoms at the comers. The gamma prime appears as small spots within the gamma
matrix and has a spherical shape. During the aging process, it begins to form at a temperature of
649°C within reasonable periods of time (about 50 minutes) and is formed homogeneously
throughout the matrix.
Although gamma double prime (𝛾 ") is an A3B compound (just like gamma prime), it has an ordered
body-centered-tetragonal structure γ″ (bct-Ni3Nb). With gamma double prime precipitation,
typical discs form at temperatures of about 649 to 760°C [128]. These discs have an average
diameter of 60 nm and a thickness of between 5 to 9 nm [129].

120

Solids with low stacking fault energies do not cross slip with screw dislocations and nickel, as the
major constituent of Inconel 718, has a low stacking-fault energy. Hence, dislocations do not
naturally cross slip around obstacles such as precipitated 𝛾 particles. Thus, they must cut through
the 𝛾 " particles rather than bypassing them. It should also be noted that over a certain particle size,
bypassing can occur either by looping or, under conditions of elevated temperature of above 0.5
Tm, by dislocation climb. Overaging can cause 𝛾 " particles to coarsen to a size sufficient for
looping. Under temperatures below 649°C, Inconel 718 will usually exhibit particle cutting by
dislocations as both particle coarsening to promote looping and elevated temperatures to promote
dislocation climb are not present [129, 130].
3.2.2.3 Phase Stability and Heat Treatments

A transformation-time-temperature diagram for IN718, which is shown in figure 3-2, was used to
select the heat treatment temperatures to possibly obtain optimal mechanical properties [131-133].

121

Figure 3-2. Transformation-time-temperature diagram of IN718 [131, 133].

122
The Inconel alloy 718 for most applications can be specified as being solutions annealed and
precipitation hardened. Inconel 718 is hardened by precipitating secondary phases (gamma prime
and gamma double-prime) within the metal matrix. Heat treatment in the temperature range of 593
to 815.5 °C induces the precipitation of these nickel-phases (aluminum, titanium, niobium) phases.
The aging constituents (aluminum, titanium, niobium) must be dissolved in the matrix in order for
the aforementioned precipitation to happen properly. In order to achieve the optimal strength of
the superalloy, a solution annealing step (solution heat treatment) must be carried out first [131-
133]. Thus the Inconel alloy 718 is generally subjected to two common heat treatments:
1. 1 hour at 982°C followed by an air cool (= solution annealing)
2. 8hr at 718°C cool 55°C /hour to 621°C, held for 8 hours and then air cooled.
It should be noted that the initial solution anneal was not necessary for AM specimens as the hot
isostatic pressing (HIP condition was 100 MPa 1175
o
C for 4 hours) replaced the necessity for
solution annealing.
3.2.3 Metal Additive Manufacturing

Additive manufacturing (AM) is the formal term for what is commonly known as 3D Printing and
it has gained considerable attention in the past few decades. Additive manufacturing is defined as
“a process of joining materials to make objects from 3D model data, usually layer upon layer, as
opposed to subtractive manufacturing methodologies” [134]. This is done with the help of a
computer, 3D modeling software (Computer Aided Design or CAD). As an alternative to
conventional metal production processes, AM has several advantages since it allows products to
be easily designed or altered, while it produces less waste throughout the entire process when
compared to the conventional metal fabrication [135,136]. Moreover, managing the microstructure

123
is also easier in additive manufacturing than in a traditional machining process. This is due to the
fact that the microstructure of AM products is mainly determined by the processing parameters
such as laser powder, laser speed and precursor feeding rate [137,138]. The laser powder bed
fusion (LPBF) process is one of the most common AM processes employed for metal fabrication
where metal powder particles are melted and fused by an intense high-energy beam. This is
followed by heat isostatic pressing (HIP), which reduces porosity caused by additive
manufacturing processes.
3.3 Material and Experimental Details
3.3.1 Inconel Superalloys

The additive manufactured IN625 and IN718 specimens were produced at the National Institute
of Science and Technology (NIST) in Gaithersburg MD, USA using an EOSINT M290 machine.
The details of the fabrication process of IN625 are described in previous, recent, articles by
Kassner’s materials research group [118,119]. The powder particle size of the IN625 was about
37 𝜇 m.
As stated earlier, the heat treatment for IN718 was 1 hour 982°C followed by an air cool (= solution
annealing) + 8hr 718°C cool 55°C /hour to 621°C), held for 8 hours and then air cooled. The initial
solution anneal was not necessary for AM specimens as the hot isostatic pressing (HIP condition
was 100 MPa 1175
o
C for 4 hours) replaced the necessity for solution annealing.
The compositions of the wrought and AM alloys are listed in Table 3-1. Notable differences in the
wrought and AM alloys are the absence of Mg in the AM alloys and lower sulfur concentrations
in wrought alloys. It was noted that with additive manufacturing of both the IN718 and IN 625
the oxygen content was much higher than in the wrought alloys. The higher oxygen is due to the
absorbed oxygen on the powder surfaces leading to the formation of Al2O3. It appears that the

124
volume fraction of alumina in the AM alloy is about 0.004. These observations can explain the
embrittlement that the Kassner’s materials research group observed in IN625 [118, 119].

125

Table 3-1. The compositions of the wrought and AM Inconel alloys.

Composition data for Inconel alloys

Ni Cr Fe Mo Nb+Ta C Mn Si Al Ti Co W
AM 718 Bal. 18.40 18.02 3.01 5.48 0.043 0.015 0.03 0.48 1.00 0.02 -
Wrought
718
Bal. 18.56 18.23 2.89 5.13 0.035 0.052 0.08 0.53 0.96 0.44 -
AM 625 Bal. 20.72 0.65 8.98 4.15 0.003 0.05 0.07 0.32 0.37 0.22 0.17
Wrought
625
Bal. 22.22 4.28 8.29 3.54 0.064 0.35 0.22 0.14 0.23 0.05 0.26

O N S P B Pb Mg Ca Zr Hf

AM 718 0.019 0.012 0.0041 <0.010 0.003 <0.0001 <0.0001 0.0015 0.0001 N/D

Wrought
718
0.003 0.007 0.0007 0.0070 0.0050 0.0003 0.0010 0.0012 0.0004 0.004

AM 625 0.020 0.01 0.0033 <0.001 0.0028 0.002 0.0008 - N/D N/D

Wrought
625
0.002 0.02 0.0017 0.002 0.0033 <0.001 0.0081 - 0.0001 N/D

126
3.3.2 Creep Instrument

The creep machine used for elevated temperature creep tests was manufactured by Arcweld
Manufacturing Company (later SATEC) and all the test samples of this study have been designed
according to ASTM E8 standard [139]. Figure 3-3 depicts the (a) actual image and (b) the
schematic of this creep machine. The schematic shows how the creep machine operates using a
lever arm design. The load is applied on the sample by a lever arm with the distance ratio of 20:1
between the distance from the input point to the fulcrum, and the distance from the output point to
the fulcrum. Therefore, the creep machine applies a load of 20 times heavier than what is applied
on the input considering the lever rule. It should be noted that external batteries were used to
protect the creep data quality by avoiding any unexpected disturbances, such as power outages
during creep testing.

127

Figure 3-3. (a) The Arcweld Manufacturing Company creep machine used for elevated
temperature creep rupture tests. (b) The creep machine schematic [140].

128

An induction furnace (Applied Test Systems Inc, Butler, U.S.A.) capable of 1100 °C is used to
achieve elevated temperature conditions for creep tests. Heat is provided by the furnace system at
heating rates of 25 – 30
o
C/min until it reaches the desired temperature. Strain was measured by a
high temperature extensometer with the linear variable differential transformer external to the
furnace: an Omega Engineering displacement reader (Omega Engineering GP 911-5 AC LVDT)
was used to measure the instantaneous length of the sample, or gauge length. Then the electrical
signal is transmitted to a data acquisition module. The module then transmits the signal data to a
computer. Then a data acquisition data module (Omega Engineering iNet-601), with a data
recording frequency of 0.1s
-1
to 10000s
-1
, records the dataset.
3.3.3 Creep Tests and Curves

Both AM IN625 and wrought IN625 were creep-tested at either 650 ˚C or 800˚C for 24 hours
subsequent to the HIP or solution annealing, or creep-tested at 650 ˚C after being (cyclically) heat
treated for 6 months and 1 year at 650 ˚C (0.6Tm). These long-term cyclic tests are unique and very
relevant to long-term high temperature service of jet engines. The cyclic temperature cycle was
intended to model aircraft turbine temperatures during service. This cycle is illustrated in figure 3-
4.

129

Figure 3-4. The one cycle of the cyclic heating profile of the longtime heat treatment cycles for 6
months and 1 year in this study [141].

One year corresponds to the target lifespan of aircraft turbine parts where IN625 has been typically
utilized. Even though the recommended aircraft engine operation time before an overhaul is
expected to be about 16000 hours (22 months) [142, 143], the setting lifespan of 1 year appears
reasonable. The target temperature of 650 ˚C is the service temperature of some turbine
components [144].
Figure 3-5 illustrates the stress versus strain behavior of wrought IN718 at ambient and 650
˚C. Embrittlement is evident at 650 ˚C. Figure 3-6 shows the true plastic strain versus time of the
high temperature creep tests at 650 ˚C on AM and wrought IN718 at 730 and 819 MPa.

130

Figure 3-5. The stress versus strain behavior of wrought IN718 at ambient and 650˚C.
Embrittlement is evident at 650˚C. LPBF corresponds to laser powder bed fusion (described in
section 3.2.3).

131

132

Figure 3-6. Representative creep curves of the high temperature creep tests at 650 ˚C on AM and
wrought IN718.

3.4 Results and Discussion

The following describes the results and discussion, from a recent conference paper entitled “The
creep and fracture behavior of additively manufactured Inconel 625 and 718” and submitted to
European Creep Collaborative Committee following the work of several graduate students
including the author of this thesis. The section for which the author was responsible will be
reported here [145].
Stresses are reported as engineering values. Figure 3-7 illustrates EBSD images of the solution
annealed wrought and HIPed AM IN718. Both specimens underwent the standard double age heat
treatment. The average grain size of AM HIPed IN718 was 68 μm. Wrought IN718 showed
equiaxed grains with average grain size of 11μm. Of course, HIPing was performed to reduce the
porosity present in all as-fabricated AM specimens.

133
A variety of microstructural instruments were used to characterize the material to understand the
creep process and the basis of the embrittlement. These included transmission electron
microscopy (TEM) which was particularly useful in characterizing the second phases and the
dislocation substructure, scanning electron microscopy (SEM), Electron Backscatter Diffraction
(EBSD) to observe whether intergranular fracture was evident for secondary cracks, Atomic Probe
Topography (APT) and Nano Secondary ion mass spectroscopy (nanoSIMS) to better understand
the sources of embrittlement by examining compositions at various interfaces in the alloy [145].

Figure 3-7. EBSD images of solution annealed wrought and HIPed AM 718. Both specimens
underwent the standard double age heat treatment. The average grain size of AM HIPed IN718
measured about 68 μm while the wrought was about 11 μm. Although the grain size is different
by a factor of six, grain size strengthening is not particularly substantial [146] at elevated
temperatures within the five power-law regime.

AM HIPed IN718
Wrought IN718

134
Figure 3-6 illustrates three of the creep tests performed on AM IN718 and wrought IN718. There
appears to be an early region of an initial relatively low (negative) creep rate in IN718 followed
by an increase in rate in the 𝛾 ” precipitation-strengthened superalloy. This has been observed in
earlier work for IN625 by Kassner’s materials research group [118,119] and is reminiscent of some
of the features of sigmoidal creep absent of an inflection to higher creep rates followed by lower
creep rates. This behavior in figure 3-6 occurs over a very small strain that is less than 10
-4
. Perhaps
it is a small error associated with our testing procedure. Curiously, the slopes of the steady-state
creep-rates versus stress in figure 3-8, suggest a slope of about (𝑛 =) 3 at 650
o
C (0.6Tm). Pure
metal and class M (metal behaving) alloys have 𝑛 values of 4-7. The explanation for this is unclear.
IN718 is precipitation-hardened rendering the explanation for such a low stress exponent is
elusive. It is interesting that in recent work by Ni and Dong [147], it was suggested that there is an
absence of a genuine steady-state. Only primary and tertiary creep are observed. Earlier work [148]
suggests a transition for diffusional creep to 5-power-law creep. Yet others [149] suggest that
deformation may be controlled by plasticity at the grain boundaries which could explain a low
stress-exponent.

In this study, in the experimental temperature range of 650 ˚C to 800 ˚C, the calculated creep
activation energies (Qc) for IN625 is about 270 kJ/mol which is close to that of lattice self-diffusion
in Ni (279 kJ/mol [150]). For wrought IN718 Qc was found to also be nearly equal to that of lattice
self-diffusion [148]. Activation energy measurements reflect the change in creep-rate with change
in temperature for a fixed (micro-) structure. Measurement would be erroneous with differences in
structure due to a different precipitation substructure as may be the case in figure 3-8(a). The stress
exponent, 𝑛 , for IN625 creep at 800
o
C falls within those for five power law creep (4-7). Thus,

135
the 𝑛 and Qc suggest that lattice self-diffusion (dislocation climb) is the rate controlling process
and five power law creep is the creep mechanism for IN625 and IN718. The compositions of AM
and wrought Inconel samples from Table 3-1 are quite similar so it is not surprising that the creep
rates at identical temperatures and applied stresses are nearly identical at 650
o
C. Figure 3-8
illustrates the nearly identical creep rates for wrought and AM IN718. However, at higher stress,
it appears that the AM IN718 is actually stronger than the wrought alloy. The basis for this
increased strength is under investigation by the authors. Figure 3-8 also illustrates that the AM
alloy is embrittled for the two stresses at which failure was observed in both AM and wrought
IN718. These higher stress tests were performed within approximately one week. Interestingly, the
ductilities of the AM and wrought IN625 were different as well, with the AM alloy being
embrittled again.

Figures 3-8 and 3-9 show that the ductility of the wrought is much higher than that of the AM for
both the IN625 and IN 718 at the elevated temperatures. Thus, embrittlement is evident in AM
specimens. APT and nanoSIMS were utilized to better understand the sources of embrittlement.
Figure 3-10 reveals that there is an absence of intergranular segregation of elements such as sulfur
or monotonic oxygen that could lead to embrittlement. Sulfur is known to embrittle Ni at higher
temperatures and oxygen has been suggested to embrittle IN718 at higher temperatures [151-153].
However, our nanoSIMS results confirmed that sulfur segregates at the Al2O3/matrix interface.
Al2O3 is evident in the AM alloy as a consequence of oxygen absorption during powder fabrication
processing. Chemical analysis of secondary cracks in a fractured specimen confirms an association
with Al2O3. We note from the composition that the AM alloy has higher sulfur concentration and
that may be the source of the embrittlement. Additionally, we note that the AM alloy, unlike the

136
wrought alloy, has an absence of Mg, which “ties-up” the sulfur (as MgS) from segregating to
interfaces that can lead to embrittlement. Mg is absent from the AM alloy as the AM processing
precludes Mg due to its volatility.

It must be pointed out that ambient temperature tensile tests reveal essentially identical strength
and ductility for both AM and wrought IN625 and IN718 [151], [154], [155]. Therefore, the loss
of ductility is a high-temperature phenomenon. The loss of ductility could not be due to residual
porosity after HIPing. The fatigue behavior and an impact toughness were recently examined by
the authors, as well [154][156] and it appears that there is a deterioration of the fatigue behavior
of both alloys [155][156]. Additionally, the 650 °C impact toughness of both AM IN625 and IN718
were lower than wrought IN625 and IN718. The fatigue behaviors of IN718 alloys are illustrated
in figure 3-11.

137

Figure 3-8. (a) and (b). The creep rate and ductility of wrought and AM 718 at 650oC. Literature
values were taken from Special Metals [143]. The creep ductility values of wrought and AM 718
at 650oC and higher stresses of 730 and 819 MPa indicate significant embrittlement of the AM
IN718. The tests with arrows indicate that failure was not observed within one day. The higher
stress tests to failure occurred over a period of about one week.

24 hour Ductility
Failure Ductility

138

Figure 3-9. Wrought and AM Inconel 625 creep test results (a) minimum (steady-state) creep rate
versus the modulus compensated (normalized) stress. (b) creep ductility at 650 ˚C (c) creep
ductility at 800 ˚C. The ductility at the two test temperatures is indicated by solid dots for 24 hour
(short creep tests or SCT) specimens, X marks for 6 months at 650 ˚C samples and + marks for 1
year at 650 ˚C specimens [119].

139

Figure 3-10. The element distribution as a function of distance from a grain boundary and a 𝛾 ’’
interface from atom probe topography (APT) for AM IN625 specimens: (a) creep deformed at 294
MPa and 650˚C within 24 hours (b) creep deformed after 6 months at 650˚C at 658 MPa (c)
concentration profile near a 𝛾 ’’ particle. The shaded area are grain boundary regions or 𝛾 ’’
precipitates [119].

10
3
10
4
10
5
10
6
10
7
10
8
300
400
500
600
700
800
900
1000
LPBF HIPed IN718
Wrought IN718
LPBF HIPed IN625
Wrought IN625
Reference wrought + standard aged 718 data [Sui et al. (2017)]
Reference LPBF + standard aged 718 data [Sui et al. (2017)]
Max. stress [MPa]
Number of cycles to failure [N
f
]
Figure 3-11. The fatigue behavior of wrought and additively manufactured IN625 and 718 at
650oC [152].

140

Figure 3-12. Strong oxygen, carbon and sulfur signals were observed at the Al2O3 particles. No
evidence of grain boundary segregation of impurities was observed in the nanoSIMS results for
AM as-HIPed 625 alloy.

Of course, it must be mentioned that oxygen embrittlement of wrought IN718 has been observed
between 550-650 ˚C [153], [157], which this study may not have observed as the present tests were
at the upper limit of the observed embrittlement in the literature studies. The mechanisms
suggested included monotonic oxygen segregation at the grain boundary leading to embrittlement
analogous to hydrogen embrittlement, or the formation of Nb carbides stimulated by oxygen

141
diffusion to the grain boundaries [158]. Regardless, it appears that for the case of IN718 at least
extra embrittlement is observed in AM IN718 over the wrought, and the explanation may be similar
to that of the AM IN625 where the higher sulfur concentration leads to embrittlement. Curiously
this is in contradiction with other IN718 work at 650 ˚C [159].
3.5 Conclusion

The creep strengths of additively manufactured Inconel 625 and 718 and wrought 625 and 718 at
650
o
C and 800
o
C appear nearly identical. However, embrittlement was observed in IN625 within
the temperatures range 650 ˚C to 800 ˚C is apparently due to sulfur segregation at the
alumina/matrix interfaces. The embrittlement appears to deteriorate the fatigue properties of both
alloys as well. The embrittlement of IN625 might be mitigated by decreasing the sulfur
concentration and adding a sulfide-forming element like Mg, by tying up the sulfur and precluding
crack formation. This may be true of AM IN718 as well. Curiously this is in contradiction by other
work [147] where AM IN718 has improved ductility over the wrought IN718 at 650 °C.

142
4 Chapter 4: Power Law Breakdown in the Creep in Single-Phase
Metals

4.1 Motivation

New analysis provides insight into the basis of power-law breakdown (PLB) in the steady-state
creep of metals and alloys. A variety of theories has been presented in the past but this new
examination suggests that there is evidence that a dramatic supersaturation of vacancies leading to
very high diffusion rates and enhanced dislocation climb is associated with the rate-controlling
process for creep in PLB. The effect of vacancy supersaturation may be enhanced by dislocation
short circuit diffusion paths at lower temperatures due to the dramatic increase in dislocation
density.

4.2 Background

Steady-state deformation of a material occurs when there is a balance between hardening and
dynamic recovery. For constant strain-rate testing, the stress is fixed during steady-state, and for
constant stress tests, the strain rate is fixed at steady-state. Generally, at temperatures above about
0.6 Tm, metals and class M alloys (behave like pure metals) obey a steady-state creep equation of
the form [160]:
𝜀 ̇ 𝑠𝑠
= 𝐴 0
(
𝜒 𝐺𝑏
)
3
(
𝐷 𝑠𝑑
𝐺𝑏
𝑘𝑇
)(
𝜎 𝑠𝑠
𝐺 )
𝑛 (4-1)
where 𝜀 ̇ 𝑠𝑠
is the steady-state strain-rate (creep rate), A0 is a constant, k is Boltzmann’s constant, G
is the shear modulus, b is the Burgers vector, n is the steady-state stress exponent, Dsd is the lattice

143
self-diffusion coefficient and χ is the stacking fault energy. The steady-state creep behavior then
has activation energy Qsd that is equal to that for lattice self-diffusion. Hence, it is usually assumed
that the five power-law creep is controlled by dislocation climb. Of course, Qsd has two
components—about half the value is the activation energy for vacancy formation and the other
half is the activation energy for the atomic jumps into the vacancy. Therefore, the creep-rate is
proportional to the vacancy concentration. It is important to emphasize that steady-state flow is a
balance between the hardening process, which appears to be dislocation network refinement [161],
and dynamic recovery; recrystallization mechanisms are not considered. Often a genuine steady-
state is confused with a minimum creep-rate which may be due to insufficient strain to reach a true
steady-state, or a material proceeding to steady-state but interrupted by stage III creep or tertiary
creep in which fracture proceeds. That is Stage I to stage III without Stage II or a steady-state. In
the case of 99.999% pure Al in figure 4-1, the highest stress points are probably not reflective of
steady-state due to DDRX, but the author believes the PLB (power-law breakdown) data just above
the highest stress of the 5 PL regime are genuine steady-states reflective of a balance between
hardening and dynamic recovery.
The stress exponent is typically a value close to five, in Equation (4-1), although variations
from four to seven have been observed. Thus, this regime of a constant stress exponent is often
referred to as five-power–law-creep. This regime is illustrated in figure 4-1 for high purity
aluminum where the stress exponent is about 4.5 in the constant stress exponent regime at
temperatures typically greater than 0.6 Tm. At very high temperatures near the melting point,
Harper–Dorn creep with n = 1 is sometimes observed, but this will not be discussed here. Below
about 0.6 Tm the stress exponent is no longer constant with changes in stress and increases with

144
increasing stress/strain-rate. This behavior is referred to as power-law breakdown (PLB). The
explanation for PLB has been elusive.

Figure 4-1. The steady-state creep behavior of high purity aluminum from temperatures ranging
from sub-ambient to near the melting point [160, 163-164].This figure is adapted from [164].

Equation (4-2) can be extended to phenomenologically describe PLB including changes in Qc, the
activation energy for creep, with temperature and stress by the hyperbolic sine function [160, 162].
ε̇ ss
= A
1
exp [
−Q
c
kT
⁄ ] [ sinhα
1
(
σ
ss
E
⁄ )]
5
(4-2)

145
4.3 Work to date

The following describes work to date, from a paper titled “Power Law Breakdown in the Creep in
Single-Phase Metals”, published in Metals [165].
4.3.1 Results and Discussion

Activation energy measurements for steady-state flow within PLB are rare. Luthy et al., [166]
attempted to measure Qc in this regime, but it now appears that the apparent “steady-state” at
ambient and near-ambient temperatures in their work was actually influenced by discontinuous
dynamic recrystallization (DDRX) in addition to dynamic recovery [167]. Thus, it is unclear
whether the Qc, values measured are reflective of the activation energy for steady-state flow. Some
of the high-stress data in figure 4-1 may be somewhat low with DRX being present.
Some have tried to measure Qc, at low temperatures by using temperature-change tests, but these
were performed at relatively small strains that were probably far from the onset of steady-state
[168]. The observed values were, nonetheless, much lower than those above 0.6 Tm. A genuine
steady-state Qc was measured with very large-strain (torsion) deformation in silver. This is
illustrated in figure 4-2.

146

Figure 4-2. The activation energy for steady-state creep of silver as a function of temperature
[169,170].

The strains to reach steady-state were typically greater than 3.0. We note that Qc decreases
with decreasing temperature and the decrease from an approximately constant value within the
five-power law regime corresponds to the onset of PLB below about 0.6 Tm. Robinson and Sherby
[171] and Luthy, Miller, and Sherby [166] suggested that as the stress exponent increases into the
PLB regime, creep is still dislocation climb controlled, but Qc may reflect dislocation-pipe
diffusion, Qp [166, 171]. Vacancy supersaturation resulting from deformation (moving
dislocations with, e.g, jogs), could also explain this decrease with decreasing temperature
(increasing stress) and still be consistent with dislocation climb control [171]. These two effects
would significantly reduce the activation energy for steady-state creep. Early modeling by
Mecking et al. [172] suggested that vacancy supersaturation by deformation in the power-law
regime was unlikely, but suggested that supersaturation may occur within PLB. More recently,
important modeling and experiments by others [173-175] also suggested that supersaturation may

147
occur with significant plasticity within PLB. Wu and Sherby [176] however, subsequently
suggested that internal stress, as did Nix and Ilschner, albeit from different sources, explains PLB
behavior and appeared to abandon vacancy superstation effects.
Aside from vacancy supersaturation and short-circuit diffusion by dislocation “pipes”,
dislocation glide mechanisms have also been suggested to be important by Arieli and Mukherjee
[177], Weertman and Weertman [178] and Ilschner and Nix [19]. Ilschner and Nix [179] elegantly
suggested that cellular dislocation structures rather than well-defined subgrains are formed in PLB
and this gives rise to lower long-range internal stresses (LRIS) leading to PLB. There may be two
difficulties with this approach. First, Kassner [180] and Levine et al. [181] report that both subgrain
and cellular structure actually have low (e.g, 0.1 σ) LRIS. Phan et al. [182] found that for severely
deformed Al at an ambient temperature where the extensive formation of well-defined subgrains
is observed, the LRIS, again, may be small. Second, recent studies observe that very well-defined
subgrain boundaries form as a result of dislocation reaction even in low stacking fault energy
metals such as silver and Zr [183] in PLB, also suggesting that substantial dislocation climb is at
least occurring [167, 169] in PLB.
Vacancy supersaturation resulting from deformation could explain the decrease in Qc with
decreasing temperature (increasing stress) and still be consistent with dislocation climb control.
Again, the diffusion activation energy has two parts—one is the activation energy for vacancy
formation, Qv, and the other is the activation energy for atomic jumps into the vacancy, Qm. If
deformation creates excess (non-equilibrium) vacancies then the Qm may dominate Qc. There can
be a further decrease in Qc if short-circuit diffusion through dislocation pipes occurs.

148
Earlier work by Kassner [169] illustrated that the increase in the steady-state stress exponent
with PLB is coincident with the increase in the (annealed) constant-structure stress exponent
defined by
N = [∂ lnε̇ / ∂σ]
T,s
(4-3)
where “s” refers to the structure and N, thus, reflects the change in creep rate with a change in
stress for a fixed structure. The steady-state stress exponent, n, refers to the change in the steady-
state creep-rate for a change in stress leading to different structures. N, the constant structure strain-
rate sensitivity, is sometimes described by m (= 1/N). Sometimes the concept of a change in creep
rate with changes in stress for a constant structure is described by an activation volume, V (= Ab
where A is an activation area and b is the Burgers vector). These terms are described in detail in
[169]. Here we utilized “N” to describe the change in creep-rate with a change in stress. N, m, V
and A are sometimes utilized in discussions of creep and are all quantitatively related.
Figure 4-3 illustrates the described coincidence. Activation area measurements on deformed silver
suggest that for strains <0.2, at low temperatures in the PLB regime, the dislocation intersection
mechanism appears reasonable; deformation does not appear to be dislocation climb controlled.
However, at much larger strains (e.g., >3.0), where a genuine steady-state is observed, the
intersection mechanism was not verified as being rate-controlling [169]. The coincidence of
increases in both N for annealed metal and n at similar temperatures and strain-rates implies that
PLB is not related to the dislocation structure. This is inconsistent with the PLB explanation based
on vacancy supersaturation and short circuit diffusion.

149

Figure 4-3. (a) the steady-state (n) and constant structure (N) stress exponents for annealed (broken
line) and steady-state (solid line) structures of AISI 304 stainless steel as a function of lattice
diffusion coefficient compensated strain-rate. (b) comparison of the constant structure stress
exponents (N) of annealed aluminum with the steady-state stress exponent (n) based on earlier

150
work by the author [184] and others [166,185,186]. Deff is the effective diffusion coefficient that
is the combined diffusivity considering dislocation pipes as described in [166].

Again, Robinson, and Sherby [171] suggested the possibility that dislocation pipe (fast) diffusion
and vacancy supersaturation from plasticity may be responsible for PLB (but as mentioned earlier,
Wu and Sherby [176] however, subsequently suggested that internal stress, as did Nix and Ilschner,
explains PLB behavior and appeared to abandon vacancy supersaturation and dislocation pipe
diffusion effects). If we assume just volume diffusion at ambient temperature for silver, which
appears to achieve steady-state after strains greater than 3.0 at ambient temperature, the predicted
vacancy concentration (based on the usual equations [187]) is roughly 10
−17
. With substantial
plasticity (e.g., >0.3) the latest experimental and theoretical estimates [172-175,188] of the
vacancy concentration at ambient temperature after large strain deformation in metals is of the
order of 10
−3
to 10
−5
(roughly a factor of 10
13
higher than the equilibrium concentration). This
suggests that the low-temperature diffusion coefficient may be relatively high giving rise to higher
than expected creep rates leading to PLB. Thus, plasticity leading to vacancy supersaturation is
certainly capable of rationalizing PLB. Also, for Al, the equilibrium vacancy concentration at
ambient temperature is about 3 × 10
−14
. Vacancy supersaturation by plasticity certainly also
rationalizes the increase in creep rate in the PLB illustrated in figure 4-1. Also, if dislocation pipe
diffusion rather than volume diffusion control climb (i.e, Deff = Dp > Dsd) then the activation energy
for diffusion would be roughly half that of volume self-diffusion and together with vacancy
supersaturation, the dislocation climb-rate would be yet higher and Qc might be a relatively small
fraction of Qsd such as in figure 4-2 (Dp is diffusion coefficient in the plastically deformed metal
at low temperatures by short-circuiting dislocations pipes and Dsd is the diffusion coefficient in the
absence of short-circuiting). Short circuit diffusion by dislocation pipes [187], at an ambient
temperature is also enough to rationalize PLB. The ratio of Dp/Dsd at ambient temperature

151
approximately equals 10
16
. Thus, PLB may be rationalized independently by dislocation pipe
diffusion.
In summary, there have been four classes of explanations for PLB:
(1) It has been proposed that there is a change in the mechanism of plastic flow from dislocation
climb-control to glide-control in PLB. However, this proposition is sometimes based on the
presence of internal stress which does not appear to be reasonable as LRIS appears to be low in
both the PL and PLB regimes. The glide mechanism at low temperatures may be relevant for low
plastic strains (e.g., <0.30), but it does not appear relevant to steady-state deformation.
(2) Only recently has vacancy supersaturation been experimentally verified. The levels of
supersaturation appear sufficient to create PLB.
(3) The coincidence between the increase in N (especially in annealed metals) and the stress
exponent, n, for steady deformation in PLB has been suggested to imply that the onset of PLB is
not structurally related. But this coincidence has been principally verified for annealed structures
for which steady-state is not relevant. Although it may be true that PLB-like behavior is
coincidentally observed for n and N, the conclusion that PLB is not structurally related may not be
justified. A detailed explanation for the coincidence is not apparent.
(4) The changes in the diffusion coefficient from Dsd to Dp with large strain plasticity within the
PLB regime may independently contribute to the observation of PLB. Another point must be made
and that is references [174,175,188] verify vacancy supersaturation at ambient temperature; other,
somewhat higher temperatures within PLB were not checked. Of course, higher temperature x-ray
diffraction experiments are more difficult than those at ambient temperature. Perhaps future

152
experiments could be performed at other temperatures within PLB to fully verify the coincidence
between the onset of PLB and vacancy supersaturation.
4.4 Conclusions

Vacancy supersaturation and/or dislocation pipe diffusion appear to be the basis for the power-
law-breakdown regime for steady-state creep in metals and alloys. Explanations based on changes
in the basic mechanism for steady-state flow do not appear to be justified. The suggestion that the
transition to PLB is not structurally related also does not appear to be justified.

153
References
[1] “SWA1380-DCA18MA142-Investigative-Update.pdf.” Accessed: Jun. 06, 2020. [Online].
Available: https://www.ntsb.gov/investigations/AccidentReports/Reports/SWA1380-
DCA18MA142-Investigative-Update.pdf.
[2] H. Mughrabi, “Dislocation wall and cell structures and long-range internal stresses in
deformed metal crystals,” Acta Metall., vol. 31, no. 9, pp. 1367–1379, Sep. 1983, doi:
10.1016/0001-6160(83)90007-X.
[3] H. Mughrabi, “A Two-parameter Description of Heterogeneous Dislocation Distributions in
Deformed Metal Crystals*t,” p. 17.
[4] D. L. Holt, “Dislocation Cell Formation in Metals,” J. Appl. Phys., vol. 41, no. 8, p. 3197,
Nov. 2003, doi: 10.1063/1.1659399.
[5] M. Sauzay and L. P. Kubin, “Scaling laws for dislocation microstructures in monotonic and
cyclic deformation of fcc metals,” Prog. Mater. Sci., vol. 56, no. 6, pp. 725–784, Aug. 2011,
doi: 10.1016/j.pmatsci.2011.01.006.
[6] P. Li, S. X. Li, Z. G. Wang, and Z. F. Zhang, “Formation mechanisms of cyclic saturation
dislocation patterns in [001], [011] and [1¯11] copper single crystals,” Acta Mater., vol. 58,
no. 9, pp. 3281–3294, May 2010, doi: 10.1016/j.actamat.2010.02.002.
[7] P. Charsley and D. Kuhlmann-wilsdorf, “Configurations of 100 dislocation walls formed
during fatigue,” Philos. Mag. A, vol. 44, no. 6, pp. 1351–1361, Dec. 1981, doi:
10.1080/01418618108235814.
[8] N. Y. Jin and A. T. Winter, “DISLOCATION STRUCTURES IN CYCLICALLY
DEFORMED [OOl] COPPER CRYSTALS,” p. 4.

154
[9] T. Fujii, T. Kajita, T. Miyazawa, and S. Arai, “Characterization of dislocation microstructures
in cyclically deformed [001] copper single crystals using high voltage scanning transmission
electron microscopy,” Mater. Charact., vol. 136, pp. 206–211, Feb. 2018, doi:
10.1016/j.matchar.2017.12.026.
[10] Z. Wang, B. Gong, and Z. Wang, “Cyclic deformation behavior and dislocation structures of
[001] copper single crystals—II. Characteristics of dislocation structures,” Acta Mater., vol.
45, no. 4, pp. 1379–1391, Apr. 1997, doi: 10.1016/S1359-6454(96)00289-3.
[11] B. Gong, Z. Wang, and Z. Wang, “Cyclic deformation behavior and dislocation structures of
[001] copper single crystals—I Cyclic stress-strain response and surface feature,” Acta Mater.,
vol. 45, no. 4, pp. 1365–1377, Apr. 1997, doi: 10.1016/S1359-6454(96)00288-1.
[12] P. Li, S. X. Li, Z. G. Wang, and Z. F. Zhang, “Fundamental factors on formation mechanism
of dislocation arrangements in cyclically deformed fcc single crystals,” Prog. Mater. Sci., vol.
56, no. 3, pp. 328–377, Mar. 2011, doi: 10.1016/j.pmatsci.2010.12.001.
[13] L. E. Levine, M. R. Stoudt, A. Creuziger, T. Q. Phan, R. Xu, and M. E. Kassner, “Basis for
the Bauschinger effect in copper single crystals: changes in the long-range internal stress with
reverse deformation,” J. Mater. Sci., vol. 54, no. 8, pp. 6579–6585, Apr. 2019, doi:
10.1007/s10853-018-03295-6.
[14] M. E. Kassner, P. Geantil, and L. E. Levine, “Long range internal stresses in single-phase
crystalline materials,” Int. J. Plast., vol. 45, pp. 44–60, Jun. 2013, doi:
10.1016/j.ijplas.2012.10.003.
[15] H. Mughrabi, “Dislocation clustering and long-range internal stresses in monotonically and
cyclically deformed metal crystals,” Rev. Phys. Appliquée, vol. 23, no. 4, pp. 367–379, 1988,
doi: 10.1051/rphysap:01988002304036700.

155
[16] M. E. Kassner, A. A. Ziaai-Moayyed, and A. K. Miller, “Some trends observed in the elevated-
temperature kinematic and isotropic hardening of type 304 stainless steel,” Metall. Trans. A,
vol. 16, no. 6, pp. 1069–1076, Jun. 1985, doi: 10.1007/BF02811676.
[17] O. B. Pedersen, L. M. Brown, and W. M. Stobbs, “The bauschinger effect in copper,” Acta
Metall., vol. 29, no. 11, pp. 1843–1850, Nov. 1981, doi: 10.1016/0001-6160(81)90110-3.
[18] A. Seeger, J. Diehl, S. Mader, and H. Rebstock, “Work-hardening and work-softening of face-
centred cubic metal crystals,” Philos. Mag. J. Theor. Exp. Appl. Phys., vol. 2, no. 15, pp. 323–
350, Mar. 1957, doi: 10.1080/14786435708243823.
[19] J. Weertman, J. R. Weertman, J. Friedel, R. G. Rhodes, and J. M. Bailey, “Elementary
Dislocation Theory, Dislocations and Imperfections and Active Centers in Semiconductors,”
Am. J. Phys., vol. 33, no. 12, pp. 1091–1093, Dec. 1965, doi: 10.1119/1.1971184.
[20] N. F. M. F.R.S, “CXVII. A theory of work-hardening of metal crystals,” Lond. Edinb. Dublin
Philos. Mag. J. Sci., vol. 43, no. 346, pp. 1151–1178, Nov. 1952, doi:
10.1080/14786441108521024.
[21] A. W. Sleeswyk, M. R. James, D. H. Plantinga, and W. S. T. Maathuis, “Reversible strain in
cyclic plastic deformation,” Acta Metall., vol. 26, no. 8, pp. 1265–1271, Aug. 1978, doi:
10.1016/0001-6160(78)90011-1.
[22] E. Orowan, “Causes and effects of internal stresses,” in Internal Stresses and Fatigue in
Metals, G.M. Rassweiler and W.L. Grube., Amsterdam: General Motors, Elsevier., 1959, p.
59_80.
[23] M. E. Kassner, M. A. Wall, and A. W. Sleeswyk, “Some observations during in-situ reversed
deformation of aluminum single crystals in the HVEM using the X-Y method,” Scr. Metall.
Mater., vol. 25, no. 7, pp. 1701–1706, Jul. 1991, doi: 10.1016/0956-716X(91)90478-J.

156
[24] Y. He et al., “Dislocation dipole-induced strengthening in intermetallic TiAl,” Scr. Mater.,
vol. 143, pp. 98–102, Jan. 2018, doi: 10.1016/j.scriptamat.2017.09.010.
[25] M. E. Kassner, M. A. Wall, and M. A. Delos-Reyes, “Microstructure and mechanisms of cyclic
deformation of aluminum single crystals at 77 K,” Metall. Mater. Trans. A, vol. 28, no. 3, pp.
595–609, Mar. 1997, doi: 10.1007/s11661-997-0045-2.
[26] M. E. Kassner and M. A. Wall, “Microstructure and mechanisms of cyclic deformation in
aluminum single crystals at 77 K: Part II. Edge dislocation dipole heights,” Metall. Mater.
Trans. A, p. 3.
[27] M. E. Kassner, M. A. Wall, and M. A. Delos-Reyes, “Primary and secondary dislocation dipole
heights in cyclically deformed copper single crystals,” Mater. Sci. Eng. A, vol. 317, no. 1–2,
pp. 28–31, Oct. 2001, doi: 10.1016/S0921-5093(01)01195-9.
[28] M. E. Kassner, M.-T. Pérez-Prado, K. S. Vecchio, and M. A. Wall, “Determination of internal
stresses in cyclically deformed copper single crystals using convergent-beam electron
diffraction and dislocation dipole separation measurements,” Acta Mater., vol. 48, no. 17, pp.
4247–4254, Nov. 2000, doi: 10.1016/S1359-6454(00)00284-6.
[29] J. G. Antonopoulos and A. T. Winter, “Weak-beam study of dislocation structures in fatigued
copper,” Philos. Mag., vol. 33, no. 1, pp. 87–95, Jan. 1976, doi: 10.1080/14786437608221093.
[30] B. Tippelt, J. Breitschneider, and P. Hähner, “The Dislocation Microstructure of Cyclically
Deformed Nickel Single Crystals at Different Temperatures,” Phys. Status Solidi A, vol. 163,
no. 1, pp. 11–26, 1997, doi: 10.1002/1521-396X(199709)163:1<11::AID-PSSA11>3.0.CO;2-
X.

157
[31] U. Essmann and H. Mughrabi, “Annihilation of dislocations during tensile and cyclic
deformation and limits of dislocation densities,” Philos. Mag. A, vol. 40, no. 6, pp. 731–756,
Dec. 1979, doi: 10.1080/01418617908234871.
[32] M. E. Kassner, M. T. Perez-Prado, and K. S. Vecchio, “Internal stress measurements by
convergent beam electron diffraction on creep-deformed Al single crystals,” Mater. Sci. Eng.
A, vol. 319–321, pp. 730–734, Dec. 2001, doi: 10.1016/S0921-5093(01)00953-4.
[33] M. Legros, O. Ferry, F. Houdellier, A. Jacques, and A. George, “Fatigue of single crystalline
silicon: Mechanical behaviour and TEM observations,” Mater. Sci. Eng. A, vol. 483–484, pp.
353–364, Jun. 2008, doi: 10.1016/j.msea.2006.10.200.
[34] H. Mughrabi * and F. Pschenitzka, “Constrained glide and interaction of bowed-out screw
dislocations in confined channels,” Philos. Mag., vol. 85, no. 26–27, pp. 3029–3045, Sep.
2005, doi: 10.1080/14786430500079975.
[35] E. Schafler et al., “A second-order phase-transformation of the dislocation structure during
plastic deformation determined by in situ synchrotron X-ray diffraction,” Acta Mater., vol. 53,
no. 2, pp. 315–322, Jan. 2005, doi: 10.1016/j.actamat.2004.09.025.
[36] M. S. Paterson, “X-ray line broadening from metals deformed at low temperatures,” Acta
Metall., vol. 2, no. 6, pp. 823–830, Nov. 1954, doi: 10.1016/0001-6160(54)90035-8.
[37] T. Ungar, H. Mughrabi, D. Rönnpagel, and M. Wilkens, “X-ray line-broadening study of the
dislocation cell structure in deformed [001]-orientated copper single crystals,” Acta Metall.,
vol. 32, no. 3, pp. 333–342, Mar. 1984, doi: 10.1016/0001-6160(84)90106-8.
[38] T. Ungár, H. Mughrabi, M. Wilkens, and A. Hilscher, “Long-range internal stresses and
asymmetric X-ray line-broadening in tensile-deformed [001]-oriented copper single crystals:

158
The correction of an erratum,” Philos. Mag. A, vol. 64, no. 2, pp. 495–496, Aug. 1991, doi:
10.1080/01418619108221201.
[39] L. E. Levine, B. C. Larson, W. Yang, M. E. Kassner, J. Z. Tischler, M. A. Delos-Reyes, R. J.
Fields, A. Liu, “X-ray microbeam measurements of individual dislocation cell elastic strains
in deformed single-crystal copper,” Nat. Mater., vol. 5, no. 8, pp. 619–622, Aug. 2006, doi:
10.1038/nmat1698.
[40] “Model 110 twin-jet electropolisher and Model 120 automatic power control service manual,
Fischione Instruments.” .
[41] C. A. Volkert and A. M. Minor, “Focused Ion Beam Microscopy and Micromachining,” MRS
Bull., vol. 32, no. 5, pp. 389–399, May 2007, doi: 10.1557/mrs2007.62.
[42] W. Zhou and Z. L. Wang, Eds., Scanning Microscopy for Nanotechnology: Techniques and
Applications. New York: Springer-Verlag, 2007.
[43] “EBSD Oxford Instruments - Principle Components of an EBSD System.”
https://www.ebsd.com/ebsd-explained/principle-components-of-an-ebsd-system (accessed
Jun. 07, 2020).
[44] D. B. Williams and C. B. Carter, Transmission Electron Microscopy: A Textbook for Materials
Science. Springer Science & Business Media, 2009.
[45] “Transmission electron microscopy in practice | MyScope.”
https://myscope.training/legacy/tem/practice/ (accessed Jun. 07, 2020).
[46] H. L. Fraser, D. W. McComb, and R. E. A. Williams, “12 - Transmission Electron Microscopy
for Physical Metallurgists,” in Physical Metallurgy (Fifth Edition), D. E. Laughlin and K.
Hono, Eds. Oxford: Elsevier, 2014, pp. 1143–1226.

159
[47] P. M. Jones, G. M. Rackham, J. W. Steeds, and F. C. Frank, “Higher order Laue zone effects
in electron diffraction and their use in lattice parameter determination,” Proc. R. Soc. Lond.
Math. Phys. Sci., vol. 354, no. 1677, pp. 197–222, May 1977, doi: 10.1098/rspa.1977.0064.
[48] R. Ermagan, M. Sauzay, and M. E. Kassner, “Assessment of Internal Stresses Using
Dislocation Dipole Heights in Cyclically Deformed [001] Copper Single Crystals,” Metals,
vol. 10, no. 4, p. 512, Apr. 2020, doi: 10.3390/met10040512.
[49] D. H. Zhang, “Electron Diffraction Patterns,” p. 35.
[50] M. E. Kassner, H. J. McQueen, J. Pollard, E. Evangelista, and E. Cerri, “Restoration
mechanisms in large-strain deformation of high purity aluminum at ambient temperature,” Scr.
Metall. Mater., vol. 31, no. 10, pp. 1331–1336, Nov. 1994, doi: 10.1016/0956-
716X(94)90113-9.
[51] M. E. Kassner, J. Pollard, E. Evangelista, and E. Cerri, “Restoration mechanisms in large-
strain deformation of high purity aluminum at ambient temperature and the determination of
the existence of ‘steady-state,’” Acta Metall. Mater., vol. 42, no. 9, pp. 3223–3230, Sep. 1994,
doi: 10.1016/0956-7151(94)90420-0.
[52] R. K. Ham, “The determination of dislocation densities in thin films,” Philos. Mag. J. Theor.
Exp. Appl. Phys., vol. 6, no. 69, pp. 1183–1184, Sep. 1961, doi: 10.1080/14786436108239679.
[53] A. S. Cheng and C. Laird, “Mechanisms of fatigue hardening in copper single crystals: The
effects of strain amplitude and orientation,” Mater. Sci. Eng., vol. 51, no. 1, pp. 111–121, Nov.
1981, doi: 10.1016/0025-5416(81)90112-9.
[54] Z. Wang, B. Gong, and Z. Wang, “Cyclic deformation behavior and dislocation structures of
[001] copper single crystals—II. Characteristics of dislocation structures,” Acta Mater., vol.
45, no. 4, pp. 1379–1391, Apr. 1997, doi: 10.1016/S1359-6454(96)00289-3.

160
[55] K. Mecke and C. Blochwitz, “Saturation Dislocation Structures in Cyclically Deformed Nickel
Single Crystals of Different Orientations,” Cryst. Res. Technol., vol. 17, no. 6, pp. 743–758,
1982, doi: 10.1002/crat.2170170610.
[56] M. Videm and N. Ryum, “Cyclic deformation of [001] aluminium single crystals,” Mater. Sci.
Eng. A, vol. 219, no. 1, pp. 1–10, Nov. 1996, doi: 10.1016/S0921-5093(96)10261-6.
[57] R. O. Scattergood and D. J. Bacon, “The Orowan mechanism in anisotropic crystals,” Philos.
Mag. J. Theor. Exp. Appl. Phys., vol. 31, no. 1, pp. 179–198, Jan. 1975, doi:
10.1080/14786437508229295.
[58] P. Veyssière and Y.-L. Chiu, “Equilibrium and passing properties of dislocation dipoles,”
Philos. Mag., vol. 87, no. 23, pp. 3351–3372, Aug. 2007, doi: 10.1080/14786430601021678.
[59] B. Fultz and J. M. Howe, Transmission Electron Microscopy and Diffractometry of Materials.
Springer Science & Business Media, 2012.
[60] M. Sauzay and L. Dupuy, Internal Stresses in Deformation-Induced Microstructures; XXIV
ICTAM: Montreal, QC, Canada, p. 2, 2016.
[61] P. D. Neumann, “The interactions between dislocations and dislocation dipoles,” Acta Metall.,
vol. 19, no. 11, pp. 1233–1241, Nov. 1971, doi: 10.1016/0001-6160(71)90057-5.
[62] A. S. Argon and C. S. Hartley, “Constitutive Equations in Plasticity,” J. Appl. Mech., vol. 44,
no. 4, pp. 801–801, Dec. 1977, doi: 10.1115/1.3424203.
[63] L. M. Brown †, “A discussion of the structure and behaviour of dipole walls in cyclic
plasticity,” Philos. Mag., vol. 84, no. 24, pp. 2501–2520, Aug. 2004, doi:
10.1080/14786430410001699791.
[64] A. Seeger, “CXXXII. The generation of lattice defects by moving dislocations, and its
application to the temperature dependence of the flow-stress of F.C.C. crystals,” Lond. Edinb.

161
Dublin Philos. Mag. J. Sci., vol. 46, no. 382, pp. 1194–1217, Nov. 1955, doi:
10.1080/14786441108520632.
[65] P. Haasen, Physical Metallurgy. London, UK: Cambridge Press, 1978.
[66] L. M. Brown, “A dipole model for the cross-slip of screw dislocations in fee metals,” Philos.
Mag. A, vol. 82, no. 9, pp. 1691–1711, Jun. 2002, doi: 10.1080/01418610208235684.
[67] C. T. Forwood and P. Humble, “Dislocation Dipoles in Elastically Anisotropic Crystals,” Aust.
J. Phys., vol. 23, no. 5, pp. 697–718, 1970, doi: 10.1071/ph700697.
[68] E. Wintner and H. P. Karnthaler, “The study of dipoles in Cu—Al alloys using weak-beam
electron microscopy,” Philos. Mag. J. Theor. Exp. Appl. Phys., vol. 36, no. 6, pp. 1317–1329,
Dec. 1977, doi: 10.1080/14786437708238519.
[69] D. R. Diercks, “Measurement of Lattice Strain and Relaxation Effects in Strained Silicon
Using X-ray Diffraction and Convergent Beam Electron Diffraction,” p. 163.
[70] S. Krämer and J. Mayer, “Using the Hough transform for HOLZ line identification in
convergent beam electron diffraction,” J. Microsc., vol. 194, no. 1, pp. 2–11, Apr. 1999.
[71] S. J. Rozeveld and J. M. Howe, “Determination of multiple lattice parameters from
convergent-beam electron diffraction patterns,” Ultramicroscopy, vol. 50, no. 1, pp. 41–56,
May 1993, doi: 10.1016/0304-3991(93)90089-G.
[72] A. D. Herron, S. P. Coleman, K. Q. Dang, D. E. Spearot, and E. R. Homer, “Simulation of
kinematic Kikuchi diffraction patterns from atomistic structures,” MethodsX, vol. 5, pp. 1187–
1203, Jan. 2018, doi: 10.1016/j.mex.2018.09.001.
[73] J. M. Zuo, “Automated lattice parameter measurement from HOLZ lines and their use for the
measurement of oxygen content in YBa2Cu3O7-δ from nanometer-sized region,”

162
Ultramicroscopy, vol. 41, no. 1, pp. 211–223, Apr. 1992, doi: 10.1016/0304-3991(92)90110-
6.
[74] P. A. Stadelmann, “EMS - a software package for electron diffraction analysis and HREM
image simulation in materials science,” Ultramicroscopy, vol. 21, no. 2, pp. 131–145, Jan.
1987, doi: 10.1016/0304-3991(87)90080-5.
[75] A. Chuvilin and U. Kaiser, “On the peculiarities of CBED pattern formation revealed by
multislice simulation,” Ultramicroscopy, vol. 104, no. 1, pp. 73–82, Aug. 2005, doi:
10.1016/j.ultramic.2005.03.003.
[76] S. Singh, F. Ram, and M. D. Graef, “EMsoft: open source software for electron
diffraction/image simulations,” Microsc. Microanal., vol. 23, no. S1, pp. 212–213, Jul. 2017,
doi: 10.1017/S143192761700174X.
[77] M. M. J. Treacy, J. M. Gibson, and A. Howie, “On elastic relaxation and long wavelength
microstructures in spinodally decomposed InxGa1−x.AsyP1−y epitaxial layers,” Philos. Mag.
A, vol. 51, no. 3, pp. 389–417, Mar. 1985, doi: 10.1080/01418618508237563.
[78] S. Straub, W. Blum, H. J. Maier, T. Ungar, A. Borbély, and H. Renner, “Long-range internal
stresses in cell and subgrain structures of copper during deformation at constant stress,” Acta
Mater., vol. 44, no. 11, pp. 4337–4350, Nov. 1996, doi: 10.1016/1359-6454(96)00104-8.
[79] H. J. Maier, H. Renner, and H. Mughrabi, “Local lattice parameter measurements in cyclically
deformed copper by convergent-beam electron diffraction,” Ultramicroscopy, vol. 51, no. 1,
pp. 136–145, Jun. 1993, doi: 10.1016/0304-3991(93)90142-K.
[80] M. E. Kassner, M. T. Pérez-Prado, M. Long, and K. S. Vecchio, “Dislocation microstructure
and internal-stress measurements by convergent-beam electron diffraction on creep-deformed

163
Cu and Al,” Metall. Mater. Trans. A, vol. 33, no. 2, pp. 311–317, Feb. 2002, doi:
10.1007/s11661-002-0092-7.
[81] L.-M. Peng, G. Ren, S. L. Dudarev, and M. J. Whelan, “Debye–Waller Factors and Absorptive
Scattering Factors of Elemental Crystals,” Acta Crystallogr. A, vol. 52, no. 3, pp. 456–470,
May 1996, doi: 10.1107/S010876739600089X.
[82] M. E. Kassner, N. Q. Nguyen, G. A. Henshall, and H. J. McQueen, “The effects of temperature
and strain rate on extended ductility of aluminum,” Mater. Sci. Eng. A, vol. 132, pp. 97–105,
Feb. 1991, doi: 10.1016/0921-5093(91)90366-U.
[83] H. J. McQueen and M.M. Myshlyaev, O.N. Senkov and V.A. Likhachev, Regularities of
Mechanical Behavior and Evolution of Structure of Aluminum and its Alloys Under
Superplasticity in Strength of Metals and Alloys. Oxford: Pergamon, 1985.
[84] V.A. Likachev, M.M. Myshlyaev and O.N. Senkov, Laws of Superplastic Behavior of Al in
Torsion. Chernogolovka, Russia: Inst. of Solid State Physics, 1981.
[85] V.A. Likhachev, M.M. Myshlyaev, O. Senkov and S.P. Belyayev, “Creep of aluminum in
torsion under superplasticity conditions,” Phys. Met. Metall, vol. 52, pp. 156–164, 1981.
[86] S.P. Belyayev, V.A. Likhachev, M.M. Myshlyaev and O. N. Senkov, “Dynamic
recrystallization of aluminum,” Phys Met Met., vol. 52, pp. 143–152, 1981.
[87] C. Perdrix, P. C, P. My, And M. F, “Comportement Mecanique Et Evolution Structurale De
L’aluminium Au Cours D’une Deformation A Chaud De Grande Amplitude,” Comport. Mec.
Evol. Struct. Alum. AU COURS UNE Deform. CHAUD Gd. AMPLITUDE, 1981.
[88] M. R. Barnett and F. Montheillet, “The generation of new high-angle boundaries in aluminium
during hot torsion,” Acta Mater., vol. 50, no. 9, pp. 2285–2296, May 2002, doi:
10.1016/S1359-6454(02)00048-4.

164
[89] S. Gourdet and F. Montheillet, “An experimental study of the recrystallization mechanism
during hot deformation of aluminium,” Mater. Sci. Eng. A, vol. 283, no. 1, pp. 274–288, May
2000, doi: 10.1016/S0921-5093(00)00733-4.
[90] H. J. McQueen and W. Blum, “Dynamic recovery: sufficient mechanism in the hot
deformation of Al (<99.99),” Mater. Sci. Eng. A, vol. 290, no. 1, pp. 95–107, Oct. 2000, doi:
10.1016/S0921-5093(00)00933-3.
[91] H. J. McQueen, W. Blum, S. Straub, and M. E. Kassner, “Dynamic grain growth: A restoration
mechanism in 99. 999 Al,” Scr. Metall. Mater. U. S., vol. 28:10, May 1993, doi: 10.1016/0956-
716X(93)90472-5.
[92] H. J. McQueen, J. K. Solberg, N. Ryum, and E. Nes, “Evolution of flow stress in aluminium
during ultra-high straining at elevated temperatures. Part II,” Philos. Mag. A, vol. 60, no. 4,
pp. 473–485, Oct. 1989, doi: 10.1080/01418618908213873.
[93] H.J. McQueen, E. Evangelista and M.E. Kassner, “The classification and determination of
restoration mechanisms in the hot working of Al alloys (Journal Article) | ETDEWEB,” Z.
Fuer Met., vol. 82, pp. 336–345, 1991.
[94] T. Pettersen and E. Nes, “On the origin of strain softening during deformation of aluminum in
torsion to large strains,” Metall. Mater. Trans. A, vol. 34, no. 12, pp. 2727–2736, Dec. 2003,
doi: 10.1007/s11661-003-0174-1.
[95] M. E. Kassner, “Large-strain deformation of aluminum single crystals at elevated temperature
as a test of the geometric-dynamic-recrystallization concept,” Metall. Trans. A, vol. 20, no. 10,
pp. 2182–2185, Oct. 1989, doi: 10.1007/BF02650307.

165
[96] M. E. Kassner, M. M. Myshlyaev, and H. J. McQueen, “Large-strain torsional deformation in
aluminum at elevated temperatures,” Mater. Sci. Eng. A, vol. 108, pp. 45–61, Feb. 1989, doi:
10.1016/0921-5093(89)90405-X.
[97] M. E. Kassner, M. Z. Wang, M.-T. Perez-Prado, and S. Alhajeri, “Large-strain softening of
aluminum in shear at elevated temperature,” Metall. Mater. Trans. A, vol. 33, no. 10, pp. 3145–
3153, Oct. 2002, doi: 10.1007/s11661-002-0300-5.
[98] M. E. Kassner and M. E. McMahon, “The dislocation microstructure of aluminum,” Metall.
Mater. Trans. A, vol. 18, no. 5, pp. 835–846, May 1987, doi: 10.1007/BF02646925.
[99] C. G. Schmidt, C. M. Young, B. Walser, R. H. Klundt, and O. D. Sherby, “The influence of
substructure on the elevated and room temperature strength of a 26 Cr-1 Mo ferritic stainless
steel,” Metall. Trans. A, vol. 13, no. 3, pp. 447–456, Mar. 1982, doi: 10.1007/BF02643353.
[100] R. D. Doherty et al., “Current issues in recrystallization: A review,” Mater. Today, vol. 1, no.
2, pp. 14–15, Jan. 1998, doi: 10.1016/S1369-7021(98)80046-1.
[101] S. Benson, “Grain size, Part II: How metal grain size affects a bending operation“,
thefabricator.com, Oct 2019.
[102] K. Huang, R.E. Logé, “A review of dynamic recrystallization phenomena in metallic
materials”, Materials & Design, vol 111, 2016, pp. 548-574.
[103] G.A. Henshall, M.E. Kassner, H.J. McQueen, “Dynamic restoration mechanisms in Al-5.8 At.
Pct Mg deformed to large strains in the solute drag regime ”Metall. Trans. A, 23, 1992 ,
pp. 881-889.
[104] A. Gholinia, F.J. Humphreys, P.B. Prangnell, “Production of ultra-fine grain microstructures
in Al-Mg alloys by coventional rolling ” Acta Mater., 50, 2002, pp. 4461-4476.

166
[105] B.D. Cullity, “Elements of X-Ray Diffraction” , 2nd ed. Reading, Mass.: Addison-Wesley,
1978.
[106] Hecker, S. S. & Stout, M. G., Strain hardening of heavily cold worked metals. In Deformation,
Processing and Structure, ed. G. Krauss. ASM, St Louis, MO, 1983, pp. 1-46.
[107] Jürgen Hirsch, Evgenij Aryshesnkij , Sergey Konovalov, “Slip system selection and Taylor
factor evolution in FCC metals“, Acta Materialia, Jun 2020.
[108] J.F.W. Bishop and R. Hill, “A Theory of the Plastic Distortion of a Polycrystalline Aggregate
Under Combined Stresses" Phil. Mag., Vol 42, 1951, 414-427.
[109] M. E. Kassner, Fundamentals of Creep in Metals and Alloys. Butterworth-Heinemann, 2015.
[110] M. E. Kassner and K. Kyle, “TAYLOR HARDENING IN FIVE POWER LAW CREEP OF
METALS AND CLASS M ALLOYS,” in Nano and Microstructural Design of Advanced
Materials, M. A. Meyers, R. O. Ritchie, and M. Sarikaya, Eds. Oxford: Elsevier Science Ltd,
2003, pp. 255–271.
[111] U. F. Kocks, C. N. Tomé, and H.-R. Wenk, Texture and Anisotropy: Preferred Orientations
in Polycrystals and Their Effect on Materials Properties. Cambridge University Press, 1998.
[112] S. C. Shrivastava, J. J. Jonas, and G. Canova, “Equivalent strain in large deformation torsion
testing : Theoretical and practical considerations,” J. Mech. Phys. Solids, vol. 30, no. 1, pp.
75–90, Apr. 1982, doi: 10.1016/0022-5096(82)90014-X.
[113] M. E. Kassner, C. S. Campbell, and R. Ermagan, “Large-Strain Softening of Aluminum in
Shear at Elevated Temperature: Influence of Dislocation Climb,” Metall. Mater. Trans. A, vol.
48, no. 9, pp. 3971–3974, Sep. 2017, doi: 10.1007/s11661-017-4166-y.
[114] M. E. Kassner and R. Ermagan, “Strain Softening of Aluminum in Shear at Elevated
Temperature,” Materials Science Forum, 2018. /MSF.941.1490 (accessed Jun. 10, 2020).

167
[115] M. E. Kassner and R. Ermagan, “Understanding Large-Strain Softening of Aluminum in Shear
at Elevated Temperatures,” in Light Metals 2018, Cham, 2018, pp. 341–345, doi: 10.1007/978-
3-319-72284-9_46.
[116] J. P. Poirier, “Is power-law creep diffusion-controlled?,” Acta Metall., vol. 26, no. 4, pp. 629–
637, Apr. 1978, doi: 10.1016/0001-6160(78)90115-3.
[117] G.Y. Chin and W.L. Mammel, “Computer solutions of the Taylor analysis for axisymmetric
flow,” Trans AIME, vol. 239, pp. 1400–1405, 1967.
[118] K. Son, M. E. Kassner, and K. A. Lee, “The Creep Behavior of Additively Manufactured
Inconel 625,” Adv. Eng. Mater., vol. 22, no. 1, p. 1900543, Jan. 2020.
[119] K. T. Son et al., “The creep and fracture properties of additively manufactured inconel 625,”
Materialia, vol. 15, p. 101021, Mar. 2021.
[120] S. Suri, G.B. Viswanathan, T. Neeraj, D.H. Hou, M.J. Mills, “Room temperature deformation
and mechanisms of slip transmission in oriented single-colony crystals of an α/β titanium
alloy”, Acta Mater; 47, 1999, pp. 1019-1034.
[121] M.E. Kassner, R.S. Rosen, G.A. Henshall, “Delayed mechanical failure of silver-interlayer
diffusion bonds“, Metall Trans A, 21A, 1990, pp. 3085–3100
[122] Z.S. Basinski, “Thermally activated glide in face-centred cubic metals and its application to the
theory of strain hardening“ Phil Mag, 4, 1959, pp. 393-432.
[123] M.E. Kassner, M.-T. Perez Prado, “Five-power-law creep in single phase metalsand alloys”,
Progress in Materials Science, 45, 2000, pp. 1-102
[124] ASTM International. B670-07(2018) Standard Specification for Precipitation-Hardening
Nickel Alloy (UNS N07718) Plate, Sheet, and Strip for High-Temperature Service. West
Conshohocken, PA; ASTM International, 2018. doi: 10.1520/B0670-07R18

168
[125] H. L. Eiseistein, "Metallurgy of a Columbium - Hardened Nickel - Chromium - Iron Alloy",
Advances in the Technology of Stainless Steels and Related Alloys, ASTM STP 369, 1965,
pp .62-79.
[126] R. C. Hall, "The Metallurgy of Alloy 7I8", Journal of Basic Engineering, September I967.
[127] L. L. Parimi, G.A. Ravi, D. Clark, M. M. Attallah, “Microstructural and texture
development in direct laser fabricated IN718”, Materials Characterization, Vol 89, 2014,
pp. 102-111.
[128] D. F. Paulonis, “Precipitation in Nickel-Base Alloy 718”, Trans. ASM, Vol 62, 1969, pp.
611-622.
[129] D.D. Keiser, H.L. Brown, “Review of the physical metallurgy of Alloy 718”. United States,
1976. doi:10.2172/4016087.
[130] D. F. Paulonis, J. M. Oblak, D. S. Duvall, “Precipitation In Nickel-Base Alloy 718”, ASM
(Amer. Soc. Metals), 62, 1969, pp. 611-22.
[131] A. Mostafa, I.R. Picazo, V. Brailovski, M. Jahazi, M. Medraj, “Structure, Texture and Phases
in 3D Printed IN718 Alloy Subjected to Homogenization and HIP Treatments” Metals,
2017; 7(6):196. https://doi.org/10.3390/met7060196
[132] M. C. Chaturvedi, Y.F. Han, “Strengthening mechanisms in Inconel 718 superalloy”, Metal
Science, 17:3, 1983, pp. 145-149, DOI: 10.1179/030634583790421032
[133] R. G. Thompson, J. R. Dobbs, D. E. Mayo, “The Effect of Heat Treatment on Microfissuring
in Alloy 718”, Welding Research Supplement, 1986, pp. 299-304
[134] ASTM F2792-12a, Standard Terminology for Additive Manufacturing
Technologies
,
(Withdrawn 2015), ASTM International, West Conshohocken, PA, 2012

169
[135] C. Qiu, N.J.E. Adkins, M.M. Attallah, Microstructure and tensile properties of selectively
laser-melted and of HIPed laser-melted Ti–6Al–4V, Mater. Sci. Eng. 578A (2013) 230- 239
[136] E.J. Bae, J.H. Kim, W.C. Kim, H.Y. Kim, Bond and fracture strength of metal-ceramic
restorations formed by selective laser sintering, J. Adv. Prosthodont. 6 (2014) 266-271. 68.
[137] R. Hague, Rapid manufacturing: an industrial revolution for the digital age, first ed., John
Wiely and Son, New York, USA (2006). 69.
[138] J.P Kruth, M.C. Leu, T. Nakagawa, Progress in Additive Manufacturing and Rapid
Prototyping, CIRP Ann. Manuf. Technol. 47 (1998) 525-540.
[139] ASTM E8 – 16a Standard Test Methods for Tension Testing of Metallic Materials, DOI:
10.1520/E0008_E0008M-16A
[140] M. Z. Khan, H. Saleem , A. M. Aleemullah , F A Khalid , S. A. A. Naqvi, T.H. Lim, R.K.
Song, “Design and Fabrication of High Temperature Creep Testing Machine”, American
Journal of Materials Engineering and Technology, Vol. 3, No. 3, 2015, pp. 51-57.
[141] Radio Technical Commission for Aeronautics (RTCA), DO-160G section 4: Temperature
and Altitude: SC-135 (2010) http://my.rtca.org.
[142] E. da C. P. Batalha, “Aircraft Engines Maintenance Costs and Reliability: An Appraisal of
the Decision Process to Remove an Engine for a Shop Visit Aiming at Minimum
Maintenance Unit Cost,” Master’s Thesis, Universidade Nova de Lisboa, 2012..
[143] “INCONEL ® alloy 625.”
[144] S. B. Lattime and B. M. Steinetz, “Turbine Engine Clearance Control Systems: Current
Practices and Future Directions,” NASA/TM-2022-211794, 2002.
[145] K.Son, “High Temperature Creep Behaviors of Additively Manufactured IN625”,
University of Southern California, May 2021.

170
[146] M.E. Kassner, “Fundamentals of Creep in Metals and Alloys - 3rd Edition.” Elsevier Ltd.,
Amsterdam, 2015.
[147] T. Ni and J. Dong, “Creep behaviors and mechanisms of Inconel718 and Allvac718plus,”
Mater. Sci. Eng. A, vol. 700, pp. 406–415, Jul. 2017.
[148] Y. Han and M. C. Chaturvedi, “Steady state creep deformation of superalloy inconel 718,”
Mater. Sci. Eng., vol. 89, no. C, pp. 25–33, May 1987.
[149] W. Chen and M. C. Chaturvedi, “Grain boundary dependent creep behaviour of inconel
718,” Can. Metall. Q., vol. 32, no. 4, pp. 363–367, Jan. 1993.
[150] R. Rodriguez, R. W. Hayes, P. B. Berbon, and E. J. Lavernia, “Tensile and creep behavior
of cryomilled Inco 625,” Acta Mater., vol. 51, no. 4, pp. 911–929, Feb. 2003.
[151] A. Kreitcberg, V. Brailovski, and S. Turenne, “Effect of heat treatment and hot isostatic
pressing on the microstructure and mechanical properties of Inconel 625 alloy processed by
laser powder bed fusion,” Mater. Sci. Eng. A, vol. 689, pp. 1–10, Mar. 2017.
[152] J. P. Beckman and D. A. Woodford, “intergranular sulfur attack in nickel and nickel-base
alloys.” Superalloys, TMS, Warrendale, pp. 795–804, 1988.
[153] J. A. Pfaendtner and J. J. McMahon, “Oxygen-induced intergranular cracking of a Ni-base
alloy at elevated temperatures - An example of dynamic embrittlement,” Acta Mater., vol.
49, no. 16, pp. 3369–3377, Sep. 2001.
[154] S. Park, K. Kim, M. Kim, M. E. Kassner, and K. Lee, “Effect of Post‐heat Treatment on the
Tensile and Cryogenic Impact Toughness Properties of Inconel 718 Manufactured by
Selective Laser Melting,” Adv. Eng. Mater., vol. 23, no. 3, p. 2001005, Mar. 2021.
[155] K.A. Lee, unpublished research, Inha Univ, Incheon, South Korea, 2021.
[156] K. S. Kim, T. H. Kang, M. E. Kassner, K. T. Son, and K. A. Lee, “High-temperature tensile

171
and high cycle fatigue properties of inconel 625 alloy manufactured by laser powder bed
fusion,” Addit. Manuf., vol. 35, Oct. 2020.
[157] L. Fournier, D. Delafosse, T. Magnin, “Oxidation induced intergranular cracking and
Portevin-Le Chatelier effect in nickel base superalloy 718.” Materials Science and
Engineering A316 pp. 166–173, 2001.
[158] C. F. Miller, G. W. Simmons, and R. P. Wei, “High temperature oxidation of Nb, NbC and
Ni3Nb and oxygen enhanced crack growth,” Scr. Mater., vol. 42, no. 3, pp. 227–232, Jan.
2000.
[159] A. De Vasconcelos Varela, H. D. De Deus, M. C. De Siqueira, M. C. Rezende, and L. H.
De Almeida, “Oxidation assisted intergranular cracking in 718 Nickel Superalloy: On the
mechanism of dynamic embrittlement,” J. Mater. Res. Technol., vol. 7, no. 3, pp. 319–325,
Jul. 2018.

[160] Kassner, M.E. Fundamentals of Creep in Metals and Alloys, 3rd ed.; Elsevier: Amsterdam,
The Netherlands, 2015; pp. 1–338.
[161] Kassner, M.E. A case for Taylor hardening during primary and steady-state creep in
aluminum and type 304 stainless steel. J. Mater. Sci. 1990, 25, 1997–2003.
[162] Straub, S.; Blum, W. Does the “Natural” Third Power Law of Steady State Creep Hold for
Pure Aluminium. Scr. Metall. Mater. 1990, 24, 1837–1842.
[163] Blum, W. Hot Deformation of Alloys; Langdon, T.G., Merchant, H.D., Morris, J.G., Zaidi,
M.A., Eds.; TMS: Warrendale, PA, USA; 1991; pp. 181–209.
[164] Kassner, M.E. Taylor hardening in five-power-law creep of metals and Class M alloys.
Acta Mater. 2004, 52, 1–9.

172
[165] M. E. Kassner, R. Ermagan, “Power Law Breakdown in the Creep in Single Phase Metals”,
Metals, 9(12), 2019, pp. 1345
[166] Luthy, H.; Miller, A.K.; Sherby, O.D. The stress and temperature dependence of steady-
state flow at intermediate temperatures for pure polycrystalline aluminum. Acta Metall.
1980, 28, 169–178.
[167] Kassner, M.E.; McQueen, H.J.; Pollard, J.; Evangelista, E.; Cerri, E. Restoration
mechanisms in large-strain deformation of high purity aluminum at ambient temperature.
Scr. Metall. Mater. 1994, 31, 1331–1336.
[168] Landon, P.R.; Lytton, J.L.; Shepard, L.A.; Dorn, J.E. The Activation Energies for Creep of
Polycrystalline Copper and Nickel. Trans. ASM 1959, 51, 900–910.
[169] Kassner, M.E. The rate dependence and microstructure of high-purity silver deformed to
large strains between 0.16 and 0.30T m. Metall. Trans. A 1989, 20, 2001–2010.
[170] Price, C.E. On the creep behavior of silver—I. The oxygen free metal. Acta Metall. 1966,
14, 1781–1786.
[171] Robinson, S.L.; Sherby, O.D. Mechanical Behavior of Polycrystalline Tungsten at
Elevated Temperature. Acta Metall. 1969, 17, 109–125.
[172] Mecking, H.; Estrin, Y. The effect of vacancy generation on plastic deformation. Scr.
Metall. 1980, 14, 15–19.
[173] Ohashi, T. Generation and accumulation of atomic vacancies due to dislocation movement
and pair annihilation. Philos. Mag. 2019, 98, 2275–2295.
[174] Ungar, T.; Schafler, E.; Hanak, P.; Bernstorff, S.; Zehetbauer, M. Vacancy Production
during Plastic Deformation in Copper Determined by in situ X-ray Diffraction. Mater. Sci.
Eng. A 2007, 462, 398–401.

173
[175] Cizek, M.M.; Janecek, M.; Vlasak, T.; Smola, B.; Melikhova, O.; Islamgaliev, R.K.;
Bobatkin, S.V. The Development of Vacancies during Severe Plastic Deformation. Mater.
Trans. 2019, 60, 1533–1542.
[176] Wu, M.Y.; Sherby, O.D. Unification of Harper-Dorn and Power Law Creep through
consideration of Internal Stress. Acta Metall. 1984, 32, 1561–1572.
[177] Arieli, A.; Mukherjee, A.K. Creep and Fracture of Engineering Materials and Structures;
Wilshire, B., Owen, D.R.J., Eds.; Pineridge Press, Swansea, UK, 1981; pp. 97–111.
[178] Weertman, J.; Weertman, J.R. Physical Metallurgy, 3rd ed.; Cahn, R.W., Haasen, P., Eds.;
North Holland: Amsterdam, The Netherlands, 1983; pp. 159–1307.
[179] Nix, W.D.; Ilschner, B. Strength of Metals and Alloys; Haasen, P., Gerold, V., Kostorz, G.,
Eds.; Pergamon, Aachen, Germany, 1980; pp. 1503–1530.
[180] Kassner, M.E.; Pérez-Prado, M.T.; Long, M.; Vecchio, K.S. Dislocation Microstructures
and Internal Stress Measurements by CBED on Creep Deformed Cu and Al. Metall. Mater.
Trans. 2002, 33, 311–317.
[181] Levine, L.E.; Stoudt, M.R.; Creuziger, A.; Phan, T.Q.; Xu, R.; Kassner, M.E. Basis for the
Bauschinger Effect in Copper Single Crystals: Changes in the Long-Range Internal Stress
with Reverse Deformation. J. Mater. Sci. 2019, 54, 6579–6585.
[182] Phan, T.Q.; Levine, L.E.; Lee, I.F.; Xu, R.; Tischler, J.Z.; Huang, Y.; Langdon, T.G.;
Kassner, M.E. Synchrotron X-Ray Microbeam Diffraction Measurements of Full Elastic
Long Range Internal Strain and Stress Tensors in Commercial-Purity Aluminum Processed
by Multiple Passes of Equal-Channel Angular Pressing. Acta Mater. 2016, 5, 231–241.
[183] {alpha}-zirconium at elevated temperatures. Acta Mater. 2004, 53, 581–591.

174
[184] Kassner, M.E Additional Data and Insight bearing on the relation between the dislocation
substructure and power law breakdown. Scr. Metall. 1982, 14, 265–266.
[185] Conway, J.B.; Stentz, R.H.; Berling, J.R. Fatigue, Tensile and Relaxation Behavior of
Stainless Steels; Reports TID-26135 United States; Atomic Energy Commission:
Washington, DC, USA, 1975; p. 194.
[186] Trozera, T.A.; Sherby, O.D.; Dorn, J.E. Effect of strain-rate and temperature on the plastic
deformation of high purity aluminum. Trans. ASM 1955, 49, 173–185.
[187] Shewmon, P.G. Diffusion in Solids; Bernstorff, S., Zehetbauer, M., Eds.; McGraw-Hill:
New York, NY, USA, 1963; p. 66.
[188] Takamura, J.I.; Takahashi, I.; Amano, M. Lattice Defects in Deformed Low-carbon Steels
and the Annealing Stage. Trans. Iron Steel Inst. Jpn. 1969, 9, 216–221.

175

Appendices
As stated in the 1.4.1.5.1 section, the EMSoft simulation codes need an Xtal file that contains the
crystal structure information including the crystal system, space group number, atomic number,
Debye-Waller factor (DWF), and of course the lattice parameter. The Bethe parameters, often
referred to as the stopping power of material in front of charged particles, will be considered in a
different set of codes before running the final simulations. It is the Bethe parameters that take the
dynamical behavior effect into account. These characteristics are listed in Table A.1 for copper.
The EMSoft codes are also shown for reference [76].
Table A.1. Crystal structure information used for the CBED simulations for copper.
Characteristic
Crystal System Space Group Number Atomic Number DWF (nm
2
)

[81]
Copper FCC 225 29 0.5505

176
Appendix A: Bethe Parameter Codes

&Bethelist
! Do not change the line above !
! Details about the particular implementation of the Bethe potentials can be
! found in the recent paper: A. Wang & M. De Graef, "Modeling dynamical electron
! scattering with Bethe potentials and the scattering matrix" Ultramicroscopy,
! vol 160, pp. 35-43 (2016)
! strong beam cutoff
c1 = 20.0,
! weak beam cutoff
c2 = 40.0,
! complete cutoff
c3 = 500.0,
! double diffraction reflections: maximum excitation error to include [nm^{-1}]
sgdbdiff = 1.00
/ [76]

177
Appendix B: EMSoft Codes
&LACBEDlist
! do not change the first line in this file !
!
! Note: All parameter values are default values for the program
!
!--------------
! the name of the crystal input file MUST be specified (full pathname)
xtalname = 'cu3589.xtal',
! microscope accelerating voltage [kV]
voltage = 200.,
! zone axis indices
k = 4 1 1,
! foil normal indices
fn = 0 1 0,
! minimum d-spacing to be taken into account [nm]
dmin = 0.115,

178
! maximum HOLZ layer number for the output file (this does not affect the number for the actual
computations)
maxHOLZ = 2,
! output will be diffraction disks inside a square of area (2*npix+1)^2
npix = 400,
! number of thickness values to be used in the output
numthick = 10,
! starting thickness value [nm]
startthick = 130.0,
! thickness increment value [nm]
thickinc = 20.0,
! min intensity to consider for output file
minten = 1.0E-6,
! name of output HDF5 file (relative to EMdatapathname)
outname = 'copper 3589.h5'
! this last line must be present!
/ [76]

Abstract (if available)

Abstract Four projects were performed and reported in the present dissertation with concentration on mechanical behavior of metals. ? The first project focuses on understanding Long Range Internal Stresses (LRIS), which is critical for explaining the basis of the Bauschinger effect, spring-back in metal forming, and plastic deformation in cyclically deformed metals. Few studies have assessed LRIS in cyclically deformed single crystals in single-slip while there are no such studies in multiple-slip. Here, we report on LRIS in a cyclically deformed copper single crystal in multiple-slip via two methods: 1- Lattice parameter determination using Convergent Beam Electron Diffraction (CBED) and 2- Measuring the maximum dipole heights. TEM micrographs show a labyrinth dislocation microstructure with high dislocation density walls and low dislocation density channels. Lattice parameters and dipole heights were assessed in the channels and walls of the labyrinth structure. Lattice parameters obtained were almost identical near the walls and in the channels. The maximum dipole heights were also approximately independent of location. Thus, a homogenous stress state within the heterogeneous dislocation structure is suggested. ? The second project describes large strains of aluminum in pure shear at elevated temperature. This pronounced softening has been attributed to a variety of phenomena. The most widely accepted early explanations involve the development of a texture leading to a decrease in the average Taylor factor. That is, there is a decrease in Schmid factors in the deformed grains. Our work suggests that the texture leads to softening through an increase in the dislocation climb stress. This appears to be particularly reasonable as dislocation climb is widely regarded the rate-controlling mechanism for high temperature plasticity. ? Additive manufacturing by selective laser powder bed fusion of alloys followed by hot isostatic pressing (HIP) has gained attention by industry as it provides economical production of complicated-configuration engine parts with fewer joining steps and greater geometric freedom. The creep behavior of additively manufactured (AM) Inconel 625 and 718 at 650 and 800 ? was compared with that of the wrought alloy after 24 hour and long-term tests up to about one year. Inconel 718 is precipitation strengthened while 625 is solid-solution strengthened. It was discovered that the creep strength of the 625 and 718 produced by additively manufacturing is essentially identical than that of the wrought alloy. Complications occur, however, with a marked loss of ductility in the AM alloys. Fatigue and impact toughness were adversely affected as well. The basis for these degradations was particularly investigated in the third project. ? The last project provides insight into the basis of power-law breakdown (PLB) in the steady-state creep of metals and alloys. A variety of theories has been presented in the past but this new examination suggests that there is evidence that a dramatic supersaturation of vacancies leading to very high diffusion rates and enhanced dislocation climb is associated with the rate-controlling process for creep in PLB. The effect of vacancy supersaturation may be enhanced by dislocation short circuit diffusion paths at lower temperatures due to the dramatic increase in dislocation density.

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Asset Metadata

Creator Ermagan, Roya (author)

Core Title Long range internal stresses in cyclically deformed copper single crystals; and, Understanding large-strain softening of aluminum in shear at elevated temperatures

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Materials Science

Degree Conferral Date 2021-08

Publication Date 07/23/2021

Defense Date 05/13/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag additive manufacturing,aluminum softening,Copper,creep,dislocations,fatigue,long range internal stresses,OAI-PMH Harvest

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Kassner, Michael Ernest (committee chair), Bermejo-Moreno, Ivan (committee member), Graham, Nicholas (committee member), Grunenfelder, Lessa (committee member)

Creator Email ermagan@usc.edu,rermagan@its.jnj.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC15618745

Unique identifier UC15618745

Legacy Identifier etd-ErmaganRoy-9847

Document Type Dissertation

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Rights Ermagan, Roya

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given.

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Repository Email cisadmin@lib.usc.edu