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Mechanical behavior of materials in extreme conditions: a focus on creep plasticity
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Mechanical behavior of materials in extreme conditions: a focus on creep plasticity
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Content
i
MECHANICAL BEHAVIOR OF MATERIALS
IN EXTREME CONDITIONS:
A Focus on Creep Plasticity
By
Kamia K. Smith
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATERIALS SCIENCE ENGINEERING)
December 2017
Copyright 2017 Kamia K. Smith
ii
DEDICATION
Whole-heartedly, and gratefully so,
I dedicate this work to my parents,
Emma and Everette
iii
ACKNOWLEDGMENTS
Graciously, I want to recognize and thank my parents, to whom this dissertation is dedicated. I am
so blessed to have parents who are, unconditionally, always in my corner with full support,
confidence and love.
It gives me great pleasure to acknowledge my research advisor, Dr. Michael Kassner. With
his guidance through the duration of my Ph.D., I have gained an exceptional amount of knowledge,
both as his student in the classroom, as well as a researcher in the lab. He, as a Professor in the
Aerospace and Mechanical Engineering department, has given me expert insight and guidance
through the duration of each project in this dissertation. The support and encouragement I received
from Dr. Kassner made the difference in my completion. I was always inspired to push the
envelope; thus, building a strong confidence in my ability as an engineer.
It also gives me great pleasure to extend my gratitude to Dr. Andrea Hodge, Professor in
Aerospace and Mechanical Engineering. I’d like to thank her for her generosity in extending her
chemical laboratory to myself and those assisting in my research. Without its access, much of the
sample preparation and chemical analysis would not have been possible. Additionally, I’d like to
thank Dr. Hodge for including me in the IRES (International Research Education for Students)
program. The program involves a group of students, graduate and undergraduate, traveling to
Karlsruhe, Germany to study at the Karlsruher Insitut für Technologie for two months. The
experience proved to be very a valuable and rewarding opportunity that enriched and increased
my knowledge in electron microscopy and provided a unique oversees opportunity.
Next, I’d like to thank my other committee members for their constructive suggestions to
this thesis. I am also extremely grateful for the time set aside to hear my defense. My three
additional committee members are Dr. Ed Goo, an Associate Professor in Materials Science; Dr.
iv
Nicholas Graham, an Assistant Professor in Chemical Engineering; and Dr. Lessa Grunenfelder, a
Lecturer in Materials Science.
Moreover, those who truly understand the rigorous nature of this program are those who
experience it with you. Thank you to my colleagues and lab members for their support, advice and
collaboration in the lab. Lastly, but surely not least, I want to lovingly thank my partner, Ebonee,
for the daily mental and emotional support that was greatly needed and appreciated during this
finishing stretch.
v
TABLE OF CONTENTS
DEDICATION ii
ACKNOWLEDGMENTS iii
TABLE OF CONTENTS v
LIST OF TABLES viii
LIST OF FIGURES x
ABSTRACT xx
CHAPTER 1. INTRODUCTION
1.1 Definition of Creep Deformation 1
1.2 Low Temperature Creep Plasticity 3
1.3 High Temperature Creep Plasticity (Harper-Dorn Theory) 6
CHAPTER 2. ROOM TEMPERATURE CREEP OF HIGH-PURITY POLYCRYSTALLINE
COPPER AND HIGH-PURITY POLYCRYSTALLINE SILVER
2.1 Literature 10
2.1.1 Titanium Alloys 10
2.1.2 Steels 11
2.1.3 Pure Metals 15
2.2 Materials 19
2.2.1 99.999% Polycrystalline Copper 19
2.2.2 99.99% Polycrystalline Silver 19
2.2.3 Vacuum Annealing 19
2.3 Microstructural Analysis of Copper and Silver Creep Samples 22
2.4 Tensile Behaviors 26
2.4.1 Polycrystalline Copper Tensile Tests 27
2.4.2 Polycrystalline Silver Tensile Tests 27
vi
2.5 Creep Machine Specifications 30
2.6 Creep Deformation Results for Copper and Silver 36
2.6.1 99.999% Copper Creep Curves 36
2.6.2 99.99% Silver Creep Curves 44
2.6.3 Unique Trend Comparison Method Using R
2
Values 49
2.7 Discussion – Phenomenological Trends 52
CHAPTER 3. LONG-TERM ANNEALING OF HIGH-PURITY SINGLE CRYSTAL
ALUMINUM: NEW INSIGHTS INTO HARPER DORN CREEP
3.1. Introduction 62
3.2. Materials 63
3.7.1 Sample Preparation 63
3.7.2 Chemical Analysis 67
3.3. Long-Term Annealing Furnaces 73
3.4. Results 76
3.4.1. Dislocation Density Measurements by Etch Pit Analysis 78
3.4.2. Dislocation Density Measurements by TEM Analysis 84
3.5. Discussion 100
CHAPTER 4. CREEP IN AMORPHOUS ALLOYS (BULK METALLIC GLASSES):
A REVIEW
4.1 Background 103
4.2 Mechanisms of Deformation 106
4.3 Homogeneous Flow at Very Low Temperatures 117
4.4 Primary and Transient Creep (Non-Steady-State Flow) 118
4.5 Summary 118
CHAPTER 5. THROUGH-THICKNESS COMPRESSION TESTING OF COMMERCIALLY
PURE (GRADE-II) TITANIUM THIN SHEET TO LARGE STRAINS
5.1 Background 120
5.2 The Present Study 130
vii
5.3 Materials 132
5.4 Sample Preparation 134
5.4.1 Machining Titanium Sheet into Individual Samples by EDM 134
5.4.2 Mechanical Polishing 137
5.4.3 Machining Titanium Samples Between Testing 140
5.4.4 Polishing Titanium Samples Between Testing 141
5.5 Mechanical Testing 144
5.5.1 Parallelism Test with Feeler Gauge 144
5.5.2 Machine Compliance 145
5.5.3 Barreling Correction 146
5.5.4 Lubrications 147
5.6 Results 151
5.7 Summary 166
CHAPTER 6. CONCLUSIONS 167
REFERENCES 169
APPENDIX. TEM ANALYSIS OF COPPER CREEP SPECIMENS 184
A.1 Electro-Polishing to Smaller Diameter TEM Disks 184
A.2 Preparation of TEM Disks 186
A.3 Jet-Polishing of thin foils for TEM Analysis 189
viii
LIST OF TABLES
TABLE 1.1. Average initial dislocation densities of aluminum in as-received and annealed
conditions. The authors of article [66] used a different Al sample for each
listed condition. PX – polycrystal, SX – single crystal. EP – etch pit, TEM –
transmission electron microscopy.
TABLE 2.1. Materials and regimes where either Logarithmic (Log) or Power-Law (PL)
description apply to low-temperature creep behavior. Note: A section with a
“--” signifies no data for those conditions were reported. The ‘Copper’ and
‘Silver’ rows are from the current research.
TABLE 2.2. Summations of values for the weights used in the creep deformation
experiments. The jagged line separates the copper (top three rows) from the
silver (bottom three rows).
TABLE 3.1. Initial thickness measurements for all aluminum samples used in the present
research.
TABLE 3.2. Initial dislocation densities averaged across samples used in this research.
TABLE 3.3. TEM dislocation density measurements for Sample S-D used in this research.
TABLE 3.4. TEM dislocation density measurements for Sample S-5 used in this research.
TABLE 3.5. TEM dislocation density measurements for Sample S-𝛾 used in this research.
TABLE 3.6. This table shows the TEM specimens’ initial disk thickness as well as their
individual preparation methods. Their final dislocation densities and final
thicknesses are listed as well.
TABLE 3.7. Dislocation densities (by etch pits) associated with each annealing time
period.
TABLE 4.1. Mechanical properties of some glassy alloys from Ref. [111, 114-134].
TABLE 4.2. Deformation data of some BMGs in the super-cooled liquid region from Ref.
[149], [150-161].
TABLE 4.3. Activation energies for creep of selected metallic glasses [139, 146, 162,
163].
TABLE 5.1. Chemical composition of the CP-II Titanium used in this study.
9
18
35
68
79
91
94
94
95
99
105
109
109
132
ix
TABLE 5.2a. Friction coefficients [185-189] of the lubricants used in the present study.
The graphite and MoS
2
aerosol lubricants were generously sprayed on the
compression platens before testing.
TABLE 5.2b. Lubricants used for each sample. In the case of the MoS
2
and Teflon
combination, the Teflon was placed on the upper and lower compression
platens first, then the MoS
2
was generously applied to the top of the Teflon.
TABLE 5.3a. Frictional Coefficients for the Different Lubrications. The Graphite and MoS
2
aerosol lubricants were sprayed on the compression platens generously
before testing. The PTFE Teflon was taped on the compression platens before
testing.
TABLE 5.3b. Lubrication(s) used for each sample. In the case of the MoS
2
and Teflon
combination, the Teflon was taped to the upper and lower compression
platens first, then the MoS
2
was generously applied on top of the paper.
TABLE 5.4. Tabulated strain-rate versus yield stress and flow stress values comparing the
present study with two literature sources previously discussed [179, 181].
136
136
149
149
165
x
LIST OF FIGURES
FIGURE 1-1. For a hypothetical material, the stress-strain curve at temperatures higher than
half its melting point is shown in (a). The creep behavior of the same material
is shown in (b). Three regimes are clearly distinguishable: Stage I, II and III
which denotes the primary, secondary (steady-state), and tertiary (fracture).
Figure taken from [2].
FIGURE 1-2. Steady-state dislocation density versus the modulus-compensated steady-
state stress at an elevated temperature of 923 K (0.99 T
m
). The data of Lin et
al. [55] and that of Barrett et al. [48] had been suggested to imply a lower
limit of the dislocation density (r).
FIGURE 2-1. Ambient-temperature creep for various Ti-alloys of different microstructures
and different compositions. From Neeraj et. al. [8] data in a log-log plot while
(b) plots the same data in a semi-log plot. Based on the R
2
values, a power-
law may better describe the creep behavior.
FIGURE 2-2. The plastic strain vs. time behavior of annealed 304 stainless steel under
different stresses from [5, 75]; (a) Linear strain vs. log time; (b) Log strain
vs. log time Based on the R
2
values, a logarithmic equation better describes
the data.
FIGURE 2-3. Cryogenic temperature (a),(b) creep behavior of pure Al based on data by
[30, 76] behavior at 77K may dominate at lower stresses. A yield stress at a
given strain-rate was not reported by either reference for their aluminum
samples.
FIGURE 2-4. Creep behavior of pure Cd, at 0.51T
m
and 0.08T
m
, based on data by [30]. At
low temperatures, logarithmic behavior (T < 0.30T
m
) may best describe the
data but power-law behavior may be better at higher temperatures. A yield
stress at a given strain-rate was not reported by Wyatt [30] for the Cadmium
data.
FIGURE 2-5. This figure shows the vacuum furnace used for the heat treatments of all
copper and silver creep specimens. The vacuum unit (on the left) reaches
pressures as low as 6.4 × 10
) *
mbar. The furnace unit (on the right) was an
auto-programmable system holding at the desired temperature for 2 hours.
FIGURE 2-6. Copper and Silver gage segments were mounted in quickset acrylic for a
microstructural study of the grain textures.
FIGURE 2-7. The EBSD scan of (a) the as-annealed 99.999% PX copper and (b) of the as-
annealed 99.99% PX silver. The average grain diameter was calculated at 310
µm for copper and 25 µm for silver.
2
8
12
13
16
17
21
24
25
xi
FIGURE 2-8. True stress vs. true strain behavior of high purity copper polycrystals used
in this research. The macroscopic yield stress is 24 MPa.
FIGURE 2-9. True stress vs. true strain behavior of high purity silver used in this research.
The macroscopic yield stress is 28 MPa.
FIGURE 2-10. The Arcweld Manufacturing Company for creep rupture testing. The setup
used is illustrated in (a) and the schematics are shown in (b).
FIGURE 2-11. A schematic of the load cell calibration is shown. The calibration of the load
cell alone with weights directly attached to it is displayed in (a) and the
schematic of how the load cell was tested in the creep machine is shown in
(b) where the weights are loaded to the machine.
FIGURE 2-12. The load cell calibration data used for the creep machine used in this study
is shown here. Three curves are plotted on a Force (in pounds, lbs) vs.
Voltage (in mV). Omega’s curve (red) is taken from the company-given
values. The curve labeled “Load Cell (alone)” was generated by the weights
directly loaded to the load cell (refer to Figure 2-11a). The curve labeled
“Load Cell (middle of the frame)” was generated by the calibration of the
load cell in Position 2 of the load frame (refer to Figure 2-11b). The best fit
lines were used to calculate the desired weight added to apply a desired load
to the sample.
FIGURE 2-13. The (a) creep deformation curve for Sample 5N-Cu-15MPa is shown along
with an (b) EBSD scan of the microstructure after creep for 266 days
performed in (a). The upper left corner shows some “noise” from the
detector.
FIGURE 2-14. Creep deformation curve for 5N-Cu-18MPa. The applied load was 18 MPa
and the total duration of the test was 5 days. The gap in data does not seem
to change the observed power-law deformation trend. At this low stress, a
power-law behavior seems to dominate. An EBSD scan was not performed
for this sample
FIGURE 2-15. Creep deformation curves for Sample 5N-Cu-20MPa shows the full
duration of data acquisition over a period of 54 days. Figure (a) shows the
sample was bumped during testing at 1 hour while taking excess tape off
the steel beams. The test was not stopped and the full deformation data is
continued and shown in (b). An EBSD scan was not performed for this
sample.
FIGURE 2-16. The (a). creep deformation curve for Sample 5N-Cu-30MPa is shown along
with an (b). EBSD scan of the microstructure after creep for 11 days
performed in (a). The scan in (b) shows some grain refinement however the
28
29
32
33
34
38
39
40
41
xii
overall microstructure seems to remain consistent to the starting
microstructure.
FIGURE 2-17. This figure displays all copper creep curves analyzed in this research. To
display the (a) linear behavior of all curves on one graph, the time scale is
“broken” into three sections. In (b), the logarithmic behavior of all copper
curves is shown. It should be noted that for 20 MPa, only the early data
(before pre-strain) is plotted on this graph.
FIGURE 2-18. The (a) creep deformation curve for Sample 4N-Ag-20MPa shows the full
duration of data acquisition over a period of 75 days. The EBSD scan shows
many twins which can most likely be attributed to the mechanical
preparation for the EBSD scan. The strain-rate finished at 2 × 10
) ,-
s
) ,
.
FIGURE 2-19. Creep deformation curve for 4N-Ag-30MPa. The applied load was 30 MPa
and the total duration of the test was 78 days. A logarithmic trend proves to
be dominate for this sample. The ending creep rate is 2×10
) ,,
s
) ,
with
strains maximized around 0.0015. An EBSD scan was not performed for
this sample.
FIGURE 2-20. Creep deformation curves for Sample 4N-Ag-40MPa shows the full
duration of data acquisition over a period of 28 days. The creep test run at
40 MPa was performed on the silver sample previously crept at 30 MPa.
FIGURE 2-21. This figure displays the linear behavior all silver creep curves analyzed in
this research. The test were performed above and below the yield stress of
the silver used in this research: 20 MPa, 30 MPa and 40 MPa with ending
creep rates equal to 2 × 10
) ,-
𝑠
) ,
, 3 × 10
) ,,
𝑠
) ,
and 2 × 10
) ,1
𝑠
) ,
,
respectively. The creep test run at 40 MPa was performed on the silver
sample previously crept at 30 MPa.
FIGURE 2-22. Here the combined R
2
-correlation analysis is shown for all copper creep
tests performed in this research. We observe a mainly power-law dominant
relationship, however for earlier times (< 30 minutes) the data is a bit
ambiguous. After a period of 30 minutes, at a stress lower than the yield
stress, a power-law dominant trend is observed, however, for higher stresses
and longer time periods, a logarithmic trend seems to be favored.
FIGURE 2-23. Here the combined R
2
-correlation analysis is shown for all silver creep tests
performed in this research. We observe a mainly logarithmic dominant
relationship for lower stresses. At higher applied loads, a power-law trend
seems to be favored. The sample loaded at the higher stress, 4N-Ag-40MPa,
was originally crept at 30 MPa, then stored at room-temperature before
being reloaded to 40 MPa.
42
45
46
47
48
50
51
xiii
FIGURE 2-24. The predicted strain vs. time plots for ambient-temperature creep of Cu
based on the dislocation intersection mechanism through Eq. (E-2.19). The
most realistic behavior is for DH
o
= 135 kJ/mol and a critical resolved shear
stress, t = 3 MPa. Power-law behavior (a) appears to better describe the
predictions of Eq. (E-2.19) than logarithmic behavior (b).
FIGURE 2-25. Activation energy for (steady-state) creep of Ag, Ni, Cu and Al as a function
of temperature. Adapted from [87, 88].
FIGURE 3-1. As-received aluminum crystal dimensions and orientation.
FIGURE 3-2. Polishing technique using SiC grit paper: a). Holding the sample in one
direction starting with 320 grit; b). Switch grit paper to next grade (e.g. 400
grit), rotate 90° and continue polishing holding in one direction; c). Previous
abrasion lines from the 320 grit are gone and only the 400 grit lines remain.
Polishing continues this way through all grades of grit papers.
FIGURE 3-3. Shown here is the electro-polishing setup for the single crystal aluminum
used in this research. A solution of 1:9 perchloric acid (70% ACS reagent)
to methanol (99.8% anhydrous), respectively, was used. The voltage was
set at 10 V leading to an approximate current of 1.6 mA. A stirring bar was
constantly running, providing smoother current flow through the solution.
FIGURE 3-4. Ramping current (A) vs. voltage (V) curve to determine the electro-
polishing conditions for single crystal aluminum at 0-5°C.
FIGURE 3-5. The results of mechanical and electro-polishing as seen under an optical
microscope; a). shows the surface of an aluminum test sample after
completing all mechanical polishing finishing with 0.25 µm diamond slurry
and b). the surface of the test sample after electro-polishing. It should be
noted that the soft-grey markings in each micrograph are from the optical
microscope lens and are not on the sample surface itself.
FIGURE 3-6. Table top furnace controlled externally by a Sigma Digital PID (three-term
regulator) MDC4E temperature control unit. This unit has a separate Type-
K thermocouple probe in the furnace. The samples again had individual
Type-K thermocouples to track the temperature readings.
FIGURE 3-7. A 3-550 Vulcan Multi-Stage Programmable Furnace was used. A ramp rate
of 20°C/min was programmed and when the temperature was reached, it
held at 646°C for as long as desired. A Type-K thermocouple was used
together with an 8-channel thermocouple data acquisition unit from Omega
Engineering. The temperature readings were recorded using a desktop
computer.
59
60
64
66
69
71
72
74
75
xiv
FIGURE 3-8. Schematics of <111> (left) and <100> (right) directions of crystal
orientation leading to pyramidal and cube, respectively, shaped etch pits
where a represents the lattice parameter and the red represents the etch pit
shapes.
FIGURE 3-9. Etch Pit micrographs taken with an optical microscope of Samples S-2 (top,
a) and S-4 (bottom, b) from Group 1 before annealing. The images were
taken after the sample surfaces had been electropolished and chemically
etched.
FIGURE 3-10. Etch Pit micrographs taken with an optical microscope of Samples S-A (top,
a) and S-C (bottom, b) from Group 2 before annealing. The images were
taken after the sample surfaces had been electropolished and chemically
etched.
FIGURE 3-11. Etch Pit micrographs taken with an optical microscope of Samples S-b (top,
a) and S-g (bottom, b) from Group 3 before annealing. The images were
taken after the sample surfaces had been electropolished and chemically
etched.
FIGURE 3-12. Etch Pit micrographs taken with an optical microscope of Sample S-6N
from Group 3 at 5X (top, a) and 10X (bottom, b) magnifications, before
annealing. The images were taken after the sample surface had been
electropolished and chemically etched.
FIGURE 3-13. Sample dimensions for S-∆. A dislocation density was taken on sides A, B
and C by etch pit analysis. The crystal orientation for each plane is notated
in the figure and was determined by EBSD.
FIGURE 3-14. Etch pit micrographs taken in SEM of (a) side A, (b) side B and (c) side C
of Sample S-∆. The dislocation densities for all sides were on the order of
10
9
m-
2
.
FIGURE 3-15. Schematic shows how a TEM sample was made from Sample S-∆. (a)
Sample S-∆ was sliced into (b) ~2-mm slabs parallel to Side C (bottom
surface of S-∆). The slabs were then ultrasonically cut into (c) 3-mm
diameter disks. Those disks were further polished to a TEM foil.
FIGURE 3-16. TEM image of S-D_TEM-1. Dislocation link lengths averaged 1 – 2µm and
the density of the disks were on the order of 10
12
lines/m
2
.
FIGURE 3-17. TEM image of S-D TEM-2. Dislocation link lengths averaged 1 µm and the
density of the disks were on the order of 10
12
lines/m
2
.
78
80
81
82
83
86
87
88
89
89
xv
FIGURE 3-18. TEM image of S-D TEM-3. This disk was mechanical polished to a thin
disk and then re-annealed for 6 hours at 650℃ (0.98T
m
) before jet-polishing.
Dislocation link lengths were 250 – 500 nm on average as seen in (a);
however there were many images where no dislocations were observed (b).
FIGURE 3-19. High Resolution TEM images taken with a JEOL 7100F of Sample S-5
(annealed 1 year) from Group 1. Dislocation link lengths averaged 50 nm
and the density of the disks were on the order of 10
13
lines/m
-2
.
FIGURE 3-20. TEM image taken with a JEOL 7100F of Sample S-𝛾 (annealed 6 months)
from Group 3.
FIGURE 3-21. Log vs log graphical representation of the dislocation densities over time
for the samples used in the present study. The graph represents the
individual dislocation measurements across a sample (colored points). The
averages across a time period for the TEM samples (red crosses) and the
etch pits (blue pluses) are represented as well.
FIGURE 3-22. Log vs log of the dislocation densities over time comparing the present
study results with literature values. The graph represents the individual
dislocation measurements across a sample (colored points). The averages
across a time period for the TEM samples (red crosses) and the etch pits
(blue pluses) are represented as well. (For reference, the yellow data point
is about 30 days.
FIGURE 3-23. Log vs log of the dislocation densities over time. This graph shows the range
of dislocation density values at a given time. The data from the current study
(red range bars) are compared with the range of literature values (green
range bar). The average of the 5N purity (red crosses) and 6N purity (purple
crosses) aluminum samples are represented along with the average literature
value (green diamond).
FIGURE 4-1. A time-temperature-transformation diagram that illustrates the important
temperature regions of BMGs from Ref. [113].
FIGURE 4-2. (a). Two-dimensional representation of a dislocation line in crystalline (left)
and amorphous (right) solids; taken from [112]; Atomistic deformation of
amorphous metals in the form of (b). Shear transformation zones (STZ);
and (c). Local atomic jump; adapted from [110].
FIGURE 4-3. Steady-state homogeneous flow data for Zr
41.2
Ti
13.8
Cu
12.5
Ni
10
Be
22.5
metallic
glass at elevated temperatures, from the work of Lu et al. [140]. Figure
based on [110].
90
92
93
96
97
98
108
110
112
xvi
FIGURE 4-4. Stress–strain rate curve for a Zr
10
Al
5
Ti
17.9
Cu
14.6
Ni glassy alloy shows
Newtonian flow at low strain rates but non-Newtonian at high strain rates
(data from Ref. [156]). Figure based on [149].
FIGURE 4-5. Deformation mechanism maps for metallic glass plotted in (a) normalized
stress versus normalized temperature. The absolute stress values indicated
in the figure are for the Zr
41.2
Ti
13.8
Cu
12.5
Ni
10
Be
22.5
metallic glass. Adapted
from [110, 112].
FIGURE 4-6. Effect of strain rate on the uniaxial stress-strain behavior of Vitreloy 1 at
643 K and strain rates of 1.0 × 10
-1
, 3.2 × 10
-2
, 5.0 × 10
-3
and 2.0 × 10
-4
s
-1
[145].
FIGURE 5-1. 99.49% Commercially-pure (Grade I) Ti tested in compression by Battaini
et al. [179] with the loading axis parallel to the plate normal. The average
grain size was 22µm. Samples with orientation, NT (green) and NR (black),
show yield stresses of 310 and 350 MPa, respectively, using a 0.2% offset.
Specimen aspect ratios are about 0.60. Annealing of the material prior to
testing was not specified.
FIGURE 5-2. 99.52% Commercially pure (Grade II) titanium tested in compression with
the loading axis parallel to extrusion direction by Podolskiy et al. [180]. The
average grain size was not reported. The yield stress was 500 MPa (with a
0.2% offset) and the aspect ratio was equal to 1.94.
FIGURE 5-3. 99.60% Commercially-pure titanium (grade not specified) was tested in
compression at a strain-rate of 10
-2
s
-1
by Long et al. [181]. The starting
grain size was 35 µm. The loading axis in relation to sample orientation was
not specified. The yield stress is 530 MPa. Please note that this study
reported the elastic region of their curve. It is unclear if this region
represents the compliance or the actual elastic regime of their sample;
however, the curve is displayed here as it was published.
FIGURE 5-4. 99.99% pure titanium was tested in compression at strain-rates of 10
-1
and
10
-3
s
-1
by Nemat-Nasser et al. [182]. The average grain size was 40µm. The
loading axis was parallel to extrusion direction. The sample yielded at 170
MPa at ε= 10
-3
s
-1
and 215 MPa at ε= 10
-1
s
-1
.
FIGURE 5-5. 99.998% pure titanium tested in compression at a strain-rate of 10
-2
s
-1
by
Salem et al. [183]. The average grain size was 30 µm. The loading axis was
parallel to the plate normal and the sample yielded at 270 MPa.
FIGURE 5-6. Summary graph of the compression stress versus strain behavior discussed
in the earlier figures in this chapter.
113
115
116
122
125
126
128
129
131
xvii
FIGURE 5-7. Optical micrographs of the plane (a) normal, (b) rolling and (c) transverse
sections of the annealed titanium sheet. Micrographs were provided by
nanoPrecision in El Segundo, CA.
FIGURE 5-8. Initial measurements of a titanium sample that was cut using the EDM
method.
FIGURE 5-9. Polishing procedure using the SiC grit paper. (a) positioning the sample in
one direction starting with 320 grit. (b) change the grit paper to the
following grade (e.g 400 grit), rotate 90° and continue polishing
maintaining same direction. (c) previous striation lines from grit 320
disappear leaving only 400 grit lines. Polishing continued in this procedure
for all grades of grit papers.
FIGURE 5-10. An example of an unfinished 1200-grit polishing step (schematically shown
in Figure 4-12b). The horizontal lines shown in the bottom right section of
the image are abrasion marks from the previous 800-grit polishing step. The
sample was polished for a longer period of time until the abrasion marks
were uniform and parallel to the new abrasive marks covering the rest of the
sample.
FIGURE 5-11. Two mounting techniques used for mechanical polishing the titanium disk
samples. (a) only one surface of the titanium sample is exposed and polished
at a time; (b) both sides of the titanium sample are exposed and can be
polished at the same time.
FIGURE 5-12. Compliance curves done to test the parallelism of the compression platens.
FIGURE 5-13. Smaller compression fixture (CF2) that did not require
compliance/parallelism testing.
FIGURE 5-14a. Sample 1 – originally cut by EDM and used dry graphite powder for
lubrication. This sample was re-machined using a conventional lathe
method after phases 1, 2 and 3 (indicated by the * next to the individual
phase t/d ratios). Lubrication was re-applied between each phase. The
average 𝜀 =3.78 × 10
) :
𝑠
) ,
.
FIGURE 5-14b. Sample 2 – originally cut by EDM and MoS
2
aerosol was used for
lubrication. This sample was not re-machined after any phases.
Lubrication was applied between each phase. The average 𝜀 =
3.53 × 10
) :
𝑠
) ,
.
FIGURE 5-14c. Sample 3 – originally cut by EDM and MoS
2
aerosol was used for
lubrication. This sample was ultrasonically re-machined after phases 5 and
6 (indicated by the * next to the individual phase t/d ratios). Lubrication
was re-applied between each phase. The average 𝜀 =5.46 × 10
) :
𝑠
) ,
.
133
135
139
139
143
148
150
152
153
154
xviii
FIGURE 5-14d. Sample 4 – originally cut by EDM and MoS
2
aerosol was used for
lubrication. This sample was ultrasonically re-machined after phases 5 and
6 (indicated by the * next to the individual phase t/d ratios). Lubrication
was re-applied between each phase. The average 𝜀 =5.77 × 10
) :
𝑠
) ,
.
The drop in stress values during the last two compression phases were not
able to be explained.
FIGURE 5-14e. Sample 7 – originally cut by EDM. MoS
2
aerosol and Teflon paper were
used for lubrication. Both faces of the sample were polished to a 0.05 µm
finish and re-polished between each phase. This sample was re-machined
ultrasonically after phase 5 (indicated by the * next to the individual phase
t/d ratio). Lubrication was re-applied between each phase. The average
𝜀 =1.11 × 10
) <
𝑠
) ,
.
FIGURE 5-14f. Sample 8 – originally cut by EDM and MoS
2
aerosol and Teflon paper
was used for lubrication. Both faces of the sample were polished to a 0.05
µm finish and re-polished between each phase. This sample was re-
machined ultrasonically after phase 9 (indicated by the * next to the
individual phase t/d ratio). Lubrication was re-applied between each
phase. The average 𝜀 =2.17 × 10
) <
𝑠
) ,
. The dashed line represents an
‘approximation’ line as to the behavior at higher strains to discount for the
extreme barreling.
FIGURE 5-14g. Sample 10 – originally cut by EDM and MoS
2
aerosol and Teflon paper
was used for lubrication. Both faces of the sample were polished to a 0.05
µm finish and re-polished between each phase. This sample was re-
machined ultrasonically after phase 5 (indicated by the * next to the
individual phase t/d ratio). Lubrication was re-applied between each
phase. The average 𝜀 =1.00 × 10
) <
𝑠
) ,
. The re-polishing of the
specimen surface may not have imported the quality of the data over non-
surface polished specimens
FIGURE 5-15. The stress versus strain behavior of the tests of this study is averaged into
two curves that assume one of the two possible frictional coefficients for
MoS
2
. The average strain-rates for each sample are also indicated with 𝜀 =
9.27 × 10
) :
𝑠
) ,
as the average for this study. Note: Sample 4 (labeled as
4 – 5.77 × 10
) :
𝑠
) ,
) is the curve that could not be explained.
FIGURE 5-16. The two “average” curves of Figure 5-15 are compared with the literature
values reported earlier in Figure 5-6. Please note that the Long et al. [181]
study reported the elastic region of their curve. It is unclear if this region
represents the compliance or the actual elastic regime of their sample;
however, the curve is displayed here as it was published.
155
157
158
159
161
162
xix
FIGURE 5-17. The stress versus strain curves of the present study and earlier studies all
normalized to a strain rate of 10
-1
s
-1
[182] through the average strain-rate
sensitivity exponent m = 0.024 [190, 191, 193-196].
FIGURE A-1. The current-voltage ramp curve was performed from 0V – 4 V at a ramp
rate of 0.1 V/s. The optimal voltage range for electro-polishing the copper
in the research is 2.4 V – 3 V.
FIGURE A-2. (a) The gage section of Sample 5N-Cu-30MPa on the left (after threaded
ends are removed) was polished in a diluted phosphoric acid solution at
2.75V to reduce the gage diameter to ~3-mm (b). The copper polished away
from the gage section was in turn plated on the opposing electrode.
FIGURE A-3. 5N-Cu-15MPa after electro-polishing the gage section to a smaller diameter
suitable for TEM disk preparation.
163
185
187
188
xx
ABSTRACT
Four projects were performed and reported in the present dissertation with concentration of
materials subjected to creep deformation (or, time-dependent plasticity).
The first project examines the dislocation density of high-purity single crystal aluminum
at high temperatures for insight into the so-called “Harper-Dorn” Creep Regime. Single crystals
of 99.999 and 99.9999% pure aluminum were annealed at high elevated temperatures (0.98T
m
) for
relatively long times of up to one year, the longest in the literature. Remarkably, the dislocation
density remains relatively constant at a value of about 10
>
m
) -
over a period of one year. The
stability suggests some sort of “frustration” limit. This has implication towards the so-called
“Harper-Dorn Creep” that generally occurs at fairly high temperatures close to the melting point
of the material (i.e. > 0.90T
m
) and at very low stresses. The observed “frustration limit” in this
study is on the order of the dislocation density for steady-state flow in the one-power law (Harper-
Dorn) creep regime. A constant dislocation density with changing applied stress may lead to one-
power law behavior. Perhaps the more recognized, 5- or 3- power law creep, occurs when the
initial dislocation density is low (e.g. << 10
9
m
-2
).
Moving from high-temperature, low-stress creep, to low-temperature, higher stress creep
in high-purity copper and silver polycrystals. Many crystalline materials are known to exhibit
creep at low temperatures (T < 0.3T
m
). Here we review and analyze the phenomenological
relationships that describe primary creep. The discussion focuses on the controversy as to whether
power-law or logarithmic descriptions better describe the experimental database. We compile data
from the literature as well as new copper data recently taken by the authors. Depending on the
material, it appears that the logarithmic form can somewhat better describe creep behavior at low
xxi
temperatures, while the power-law behavior manifests at intermediate temperatures. The basic
mechanism(s) of low-temperature creep plasticity is examined, as well.
The third reviews and assesses the work on creep behavior in amorphous metals (also
called, bulk metallic glasses). There have been, over the past several years, a few reviews of the
mechanical behavior of amorphous metals. Of these, the review on the creep properties of
amorphous metals by Schuh et al, though oldest of the three, is particularly noteworthy and the
reader is referred to this article published in 2007. The current review of creep of amorphous metals
particularly focuses on those works since that review and places the work prior to 2007 in a
different context where new developments warrant.
The last project looks at unique compression experiments on commercially-pure grade-II
titanium thin sheets. This research examined the through-thickness (z-direction) compressive
stress versus strain behavior of 99.76% commercially-pure (Grade II) titanium sheet with relatively
small grain size. The low aspect ratio of the compression specimens extracted from the sheet, led
to frictional effects that can create high triaxial stresses complicating the uniaxial stress versus
strain behavior analysis. Nonetheless, reasonable estimates were made of the through-thickness
large strain behavior of a commercially-pure (Grade II) thin Ti sheet to relatively large true strains
of about 1.0.
1
CHAPTER 1. INTRODUCTION
The mechanical behavior of materials has been studied under a wide range of categories. The most
relevant to the research presented in this dissertation is creep deformation, or time-dependent
plasticity.
1.1 DEFINITION OF CREEP DEFORMATION
Creep is a time-dependent process that is governed by dislocation movement or vacancy diffusion
[1, 2] during an applied load over a certain period of time. Generally, tests are performed at
temperatures that are greater than 0.5 T
A
where the applied load is less than the yield stress (σ<
σ
D
) of the given material being tested [2, 3]. A material can creep at any temperature however the
processes are more noticeable at higher temperatures. There are three stages associated with creep
plasticity. Referring to Figure 1-1 (a), an engineering stress-strain curve of a hypothetical material
is shown at T>0.5 T
A
and a constant strain-rate (ε). Though the strain-rate is not specified in the
figure, a general strain-rate tends to be on the order of 10
-3
– 10
-4
s
-1
. Figure 1-1 (b) shows the
creep strain vs. time behavior of the same hypothetical material as Fig 1-1 (a). It should be
specified that the strain-rate is constant in Fig 1-1 (a), but changes in Fig 1-1 (b).
The first regime, Stage I, or primary creep, occurs immediately after applying a load to the sample.
The strain-rate (ε) rapidly increases but at a decreasing rate. Once the strain-rate reaches a steady-
state, this is when the material enters secondary creep (Stage II). During secondary creep, the
strain-rate is constant and the material is deforming uniformly.
2
(a).
(b).
FIGURE 1-1. For a hypothetical material, the stress-strain curve at temperatures higher than half
its melting point is shown in (a). The creep behavior of the same material is shown
in (b). Three regimes are clearly distinguishable: Stage I, II and III which denotes
the primary, secondary (steady-state), and tertiary (fracture). Figure taken from [2].
σ
σ
ss
ε
ε
p
t
T > 0.5 Tm
ε ̇ = constant = ε ̇ ss
T > 0.5 Tm
σ = constant = σ
ss
I
I
II
II
III
III
σ
σ
ss
ε
ε
p
t
T > 0.5 Tm
ε ̇ = constant = ε ̇ ss
T > 0.5 Tm
σ = constant = σ
ss
I
I
II
II
III
III
3
Lastly, there is a point when the material can no longer compensate for the applied load. This is
when the material enters the tertiary creep regime (Stage III, or fracture). In this regime, the strain-
rate increases at an increasing rate, which in turn sends most materials to failure.
1.2 LOW TEMPERATURE CREEP PLASTICITY
Creep of metals and ceramics occurs over three broad temperature ranges: high (T > 0.6T
m
),
intermediate (0.3T
m
< T < 0.6T
m
), and low (T < 0.3T
m
). This section concerns itself largely with
the low-temperature range. Less attention has been paid to this range because many materials
generally do not experience significant time-dependent plasticity at lower temperatures. However,
some materials do demonstrate significant creep at T < 0.3T
m
and, furthermore, at stresses both
above and below the macroscopic yield stress (σ
D
1.11-
). These materials include Ti-alloys and steels
[4-13], Al-Mg[14], α-Brass [15], ionic solids [16], pure Au, Cd, Cu, Al, Ti, Hg, Ta, Pb, Zn [17-
31] and precipitation hardened alloys [32] (as well as glass and rubber [26]). Materials may
undergo plasticity that affects its intended (e.g. structural and electrical [33-35]) performance.
However, the mechanism(s) of low-temperature creep are not yet established. Seeger et al. [19, 36,
37] proposed a dislocation intersection mechanism for pure FCC metals under poly-slip. Others
have suggested that creep occurs due to quantum mechanical tunneling of dislocations at very low
temperature [16-18, 20, 24, 25, 28], but there is no consensus as to the actual mechanisms involved.
In an empirical analysis, Neeraj et al. [8] assumed the Dorn and Holloman flow equations:
σ=Kε
H
ε
A
(E-1.1)
4
Where K is the strength parameter, n is the strain-hardening exponent, and m is the strain-rate
sensitivity exponent. This equation was found to reasonably describe some Ti-alloy behavior. They
showed:
ε=
IJ
IK
=
L
M
N
O
ε
)
P
O
(E-1.2)
∫ε
P
O
dε=∫
L
M
N
O
(E-1.3)
So that under constant stress,
ε=
L
M
N
OSP
A TH
A
O
OSP
t
O
OSP
(E-1.4)
Equation (E-2.4) now fits the general form, 𝜀 =𝑎𝑡
X
. This represents the power-law behavior,
which will be discussed at length, later. This model suggests more pronounced low-temperature
creep in materials with low values of n and moderate values of m.
This conclusion makes sense based on the arguments concerning the mechanisms of creep, in
general [2]. Creep is often associated with constant stresses, as opposed to constant strain-rates.
For a constant strain-rate test there is an observed yield stress. A “drop” in the (constant) stress
below this yield stress will still cause time-dependent plasticity, or “creep”, at the lower stress and
lower creep-rate, which becomes more substantial for higher strain-rate sensitivity (m) and lower
strain-hardening (n).
Generally, but not always, low temperature creep is considered to be primary creep that does not
usually reach a genuine mechanical steady state. Long ago, Andrade [38] and Orowan [39]
5
suggested that primary creep at high temperatures obeys a power law behavior with an exponent
of
1
3
:
ε=bt
, Z
+c
,
(E-1.5)
Evans and Wilshire [40] reviewed the high-temperature primary creep equations and suggested a
refinement that led to an equation of the form:
ε=at
, Z
+ct+dt
< Z
(E-1.6)
Variations to this equation include [41]:
ε=at
, Z
+ct (E-1.7)
and [42],
ε=at
, Z
+bt
- Z
+ct (E-1.8)
or [43]
ε=at
^
+c
K
(E-1.9)
where, 0 < b < 1
or simply,
𝜀 =𝑎𝑡
X
(E-1.10)
where, 0 < b < 1
Note that Equations (E-1.5) – (E-1.10) are all of a similar, power-law form, as suggested in the
Neeraj et al. [8] work discussed earlier.
Another set of phenomenological equations, with a logarithmic rather than power-law form, was
suggested by Phillips [26], Laurent and Eudier [44] and Chévenard [45] for low temperatures,
𝜀 =𝛼ln𝑡+𝑐
-
(E-1.11)
Wyatt [30], long ago, suggested that for pure metals, such as Al, Cd and Cu, that at higher
temperatures, Eq. (E-1.10) was the proper descriptive equation, but at lower temperatures, Eq. (E-
6
1.11) was the proper form. This contention is not well established, and will be discussed in detail
in this thesis.
1.3 HIGH TEMPERATURE CREEP PLASTICITY (HARPER-DORN THEORY)
It has been suggested by several investigators of very high temperature creep experiments (0.98
T
m
) and at very low stress (< s
ss
/G = 10
-5
) that the dislocation density becomes independent of the
stress level and creep changes from a five-power law type to that of a stress exponent of one [46-
49] and Harper-Dorn creep is observed. As just mentioned, others [50-54] have observed the
dislocation density to continually decrease. It was suggested by Lin et al. [55] that due to a
dislocation network frustration within the material, the dislocation density would only decrease to
a constant value. This limitation was suggested to be due to an inability of Frank’s rule (E-1.12)
[56] to be satisfied with additional coarsening of the dislocation network.
𝑏
d
e
,
=0 (E-1.12)
Long-term annealing experiments could confirm or refute this frustration limit for the starting
dislocation density. Harper and Dorn, in 1957 [57], observed a low stress exponent (n) creep
mechanism at very low stresses to take place according to the equation,
𝜀
ff
=A
hi
i
jk
l^
mn
L
l
H
(E-1.13)
where A
HD
is the Harper-Dorn coefficient, D
sd
the lattice self-diffusion coefficient, G is the shear
modulus, b is the Burgers vector, s is the “effective” stress (a threshold stress is subtracted from
the applied stress to give this s value [53] in their study) and n is a value of 1. [Had this (probably
7
fictitious) threshold not been subtracted, normal 3- to 5-power law creep is observed which most
investigators have overlooked]. Theoretically, a dislocation network creep model developed by the
authors [52] in an earlier article, suggests that if the dislocation density varies with stress in a
predictable way in the low stress regime (see Figure 1-2), n, is slightly larger than 3. On the other
hand, for a constant dislocation density, n is about 1, as Harper and Dorn observed. Hence, for
Harper-Dorn proponents, the observations of a constant dislocation density with decreasing stress,
and the observation of a stress exponent of 1, appear consistent. It should be mentioned that it is
assumed that the strength of the structure within the Harper-Dorn regime is provided by the Frank
dislocation network, just as at higher stresses and lower temperatures within the five-power-law
regime [58]. Fig. 1-2 is compiled from creep studies by [48, 55, 59-63]. Lin et al. [55] and Barrett
et al. [48] both suggested that the dislocation density remains fixed with decreasing stress at about
10
8
m
-2
. The shear modulus (G) used was 16.96 GPa at 920K (0.98T
m
) [64]. Some earlier work by
the authors [51-53, 61] suggest that the dislocation values in fact continue to decrease with stress
in a manner as with five-power-law creep of aluminum. This work attempts to verify this trend by
decreasing the stress to zero in long-term annealing experiments.
Note that, from Table 1.1, the range of annealed dislocation density values includes this value (10
8
m
-2
) but are generally much higher. The question remains as to whether longer annealing times
(up to one year) can lead to even lower dislocation densities below the range observed. Some
common dislocation densities from established aluminum annealing studies [47, 48, 60, 62, 65-
67] are listed in Table 1.1.
8
FIGURE 1-2. Steady-state dislocation density versus the modulus-compensated steady-state
stress at an elevated temperature of 923 K (0.99 T
m
). The data of Lin et al. [55] and
that of Barrett et al. [48] had been suggested to imply a lower limit of the dislocation
density (r). The red bracket represents the range of initial dislocation density values
reported in literature which includes the initial value reported by the present
research in Chapter 3.
"Harper-Dorn" regime
99.999% Pure Al SX
1
Barrett et al. Average
2
5-Power Law regime
99.999% Pure Al - 5%Zn
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1.0E+08
1.0E+09
1.0E+10
1.0E+11
1.0E+12
1.0E+13
1.0E+14
1.0E-10 1.0E-09 1.0E-08 1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02
Dislocation Density (line length/m
2
)
s
ss
/G
Blum (1993)
Kumar et al. (2008)
Barrett et al. (1972)
Mohamed et al. (1973)
Mohamed and Ginter (1982)
Kassner and McMahon (1987)
Lin, Lee and Ardell (1989)
Stress due to the weight at
the bottom of our specimens
10
14
10
13
10
12
10
11
10
10
10
9
10
8
10
7
10
6
10
5
10
4
10
3
10
2
10
1
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
Lin et al. Average
9
TABLE 1.1. Average initial dislocation densities of aluminum in as-received and annealed
conditions. The authors of article [66] used a different Al sample for each listed
condition. PX – polycrystal, SX – single crystal. EP – etch pit, TEM – transmission
electron microscopy.
Material
Dislocation
Density (r)
Conditions Reference
1. 99.99% coarse-
grained PX Al
6 x 10
10
m
-2
As received
[66]
5 x 10
10
m
-2
Annealed
773 K for 10 hours
6 x 10
9
m
-2
Annealed
823 K for 1 hour
4.5 x 10
9
m
-2
Annealed
903 K for 10 hours
2. 99.99% PX Al 2.4 x 10
11
m
-2
Annealed (in vacuum)
773 K for 25 minutes
[67]
3. 99.994% SX Al 1.32 x 10
7
m
-2
Annealed
(in creep machine under <
0.00275 MPa)
823 K for 36 hours
[48]
4. 99.9995% SX Al 1 x 10
8
m
-2
Annealed
926 K for 50 hours
[65]
5. 99.99% SX Al 3 x 10
11
m
-2
Annealed
923 K for 48 hours
[47]
6. 99.999% SX Al 6 x 10
10
m
-2
Annealed
926 K for 50 hours
[62]
7. 99.999% PX Al 4.3 x 10
11
m
-2
Annealed 698 K for 1
hour (TEM)
[60]
8. 99.999% SX Al 6.5 x 10
7
m
-2
Annealed 50 hrs. at
913K
[64]
10
CHAPTER 2. ROOM TEMPERATURE CREEP OF HIGH-PURITY
POLYCRYSTALLINE COPPER AND HIGH-PURITY
POLYCRYSTALLINE SILVER
2.1. LITERATURE
In this section, we examine the phenomenological trends in the literature for various materials with
an objective towards trying to understand how well, and to what extent, they obey power-law or
logarithmic behavior.
2.1.1. Titanium Alloys
Neeraj et al. [8] carefully described the low-temperature creep behavior of Ti-alloys. Figure 2-1
(taken from Neeraj et al.) plots ambient-temperature creep data [9, 68, 69] for Ti-alloys with
different microstructures and different compositions. A particularly interesting characteristic of
this data is the extension of creep tests to longer times (sometimes over a month). Equation (E-
1.10), the power-law relationship, appears to better describe the data. In contradiction to some of
the earliest phenomenological equations (albeit at higher temperatures) [38], the exponent of b =
⅓ used in Equations (E-1.5) – (E-1.8) does not fit the data, and a smaller value b = 0.2 seems to
be better. Also, note that the applied stresses are all below the macroscopic yield stress (determined
at a conventional strain-rate) where creep is less expected. As Neeraj et al. point out, other
literature confirms that 0.03 < b < 1. Cottrell [70-72] and Nabarro [73] recently tried to
theoretically justify a value for b = ⅓ for power-law creep at low temperatures but that value does
not appear to be universally valid. Other creep work on titanium has been performed by others as
well [74].
11
2.1.2. Steels
AISI 4340 Steel
Oehlert and Atrens [10] performed creep studies on ferritic steel at ambient temperature where the
applied stress is below the macroscopic yield stress. One of the tests was conducted at an applied
stress of just half the yield stress (again, assessed at a conventional strain-rate between 10
-3
– 10
-4
s
-1
). The duration of the tests is relatively short, performed over, at most, 20 min and this is
unfortunately short. The nature of the phenomenological equations may not be fully assessed with
such short testing times. Nonetheless, the creep behaviors were best described by a logarithmic
equation (i.e., Equation E-2.1):
ε
o
=ε
oD
+αln (t) (E-2.1)
where ε
oD
is the plastic strain on loading and is a function of the applied stress. One clumsiness
with Equation E-2.2 is that, at t = 0, an infinite creep rate is predicted although the strain should
be equal to ε
oD
. The clumsiness was eliminated by modifying the equation to:
ε
o
=ε
oD
+αln (1+βt) (E-2.2)
where α and β are constants. Oehlert and Atrens found that, over the range of stresses studied, the
constant α varied by a factor of nearly 20 and β by a factor of 2. They also examined 3.5NiCrMoV
and AeroMet100 over similar time ranges and observed a similar logarithmic phenomenology. By
contrasting the differences in the logarithmic and power-law fits, our R
2
correlation factor analysis
is consistent with their conclusions.
12
a).
b).
FIGURE 2-1. Ambient-temperature creep for various Ti-alloys of different microstructures and
different compositions. From Neeraj et. al. [8] data in a log-log plot while (b) plots
the same data in a semi-log plot. Based on the R
2
values, a power-law may better
describe the creep behavior.
σ = 637 MPa
σ
y
= 796 MPa
ε = 5 x 10
-3
s
-1
σ = 667 MPa
σ
y
= 834 MPa
ε = 8.3 x 10
-4
s
-1
σ = 608 MPa
σ
y
= 760 MPa
ε = 5 x 10
-3
s
-1
σ = 870 MPa
σ
y
= 1088 MPa
ε = 8.3 x 10
-4
s
-1
R² = 0.82161
R² = 0.93447
R² = 0.92414
R² = 0.98529
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
1 10 100 1,000 10,000 100,000 1,000,000 10,000,000
Strain
Time (sec)
Logarithmic
Ti-Alloys
T = 0.15T
m
Ti-6211 Basketweave
Ti-5-2.5
Ti-6211 Colony
Ti-6-4 Colony
σ = 637 MPa
σ
y
= 796 MPa
ε = 5 x 10
-3
s
-1
σ = 667 MPa
σ
y
= 834 MPa
ε = 8.3 x 10
-4
s
-1
σ = 608 MPa
σ
y
= 760 MPa
ε = 5 x 10
-3
s
-1
σ = 870 MPa
σ
y
= 1088 MPa
ε = 8.3 x 10
-4
s
-1
R² = 0.8773
R² = 0.86482
R² = 0.94396
R² = 0.98529
0.0001
0.001
0.01
0.1
1 10 100 1,000 10,000 100,000 1,000,000 10,000,000
Strain
Time (sec)
Power-Law
Ti-Alloys
T = 0.15T
m
Ti-6211 Basketweave
Ti-5-2.5
Ti-6211 Colony
Ti-6-4 Colony
13
a).
b).
FIGURE 2-2. The plastic strain vs. time behavior of annealed 304 stainless steel under different
stresses from [5, 75]; (a) Linear strain vs. log time; (b) Log strain vs. log time Based
on the R
2
values, a logarithmic equation better describes the data.
259 MPa
276 MPa
293 MPa
138 MPa
145 MPa
155 MPa
190 MPa
172 MPa
207 MPa
224 MPa
241 MPa
R² = 0.99999
R² = 0.99999
R² = 0.99996
R² = 0.99998
R² = 0.9937
R² = 0.99958
R² = 0.98928
R² = 0.99373
R² = 0.9898
R² = 0.99973
R² = 0.99976
R² = 0.99994
R² = 0.99955
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
1 10 100 1,000 10,000 100,000 1,000,000 10,000,000
Strain
Time (sec)
Logarithmic
304 Stainless Steel
T = 0.17T
m
σ
y
= 221 MPa
ε = 3 x 10
-4
s
-1
Extensometers
Optical Comparator
Measurements
259 MPa
276 MPa
293 MPa
138 MPa
145 MPa
155 MPa
190 MPa 172 MPa
207 MPa
224 MPa
241 MPa
R² = 0.98635
R² = 0.98101
R² = 0.99743
R² = 0.96104
R² = 0.98409
R² = 0.96687
R² = 0.95812
R² = 0.94276 R² = 0.99658
R² = 0.92614
R² = 0.95734
R² = 0.97846
R² = 0.99077
0.0001
0.001
0.01
0.1
1 10 100 1,000 10,000 100,000 1,000,000 10,000,000
Strain
Time (sec)
Power-Law
304 Stainless Steel
T = 0.17T
m
σ
y
= 221 MPa
ε = 3 x 10
-4
s
-1
Extensometers
Optical Comparator
Measurements
14
304 Stainless Steel
Figure 2-2 shows the creep curves for annealed 304 stainless steel [5], a material that evinces some
typical features of low-temperature creep. The applied stresses are both above and below the yield
stress (here, 𝜎
t
= 221 MPa at 𝜀 = 3 x 10
-4
s
-1
at room temperature). Fig. 2-2(a) is a semi-log plot
and shows the full time-span of the tests. Note that times, in some cases, extend to nearly a year,
although tests above the yield stress were generally performed over shorter times (a week or less).
For the shorter tests above the yield stress, strains were measured using an extensometer and for
the longer tests, below the yield stress, strain was measured using an optical comparator. Based on
the R
2
quality-of-fit factor, it appears that the creep data for this material, at ambient temperature,
best follows a logarithmic behavior at a fixed stress, . This is similar to E-2.2 that represented
the ferritic steel data of Ohlert and Atrens. can be approximated by the usual relationship,
ε
oD
=a+bσ (E-2.3)
where a and b are constants, and a basically reflects the strain on loading. Also, is the slope in
Fig. 2-2(b) and appears to decrease with decreasing stress, approximated by,
β=−kσ+C
x
(E-2.4)
where k and C
o
are constants[5].
σ
ε
py
β
15
2.1.3. Pure Metals
Aluminum
The creep behavior of high-purity aluminum investigated by Wyatt [30] and Sherby et. al [76] is
shown in Figure 2-3. The 300 K (0.32 T
m
) data shown in Figure 2-3 (a-b), exhibits power-law
behavior (Eq. 10) perhaps because the temperature is high enough to transition from a low-
temperature to intermediate-temperature creep regime as also observed in Cu (discussed later).
Again, test times are unfortunately relatively short. However, the 77 K (0.08 T
m
) data [76] also
exhibits a somewhat better fit with a power-law equation from the Sherby et al. data for longer
times. However, the Wyatt data for shorter times shows logarithmic behavior. The lower
temperature tests may be more reliable in assessing low-temperature creep, in that this data lies
purely in the low temperature regime.
Cadmium
Figure 2-4 [30] illustrates the creep behavior of pure Cd at 77 K (0.13 T
m
) and 300 K (0.51 T
m
).
The low-temperature behavior over a period of just one hour may evince logarithmic behavior,
while the higher temperature data seems better described by a power-law, again, consistent with
Al, just discussed.
Table 2.1 summarizes the overall phenomenological behavior of the metals and alloys discussed.
Trends are described for both above and below the conventional yield strength of the metals as
well as for shorter and longer creep times (strains). Obviously, more data is desirable, especially
for longer times. Overall, logarithmic behavior appears to be more commonly observed.
16
a).
b).
FIGURE 2-3. Cryogenic temperature (a),(b) creep behavior of pure Al based on data by [30, 76]
behavior at 77K may dominate at lower stresses. A yield stress at a given strain-
rate was not reported by either reference for their aluminum samples.
Sherby et al.
σ = 152 MPa
Wyatt et al.
σ = 59 MPa
σ = 19.5 MPa
R² = 0.92887
R² = 0.9953
R² = 0.99324
0.00
0.05
0.10
0.15
0.20
0.25
0.30
1 10 100 1,000 10,000 100,000
Strain
Time (sec)
Logarithmic
Aluminum
T = 0.08 T
m
(77K)
Sherby et al.
σ = 152 MPa
Wyatt et al.
σ = 59 MPa
σ = 19.5 MPa
R² = 0.95109
R² = 0.97022
R² = 0.98094
0.001
0.01
0.1
1
1 10 100 1,000 10,000 100,000
Strain
Time (sec)
Power-Law
Aluminum
T = 0.08 T
m
(77K)
17
a).
b).
FIGURE 2-4. Creep behavior of pure Cd, at 0.51T
m
and 0.08T
m
, based on data by [30]. At low
temperatures, logarithmic behavior (T < 0.30T
m
) may best describe the data but
power-law behavior may be better at higher temperatures. A yield stress at a given
strain-rate was not reported by Wyatt [30] for the Cadmium data.
σ = 59 MPa
T = 0.13T
m
σ = 20 MPa
T = 0.51T
m
R² = 0.98352
R² = 0.83012
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
1 10 100 1,000 10,000
Strain
Time (sec)
Logarithmic
99.999% Cadmium
(Data from Wyatt et al.)
σ = 59 MPa
T = 0.13T
m
σ = 20 MPa
T = 0.51T
m
R² = 0.89796
R² = 0.99831
0.001
0.01
0.1
1
1 10 100 1,000 10,000
Strain
Time (sec)
Power-Law
99.999% Cadmium
(Data from Wyatt et al.)
18
TABLE 2.1. Materials and regimes where either Logarithmic (Log) or Power-Law (PL)
description apply to low-temperature creep behavior. Note: A section with a “--”
signifies no data for those conditions were reported. The ‘Copper’ and ‘Silver’ rows
are from the current research.
T
creep
< (0.30)T
melting
Overall
Trend
t
creep
< 1 hour t
creep
> 1 hour
s
creep
< s
y
s
creep
> s
y
s
creep
< s
y
s
creep
> s
y
ALLOYS
Titanium Alloys Mostly PL -- -- Mostly PL --
AISI 4340 Steel Log Log Log Log Log
304 Stainless Steel Log Log -- -- --
PURE METALS
Aluminum Log/PL Log Log -- PL
Cadmium Log Log -- -- --
Copper Log/PL PL Log/PL Theo-PL Log/PL
Silver Log Log Log Theo-Log Log
19
2.2 MATERIALS
2.2.1 99.999% POLYCRYSTALLINE COPPER
The 1.219 m x 15.875 mm 99.999% Copper rod (Cu) was ordered from The Metron Group in
Scottsville, Virginia. The copper rod was cut into 14 samples at the University of Southern
California machine shop. The samples had measurements consistent with ASTM [77] tensile
testing parameters of 5:1 gage length to gage diameter ratio. The gage length was 34-mm in length
and 6.55-mm in diameter.
2.2.2 99.99% POLYCRYSTALLINE SILVER
The 305-mm (1-ft) x 12.7-mm 99.99% Silver (Ag) rod was ordered through ESPI Metals in
Ashland, Oregon. The silver rod was machined into four (4) samples. The samples had
measurements consistent with the ASTM [77] tensile testing parameters of 6:1 gage length to gage
diameter (l:d) ratio. Two (2) samples had a l:d ratio of 30:5 (in mm); One (1) sample had a l:d ratio
of 24:4 (in mm); One (1) sample had a l:d ratio of 18:3 (in mm).
2.2.3 VACUUM ANNEALING
Following machining, all samples were annealed in a vacuum furnace for 2 hours to remove any
mechanical damage. Figure 2-5 shows the vacuum furnace used. The copper samples were heat-
treated at 420°C and the silver samples were heat-treated at 620°C. The samples were loaded into
the furnace at atmospheric pressure (10
3
mbar). It took about 2 hours for the vacuum to reach a
low enough pressure to begin the heat treatment. This pressure was 5.4 × 10
) *
mbar. The furnace
was turned on at a heating rate of 10 °C/min. Once the desired temperature was reached, the
vacuum pressure slightly increased to 1.9 × 10
) :
mbar due to sample expansion and the release
20
of oxygen from the sample. This increase was negligible and had no effect on the annealing
process. Towards the end of the heat treatment, the vacuum had reached a pressure of 6.4 × 10
) *
mbar.
The furnace was programed to ramp down in temperature after 2 hours while keeping the vacuum
pressure consistent. After the furnace was fully cooled, the samples were left in vacuum for 2 days
to prevent and ensure there was no oxidation. Samples were individually wrapped in scratch-free
TX 4004 MiracleWipe
®
synthetic wipers made by ITW Texwipe
®
and stored at room temperature
and atmospheric pressure until further use. The detailed history of each sample is presented in the
following sections.
21
FIGURE 2-5. This figure shows the vacuum furnace used for the heat treatments of all
copper and silver creep specimens. The vacuum unit (on the left) reaches
pressures as low as 6.4 × 10
) *
mbar. The furnace unit (on the right) was an
auto-programmable system holding at the desired temperature for 2 hours.
Furnace
For Cu: Set to 420 ℃
For Ag: Set to 620 ℃
Vacuum Unit
Pressure = 6.4 × 10
−6
mbar
Sample
(inside)
22
2.3 MICROSTRUCTURAL ANALYSIS OF COPPER AND SILVER CREEP SAMPLES
The threaded section, of a tested copper and silver sample, was cut from the gage section using a
0.012˝-thick diamond saw. The threaded section was used as an ‘as-annealed’ sample because the
threaded sections are not known to experience plasticity during testing. The segments were
mounted using a quickset acrylic powder and water mixture. A schematic of the sample mount is
shown below in Figure 2-6. The surfaces were polished to a 0.25µm surface finish with diamond
slurry paste.
Chemical Etching
The copper etchant used to expose the grain boundaries was a mixture of 7.5 mL 35 wt%
Ammonium Hydroxide (NH
4
OH), 7.5 mL DI H
2
O and 1.5 mL 3 wt% Hydrogen Peroxide (H
2
O
2
).
This solution was taken from [78]. The NH
4
OH and DI H
2
O were mixed first and sat for a few
minutes while the sample was being prepared to etch. The H
2
O
2
was added and swirled into the
pre-mixture immediately before etching. This helped to maintain a fresh solution. The silver
etchant used to expose the grain boundaries was a mixture of 15-mL of 35 wt% Ammonium
Hydroxide (NH
4
OH), and 15-mL of 30 wt% Hydrogen Peroxide (H
2
O
2
). The 30 wt% Hydrogen
Peroxide (H
2
O
2
) was made from diluted 50 wt% (H
2
O
2
) with water. This etchant was adapted from
a book of Metallographic Etchants [79].
Tweezers were used to submerge the surface of the sample into the etchant for
approximately 15 seconds. The sample was quickly dipped into a DI water bath to stop the reaction
and then flushed with DI water for 30 seconds in all directions to remove any residual etchant. An
23
optical microscope was used to verify that the grains were etched; a scanning electron microscope
(SEM) was used to perform Electron Back Scatter Diffraction (EBSD) scans, shown in Figure 2-
7 (a-b). The scans were utilized instead of optical images because the quality of imaging is better.
Also, the randomly oriented texture of the grains is easily seen with an EBSD scan as opposed to
optical microscope images. EBSD scans were also done for samples after being deformed and
those are shown in the results section of this chapter.
The average grain diameter was calculated using the line intersection method:
d=
ℓ
z
(E-2.5)
where d represents the average grain diameter, ℓ represents the line length and N represents the
number of grain boundary intersections with the line.
The average grain diameter for copper was calculated at 310 µm and the average grain diameter
for silver was calculated at 25 µm.
24
FIGURE 2-6. Copper and Silver gage segments were mounted in quickset acrylic for a
microstructural study of the grain textures.
25
(a).
(b).
FIGURE 2-7 (a-b). The EBSD scan of (a) the as-annealed 99.999% PX copper and (b) of the
as-annealed 99.99% PX silver. The average grain diameter was calculated
at 310 µm for copper and 25 µm for silver.
26
2.4 TENSILE BEHAVIORS
To obtain a specific yield stress for the copper and silver used in this research, a stress-strain curve,
for each, was performed. Determining the yield stress allowed estimation of the desired applied
stress used during the creep experiments. The stress-strain test was done in tension. The testing
was done on a Model Series 5900 electrochemical Instron Testing System and the data acquisition
was done through BlueHill software. The strain was measured using an extensometer.
The curves were recorded as an Engineering stress (s) vs. Engineering strain (e) behavior but these
were converted true stress (s) vs. true strain (e) behavior using the below relations, where L
o
is the
initial gage length, DL is the change in length (L
f
–L
i
), F represents the load applied, and A
i
represents the instantaneous area:
Engineering Strain (e) = True Strain (e) =
Engineering Stress (s) = True Stress (s) =
Calculation of the true stress assumes the volume relationship, , to find the
instantaneous area.
ΔL
L
o
ln(1+e)
F
A
o
F
A
i
(A−ΔA)×(L+ΔL)
27
2.4.1 POLYCRYSTALLINE COPPER TENSILE TESTS
The measurements of the tensile specimens are the same as those machined for the creep
experiments:
Cross-sectional area(from a 6.35-mm diameter) = 31.67 mm
2
Gage length = 32 mm
Strain-rate = 10
-4
s
-1
Crosshead speed = 0.0032 mm/s
Three samples (Sample 1, Sample 2 and Sample 3) were used for the stress-strain measurements
but only the third showed the true stress-strain behavior. The yield stress was very low. The first
two samples were loaded into the machine with a pre-load of 500 N (15.8 MPa) to take the slack
out of the attachment beams to the sample. This pre-strained the first two samples. However, with
the third sample, a lower load was applied to remove the slack and this proved to be sufficient.
The third sample started at 8 N (0.25 MPa). The curve is shown in Figure 2-8.
2.4.2 POLYCRYSTALLINE SILVER TENSILE TESTS
The tensile specimen measurements are the same as those machined for the creep experiments:
Cross-sectional area (from a 5-mm diameter) = 19.63 mm
2
Gage length = 30 mm
Strain-rate = 10
-4
s
-1
Crosshead speed = 0.0019 mm/s
The true stress strain behavior for the silver used in this study is shown in Figure 2-9.
28
FIGURE 2-8. True stress vs. true strain behavior of high purity copper polycrystals used
in this research. The macroscopic yield stress is 24 MPa.
0
50
100
150
200
250
0 0.02 0.04 0.06 0.08 0.1 0.12
True Stress (MPa)
True Strain
99.999% Polycrystalline Copper (Cu)
True Stress Strain (s-e) Behavior
σ
"
#. %%
ε̇ =1 × 10
./
s
.1
0.2% offset line
2
3
#. %%
= 24 MPa; (σ
"
= 18 MPa)
29
FIGURE 2-9. True stress vs. true strain behavior of high purity silver used in this research.
The macroscopic yield stress is 28 MPa.
0
50
100
150
200
0 0.02 0.04 0.06 0.08 0.1 0.12
True Stress (MPa)
True Strain
99.99% Polycrystalline Silver (Ag)
True Stress Strain (s-e) Behavior
σ
"
#. %%
ε̇ =1 × 10
./
s
.1
0.2% offset line
2
3
#. %%
= 28 MPa; (σ
"
= 22 MPa)
30
2.5 CREEP MACHINE SPECIFICATIONS
The creep machine used was an older model made by the Arcweld Manufacturing Company. A
simple schematic of the machine’s functioning is shown below in Figure 2-10. A lever arm design
is used where W represents the free weights.
Calibration
The calibration of the machine was done using an S-Type tensile load cell purchased from Omega
Engineering. An excitation voltage of at least 10 V was required and supplied by a DC power
supply purchased from BK Precision®. The output was read with a hand-held voltmeter.
First, the load cell was calibrated by itself with weight directly hung from it in order to match and
confirm the calibration done by Omega (schematic shown in Figure 2-11a). The voltage output
read from the load cell.
Next, the load cell was then attached to the load column of the creep frame (schematic shown in
Figure 2-11b) and weights were loaded in the machine to simulate a creep test. The load cell was
placed in three different spots along the frame. This was done by attaching the load cell to the top
(position 1 in Fig 2-11b), the middle (position 2 in Fig 2-11b) and towards the bottom (position 3
in Fig 2-11b) of the load column. The middle of the frame (position 2) showed a more accurate
reading of the load imposed on the copper sample. Please note that each position was analyzed
individually; there were not three different load cells used. The voltage output read from the load
cell was then compared to the voltage outputs read from direct loading of weights to the load cell,
previously described.
31
A schematic of the calibration system and a graph of the data is shown in Figures 2-11(a-b) and 2-
12, respectively. Using the “best-fit” lines from Figure 2-12, one can calculate the weight that
needed to be loaded for the copper, or silver, sample to experience the desired load for creep
testing. Testing was done at three stresses relative to the yield stress of the copper (s
y
≈ 24 MPa),
or silver (s
y
≈ 28 MPa), samples. A tabulated summary of values for the weights used are shown
in Table 2.2.
All the creep experiments were performed at room temperature (273-300K), which equates to
0.22T
m
of copper and 0.24T
m
of silver. To classify the experiments as “low-temperature” creep,
the experiments must be less than 0.30T
m
. Two types of strain gauges were used, from Omega
Engineering Inc., to measure the extension of each sample. Each strain gauge had a different
Wheatstone bridge configuration: quarter bridge and full bridge. Four copper samples and three
silver samples were tested in the present creep study. An analysis of each sample is outlined in the
following subsections.
32
(a).
(b).
FIGURE 2-10. The Arcweld Manufacturing Company for creep rupture testing. The setup
used is illustrated in (a) and the schematics are shown in (b).
!
W
26 inches 9 inches
Universal Threaded
Double Hinged
Joints
Copper
Sample
Steel Rods
33
(a).
(b).
FIGURE 2-11(a-b). A schematic of the load cell calibration is shown. The calibration of the load
cell alone with weights directly attached to it is displayed in (a) and the
schematic of how the load cell was tested in the creep machine is shown in
(b) where the weights are loaded to the machine.
!
!
DC Power
Supply
Voltmeter
Steel rods connected to
creep machine
Tensile displacement
sensor
Weight
Position 1
Position 2
Position 3
S-Type Load
Cells
Creep Machine
34
FIGURE 2-12. The load cell calibration data used for the creep machine used in this study
is shown here. Three curves are plotted on a Force (in pounds, lbs) vs.
Voltage (in mV). Omega’s curve (red) is taken from the company-given
values. The curve labeled “Load Cell (alone)” was generated by the weights
directly loaded to the load cell (refer to Figure 2-11a). The curve labeled
“Load Cell (middle of the frame)” was generated by the calibration of the
load cell in Position 2 of the load frame (refer to Figure 2-11b). The best fit
lines were used to calculate the desired weight added to apply a desired load
to the sample.
Omega® Calculations
y = 33.326x - 2.4048
R² = 1
Load Cell (middle of frame)
y = 12.401x + 17.248
R² = 0.97048
Load Cell (alone)
y = 32.806x + 0.1064
R² = 0.99998
0
200
400
600
800
1000
1200
0 5 10 15 20 25 30 35
Force (lbs)
Voltage (mV dc)
Load Cell Calibration
Omega® Calculations
Load Cell (middle of frame)
Load Cell (alone)
35
TABLE 2.2. Summations of values for the weights used in the creep deformation experiments.
The jagged line separates the copper (top three rows) from the silver (bottom three
rows).
Sample
Total time
tested
Gage
diameter
Loaded to
the creep
machine
Weight on the
sample while
loaded
Resulting
Specimen
Stress
5N-Cu-15MPa 266 days 6.35-mm 58 lbs. 107 lbs. 15 MPa
5N-Cu-18MPa 5 days 6.35-mm 66 lbs. 128 lbs. 18 MPa
5N-Cu-20MPa 54 days 6.35-mm 71 lbs. 142 lbs. 20 MPa
5N-Cu-30MPa 12 days 6.35-mm 98 lbs. 214 lbs. 30 MPa
4N-Ag-20MPa 75 days 3.00-mm 16 lbs. 32 lbs. 20 MPa
4N-Ag-30MPa 78 days 5.00-mm 67 lbs. 132 lbs. 30 MPa
4N-Ag-40MPa Ongoing 5.00-mm 85 lbs. 176 lbs. 40 MPa
36
2.6 CREEP DEFORMATION RESULTS FOR COPPER AND SILVER
2.6.1 99.999% COPPER CREEP CURVES
Individual copper creep experiments are described in the following subsections with the curves
shown in Figures 2-13 – 2-16. The combined curves are shown in Figure 2-17 and the R
2
-
correlation for all copper curves are displayed and described later in Figure 2-22.
2.6.1.1 5N-Cu-15MPa
A quarter bridge was used for this sample and the applied load was 15 MPa. The total duration of
this test was 266 days, the longest creep test performed in this research. Analyzing the R
2
-
correlation, the deformation trend shows an undisputed power-law behavior after 30 minutes of
testing. The creep curve for this load is shown in Figure 2-13(a), and an EBSD scan of the grain
texture is represented in Figure 2-13(b). The grain size seems to stay relatively consistent with the
original microstructure (Figure 2-7(a)). An ending creep-rate (𝜀) on the order of 10
) ,-
s
) ,
was
calculated from the data.
2.6.1.2 5N-Cu-18MPa
A quarter bridge was used for this sample. The applied load was 18 MPa, and the total duration of
the test was 5 days. At this low stress, a power-law behavior seems to dominate; however, if the
R
2
-correlation is analyzed, it does not seem that a consistent trend can be solidified at such short
times. Data acquisition started at 100 seconds and regular data points were taken for a period of
30 minutes. The last data point was taken after a total time of 5 days. The reason for this error in
data was caused by testing of new data acquisition software. The data must still be reported;
however, the gap in data does not seem to change the observed power-law deformation trend.
37
2.6.1.3 5N-Cu-20MPa
A full bridge was used for this sample. The applied load was 20 MPa, and the total duration of the
test was about 54 days. This sample was bumped during testing at 1 hour while taking excess tape
off the steel beams. At that point, the test was not stopped but the sample experienced a pre-strain.
This resulted in the total amount of strain on the sample to jump significantly. The creep curve
for this load is shown in Figure 2-15(a-b). The power law behavior seems to dominate, however,
conclusions cannot be drawn from the specimen data alone. A more in-depth analysis of the R
2
values is discussed in the later sections.
2.6.1.4 5N-Cu-30MPa
A full bridge was used for this sample. The applied load was 30 MPa and the total duration of the
test was about 11 days. At these higher loads, the R
2
-correlation shows an undisputed logarithmic
behavior after approximately 1 day of testing. The creep curve for this load is shown in Figure 2-
16(a), and an EBSD scan of the grain texture is represented in Figure 2-16(b). The EBSD scan
shows some grain refinement. Smaller grains in the range of 50 – 100 µm are seen; however, the
majority of the grains stayed in the 300-µm range. microstructure seems to remain consistent to
the starting microstructure, shown in Figure 2-7(a). The average grain size seems to stay relatively
consistent with the original microstructure. An ending creep-rate (𝜀) on the order of 10
) ,,
s
) ,
was calculated from the data.
38
(a).
(b).
FIGURE 2-13 (a-b). The (a). creep deformation curve for Sample 5N-Cu-15MPa is shown along
with an (b). EBSD scan of the microstructure after creep for 266 days
performed in (a). The upper left corner shows some “noise” from the
detector.
0
0.00005
0.0001
0.00015
0.0002
0.00025
0.0003
0 5,000,000 10,000,000 15,000,000 20,000,000 25,000,000
Strain
Time (sec)
Room Temperature Creep of 99.999% Polycrystalline Cu
Test Stress = 15 MPa
Linear Behavior
ε̇ =3 × 10
)*+
s
)*
10
-
5 × 10
/
1 × 10
0
1.5 × 10
0
2 × 10
0
2.5 × 10
0
39
FIGURE 2-14. Creep deformation curve for 5N-Cu-18MPa. The applied load was 18 MPa
and the total duration of the test was 5 days. The gap in data does not seem
to change the observed power-law deformation trend. At this low stress, a
power-law behavior seems to dominate. An EBSD scan was not performed
for this sample.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0 1,000 2,000 3,000
Strain
300,000 350,000 400,000 450,000
Time (sec)
Room Temperature Creep of 99.999% Polycrystalline Cu
Test Stress = 18 MPa
Linear Behavior
10
#
1 × 10
&
2 × 10
&
3 × 10
&
3 × 10
)
3.5 × 10
)
4 × 10
)
4.5 × 10
)
40
(a).
(b).
FIGURE 2-15 (a-b). Creep deformation curves for Sample 5N-Cu-20MPa shows the full
duration of data acquisition over a period of 54 days. Figure (a) shows the
sample was bumped during testing at 1 hour while taking excess tape off
the steel beams. The test was not stopped and the full deformation data is
continued and shown in (b). An EBSD scan was not performed for this
sample.
0
0.0005
0.001
0.0015
0.002
0.0025
0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000
Strain
Time (sec)
Room Temperature Creep of 99.999% Polycrystalline Cu
Test Stress = 20 MPa
Linear Behavior
Sample was bumped
which resulted in a jump
in strain. Only the data
before this error is
analyzed.
0
0.0005
0.001
0.0015
0.002
0.0025
0 1,000 2,000 3,000 4,000
Strain
0 1,000,000 2,000,000 3,000,000 4,000,000 5,000,000
Time (sec)
Room Temperature Creep of 99.999% Polycrystalline Cu
Test Stress = 20 MPa
Linear Behavior
The time scale
changes at this point
in order to fit all data
on the graph.
41
(a).
(b).
FIGURE 2-16. The (a). creep deformation curve for Sample 5N-Cu-30MPa is shown along
with an (b). EBSD scan of the microstructure after creep for 11 days
performed in (a). The scan in (b) shows some grain refinement however the
overall microstructure seems to remain consistent to the starting
microstructure.
0
0.001
0.002
0.003
0.004
0.005
0.006
0 200,000 400,000 600,000 800,000 1,000,000
Strain
Time (sec)
Room Temperature Creep of 99.999% Polycrystalline Cu
Test Stress = 30 MPa
Linear Behavior
ε̇ =2 × 10
)**
s
)*
10
,
2 × 10
-
4 × 10
-
6 × 10
-
8 × 10
-
1 × 10
1
42
(a).
(b).
FIGURE 2-17. This figure displays all copper creep curves analyzed in this research. To
display the (a) linear behavior of all curves on one graph, the time scale is
“broken” into three sections. In (b), the logarithmic behavior of all copper
curves is shown. It should be noted that for 20 MPa, only the early data
(before pre-strain) is plotted on this graph.
1.E+04 5.E+05
Strain
Time (sec)
10
#
5 × 10
'
1 × 10
(
5 × 10
)
1 × 10
*
1 × 10
+
2 × 10
+
3 × 10
+
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.E+00 5.E+03 1.E+04 1.E+06 1.E+07 2.E+07 3.E+07
Room-Temperature Creep of 99.999% Polycrystalline Cu
Linear Behavior (with changing time scales)
30 MPa (5N-PX-Cu)
20 MPa (5N-PX-Cu)
18 MPa (5N-PX-Cu)
15 MPa (5N-PX-Cu)
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08
Strain
Time (sec)
Room Temperature Creep of 99.999% Polycrystalline Cu
Logarithmic Scale
15 MPa (5N-PX-Cu)
18 MPa (5N-PX-Cu)
20 MPa (5N-PX-Cu)
30 MPa (5N-PX-Cu)
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
43
2.6.2 99.99% SILVER CREEP CURVES
Individual silver creep experiments are described in the following subsections with their
corresponding curves shown in Figures 2-18 – 2-20. A quarter bridge strain gage was used for all
silver creep tests. The combined curves are shown in Figure 2-21 and the R
2
-correlation for all
silver curves are graphed in Figure 2-23.
2.6.2.1 4N-Ag-20MPa
The total duration of this creep test was 75 days; the curves for this load are shown in Figure 2-
18(a). Analyzing the R
2
-correlation (Figure 2-23), the deformation trend shows an undisputed
logarithmic behavior for the duration of testing. Interestingly, the trend looks to potentially peek
into a power law behavior after a period of almost 3 months. Longer creep tests would need to be
performed to confirm or refute that speculation. An ending creep-rate (𝜀), on the order of
10
) ,-
s
) ,
, was calculated from the data.
An EBSD scan of the grain texture after deformation is represented in Figure 2-18(b). As compared
to the original microstructure (refer to Figure 2-7(b)), this sample’s scan shows an excess of twin
boundaries. These are characteristic of a sample experiencing plastic strain. Because of the very
low applied stress, the EBSD scan of 4N-Ag-20MPa most likely shows damage artificially
introduced from the mechanical polishing done to prepare the sample for EBSD. If compared to
the higher loaded sample (4N-Ag-40MPa, discussed in the next two subsections), in which its
EBSD scan (refer to Figure 2-20(b)) shows a microstructure very similar to that of the original
equiaxed microstructure with grain sizes averaged 25µm. The preparation process was the same
for all samples, so an explanation for the damage could be that this sample had smaller diameter,
d = 3 mm, than the other samples. The other creep samples, copper and silver, had gage diameters
44
in the range of 4 – 6.35 mm. High-purity silver is a soft metal, therefore one can conclude that the
EBSD scan in Figure 2-18(b) is not representative of the actual microstructure after being
deformed at 20 MPa.
2.6.2.2 4N-Ag-30MPa
The total duration of this creep test was 78 days; the curves for this load are shown in Figure 2-19.
No EBSD scan was done for this sample. Consistent with the previous test done at 20 MPa, the
deformation trend, at 30 MPa, shows an undisputed logarithmic behavior. Analyzing the R
2
-
correlation for this sample, though a logarithmic trend looks to dominate, the curve could peek
into a power law dominated behavior at longer times. Extended creep tests would need to be
performed to confirm or refute that speculation as well. The ending creep-rate (𝜀) of this sample
is an order of magnitude faster than the previous silver creep test and is on the order of 10
) ,,
s
) ,
.
2.6.2.3 4N-Ag-40MPa
The total duration of this creep test was 28 days. The curve for this load is shown in Figure 2-
20(a). The EBSD scan for this sample is shown in Figure 2-20(b). It should be noted that the
current sample to be discussed is the 4N-Ag-30MPa sample, previously loaded to 30 MPa,
reloaded to 40 MPa. The creep specimen has been stored at room temperature for 6 months before
being reloaded to 40 MPa. Based on the EBSD scan of the gage section, after reloading to 40 MPa,
we observe a microstructure very similar to that of the original microstructure (refer to Figure 2-
7(b)) The trend shows to be power-law dominant, according to the R
2
-correlation (refer to Figure
2-23) for this sample. If the test were to be extended to longer times, the trend would appear to
flatten in the power-law dominant region. The ending creep-rate (𝜀) of this sample is an order of
magnitude faster than the previous silver creep test and is on the order of 10
) ,1
s
) ,
.
45
(a).
(b).
FIGURE 2-18. The (a) creep deformation curve for Sample 4N-Ag-20MPa shows the full
duration of data acquisition over a period of 75 days. The EBSD scan shows
many twins which can most likely be attributed to the mechanical
preparation for the EBSD scan. The strain-rate finished at 2 × 10
) ,-
s
) ,
.
0.0005
0.00055
0.0006
0.00065
0.0007
- 1,000,000 2,000,000 3,000,000 4,000,000 5,000,000 6,000,000 7,000,000 8,000,000
Strain
Time (sec)
Creep of 99.99% Polycrystalline Silver (Ag)
Creep Stress = 20 MPa
!̇ =2 × 10
)*+
,
)*
10
-
1 × 10
.
2 × 10
.
3 × 10
.
4 × 10
.
5 × 10
.
6 × 10
.
7 × 10
.
8 × 10
.
46
FIGURE 2-19. Creep deformation curve for 4N-Ag-30MPa. The applied load was 30 MPa
and the total duration of the test was 78 days. A logarithmic trend proves to
be dominate for this sample. The ending creep rate is 2×10
) ,,
s
) ,
with
strains maximized around 0.0015. An EBSD scan was not performed for
this sample
0
0.0005
0.001
0.0015
0.002
0.0025
- 1,000,000 2,000,000 3,000,000 4,000,000 5,000,000 6,000,000 7,000,000 8,000,000
Strain
Time (sec)
Creep of 99.99% Polycrystalline Silver (Ag)
Creep Stress = 30 MPa
!̇ =2 × 10
)**
+
)*
10
,
1 × 10
-
2 × 10
-
3 × 10
-
4 × 10
-
5 × 10
-
6 × 10
-
7 × 10
-
8 × 10
-
47
(a).
(b).
FIGURE 2-20. Creep deformation curves for Sample 4N-Ag-40MPa shows the full
duration of data acquisition over a period of 28 days. The creep test run at
40 MPa was performed on the silver sample previously crept at 30 MPa.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
- 400,000 800,000 1,200,000 1,600,000 2,000,000
Strain
Time (sec)
Creep of 99.99% Polycrystalline Silver (Ag)
Creep Stress = 40MPa
(This sample was previously crept at 30 MPa for 78 days; Sample 4N-Ag-30)
!̇ =2 × 10
)*+
,
)*
10
+
4 × 10
.
8 × 10
.
1.2 × 10
1
1.6 × 10
1
2 × 10
1
48
FIGURE 2-21. This figure displays the linear behavior all silver creep curves analyzed in
this research. The test were performed above and below the yield stress of
the silver used in this research: 20 MPa, 30 MPa and 40 MPa with ending
creep rates equal to 2 × 10
) ,-
𝑠
) ,
, 3 × 10
) ,,
𝑠
) ,
and 2 × 10
) ,1
𝑠
) ,
,
respectively. The creep test run at 40 MPa was performed on the silver
sample previously crept at 30 MPa.
0
0.00025
0.0005
0.00075
0.001
0.00125
0.0015
0.00175
0.002
0.00225
0.0025
0.00275
0.003
0.00325
0.0035
- 1,000,000 2,000,000 3,000,000 4,000,000 5,000,000 6,000,000 7,000,000 8,000,000 9,000,000
Strain
Time (sec)
Creep of 99.99% Polycrystalline Silver (Ag)
Linear Behavior
s
y
(Ag) = 25 MPa
20 Mpa
30 MPa
40 MPa*
!̇ =3 × 10
)**
+
)*
!̇ =2 × 10
)*-
+
)*
!̇ =2 × 10
)*.
+
)*
49
2.6.3 A UNIQUE TREND COMPARISON METHOD USING R
2
– CORRELATION VALUES
In the literature (Section 2.1), all R
2
values listed next to their respective creep curves (Figure 2-1
– Figure 2-4), represent the ending trend behavior, logarithmic or power-law. However, they do
not show how the behavior changes with time which would provide a better insight into the
preferred deformation behavior for low-temperature creep. For each creep curve, in the present
research, a comparison method between power-law and logarithmic behavior was created to
observe how the trend varies over time.
The trend comparison method was created by taking a R
2
– value at a point in time and subtracting
that value from the immediately previous R
2
– value. The comparison relationship at one point in
time is shown below,
ΔR
-
Logarithmic −ΔR
-
Power−Law = ±(value)
where ‘ΔR
-
Logarithmic ’ and ‘ΔR
-
Power−Law ’ represents the logarithmic correlation
value and power-law correlation value, respectively. The ‘± value’ distinguishes the dominant
trend at that specific point in time. A positive (+) value, that represents a logarithmic dominant
behavior; contrariwise, a negative (−) value, represents a power-law dominant trend, at that
specific point in time.
Regarding copper (refer to Figure 2-22), we observe a mainly power-law dominant relationship
for copper at a stress lower than the yield stress. However, for higher stresses and longer time
periods, a logarithmic trend seems to be favored. For silver, (refer to Figure 2-23), we observe a
mainly logarithmic dominant relationship for lower stresses. At higher applied loads, a power-law
trend seems to be favored. The sample loaded at the higher stress, 4N-Ag-40MPa, was originally
crept at 30 MPa, then stored at room-temperature before being reloaded to 40 MPa.
50
FIGURE 2-22. Here the combined R
2
-correlation analysis is shown for all copper creep
tests performed in this research. We observe a mainly power-law dominant
relationship, however for earlier times (< 30 minutes) the data is a bit
ambiguous. After a period of 30 minutes, at a stress lower than the yield
stress, a power-law dominant trend is observed, however, for higher stresses
and longer time periods, a logarithmic trend seems to be favored.
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000
DR
2
(Log) -DR
2
(PL)
Time (sec)
Analysis of Logarithmic and Power-Law Best-Fit Trends
[based on the change in Correlation Factor (R
2
)]
15 MPa (5N-PX-Cu)
18 MPa (5N-PX-Cu)
20 MPa (5N-PX-Cu)
30 MPa (5N-PX-Cu)
1 hour 1 day 1 month 1 year
10
0 10
1
10
2 10
3
10
4 10
5
10
6
10
7
10
8
Power-Law Dominant Logarithmic Dominant
51
FIGURE 2-23. Here the combined R
2
-correlation analysis is shown for all silver creep tests
performed in this research. We observe a mainly logarithmic dominant
relationship for lower stresses. At higher applied loads, a power-law trend
seems to be favored. The sample loaded at the higher stress, 4N-Ag-40MPa,
was originally crept at 30 MPa, then stored at room-temperature before
being reloaded to 40 MPa.
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 1,000,000 2,000,000 3,000,000 4,000,000 5,000,000 6,000,000 7,000,000 8,000,000
DR
2
(Log) -DR
2
(PL)
Time (sec)
Analysis of Logarithmic and Power-Law Best-Fit Trends
[based on the change in Correlation Factor (R
2
)]
20 MPa (4N-PX-Ag)
30 MPa (4N-PX-Ag)
Reloaded to 40 MPa (4N-PX-Ag)
(Previously crept at 30 MPa for 78 days)
1 month
10
0
1 x 10
6
2 x 10
6
3 x 10
6
4 x 10
6
5 x 10
6
6 x 10
6
7 x 10
6
8 x 10
6
2 months 3 months
Power-Law Dominant Logarithmic Dominant
52
2.7 DISCUSSION – PHENOMENOLOGICAL TRENDS
The earliest explanation of low-temperature creep by Mott and Nabarro [73, 81] was based on
dislocation glide and dislocation exhaustion. The term “exhaustion creep” is used by a variety of
investigators but the meaning is unclear. It appears that some common descriptions of exhaustion
creep are that at low temperatures we generally observe primary creep that never reaches a
mechanical steady state, a balance of hardening and recovery processes. It is often suggested that
the decrease in creep-rate is a result of a decrease in the mobile dislocation density. This could be
a consequence of an increase in the energy required to surmount obstacles because of dislocation
multiplication (i.e. hardening). Alternatively, the dislocation density may not change and the
distribution of the obstacle energy increases with plasticity without multiplication, instead arising
from a statistical change in the distribution of obstacles.
These descriptions often are placed in the context of the dislocation-intersection mechanism by
Seeger et al. [36, 37] where the concept of activation volume and area were probably first used.
The product of the activation area, A, Burgers vector, b, the difference between the applied stress,
t and the “back”, or long-range internal stress (LRIS), t
G
, due to other dislocations, is the energy
supplied. The activation area is usually defined as the product of the width of the obstacle, d, and
the obstacle spacing, ℓ
. Alternatively, the activation volume, V, equals Ab = ℓ b
2
. This leads to
the classic rate equation [19, 36, 82],
γ=NAbν
x
exp
) h
) )
mn
(E-2.6)
where = strain-rate, N = number of dislocation segments per unit volume held up at the
intersection points of mean spacing, ℓ, n
o
is an atomic frequency of the order of the Debye
! γ
53
frequency, = energy required for the intersection process, i.e. the energy for jog formation ≈
[56].
According to Conrad [19],
τ
l
=τ
l
x
+ hdγ
1
(E-2.7)
where is the stress due to the dislocations initially in the crystal and h is the strain-hardening
coefficient, which is defined as .
Conrad further suggested:
γ=NAbν
x
exp
) h
) )
)
mn
(E-2.8)
Integrating:
mn
exp
) h
) )
)
mn
=NAbν
x
t+D (E-2.9)
So:
) h
) )
)
mn
=ln (ν′t+D′) (E-2.10)
Where,
ν′=
z^
mn
Rearranging gives:
γ=
mn
ln ν′t+D′ −
h
T )
The constant D’ should be chosen so that 𝛾 0 =0. This requires:
ΔH
o
1
10
Gb
2
τ
G
o
dτ
dγ
54
D′=exp
ΔH
x
+ V τ−τ
l
kT
This can be manipulated into the form found by Conrad [19]:
γ=
mn
¡
ln (Ct+1) (E-2.11)
Where,
C=ν′exp −
ΔH
x
+ V τ−τ
l
kT
Equation E-2.11 is in the form of Eq. (E-1.11) and suggests logarithmic behavior throughout.
However, there are troubling aspects to this analysis. The logarithmic behavior stems from
Equation (E-2.7) which leads to the −ℎ𝛾 term in the numerator of the exponential in E-2.8. in
Eq. E-2.7 must be about zero [72, 83] and the inclusion of −ℎ𝛾 in Eq. 18 is unnecessary. Without
the inclusion of −ℎ𝛾, E-2.11 is impossible. Greater detail for the derivation of Eq. (E-2.7) is
presented [83] by the authors. Other attempts to justify this equation were made by Wyatt [30],
but the methodology was unclear. Welch et al. [84] appeared to attempt a similar approach to that
of Wyatt.
Perhaps, a more realistic approach may be found using the Seeger intersection mechanism equation
in a different way. The plastic shear strain is defined as:
γ
o
=xρ
A
b (E-2.12)
Assume γ
o
≈ 0.01.
Here:
ρ
A
≡ mobile dislocation density
τ
G
55
x ≡ average distance a dislocation moves
Assume x=
,
¥
¦
, where ρ
n
≡ total dislocation density.
ρ
A
=
§
¨^
=
1.1, ¥
¦
^
(E-2.13)
The Burgers vector (b) for Cu =2.55 × 10
) ©
cm.
In annealed copper, assume ρ
n
=ρ
x
≈2 × 10
©
cm
) -
.
Multiplication of dislocations is likely occurring so we can assume:
ρ
n
=M
x
γ+ρ
x
(E-2.14)
where M
o
is the dislocation multiplication constant and ρ
n
= 0.84 × 10
,,
cm
) -
𝛾+ρ
x
for
copper [85].
Starting with Seeger’s Equation (E-2.6), we can replace many of the quantities in terms of the
more fundamental quantities. Consider again Eq. (E-2.6) where,
A = activation area = d′ℓ≈ℓb
V = activation volume = Ab=ℓb
-
ℓ = average dislocation segment length =
,
¥
¦
=
,
¬
T¥
From [83, 86] we must assume 𝜏
®
= back-stress ≈ 0.
N ≡ number of dislocation links ≈
¥
¦
ℓ
ν
x
≡ vibrational frequency ≈ Debye Frequency ≈ 10
13
(for Cu)
The Debye Frequency for Cu can be calculated from the equation, ν
x
=
Zz
<¯
, Z
ν
°
.
Where (in copper),
ν
°
=4.6 × 10
,-
e±
f²³
,
56
z
, /Z
=
<
µ
¶
, /Z
=
<
(1.Z*, HA)
¶
, /Z
=4.397 nm
) ,
,
Z
<·
, /Z
=0.620
Leading to a Debye Frequency (ν
x
)=1.254 × 10
,Z
s
) ,
.
Substituting the above terms, into the Seeger Equation (E-2.6), leads to:
γ=b
-
ν
x
M
x
γ+ ρ
x
) , /-
exp
) h
) ^
¸
¬
T¥
¹N/¸
mn
(E-2.15)
Here the hardening occurs due to variation in the average dislocation segment length through the
M
x
γ+ρ
x
) , /-
term that appears in both the pre-factor and in the exponential. In both cases, due
to the negative exponent, it works to slow the strain rate as γ increases and thus acts as a hardening
term. Therefore, in this analysis, hardening is specifically linked to dislocation multiplication, as
at elevated temperatures.
The solution for γ(t) is most easily found numerically. However, while complicated, there is an
analytic solution. First, while one cannot integrate (E-2.15) to get γ(t), one can easily obtain the
inverse solution for t(γ):
t=
-º¨o
»¼
½¦
¬
^
¸
Ei −
^
¸
¥
¹N/¸
¿À
−Ei −
^
¸
¬
T ¥
¹N/¸
¿À
(E-2.16)
Where 𝐸𝑖 is the exponential integral:
Ei x =
º
¹Ã
D
Ä
) ¨
(E-2.17)
Substituting Ei x into t(γ):
t=
-º¨o
»¼
½¦
¬
^
¸
º
¹Ã
D
)
Å
¸
ÆÇ
¹N/¸
ÈÉ
)
Å
¸
Æ Ê
ËS Ç
¹N/¸
ÈÉ
(E-2.18)
57
Since Ei x is multiple valued, it is not invertible. However, in this problem the argument is always
negative and in that range, Ei x is single valued and invertible. Therefore, it is proper in this
context to assume that Ei
) ,
x exists, especially as Ei x and its inverse for negative arguments,
Ei
) ,
x , must ultimately be determined numerically:
γ=
,
¬
mn
^
¸
Ei
) ,
¬
^
¸
K
-º¨o
»¼
½¦
−Ei −
^
¸
¥
¹N/¸
mn
) -
−
¥
¬
(E-2.19)
Figure 2-24(a-b) shows creep predictions based on Eq. E-2.19 for combinations of applied stress
(𝜏) and activation energy (ΔH
x
), the two principal variables in Eq. E-2.19. If a reasonable value of
ΔH
x
is used, reasonable creep strains and times are predicted for a resolved shear stress of 3 MPa.
This translates to an applied tensile stress for polycrystalline Cu of about ,
which is roughly a factor of 4 lower than expected (3.06 is the Taylor factor to convert a perfectly
oriented shear strain to uniaxial tension of a polycrystal). If a more reasonable resolved shear stress
is used, such as 35 MPa, unreasonably high (factor of 10) activation energies of 1350 kJ/mol are
produced to achieve the experimental observations. Of course, dislocation pile-ups with just 2-3
dislocations could rationalize the lower than predicted stress based on the activation energy.
Additionally, we note for the most reasonable case, power-law behavior is predicted. Of course,
the power-law behavior is merely an approximation of the fundamental equation (E-2.19). The
Seeger model appears, overall, to very reasonably predict the Cu room temperature creep behavior.
If other metals, such as aluminum and silver, are examined, unrealistic stresses and/or strains are
predicted for Eq. (E-2.15). In aluminum, cross-slip may be an important thermally activated
mechanism for plasticity.
(3.06×3MPa)≈ 9MPa
58
Other creep mechanisms considered include quantum mechanical tunneling, which predicts an
athermal creep behavior at low temperatures. Early proponents included [20, 24, 25, 28].
Conversely, subsequent work by [16-18] suggests that, even to 4 K, creep is time-dependent and
is a result of the thermal activation of dislocations. Dislocation kink mechanisms were suggested
for BCC metals [18], however, it is unclear how this mechanism, by itself, explains the observed
creep behavior. In materials with solutes, it is suggested that creep occurs through thermal
activation past pinning solutes [14] in Al-Mg. The precise mechanism by which the strengthening
variables superimpose is unclear. In HCP metals, single slip may occur and the Seeger intersection
model may not apply.
Thus, poly-slip Cu may represent one of the few cases examined in this study where the Seeger
mechanism is applicable. For Cu, the fundamental physics may be the dislocation intersection
mechanism as in Eq. (E-2.19). Equations (E-1.10) and (E-1.11) are merely simple forms that
approximately fit the data. Nonetheless, equations (E-1.10) and especially (E-1.11), reasonably
describe and predict low-temperature creep behavior in all the metal and alloy systems examined.
Increasing creep resistance at low temperatures appears to be accomplished in similar ways as at
elevated temperatures. For example, cold work increased the creep resistance in 304-stainless
steel at both low and elevated temperatures [34]. Others have suggested the role of other features
[4] such as twin boundaries.
59
a).
b).
FIGURE 2-24(a-b). The predicted strain vs. time plots for ambient-temperature creep of Cu
based on the dislocation intersection mechanism through Eq. (E-2.19). The
most realistic behavior is for DH
o
= 135 kJ/mol and a critical resolved shear
stress, t = 3 MPa. Power-law behavior (a) appears to better describe the
predictions of Eq. (E-2.19) than logarithmic behavior (b).
!
R² = 0.89172
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000
γ
Time (sec)
Logarithmic
ΔH
o
= 135 kJ/mol
τ = 3 MPa
Predicted Cu Data Line based on Seeger Mechanism
!
R² = 0.9846
0.0001
0.0010
0.0100
0.1000
1.0000
1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000
γ
Time (sec)
Power-Law
ΔH
o
= 135 kJ/mol
τ = 3 MPa
Predicted Cu Data Line based on Seeger Mechanism
60
FIGURE 2-25. Activation energy for (steady-state) creep of Ag, Ni, Cu and Al as a function
of temperature. Adapted from [87, 88].
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Q
creep
/Q
SD
T/T
m
Activation Energy Ratios for Creep and
Self-Diffusion vs. Melting Temperature Ratios
of Pure Metals
Silver
Nickel
Aluminum
Copper
61
Figure 2-25 [87, 88] shows that the activation energy decreases with temperature below 0.30 T
m
.
The values are much lower than at higher temperatures where the activation energy, Q, is
associated with lattice self-diffusion (dislocation climb). The lower temperature activation
energies suggest some mechanism other than dislocation climb may be active.
The different metals may be associated with different rate-controlling processes. In Cu, for
example, the dislocation intersection mechanism (Seeger) where poly-slip is expected in grains
and negligible solute strengthening is likely occurring. In other poly-slip materials (e.g. steels)
solute strengthening in addition to dislocation intersection mechanisms may be active. How these
superimpose in a mathematical description is unclear. If single slip occurs (e.g. pure HCP at low
temperatures) glide-controlled mechanisms may be relevant.
The experimental literature on low-temperature creep (T < 0.30 T
m
) of a variety of metals and
alloys was examined and combined with new low-temperature creep tests performed on pure Cu.
In addition, theoretical studies were examined and a new derivation, based on the dislocation
intersection model by Seeger was performed that lead to power law behavior, rather than the
logarithmic behavior found by Conrad. Although both power law and logarithmic models can
reasonably model the creep behavior in various conditions, neither is perfect.
The logarithmic form generally appears to more reliably describe the low-temperature creep
behavior of most of the metals and alloys analyzed (Cd, ferritic steel, stainless steel). Most of the
Ti-alloys are better described by power-law equations and Cu and Al may be approximately
equally well described by logarithmic and power law equations. The equation forms may allow
the extrapolation of reliable creep behavior to longer times. Nonetheless, the power-law and
logarithmic behavioral trends are simple curve fits and not reflections of the fundamental physics.
62
CHAPTER 3. LONG-TERM ANNEALING OF HIGH-PURITY
ALUMINUM SINGLE CRYSTALS: NEW INSIGHTS
INTO HARPER-DORN CREEP
3.1. INTRODUCTION
Annealing times are relatively short (< 50 hours and usually on the order of just one hour) in
comparison to the present study (up to 1 year). Typical annealed dislocation densities for short
anneals (1-2 hours) are in the range of 10
9
– 10
11
m
-2
and for longer (10-50 hours) anneals, densities
are in the range of ~10
8
– 10
10
m
-2
. The dislocation density decreases with annealing time at high
temperatures, but very long annealing times of up to a year do not appear to have ever been
performed.
Long-term annealing at 0.98T
m
of <100> and <111> oriented 99.999% (5N) and 99.9999% (6N)
pure aluminum single-crystals was investigated in order to analyze the decrease in dislocation
density with various times up to one year. These are the longest annealing times in the literature.
The existence of a “frustration limit” of the dislocation density, suggested by Ardell and coworkers
[46, 47, 49] for Harper-Dorn creep (low-stress and generally very high temperatures (e.g. 0.98T
m
)
[57]), in which the dislocation density does not decrease below a certain value (even at very low
stress), is, thus, also examined in this work. Many investigators have observed a dislocation density
that is independent of stress at low dislocation density values in the so-called Harper-Dorn creep
regime [47, 58, 60]. Others have observed a density that continually decreases with stress
consistent with traditional “five-power law” creep [50, 52, 54, 55, 67]. Thus, an ancillary purpose
of this annealing study is to check the concept of a frustration limit for low stress, high-
temperature, creep.
63
3.2. MATERIALS
Single crystal aluminum samples were bought from Material–Technologie & Kristalle GmbH
(MaTecK), Jülich, Germany with a purity of 99.999%, with cylindrical dimensions of 12.7 cm
length x 25.4 mm diameter and an orientation of <100> or <111> along the length of the cylinder
(Figure 3-1). The crystals were not pre-strained in any way.
3.7.1 SAMPLE PREPARATION
The Al single crystals were cut into individual pieces of 6-19 mm length x 25.4 mm diameter using
the IsoMet® 1000 Precision Saw from Buehler. A 0.305 mm thick diamond saw wafer-blade was
used at a slow speed of 100 rpm with oil-based lubricants. Seven specimens were used in the
annealing experiments. Polishing the samples consisted of using both mechanical and electro-
polishing techniques. Dislocation densities were then determined by surface etching.
The sample cutting done with the diamond saw (even at its lowest speed), causes considerable
mechanical damage as the Al is soft. The amount of material initially taken off from polishing is
recorded in the following subsections.
Mechanical Polishing
Two mechanical polishing techniques were utilized to remove the cutting damage from the IsoMet
on each sample: 1). silicon carbide grit paper and 2). diamond suspension slurry.
The Ecomet III Wet Polisher/Grinder was used with varying grades (320-1200) of 20 cm diameter
CarbiMet silicon carbide grit paper discs from Buehler.
64
FIGURE 3-1. As-received aluminum crystal dimensions and orientation.
127-mm
25-mm
[111]
or
[100]
Al-5N
As-received crystal
Diamond blade saw
was used to cut
perpendicular to the
primary axis.
[NOT TO SCALE]
65
Deionized (DI) water was constantly flowing to wet the grit paper to avoid more mechanical
damage to the sample as a result of excess friction. The samples were washed with DI water in
between each step and compression air-dried to prevent any water spots or small amounts of
surface oxidation.
During grit paper polishing, the samples were kept in the same position to create unidirectional
striations. Samples were rotated 90° when changing grit paper size. When previous striations were
no longer visible then the next grit paper size was used and the process was repeated as shown in
Figure 3-2. This allowed for clear indication that the mechanical damage layer was being removed
as polishing continued.
Buehler MetaDi™ monocrystalline diamond suspensions, from 15 µm to 0.25 µm particle sizes,
on a Struers LaboPol-2 230 mm diameter magnetic polishing wheel, were sequentially used to
further remove damage from the previous grit paper. DI water was also used with the diamond
slurries. The samples were washed with DI water between each grinding step and compression air-
dried to prevent any water spots or small amounts of local surface oxidation. Specimens were
stored in vacuum.
For diamond slurry polishing, the samples were moved in a non-uniform manner to assure that the
diamond particles were always hitting the sample surface at a different angle. This allowed
smoother and more uniformity on the surface. After the last stage of mechanical polishing (0.25
µm slurry), the crystals had a semi mirror-like appearance. An optical microscope was used to
verify if more polishing needed to be done based on the width of scratches on the surface. Any
scratches with a width greater than 0.25 µm would suggest that the sample would have to be re-
polished based on their size (i.e. a sample with a scratch of width 8 µm would have to be polished
again starting at the 9 µm slurry).
66
FIGURE 3-2. Polishing technique using SiC grit paper: a). Holding the sample in one
direction starting with 320 grit; b). Switch grit paper to next grade (e.g. 400
grit), rotate 90° and continue polishing holding in one direction; c). Previous
abrasion lines from the 320 grit are gone and only the 400 grit lines remain.
Polishing continues this way through all grades of grit papers.
!
Polishing direction
Change of grit paper
Rotate 90°
Polishing direction Polishing direction
Abrasion Lines
a). b). c).
67
3.7.2 CHEMICAL ANALYSIS
Electro-polishing
Electro-polishing was used to remove any remaining traces of surface damage from mechanical
polishing and give the aluminum crystals a virtually damage free surface. Electropolishing is the
process in which a galvanic cell is used to remove electrons from the surface of the desired sample
(anode) and plate them on a “dummy” sample (cathode). Generally, the cathode is another piece
of aluminum. The setup is shown in Figure 3-3.
The solution used was a 1:9 perchloric acid (70% ACS reagent) to methanol (99.8% anhydrous)
mixture, respectively [53, 89-91]. Both chemicals were purchased from Sigma-Aldrich. The
conditions for electro-polishing were determined by performing a ramping curve from 5 V to 11
V at a ramp rate of 0.1 V/s and an acquisition frequency of 300 Hz. Electrical tape was used to
wrap around the sample leaving only the face of the sample needing to be polished. This helped in
reducing the amount of current needed due to the lack of surface area to be polished. A common
current density (amps/cm
2
) vs. voltage (volts) curve includes four (sometimes three) distinct
sections in which the plateau region represents the voltage and current necessary for electrolytic
polishing of a sample. The curve done in our laboratory was performed at the temperature used
during actual polishing (0-5°C). The ramping curve (Figure 3-4) showed optimal polishing
operating conditions. The tests were run at 10 V, specifically.
It was also determined through various trials that running the experiment at cold temperatures (0°C
– 5°C) worked best because it kept a fresher solution for longer periods. Previous studies used
liquid nitrogen to keep a cool temperature (< 0°C). However, for my purposes, packing ice around
the galvanic cell worked well. Table 3.1 shows the amount of material removed through
mechanical and electro-polishing of each sample.
68
TABLE 3.1. Initial thickness measurements for all aluminum samples used in the present
research.
Sample
Purity and
Orientation
Initial
thickness after
cutting (mm)
Final thickness
before annealing
(mm)
Total material
removed from
polishing
(mm)
Group 1
S-1
99.999%
<111>
6.50 4.53 1.97
S-2 6.24 4.52 1.72
S-3 7.69 5.53 2.16
S-4 5.80 5.02 0.78
S-5 5.85 5.29 0.56
Group 2
S-A
99.999%
<111>
19.00 16.46 2.53
S-B 19.00 18.12 0.88
S-C 130.05 125.46 4.59
Group 3
S-a
99.999%
<111>
18.13
S-b 19.76
S-g 88.79
S-6N
99.9999%
<100>
20.00 18.90 1.10
69
FIGURE 3-3. Shown here is the electro-polishing setup for the single crystal aluminum
used in this research. A solution of 1:9 perchloric acid (70% ACS reagent)
to methanol (99.8% anhydrous), respectively, was used. The voltage was
set at 10 V leading to an approximate current of 1.6 mA. A stirring bar was
constantly running, providing smoother current flow through the solution.
DC Power Supply
(~10 V, ~1.6 mA)
HClO4 and HCl polishing solution
e-
Stirring Bar
e-
Ice Bath
Anode Cathode
Electrical Tape
Aluminum Sample
70
Vigorous stirring with a magnetic stir bar was used to ensure even dispersal of ions from the anode
surface (i.e. polishing). Samples exuded a mirror-like appearance and an optical microscope was
used to ensure they were free of any scratches or pits from mechanical polishing. The comparison
of surface images before and after electropolishing is show in Figure 3-5.
Chemical Etching
The etching solution, adapted from [47, 48, 53, 62, 92, 93] is a 50:47:3 mixture of hydrochloric
acid (37 vol% ACS Reagent), nitric acid (70 vol% ACS Reagent) and hydrofluoric acid (48 vol%
ACS Reagent), respectively. The mixing of the chemicals is very volatile and had to be done very
carefully and quickly so as not to ruin the aluminum crystals. Mixing HCl and HNO
3
was done
first in a small beaker encapsulated in a larger beaker with ice. This was done to slow down the
exothermic reaction known as ‘Aqua Regia’ and take on a dark reddish color. If that happened the
solution was unacceptable and would cause corrosion to the sample surface. The HCl-HNO
3
solution was slowly stirred to allow full mixing. After about 2-3 minutes, the hydrofluoric acid
was added last immediately before etching the samples.
Using metal tongs, the samples were etched 5-7 seconds and only submerged enough for the
surface to slightly penetrate the etchant. After a sample was etched it was immediately washed
with DI water and compressed-air dried to prevent even slight surface oxidation. This process was
repeated for all samples. It should also be noted that sometimes the etchant would expire before
all samples were finished and therefore a fresh etchant would need to be made before finishing. If
a sample were corroded (from etching too long), electro-polishing would need to be performed
again, to remove the corroded layer, then re-etched.
71
FIGURE 3-4. Ramping current (A) vs. voltage (V) curve to determine the electro-
polishing conditions for single crystal aluminum at 0-5°C.
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
5 6 7 8 9 10 11 12
Current Density (A/cm
2
)
Voltage (V)
Current(I) - Voltage(V) Ramping Curve
99.999% Single-Crystal Aluminum
Passivation
Etching
Polishing
Pitting
72
(a).
(b).
FIGURE 3-5. The results of mechanical and electro-polishing as seen under an optical
microscope; a). shows the surface of an aluminum test sample after completing all
mechanical polishing finishing with 0.25 µm diamond slurry and b). the surface of
the test sample after electro-polishing. It should be noted that the soft-grey
markings in each micrograph are from the optical microscope lens and are not on
the sample surface itself.
73
3.3. LONG-TERM ANNEALING FURNACES
The melting point of aluminum is 661°C. The temperature on each sample varied ±6˚C and
therefore the range of temperatures is 640˚C to 653˚C (0.96 – 0.99 T
m
) with the average
temperature about equal to 646˚C. A Type-K thermocouple on each sample was used together with
an 8-channel thermocouple data acquisition unit from Omega Engineering. The temperature
readings were recorded using a desktop computer and each sample had an individual soldered
thermocouple. A description of each annealing group is outlined in the results sections.
Two non-vacuumed furnaces were used in the present research:
1). For the last nine months of Group 1 and the entirety of Group 2, the smaller desktop furnace
(Figure 3-6) was used and controlled externally by a Sigma Digital PID (three-term regulator)
MDC4E temperature control unit. This unit has a separate Type-K thermocouple probe in the
furnace. The samples again had individual Type-K thermocouples to track the temperature.
2). A 3-550 Vulcan Multi-Stage Programmable Furnace was used (Figure 3-7) for the first 3
months of Group 1 as well as the entirety of Group 3. A ramp rate of 20°C/min was programmed
and when the temperature was reached, it held at 646°C for as long as desired.
74
FIGURE 3-6. Table top furnace controlled externally by a Sigma Digital PID (three-term
regulator) MDC4E temperature control unit. This unit has a separate Type-K
thermocouple probe in the furnace. The samples again had individual Type-K
thermocouples to track the temperature readings.
75
FIGURE 3-7. A 3-550 Vulcan Multi-Stage Programmable Furnace was used. A ramp rate of
20°C/min was programmed and when the temperature was reached, it held at 646°C
for as long as desired. A Type-K thermocouple was used together with an 8-channel
thermocouple data acquisition unit from Omega Engineering. The temperature
readings were recorded using a desktop computer.
76
3.4. RESULTS
Group 1 (Samples S-1 – S-6) had multiple dislocation density checkpoints: (1) 0 days (initial
count), (2) 3 days, (3) 10 days, (4) 1 months, (5) 3 months, (6) 6 months and (7) 1 year. Electro-
polishing was performed at each checkpoint to remove previous etch pits and the oxidation layer
from heat treating in air, then re-etched to acquire a new dislocation density.
Group 2 (Samples S-A, S-B, S-C) left the samples in the furnace uninterrupted for a period of time.
Dislocation density measurements were only taken at the beginning. These were done in order to
confirm, or refute, the results of the first cycle. Samples A and B were considerably thicker than
the samples from the first cycle. Sample C was an uncut, as-received aluminum single crystal
cylinder. The polishing and etching routine for this cycle stayed consistent as in the first cycle.
Sample C is used to determine the dislocation density difference between the top and underside of
the cylinder. Both sides were polished and the total sample mass (after polishing) is 155.8678
grams. The mass of the sample before polishing was 157.4494 grams. Therefore, using equation
(2), the pressure on the underside of Sample C is 3.314 kPa. However, Group 2’s final density
measurements could not be taken because, the samples melted do to an arc in the table-top furnace.
It caused the furnace to spike in temperature and, as a result, the samples melted.
Group 3 (Samples S-a, S-b, S-g, S-6N) was left in the furnace, uninterrupted, for a period of 6
months. Dislocation density measurements were only taken at the beginning and at the end.
77
3.4.1. DISLOCATION DENSITY MEASUREMENTS BY ETCH PIT ANALYSIS
The geometry of the etch pits are determined by the orientation of the crystals when they are grown.
In this case, the aluminum single crystals were a <111> orientation and therefore, their etch pits
are visible as triangles. Refer to Figure 3-8, where a is the lattice parameter of the unit cell. Upon
analysis, a desirable etch pit is classified by a near-perfect geometric triangle. If the etch pit is
jagged on the edges or circular, then this could be characteristic of corrosion pits or over-etching
of the sample. This is problematic because it can result in inaccurate density measurements.
The dislocation density is calculated by the below equation (E-3.1),
ρ=
-z
(E-3.1)
where N is the number of etch pits and A is the area in which the etch pits are counted. The etch
pits can only be accurately determined after etching the crystal surface following electro-polishing.
The image area was calculated using the scale bar on each of the images. Some of the images used
for calculations are displayed in Figures 3-9 – 3.12. All images cannot be shown because of the
great amount; however the images shown, are representative of all samples. An initial density
measurement was taken with an optical microscope, on the aluminum crystals. Measurements were
taken at multiple magnifications and with multiple images to achieve an accurate average across a
sample.
A summary of the tabulated values for the average dislocation density measurements for each
sample is listed in Table 3.2. Please note that those values are based on all the images taken for a
sample and not just the ones shown in this thesis.
78
FIGURE 3-8. Schematics of <111> (left) and <100> (right) directions of crystal orientation
leading to pyramidal and cube, respectively, shaped etch pits where a represents
the lattice parameter and the red represents the etch pit shapes.
!
x
y
z
1
1
1
a
1
1
1
a
79
TABLE 3.2. Initial dislocation densities averaged across samples used in this research.
Etch Pit Density (etch pits/m
2
)
Sample Small Etch Pits Large Etch Pits Group Averages
Group 1
S-1 1.45 × 10
>
3.4 × 10
>
S-2 1.02 × 10
,1
3.52 × 10
>
S-3 1.06 × 10
>
S-5 7.39 × 10
©
Group 2
S-A 9.30 × 10
,1
2.74 × 10
>
4 × 10
,1
S-B 1.38 × 10
,,
6.93 × 10
>
S-C 2.04 × 10
>
Group 3
S-a 3.04 × 10
>
3 × 10
>
S-b 6.30 × 10
©
S-g 1.17 × 10
,1
1.65 × 10
>
S-6N 2.42 × 10
>
5.22 × 10
©
*1 × 10
>
*The average value for S-6N is not included in the Group 3 average and must be analyzed
seperately from the 5N samples because of the difference in purity.
Note: A blank box represents a value of ‘0’.
80
(a).
(b).
FIGURE 3-9 (a-b). Etch Pit micrographs taken with an optical microscope of Samples S-2 (top,
a) and S-4 (bottom, b) from Group 1 before annealing. The images were
taken after the sample surfaces had been electropolished and chemically
etched.
81
(a).
(b).
FIGURE 3-10 (a-b). Etch Pit micrographs taken with an optical microscope of Samples S-A (top,
a) and S-C (bottom, b) from Group 2 before annealing. The images were
taken after the sample surfaces had been electropolished and chemically
etched.
82
(a).
(b).
FIGURE 3-11 (a-b). Etch Pit micrographs taken with an optical microscope of Samples S-b (top,
a) and S-g (bottom, b) from Group 3 before annealing. The images were
taken after the sample surfaces had been electropolished and chemically
etched.
83
(a).
(b).
FIGURE 3-12 (a-b). Etch Pit micrographs taken with an optical microscope of Sample S-6N
from Group 3 at 5X (top, a) and 10X (bottom, b) magnifications, before
annealing. The images were taken after the sample surface had been
electropolished and chemically etched.
84
3.4.2. DISLOCATION DENSITY MEASUREMENTS BY TEM ANALYSIS
Transmission electron microscopy (TEM) was done to either confirm or refute the etch pit data.
Etch pits have been speculated [94-97] to show only a fraction of the dislocations, in some cases,
as low as 10% of the dislocations were shown in, but not limited to, beryllium, silver and iron
alloys through etch pit analysis. In aluminum, specifically, 60-70% of the dislocations create an
etc pit. However, TEM shows all dislocations. Three samples were analyzed using TEM.
A Model 110 Fischione twin jet electro-polisher was used for electro-polishing. Both the sample
disk and the nozzles are submerged in a 25 vol% of Nitric Acid (HNO
3
; 70% ACS reagent) in 200-
proof ethanol (C
2
H
5
OH) solution. Both chemicals were purchased from Sigma-Aldrich A
description of the twin jet polisher is described in the previous chapter.
Sample Delta (S-∆) (as-received; unannealed)
One sample is a part of the as-received <111> 99.999% (5N) aluminum rod (refer to Section 3.2).
This as-received sample is also unannealed and will be labeled S-∆. Its dimensions are shown in
Figure 3-14. A dislocation density was taken on sides A, B and C (Figure 3-15) by etch pit analysis
before slicing it for TEM. Figure 3-16 shows the process of making a TEM sample from S-∆.
Sample S-∆ was sliced into ~2-mm slabs parallel to Side C (bottom surface of S-∆; refer to Figure
3-13). The disks were ultrasonically cut into 3-mm diameter disks. Those disks were further
mechanically polished to 1200 SiC grit. The final thicknesses of the TEM disks were fairly thick,
averaging 500 𝜇m. The foils were finally jet polishing to penetration before being viewed in the
85
TEM. The jet-polishing solution was a 20–25 vol% solution of 70wt% nitric acid (HNO
3
) in 200-
proof ethanol (C
2
H
5
OH).
A total of three TEM foils were made from Sample S-∆. The TEM images of these foils are shown
in Figures 3-16 – 3-18. Additionally, a summary of their dislocation densities are shown in Table
3.3.
Sample S-5 (Group 1; annealed 1 year) and Sample S-𝛾 (Group 2; annealed 6 months)
The other two samples analyzed by TEM are S-5 and S-𝛾. A Model 170 Ultrasonic Disk Cutter
was used with 800-SiC grit abrasive slurry to cut 3-mm cylindrical sections out of the samples. A
wire saw was used to slice thinner disks from the cylindrical cutout. Those disks were further
mechanically polished to 1200 SiC grit. The final thicknesses of the TEM disks were fairly thick,
averaging 500 𝜇m. The foils were finally jet polishing to penetration before being viewed in the
TEM. The jet-polishing solution was a 20–25 vol% solution of 70wt% nitric acid (HNO
3
) in 200-
proof ethanol (C
2
H
5
OH).
86
FIGURE 3-13. Sample dimensions for S-∆. A dislocation density was taken on sides A, B
and C by etch pit analysis. The crystal orientation for each plane is notated
in the figure and was determined by EBSD.
B
C
Sample S – ∆
Sample Dimensions after Polishing
20-mm
15-mm
10-mm
11-mm
A
[NOT TO SCALE]
{ 7
̅
25 }
{ 7 29 }
{ 111 }
87
(a).
(b).
(c).
FIGURE 3-14. Etch pit micrographs taken in SEM of (a) side A, (b) side B and (c) side C
of Sample S-∆. The dislocation densities for all sides were on the order of
10
9
m-
2
.
88
FIGURE 3-15. Schematic shows how a TEM sample was made from Sample S-∆. (a)
Sample S-∆ was sliced into (b) ~2-mm slabs parallel to Side C (bottom
surface of S-∆). The slabs were then ultrasonically cut into (c) 3-mm
diameter disks. Those disks were further polished to a TEM foil.
Sample S – ∆
TEM Specimens:
S –∆_TEM-(1-4)
A
∥ to C
Slab rotated ⊥ to
Surfaces ‘A’ and ‘C’,
and ∥ to Surface ‘B’
3-mm
1-mm
[NOT TO SCALE]
(a).
(b).
(c).
89
FIGURE 3-16. TEM image of S-D_TEM-1. Dislocation link lengths averaged 1 – 2µm and
the density of the disks were on the order of 10
12
lines/m
2
.
FIGURE 3-17. TEM image of S-D TEM-2. Dislocation link lengths averaged 1 µm and the
density of the disks were on the order of 10
12
lines/m
2
.
90
(a).
(b).
FIGURE 3-18 (a-b). TEM image of S-D TEM-3. This disk was mechanical polished to a thin
disk and then re-annealed for 6 hours at 650℃ (0.98T
m
) before jet-polishing.
Dislocation link lengths were 250 – 500 nm on average as seen in (a);
however there were many images where no dislocations were observed (b).
91
TABLE 3.3. TEM dislocation density measurements for Sample S-D used in this research.
Dislocation Line Density (lines/m
2
)
Samples
Annealing
Time
Dislocation Line
Density
(dislocation lines /
m
2
)
TEM disk
Specimen
Averages
Overall TEM
Sample
Averages (m
-2
)
S-D_TEM-1 Unannealed
1.41 × 10
,Z
1 × 10
,Z
𝟓 × 𝟏𝟎
𝟏𝟐
1.48 × 10
,Z
7.32 × 10
,-
8.13 × 10
,-
S-D_TEM-2 Unannealed
1.92 × 10
,-
5 × 10
,-
3.75 × 10
,-
3.00 × 10
,-
1.05 × 10
,Z
S-D_TEM-3
6 hours
(after TEM prep)
3.09 × 10
,-
4 × 10
,-
9.74 × 10
,-
1.01 × 10
,Z
4.18 × 10
,-
3.14 × 10
,-
0
0
S-D_TEM-4
6 hours
(after TEM prep)
0 0
92
(a). (b).
(c).
FIGURE 3-19 (a-c). High Resolution TEM images taken with a JEOL 7100F of Sample S-5
(annealed 1 year) from Group 1. Dislocation link lengths averaged 50 nm
and the density of the disks were on the order of 10
13
lines/m
-2
.
93
FIGURE 3-20. TEM image taken with a JEOL 7100F of Sample S-𝛾 (annealed 6 months)
from Group 3.
94
TABLE 3.4. TEM dislocation density measurements for Sample S-5 used in this research.
Dislocation Line Density (lines/m
2
)
Samples
Annealing
Time
(dislocation
lines / m^2)
TEM disk
Specimen
Averages
Overall TEM
Sample Averages
S-5_TEM-1 1 year
2.45 × 10
,Z
3 × 10
,Z
𝟏 × 𝟏𝟎
𝟏𝟑
0
2.74 × 10
,Z
1.37 × 10
,Z
1.88 × 10
,Z
0
1.42 × 10
,<
3.79 × 10
,Z
S-5_TEM-2 1 year
4.24 × 10
,,
4 × 10
,-
4.42 × 10
,-
0
4.32 × 10
,-
8.64 × 10
,-
3.68 × 10
,-
3.82 × 10
,-
2.69 × 10
,-
S-5_TEM-3 1 year
4.00 × 10
,,
7 × 10
,1
0
0
0
0
0
5.87 × 10
,-
4.35 × 10
,,
TABLE 3.5. TEM dislocation density measurements for Sample S-𝛾 used in this research.
Dislocation Line Density (lines/m
2
)
Samples
Annealing
Time
(dislocation
lines / m^2)
TEM disk
Specimen
Averages
Overall TEM
Sample Averages
S-𝛾_TEM-1 6 months 0 0
𝟐 × 𝟏𝟎
𝟏𝟑
S-𝛾_TEM-2 6 months
7.69 × 10
,Z
3 × 10
,Z
2.31 × 10
,Z
9.23 × 10
,Z
5.00 × 10
,-
1.23 × 10
,Z
0
0
95
TABLE 3.6. This table shows the TEM specimens’ initial disk thickness as well as their
individual preparation methods. Their final dislocation densities and final
thicknesses are listed as well.
Sample Name
Initial Slab
Thickness (mm)
Initial Disk
Thickness (𝛍m)
Grinding
Method
Final Disk
Thinness
(𝛍m)
Dislocation
Density
(lines/m
2
)
S-D_TEM-1 2.49
Wheel
(1 µm)
Then TEM punch
250 1 × 10
,Z
S-D_TEM-2 1.70
Wheel
(1 µm)
490 5 × 10
,-
S-D_TEM-3
(reannealed after polishing)
1.70
Wheel
(1 µm)
550 4 × 10
,-
S-𝛾_TEM-1 450 None 450 0
S-𝛾_TEM-2 2040
Hand
(1200 grit)
510 3 × 10
,Z
S-5_TEM-1 3140
Wheel
(1 µm)
210 3 × 10
,Z
S-5_TEM-2 2450
Wheel
(9 µm)
500 4 × 10
,-
S-5_TEM-3a 660
Hand
(800 grit)
590 7 × 10
,1
96
FIGURE 3-21. Log vs log graphical representation of the dislocation densities over time
for the samples used in the present study. The graph represents the
individual dislocation measurements across a sample (colored points). The
averages across a time period for the TEM samples (red crosses) and the
etch pits (blue pluses) are represented as well.
1E+0
1E+1
1E+2
1E+3
1E+4
1E+5
1E+6
1E+7
1E+8
1E+9
1E+10
1E+11
1E+12
1E+13
1E+14
0 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000100,000,000
Dislocation Density (lines/m
2
)
Time (sec)
Dislocation Densities in Single Crystal Aluminum vs. Annealing Time
(Etch Pit vs TEM Measurements)
5N (Small Etch Pits)
5N (Large Etch Pits)
6N (All Etch Pits)
TEM Samples
Sample Averages (TEM)
Sample Averages (Etch Pits)
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
97
FIGURE 3-22. Log vs log of the dislocation densities over time comparing the present
study results with literature values. The graph represents the individual
dislocation measurements across a sample (colored points). The averages
across a time period for the TEM samples (red crosses) and the etch pits
(blue pluses) are represented as well. (For reference, the yellow data point
is about 30 days.
1E+0
1E+1
1E+2
1E+3
1E+4
1E+5
1E+6
1E+7
1E+8
1E+9
1E+10
1E+11
1E+12
1E+13
1E+14
0 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000100,000,000
Dislocation Density (lines/m
2
)
Time (sec)
Dislocation Densities in Single Crystal Aluminum in Relation to Annealing Time
(The Present Study compered to Literature Values)
5N (Small Etch Pits) 5N (Large Etch Pits)
6N (All Etch Pits) TEM Samples
Sample Averages (TEM) Sample Averages (Etch Pits)
[48] [49]
[44] [45]
[43] [47]
[46]
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
98
FIGURE 3-23. Log vs log of the dislocation densities over time. This graph shows the range
of dislocation density values at a given time. The data from the current study
(red range bars) are compared with the range of literature values (green
range bar). The average of the 5N purity (red crosses) and 6N purity (purple
crosses) aluminum samples are represented along with the average literature
value (green diamond).
1E+0
1E+1
1E+2
1E+3
1E+4
1E+5
1E+6
1E+7
1E+8
1E+9
1E+10
1E+11
1E+12
0 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000
Dislocation Density (line length/m
2
)
Time (sec)
Dislocation Densities in Single Crystal Aluminum vs. Annealing Time
Present Study - 5N Sample Averages
Present Study - 6N Sample Averages
Literature Averages
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
Literature
Averages
Present
Study
3 days 10 days 1 month 6 months 1 year
99
TABLE 3.7. Dislocation densities (by etch pits) associated with each annealing time period.
Time
Average Dislocation Densities (r)
5N
Average Dislocation Densities (r)
6N
Initial
8.76 x 10
7
m
-2
(avg.)
range: (8.76 x 10
7
– 1.02 x 10
10
m
-2
)
1.28 x 10
9
m
-2
(avg.)
range: (3.54 x10
8
– 4.01 x10
9
m
-2
)
3 days 6.55 x 10
7
m
-2
10 days 7 x 10
8
m
-2
30 days 3 x 10
9
m
-2
3 months 1.88 x 10
7
m
-2
6 months 5.05 x 10
8
m
-2
1.98 x 10
9
m
-2
1 year 3.7 x 10
9
m
-2
100
3.5. DISCUSSION
Single crystals of 99.999 and 99.9999% pure aluminum were annealed at high elevated
temperatures (0.98T
m
) for relatively long times of up to one year, the longest in the literature.
Remarkably, the dislocation density remains relatively constant at a value of about 10
9
m
-2
over a
period of one year. The stability suggests some sort of “frustration” limit. This may have
implications towards the so-called “Harper-Dorn creep” that generally occurs at fairly high
temperatures (e.g. > 0.90T
m
) and very low stresses. The observed “frustration limit” in this study
is on the order of the dislocation density of literature steady-state flow in the one-power law
(Harper-Dorn) creep regime. A constant dislocation density with changing applied stress may lead
to one-power law behavior. Perhaps more normal five or three power law creep occurs when the
initial dislocation density is low (e.g. << 10
9
m
-2
) and increases to a steady-state value. Perhaps
starting with a higher, frustrated, dislocation density leads to a constant dislocation density with
decreasing stress.
Table 3.8 lists the dislocation densities that are also plotted in Figures 3-22 and 3-23. The data
suggests that the dislocation density is relatively stable during very long-time high-temperature
annealing. Obviously, very high dislocation densities will recover, but coarsening of the Frank
network is not observed for at least some low (often annealed) dislocation densities (and associated
Burgers vector distributions). This has also been very recently observed in unpublished
experiments [98] on high purity 99.999% lithium fluoride single crystals. That study examined the
effect of static annealing at 0.92T
m
on three LiF crystals, including one which was 2% pre-strained,
over a period of 42 days. Apparently, the as received and pre-strain dislocation structure appears
relatively resistant to annealing over a 42-day period.
101
The literature (as well as our data) suggests that the “starting” dislocation density varies
substantially in Al and often can be of the order of the expected creep-deformed steady-state
dislocation density in the Harper-Dorn regime. These same densities may be resistant to
coarsening leading to a fixed (HD) dislocation density versus stress. The dislocation densities
observed in the Harper-Dorn regime are comparable to that at which we observed frustration or
resistance to recovery under static annealing conditions. As the applied stress drops, the dislocation
density cannot correspondingly decrease. Thus, a power of one may be observed. On the other
hand, if the initial density is relatively low, as in the case of slowly cooled single crystals of Nes
and Nöst [99] (e.g. 3 x 10
5
m
-2
), then a stress dependence, of the dislocation density is observed
[54].
Here, the application of a stress in the HD regime is only expected to increase the low initial
dislocation density. In the study of low-stress creep by the authors, in which we did not observe 1-
power (HD) law and observed 3+-power instead of HD], the initial dislocation density was also
relatively low at 6.5 x 10
7
m
-2
. However, studies in which power one (i.e. HD) behavior was
observed had, in at least one case, an initial dislocation density comparable to our earlier study,
but not nearly as low as that of the Nes, and Nöst and Nes [54, 99] work that also failed to observe
1-power HD behavior.
In summary, the aluminum single crystals were annealed for various times up to one year at 920
K (0.98 T
m
). Our results show that the dislocation density is relatively constant from the starting
density up to long annealing times of one year. This may suggest that the concept of dislocation
“frustration” suggested by Ardell and coworkers may be valid. This potentially explains some of
the Harper-Dorn observation in aluminum with respect to the observed dislocation densities and
102
stress exponents. Perhaps the “frustration limit” observed in this study precludes a variation in
dislocation density with steady-state stress when the initial dislocation density is of the order of
10
8
– 10
9
m
-2
, and one-power law (Harper-Dorn) creep is observed. Perhaps more normal five or
three power law creep occurs when the initial dislocation density is low. The initial dislocation
density ranges widely in Al and both cases (high and low) are common. The dislocation densities
of this study are about an order of magnitude higher than in an earlier study [53] by Kassner et al.
where Harper-Dorn was not observed (for reasons probably relating to crystal growth procedures)
and this may explain our observation of frustration in this study and an absence in the earlier
Kassner et al study. Of course, our current study did not involve the application of a stress.
In order to observe Harper-Dorn Creep in aluminum, we postulate that the dislocation density must
remain constant:
1. If a material starts with a high dislocation density (𝜌), and loaded in the H-D regime, then
the density will remain constant and the strain-rate will be linearly proportional to the stress
(𝜎)with a stress exponent (n) equal to 1.
2. If a material begins with a very low dislocation density(𝜌), and a low stress is applied in
the H-D regime, then the density will increase with strain-rate (𝜀) and stress (𝜎). H-D creep
will not occur and a stress exponent (n) closer to 3 or 4 is observed.
103
CHAPTER 4. CREEP IN AMORPHOUS ALLOYS (BULK METALLIC
GLASSES): A REVIEW
4.1 BACKGROUND
Amorphous metals are a relatively new class of alloy, originating in about 1960 with the discovery
of thin metallic ribbons by splat cooling [100]. These are nearly always alloys, and pure metal
glasses are rarely produced (Selenium is an exception). As these alloys are non-crystalline, they
have no dislocations, at least in the sense normally described in crystalline materials. Thus,
amorphous metals have yield stresses that are higher than crystalline alloys. High fracture stress,
low elastic moduli, and sometimes-favorable fracture toughness are observed. Often, favorable
corrosion properties were observed, as well, partly due to an absence of grain boundaries. Towards
1990, alloys with deep eutectics were developed that allowed liquid structures to be retained in
thicker sections in the amorphous state on cooling to ambient temperature [101-112]. With this
development, there has been fairly intensive study of bulk metallic glasses (BMG) for possible
structural applications. Most of the alloys in this chapter are relevant to BMGs. Table 1 lists some
of the short-term mechanical properties of some BMGs taken from [112] and some of the
impressive properties are listed.
Figure 4-1, based on an illustration by [113], is a time-temperature-transformation (T-T-T)
diagram that illustrates some of the important temperatures for metallic glasses. First, there is the
equilibrium liquid to solid transition at the melting temperature T
m
where, of course, multiple solid
crystalline phases form on cooling. Below this temperature, a T-T-T curve is illustrated. Cooling
below T
m
must be sufficiently rapid to avoid intersecting the “nose” of the curve. Also illustrated
104
is the glass transition temperature, T
g
. T
x
is the onset temperature of the crystallization event. This
is generally assigned to that temperature where there is as discontinuity in the change of a property
(e.g. heat capacity, thermal expansion coefficient, etc.) with temperature. The region between T
m
and T
g
is generally referred to as the super-cooled liquid regime. Some values for various BMGs
are listed in Table 4.2.
The discussions in this chapter will be largely confined to temperatures above 0.70T
g
. As will be
discussed subsequently, this is the regime in which homogeneous deformation is observed and is
also referred to as the “creep regime” for amorphous metals. A practical importance of this regime
is that this is where forming of a metallic glass is frequently performed. This regime is contrasted
by the regime of lower temperatures where heterogeneous deformation or shear banding is often
(but not always) observed.
105
TABLE 4.1. Mechanical properties of some glassy alloys from Ref. [111, 114-134].
Material E (GPa)
𝛔
𝐲
(MPa)
𝛔
𝐟
(MPa)
𝛆
𝐲
(%)
𝛆
𝐟
(%)
Reference
Co
43
Fe
20
Ta
5.5
B
31.5
268 5185 2 [122]
Cu
60
Hf
25
Ti
15
124 2024 2088 1.6 [129]
(Cu
60
Hf
25
Ti
15
)
96
Nb
4
130 2405 2.8 [128]
Cu
47
Ti
33
Zr
11
Ni
6
Sn
2
Si
1
1930 2250 [119]
Cu
50
Zr
50
84 1272 1794 1.7 6.2 [118]
Cu
64
Zr
36
92.3 2000 2.2 [132]
(Fe
0.9
Co
0.1
)
64.5
Mo
14
C
15
B
6
Er
0.5
192 3700 4100 0.55 [121]
Fe
71
Nb
6
B
23
4850 1.6 [133]
Fe
72
Si
4
B
20
Nb
4
200 4200 2.1 1.9 [114]
Fe
74
Mo
6
P
10
C
7.5
B
2.5
3330 3400 2.2 [124]
[(Fe
0.6
Co
0.4
)
0.75
B
0.2
Si
0.05
]
96
Nb
4
210 4100 4250 2 2.25 [123]
Fe
49
Cr
15
Mo
14
C
15
B
6
Er
1
220 3750 4140 0.25 [120]
Gd
60
Co
15
Al
25
70 1380 1.97 [115]
Ni
61
Zr
22
Nb
7
Al
4
Ta
6
3080 5 [127]
Pd
77.5
Cu
6
Si
16.5
1476 1600 11.4 [134]
Pd
79
Cu
6
Si
10
P
5
82 1475 1575 3.5 [125]
Pt
57.5
Cu
14.7
Ni
5.3
P
22.5
1400 1470 2 20 [130]
Ti
41.5
Zr
2.5
Hf
5
Cu
42.5
Ni
7.5
Si
1
95 2040 0 [126]
Zr
55
Cu
30
Al
10
Ni
5
1410 1420 [131]
Zr
41.25
Ti
13.75
Cu
12.5
Ni
10
Be
22.5
96 1900 1900 2 [117]
Zr
57
Nb
5
Al
10
Cu
15.4
Ni
12.6
86.7 1800 1800 2 [116]
Note: E : Young’s Modulus; σ
D
: yield strength; σ
Ø
: fracture stress; ε
D
: elongation at yielding;
ε
o
: plastic elongation.
All the tests were conducted under compression, generally, at strain-rates (ε) from
1 – 5 x 10
-4
s
-1
.
106
4.2 MECHANISMS OF DEFORMATION
The suggested mechanisms for deformation in bulk metallic glasses have generally fallen into
three categories: a.) Dislocation-like defects [7, 135, 136], diffusion type deformation [137], and
shear transformation zones (STZs) [138, 139]. These are illustrated in Figure 5-2 of Lu et al. [140],
as well as Schuh et al. [141], and are all early explanations for plasticity but it appears that the
amorphous metals community has generally embraced the third, STZs [111, 112, 142].
The essence of this latter mechanism is that there is a so-called “free volume” in amorphous
metals. Free volume is a "concept" and it has no absolute definition. The starting state is the
baseline; only the difference has meaning, so a change in density after deformation defines the free
volume. The exact form and shape of these free volumes is not known. Increasing free volume
would be associated with decreased density. During deformation, there are changes to the density
of a BMG based (partly) on the change in free volume. The Tan et al. [143] group showed a
correlation between the decrease in density with the increase of free volume. Consequently, an
increase in free volume improves plasticity but lowers its yield strength. A dramatic change in
free volume can be on the order of 2.31% from an original value closer to 1%.
Estimates for free volume for Zr
41.2
Ti
13.8
Cu
12.5
Ni
10
Be
22.5
(Vitreloy 1) is about 3% [144].
Free volume decreases (tighter packing) appear to increase ductility in homogeneous deformation
at ambient temperature [145]. With an applied stress, groups of atoms (e.g. few to 100 [110, 142,
146]), under an applied shear stress, t, move and perform work. This constitutes an STZ. Argon
et al. [138, 139] considered that the STZ operation takes place within the elastic confinement of a
surrounding glass matrix, and the shear distortion leads to stress and strain redistribution around
the STZ region [110, 138, 139]. When the STZs exist throughout the alloy we have homogeneous
deformation. STZs also occur in shear bands leading to heterogeneous deformation. STZs have
107
been observed to create free volume during homogeneous deformation [147, 148]. Steady-state
flow within the homogeneous regime can be considered a case where there is a balance between
free volume creation and annihilation.
Figure 4-2(b) illustrates the STZ mechanism. Argon et al. described the activation energy for this
process and Schuh et al. estimated the predicted activation energy, Q, as 100 – 500 kJ/mol. Table
4.3 lists some of the experimentally observed activation energies which are consistent with
Argon’s predictions.
The equations that have been used to describe the creep-rate based on STZ have used the classic
rate equation formalism leading to [110]:
γ= α
x
ν
x
γ
x
∗exp −
Ú
mn
sinh
mn
(E-4.1)
where 𝛼
Û
is a constant that includes the fraction of material deforming by activation, 𝜈
Û
is an
attempt frequency, and 𝛾
Û
is the characteristic strain of an STZ, and V is the activation volume.
The hyperbolic sine function arises, as there can be both a forward and reverse “reaction”.
At low stresses (τ << kT/V), this equation reduces to the Newtonian:
γ=
Ý
¿À
∗exp −
Ú
mn
τ (E-4.2)
since “reverse” deformation is irrelevant.
108
FIGURE 4-1. A time-temperature-transformation diagram that illustrates the important
temperature regions of BMGs from Ref. [113].
! 3!
Temp
Time
Figure 1. A time-temperature-transformation diagram that illustrates the important temperature
regions of BMGs from Ref. [12]
This is generally assigned to that temperature where there is as discontinuity in the
change of a property (e.g. heat capacity, thermal expansion coefficient, etc.) with
temperature. The region between T
m
and T
g
is generally referred to as the super-cooled
liquid regime.
Table 2. Deformation data of some BMGs in the super-cooled liquid region from Ref. [5]
Alloys (in at.%) T
g
(K) T
x
(K) m value Ductility
a
Reference
La
55
Al
25
Ni
20
480 520 1.0 1800 (T) [21]
Zr
65
Al
10
Ni
10
Cu
15
652 757 0.8–1.0 340 (T) [7]
Zr
52.5
Al
10
Ti
5
Cu
17.9
Ni
14.6
358 456 0.45–0.55 650 (T) [18]
Zr
55
Cu
30
Al
10
Ni
5
683 763 0.5–1.0 N/A (C) [22]
La
60
Al
20
Ni
10
Co
5
Cu
5
451 523 1.0 N/A [23]
Pd
40
Ni
40
P
20
589 670 0.5–1.0 0.94 (C) [24]
Zr
65
Al
10
Ni
10
Cu
15
652 757 0.83 750 (T) [25]
Zr
55
Al
10
Cu
30
Ni
5
670 768 0.5–0.9 800 (T) [26]
Ti
45
Zr
24
Ni
7
Cu
8
Be
16
601 648 N/A 1.0 (T) [27]
Cu
60
Zr
20
Hf
10
Ti
10
721 766 0.3–0.61 0.78 (C) [28]
Zr
52.5
Al
10
Cu
22
Ti
2.5
Ni
13
659 761 0.5–1.0 >1.0 (C) [29]
Zr
41.25
Ti
13.75
Ni
10
Cu
12.5
Be
22.5
614 698 0.4–1.0 1624 (T) [30]
a
: “T” and “C” stand for tension and compression, respectively.
Crystalline
Amorphous
T
l
T
g
109
TABLE 4.2. Deformation data of some BMGs in the super-cooled liquid region from Ref. [149],
[150-161].
Alloys (in at.%) T
glass
(K) T
rx
(K) m-value Ductility
a
Reference
La
55
Al
25
Ni
20
480 520 1.0 1800 (T) [155]
Zr
65
Al
10
Ni
10
Cu
15
652 757 0.8 – 1.0 340 (T) [156]
Zr
52.5
Al
10
Ti
5
Cu
17.9
Ni
14.6
358 456 0.45 – 0.55 650 (T) [158]
Zr
55
Cu
30
Al
10
Ni
5
683 763 0.5 – 1.0 N/A (C) [159]
La
60
Al
20
Ni
10
Co
5
Cu
5
451 523 1.0 N/A [160]
Pd
40
Ni
40
P
20
589 670 0.5 – 1.0 0.94 (C) [154]
Zr
65
Al
10
Ni
10
Cu
15
652 757 0.83 750 (T) [157]
Zr
55
Al
10
Cu
30
Ni
5
670 768 0.5 – 0.9 800 (T) [153]
Ti
45
Zr
24
Ni
7
Cu
8
Be
16
601 648 N/A 1.0 (T) [150]
Cu
60
Zr
20
Hf
10
Ti
10
721 766 0.3 – 0.61 0.78 (C) [152]
Zr
52.5
Al
10
Cu
22
Ti
2.5
Ni
13
659 71 0.5 – 1.0 > 1.0 (C) [151]
Zr
41.25
Ti
13.75
Ni
10
Cu
12.5
Be
22.5
614 698 0.4 – 1.0 1624 (T) [161]
a
: (T) and (C) are defined as tension and compression, respectively.
TABLE 4.3. Activation energies for creep of selected metallic glasses [139, 146, 162, 163].
Composition T
glass
(K) T
test
(K) ∆𝐐 (kJ/mol)
Al
20
Cu
25
Zr
55
740 573 230.12
Cu
4
0Zr
60
677 543 218.82
Cu
56
Zr
44
727 573 217.57
Cu
60
Zr
40
750 573 228.45
Pd
80
Si
20
673 546 191.63
Zr
55
Cu
30
Al
10
Ni
5
410
Au
49
Ag
5.5
Pd
2.3
Cu
26.9
Si
16.3
103
Zr
44
Ti
11
Cu
10
Ni
10
Be
25
625/632 366
110
FIGURE 4-2(a-c). (a). Two-dimensional representation of a dislocation line in crystalline (left)
and amorphous (right) solids; taken from [112]; Atomistic deformation of
amorphous metals in the form of (b). Shear transformation zones (STZ);
and (c). Local atomic jump; adapted from [110].
! 4!
The discussions in this chapter will be largely confined temperatures above 0.7T
g
. As will
be discussed subsequently, this is the regime in which homogeneous deformation is
observed. This review is referring to this regime as a “creep” regime of amorphous
metals. A practical importance of this regime is that this is where forming of a metallic
glass is frequently performed. This regime is contrasted by the regime of lower
temperatures where heterogeneous deformation or shear banding is often (but not always)
observed.
Mechanisms of Deformation
The suggested mechanisms have generally fallen into three categories: a.) Dislocation-
like defects [6,36,37], diffusion type deformation [8,35], and shear transformation zones
(STZs)[9,52]. These are illustrated in Fig. 2 [54] of Lu et al., and are all early
explanations for plasticity but it appears that the amorphous metals community has
generally embraced the third, STZ [2,34,39].
a).
b). c).
! 4!
The discussions in this chapter will be largely confined temperatures above 0.7T
g
. As will
be discussed subsequently, this is the regime in which homogeneous deformation is
observed. This review is referring to this regime as a “creep” regime of amorphous
metals. A practical importance of this regime is that this is where forming of a metallic
glass is frequently performed. This regime is contrasted by the regime of lower
temperatures where heterogeneous deformation or shear banding is often (but not always)
observed.
Mechanisms of Deformation
The suggested mechanisms have generally fallen into three categories: a.) Dislocation-
like defects [6,36,37], diffusion type deformation [8,35], and shear transformation zones
(STZs)[9,52]. These are illustrated in Fig. 2 [54] of Lu et al., and are all early
explanations for plasticity but it appears that the amorphous metals community has
generally embraced the third, STZ [2,34,39].
Figure 2. a). Two-dimensional representation of a dislocation line in crystalline (left) and
amorphous (right) solids; taken from [34]; Atomistic deformation of amorphous metals in the
form of b). Shear transformation zones (STZ); and c). Local atomic jump; adapted from [1].
! 5!
a).
τ τ
b). c).
Figure 2. a). Two-dimensional representation of a dislocation line in crystalline (left) and
amorphous (right) solids; taken from [34]; Atomistic deformation of amorphous metals in the
form of b). Shear transformation zones (STZ); and c). Local atomic jump; adapted from [1].
The essence of this latter mechanism is that there is a so-called “free volume” in
amorphous metals. Free volume is a "concept" and it has no absolute definition. The
starting state is the baseline; only the difference has meaning, so a change in density after
deformation defines the free volume. The exact form and shape of these free volumes is
not known [38]. Increasing free volume would be associated with decreased density.
Estimates for free volume is about 3% for Zr
41.2
Ti
13.8
Cu
12.5
Ni
10
Be
22.5
(Vitreloy 1) [55].
Free volume decreases (tighter packing) appears to increase ductility in homogeneous
deformation at ambient temperature [51]. With an applied stress, groups of atoms (e.g.
few to 100 [1,2,62]), under an applied shear stress, τ, move and perform work. This
constitutes an STZ. Argon et al. [9,48] considered that the STZ operation takes place
within the elastic confinement of a surrounding glass matrix, and the shear distortion
leads to stress and strain redistribution around the STZ region [1,9,48]. When the STZs
exist throughout the alloy we have homogeneous deformation. STZs also occur in shear
large free volume, as illustrated in Fig. 2b [33]. Correspondingly, the activation energy for strain
accommodation via the diffusion-like atomic jump model is lower than that for that involving distor-
tion of the STZ. In spite of differences in local atomic motions, the STZ distortion model and the atom-
ic-jump free volume models share common features, which according to Schuh et al. [2] have
implications for plastic deformation of the glass, its temperature- and pressure-dependence and flow
localization.Steifetal.[47]modifiedSpaepen’smodelbyincludingadditionalfreevolumechangedue
to pressure. Khonik [48] proposed a directional structural relaxation model suggesting that each rear-
rangement event can be interpreted as a thermally-activated shear due to local atomic structures and
subsequentlynearlyathermalviscousflowbyexternalstress.Alternatively,acooperativeshearmodel
has also been proposed by Johnson and Samwer [49] in which yielding of metallic glasses displays a
(T/T
g
)
2/3
temperature dependence. However, Spaepen’s [33] and Argon’s [46] models remain most
popular for describing deformation of metallic glasses.
The plastic flow and deformation behavior of metallic glasses has also been explained on the basis
ofdislocation models [50,51]. Although no lineardefects can be considered to exist in amorphoussol-
ids, the boundary between an un-deformed and deformed region, such as the propagating front of a
shear band, can be conceptualized as the dislocation. Fig. 3 illustrates a model of a dislocation in a
crystalline and glassy solid, represented by two-dimensional arrays of polyhedra. The dislocation line
is represented as the bottom row of atoms of extra-half plane perpendicular to the surface. Burger’s
vectors which are parallel and of same magnitude for the crystalline solid are different in magnitude
and direction in the case of the glassy solid. Hence, the dislocation line is not forced to remain on a
crystallographic plane, which does not exist in the glass. It is conceptualized that stress concentration
at the shear band front, i.e., the dislocation line, can activate STZs and result in growth and propaga-
tion of the shear band producing macroscopic strain. However, unlike the interaction of dislocations
withothermicrostructuralfeaturesincreasingtheresistance todeformationand influencingmechan-
ical properties of crystalline solids, no such resistance to plastic flow and associated strain hardening
is observed in the case of metallic glasses.
Themacroscopicdeformationresponseofmetallicglassesthroughaccumulationoflocalstrainsvia
operation of ‘‘shear transformation zones” or ‘‘atomic jumps” into free volume spaces, can occur
homogeneously or inhomogeneously depending on the temperature, applied stress, and strain rate.
Spaepen proposed a deformation mechanism map describing these two basic modes of deformation
in metallic glasses: homogeneous flow in which each volume element of the specimen contributes
to the strain, and inhomogeneous flow in which the strain is localized in few very thin shear bands
[33]. Fig. 4 shows the schematic map plotted with normalized stress versus homologous temperature
(T/T
m
), illustrating how the deformation behavior transitions as a function of strain rate and temper-
ature.Homogeneousflow,whichisclosetoNewtonianviscous ð _ c/ sÞflow,occursatlowstressesand
high temperatures. In this deformation mode, the glass thins uniformly and fracture occurs when
some section of the specimen has narrowed to zero thickness. Inhomogeneous flow occurs at high
Fig. 3. Schematic showing dislocation represented by 2-D arrays of polyhedra in crystalline (left) and glassy (right) solids.
Dislocation in glass is conceptualized as boundary between un-deformed and deformed region, such as propagating front of a
shear band. (Adapted from [31].)
764 M.M. Trexler, N.N. Thadhani/Progress in Materials Science 55 (2010) 759–839
glass-based materials have also been developed – including
foams, composites and nanocrystal-reinforced alloys –
whose mechanical properties are just beginning to be stud-
ied seriously. There are a number of recent papers which
outline the mechanical properties of metallic glasses in
broad strokes, which tabulate and compare the properties
among glasses of different and varied compositions, or
which review some specific properties in detail [2–7]. On
the other hand, the fundamental principles and mecha-
nisms that underpin the mechanical properties of amor-
phous metals have not yet been holistically synthesized
with the accumulation of new data over the past decade
or so.
Our purpose in this article is to present an overview of
the mechanical properties of metallic glasses with a specific
focus upon fundamentals and mechanisms of deformation
andfracture.Thissynthesisfollowsintheveinoftheclassic
review articles of Pampillo (1975) [8] and Argon (1993) [9];
while many of the concepts laid out in these earlier reviews
remain equally valid today, we incorporate here what we
view as the most important refinements, revisions and
recent advances in understanding the deformation of
metallic glasses and their derivatives. Beginning from an
atomistic picture of deformation mechanisms in amor-
phous metal, we proceed to review elastic, plastic and frac-
ture behavior in light of these mechanisms. We then
explore the importance of glass structure and its evolution
during deformation, and survey the growing literature on
ductilization of metallic glasses. The paper concludes with
a view of important unresolved questions for what is a rap-
idly expanding field of research.
2. Deformation mechanisms
Because the bonding in amorphous alloys is of primarily
metallic character, strain can be readily accommodated at
the atomic level through changes in neighborhood; atomic
bonds can be broken and reformed at the atomic scale
without substantial concern for, e.g. the rigidity of bond
angles as in a covalent solid, or the balance of charges as
in an ionic solid. However, unlike crystalline metals and
alloys, metallic glasses do not exhibit long-range transla-
tional symmetry. Whereas crystal dislocations allow
changes in atomic neighborhood at low energies or stresses
in crystals, the local rearrangement of atoms in metallic
glasses is a relatively high-energy or high-stress process.
The exact nature of local atomic motion in deforming
metallic glasses is not fully resolved, although there is gen-
eral consensus that the fundamental unit process underly-
ing deformation must be a local rearrangement of atoms
that can accommodate shear strain. An example of such
a local rearrangement is depicted in the two-dimensional
schematic of Fig. 1a, originally proposed by Argon and
Kuo [10] on the basis of an atomic-analog bubble-raft
model. The event depicted in Fig. 1a has been referred to
as a ‘‘flow defect’’ or ‘‘s defect’’ [11,12], a ‘‘local inelastic
transition’’ [13–15] and, increasingly commonly, a ‘‘shear
transformation zone’’ (STZ) [12,16–22]. The STZ is essen-
tially a local cluster of atoms that undergoes an inelastic
shear distortion from one relatively low energy configura-
tion to a second such configuration, crossing an activated
configuration of higher energy and volume. Since the origi-
nal analog model of Argon et al. [10,23], more sophisti-
cated computer models have been employed to study
glass deformation in both two and three dimensions
[11,12,16,18,24–38]. STZs comprising a few to perhaps
!100 atoms are commonly observed in such simulation
works, which span a variety of simulated compositions
and empirical interatomic potentials; this suggests that
STZs are common to deformation of all amorphous met-
als, although details of the structure, size and energy scales
of STZs may vary from one glass to the next.
It is important to note that an STZ is not a structural
defect in an amorphous metal in the way that a lattice dis-
location is a crystal defect. Rather, an STZ is defined by its
transience – an observer inspecting a glass at a single
instant in time cannot, a priori, identify an STZ in the
structure, and it is only upon inspecting a change from
one moment in time (or strain) to the next that STZs
may be observed and cataloged. In other words, an STZ
is an event defined in a local volume, not a feature of the
glass structure. This is not to suggest that the operation
of an STZ is independent of the glass structure; indeed,
STZ operation is strongly influenced by local atomic
arrangements, and also has important consequences for
structural evolution of a deforming glass. In a metallic
glassbodyexperiencinguniformstress,theSTZthatisacti-
vated first is selected from among many potential sites on
the basis of energetics, which vary with the local atomic
arrangements [11,27,36–38]. For example, the local distri-
bution of free volume is widely believed to control defor-
mation of metallic glasses [10,23,39–42], and it is easy to
envision that sites of higher free-volume would more read-
ily accommodate local shear. Atomistic simulations have
also correlated other structural state variables with local
shearing, including short-range chemical or topological
order [11,43,44].
Fig. 1. Two-dimensional schematics of the atomistic deformation mech-
anisms proposed for amorphous metals, including (a) a shear transfor-
mation zone (STZ), after Argon [40], and (b) a local atomic jump, after
Spaepen [39].
C.A. Schuh et al. / Acta Materialia 55 (2007) 4067–4109 4069
glass-based materials have also been developed – including
foams, composites and nanocrystal-reinforced alloys –
whose mechanical properties are just beginning to be stud-
ied seriously. There are a number of recent papers which
outline the mechanical properties of metallic glasses in
broad strokes, which tabulate and compare the properties
among glasses of different and varied compositions, or
which review some specific properties in detail [2–7]. On
the other hand, the fundamental principles and mecha-
nisms that underpin the mechanical properties of amor-
phous metals have not yet been holistically synthesized
with the accumulation of new data over the past decade
or so.
Our purpose in this article is to present an overview of
the mechanical properties of metallic glasses with a specific
focus upon fundamentals and mechanisms of deformation
andfracture.Thissynthesisfollowsintheveinoftheclassic
review articles of Pampillo (1975) [8] and Argon (1993) [9];
while many of the concepts laid out in these earlier reviews
remain equally valid today, we incorporate here what we
view as the most important refinements, revisions and
recent advances in understanding the deformation of
metallic glasses and their derivatives. Beginning from an
atomistic picture of deformation mechanisms in amor-
phous metal, we proceed to review elastic, plastic and frac-
ture behavior in light of these mechanisms. We then
explore the importance of glass structure and its evolution
during deformation, and survey the growing literature on
ductilization of metallic glasses. The paper concludes with
a view of important unresolved questions for what is a rap-
idly expanding field of research.
2. Deformation mechanisms
Because the bonding in amorphous alloys is of primarily
metallic character, strain can be readily accommodated at
the atomic level through changes in neighborhood; atomic
bonds can be broken and reformed at the atomic scale
without substantial concern for, e.g. the rigidity of bond
angles as in a covalent solid, or the balance of charges as
in an ionic solid. However, unlike crystalline metals and
alloys, metallic glasses do not exhibit long-range transla-
tional symmetry. Whereas crystal dislocations allow
changes in atomic neighborhood at low energies or stresses
in crystals, the local rearrangement of atoms in metallic
glasses is a relatively high-energy or high-stress process.
The exact nature of local atomic motion in deforming
metallic glasses is not fully resolved, although there is gen-
eral consensus that the fundamental unit process underly-
ing deformation must be a local rearrangement of atoms
that can accommodate shear strain. An example of such
a local rearrangement is depicted in the two-dimensional
schematic of Fig. 1a, originally proposed by Argon and
Kuo [10] on the basis of an atomic-analog bubble-raft
model. The event depicted in Fig. 1a has been referred to
as a ‘‘flow defect’’ or ‘‘s defect’’ [11,12], a ‘‘local inelastic
transition’’ [13–15] and, increasingly commonly, a ‘‘shear
transformation zone’’ (STZ) [12,16–22]. The STZ is essen-
tially a local cluster of atoms that undergoes an inelastic
shear distortion from one relatively low energy configura-
tion to a second such configuration, crossing an activated
configuration of higher energy and volume. Since the origi-
nal analog model of Argon et al. [10,23], more sophisti-
cated computer models have been employed to study
glass deformation in both two and three dimensions
[11,12,16,18,24–38]. STZs comprising a few to perhaps
!100 atoms are commonly observed in such simulation
works, which span a variety of simulated compositions
and empirical interatomic potentials; this suggests that
STZs are common to deformation of all amorphous met-
als, although details of the structure, size and energy scales
of STZs may vary from one glass to the next.
It is important to note that an STZ is not a structural
defect in an amorphous metal in the way that a lattice dis-
location is a crystal defect. Rather, an STZ is defined by its
transience – an observer inspecting a glass at a single
instant in time cannot, a priori, identify an STZ in the
structure, and it is only upon inspecting a change from
one moment in time (or strain) to the next that STZs
may be observed and cataloged. In other words, an STZ
is an event defined in a local volume, not a feature of the
glass structure. This is not to suggest that the operation
of an STZ is independent of the glass structure; indeed,
STZ operation is strongly influenced by local atomic
arrangements, and also has important consequences for
structural evolution of a deforming glass. In a metallic
glassbodyexperiencinguniformstress,theSTZthatisacti-
vated first is selected from among many potential sites on
the basis of energetics, which vary with the local atomic
arrangements [11,27,36–38]. For example, the local distri-
bution of free volume is widely believed to control defor-
mation of metallic glasses [10,23,39–42], and it is easy to
envision that sites of higher free-volume would more read-
ily accommodate local shear. Atomistic simulations have
also correlated other structural state variables with local
shearing, including short-range chemical or topological
order [11,43,44].
Fig. 1. Two-dimensional schematics of the atomistic deformation mech-
anisms proposed for amorphous metals, including (a) a shear transfor-
mation zone (STZ), after Argon [40], and (b) a local atomic jump, after
Spaepen [39].
C.A. Schuh et al. / Acta Materialia 55 (2007) 4067–4109 4069
! 4!
The discussions in this chapter will be largely confined temperatures above 0.7T
g
. As will
be discussed subsequently, this is the regime in which homogeneous deformation is
observed. This review is referring to this regime as a “creep” regime of amorphous
metals. A practical importance of this regime is that this is where forming of a metallic
glass is frequently performed. This regime is contrasted by the regime of lower
temperatures where heterogeneous deformation or shear banding is often (but not always)
observed.
Mechanisms of Deformation
The suggested mechanisms have generally fallen into three categories: a.) Dislocation-
like defects [6,36,37], diffusion type deformation [8,35], and shear transformation zones
(STZs)[9,52]. These are illustrated in Fig. 2 [54] of Lu et al., and are all early
explanations for plasticity but it appears that the amorphous metals community has
generally embraced the third, STZ [2,34,39].
Figure 2. a). Two-dimensional representation of a dislocation line in crystalline (left) and
amorphous (right) solids; taken from [34]; Atomistic deformation of amorphous metals in the
form of b). Shear transformation zones (STZ); and c). Local atomic jump; adapted from [1].
! 5!
a).
τ τ
b). c).
Figure 2. a). Two-dimensional representation of a dislocation line in crystalline (left) and
amorphous (right) solids; taken from [34]; Atomistic deformation of amorphous metals in the
form of b). Shear transformation zones (STZ); and c). Local atomic jump; adapted from [1].
The essence of this latter mechanism is that there is a so-called “free volume” in
amorphous metals. Free volume is a "concept" and it has no absolute definition. The
starting state is the baseline; only the difference has meaning, so a change in density after
deformation defines the free volume. The exact form and shape of these free volumes is
not known [38]. Increasing free volume would be associated with decreased density.
Estimates for free volume is about 3% for Zr
41.2
Ti
13.8
Cu
12.5
Ni
10
Be
22.5
(Vitreloy 1) [55].
Free volume decreases (tighter packing) appears to increase ductility in homogeneous
deformation at ambient temperature [51]. With an applied stress, groups of atoms (e.g.
few to 100 [1,2,62]), under an applied shear stress, τ, move and perform work. This
constitutes an STZ. Argon et al. [9,48] considered that the STZ operation takes place
within the elastic confinement of a surrounding glass matrix, and the shear distortion
leads to stress and strain redistribution around the STZ region [1,9,48]. When the STZs
exist throughout the alloy we have homogeneous deformation. STZs also occur in shear
large free volume, as illustrated in Fig. 2b [33]. Correspondingly, the activation energy for strain
accommodation via the diffusion-like atomic jump model is lower than that for that involving distor-
tion of the STZ. In spite of differences in local atomic motions, the STZ distortion model and the atom-
ic-jump free volume models share common features, which according to Schuh et al. [2] have
implications for plastic deformation of the glass, its temperature- and pressure-dependence and flow
localization.Steifetal.[47]modifiedSpaepen’smodelbyincludingadditionalfreevolumechangedue
to pressure. Khonik [48] proposed a directional structural relaxation model suggesting that each rear-
rangement event can be interpreted as a thermally-activated shear due to local atomic structures and
subsequentlynearlyathermalviscousflowbyexternalstress.Alternatively,acooperativeshearmodel
has also been proposed by Johnson and Samwer [49] in which yielding of metallic glasses displays a
(T/Tg)
2/3
temperature dependence. However, Spaepen’s [33] and Argon’s [46] models remain most
popular for describing deformation of metallic glasses.
The plastic flow and deformation behavior of metallic glasses has also been explained on the basis
ofdislocation models [50,51]. Although no lineardefects can be considered to exist in amorphoussol-
ids, the boundary between an un-deformed and deformed region, such as the propagating front of a
shear band, can be conceptualized as the dislocation. Fig. 3 illustrates a model of a dislocation in a
crystalline and glassy solid, represented by two-dimensional arrays of polyhedra. The dislocation line
is represented as the bottom row of atoms of extra-half plane perpendicular to the surface. Burger’s
vectors which are parallel and of same magnitude for the crystalline solid are different in magnitude
and direction in the case of the glassy solid. Hence, the dislocation line is not forced to remain on a
crystallographic plane, which does not exist in the glass. It is conceptualized that stress concentration
at the shear band front, i.e., the dislocation line, can activate STZs and result in growth and propaga-
tion of the shear band producing macroscopic strain. However, unlike the interaction of dislocations
withothermicrostructuralfeaturesincreasingtheresistance todeformationand influencingmechan-
ical properties of crystalline solids, no such resistance to plastic flow and associated strain hardening
is observed in the case of metallic glasses.
Themacroscopicdeformationresponseofmetallicglassesthroughaccumulationoflocalstrainsvia
operation of ‘‘shear transformation zones” or ‘‘atomic jumps” into free volume spaces, can occur
homogeneously or inhomogeneously depending on the temperature, applied stress, and strain rate.
Spaepen proposed a deformation mechanism map describing these two basic modes of deformation
in metallic glasses: homogeneous flow in which each volume element of the specimen contributes
to the strain, and inhomogeneous flow in which the strain is localized in few very thin shear bands
[33]. Fig. 4 shows the schematic map plotted with normalized stress versus homologous temperature
(T/Tm), illustrating how the deformation behavior transitions as a function of strain rate and temper-
ature.Homogeneousflow,whichisclosetoNewtonianviscous ð _ c/ sÞflow,occursatlowstressesand
high temperatures. In this deformation mode, the glass thins uniformly and fracture occurs when
some section of the specimen has narrowed to zero thickness. Inhomogeneous flow occurs at high
Fig. 3. Schematic showing dislocation represented by 2-D arrays of polyhedra in crystalline (left) and glassy (right) solids.
Dislocation in glass is conceptualized as boundary between un-deformed and deformed region, such as propagating front of a
shear band. (Adapted from [31].)
764 M.M. Trexler, N.N. Thadhani/Progress in Materials Science 55 (2010) 759–839
glass-based materials have also been developed – including
foams, composites and nanocrystal-reinforced alloys –
whose mechanical properties are just beginning to be stud-
ied seriously. There are a number of recent papers which
outline the mechanical properties of metallic glasses in
broad strokes, which tabulate and compare the properties
among glasses of different and varied compositions, or
which review some specific properties in detail [2–7]. On
the other hand, the fundamental principles and mecha-
nisms that underpin the mechanical properties of amor-
phous metals have not yet been holistically synthesized
with the accumulation of new data over the past decade
or so.
Our purpose in this article is to present an overview of
the mechanical properties of metallic glasses with a specific
focus upon fundamentals and mechanisms of deformation
andfracture.Thissynthesisfollowsintheveinoftheclassic
review articles of Pampillo (1975) [8] and Argon (1993) [9];
while many of the concepts laid out in these earlier reviews
remain equally valid today, we incorporate here what we
view as the most important refinements, revisions and
recent advances in understanding the deformation of
metallic glasses and their derivatives. Beginning from an
atomistic picture of deformation mechanisms in amor-
phous metal, we proceed to review elastic, plastic and frac-
ture behavior in light of these mechanisms. We then
explore the importance of glass structure and its evolution
during deformation, and survey the growing literature on
ductilization of metallic glasses. The paper concludes with
a view of important unresolved questions for what is a rap-
idly expanding field of research.
2. Deformation mechanisms
Because the bonding in amorphous alloys is of primarily
metallic character, strain can be readily accommodated at
the atomic level through changes in neighborhood; atomic
bonds can be broken and reformed at the atomic scale
without substantial concern for, e.g. the rigidity of bond
angles as in a covalent solid, or the balance of charges as
in an ionic solid. However, unlike crystalline metals and
alloys, metallic glasses do not exhibit long-range transla-
tional symmetry. Whereas crystal dislocations allow
changes in atomic neighborhood at low energies or stresses
in crystals, the local rearrangement of atoms in metallic
glasses is a relatively high-energy or high-stress process.
The exact nature of local atomic motion in deforming
metallic glasses is not fully resolved, although there is gen-
eral consensus that the fundamental unit process underly-
ing deformation must be a local rearrangement of atoms
that can accommodate shear strain. An example of such
a local rearrangement is depicted in the two-dimensional
schematic of Fig. 1a, originally proposed by Argon and
Kuo [10] on the basis of an atomic-analog bubble-raft
model. The event depicted in Fig. 1a has been referred to
as a ‘‘flow defect’’ or ‘‘s defect’’ [11,12], a ‘‘local inelastic
transition’’ [13–15] and, increasingly commonly, a ‘‘shear
transformation zone’’ (STZ) [12,16–22]. The STZ is essen-
tially a local cluster of atoms that undergoes an inelastic
shear distortion from one relatively low energy configura-
tion to a second such configuration, crossing an activated
configuration of higher energy and volume. Since the origi-
nal analog model of Argon et al. [10,23], more sophisti-
cated computer models have been employed to study
glass deformation in both two and three dimensions
[11,12,16,18,24–38]. STZs comprising a few to perhaps
!100 atoms are commonly observed in such simulation
works, which span a variety of simulated compositions
and empirical interatomic potentials; this suggests that
STZs are common to deformation of all amorphous met-
als, although details of the structure, size and energy scales
of STZs may vary from one glass to the next.
It is important to note that an STZ is not a structural
defect in an amorphous metal in the way that a lattice dis-
location is a crystal defect. Rather, an STZ is defined by its
transience – an observer inspecting a glass at a single
instant in time cannot, a priori, identify an STZ in the
structure, and it is only upon inspecting a change from
one moment in time (or strain) to the next that STZs
may be observed and cataloged. In other words, an STZ
is an event defined in a local volume, not a feature of the
glass structure. This is not to suggest that the operation
of an STZ is independent of the glass structure; indeed,
STZ operation is strongly influenced by local atomic
arrangements, and also has important consequences for
structural evolution of a deforming glass. In a metallic
glassbodyexperiencinguniformstress,theSTZthatisacti-
vated first is selected from among many potential sites on
the basis of energetics, which vary with the local atomic
arrangements [11,27,36–38]. For example, the local distri-
bution of free volume is widely believed to control defor-
mation of metallic glasses [10,23,39–42], and it is easy to
envision that sites of higher free-volume would more read-
ily accommodate local shear. Atomistic simulations have
also correlated other structural state variables with local
shearing, including short-range chemical or topological
order [11,43,44].
Fig. 1. Two-dimensional schematics of the atomistic deformation mech-
anisms proposed for amorphous metals, including (a) a shear transfor-
mation zone (STZ), after Argon [40], and (b) a local atomic jump, after
Spaepen [39].
C.A. Schuh et al. / Acta Materialia 55 (2007) 4067–4109 4069
glass-based materials have also been developed – including
foams, composites and nanocrystal-reinforced alloys –
whose mechanical properties are just beginning to be stud-
ied seriously. There are a number of recent papers which
outline the mechanical properties of metallic glasses in
broad strokes, which tabulate and compare the properties
among glasses of different and varied compositions, or
which review some specific properties in detail [2–7]. On
the other hand, the fundamental principles and mecha-
nisms that underpin the mechanical properties of amor-
phous metals have not yet been holistically synthesized
with the accumulation of new data over the past decade
or so.
Our purpose in this article is to present an overview of
the mechanical properties of metallic glasses with a specific
focus upon fundamentals and mechanisms of deformation
andfracture.Thissynthesisfollowsintheveinoftheclassic
review articles of Pampillo (1975) [8] and Argon (1993) [9];
while many of the concepts laid out in these earlier reviews
remain equally valid today, we incorporate here what we
view as the most important refinements, revisions and
recent advances in understanding the deformation of
metallic glasses and their derivatives. Beginning from an
atomistic picture of deformation mechanisms in amor-
phous metal, we proceed to review elastic, plastic and frac-
ture behavior in light of these mechanisms. We then
explore the importance of glass structure and its evolution
during deformation, and survey the growing literature on
ductilization of metallic glasses. The paper concludes with
a view of important unresolved questions for what is a rap-
idly expanding field of research.
2. Deformation mechanisms
Because the bonding in amorphous alloys is of primarily
metallic character, strain can be readily accommodated at
the atomic level through changes in neighborhood; atomic
bonds can be broken and reformed at the atomic scale
without substantial concern for, e.g. the rigidity of bond
angles as in a covalent solid, or the balance of charges as
in an ionic solid. However, unlike crystalline metals and
alloys, metallic glasses do not exhibit long-range transla-
tional symmetry. Whereas crystal dislocations allow
changes in atomic neighborhood at low energies or stresses
in crystals, the local rearrangement of atoms in metallic
glasses is a relatively high-energy or high-stress process.
The exact nature of local atomic motion in deforming
metallic glasses is not fully resolved, although there is gen-
eral consensus that the fundamental unit process underly-
ing deformation must be a local rearrangement of atoms
that can accommodate shear strain. An example of such
a local rearrangement is depicted in the two-dimensional
schematic of Fig. 1a, originally proposed by Argon and
Kuo [10] on the basis of an atomic-analog bubble-raft
model. The event depicted in Fig. 1a has been referred to
as a ‘‘flow defect’’ or ‘‘s defect’’ [11,12], a ‘‘local inelastic
transition’’ [13–15] and, increasingly commonly, a ‘‘shear
transformation zone’’ (STZ) [12,16–22]. The STZ is essen-
tially a local cluster of atoms that undergoes an inelastic
shear distortion from one relatively low energy configura-
tion to a second such configuration, crossing an activated
configuration of higher energy and volume. Since the origi-
nal analog model of Argon et al. [10,23], more sophisti-
cated computer models have been employed to study
glass deformation in both two and three dimensions
[11,12,16,18,24–38]. STZs comprising a few to perhaps
!100 atoms are commonly observed in such simulation
works, which span a variety of simulated compositions
and empirical interatomic potentials; this suggests that
STZs are common to deformation of all amorphous met-
als, although details of the structure, size and energy scales
of STZs may vary from one glass to the next.
It is important to note that an STZ is not a structural
defect in an amorphous metal in the way that a lattice dis-
location is a crystal defect. Rather, an STZ is defined by its
transience – an observer inspecting a glass at a single
instant in time cannot, a priori, identify an STZ in the
structure, and it is only upon inspecting a change from
one moment in time (or strain) to the next that STZs
may be observed and cataloged. In other words, an STZ
is an event defined in a local volume, not a feature of the
glass structure. This is not to suggest that the operation
of an STZ is independent of the glass structure; indeed,
STZ operation is strongly influenced by local atomic
arrangements, and also has important consequences for
structural evolution of a deforming glass. In a metallic
glassbodyexperiencinguniformstress,theSTZthatisacti-
vated first is selected from among many potential sites on
the basis of energetics, which vary with the local atomic
arrangements [11,27,36–38]. For example, the local distri-
bution of free volume is widely believed to control defor-
mation of metallic glasses [10,23,39–42], and it is easy to
envision that sites of higher free-volume would more read-
ily accommodate local shear. Atomistic simulations have
also correlated other structural state variables with local
shearing, including short-range chemical or topological
order [11,43,44].
Fig. 1. Two-dimensional schematics of the atomistic deformation mech-
anisms proposed for amorphous metals, including (a) a shear transfor-
mation zone (STZ), after Argon [40], and (b) a local atomic jump, after
Spaepen [39].
C.A. Schuh et al. / Acta Materialia 55 (2007) 4067–4109 4069
! 4!
The discussions in this chapter will be largely confined temperatures above 0.7T
g
. As will
be discussed subsequently, this is the regime in which homogeneous deformation is
observed. This review is referring to this regime as a “creep” regime of amorphous
metals. A practical importance of this regime is that this is where forming of a metallic
glass is frequently performed. This regime is contrasted by the regime of lower
temperatures where heterogeneous deformation or shear banding is often (but not always)
observed.
Mechanisms of Deformation
The suggested mechanisms have generally fallen into three categories: a.) Dislocation-
like defects [6,36,37], diffusion type deformation [8,35], and shear transformation zones
(STZs)[9,52]. These are illustrated in Fig. 2 [54] of Lu et al., and are all early
explanations for plasticity but it appears that the amorphous metals community has
generally embraced the third, STZ [2,34,39].
Figure 2. a). Two-dimensional representation of a dislocation line in crystalline (left) and
amorphous (right) solids; taken from [34]; Atomistic deformation of amorphous metals in the
form of b). Shear transformation zones (STZ); and c). Local atomic jump; adapted from [1].
! 5!
a).
τ τ
b). c).
Figure 2. a). Two-dimensional representation of a dislocation line in crystalline (left) and
amorphous (right) solids; taken from [34]; Atomistic deformation of amorphous metals in the
form of b). Shear transformation zones (STZ); and c). Local atomic jump; adapted from [1].
The essence of this latter mechanism is that there is a so-called “free volume” in
amorphous metals. Free volume is a "concept" and it has no absolute definition. The
starting state is the baseline; only the difference has meaning, so a change in density after
deformation defines the free volume. The exact form and shape of these free volumes is
not known [38]. Increasing free volume would be associated with decreased density.
Estimates for free volume is about 3% for Zr
41.2
Ti
13.8
Cu
12.5
Ni
10
Be
22.5
(Vitreloy 1) [55].
Free volume decreases (tighter packing) appears to increase ductility in homogeneous
deformation at ambient temperature [51]. With an applied stress, groups of atoms (e.g.
few to 100 [1,2,62]), under an applied shear stress, τ, move and perform work. This
constitutes an STZ. Argon et al. [9,48] considered that the STZ operation takes place
within the elastic confinement of a surrounding glass matrix, and the shear distortion
leads to stress and strain redistribution around the STZ region [1,9,48]. When the STZs
exist throughout the alloy we have homogeneous deformation. STZs also occur in shear
large free volume, as illustrated in Fig. 2b [33]. Correspondingly, the activation energy for strain
accommodation via the diffusion-like atomic jump model is lower than that for that involving distor-
tion of the STZ. In spite of differences in local atomic motions, the STZ distortion model and the atom-
ic-jump free volume models share common features, which according to Schuh et al. [2] have
implications for plastic deformation of the glass, its temperature- and pressure-dependence and flow
localization.Steifetal.[47]modifiedSpaepen’smodelbyincludingadditionalfreevolumechangedue
to pressure. Khonik [48] proposed a directional structural relaxation model suggesting that each rear-
rangement event can be interpreted as a thermally-activated shear due to local atomic structures and
subsequentlynearlyathermalviscousflowbyexternalstress.Alternatively,acooperativeshearmodel
has also been proposed by Johnson and Samwer [49] in which yielding of metallic glasses displays a
(T/Tg)
2/3
temperature dependence. However, Spaepen’s [33] and Argon’s [46] models remain most
popular for describing deformation of metallic glasses.
The plastic flow and deformation behavior of metallic glasses has also been explained on the basis
ofdislocation models [50,51]. Although no lineardefects can be considered to exist in amorphoussol-
ids, the boundary between an un-deformed and deformed region, such as the propagating front of a
shear band, can be conceptualized as the dislocation. Fig. 3 illustrates a model of a dislocation in a
crystalline and glassy solid, represented by two-dimensional arrays of polyhedra. The dislocation line
is represented as the bottom row of atoms of extra-half plane perpendicular to the surface. Burger’s
vectors which are parallel and of same magnitude for the crystalline solid are different in magnitude
and direction in the case of the glassy solid. Hence, the dislocation line is not forced to remain on a
crystallographic plane, which does not exist in the glass. It is conceptualized that stress concentration
at the shear band front, i.e., the dislocation line, can activate STZs and result in growth and propaga-
tion of the shear band producing macroscopic strain. However, unlike the interaction of dislocations
withothermicrostructuralfeaturesincreasingtheresistance todeformationand influencingmechan-
ical properties of crystalline solids, no such resistance to plastic flow and associated strain hardening
is observed in the case of metallic glasses.
Themacroscopicdeformationresponseofmetallicglassesthroughaccumulationoflocalstrainsvia
operation of ‘‘shear transformation zones” or ‘‘atomic jumps” into free volume spaces, can occur
homogeneously or inhomogeneously depending on the temperature, applied stress, and strain rate.
Spaepen proposed a deformation mechanism map describing these two basic modes of deformation
in metallic glasses: homogeneous flow in which each volume element of the specimen contributes
to the strain, and inhomogeneous flow in which the strain is localized in few very thin shear bands
[33]. Fig. 4 shows the schematic map plotted with normalized stress versus homologous temperature
(T/Tm), illustrating how the deformation behavior transitions as a function of strain rate and temper-
ature.Homogeneousflow,whichisclosetoNewtonianviscous ð _ c/ sÞflow,occursatlowstressesand
high temperatures. In this deformation mode, the glass thins uniformly and fracture occurs when
some section of the specimen has narrowed to zero thickness. Inhomogeneous flow occurs at high
Fig. 3. Schematic showing dislocation represented by 2-D arrays of polyhedra in crystalline (left) and glassy (right) solids.
Dislocation in glass is conceptualized as boundary between un-deformed and deformed region, such as propagating front of a
shear band. (Adapted from [31].)
764 M.M. Trexler, N.N. Thadhani/Progress in Materials Science 55 (2010) 759–839
glass-based materials have also been developed – including
foams, composites and nanocrystal-reinforced alloys –
whose mechanical properties are just beginning to be stud-
ied seriously. There are a number of recent papers which
outline the mechanical properties of metallic glasses in
broad strokes, which tabulate and compare the properties
among glasses of different and varied compositions, or
which review some specific properties in detail [2–7]. On
the other hand, the fundamental principles and mecha-
nisms that underpin the mechanical properties of amor-
phous metals have not yet been holistically synthesized
with the accumulation of new data over the past decade
or so.
Our purpose in this article is to present an overview of
the mechanical properties of metallic glasses with a specific
focus upon fundamentals and mechanisms of deformation
andfracture.Thissynthesisfollowsintheveinoftheclassic
review articles of Pampillo (1975) [8] and Argon (1993) [9];
while many of the concepts laid out in these earlier reviews
remain equally valid today, we incorporate here what we
view as the most important refinements, revisions and
recent advances in understanding the deformation of
metallic glasses and their derivatives. Beginning from an
atomistic picture of deformation mechanisms in amor-
phous metal, we proceed to review elastic, plastic and frac-
ture behavior in light of these mechanisms. We then
explore the importance of glass structure and its evolution
during deformation, and survey the growing literature on
ductilization of metallic glasses. The paper concludes with
a view of important unresolved questions for what is a rap-
idly expanding field of research.
2. Deformation mechanisms
Because the bonding in amorphous alloys is of primarily
metallic character, strain can be readily accommodated at
the atomic level through changes in neighborhood; atomic
bonds can be broken and reformed at the atomic scale
without substantial concern for, e.g. the rigidity of bond
angles as in a covalent solid, or the balance of charges as
in an ionic solid. However, unlike crystalline metals and
alloys, metallic glasses do not exhibit long-range transla-
tional symmetry. Whereas crystal dislocations allow
changes in atomic neighborhood at low energies or stresses
in crystals, the local rearrangement of atoms in metallic
glasses is a relatively high-energy or high-stress process.
The exact nature of local atomic motion in deforming
metallic glasses is not fully resolved, although there is gen-
eral consensus that the fundamental unit process underly-
ing deformation must be a local rearrangement of atoms
that can accommodate shear strain. An example of such
a local rearrangement is depicted in the two-dimensional
schematic of Fig. 1a, originally proposed by Argon and
Kuo [10] on the basis of an atomic-analog bubble-raft
model. The event depicted in Fig. 1a has been referred to
as a ‘‘flow defect’’ or ‘‘s defect’’ [11,12], a ‘‘local inelastic
transition’’ [13–15] and, increasingly commonly, a ‘‘shear
transformation zone’’ (STZ) [12,16–22]. The STZ is essen-
tially a local cluster of atoms that undergoes an inelastic
shear distortion from one relatively low energy configura-
tion to a second such configuration, crossing an activated
configuration of higher energy and volume. Since the origi-
nal analog model of Argon et al. [10,23], more sophisti-
cated computer models have been employed to study
glass deformation in both two and three dimensions
[11,12,16,18,24–38]. STZs comprising a few to perhaps
!100 atoms are commonly observed in such simulation
works, which span a variety of simulated compositions
and empirical interatomic potentials; this suggests that
STZs are common to deformation of all amorphous met-
als, although details of the structure, size and energy scales
of STZs may vary from one glass to the next.
It is important to note that an STZ is not a structural
defect in an amorphous metal in the way that a lattice dis-
location is a crystal defect. Rather, an STZ is defined by its
transience – an observer inspecting a glass at a single
instant in time cannot, a priori, identify an STZ in the
structure, and it is only upon inspecting a change from
one moment in time (or strain) to the next that STZs
may be observed and cataloged. In other words, an STZ
is an event defined in a local volume, not a feature of the
glass structure. This is not to suggest that the operation
of an STZ is independent of the glass structure; indeed,
STZ operation is strongly influenced by local atomic
arrangements, and also has important consequences for
structural evolution of a deforming glass. In a metallic
glassbodyexperiencinguniformstress,theSTZthatisacti-
vated first is selected from among many potential sites on
the basis of energetics, which vary with the local atomic
arrangements [11,27,36–38]. For example, the local distri-
bution of free volume is widely believed to control defor-
mation of metallic glasses [10,23,39–42], and it is easy to
envision that sites of higher free-volume would more read-
ily accommodate local shear. Atomistic simulations have
also correlated other structural state variables with local
shearing, including short-range chemical or topological
order [11,43,44].
Fig. 1. Two-dimensional schematics of the atomistic deformation mech-
anisms proposed for amorphous metals, including (a) a shear transfor-
mation zone (STZ), after Argon [40], and (b) a local atomic jump, after
Spaepen [39].
C.A. Schuh et al. / Acta Materialia 55 (2007) 4067–4109 4069
glass-based materials have also been developed – including
foams, composites and nanocrystal-reinforced alloys –
whose mechanical properties are just beginning to be stud-
ied seriously. There are a number of recent papers which
outline the mechanical properties of metallic glasses in
broad strokes, which tabulate and compare the properties
among glasses of different and varied compositions, or
which review some specific properties in detail [2–7]. On
the other hand, the fundamental principles and mecha-
nisms that underpin the mechanical properties of amor-
phous metals have not yet been holistically synthesized
with the accumulation of new data over the past decade
or so.
Our purpose in this article is to present an overview of
the mechanical properties of metallic glasses with a specific
focus upon fundamentals and mechanisms of deformation
andfracture.Thissynthesisfollowsintheveinoftheclassic
review articles of Pampillo (1975) [8] and Argon (1993) [9];
while many of the concepts laid out in these earlier reviews
remain equally valid today, we incorporate here what we
view as the most important refinements, revisions and
recent advances in understanding the deformation of
metallic glasses and their derivatives. Beginning from an
atomistic picture of deformation mechanisms in amor-
phous metal, we proceed to review elastic, plastic and frac-
ture behavior in light of these mechanisms. We then
explore the importance of glass structure and its evolution
during deformation, and survey the growing literature on
ductilization of metallic glasses. The paper concludes with
a view of important unresolved questions for what is a rap-
idly expanding field of research.
2. Deformation mechanisms
Because the bonding in amorphous alloys is of primarily
metallic character, strain can be readily accommodated at
the atomic level through changes in neighborhood; atomic
bonds can be broken and reformed at the atomic scale
without substantial concern for, e.g. the rigidity of bond
angles as in a covalent solid, or the balance of charges as
in an ionic solid. However, unlike crystalline metals and
alloys, metallic glasses do not exhibit long-range transla-
tional symmetry. Whereas crystal dislocations allow
changes in atomic neighborhood at low energies or stresses
in crystals, the local rearrangement of atoms in metallic
glasses is a relatively high-energy or high-stress process.
The exact nature of local atomic motion in deforming
metallic glasses is not fully resolved, although there is gen-
eral consensus that the fundamental unit process underly-
ing deformation must be a local rearrangement of atoms
that can accommodate shear strain. An example of such
a local rearrangement is depicted in the two-dimensional
schematic of Fig. 1a, originally proposed by Argon and
Kuo [10] on the basis of an atomic-analog bubble-raft
model. The event depicted in Fig. 1a has been referred to
as a ‘‘flow defect’’ or ‘‘s defect’’ [11,12], a ‘‘local inelastic
transition’’ [13–15] and, increasingly commonly, a ‘‘shear
transformation zone’’ (STZ) [12,16–22]. The STZ is essen-
tially a local cluster of atoms that undergoes an inelastic
shear distortion from one relatively low energy configura-
tion to a second such configuration, crossing an activated
configuration of higher energy and volume. Since the origi-
nal analog model of Argon et al. [10,23], more sophisti-
cated computer models have been employed to study
glass deformation in both two and three dimensions
[11,12,16,18,24–38]. STZs comprising a few to perhaps
!100 atoms are commonly observed in such simulation
works, which span a variety of simulated compositions
and empirical interatomic potentials; this suggests that
STZs are common to deformation of all amorphous met-
als, although details of the structure, size and energy scales
of STZs may vary from one glass to the next.
It is important to note that an STZ is not a structural
defect in an amorphous metal in the way that a lattice dis-
location is a crystal defect. Rather, an STZ is defined by its
transience – an observer inspecting a glass at a single
instant in time cannot, a priori, identify an STZ in the
structure, and it is only upon inspecting a change from
one moment in time (or strain) to the next that STZs
may be observed and cataloged. In other words, an STZ
is an event defined in a local volume, not a feature of the
glass structure. This is not to suggest that the operation
of an STZ is independent of the glass structure; indeed,
STZ operation is strongly influenced by local atomic
arrangements, and also has important consequences for
structural evolution of a deforming glass. In a metallic
glassbodyexperiencinguniformstress,theSTZthatisacti-
vated first is selected from among many potential sites on
the basis of energetics, which vary with the local atomic
arrangements [11,27,36–38]. For example, the local distri-
bution of free volume is widely believed to control defor-
mation of metallic glasses [10,23,39–42], and it is easy to
envision that sites of higher free-volume would more read-
ily accommodate local shear. Atomistic simulations have
also correlated other structural state variables with local
shearing, including short-range chemical or topological
order [11,43,44].
Fig. 1. Two-dimensional schematics of the atomistic deformation mech-
anisms proposed for amorphous metals, including (a) a shear transfor-
mation zone (STZ), after Argon [40], and (b) a local atomic jump, after
Spaepen [39].
C.A. Schuh et al. / Acta Materialia 55 (2007) 4067–4109 4069
111
Conversely, at stresses, τ >> kt/V,
γ=
,
-
α
x
ν
x
γ
x
∗exp −
Ú)
mn
(E-4.3)
Schuh et al. [110] point out that equation (E-4.1) suggests a Newtonian region followed by, with
increasing stress, continual increase in stress exponent. Examples of BMGs that have evinced
equations E4.1 – E-4.3 behaviors are illustrated in Figures 4-3 and 4-4, which plot the steady-state
creep behavior of several BMGs [110, 149].
The figures illustrate steady-state behavior that with increasing strain-rate and/or decreasing
temperature there is a breakdown in Newtonian behavior and the apparent stress-exponent
increases. Generally, this has been regarded as a natural consequence of Eq. E-4.2, the rate
equation that predicts Newtonian behavior at low stresses (higher temperature and lower strain-
rates) but increased exponents with higher stresses (low temperatures and higher strain-rates). This
explanation does not appear to be unanimously embraced [149]. For some, an important question
is whether the non-Newtonian homogeneous deformation region is actually a reflection of nano-
crystallization.
These equations suggest that free volume is largely responsible for plastic flow; larger free
volumes would appear to more easily lead to regions of plastic flow. Schuh et al point out that
atomic simulations have suggested that other variables such as short-range chemical ordering can
affect plasticity as well, which is not explicitly included in the above equations [164-166]. The
pressure sensitivity of these equations was addressed by Sun et al. [167].
112
FIGURE 4-3. Steady-state homogeneous flow data for Zr
41.2
Ti
13.8
Cu
12.5
Ni
10
Be
22.5
metallic glass
at elevated temperatures, from the work of Lu et al. [140]. Figure based on [110].
1
10
100
1000
0.000001 0.00001 0.0001 0.001 0.01 0.1 1
Stress (MPa)
Strain-Rate (s
-1
)
Zr
41.2
Ti
13.8
Cu
12.5
Ni
10
Be
22.5
683 K
663 K
643 K
623 K
613 K
603 K
593 K
573 K
10
#$
10
#&
10
#'
10
#(
10
#)
10
#*
1
Strain-Rate (s
-1
)
10
(
10
)
10
*
113
FIGURE 4-4. Stress–strain rate curve for a Zr
10
Al
5
Ti
17.9
Cu
14.6
Ni glassy alloy shows Newtonian
flow at low strain rates but non-Newtonian at high strain rates (data from Ref.
[156]). Figure based on [149].
10
100
1000
0.0001 0.001 0.01
Stress (MPa)
Strain Rate (s
-1
)
Zr
10
Al
5
Ti
17.9
Cu
14.6
Ni
T = 683 K
m = 1
10
−4
10
−3
10
−2
10
(
10
)
10
*
114
Nieh and Wadsworth [149] found that nano-crystallization occurred in Zr
10
Al
5
Ti
19.9
Cu
14.6
Ni BMG
coincident with the deviation from Newtonian behavior. Nieh rationalized the nano-crystalline
precipitates as akin to dispersion strengthening. Suryanarayana and Inoue [111] appear to suggest
that the stress exponent increases due to second phase strengthening of the nanoparticles by a
straightforward rule of mixtures for the flow strength. Schuh et al. [110] referenced the Nieh and
Wadsworth work and certainly acknowledged the observation that deformation can induce
crystallization (as have others [168-171] favor the rate equation as an explanation for the deviation
from Newtonian behavior at higher stresses. Wang et al [172] found only non-linear creep behavior
in Vitreloy 1 if some crystallization occurred. Whereas Newtonian conditions led to elongations
in La
55
Al
25
Ni
20
in excess of 20,000% [149], those at higher rates with non-Newtonian behavior
exhibited dramatically reduced values. Many authors [110-112] proposed metallic glass
deformation maps, similar to the construct by Ashby and Frost for crystalline materials. A metallic
glass deformation–map is illustrated in Figure 4-5.
Newtonian deformation appears to reflect a fully amorphous alloy, but at least in other regions,
including heterogeneous deformation, nano-crystallization may be occurring [168-171].
Furthermore, as will be discussed in a subsequent section, homogeneous deformation may extend
to low temperature, in at least some cases.
115
FIGURE 4-5. Deformation mechanism maps for metallic glass plotted in (a) normalized stress
versus normalized temperature. The absolute stress values indicated in the figure
are for the Zr
41.2
Ti
13.8
Cu
12.5
Ni
10
Be
22.5
metallic glass. Adapted from [110, 112].
Applied Pressure:
0.01µ (0.4 GPa)
Applied Pressure:
0µ (0 GPa)
ELASTIC
DEFORMATION
SHEAR LOCALIZATION
(inhomogeneous deformation)
Non-Newtonian
Newtonian
0.03
0.3
0.001
0.01
0.1
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Shear Stress at 300 K (GPa)
Normalized Stress (t/µ)
T/T
g
Homogeneous
Deformation
Non-Newtonian
Newtonian
SHEAR LOCALIZATION
(inhomogeneous deformation)
10
#$
10
#%
10
#&
116
FIGURE 4-6. Effect of strain rate on the uniaxial stress-strain behavior of Vitreloy 1 at
643 K and strain rates of 1.0 × 10
-1
, 3.2 × 10
-2
, 5.0 × 10
-3
and 2.0 × 10
-4
s
-1
[145].
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Stress (MPa)
Strain
0.1 (1/s)
0.032 (1/s)
0.005 (1/s)
0.0002 (1/s)
s
"#
s
"#
s
"#
s
"#
Viterloy 1
Zr
41.2
Ti
13.8
Cu
12.5
Ni
10
Be
22.5
T = 643 K
117
4.3 HOMOGENEOUS FLOW AT VERY LOW TEMPERATURES
Recent work [148, 173-175] shows that, given sufficient time, homogeneous deformation can be
detected under “electrostatic” (i.e. at a stress less than the yield stress, σ
D
) loading at room
temperature (RT). The stress exponent was not assessed, so it was unclear whether Newtonian
flow was observed. Alloys include Zr
46.75
Ti
8.25
Cu
7.5
Ni
10
Be
27.5
, Ni
62
Nb
38
, Cu
50
Zr
50
, Cu
57
Zr
43
and
Cu
65
Zr
35
. Of course, some BMGs, such as Zn
20
Cu
20
Tb
20
(Li
0.55
Mg
0.45
)
20
, may have low T
g
(323K)
allowing homogeneous deformation at RT [176]. Alloys with higher packing densities exhibit
greater plastic strain during homogeneous deformation at room temperature, but show less global
plasticity during inhomogeneous deformation in a typical compression test [145]. Park et al.
suggest deformation induced structural disordering by MD simulations, although others [173]
simply STZ as the mechanism. Compression tests on Pd
77
Si
23
with 8µm and 140mm showed that
as the sample size decreased to the submicron range, homogeneous deformation occurs and was
suggested to occur due to the necessity of a critical size volume for shear bands [177].
Anelasticity
In the above discussion of the so-called “electrostatic” regime, a substantial fraction of the (small)
non-elastic strain is anelastic. It should be noted that the STZ model naturally predicts some
anelasticity. An isolated STZ, by the Argon model, is elastically constrained during activation.
This implies that even at low applied stress (where backflow according to Eq. (E-4.2) is negligible),
there is, nonetheless, a back stress which on unloading leads to anelastic back flow. It was
additionally pointed out by Ke et al. [174] that there is a range of atomic environments in a glass
is such that some atoms reside in regions where the local topology is unstable. In these regions,
the response to shear stress may include not only atomic displacements but also an anelastic
118
reshuffling of the atomic near-neighbors (i.e. an anelastic STZ operation). Even though the fraction
of atoms involved in these events may be small, the local strains are large enough that their
cumulative effect makes a significant contribution to the macroscopic strain [174].
4.4 PRIMARY AND TRANSIENT CREEP (NON-STEADY-STATE FLOW)
Steady-state flow has principally been discussed, so far. It has been presumed that STZs create
free volume (leading to softening) and that recovery processes or the annihilation of free volume
(leading to hardening). Therefore, steady state has been regarded as a balance between free volume
creation and annihilations. Other hardening effects such as chemical ordering have not been
explicitly considered for steady state. It has been suggested that there can be a net free volume
increase or decrease during deformation that precedes a steady state. Figure 4-6 from Lu et al.
[140] shows hardening at the onset of deformation that continues beyond the eventual steady state.
The interpretation of this peak stress followed by softening to a steady state is unclear.
4.5 SUMMARY
This review of creep (above 0.7T
g
) in amorphous alloys emphasizes a variety of conclusions. First,
the mechanism of creep appears to largely be explained by shear transformation zones where
deformation is homogeneous. The descriptive equation for STZs suggests a Newtonian region
followed by, with increasing stress, a continual increase in the stress exponent. It is not clear, as
some suggest, that the non-Newtonian behavior is due to nano-crystallization. Second,
homogeneous deformation at room temperature has recently been observed. Third, a substantial
119
fraction of the small non-elastic strain at room temperature is anelastic. Fourth, during primary
creep, STZs create free volume, leading to softening. Furthermore, recovery processes or
annihilation of free volume leads to hardening.
120
CHAPTER 5. THROUGH-THICKNESS COMPRESSION TESTING OF
COMMERCIALLY PURE (GRADE-II) TITANIUM THIN
SHEET TO LARGE STRAINS.
5.1 BACKGROUND
The purpose of this study was to assess the through-thickness compressive mechanical-behavior
of a thin commercial-purity titanium sheet, to relatively large strains. Such sheet is often used in
metal forming operations. Several studies have previously assessed the mechanical properties of
Commercially-Pure (CP) titanium in compression, usually as an extruded rod or sheet. Sheet
thicknesses were much larger than the current study and strains were relatively small in
comparison to the current study as well. Adiabatic heating may have been an additional issue
complicating the stress versus strain behavior in these earlier studies. Large strain studies of metals
reveal substantially different hardening behaviors in tension, compression and torsion, largely due
to different textural evolutions (differences in average M or Taylor factors) [178]. The focus of the
current study was large-strain deformation in compression. Earlier studies are reviewed first
below.
It should also be mentioned that throughout this chapter, both Grade-I and Grade-II
titanium are discussed. The difference between the two are directly related to low (≤ 0.18%) and
standard (≤ 0.25%) oxygen content, respectively, according to the ASTM B265 Standards. Grade-
II generally has a higher yield strength.
Compression experiments on CP (Grade-I) Ti performed by Battaini et al. [179] focused on
orientation effects. Five differently oriented samples were prepared from a 99.49% pure hot-rolled
plate with a 10-mm thickness. However, only samples with the compression axis parallel to the
121
plate normal (denoted as NT and NR) are discussed in this thesis. The notation given to the samples
are defined by the compression direction (1
st
letter) and the extension direction (2
nd
letter), where
‘N’ denotes the normal direction, ‘R’ denotes the rolling direction, and ‘T’ denotes the transverse
direction. The stress-strain curves for NT and NR are illustrated in Figure 1a and were deformed
at a strain-rate (𝜀) of 0.1 s
-1
. It is unclear whether the samples were annealed prior to testing. A
grain size of 22 µm was reported. No lubrication was specified. Their orthorhombic samples had
dimensions of 8-mm length, 12-mm width and 5.95-mm height. To be consistent with the aspect
ratio annotation in the present study, that will be discussed later, the above dimensions can be
translated to an aspect ratio annotation of N/ 𝑇𝑅 , where 𝑇𝑅 denotes the average of the length
(T) and width (R). Therefore, sample NR and NT have an aspect ratio of 0.595 (rounded to 0.60
in Figure 5-1. Specimens deformed in the normal direction showed the highest yield stress. A
lower yield and lower flow stresses were observed along the axis parallel to the rolling direction.
Twinning was also observed in their deformed specimens as confirmed with EBSD. The specimens
were only deformed to a modest strain of about 0.12.
Podolskiy et al. performed compression tests on 99.52% commercially pure (Grade-II) tetragonal-
shaped titanium [180] specimens deformed at 𝜀 = 3 x 10
-4
s
-1
. The samples were cut from an
extruded rod and loaded parallel to the extrusion direction. Samples were not specified as having
been annealed, but were dipped in liquid nitrogen immediately following machining of individual
samples. The dimensions were reported as 1.8 mm x 1.8 mm x 3.5 mm (length x width x height)
with an aspect ratio of 1.94.
122
FIGURE 5-1. 99.49% Commercially-pure (Grade I) Ti tested in compression by Battaini et al.
[179] with the loading axis parallel to the plate normal. The average grain size was
22µm. Samples with orientation, NT (green) and NR (black), show yield stresses
of 310 and 350 MPa, respectively, using a 0.2% offset. Specimen aspect ratios are
about 0.60. Annealing of the material prior to testing was not specified.
0
100
200
300
400
500
600
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
True Stress (MPa)
True Strain
99.49% CP-(I) Titanium
(M. Battaini et. al.)
! = 10
-1
s
-1
s
y
(NR,0.2%)
= 350 MPa
s
y
(NT ,0.2%)
= 310 MPa
average grain size = 22µm
NT and NR (h/w) = 0.60
(Loading axis parallel to plate normal)
NR
NT
123
An engineering stress-strain curve is shown for the course-grained (CG) sample. Podolskiy et al.
used the CG notation to denote a sample that was not subjected to severe plastic deformation (SPD)
through ECAP. Though an initial grain size is not reported, it can be presumed that a CG sample
is coarse as compared to the ultra-fine grain (UFG) sized samples subjected to SPD. The ambient
temperature Ti data is illustrated in Figure 5-2. A noticeable upward parabolic shape characteristic
of an increasing hardening-rate (ds/de) is observed, which is unexpected. As discussed more
subsequently, a surprisingly high strain-hardening rate was observed at uncharacteristically low
strain levels around 0.15. No lubrication was stated to have been used in this study. Perhaps the
rapid increase in stress over a small strain interval is a result of frictional effects leading to high
triaxiality. This topic is discussed more in the following subsection. The increase in stress could
appear exaggerated when plotted as engineering stress-strain rather than true stress-strain,
however, the parabolic shape would still be apparent even if the conversion is taken into account.
A rather high 0.2% offset yield stress was observed at 500 MPa. This may be an indication of pre-
straining from the prior extrusion in addition to any frictional effects.
Coarse-grained (CG) and ultra-fine grained (UFG) titanium were compared at cryogenic, ambient
and elevated temperatures by Long et al. [181]. A 99.60% pure Ti cylindrical billet with an initial
diameter of 25 mm and length equal to 100 mm was used as the starting material. Grain sizes for
the CG-Ti and UFG-Ti are 30 µm and 250 nm, respectively. The CG-Ti samples were tested from
extruded rod, where the UFG-Ti was additionally refined by ECAP. The sample dimensions were
tetragonal and had length:width:height ratios equal to 1:1:1.5 (an aspect ratio of 1.50). Specific
sample dimensions were not reported. The ambient temperature CG-Ti data is reported in Figure
5-3. A large amount of strain-softening was observed in the UFG-Ti in contrast to the CG-Ti
124
(UFG-Ti plot is not shown for comparison due to lack of relevance to the present study). A lower
yield stress was observed in the CG-Ti as opposed to the UFG-Ti. Relatively large strains of 0.55
were achieved. A 0.2% offset yield strength equal to 530 MPa for the CG-Ti was reported. Please
note that the Long et al. [181] study reported the elastic region of their curve. It is unclear if this
region represents the compliance or the actual elastic regime of their sample; however, the curve
is displayed here as it was published.
Nemat-Nasser et al. [182] performed compression studies on 99.99% high-purity titanium at
temperatures ranging from 77 K – 1000 K under lower (10
-3
– 10
-1
s
-1
) and dynamic (2200 – 8000
s
-1
) strain-rate conditions. Only experiments done at strain-rates (𝜀) of 10
-1
and 10
-3
s
-1
at ambient
temperature are discussed here. Cylindrical samples with an aspect ratio of 1.0 were annealed at
704°C before testing. The sample height was 4.8 mm and the average grain size after annealing
was 40 µm. The compression axis was parallel to the extrusion direction of the extruded rod.
Strains up to 0.43 and 0.41 were achieved for 𝜀 = 10
-3
s
-1
and 𝜀 = 10
-1
s
-1
, respectively, at 296 K.
The behavior is illustrated in Figure 5-4.
Mechanical twins were observed in this study, as well. This data suggests a strain-rate sensitivity
(m) between 0.027 and 0.044 [182]. The strain-rate sensitivity is defined by the expression:
𝑚=
ãäH (å)
ãäH (æ)
f,À
(E-5.1)
where s denotes a constant dislocation structure and T is temperature. The lower strain-rate shows
a lower yield stress as expected.
125
FIGURE 5-2. 99.52% Commercially pure (Grade II) titanium tested in compression with the
loading axis parallel to extrusion direction by Podolskiy et al. [180]. The average
grain size was not reported. The yield stress was 500 MPa (with a 0.2% offset) and
the aspect ratio was equal to 1.94.
0
200
400
600
800
1000
1200
1400
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Engineering Stress (MPa)
Engineering Strain
99.52% CP-(II) Titanium
(A.V . Podolskiy et. al.)
! = 3 x 10
-4
s
-1
σ
y
= 500 MPa;
h/w = 1.94
(Loading axis parallel to extrusion direction)
126
FIGURE 5-3. 99.60% Commercially-pure titanium (grade not specified) was tested in
compression at a strain-rate of 10
-2
s
-1
by Long et al. [181]. The starting grain size
was 35 µm. The loading axis in relation to sample orientation was not specified.
The yield stress is 530 MPa. Please note that this study reported the elastic region
of their curve. It is unclear if this region represents the compliance or the actual
elastic regime of their sample; however, the curve is displayed here as it was
published.
0
100
200
300
400
500
600
700
800
900
1000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True Stress (MPa)
True Strain
99.60% CP-Titanium
(F.W.Long et. al.)
! = 10
-2
s
-1
σ
y
(0.2%)
= 530 MPa;
average grain size = 35 µm
h/l = 1.50
(Loading axis orientation to sample not specified)
127
Strains greater than 1.0 were achieved by Salem et al. [183] in 99.998% high-purity titanium. Re-
lubrication, with high-pressure grease and Teflon sheets, was performed every 0.3~0.4 strain.
This was intended to reduce barreling and allow for higher strain accumulation by precluding
excessive triaxiality. Samples were machined from a plate of unspecified thickness and were
annealed for 1 hour at 800˚C; then oil-quenched. The behavior of this material is shown in Figure
5-5.
The deformed samples were prepared for TEM in order to observe the onset of twinning and Salem
et al. reports this as the primary mode of deformation for high-purity a-titanium. Twins were not
observed until about 0.05 strain, resulting in an increased strain-hardening rate.
Figure 5-6 is a composite of the previously discussed literature studies [179-183]. The lower two
reported yield stresses are the higher purity compression experiments. The highest yield stress
appears to represent a pre-strained, un-annealed (and possibly unlubricated) specimen. All, but
one, of the studies are for strains of 0.12 to 0.55. There appears to be significant variation in the
yield stress of the specimens, even with three studies of comparable purity. Additionally, the tests
are at higher strain-rates where adiabatic heating (particularly in CP Ti) can occur and obfuscate
the isothermal mechanical behavior [184]. The grain sizes appear to be generally similar.
128
FIGURE 5-4. 99.99% pure titanium was tested in compression at strain-rates of 10
-1
and 10
-3
s
-1
by Nemat-Nasser et al. [182]. The average grain size was 40µm. The loading axis
was parallel to extrusion direction. The sample yielded at 170 MPa at ε= 10
-3
s
-1
and 215 MPa at ε= 10
-1
s
-1
.
0
100
200
300
400
500
600
700
800
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
True Stress (MPa)
True Strain
99.99% Titanium
(S. Nemat-Nasser et al.)
! =10
-1
s
-1
& 10
-3
s
-1
σ
y
(0.2%)
= 215 MPa & 170 MPa;
average grain size = 40 µm
t/d = 1.0
(Loading axis parallel to extrusion direction)
!̇ = 10
-1
s
-1
!̇ = 10
-3
s
-1
129
FIGURE 5-5. 99.998% pure titanium tested in compression at a strain-rate of 10
-2
s
-1
by Salem et
al. [183]. The average grain size was 30 µm. The loading axis was parallel to the
plate normal and the sample yielded at 270 MPa.
0
100
200
300
400
500
600
700
800
900
1000
0 0.2 0.4 0.6 0.8 1 1.2
True Stress (MPa)
True Strain
99.998% High-Purity α-Titanium
(A.A. Salem et. al.)
! = 10
-2
s
-1
σ
y
(0.2%)
=270 MPa;
average grain size = 30 µm
t/d unspecified
(Loading axis parallel to plate normal)
130
5.2 PRESENT RESEARCH
This study attempted to characterize the large-strain stress versus strain behavior of very thin
commercial purity Ti sheet in the through-thickness direction at strain-rates of 10
-4
s
-1
where
adiabatic heating is not expected and isothermal behavior can be characterized. Higher strains than
observed in the literature would be attempted. This would represent new data in the literature, but
this work had several experimental complications, as discussed subsequently.
As noted in the previous literature studies, titanium compression tests were performed on
specimens that, in all cases, were 8 mm or greater in height for commercial purity (about 5 mm for
high purity). Furthermore, as mentioned previously, the total strains were only about 0.12 to 0.55
(with only one exception). As discussed subsequently, these are relatively thick. Also, the strains
are typically much larger for metal forming of thin sheets and the sheet thicknesses are often below
2mm. Strain-rates for the literature studies, discussed previously, were also typically relatively
high at 10
-2
s
-1
to 10
-1
s
-1
which, as mentioned earlier, can be associated with adiabatic heating,
particularly in higher strength CP-Ti as compared to high-purity Ti, and temperature rises in the
specimen can occur.
Barreling and frictional stresses are a major complication in all compression testing.
Compression experiments especially on sheet metals with an initially low height, as in this study,
encounter frictional effects which can significantly complicate the assessment of the uniaxial stress
versus strain behavior. Bly [185] provided an analysis of the frictional effects in compression
testing and estimated a correction factor for frictional effects to the observed flow stress in a
material. Bly’s correction factor is used in the analysis of the present work. Barreling is typical of
a sample that has a very small aspect ratio (l/d << 0.8, i.e. l << d) [77], where ‘l’ is the specimen
height and ‘d’ is the specimen diameter.
131
FIGURE 5-6. Summary graph of the compression stress versus strain behavior discussed in the
earlier figures in this chapter.
0
200
400
600
800
1000
1200
1400
0 0.2 0.4 0.6 0.8 1 1.2
True Stress (MPa)
Engineering Stress (MPa) for A.V. Podolskiy et al.
True Strain
Engineering Strain for A.V. Podolskiy et al.
Commercially-Pure and High-Purity Titanium
(Combined Literature)
99.60% CP-Ti (F.W. Long et. al.), t = not given, t/d = 1.50
99.52% CP-(II) Ti (A.V. Podolskiy et al.), t = 35 mm, t/d = 3.50
99.49% CP-Ti [NR] (M. Battaini et. al.), t = 8 mm, t:w:d = 0.75
99.49% CP-Ti [NT] (M. Battaini et. al.), t = 8 mm, t:w:d = 0.75
99.998% High-Purity Ti (A.A. Salem et. al.), dimensions not given
99.99% High-Purity Ti (S. Nemat-Nasser et al.), t = 4.8 mm, t/d = 1.00
strain-rate = 10
-2
s
-1
grain size = 30µm
strain-rate = 3 x 10
-4
s
-1
grain size not reported
strain-rate = 10
-1
s
-1
grain size = 22µm
strain-rate = 10
-2
s
-1
grain size = 35µm
strain-rate = 10
-3
s
-1
grain size = 40µm
132
5.3 MATERIALS
The Commercially-Pure (Grade-II) titanium (CP-II Ti) material used in this study was supplied by
UniTi Titanium LLC in Coraopolis, PA. The chemical composition, shown in Table 5.1, was
performed using inductively coupled plasma atomic emission spectroscopy (IC-AES). This
analysis was done by Westmoreland Mechanical Test & Research, Inc. (WMT&R) in accordance
with ASTM E2371-13.
TABLE 5.1. Chemical composition of the CP-II Titanium used in this study
Element Carbon Oxygen Nitrogen Hydrogen Iron Ti (balance)
Weight (in %) 0.01 % 0.13 % 0.002 % 0.0018 % 0.09 % 99.76 %
The titanium sheet was 1.60 mm in thickness, much thinner than that of sheet/plate of the
previously discussed studies. It was provided as annealed and the average grain size was about 14
µm. The average grain diameter was calculated using the average grain intercept method. A
micrograph of the normal, rolling and transverses faces, respectively, are illustrated in Figure 5-7
(a-c). These were provided by NanoPrecision in El Segundo CA.
The titanium sheet was sent to Able Wire EDM Inc, in Brea, CA where electrical discharge
machining (EDM) was used to cut the sheet into smaller cylindrical samples. All samples were
spark cut to a diameter of 3 mm. There were many samples made but only seven of them were
compression tested and named sequentially as Samples 1, 2, 3, 4, 7, 8 and 10. As discussed in the
Introduction, some of the specimens were re-machined after strains of approximately 0.4 – 0.5.
133
(a).
(b).
(c).
FIGURE 5-7 (a-c). Optical micrographs of the plane (a) normal, (b) rolling and (c) transverse
sections of the annealed titanium sheet. Micrographs were provided by
nanoPrecision in El Segundo, CA.
134
5.4 SAMPLE PREPARATION
5.4.1 MACHINING TITANIUM SHEET INTO INDIVIDUAL SAMPLES BY EDM
The titanium sheet (specification analysis in Section 5.3), provided by NanoPrecision to USC, was
sent to Able Wire EDM Inc., in Brea, CA. The company used Electrical Discharge Machining
(EDM) to cut 15 smaller cylindrical samples of equal diameter. The height of the titanium sheet
was unaffected during cutting. A diameter of 3.0 mm was chosen yet each sample’s measurements
vary within +/- 0.0002 mm. A schematic of the sample dimensions is shown in Figure 5-8.
Individual measurements were taken with 4-decimal place accuracy and are detailed in the
following sections. All thicknesses of the specimens were on the order of the original sheet
thickness: 1.60 mm.
EDM is a method used to cut samples with a thin wire electrode, generally, 0.1–0.3 mm diameter.
The wire electrode is charged to an extremely high voltage while surrounded by continuously
running deionized water. A spark occurs (~8,000-12,000°C) which locally melts the sample
surface and then is rapidly cooled by flowing water. There is no physical contact between the
actual wire and specimen surface leading to more precise cutting and little to no damage layer. The
cutting accuracy is within +/- 0.00254 mm.
135
FIGURE 5-8. Initial measurements of a titanium sample that was cut using the EDM method.
~ 1.6002 (+/- 0.0002) mm
~ 3.0000 (+/- 0.0002) mm
x
y
z
Rolling Direction
136
TABLE 5.2a: Friction coefficients [185-189] of the lubricants used in the present study. The
graphite and MoS
2
aerosol lubricants were generously sprayed on the compression
platens before testing.
Lubricant Form of Lubricant Friction Coefficient (µ)
Graphite Dry Aerosol 0.123 [189]
Molybdenum Sulfide (MoS
2
) Dry Aerosol 0.03 or 0.23 [187, 188]
PTFE Teflon 0.0762 mm-thick paper 0.05 – 0.08 [186]
TABLE 5.2b: Lubricants used for each sample. In the case of the MoS
2
and Teflon combination,
the Teflon was placed on the upper and lower compression platens first, then the
MoS
2
was generously applied to the top of the Teflon.
Sample Lubricant
Sample 1 Graphite
Sample 2 MoS
2
Sample 3 MoS
2
Sample 4 MoS
2
Sample 7 MoS
2
and Teflon
Sample 8 MoS
2
and Teflon
Sample 10 MoS
2
and Teflon
137
5.4.2 MECHANICAL POLISHING
Silicon Carbide (SiC) grit paper
The III Wet Polisher/Grinder was used with several grades ranging from 320-1200 of 20 cm
diameter CarbiMet SiC grit paper discs from Buehler. Cool deionized water was continuously
flowing to keep the grit paper wet. This was thought to reduce any mechanical damage done to the
sample because of excess friction. During polishing with grit paper, the sample is held in a
constant/steady position to create unidirectional striations (refer to Figure 5-9). Between every
increment of grit paper, the sample was rinsed thoroughly with deionized water then dried with
high-pressure compressed air. The sample was then rotated 90 degrees with respect to the previous
position before continuing the polishing process (refer to Figure 5-9b). When the preceding
striations are no longer visible and a new set of striations appear in a 90-degree orientation with
respect to the previous striations (refer to Figure 5-9c), then it is clear that the damage layer from
the previous grit level is removed. The process is repeated with the next level of grit paper until
1200 grit is reached. An optical micro-scope was used to check the striations between each step.
An example of a sample surface with incomplete polishing is shown in Figure 5-10.
Diamond/Alumina Suspension slurry
Buehler MetaDi™ monocrystalline diamond suspensions were used, after the 1200 grit paper, on
a Struers LaboPol-2 203 mm diameter magnetic polishing wheel to further remove the damage
from the grit paper and further polish the sample to a finer level. The diamond slurries range from
9 µm – 0.25 µm. The last step was to use 0.05 µm alumina paste. Deionized water was used to
keep the slurry moist during polishing and to keep a soapy texture.
138
During the diamond suspension polishing, the specimen was moved in a random manner to ensure
that the diamond particles were continuously hitting the sample surface at different angles. This
ensures a uniform, smooth surface. After the final steps of polishing the sample surface had a
mirror-like appearance.
Between polishing diamond slurry polishing steps, the microscope was used to identify if further
polishing is needed by measuring the width of scratches on the surface. If the scratches were a
width greater than the diamond slurry particle size just used, then the specimen would require
further polishing based on the size of scratch. For example, if the width of a scratch was 5 µm the
specimen required re-polishing starting with 6 µm slurry.
The below table denotes the polishing routine that took place in between the compression testing
phases. Note that, not all samples were subjected to the same polishing routine. Individual sample
specifications are detailed in the results section.
Polisher Used Polishing Routine
Compression
Platens
Diamond Slurry 0.25 µm
and Alumina 0.05 µm
Manually scrubbed for 5 mins using Kimwipes®
and deionized water
Specimen Alumina 0.05 µm
Manually polished by carefully moving sample
around with a finger for 5 mins. using a mini
Struers LaboPol-2 microfiber felt pad
139
FIGURE 5-9 (a-c). Polishing procedure using the SiC grit paper. (a) positioning the sample in
one direction starting with 320 grit. (b) change the grit paper to the
following grade (e.g 400 grit), rotate 90° and continue polishing
maintaining same direction. (c) previous striation lines from grit 320
disappear leaving only 400 grit lines. Polishing continued in this procedure
for all grades of grit papers.
FIGURE 5-10. An example of an unfinished 1200-grit polishing step (schematically shown
in Figure 5-9b). The horizontal lines shown in the bottom right section of
the image are abrasion marks from the previous 800-grit polishing step. The
sample was polished for a longer period of time until the abrasion marks
were uniform and parallel to the new abrasive marks covering the rest of the
sample.
!
Polishing direction
Change of grit paper
Rotate 90°
Polishing direction Polishing direction
Abrasion Lines
a). b). c).
140
5.4.3 MACHINING TITANIUM SAMPLES BETWEEN TESTING
Barreling is a result of excess friction in which the material is restricted from moving uniformly.
The sample becomes pinned at the specimen surfaces in contact with the moving platens during
testing. This creates a triaxial state of stress, which reduces the applied shear stress which creates
an observed uniaxial flow stress that is higher than that which would be observed for pure uniaxial
loading. One way to mitigate the frictional effects is to use lubricants on both surfaces of the
sample before testing. Bly [185] suggested that the triaxiality leads to a necessary correction to the
observed load for deformation to the equivalent uniaxial stress for plasticity in the absence of
friction even with a lubricant. This requires knowledge of the l/d and the frictional coefficient. A
small l/d (as in the present study) tends to lead to larger corrections. Furthermore, during testing,
the frictional coefficient may change (increase) leading to increasing triaxiality resulting from the
barreling. This prescribes a repeating sequence of unloading and reloading with new lubrication
of the specimen and platen surfaces after varying incremental strains. It should be noted that the
first four specimens (Samples 1-4) were not finely polished at the surface prior to compression
testing and were tested “as-machined”. The last three specimens (Samples, 7, 8 & 10) were finely
polished to 0.05-µm alumina. Additionally, the compression platens were polished before every
test and between testing phases, as well as a new application of lubricant.
Re-machining decreased the diameter allowing the aspect ratio to decrease, and decrease the
triaxiality. The cylindrical samples were re-machined using an ultrasonic disk cutter (described
below:
141
Ultrasonic Disk Cutter
A Model 170 Ultrasonic Disk Cutter made by E.A. Fischione Instruments Inc. was used to cut
Samples 3, 4, 7 and 8 after deformation (after the initial EDM cutting). A titanium cylindrical tube
is used to effectively cut through a sample with minimal mechanical and thermal damage. This
cuts disks of the size of the inner diameter of the drill. Two cylinders with inner diameter sizes of
2.3-mm and 3-mm were available. The 2.3-mm cylindrical drill was used to re-machine samples
3, 4, 7 and 10. The 3-mm cylindrical drill was used to re-machine sample 8. The reason for two
different diameters was to hopefully further increase the aspect ratio to its original ratio (0.54).
This was intended to improve the continuity of the stress versus strain curves upon reloading. The
ultrasonic cutter uses a high oscillating frequency (~ 26 kHz) with an abrasive 800 grit silicon
carbide (SiC) slurry.
5.4.4 POLISHING TITANIUM SAMPLES BETWEEN TESTING
Due to the small size of the titanium samples (refer to Figure 5-8), mounting them in acrylic was
the only way to obtain an expectable surface polish. In a compression test, since both sides of the
sample should deform as uniformly as possible, both sides should be polished to the same degree.
There were two different mounting techniques attempted in this study. The first, shown in Figure
5-11 (a-b), required the whole sample to be immersed in quickset acrylic with only the face of one
surface exposed. That surface would be mechanically polished, as outlined in the next section, then
removed from the mount and then remounted to have the other face of the sample exposed to be
polished. The sample would be removed from the acrylic by submerging the sample/mount in
acetone in a beaker. Placing the beaker in an ultrasonic cleaner for a few minutes, dissolves and
142
loosens the acrylic enough to carefully tweeze the sample out of the mount. This method took more
time but was more effective in producing a well-polished sample on both sides.
The second mounting method, shown in Figure 5-11b, was less successful that the first. Acrylic
was only poured just enough to have a tiny disk of mount around the sample with both sample
faces exposed. This method took less time to polish a sample but did not provide a good grip area
to perform a decent, parallel polishing job. Another flaw with this method is that it can only be
performed on un-deformed samples with a thickness of about 1.6 mm. Deformed samples are too
thin to be polished using the second method.
143
FIGURE 5-11 (a-b): Two mounting techniques used for mechanical polishing the titanium disk
samples. (a) only one surface of the titanium sample is exposed and polished
at a time; (b) both sides of the titanium sample are exposed and can be
polished at the same time.
a).
b).
(Not to Scale)
Titanium Sample
Titanium Sample
Acrylic Mount
Acrylic Mount
144
5.5 MECHANICAL TESTING
The Instron
®
5585H Series Floor Model Testing System was used with a +/- 250 kN load cell.
Bluehill
®
Software was used for collecting the raw data and controlling the machine load frame
positioning.
Compression testing utilized an Instron
5585H Series Floor Model Testing System with a +/- 250
kN load cell. Bluehill
®
Software was used for collecting the data. An OEM
®
Tools 25025 26-blade
Master Feeler Gauge was used to measure the space between platens to test parallelism of the
compression platens. A machine compliance test was done before each test to ensure that the
specimen strain was accurately assessed.
Seven samples were tested (i.e. 1, 2, 3, 4, 7, 8 and 10). Three different types of lubrication were
used during compression testing to eliminate as much friction as possible: (1) B’laster
®
Graphite
Dry Lubrication, (2) CRC
®
Industrial Inc. dry Moly Lube – Molybdenum Sulfide (MoS
2
) and (3)
PTFE Teflon paper (thickness = 0.0762 mm) from McMaster-Carr
®
Supply Company. The
frictional coefficients for each lubrication, and which lubrication was used on each sample, are
listed in Table 5.2 (a-b).
5.5.1 PARALLELISM TEST WITH A FEELER GAUGE
An OEM
®
Tools 25025 26-blade Master Feeler Gauge was used to test parallelism of the
compression platens by measuring the space between them. The tempered stainless steel blades
ranged in thicknesses from 0.0015” (0.038 mm) to 0.025” (0.635 mm). The fixtures were lowered
close to one another using the built-in machine controls. Starting with the 0.006” (0.1524 mm)
blade, the feeler gauge was inserted between the platens and slowly moved around the
145
circumference. The upper compression fixture of CF1 was adjusted using three equally spaced
screws (120° apart from one another) around the top of the upper platen. If the feeler gauge blade
cannot pass between the platens at any point, then the screws were adjusted accordingly to ensure
a level surface. When the blade fits uniformly between the platens, around the whole
circumference, then the fixtures are lowered closer to each other and a smaller/thinner feeler gauge
blade is used. The same leveling process is used. Eventually, a minimal separation of 0.0015” was
able to be attained.
In should be noted that, the Instron testing machine was shared between research groups on the
University of Southern California campus. Therefore, a parallelism test was done before each test
in which the fixtures had to be re-attached to the machine.
5.5.2 MACHINE COMPLIANCE TESTING
There were two (2) different compression fixtures used in the study. The first is much bigger than
the second and requires adjustment using screws. Three screws, spaced apart equally, were used
to lower and/or raise the platen in different areas around the platen to ensure levelness. The second
fixture (displayed in Figure 5-13) required no adjustments to the levelness because it had an
internal spring system that allowed it to stay level. A machine compliance test was done as a
precaution to confirm the plates are touching each other uniformly.
As aforementioned, because the machine is being shared between multiple research groups, a
machine compliance test was done before each compression experiment to ensure the
machine/fixtures are performing correctly.
146
This is done by performing a “blank” run compressing the platens together with no sample. This
helps to double-check that the compression platens are level, parallel and make contact with each
other at all surface points. Due to the small size of our samples, any tilt or angle between the plates
can cause the sample to become slanted after testing; therefore, altering its through thickness
orientation and altering the stress strain curve.
The test was programmed to stop at a load of 1000 N. Referring to Figure 5-12, twelve (12) curves
are plotted and each one represents a machine compliance test. For the eleven (11) curves grouped
together on the left hand side of the graph we can see that the plates touch each other quicker and
at lower extensions than the 12
th
curve farthest to the right. This could be due to an unsatisfactory
parallelism. Once the curves have reached a constant slope then it can be assumed that both of the
platens are fully in contact with each other. The better prepared the plates to be as level as possible,
the easier and quicker it is for the load to reach 1000 N. The compliance test may differ depending
on the fixture platens used. The crosshead speed varies for every compliance test performed and
is consistent with the crosshead speed of the compression test that will be subsequently performed.
5.5.3 BARRELING CORRECTION
Corrections to the Load Area versus strain due to friction (leading to barreling) utilized
Schroeder and Webster’s equation, previously mentioned as the equation used by Bly [185]:
𝜎 =
è
é
¸
êë
ì
¸
º¨o
êë
ì
)
êë
ì
) ,
, (E-4.2)
147
where s
m
denotes the measured (uniaxial) flow stress (load/area), µ is the lubricant friction
coefficient, d is the sample diameter, l is the sample height (length), and s is the resulting corrected
flow stress. This equation assumes that the friction coefficients (µ) are constant with strain,
however, this may not be the case. In fact, specimens were unloaded and reloaded after reapplying
lubricant as the friction may increase with strain during a loading phase.
5.5.4 LUBRICATIONS
Three (3) different types of lubrication were used during compression testing to eliminate as much
friction as possible: (1) B’laster
®
Graphite Dry Lubrication, (2) CRC
®
Industrial Inc. Dry Moly
Lube – Molybdenum Sulfide (MoS
2
) and (3) PTFE Teflon paper (thickness = 0.0762 mm) from
McMaster-Carr
®
Supply Company. The frictional coefficients for each lubrication, and which
lubrication was used on each sample, are given in Table 5.3 (a-b).
148
FIGURE 5-12. Compliance curves done to test the parallelism of the compression platens.
0
200
400
600
800
1000
1200
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Load (N)
Extension (mm)
Machine Compliance Curves
Loading Rate = 1 X 10
-4
s
-1
Elastic Modulus = 10 GPa
149
TABLE 5.3a: Frictional Coefficients for the Different Lubrications. The Graphite and MoS
2
aerosol lubricants were sprayed on the compression platens generously before
testing. The PTFE Teflon was taped on the compression platens before testing.
Lubrication Name Type of Lubricant Friction Coefficient (µ)
Graphite Dry Aerosol 0.12
Molybdenum Sulfide (MoS
2
) Dry Aerosol 0.23
PTFE Teflon 0.003’’ Paper 0.05-0.08
TABLE 5.3b: Lubrication(s) used for each sample. In the case of the MoS
2
and Teflon
combination, the Teflon was taped to the upper and lower compression platens first,
then the MoS
2
was generously applied on top of the paper.
Sample Lubricant Used
Sample 1 Graphite
Sample 2 MoS
2
Sample 3 MoS
2
Sample 4 MoS
2
Sample 7 MoS
2
and Teflon
Sample 8 MoS
2
and Teflon
Sample 10 MoS
2
and Teflon
150
FIGURE 5-13. Smaller compression fixture (CF2) that did not require
compliance/parallelism testing.
151
5.6 RESULTS
The true stress versus true strain behaviors are illustrated below in Figure 5-16(a-g) for all samples.
The term “phase” in the accompanying tables refers to the (re)loading stages. A best fit line is used
to describe the stress versus strain behavior after barreling (from friction) corrections. Note that
re-machining was performed to keep the t/d ratios as low as possible without cutting the specimen
to impractical testing diameters. Samples 1, 2, 3 and 4 surfaces were not polished before initial
testing nor reloading. New lubrication was applied before each compression phase. Only samples
7, 8 and 10 were re-polished prior to each reload to minimize frictional effects. As will be discussed
later, polished specimens did not necessarily produce better stress versus strain curves than
specimens with unpolished surfaces. But the platens in all tests were polished before testing.
It should be mentioned that texture was not considered to affect the mechanical properties.
Previous work [178, 179, 190-192] on thicker sheets have been done to confirm that when working
in a preferred loading axis, the crystal will develop a preferred grain texture. But the stress versus
strain behavior is not affected.
The coefficient of friction of MoSi
2
is listed in some sources as just 0.03 [187] but the
manufacturer of the spray utilized in this work listed the frictional coefficient as 0.23 [188]. Both
values were considered in the Sample 2 plot.
152
FIGURE 5-14a. Sample 1 – originally cut by EDM and used dry graphite powder for
lubrication. This sample was re-machined using a conventional lathe
method after phases 1, 2 and 3 (indicated by the * next to the individual
phase t/d ratios). Lubrication was re-applied between each phase. The
average 𝜀 =3.78 × 10
) :
𝑠
) ,
.
Sample 1
Phase #
t
o
(mm)
t
f
(mm)
d
o
(mm)
d
f
(mm)
t
o
/d
o
t
f
/d
f
volume
(mm
3
)
crosshead
speed
(mm/min)
strain-rate
(s
-1
)
1* 1.61 1.43 2.98 3.16 0.54 0.45 11.23 0.003 3.11 x 10
-5
2* 1.53 1.36 2.86 3.03 0.53 0.45 9.83 0.003 3.27 x 10
-5
3* 1.43 1.16 2.87 3.15 0.50 0.37 9.25 0.003 3.50 x 10
-5
4 1.19 1.06 1.72 1.80 0.69 0.59 2.76 0.003 4.20 x 10
-5
5 1.04 0.94 1.78 1.93 0.59 0.48 2.76 0.003 4.81 x 10
-5
0
200
400
600
800
1000
1200
0 0.1 0.2 0.3 0.4 0.5 0.6
True Stress (MPa)
True Plastic Strain
Sample 1
yield stress = 395 MPa
Initially cut with EDM; Dry graphite powder used for lubrication
(*indicates machining done after that given compression phase)
Best Fit Line
Corrected Flow Stress (u = 0.123)
True Stress (MPa) - no correction
t/d = 0.540*
t/d = 0.535*
t/d = 0.498*
t/d = 0.692
t/d = 0.586
(µ
153
FIGURE 5-14b. Sample 2 – originally cut by EDM and MoS
2
aerosol was used for
lubrication. This sample was not re-machined after any phases. Lubrication
was applied between each phase. The average 𝜀 =3.53 × 10
) :
𝑠
) ,
.
Sample 2
Phase #
t
o
(mm)
t
f
(mm)
d
o
(mm)
d
f
(mm)
t
o
/d
o
t
f
/d
f
volume
(mm
3
)
crosshead
speed
(mm/min)
strain-rate
(s
-1
)
1 1.6256 1.4986 2.9972 3.1242 0.5424 0.4797 11.4689 0.003 3.08 x 10
-5
2 1.4986 1.4097 3.1242 3.2258 0.4797 0.4370 11.4689 0.003 3.27 x 10
-5
3 1.4097 1.3335 3.2258 3.3274 0.4370 0.4008 11.4689 0.003 3.55 x 10
-5
4 1.3335 1.2700 3.3274 3.4160 0.4008 0.3718 11.4689 0.003 3.75 x 10
-5
5 1.2700 1.1557 3.4160 3.5687 0.3718 0.3238 11.4689 0.003 3.94 x 10
-5
0
100
200
300
400
500
600
700
800
900
1000
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
True Stress (MPa)
True Plastic Strain (mm/mm)
Sample 2
yield stress 380 MPa
Initially cut with EDM; No re-maching done to this sample
MoS
2
aerosol used for lubrication
Best Fit Line (0.03)
Corrected Flow Stress (u = 0.23)
True Stress (MPa) - no correction
Corrected Flow Stress (u = 0.03)
Best Fit Line (0.23)
t/d = 0.542
t/d = 0.466
t/d = 0.438
t/d = 0.401
t/d = 0.372
(µ
(µ
154
FIGURE 5-14c. Sample 3 – originally cut by EDM and MoS
2
aerosol was used for
lubrication. This sample was ultrasonically re-machined after phases 5 and
6 (indicated by the * next to the individual phase t/d ratios). Lubrication was
re-applied between each phase. The average 𝜀 =5.46 × 10
) :
𝑠
) ,
.
Sample 3
Phase #
t
o
(mm)
t
f
(mm)
d
o
(mm)
d
f
(mm)
t
o
/d
o
t
f
/d
f
volume
(mm
3
)
crosshead
speed
(mm/min)
strain-rate
(s
-1
)
1 1.6002 1.4860 2.9845 3.1370 0.5362 0.4737 11.1942 0.003 3.13 x 10
-5
2 1.4860 1.3716 3.1370 3.2572 0.4737 0.4211 11.1942 0.003 3.37 x 10
-5
3 1.3716 1.2827 3.2572 3.3655 0.4211 0.3811 11.1942 0.003 3.65 x 10
-5
4 1.2827 1.2065 3.3655 3.4925 0.3811 0.3455 11.1942 0.003 3.90 x 10
-5
5* 1.2065 1.1050 3.4925 3.6070 0.3455 0.3063 11.1942 0.003 4.14 x 10
-5
6* 1.0600 0.7800 2.3000 2.6600 0.4608 0.2932 4.4043 0.0064 1.00 x 10
-4
7 0.7900 0.6100 2.6600 3.1000 0.2969 0.1968 4.4043 0.0047 1.00 x 10
-4
0
200
400
600
800
1000
1200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
True Stress (MPa)
True Plastic Strain
Sample 3
yield stress 401 MPa
Initially cut with EDM
MoS
2
aerosol used for lubrication
(* indicates re-machining done after that given compression phase)
Best Fit Line (0.03)
Corrected Flow Stress (u = 0.03)
True Stress (MPa) - no correction
Corrected Flow Stress (u = 0.23)
Best Fit Line (0.23)
t/d = 0.536
t/d = 0.480
t/d = 0.425
t/d = 0.385
t/d = 0.351*
(µ
(µ
t/d = 0.461*
t/d = 0.297
155
FIGURE 5-14d. Sample 4 – originally cut by EDM and MoS
2
aerosol was used for
lubrication. This sample was ultrasonically re-machined after phases 5 and
6 (indicated by the * next to the individual phase t/d ratios). Lubrication was
re-applied between each phase. The average 𝜀 =5.77 × 10
) :
𝑠
) ,
. The
drop in stress values during the last two compression phases were not able
to be explained.
Sample 4
Phase #
t
o
(mm)
t
f
(mm)
d
o
(mm)
d
f
(mm)
t
o
/d
o
t
f
/d
f
volume
(mm
3
)
crosshead
speed
(mm/min)
strain-rate
(s
-1
)
1 1.6256 1.3716 2.9972 3.2893 0.5424 0.4169 11.4689 0.003 3.08 x 10
-5
2 1.3716 1.1684 3.2893 3.5687 0.4169 0.3274 11.4689 0.003 3.65 x 10
-5
3 1.1684 1.0668 3.5687 3.7084 0.3274 0.2877 11.4689 0.003 4.28 x 10
-5
4 1.0668 0.9652 3.7084 3.9497 0.2877 0.2444 11.4689 0.003 4.69 x 10
-5
5 0.9652 0.9525 3.9497 3.9525 0.2444 0.2410 11.4689 0.003 5.18 x 10
-5
6* 0.9525 0.8763 3.9525 4.1656 0.2410 0.2104 11.4689 0.003 5.25 x 10
-5
7* 0.8700 0.7500 2.3500 2.5750 0.3702 0.2913 3.7735 0.0052 1.00 x 10
-4
8 0.7500 0.4500 2.5750 3.1800 0.2913 0.1415 3.7735 0.0045 1.00 x 10
-4
0
200
400
600
800
1000
1200
0 0.2 0.4 0.6 0.8 1 1.2
True Stress (MPa)
True Plastic Strain
Sample 4
yield stress 345 MPa
Initially cut with EDM; MoS
2
aerosol used for lubrication
(* indiciates re-machining after that given compression phase)
Best Fit Line (0.03)
True Stress (MPa) - no correction
Corrected Flow Stress (u = 0.03)
Corrected Flow Stress (u = 0.23)
Best Fit Line (0.23)
Best Fit Line (0.03)
Best Fit Line (0.23)
t/d = 0.542
t/d = 0.420
t/d = 0.331
t/d = 0.288
t/d = 0.248*
(µ
(µ
t/d = 0.368*
t/d = 0.294
(e < 0.60)
(0 < e < 1.15)
(e < 0.60)
(0 < e < 1.15)
156
The results of Sample 4 indicate an anomalous drop in flow stress on reapplication of loading at a
strain of about 0.6. The source of this anomaly is unknown. However, the large “drop” in this
specimen and the observed (smaller) drops and other anomalous behaviors in other tests may
reflect the complications associated with very low aspect ratio compression tests. One of the
authors of this study performed earlier compression tests on CP-Al [193] and maintained a 1.5
aspect ratio and found none of the complications found in this work using nearly identical testing
procedures.
All of the tested samples are graphically summarized in Figure 5-17. The curves shown represent
the best fit line for each sample. The dashed pattern of each curve denotes the frictional coefficient
belonging to the lubrication used for that sample, and the average strain-rate for each sample is
also noted. Because both frictional coefficients are being considered for MoS
2
, the red and green
curves represent the average stress-strain curve over all samples using 𝜇 =0.03,0.05,0.123 and
𝜇 =0.05,0.123,0.23, respecitively.
157
FIGURE 5-14e. Sample 7 – originally cut by EDM. MoS
2
aerosol and Teflon paper were
used for lubrication. Both faces of the sample were polished to a 0.05 µm
finish and re-polished between each phase. This sample was re-machined
ultrasonically after phase 5 (indicated by the * next to the individual phase
t/d ratio). Lubrication was re-applied between each phase. The average 𝜀 =
1.11 × 10
) <
𝑠
) ,
.
Sample 7
Phase #
t
o
(mm)
t
f
(mm)
d
o
(mm)
d
f
(mm)
t
o
/d
o
t
f
/d
f
volume
(mm
3
)
crosshead
speed
(mm/min)
strain-rate
(s
-1
)
1 1.6000 1.5748 2.9720 3.0353 0.5384 0.5188 11.0993 0.0096 1.00 x 10
-4
2 1.5748 1.3970 3.30353 3.2893 0.5188 0.4247 11.0993 0.0096 1.02 x 10
-4
3 1.3970 1.0922 3.2893 3.7211 0.4247 0.2935 11.0993 0.0096 1.15 x 10
-4
4* 1.0922 0.9652 3.7211 4.0386 0.2935 0.2390 11.0993 0.0096 1.46 x 10
-4
5* 0.8800 0.6300 2.2800 2.7800 0.3860 0.2266 3.5929 0.0053 1.00 x 10
-4
6 0.6400 0.5600 2.7100 2.9500 0.2362 0.1898 3.5929 0.0038 1.00 x 10
-4
0
200
400
600
800
1000
1200
1400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
True Stress (MPa)
True Plastic Strain
Sample 7
yield stress = 335 MPa
Initially cut with EDM
MoS
2
aerosol and Teflon paper used for lubrication
(* indicates re-machining done after that given compression test)
Best Fit Line
Corrected Flow Stress (u = 0.05)
True Stress (MPa) - no correction
t/d = 0.538
t/d = 0.519
t/d = 0.425
t/d = 0.294*
(µ
t/d = 0.386*
t/d = 0.236
158
FIGURE 5-14f. Sample 8 – originally cut by EDM and MoS
2
aerosol and Teflon paper was
used for lubrication. Both faces of the sample were polished to a 0.05 µm
finish and re-polished between each phase. This sample was re-machined
ultrasonically after phase 9 (indicated by the * next to the individual phase
t/d ratio). Lubrication was re-applied between each phase. The average 𝜀 =
2.17 × 10
) <
𝑠
) ,
. The dashed line represents an ‘approximation’ line as to
the behavior at higher strains to discount for the extreme barreling.
Sample 8
Phase #
t
o
(mm)
t
f
(mm)
d
o
(mm)
d
f
(mm)
t
o
/d
o
t
f
/d
f
volume
(mm
3
)
crosshead
speed
(mm/min)
strain-rate
(s
-1
)
1 1.5799 1.4935 2.9921 3.1242 0.5280 0.4780 11.1086 0.0096 1.01 x 10
-4
2 1.4935 1.1406 3.1242 3.6169 0.4780 0.3153 11.1086 0.0096 1.07 x 10
-4
3 1.1406 0.9449 3.6169 4.0386 0.3153 0.2339 11.1086 0.0096 1.15 x 10
-4
4 0.9449 0.8636 4.0386 4.2469 0.2339 0.2033 11.1086 0.0096 1.69 x 10
-4
5 0.8636 0.8001 4.2469 4.5187 0.2033 0.1770 11.1086 0.0096 1.85 x 10
-4
6 0.8001 0.6985 4.5187 4.8006 0.1770 0.1455 11.1086 0.0096 1.99 x 10
-4
7 0.6985 0.6375 4.8006 5.2070 0.1455 0.1224 11.1086 0.0096 2.29 x 10
-4
8 0.6375 0.5969 5.2070 5.5118 0.1224 0.1083 11.1086 0.0096 2.51 x 10
-4
9* 0.5969 0.5055 5.5118 5.7531 0.1083 0.0878 11.1086 0.0096 2.68 x 10
-4
10 0.2921 0.2870 2.9464 3.4544 0.0991 0.0831 1.9920 0.0096 5.48 x 10
-4
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 0.2 0.4 0.6 0.8 1 1.2
True Stress (MPa)
True Plastic Strain
Sample 8
yield stress = 351 MPa
Initially cut with EDM
MoS
2
aerosol and Teflon paper used for lubrication
(* indicates re-machining after that given compression test)
Best Fit Line
Corrected Flow Stress (u = 0.05)
True Stress (MPa) - no correction
t/d = 0.528
t/d = 0.478
t/d = 0.315
t/d = 0.234
t/d = 0.203
t/d = 0.177 t/d = 0.146
t/d = 0.122
t/d = 0.108*
t/d = 0.099
(µ
159
FIGURE 5-14g. Sample 10 – originally cut by EDM and MoS
2
aerosol and Teflon paper was
used for lubrication. Both faces of the sample were polished to a 0.05 µm
finish and re-polished between each phase. This sample was re-machined
ultrasonically after phase 5 (indicated by the * next to the individual phase
t/d ratio). Lubrication was re-applied between each phase. The average 𝜀 =
1.00 × 10
) <
𝑠
) ,
. The re-polishing of the specimen surface may not have
imported the quality of the data over non-surface polished specimens
Sample 10
Phase #
t
o
(mm)
t
f
(mm)
d
o
(mm)
d
f
(mm)
t
o
/d
o
t
f
/d
f
volume
(mm
3
)
crosshead
speed
(mm/min)
strain-rate
(s
-1
)
1* 1.4935 0.9398 3.0226 3.8862 0.4940 0.2418 10.7163 0.00896 1.00 x 10
-4
2 0.8800 0.6807 1.7100 1.9812 0.5146 0.3436 2.3630 0.00528 1.00 x 10
-4
0
200
400
600
800
1000
1200
1400
0 0.1 0.2 0.3 0.4 0.5 0.6
True Stress (MPa)
True Plastic Strain
Sample 10
yield stress = 469 MPa
MoS
2
aerosol and Teflon paper used for lubrication
(* indicates re-machining done after that given compression phase)
Best Fit Line
Corrected Flow Stress (u = 0.05)
True Stress (MPa) - no correction
t/d = 0.494*
t/d = 0.476
(µ
160
The complication of Figure 5-15 is that the data of the various studies were obtained at different
strain rates. Ideally, comparisons are made at a fixed strain-rate. Thus, the data of Figure 5-15 was
normalized to a strain-rate equal to 10
-1
s
-1
(the maximum of any study) using a conversion
procedure based on an average strain-rate sensitivity exponent (m) (E-4.1). Several studies [194-
199] measured the room temperature strain-rate sensitivity of Ti-alloys and determined m-values
ranging from 0.007 – 0.04 for CP-Ti. The average of 0.024 is somewhat lower the average m-value
reported in the Nemat-Nasser et al. study [182] of Figure 5-4 which suggested a value of about
0.035. The higher value may reflect a lack of a fixed structure at a fixed strain, thus artificially
increasing m in this case. Figure 5-19 is plotted using a m-value of 0.024 and all stress versus strain
curves of the various studies are normalized to a strain rate of 0.1 s
-1
.
It can be observed that the behavior in the current study appears similar to earlier work [184] for
the cases of similar purity, and annealed Ti. The exception to the similarity appears that our thin
sheet has somewhat higher flow stresses after the strain-rate correction. The origin of the relative
increase in normalized stress is unclear.
This might be at least partially due to adiabatic heating for the tests greater than 10
-3
s
-1
. One
advantage of the current study is that these effects are expected to be absent because only slow
strain-rates were achieved. The other compression studies referenced, at least at lower purities
where stresses are higher, have stress versus strain behaviors that may be influenced by adiabatic
heating. Heat would decrease the flow stress of the higher strain-rate test thus only appearing less
strong than our thin sheet. The Long et al. [181] curve shifts downward with respect to our thin
sheet with strain-rate normalization.
161
FIGURE 5-15. The stress versus strain behavior of the tests of this study is averaged into
two curves that assume one of the two possible frictional coefficients for
MoS
2
. The average strain-rates for each sample are also indicated with 𝜀 =
9.27 × 10
) :
𝑠
) ,
as the average for this study. Note: Sample 4 (labeled as 4
– 5.77 × 10
) :
𝑠
) ,
) is the curve that could not be explained.
.
0
200
400
600
800
1000
1200
0 0.2 0.4 0.6 0.8 1 1.2
True Stress (MPa)
True Plastic Strain
Current Study
99.76% CP-II Titanium
Average e = 9.27 x 10
-5
s
-1
initial sample thicknesses = 1.6 mm
average yield stress = 382 MPa
Average Stress (with u = 0.03, 0.05, 0.123)
Best Fit Line (u = 0.123)
Best Fit Line (u = 0.03)
Best Fit Line (u = 0.05)
Best Fit Line (u = 0.23)
Average Stress (with u = 0.05, 0.123, 0.23)
(µ
(µ
µ
µ
(µ
µ
4 - (5.77 x 10
-5
s
-1
)
4 - (5.77 x 10
-5
s
-1
)
1 - (3.78 x 10
-5
s
-1
)
7 - (1.11 x 10
-4
s
-1
)
3 - (5.46 x 10
-5
s
-1
)
8 - (2.17 x 10
-4
s
-1
)
3 - (5.46 x 10
-5
s
-1
)
2 - (3.53 x 10
-5
s
-1
)
10 - (1.00 x 10
-4
s
-1
)
2 - (3.53 x 10
-5
s
-1
)
162
FIGURE 5-16. The two “average” curves of Figure 5-15 are compared with the literature
values reported earlier in Figure 5-6. Please note that the Long et al. [181]
study reported the elastic region of their curve. It is unclear if this region
represents the compliance or the actual elastic regime of their sample;
however, the curve is displayed here as it was published.
0
200
400
600
800
1000
1200
0 0.2 0.4 0.6 0.8 1 1.2 1.4
True Stress (MPa)
Engineering Stress (MPa) for A.V. Podolskiy et al.
True Plastic Strain
Engineering Strain for A.V. Podolskiy et al.
Commercially Pure and High-Purity Titanium
Temperature = 273K - 300K
99.60% CP-Ti (F.W. Long et. al.), t = not given, t/d = 1.50
99.52% CP-(II) Ti (A.V. Podolskiy et al.), t = 35 mm, t/d = 3.50
99.49% CP-Ti [NR] (M. Battaini et. al.), t = 8 mm, t:w:d = 0.75
99.49% CP-Ti [NT] (M. Battaini et. al.), t = 8 mm, t:w:d = 0.75
99.998% High-Purity Ti (A.A. Salem et. al.), dimensions not given
99.99% High-Purity Ti (S. Nemat-Nasser et al.), t = 4.8 mm, t/d = 1.00
99.76% Ti (Current Study); t = 1.6 mm, t/d = 0.52
99.76% Ti (Current Study); t = 1.6 mm, t/d = 0.52
163
FIGURE 5-17. The stress versus strain curves of the present study and earlier studies all
normalized to a strain rate of 10
-1
s
-1
[182] through the average strain-rate
sensitivity exponent m = 0.024 [190, 191, 193-196].
0
200
400
600
800
1000
1200
1400
0 0.2 0.4 0.6 0.8 1 1.2 1.4
True Stress (MPa)
Engineering Stress (MPa) for A.V. Podolskiy et al.
True Plastic Strain
Engineering Strain for A.V. Podolskiy et al.
Commercially-Pure and High-Purity Titanium
Normalized to Strain-Rate = 0.1 s
-1
(Combined Literature)
99.60% CP-Ti (F.W. Long et. al.), t = not given, t/d = 1.50
99.52% CP-(II) Ti (A.V. Podolskiy et al.), t = 35 mm, t/d = 3.50
99.49% CP-Ti [NR] (M. Battaini et. al.), t = 8 mm, t:w:d = 0.75
99.49% CP-Ti [NT] (M. Battaini et. al.), t = 8 mm, t:w:d = 0.75
99.998% High-Purity Ti (A.A. Salem et. al.), dimensions not given
99.99% High-Purity Ti (S. Nemat-Nasser et al.), t = 4.8 mm, t/d = 1.00
99.76% Ti (Current Study); t = 1.6 mm, t/d = 0.52
99.76% Ti (Current Study); t = 1.6 mm, t/d = 0.52
164
We note from Table 5.4 that the yield stresses of the Long et al. study is much higher (530 MPa)
then our study (380 MPa average). This might be partly expected based on the reported m-values
and the realization that adiabatic heating effect would be minimal at the yield stress. However, we
also note from Table 5.4 that at strains 0.1 to 0.4, the stress differences are much less substantial.
This loss of apparent strength may actually be due to adiabatic heating. In the absence of this
heating, the flow curves might actually be less disparate. However, work by Horiuchi et al. [200]
in specimens with a low length to diameter ratio (i.e. <1.5) adiabatic heating even at a high rate
was not substantial in aluminum (high thermal conductivity). Aluminum has higher thermal
conductivity than Ti, and the Horiuchi study had large thermal “sinks” as grips, which may be
absent in the Ti compression specimens where lubrication (thermal barriers) may be present.
Therefore, in the absence of a very detailed thermal analysis, the effects of adiabatic heating on
the flow stress in the Ti compression tests referenced are unclear.
Also, the fact that the sheet of the current study is the thinnest may suggest greater strain in our
case through rolling. It appears that the anneal, as a consequence, resulted in a smaller grain size
at 14µm (compared with 35µm of the Long et al. study) that leads to a stress increase. The Hall-
Petch relationship for Ti [183] suggests the constant (k
y
) to be 0.671MN/m
3/2
. A stress increase
with this level of grain size refinement is expected to be about 67 MPa. This could explain some
of the observed normalized differences (33-67%) in strength of the thin sheet versus the thicker
plate, such as the Long et al. [181] study.
165
TABLE 5.4. Tabulated strain-rate versus yield stress and flow stress values comparing the
present study with two literature sources previously discussed [179, 181].
Study
Strain-Rate
(𝜺, s
-1
)
Yield Stress
(𝝈
𝒚
𝟎.𝟐%
, MPa)
Flow Stress (𝝈
𝒇
, MPa)
10% strain 20% strain 40% strain
Present Study 9.27 x 10
-5
𝜇
,
,𝜇
-
𝜇
,
,𝜇
-
𝜇
,
,𝜇
-
𝜇
,
,𝜇
-
373, 388 660, 700 740, 780 820, 880
Battaini et al.
[179]
1.00 x 10
-1
NT, NR NT, NR
310, 350 535, 525
Long et al. [181] 1.00 x 10
-2
530 640 720 820
𝜇
,
: represents average curve for µ = 0.05, 0.123, 0.23
𝜇
-
: represents average curve for µ = 0.03, 0.05, 0.123
166
5.7 SUMMARY
This study assessed the through-thickness compressive stress versus strain behavior of
commercially pure (Grade II) Ti thin-sheet to relatively large true strains of about 1.0. The data is
unique and is valuable for a variety of applications including Ti sheet metal forming operations.
The relatively high strength of the thin sheet revealed through strain-rate normalization may be
due to softening by adiabatic heating of higher strain rate tested Ti with which the Ti of this study
is compared and/or the refined grain size of the thin titanium sheet of this study resulting from the
annealing treatment.
167
CHAPTER 6. CONCLUSIONS
1. The logarithmic form generally appears to more reliably describe the low-temperature creep
behavior of most of the metals and alloys analyzed (Cd, ferritic steel, stainless steel). Most of
the Ti-alloys are better described by power-law equations and Cu and Al may be approximately
equally well described by logarithmic and power law equations. The equation forms may allow
the extrapolation of reliable creep behavior to longer times. Nonetheless, the power-law and
logarithmic behavioral trends are simple curve fits and not reflections of the fundamental
physics.
2. In order to observe Harper-Dorn Creep in aluminum, we postulate that the dislocation density
must remain constant. If a material starts with a high dislocation density (𝜌), and loaded in the
H-D regime, then the density will remain constant and the strain-rate will be linearly
proportional to the stress (𝜎)with a stress exponent (n) equal to 1. If a material begins with a
very low dislocation density(𝜌), and a low stress is applied in the H-D regime, then the density
will increase with strain-rate (𝜀) and stress (𝜎). H-D creep will not occur and a stress exponent
(n) closer to 3 or 4 is observed.
3. For the literature review of Bulk Metallic Glasses (amorphous alloys), the mechanism of creep
appears to largely be explained by homogeneous deformation caused through shear
transformation zones. For low strain-rates (𝜀) and low stresses (𝜎), a Newtonian flow (n = 1)
is suggested; however, with increasing stresses and strain-rates, an increasing stress exponent
(n > 1) is observed. The non-Newtonian behavior, postulated to result from nano-
crystallization, is unclear. During primary creep, STZs create free volume, leading to softening.
168
Furthermore, recovery processes or annihilation of free volume leads to hardening.
4. Lastly, the relatively high strength of the CP-Grade-II thin sheet Titanium, may be due to
softening by adiabatic heating of higher strain-rate tested titanium. The titanium of this study
is compared to those in literature (through strain-rate normalization) leading to a conclusion
that implies the refined grain size of the thin titanium sheet of this study resulting from the
annealing treatment is the cause of this heightened strength.
169
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APPENDIX. TEM PREPARATION OF COPPER CREEP SPECIMENS
To analyze the grain microstructure after creep, small disks from the gage section of some creep
samples were prepared for electron microscopy analysis. Disks from as-annealed material was also
analyzed for comparison. The silver creep samples were not analyzed by TEM.
A.1 ELECTRO-POLISHING TO SMALLER DIAMETER TEM DISKS
Electro-polishing was used to reduce the gage section diameter of certain creep samples. This
method was used because it was the least invasive method to reduce the 6.35-mm diameter to ~3-
mm TEM disks. The creep experiments were performed at low loads and the specimens were still
relatively soft. Therefore, to ensure an absence of damage, electro-polishing was considered the
safer route.
A 6-inch diameter and 0.012-inch thick diamond saw blade was used to cut off the threaded end
pieces of each sample in order to isolate the gage sections. The slowest speed of 100 rpm was used
with no added weight to pressure the sample on the blade. An electrical cord was stripped and the
inner wire was wrapped around the upper circumference of the gage section. Electrical tape was
then wrapped around the top to keep the wire in place as well as to control the area to be
electropolished, leaving approximately 23 mm of the gage length exposed. A ramping curve was
first done in order to determine the optimal polishing voltage. Shown in Figure A-1, the optimal
voltage was within the 2.4V – 3V range.
A diluted phosphoric acid solution was used to electropolish the samples from an initial diameter
of 6.35 mm to an approximate diameter of ≤ 3 mm (this diameter is chosen to conform to most
TEM holder size specifications).
185
FIGURE A-1. The current-voltage ramp curve was performed from 0V – 4 V at a ramp rate of 0.1
V/s. The optimal voltage range for electro-polishing the copper in the research is
2.4 V – 3 V.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Current Density (mA/cm
2
)
Voltage (V)
Current(I) - Voltage(V) Ramping Curve
99.999% Polycrystalline Copper
Passivation
Etching
Polishing
Pitting
186
The 85w/w% phosphoric acid was diluted with deionized water to a 70w/w% solution. The plating
electrode (cathode) had dimensions of 15.89-mm diameter and 6.38-mm height. The exposed
surface was ground with 180 SiC grit paper to create a rougher surface area. The electro-polishing
setup is shown in Figure A-2(a-b). Figure A-2(a) shows the beginning of the process, that
ultimately took 36 hours. The galvanic cell was positioned on a stir plate that kept the diluted
phosphoric acid solution vigorously stirring for the duration of the polishing process. Figure A-
2(b) displays the resulting gage section on the left and the plated counter electrode on the right.
The polished gage section shows an area that is “raised” and not atomically smooth as the rest of
the sample surface. This was due to glue from the strain gage having been left on the surface and
therefore blocked that material from being polished.
An image of 5N-Cu-15MPa after electro-polishing is shown in Figure A-3. It’s clear to see that
the gage length is not uniform from top to bottom. This is simply due to prential polishing of the
side parallel and facing the opposing electrode. The material that is furthest away from the
opposing electrode experiences less current density and therefore polishes slower.
A.2 PREPARATION OF TEM DISKS
A high-precision wire saw from Princeton Scientific Corp. was used to slice the reduced gage
section into disks. The wire saw uses a 50-µm thin tungsten wire that utilizes 800 SiC grit oil-
based abrasive automatically dispensed onto the sample. The wire oscillates rapidly up and down
while the sample in moving rapidly back and forth. The machine is designed to reduce the amount
of damage exerted onto the sample while also reducing the amount of material lost.
Disks were cut to an approximate 1-mm thickness and then mechanically polished using only 800
SiC grit to a fairly thick TEM disk of 400 – 450-µm thicknesses.
187
(a).
(b).
FIGURE A-2. (a) The gage section of Sample 5N-Cu-30MPa on the left (after threaded ends are
removed) was polished in a diluted phosphoric acid solution at 2.75V to reduce the
gage diameter to ~3-mm (b). The copper polished away from the gage section was
in turn plated on the opposing electrode.
188
FIGURE A-3. 5N-Cu-15MPa after electro-polishing the gage section to a smaller diameter
suitable for TEM disk preparation. A high-precision wire saw from Princeton
Scientific Corp. was used to slice the reduced gage section into disks.
27-mm
3-mm
189
A.3 JET-POLISHING OF THIN FOILS FOR TEM ANALYSIS
A Model 110 Fischione twin jet electro-polisher was used for electro-polishing. Both the sample
disc and the nozzles are submerged in the same electrolytic solution used in Section 2.7.1 to polish
the gage sections to a small diameter. A small submersible pump, which is a part of the equipment,
pumps jets of the electrolyte through these nozzles on both sides of the sample. The perforation
was identified using a light source and a light sensor that are placed on opposite sides of the sample
[80]. An alarm would sound once the sample thinned enough for light to penetrate through it
received by the light sensor. This confirms that the sample is electron transparent and can possibly
viewed in the TEM.
Abstract (if available)
Abstract
Four projects were performed and reported in the present dissertation with concentration of materials subjected to creep deformation (or, time-dependent plasticity). ❧ The first project examines the dislocation density of high-purity single crystal aluminum at high temperatures for insight into the so-called ""Harper-Dorn"" Creep Regime. Single crystals of 99.999 and 99.9999% pure aluminum were annealed at high elevated temperatures (0.98Tm) for relatively long times of up to one year, the longest in the literature. Remarkably, the dislocation density remains relatively constant at a value of about 10⁹ m⁻² over a period of one year. The stability suggests some sort of 'frustration"" limit. This has implication towards the so-called ""Harper-Dorn Creep"" that generally occurs at fairly high temperatures close to the melting point of the material (i.e. > 0.90Tm) and at very low stresses. The observed “frustration limit” in this study is on the order of the dislocation density for steady-state flow in the one-power law (Harper-Dorn) creep regime. A constant dislocation density with changing applied stress may lead to one-power law behavior. Perhaps the more recognized, 5- or 3- power law creep, occurs when the initial dislocation density is low (e.g. << 10⁹ m⁻²). ❧ Moving from high-temperature, low-stress creep, to low-temperature, higher stress creep in high-purity copper and silver polycrystals. Many crystalline materials are known to exhibit creep at low temperatures (T < 0.3Tm). Here we review and analyze the phenomenological relationships that describe primary creep. The discussion focuses on the controversy as to whether power-law or logarithmic descriptions better describe the experimental database. We compile data from the literature as well as new copper data recently taken by the authors. Depending on the material, it appears that the logarithmic form can somewhat better describe creep behavior at low temperatures, while the power-law behavior manifests at intermediate temperatures. The basic mechanism(s) of low-temperature creep plasticity is examined, as well. ❧ The third reviews and assesses the work on creep behavior in amorphous metals (also called, bulk metallic glasses). There have been, over the past several years, a few reviews of the mechanical behavior of amorphous metals. Of these, the review on the creep properties of amorphous metals by Schuh et al, though oldest of the three, is particularly noteworthy and the reader is referred to this article published in 2007. The current review of creep of amorphous metals particularly focuses on those works since that review and places the work prior to 2007 in a different context where new developments warrant. ❧ The last project looks at unique compression experiments on commercially-pure grade-II titanium thin sheets. This research examined the through-thickness (z-direction) compressive stress versus strain behavior of 99.76% commercially-pure (Grade II) titanium sheet with relatively small grain size. The low aspect ratio of the compression specimens extracted from the sheet, led to frictional effects that can create high triaxial stresses complicating the uniaxial stress versus strain behavior analysis. Nonetheless, reasonable estimates were made of the through-thickness large strain behavior of a commercially-pure (Grade II) thin Ti sheet to relatively large true strains of about 1.0.
Linked assets
University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Smith, Kamia K.
(author)
Core Title
Mechanical behavior of materials in extreme conditions: a focus on creep plasticity
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Materials Science
Publication Date
11/28/2017
Defense Date
07/20/2017
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Aluminum,bulk metallic glasses,Copper,creep,deformation,dislocation density,long-term annealing,low-aspect ratio compression,OAI-PMH Harvest,plasticity,room temperature creep,titanium
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Kassner, Michael (
committee chair
), Goo, Edward (
committee member
), Graham, Nicholas (
committee member
), Grunenfelder, Lessa (
committee member
), Hodge, Andrea (
committee member
)
Creator Email
kamia.smith.88@gmail.com,kamiasmi@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c40-460198
Unique identifier
UC11268086
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etd-SmithKamia-5935.pdf (filename),usctheses-c40-460198 (legacy record id)
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etd-SmithKamia-5935.pdf
Dmrecord
460198
Document Type
Dissertation
Rights
Smith, Kamia K.
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
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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 a...
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Tags
bulk metallic glasses
creep
deformation
dislocation density
long-term annealing
low-aspect ratio compression
plasticity
room temperature creep
titanium