University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange
Doctoral Dissertations Graduate School
5-1997
An Experimental Investigation of Creep and Viscoelastic Properties Using Depth-Sensing Indentation Techniques
Barry Neal Lucas University of Tennessee - Knoxville
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Recommended Citation Lucas, Barry Neal, "An Experimental Investigation of Creep and Viscoelastic Properties Using Depth- Sensing Indentation Techniques. " PhD diss., University of Tennessee, 1997. https://trace.tennessee.edu/utk_graddiss/1255
This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council:
I am submitting herewith a dissertation written by Barry Neal Lucas entitled "An Experimental Investigation of Creep and Viscoelastic Properties Using Depth-Sensing Indentation Techniques." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Materials Science and Engineering.
Carl J. McHargue, Major Professor
We have read this dissertation and recommend its acceptance:
Warren C. Oliver, Ben F. Oliver, Thomas G. Carley
Accepted for the Council:
Carolyn R. Hodges
Vice Provost and Dean of the Graduate School
(Original signatures are on file with official studentecor r ds.) To the Graduate Council:
I am submitting herewith a dissertation written by Barry Neal Lucas entitled" An Experimental Investigation of Creep and Viscoelastic Properties Using Depth Sensing Indentation Techniques." I have examined the final copy of the dissertation for fonn and content and recommend thatit be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Metallurgical Engineering.
u� Carl�. J. McHargue, Chairperson
We have read this dissertation and recommend its acceptan
Warren C. Oliver, Technical Advisor :;7 ��,if�.� ve
-=- Thomas G. C�r
Accepted for the Council:
Associate Vice Chancellor and Dean of the Graduate School AN EXPERIMENTAL INVESTIGATION OF CREEP AND
VISCOELASTIC PROPERTIES USING DEPTH-SENSING
INDENTATION TECHNIQUES
A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville
Barry Neal Lucas May 1997 Dedication
This dissertation is lovingly dedicated to my wife and sons;
Becky, Noah, and CalebLucas
and to my parents;
Arviland Jane Lucas
Thank you all.
11 Acknowledgments
There are many people to whom lowe a debt of gratitude for making this work possible. First and foremost, my technical advisor throughout my graduate career, Dr.
Warren C. Oliver. It is from his creative imagination that the idea for the hardware built for this study evolved. Not only has he acted as my advisor and mentor throughout the entirety of this work, he has also been my friend. I will always consider him my greatest teacher for it is from him that I have not only learned a great deal of materials science, but also a great deal about what it takes to be a materials scientist.
I also wish to thank the other members of my committee, Dr. Carl J. McHargue,
Dr. Ben F. Oliver and Dr. Thomas G. Carley. In addition to these official members, special thanks are owed to Dr. George M. Pharr of Rice University for his valuable contributions to the indentation creep portion of this work, and to Dr. Jean-Luc Loubet of the Ecole Centrale de Lyon for the opportunity to work with him on the measurement of the viscoelastic properties of materials with indentation techniques.
There are many others who have helped in many ways during the course of this work. Dr. Ting Y. Tsui and Dr. Alexeii Bolshakov, two fellow Ph. D. students who have gone on to great things, are thanked for their assistance at various times during this work. I would also like to thank all of the people at the Oak Ridge National Laboratory with whom I have had the opportunity to work but are too numerous to name. These are the people who have quite often given very unselfishly of both their time and resources with no reason for doing so other than their willingness to help a friend. All of you are graciously thanked and you will be sorely missed.
Saving the best for last, lowe everything that I have accomplished to my family to whom this work is dedicated. My wife Rebecca has stood by me through all of the
iii long days and nights and never wavered in her love and support. She is quite truly the most remarkable woman that I know. Not only is she a wonderful wife, but also a wonderful mother to my two sons Noah and Caleb. The three of them have been a constant reminder that there are more important things in life than ones work. And finally to my parents, thank you for giving me every imaginable opportunity, which is all anyone can ever ask for. Your never ending support for me throughout every endeavor of my life will never be forgotten.
Financial support for this work from Nano Instruments, Inc. and the division of
Materials Sciences, U.S. Department of Energy, under contract DE-AC05-840R21400 with Lockheed-Martin Energy Systems, Inc. is gratefully acknowledged.
iv Abstract
Broad band, quasi-static and frequency specific dynamic techniques adapted to depth-sensing indentation testing have been utilized to measure both time dependent plasticity (creep) at both room and elevated temperatures as well as time dependent elasticity (viscoelasticity) at room temperature.
Indentation Creep
Using a variety of depth-sensing indentation techniques at both room and elevated temperatures, the dependency of the indentation hardness on the variables of indentation strain rate (stress exponent for creep, n) and temperature (apparent activation energy for creep, Q), and the existence of a steady state behavior in an indentation creep test with a Berkovich indenter were investigated. The indentation creep response of five materials, Pb-65 at% In (at RT), high purity indium (from RT to 75°C), high purity aluminum (from RT to 250°C), a vapor deposited amorphous alumina film (at RT), and single crystal alumina (sapphire) (at RT), was measured. It was shown for the first time that the indentation strain rate, def ined as it/h, could be held constant during an experiment using a Berkovich indenter by controlling the loading rate such that the loading rate divided by the load, Pip, remained constant. This technique yields the most unambiguous determination of the stress exponent for creep and seems to most closely approximate the steady state results from uniaxial testing. The results from the constant P/ p experiments were compared to the results from conventional indentation creep experiments where the load is ramped on at a high rate and then held constant for a period of time. It was shown that the transition from the loading segment to hold
v segment shows a transient period with an apparent higher stress exponent for creep which has previously been mistaken for power law breakdown type behavior. The apparent activation energy for indentation creep in indium was found to be approximately 78 kJ/mol, in excellent agreement with the activation energy for self diffusion in the material. The temperature dependence of the indentation creep process in aluminum was found to be best described by an effective diffusion coefficient as described in the literature for bulk aluminum at intermediate temperatures. By performing Pip change experiments it was shown that a steady state path independent hardness could be reached in an indentation test with a geometrically similar indenter.
The arrival at a new steady state value of hardness seems to depend on the accumulation of strain rather than a relaxation time. The measurements on amorphous alumina and sapphire demonstrate the technique's ability to measure differences in the time dependent response of materials that can not be tested with other techniques.
Viscoelasticity
Using a frequency specific dynamic indentation technique, a general method to measure the linear viscoelastic properties of polymers was determined. The polymer tested was an amorphou s unvu1canized natural rubber, poly-cis l,4-isoprene. By imposing a small harmonic force excitation on the specimen during the indentation process and measuring the displacement response at the same frequency, the complex elastic modulus, G*=G'+iG" , of the polymer was determined. The portion of the displacement signal which is "in phase" with the excitation represents the elastic response of the contact and is related to the stiffness (S) of the contact and to the elastic modulus, or storage modulus (G') , of the material. The "out of phase" portion of the displacement signal represents the energy being absorbed by the material, i.e. the
VI damping (Coo where 00=2 1t t) of the contact, and thus the loss modulus (G") of the material. It was shown that the storage, S, and loss, Coo, components of the displacement response scale as the respective component of the complex modulus multiplied by the square root of the contact area.
Vll Table of Contents
Chapter 1 Introduction ...... 1
1. Introduction...... 1
1.1 Time Dependent Plasticity - Creep ...... 2
1.1.1 Introduction ...... 2
1.1.2 Phenomenological Description ...... 3
1.1.3 Mechanisms of Creep ...... 3
1.2 Time Dependent Elasticity - Viscoelasticity ...... 5
1.2.1 Introduction ...... 5
1.2.2 Phenomenological Description ...... 5
1.2.3 Mechanisms of Viscoelastic Behavior ...... 7
Chapter 2 Review of Literature ...... 8
2. Introduction ...... 8
2.1 Time Dependent Plasticity - Creep ...... 8
2.1.1 Introduction ...... 8
2.1.2 Review of Uniaxial Literature ...... 9
2.1.3 Review of Indentation Creep Literature ...... 18
2.2 Time Dependent Elasticity - Viscoelasticity ...... 33
2.2.1 Definition and Terminology ...... 33
2.2.2 Phenomenological Description ...... 35
2.2.3 Mechanisms of Viscoelastic Relaxation ...... 42
Chapter 3 Experimental Equipment ...... 44
3. Introduction ...... 44
viii ® 3.1 Description of Nano Indenter II ...... 44
3.2 Description of the High Temperature Mechanical Properties
Microprobe ...... 46
3.2.1 Introduction ...... 46
3.2.2 Loading System ...... 48
3.2.3 Displacement Sensing System ...... 48
3.2.4 Vacuum I Manipulation System ...... 50
3.2.5 Temperature Monitoring and Control ...... 51
Chapter 4 Developing a Set of Indentation Creep Constitutive
Equations and DefiningIndentation Creep Terminology ...... 52
4. Introduction ...... 52
4.1 Differences Between Uniaxial and Indentation Creep ...... 52
4.1.1 Introduction ...... 52
4. 1.2 Steady State versus Transient Behavior ...... 53
4.1.3 Homogeneous Stress Distribution versus Radial
Stress Distribution ...... 54
4.1.4 Constant Volume versus Expanding Volume ...... 54
4.1.5 User defined Geometry versus Material Defined
Geometry ...... 55
4.2 Analogy Between Indentation andUniaxial Geometries...... 55
4.3 Definition of Hardness as Indentation Stress ...... 57
4.3 "Steady State Indentation Strain Rate and Hardness" ...... 58
Chapter 5 Indentation Creep. Materials/Sample Preparation...... 60
5.1 Lead-65 at% Indium...... 60
5.2 Indium ...... 60
ix 5.3 Aluminum ...... 61
5.4 A1203 ...... 62
Chapter 6 Indentation Creep - Experimental Techniques and Details ...... 63
6. Introduction ...... 63
6.1 Load Time Histories...... 63
6.1.1 Step LoadIHold (Step/Hold) ...... 63
6.1.2 %Step/Hold ...... 64
6.1.3 Constant Rate of Loading/Hold (CRLlHold)...... 67
6.1.4 Constant pip ...... 69 6.1.4.1 Development of a Constant "Indentation
Strain Rate" Experiment ...... 69
6. 1.4.2 Experimental ...... 70
6.1.5 pip Change ...... 72
6.2 Indenter Geometry ...... 72
6.3 Thermal Drift ...... 74
6.3.1 Nano Indenter® II ...... 74
6.3.2 HTMPM...... 76
Chapter 7 Indentation Creep - Results and Discussion ...... 77
7.1 Effects of Stress ...... 77
7.1.1 Introduction ...... 77
7.1.2 Results and Discussion ...... 77
7.1.2.1 Pb-65 at% In ...... 77
7.1.2.2 Indium ...... 79
7.1.2.3 Aluminum ...... 105
7.1 .2.4 Al203 ...... 114
x 7.2 Effects of Temperature ...... 119
7.2.1 Introduction ...... 119
7.2.2 Results and Discussion ...... 119
7.2.2.1 Indium ...... 119
7.2.2.2 Aluminum ...... 127
7.3 Steady State Path Independent Hardness...... 136
7.3.1 Introduction ...... 136
7.3.2 Pip change Tests on Indium ...... 137
7.4 Comparison to Uniaxial Data ...... 140
7.4.1 Introduction ...... 140
7.4.2 Pb-65 at% In ...... 141
7.4.3 Indium...... 141
7.4.4 Aluminum ...... 153
Chapter 8 Indentation Creep. Conclusions and Suggestion for
Future Research ...... 157
8.1 Conclusions ...... 157
8.2 Suggestions for Future Research...... 162
Chapter 9 Developing Indentation Viscoelasticity Constitutive
Equations ...... 164
9. Introduction ...... '" ...... 164
9.1 Analysis of Indentation Data ...... 164
Chapter 10 Indentation Viscoelasticity - Materials/Sample
Preparation ...... 168
Chapter 11 Indentation Viscoelasticity· Experimental Techniques ...... 169
11.1 Introduction ...... 169
Xl 11.2 ConstantContact AreaIVariable Frequency ...... 169
11.3 ConstantFre quencyNariable Contact Area ...... 172
Chapter 12 Indentation Viscoelasticity - Results and Discussion ...... 173
12.1 Introduction ...... 173
12.2 ConstantContact ArealVariable Frequency ...... 173
12.3 ConstantFrequencyNariable Contact Area ...... 180
Chapter 13 Indentation Viscoelasticity - Conclusions and Suggestions
for Future Research ...... 187
13.1 Conclusions ...... 187
13.2 Suggestions for Future Research ...... 187
Bibliography ...... 189
Appendices...... 195
Appendix A. Dynamic Modeling ...... 196 ® A. l Calibration and Characterization of Nano Indenter II...... 197
A.2 Modeling the Elastic Contact ...... 203
A.3 Calibration and Characterization of HTMPM...... 204
Appendix B. Area Verification ...... 222
B.l Indium ...... 222
B.2 Aluminum ...... 222
Vita ...... 225
xii List of Figures
Figure 1. A comparison of activation energies for creep and self diffusion for a
wide range of materials. From Sherby[35]...... 12
Figure 2. A plot of stress change data for AI from Horiuchi and Otsuka[45] replotted by Sherby et al[26]...... 17
Figure 3. A schematic of the strain response to an applied stress for a purely
elastic material ...... 36
Figure 4. A schematic of the strain response to an applied stress for a
viscoelastic material ...... 37
Figure 5. A schematic of the Nano Indenter® II ...... 45
Figure 6. A schematic of the High Temperature Mechanical Properties
Microprobe ...... 47
Figure 7. Geometric comparison of uniaxial compression specimen (a) and
indentation creep 'specimen' (b) and equations for describing each ...... 56
Figure 8. A plot of the load time history for step load experiments conducted on
Pb-65 at% In, amorphous A1203, and sapphire ...... 65 Figure 9. A plot of the load time history for the study of step loading effects on
In...... 66 Figure 10. A plot of load versus time for the constant rate of loading experiments ...... 68
Figure 11. A plot of load versus time for the experiments, conducted on In, during which pip was held constant...... 71
Figure 12. A plot of pip versus displacement for pip change experiments
conducted on In ...... 73
xiii Figure 13. A typical plot of displacement versus time during 3600 second hold
period under constant load following an 80 mNstep load for Pb-65
at% In...... 78
Figure 14. A log-log plot of indentation strain rate versus hardness for step load
experiments conducted on Pb-65at% In ...... 80
Figure 15. A semi-log plot of displacement versus time for In showing the
response of the material to a variety of step loads ...... 81
Figure 16. A plot of displacement versus time showing the oscillatory response
of the indenter duringthe 80 ms immediately following a step load of
9 mN on In...... 83
Figure 17. A plot of indentation strain rate versus displacement for CRLlHold
experiments conducted on In...... 86
Figure 18. A plot of hardness versus displacement for CRLlHold experiments
conducted on In...... 87
Figure 19. A log-log plot of the average indentation strain rate versus the
average hardness for series of constant loading ratelhold experiments
on In...... 88
Figure 20. A plot of displacement versus time during the hold period under
constant load following a 1 second ramp to 10 mN for In at temperatures ranging from -100°C to 75°C ...... 90
Figure 21. A log-log plot of indentation strain rate versus hardness for indentation creep experiments on In at temperatures of 28, 50, and 75°C ...... 91
xiv Figure 22. A plot of indentation strain rate versus displacement for three different constant Pip experiments conducted on In showing
constant indentation strain rates equal to 0.5 Pip ...... 93 Figure 23. A plot of hardness versus displacement for five different constant Pip experiments conducted on In showing indentation steady state behavior in response to constant Pip load ramp ...... 94 Figure 24. A log-log plot of average indentation strain rate versus average hardness for constant Pip experiments conducted on In ...... 95
Figure 25. A log-log plot of hardness versus time under load for In...... 97
Figure 26. A log-log plot of hardness versus time for In for experiments
conducted using 10 second constant rate of loading ramp to 10 mN
followed by a hold period under constant load ...... 100
Figure 27. A log-log plot of indentation strain rate versus hardness for
experiments conducted using 10 second loading ramp to 10 mN
followed by a hold periodunder constant load...... 10 1
Figure 28. A plot of indentation strain rate versus time for 1, 10, and 30 second
load ramps to 10 mN followed by hold period under constant load for
In...... 102
Figure 29. A log-log plot of indentation strain rate versus hardness comparing the results of the CRUH oid experimentsto the results of the constant Pip experiments...... 103 Figure 30. A plot of displacement versus time during hold period under constant
load following 1 second ramp to 10 mN for Al at temperatures
ranging from 100°C to 250°C ...... 106
xv Figure 31. A log-log plot of indentation strain rate versus hardness for
indentation creep experiments on Al conducted at 21°e ...... 107 Figure 32. A log-log plot of indentation strain rate versus hardness for
indentation creep experiments on AI conducted at 100° e ...... 108 Figure 33. A log-log plot of indentation strain rate versus hardness for
indentation creep experiments on AI conducted at l50°C...... 109 Figure 34. A log-log plot of indentation strain rate versus hardness for
indentation creep experiments on Al conducted at 200°C...... 110
Figure 35. A log-log plot of indentation strain rate versus hardness for
indentation creep experiments on Al conducted at 250°C...... 111
Figure 36. A log-log plot of indentation strain rate versus hardness for
indentation creep experiments on Al conducted at 21, 100, 150, 200,
and 250°C ...... 113
Figure 37. A plot of the displacement versus time response to 80 mN step load
for a 1.9/lmamorphou s alumina filmon sapphire ...... 115
Figure 38. A log-log plot of indentation strain rate versus hardness for 1.9 /lm
amorphous alumina filmon sapphire ...... 116
Figure 39. A plot of displacement versus time response to 80 mN step load for
sapphire ...... 117 Figure 40. A log-log plot of indentation strain rate versus hardness for sapphire ...... 118 Figure 41. A plot of hardness versus homologous temperature for In over the homologous temperature range of 0.4 to 0.8 ...... 120
Figure 42. A plot of natural log of indentation strain rate at constant values of
hardness versus the reciprocal absolute temperature for In ...... 122
xvi Figure 43. A log-log plot of temperature compensated indentation strain rate
versus hardness for 28, 50 and 75°C experiments on In ...... 123
Figure 44. A plot of the shear modulus and Young's modulus of In as a function
of temperature[66] ...... 125
Figure 45. A plot of the natural log of the indentation strain rate at constant
values of HlG and HIE versus the reciprocal absolute temperaturefor
In...... 126
Figure 46. A plot of natural log hardness at constant time under load versus the
reciprocal homologous temperaturefor In ...... 128
Figure 47. A plot of average hardness versus homologous temperature for Al
over the homologous temperature range of 0.31 to 0.56...... 129
Figure 48. A plot of Young's modulus versus temperature for A1[40] ...... 132 Figure 49. Plot of dislocation density calculated from the Taylor relation by
replacing the normalized stress dependence, alE, of the Taylor
relation with (Hl3E)...... 133
Figure 50. A log-log plot of temperature compensated indentation strain rate (indentation strain rate divided by the effective diffusion coefficient)
versus HIEfor AI...... 135 Figure 51. A plot of indentation strain rate versus displacement for Pip change
experiments on In...... 138 Figure 52. A plot of hardness versus displacement for Pip changetests on In ...... 139
Figure 53. A plot of stressversus strain from uniaxial compression experiments run on Pb-65 at% In alloy at a variety of crosshead speeds ...... 142
Figure 54. A log-log plot of stress versus strain rate a function of strain for as uniaxial compression tests on Pb-65 at% In ...... 143
xvii Figure 55. A log-log plot of indentation strain rate versus hardness and uniaxial
strain rate versus stress at 10 and 80% strain for Pb-65 at% In ...... 144
Figure 56. A log-log plot of indentation strain rate versus hardnessl3.3 and
uniaxial strain rate versus stress at 10 and 80% strain for Pb-65 at%
In...... 145
Figure 57. A log-log plot of indentation strain rate versus hardness for constant Pip experiments on Inplotted with tensile strain rate versus stress data fromWeertman [39] ...... 147
Figure 58. A log-log plot of temperature compensated indentation strain rate versus hardnessfor constant Pip experiments on In plottedwith temperature compensated tensile strain rate versus stress data on In
from Weertman[39] ...... 148
Figure 59. A log-log plot of indentation strain rate versus hardness/3.3 for constant Pip experiments on In plotted with tensile strain rate versus stress data from Weertman[39] ...... 149
Figure 60. A plot of the Bower factor[76] for relating indentation strain rate to
uniaxial strain rate versus stress exponent for creep ...... 151
Figure 61. Log-log plot of indentation strain ratelBowerfac tor versus hardness fo r constant Pip experiments on In plotted with tensile strain rate versus stress data from Weertman[39]...... 152 Figure 62. A log-log plot of indentation strain rate divided by the effective diffusion coefficientversus HIEfor Al plotted with bulk data obtained in torsion and tension from Luthy et al[40] ...... 154
XVlll Figure 63. A log-log plot of indentation strain rate divided by the effective
diffusioncoefficient versus Hl3.3E for AI plotted with bulk: data obtained in torsion and tension from Luthyet al[40] and Trozera et
al[77] ...... 155
Figure 64. A plot of a typical load time history for constant contact area/variable
frequency experiment conducted on polyisoprene...... 170
Figure 65. A typical plot of load versus displacement for polyisoprene during
constant contact area/variablefr equency experiment...... 174
Figure 66. A log-log plot of dynamic compliance, h1F, versus radial frequency showing response of freehanging indenter as well as the response of
the indenter in contact with polyisoprene under conditions of three
different constant contactareas ...... 175 Figure 67. A semi-log plot of phase angle, $, versus radial frequency showing
response of free hanging indenter well as the response of the as indenter in contact with polyisoprene under conditions of three
different constant contact areas...... 177 Figure 68. A log-log plot of stiffness, S, and damping, COl, versus radial
frequency for polyisoprene under conditions of constant contact area ...... 178
Figure 69. A plot of the storage modulus, G', and loss modulus, Gil, as a function of radial frequency for polyisoprene under conditions of constant contact area ...... 179 Figure 70. A plot of dynamic compliance versus absolute displacement of capacitive plate in gap for conditions of no contact between the
indenter and sample and conditions of increasing contact area
between indenter and polyisoprene ...... 181
xix Figure 71. A plot of the phase angle versus absolute displacement of capacitive
plate in gap for conditions of no contact between the indenter and
sample and conditions of increasing contact area between indenter
and polyisoprene ...... 182
Figure 72. A plot of contact stiffness,S, versus displacement for polyisoprene...... 184
Figure 73. A plot of the contact damping coefficient, C, versus displacement for
polyisoprene ...... 185
Figure 74. A log-log plot of the storage modulus, G', and loss modulus, Gil, of
polyisoprene as a function of radial frequency showing data from
constant contact area/variable frequency experiments and constant
frequency/variablecontact area experiments ...... 186
Figure 75. Log-log plot of dynamiccompliance versus radial frequency for the
Nano Indenter® II free hanging indenter...... 199 Figure 76. A plot of Nano Indenter® II support spring stiffness as a function of
the absolute position of the indenter with respect to the capacitive
displacement gauge ...... 201
Figure 77. A plot of Nano Indenter® II dampingcoeff icient as a function of the absolute position of the indenter with respect to the capacitive
displacement gauge...... 202 Figure 78. A schematic of the components of the HTMPM for consideration in
the dynamic model...... 205
Figure 79. A plot of displacement versus time showing free vibration of
HTMPM indenter in response to step load ...... 207
Figure 80. A plot of cycle number versus time yielding the damped natural
frequency of the HTMPM indenter mass, spring,dashpot system ...... 208
xx Figure 81. A plot of the absolute peak: displacement of the HTMPM indenter
versus time during free vibration showing exponential decay of the
vibrational amplitude ...... 209
Figure 82. A plot of displacement versus time showing free vibration of the
HTMPM specimen stage in response to a step load...... 211
Figure 83. A plot of cycle number versus time yielding the damped natural
frequency of the HTMPM stage mass, spring, dashpot system...... 212
Figure 84. A plot of the absolute peak: displacement of the HTMPM stage versus
time during free vibration showing exponential decay of the
vibrational amplitude ...... 213
Figure 85. A plot of amplitude versus frequency for forced vibration of the
HTMPM specimen stage ...... 215
Figure 86. A plot of (Xindenter-Xstage)lXstage showing predicted response
calculated from dynamic model solution for contact conditions
determined from %Step Load Experiments ...... 22 1
Figure 87. A plot of the optically measured contact area divided by the contact
areacalculated from total depth of the indent andthe perfect
Berkovich indenter area function as a function of loading time for In...... 223
Figure 88. A plot of the optically measured contact area divided by the area
calculated from total depth of indent and perfect Berkovich indenter
geometry as a function of homologous temperature for AI...... 224
xxi List of Tables
Table (1). Summary of Stress Exponents Determined for Indium...... 105
Table (2). Range of temperatures for indentation creep of Aluminum...... 112
Table (3). Summary of dynamic constants for the Nano Indenter® IT obtained
from fitof equation (64)to the dynamic compliance versus radial
frequency...... 199
Table (4). Coefficients of third order polynomial fit to the Nano Indenter® IT support spring stiffness as a function of absolute position within the
capacitive gap ...... 203
Table (5). Coefficients offourth order polynomial fitto indenter damping
coefficient as a function of position within the capacitive gap ...... 203 Table (6). Summary of dynamic constants for HTMPM indentation head ...... 206
Table (7). Summary of dynamic constants for HTMPM specimen stage ...... 210
xxii Chapter 1 Introduction
1. Introduction
The purpose of this study is to contribute to the understanding of how to use depth-sensing indentation techniques to measure the time and temperature dependent mechanical properties of materials. The basis for all studies of mechanical properties is the response of a material to an applied force. This response can generally be thought of as occurring by three different means: (1) elastic (recoverable) deformation where the strains are nearly instantaneous; (2) elastic deformation where the strains are not instantaneous but rather depend on time and/or temperature, and; (3) plastic deformation where the material undergoes non-recoverable strain the extent of which may depend on time, temperature and the prior strain/time history the material has undergone. As the problem of measuring time insensitive elastic and athermal plastic phenomena using depth-sensing indentation techniques has been addressed by previous authors[I-3], it is the study of time and temperature dependent mechanical properties that will be the topic of this work.
The techniques for the measurement of time dependent phenomena can be conveniently divided into two classes: (1) broad-band, quasi-static or creep techniques where the load, stress or strain-rate is held constant for a period of time ranging from seconds to hours while the response of the material is monitored, and; (2) frequency specificdynamic techniques where the load or stress is varied at a single frequency and the response of the material is measured. While standard bulk testing techniques exist for these types of measurements, in many instances the volumes of material of interest may be on the scale that bulk techniques become impractical. Such is the case in the
I thin film devices found in the microelectronics industry, the protective coatings and aesthetic finishes found in the automotive industry, the adhesives and backing materials found in the paper industry, the skin effects due to the solidification process in injection molded thermoplastics, the surface layers formed by techniques such as ion implantation or irradiation , the individual cell walls in trees or even the diseased arterial tissue found within the human body, to name just a few. While in some of these cases specimens may be prepared in a form that allows testing with standard uniaxial techniques, these preparation processes may be very tedious or in some instances may very well alter those properties that are of interest. It then becomes necessary to find an alternate means for mechanical characterization. Depth-sensing indentation testing has proven to be a reliable technique for measuring many of the important mechanical properties of such small volumes of materials.
In this work, a variation of both the broad band, quasi-static and frequency specific dynamic techniques described above will be utilized to measure both time dependent plasticity or indentation creep at several temperatures and time dependent elasticity or viscoelasticity at room temperature.
1.1 Time Dependent Plasticity - Creep
1.1.1 Introduction
The goal of this study of indentation creep is to explore the dependency of the indentation stress, or hardness, as measured using depth-sensing indentation techniques, on the variables of strain rate and temperature as well as to investigate the existence of a steady state behavior in indentation creep. These relationships have been very thoroughly studied in bulk materials and are fairly well understood.
2 1.1.2 Phenomenological Description
One equation used to describe the behavior of materials under creep conditions is
e = AcT' (1)
where e and CT are the strain-rate and stress, respectively. The quantity n is known as the stress exponent for creep. When a material behavior is well described by this equation using a constant value of n, the phenomenon is referred to as power law creep.
The stress exponent of the steady state creep rate in pure metals at elevated temperatures
(T > 0.5Tm) in many cases has a value of about 5[4]. The majority of indentation creep work has centered around measuring the strain rate dependence of the hardness at room temperature. Here, analogies are typically drawn between a uniaxial flow stress and the hardness, and the uniaxial strain rate is related to the instantaneous displacement rate of the indenter divided by the instantaneous displacement, a quantity known in the literature as the indentation strain rate. Using these analogies, an attempt is made to gain information into the stress dependence of deformation, i.e., the stress exponent for creep. Reasonably good agreement has been found between the stress exponents measured with depth-sensing indentation and those measured using uniaxial techniques although questions still exist about the appearance of deviation from power law behavior at lower than expected strain rates.
1.1.3 Mechanisms of Creep
At temperatures above 0.5 Tm, creep in pure metals for which dislocation glide is easy is thought to be controlled by the climbing of dislocations out of their glide planes to overcome obstacles to motion. The climb velocities are limited by the rate at which
3 vacancies can diffuse in the material. As such, creep at elevated temperatures is expected to follow an Arrhenius type of behavior of the form
(2)
where Q is the apparent activation energy for creep, R is the gas constant and T is the absolute temperature and the remaining variables are as previously defined. A semi-log plot of strain-rate versus Iff will then yield the activation energy for the rate controlling mechanism. The apparent activation energies for high temperature creep in pure metals as determined by uniaxial techniques have been shown to be in general agreement with the activation energy for self diffusion[5]. Using the same analogies for stress andstrain rate described above, the temperature dependence of indentation creep has received limited attention, mainly being investigated through the use of impression creep experiments at elevated temperatures using a flat-ended punch or hot hardness tests where hardness as a function of dwell time under load has been measured[6-13].
Another concept developed to describe the creep process is the idea of a steady state microstructure. At a given temperature and stress, given enough time, there is strong evidence that a steady state strain rate and microstructure canbe reached in some materials[14]. Any perturbations in these parameters will typically result in a transient period of time during which the microstructure will evolve to a new state representative of the new set of conditions. Following this transient period, steady state conditions can again be reached. Experimental evidence has shown that the microstructure that is developed in a given metal undergoing steady-state creep is independent of its previous work-hardened or pre-crept condition and depends only on the value of the steady-state stress[I5]. The notion of a steady-state stress or microstructure during indentation creep
4 tests incorporating a sharp indenter has not been investigated to date due to the lack of a test technique capable of yielding a constant indentation strain-rate. A new technique which allows the indentation strain rate not only to be controlled but varied systematically is introduced in this work. This type of test makespossible, for the first time, the investigation of a path independent steady state being reached in an indentation test.
1.2 Time Dependent Elasticity . Viscoelasticity
1.2.1 Introduction
The goal of this study of viscoelasticity is to attempt the first quantitative measurement of viscoelastic properties using dynamic depth-sensing indentation techniques. The object of a phenomenological study such as this is not to discover the relaxation properties in a complete fashion, but rather to be able to predict behavior under certain circumstances having observed it under others.
1.2.2 Phenomenological Description
According to McCrum et al or Nowick and Berry[16, 17], if the stress is expressed in complex form,
(3)
where 0"0 is the stress amplitude and m is the angular frequency, then the complex strain is given by
5 (4) where ro is the strain amplitude. The stress-strain relationship for such a material may be expressed as
.. '* '* .. G = G r = (G '-iG " ) r (5)
where
G' (m) = Gol = Gol cosl/J (6) Irl Iro
is the storage modulus of the material and
il G (m) = Gol = Gol sin l/J (7) In Iro
is the loss modulus of the material. More detailed information concerning this type of material behavior may be found in the literaturereview section of this work.
A variety of bulk techniques incorporating either a quasi-static or a dynamic mechanical testing technique exist for measurement of these properties. Recent efforts to measure these properties using both Surface Force Apparatuses[18, 19] and Atomic
Force Microscopy[20] have been made. While a limited amount of qualitative work has been performed using dynamic contact techniques a quantitative model has not been developed due to these systems inabilities to accurately determine the contact area between the tip and the sample or the net applied strain amplitUde. This work represents the first attempt at measuring the viscoelastic properties of materials using a depth-
6 sensing indentation system where a precise knowledge of both the contact area and the applied force are known.
1.2.3 Mechanisms of Viscoelastic Behavior
Unlike phenomenological theories, molecular theories attempt to explain the relaxation behavior of polymers in terms of proposed mechanisms for the motions of polymer molecules [ 16]. The mechanisms of viscoelastic behavior differ greatly not only from material system to material system, but within a specific material system as well evidenced by the fact that most polymers exhibit more than one mechanical relaxation [16] . There are three theories that are typically considered to be generally applicable to a wide range of polymer systems: (I) the large scale motions of a polymer chain in a viscous medium; (2) the interaction of a moving unit with a neighboring unit by means of jumping across a potential energy barrier from one equilibrium position to another; and, (3) local mode theories which consider relaxations resulting fromdamped oscillations of molecular groups close to their equilibrium positions. While not a specific topic of this work, a brief discussion of these phenomena is included in the review of the literature.
7 Chapter 2 Review of Literature
2. Introduction
An important aspect of any scientific endeavor is to understand the work that has been performed by other authors in the subject area and to use this knowledge as a rational starting point for further work. As the goal of this work is to further the understanding of how time and temperature dependent properties measured by means of depth-sensing indentation can be compared and related to those same properties measured using standard bulk techniques, this review of literature is intended to acquaint the reader with the current base of knowledge in the areas of mechanical properties that are pertinent to the work at hand and entails a review of the fundamentals of time- dependent plastic and elastic behavior and the techniques and models that currently exist for their measurement and understanding. This review is by no means meant to be a complete review of the field of the study of mechanical properties, instead readers are urged to consult the numerous excellent documents referenced herein for more detailed information about any of the specificareas that may be of interest.
2.1 Time Dependent Plasticity - Creep
2.1.1 Introduction
The progressive deformation of a material at constant stress is called creep[21].
Central to the understanding of creep properties is the dependence of the creep rate on the variables of stress and temperature[22] or conversely the dependence of the creep stress on the variables of strain-rate and temperature. These relationships have been
8 very thoroughly studied in bulk materials and are fairly well understood. A third idea central to the understanding of the creep process is the development and role of the creep microstructure. This review will focus on the fundamentals of these three parameters.
2.1.2 Review of Uniaxial Literature
2.1.2.1 Effect of Stress
Power-Law Creep
The central idea of power law creep is that the creep rate, e, at a given temperature depends both upon the applied stress, cr, and atleast one strength factor, 't':
e = t(O', "C). (8)
Under constant stress, the creep rate depends on how the strength parameter changes in the course of creep. Bailey[23] and Orowan[24] have suggested that strain hardening and recovery serve to increase and decrease the strength respectively and that at steady state the rate of hardening is equal to the rate of recovery[25] and propose an evolutionary law for the strength parameter during creep of the form
(9)
By taking into account the appropriate expressions for hardening and recovery, it is possible to arrive at the Dom equation that expresses the correlation of creep rates
9 among various metals as a function of stress and temperature[22] which can be expressed as:
(10)
where f(T) is the temperature dependence which will be considered in the fo llowing section, G is the shear modulus , n is the stress exponent which is definedas dinEldlnaat a constant temperature and A is a parameter which accounts for any effects of structure not explicitly included in the derivationof the Dom equation.
For certain pure metals under the appropriate conditions, n is generally found to range from 4-6 with an average of about 5. It is interesting to note that the actual derivation of the natural creep law fo r diffusion controlled creep yields a power law 3 dependence rather than the power law 5 dependence as suggested from a large body of experimental evidence[14].
The appearance of the shear modulus in equation (10) is a factor that warrants consideration. According to Sherby et al[26], the elastic modulus is the second of two main principal factors (the first being the diffusion coefficient) influencing the creep rate at a given level of stress. This factor is important because the speed of dislocation glide and climb are strongly influenced by the elastic stress fields of dislocations obstructing the path of moving dislocations. This hypothesis is demonstrated by a log-linear relation between the steady state flow stress at a given value of e/ D, where D is the appropriate diffusioncoeff icient, and the corresponding elastic modulus.
10 Power-Law Breakdown
Although high-temperature creep can often be described by a power-law creep equation with a constant stress exponent of 4-6, this relationship is unable to appropriately describe the creep response at low temperatures or high stresses. This failure of the power-law creep equation is typically known as power law breakdown[27].
According to Ashby and Frost[28], power law breakdown represents a transition from a diffusion limited dislocation motion to an obstacle limited glide regime.
However, others have pointed out that this trend can perhaps be explained by considering the effects of diffusion along dislocation cores[29]. It is possible to empirically describe the phenomenon of power law breakdown using a hyperbolic sine functionof the form[30]
(11)
but this expression has not yet been derived from the consideration of mechanisms involving dislocation motion.
2.1.2.2 Effect of Temperature
Creep is possible only because of thermal activation of deformation processes and in the absence of thermal fluctuations, creep could not occur[3 1]. It is well established that atom mobility, i.e., the diffusion coefficient, is of primary importance in influencing the creep rate in metals above O.4T m. This fact is convincingly demonstrated by the equality of the activation energies for creep and lattice self diffusion at temperatures above O.6Tm[32-34] and is supported by a large body of experimental evidence as shown in Figure (1) which is a plot comparing the activation
11 tN, mm3/mole GJ - '0 10 10 0 , E - .., ::!: w/• � Nb .6A1203 >- uo ...... " Ta l!) a::: NI 2 Mo W aFt elFt Z Q Q Mt�� 0 C = L N,Ct W -:7t9Br -GJ Z 9 0 0 M,.,..,'At A E .- �.-�Zn ")TI . '" « 0.1 ,'- Pb > o-TI'- 1Of: 6VC.? E .- - U 7'c.s" � « eLI <:J Z N. .. Age,. 0 � C/) :J P U. U. 0 P./ U. '-AI ...J N.. UJ 0.01 10 C/) 0.01 0.1 1 CREEP ACT IVATION ENERGY, Q c , MJ/mole
Figure 1. A comparison of activation energies for creep and self diffusion for a wide range of materials. From Sherby[35].
Source: O. D. Sherby and A. K. Miller, ASME Journal of Materials and
Technology, Vol. 101, 1979, 387.
12 energies for steady-state creep and lattice self diffusion for a large number of
materials[35]. The narrow temperature regime for which the correlation between power
law creep and diffusion holds for some metals led some authors such as Poirier[36] to question the role of diffusion in power law creep and to suggest that nondiffusional processes such as cross slip are more important. However, the evidence for diffusion
controlled creep remains strong for many metals and the description of high temperature
creep in terms of diffusion is far more complete and predictive than any other approach yet devised.
Given that at temperatures above 0.5 T m, creep in pure metals is thought to be controlled by the climbing of dislocations out of their glide planes to overcome obstacles
to motion, the climb velocities are limited by the rate at which vacancies can diffuse in the material. As such, creep at elevated temperatures is expected to follow an Arrhenius type of behavior of the form
e..= eo exp (Q--) (12) RT
where Q is the apparent activation energy for creep, R is the gas constant and T is the absolute temperature. A semi-log plot of strain-rate at constant stress versus 1fT will then yield the activation energy for the rate controlling mechanism. This temperature dependence may conversely be presented in terms of the appropriate diffusion coefficient, Le., the self-diffusion coefficientfor pure metals, as
(13)
13 where DL is the lattice self-diffusion coefficient, G is the shear modulus, b is the
Burger's vector andT is the absolute temperature.
In calculating the temperature compensated creep rate it is often necessary to
take into account the temperature dependence of the shear modulus in order to arrive at
an appropriate activation energy[37, 38]. It has been shown for a number of materials that taking values at a constant stress rather than at a constant value of a/ G(T) can lead
to overestimation of the activation energy for creep[39].
At lower temperatures, corresponding to higher stresses and strain rates, the
stress exponents for steady state creep become much greater than 5. In addition to
higher stress exponents, the activation energies for creep in this regime lower than are those expected for lattice self diffusion. This observation has been explained by an
additional dislocation creep mechanism where transport occurs via dislocation core
diffusion[28]. This contribution can be accounted for by defining an effective diffusion coefficient as
(14)
where Dc is the core diffusion coefficient and fc and fv are the fraction of atom sites
associated with each type of diffusion. The value of fv is essentially unity. The value of
fc is determined by the dislocation density, p, as fc = acp where ac is the cross-sectional area of the core. At high temperatures (>0.5 T m), the lattice diffusion contribution is dominant (high temperature creep) while at intermediate temperatures (0.3-0.5 Tm)the core diffusion contribution becomes dominant (low temperature creep). This
mechanism has been observed as the dominant mechanism in high purity aluminumin the temperature range 0.29 to 0.48 T m by Luthy, Miller and Sherby[40] .
14 One major limitation of the effective diffusivity concept is that it fails to provide a natural explanation of the steadily increasing stress exponent in power-law breakdown[27].
2.1.2.3 Formation of Steady State Microstructure
The final idea central to the understanding of the creep process is the development and role of the creep microstructure. The formation of subgrains dominates the microstructural changes that occur during creep flow in most annealed poly crystalline metals at temperatures above 0.4 Tm[32-34]. The first researchers to observe the formation of subgrains were Jenkins and Mellor in 1935 [4 1]. An explanation of subgrain development however was not proposed until the late 1940's when Wood and co-workers[42, 43] suggested that these cells were due to accumulation of edge dislocations by climb (polygonization) leading to the development of low-angle boundaries which are now universally called "subgrains." The "cell" or subgrain structures that associated with the creep process do not are form instantaneously, rather, they form gradually throughout the course of primary creep[ 14]. During the initial stages of deformation, dislocations move through a well annealed crystal at speeds approaching the speed of sound in the material[44]. During this period of time, the dislocation density increases abruptly and the dislocations are typically distributed in a homogeneous fashion[14]. This rapid deformation continues until the interdislocation stresses approach the applied stress. If there were no thermal activation, deformation would stop at this point. However, at elevated temperatures, thermal activation allows continued dislocation motion past obstacles and deformation can continue. Steady state creep coincides with the development of a steady state substructure which is characterized by the condition that the average dislocation density
15 and the subgrain size remain constant with further creepstrain [27].
These substructural changes that occur during the creep process known to be are insensitive to temperature and to only depend on time and stress[22] according to the relationship
(15)
where A is the subgrain size, A is a constant equal to approximately 4 for a number of materials but appearing to be a function of the stacking fault energy, b is the Burger's vector, is the stress, and E is the temperature compensated Young's modulus. This cr constancy of microstructure for a given level of stress has been shown by incorporating stress-change tests where the applied stress is abruptly changed once the steady state has been reached. The data of Horiuchi and Otsuka[45] as analyzed by Sherby, Klundt and
Miller[26] show that when the stress is suddenly decreased, an instantaneous drop in the strain rate occurs where the rate of strain is below the steady-state value for the lower stress. This significantly lower value of the creep rate is attributed to the finer subgrain size developed at the higher stress. These data are shown in Figure (2). As further strain is incurred, the creep rate is observed to increase. This increase in creep rate is accompanied by an increase in subgrain size. Eventually a new steady state creep rate, identical to the steady state creep rate at the lower stress, is obtained as the new subgrain size for the low creep stress is attained. An increase in the creep stress is observed to have the opposite effect as that observed in the stress-drop test.
Assuming the constancy of structure at a given level of stress, Sherby et al[26] developed an equation which predicts the creep rate as a function of the subgrain size.
16 HiOh purity aluminum 683°K
..... -
'u • ..
.08 .16 .24 CREEP STRAIN Figure 2. A plot of stress change data for Al from Horiuchi and Otsuka[45] replotted by Sherby et al[26].
Source: O. D. Sherby, R. H. Klundt, and A. K. Miller, Met. Trans. A, 8A, 1977,
843.
17 Based on the stress change tests as described above, Sherby et al arrive at a relationship for the flow stress-instantaneous creeprate which pertains to a constant structure. They find that the stress exponent, n, for constant structure creep tests is closer to 8 than to 5 as determined under conditions of constant stress. They summarize the instantaneous constant structure creep rate for polycrystalline metals as
(16)
where S is a constant equal to about 1.5x 109 for high stacking fault energy metals and all other quantities are as previously defined. Equation (16) is seen to reduce to the 5 power law equation normally given for describing the steady state creep rate of many pure polycrystalline metals as the microstructural dependence is of reciprocal third order dependence on stress.
2.1.3 Review of Indentation Creep Literature
2.1.3.1 Introduction
Since the flow stress of a material is a function of strain rate and temperature, the hardness of a material should be expected to vary in an analogous way[lO]. This fact has led to numerous studies attempting to correlate the measured hardness or indentation stress with the variables of time (or indentation strain rate) and temperature. The techniques used to measure the time and temperature dependent properties of materials with indentation testing may be divided into two classes: (1) conventional microhardness and "hot hardness" experiments where an attempt to gain insight into the time and temperature dependent properties was made by monitoring the optically
18 measured hardness as a function of dwell time under load and temperature, and; (2) depth-sensing indentation techniques at both room and elevated temperatures where the load on the indenter or the displacement of the indenter is controlled dynamically while the other is monitored in such a way to define both an indentation strain rate and an indentation stress or hardness.
2.1.3.2 Conventional Indentation Techniques
As the techniques for measuring the time and temperature dependence of materials using these techniques are quite often intermingled, this section of the review will attempt to follow the chronological history of this technique rather than attempting to separate the work into separate areas of time and temperature dependence. The subject of microstructure will not explicitly be discussed in this section.
The earliest technique used in an attempt to gain information into the time and temperature dependence of hardness was the hot hardness test. This type of test utilized a standard micro indentation system modified to allow for elevated temperature measurements. Experiments were typically conducted to study the effects of dwell time and temperature on hardness.
One of the original works in this area was that of Mulhearn and Tabor in
1960[9]. Starting with a power law equation for viscous creep they arrived at an expression relating the measured hardness to time under load. For loading times exceeding a few seconds they found, at any one temperature, a linear relationship between log hardness and log time. This relationship was found to be independent of temperature and the results were explained in terms of viscous creep.
Coincident to this work by Mulhearn and Tabor was an independent study by
Pomey and co-workers[46] that found evidence showing that the relationship between
19 log hardness and log time was in fact a function of temperature, the slope of a plot of log hardness versus log time increasing with increasing temperature.
These conflicting results led to a study of the indentation hardness and the creep of solids by Atkins et al in 1966[10]. They began by reviewing the earlier study of
Mulhearnand Tabor[9] and stated that rather than use a steady-state creep equation, it is more appropriate to use a transient equation of state due to the fact that the growth of the indentation with loading time is determined by the rate at which the elastic plastic boundary can diffuse into the specimen. Using an expression for transient creep from
Mott[47] developed for conditions of constant stress,Atkins et al arrive at an expression relating the measured hardness to the time under load that is strikingly similar to that obtained by Mulhearn and Tabor[9]. They assume that even though the expression of
Mott is explicitly for conditions of homogeneous deformation under conditions of constant stress, it can be applied to an indentation test if the stress exponent is large and the stress is not changing rapidly.
Atkins et al began by examining the creep response of indium using a variety of indenter geometries. An important finding of their work is that the rate at which the deformation proceeds, i.e., the rate of progression of the elastic/plastic boundary, does not depend upon the geometry of the indenter unless the change in geometry is accompanied by a change in the mode of deformation. This leads them to suggest that the rate of deformation cannot be associated with the geometry of the indenter itself but rather the rate at which the elastic/plastic boundary can diffuseinto the material.
Atkins et al examined data from indium, lead, aluminum, MgO, and we obtained by performing 1 second load ramps fo llowed by holds under constant load at
Pmax for a variety of times. The load was then removed and the size of indentation measured.
20 Atkins et al found a linear log hardness-log time relationship for longer loading
times but found that the slope of the line increases, i.e., there is a deviation from
linearity, for short loading times. In the linear region they find that the stress exponent for creep, n, is approximately 10 for the stresses involved in hardness measurements.
Atkins et al also found, in accord with a number of other authors, that the slope of a plot of log hardness versus log time increases with increasing temperature.
By analyzing the data according to their assumed equation of state, Atkins et al calculate an apparent activation energy for creep from the data at various temperatures.
They found values near that for self-diffusion for Sn (Trrm=O.54-0.78) and Al
(Trrm=O.64-0.81). Atkins et al concluded that the observed hot hardness behavior closely parallels the creep characteristics of the materiaL
A major problem associated with the technique of Atkins et al is that the result depends on a measure of absolute time under load. An important question is how time zero is appropriately defined, i.e., at the beginning of the loading segment or at the beginning of the constant load segment as is done by Atkins et al. If the loading time is short, the effects of the choice of time zero become less pronounced very quickly, however, as Atkins et al point out, there seems to be a dependence of the result on the rate at which the load was applied. This seems a logical result as the strain rate experienced by the material will depend upon how the load was applied.
In another study of hot hardness data obtained under conditions of constant time
of loading, Larsen-Badse[11] determined "apparent activation energies for indentation hardness" using a relationship of the form
(17)
21 where H is the hardness, Q' is the apparent activation energy, and T is the absolute temperature. From the values obtained for 23 pure metals, Larsen-Badse concluded that the apparent activation energy for the hot hardness tests was proportional to but not equal to the activation energy for self-diffusionor steady-state creep.
The data of Larsen-Badse were subsequently reanalyzed by Sherby and
Armstrong [ 12] who show that rather than an expression such as equation (17) as used by
Larsen-Badse[l1], an expression of the form
(18)
where QL is the activation energy for self-diffusionand n is the stress exponent for creep is more appropriate. Comparison of this expression with that of Larsen-Badse reveals that the apparent activation energy obtained by Larsen-Badse is equal to the activation energy for self-diffusion divided by the quantity nR. Since n=5-7 for pure metals at elevated temperatures and R = 2 CalIK mol, QLis roughly 10-14 times Q' which agrees well with the experimental results.
Sherby and Armstrong go on to analyze the hot hardness data in terms of the creep characteristics of pure metals. They begin by introducing an expression for the temperature dependence of the hardness that incorporates the temperature dependence of the elastic modulus, E. This equation is then used to examine the hot hardness behavior of Al and Cu at temperatures ranging from 0.55 to 0.95 Tm. They show that a plot of log (HIE) versus the reciprocal homologous temperature, T mIT, shows two linear regions with a break at 0.75 Tm. At temperatures greater than 0.75 Tm, they find activation energies nearly equal to those for self-diffusionin the two materials.
22 Sherby and Armstrong associate the break in the curve of log (HIE) versus log
(TmlT) with the onset of power-law breakdown behavior based on an analysis of the average strain rate occurring under the indenter and the high temperature diffusivity.
They premise this argument with an important statement, that the strain rate in a hardness test is not constant. They assume that an average strain rate can be defined by considering that a characteristic strain of about 10% for a Vickers indenter occurs during the 30 seconds during which the load is applied. (The idea of defining a characteristic strain rate based on the time under load and a characteristic strain will be subsequently discussed further in light of this study.) Based on this analysis and a typical high temperature diffusivity, Sherby and Armstrong arrive at a value of e I D, where D is the diffusion coefficient, of 109, a value where they claim power-law breakdown is observed to occur for most metals.
Two conclusions are drawn by Sherby and Armstrong from these results regarding the hot hardness testing of pure metals. Between 0.75 and 1.0 Tm, the deformation process is associated with lattice self-diff usion and creep flow in the power law region. Between 0.5 and 0.75 Tm, the rate determining process is associated with dislocation pipe diffusion and creep flow in the power law breakdown region. It is important to bear in mind that implicit to their determination of activation energies is the assumption of a constant value of 5-7 for the stress exponent for creep at elevated temperature in pure metals.
In the most recent analysis of hot hardness testing by Sargent and Ashby[13], a variety of analysis techniques are outlined by which the stress exponent for creep and the apparent activation energy for creep can be determined.
Sargent and Ashby begin by performing a dimensional analysis of creep indentation from which several analysis techniques can be surmised. According to
23 Sargent and Ashby, when an indenter is pressed into a "hot" surface, it penetrates the material first by yielding and subsequently by creep. The load on the indenter, P, is carried by a projected contact area, A, such that the indentation pressure, H =PIA. The stresses cause the material to creep and the indenter penetrates the material. As the deformation proceeds, the pressure is reduced as the contact area grows. Sargent and
Ashby derive a relationship for the hardness as a function of time of the form
(19)
where
(20)
0'0 and C are constants, t is the time under load, and n is the stress exponent for creep. The stress exponent for creep can be determined by one of a number of methods from the Sargent and Ashby analysis. From equation (19), the slope of a plot of log
(H/G) versus log time is -lin. If this equation is appropriate, the data should be a family of straight lines with a change in slope being accounted for by the dependence of the stress exponent on temperature.
Alternatively, differentiation of equation (19) gives the rate of indentation in terms of the time rate of change of the hardnessas
1 dH 1 = (21) H dt nt
24 such that a plot of log (l1H dHldt) versus log time superimposes all of the data for one material on a master curve having an intercept of lInt at the specific value of t.
The apparent activation energy of the creep process is determined either via a plot of In(HlG) versus (TlTm) at constant time having a slope of Q/nRTm or conversely by a plot of In time versus (TmIT) at constant (HlG) having a slope of QIRTm which suggests a constant structure.
Sargent and Ashby apply their analysis to hot hardness data for Armco iron and copper. Reasonably good agreement between the stress exponent for creep and the activation energy obtained from their analysis and that obtained from uniaxial data for the iron is obtained. The large scatter in the indentation data for the copper prevented any conclusions from being drawn from that analysis.
One potential problem associated with the analysis of Sargent and Ashby is that the stress and displacement fields beneath the indenter are assumed to be self-similar as the deformation proceeds and to change only in scale and not in shape. This would not seem to be entirely true as the basis for this analysis is that as the deformation proceeds the stress and strain rate are changing in such a way that the relationship between the two is related to the stress exponent of the material. The stress and displacement fields might be expected to vary as the deformation proceeds due to these changes.
2.1.3.3 Depth Sensing Indentation Techniques
Introduction
The use of depth-sensing indentation techniques to study the time-dependent properties of materials is an area of recent undertaking, having only seen a large amount of work within the last lO years. The majority of this work has concentrated on
25 measuring the relationship between the hardness and an experimentally defined indentation strain rate. While a limited amount of work has been performed at elevated temperatures to investigate the temperature dependence of the indentation creep process, the majority of the work has been at or below room temperature[48] or utilized a flat ended punch[6] rather that a geometrically similar indenter.
Effect of Stress
There are four types of tests that have been employed using depth sensing indentation systems to gain insight into the relationship between the indentation strain rate and the indentation stress or hardness: indentation load relaxation (ILR) tests[49,
50], constant rate of loading (CRL) tests[51, 52], constant load hold indentation creep tests[53-58] and impression creep tests[6, 8].
Indentation Load Relaxation (ILR)
The idea of using ILR techniques was first presented by Hannula et al[49]. In this type of experiment, an indenter is pushed into a sample surface with the aid of a crosshead. When the desired load or displacement is achieved, the position of the indenter is fixed. Load relaxation occurs by conversion of the elastic strain in the machine and the specimen into plastic strain in the specimen. This technique has been used[50] to study time dependent deformation in small volumes. Using a Vickers indenter with a 1!lm2 flatarea on the apex, an effective stress was defined as
(22)
where P is the load on the indenter and A is the projected area of contact. An effective
26 strain rate was defined as
. h E=C- (23) h where c is a constant, it is the instantaneous descent rate of the indenter and h is the indenter depth (total depth or plastic depth), The stress-exponent for creep is then defined din e / din H at constant temperature. as
Constant Rate of Loading (CRL)
The constant rate of loading (CRL) test is a multi-indentation test that was developed for determining stress exponents using depth sensing indentation systems[51,
52]. This type of test utilized the ability of the Nano Indenter® to incrementally apply the load on the indenter at rates specified by the experimenter. In a CRL test, the indenter is loaded at a constant loading rate until the indenter has reached a prescribed depth in the material. A complete series of experiments would involve this procedure utilizing a different loading rate for each indentation made and calculating an effective stress for that loading rate from the applied load and the achieved depth. The stress exponent for creep is then defined in ananalogous way as dIn e / din H ,
The significant problem associated with performing experiments in this fashion is that the indentation strain rate is continually changing throughout the loading segment. While the changes in the stress and strain rate may be small, this type of experiment is nonetheless a far more dynamic test than the others.
27 Constant Load - Hold
The idea of measuring creep properties of materials with depth-sensing indentation systems under conditions of constant load was first introduced by Pollock et al[53]. Following the ideas of Atkins, Silverio and Tabor[10], Pollock makes the approximation that the strain rate, t.,as predicted by any physical model for creep, should scale according to
h . k2- (24) 8= h where h is the instantaneous descent rate of the indenter, h is the instantaneous displacement and k is a geometric constant. Using this analysis Pollock investigated the time dependent response of Ni films on Si at room temperature.
Mayo and Nix developed a similar technique for determining time dependent properties with the Nano Indenter® [54, 55]. Their experimental procedure involved loading the indenter at a high loading rate and then holding the load on the indenter constant and monitoring the displacement of the indenter as a function of time. Since the contact area changes as the indenter penetrates the material while the load is held constant, the hardness or effective stress changes as the test proceeds. This allows one to obtain several effective stress-effective strain rate pairs from one indentation experiment.
Raman and Berriche[56, 57] utilized this technique to investigate the creep behavior of sputtered Sn and Al films on Si. They found that the stress exponents obtained from the indentation tests were in reasonable agreement with those obtained from conventional uniaxial tests.
28 This technique was also used by Poisl et al[59] to empirically determine the relationship between the indentation strain rate and the uniaxial strain rate as measured for amorphous Se. They fo und that at temperatures above the glass transition temperature Newtonian viscous or power law 1 flow was observed ( i.e., the slope of a plot of log hardness versus log indentation strain rate yields a straight line with a slope of unity). Assuming that the coefficient relating the hardness and the flow stress, 0', of Se was equal to 3, i.e., H=3cr,the relationship between the indentation strain rate and the uniaxial strain rate was determined. Poisl et al fo und that the coefficient relating the indentation strain rate to the effective strain rate is equal to 0.09, or £ = 0.09£/ where
£/ is the indentation strain rate as previously defined in equation (24).
This type of experiment suffers from the same problem however that the CRL experiment experiences in that the indentation strain rate is continually changing throughout the loading segment and the subsequent hold segment.
Impression Creep
As discussed above, one of the problems associated with the current state of indentation creep testing with sharp indenters is that when the hardness decreases with time, the stress driving the creep process also decreases and no steady state can be achieved. To avoid these difficulties, Chu and Li[6, 8] proposed a new type of test for bulk materials in which the shape of the indenter is changed from a sphere or pyramid to a circular cylinder with a flat end. As the contact area remains constantduring the test, the punching or indenting stress is thus constant at constant load. By incorporating fi nite element analysis, Yu and Li[60] show in a concurrent paper that the effective Von Mises stress available for creep at any distance relative to the punch radius is proportional to the applied punching stress. As a result, the steady
29 state impressing velocity is proportional to the punch radius, and has the same stress dependence as the steady state creep rate. Using a power law constitutive equation, Chu and Li therefore arrive at anexpr ession relating the punch velocity to the effective stress.
Following a short transient period, they observe a "steady-state" velocity during which the descent rate of the indenter is found to have the same stress dependence as that observed during conventional unidirectional creep tests using bulk specimens for LiF single crystals[7], succinonitrile crystals[6], and 8-tin single crystals[8]. The only significant drawback of this technique is that the volume of material being deformed is essentially defined by the radius of the punch[7]. To perform these experiments in such a way as to constrain the plastically deformed volume to shallow depths it becomes necessary to use punches of smaller and smaller diameters. As the size of the punch is decreased, the errors associated with the contact area of the flat punch become more pronounced eventually limiting the useful range of the technique.
These problems can be overcome by using geometrical similar indenters such as a pyramid or cone if a technique can be developed that allows a similar "steady-state" to be reached.
Comparison of Indentation Techniques for Determining Stress Exponent
The ability of the above mentioned approaches to obtain consistent and technique independent data has been examined by Stone and Yoder[48] in investigating the rate dependent and rate independent components of the hardness of Mo. Using constant load-hold, indentation load relaxation, and indentation rate changeexperi ments, they found that the strain rate sensitivity, m (where m=lIn), was independent of the method of measurement and in general agreement with values obtained from uniaxial techniques
30 Effect of Temperature. Q
The temperature dependence of indentation creep data is an area that has received little attention in terms of depth-sensing techniques due mainly to the inherent difficulties in measuring the small displacements associated with the technique at elevated temperatures. The work that has been accomplished has typically been conducted around room temperature or using techniques that incorporate large displacements such that the thermal effects become small in comparison to the creep displacements.
The earliest investigations into the temperature dependence of indentation creep were carried out using the impression creep test described previously. Yu and Li and
Chu and Li investigated the temperature dependence of the impression creep process in LiF single crystals[7], succinonitrile crystals[6] and �-tin single crystals[8]. They reported activation energies for high temperature impression creep that were in good agreement with the activation energies reported for high temperature uniaxial creep as well as the activation energies for self diffusion.
Poisl et al[59] investigated the temperature dependence of the indentation creep process using a Berkovich indenter by changing the ambient temperature of the laboratory near room temperature in their investigation into the relationship between indentation creep and uniaxial creep in amorphous Se. From the Newtonian portions of the curves of log hardness versus log indentation strain rate they were able to calculate an activation energy for the indentation creep process. Using the indentation strain rates at a constant hardness, they obtained an activation energy of 495 kJ/mol which compares favorably with the activation energy for viscous flow of 535 kJ/moL
Stone and Yoder[48] used an indentation system capable of operating in the 160-
298K temperature range to investigate the rate dependent and rate independent
31 components of the hardness of Mo. They reported activation volumes , v*, defined as
v* = 9kT/mH (25)
where m is the strainrate sensitivity of the hardness, H. Using this technique they found activation volumes in agreement with those reported in the literature given 50% typical scatter in the data.
Indentation SteadyState
A final concept developed to describe the creep process is the idea of a steady state stress or strain rate and microstructure. At a given temperature and stress (or strain rate), given enough time, there is strongevidence that a steady state strain rate (or stress) and microstructure can be reached in some materials[14]. The onset of steady state creep is believed to occur when the rate of hardening in the material is balanced equally by the rate of recovery.
In an indentation test, the dynamics of deformation are very different than that occurring in the previously described uniaxial creep test. The deformed volume of material under the indenter is continually expanding to encompass previously undeformed materiaL As the material strains under the indenter, the material underneath the indenter is very often likened to an expanding cavity with a hydrostatic core where no deformation is occurring and an expanding elastic/plastic boundary. The creep process is believed to depend upon the rate at which the elastic/plastic boundary can proceed into the materiaL The question then becomes what steady state entails in an indentation creep experiment.
32 One indentation technique which has been shown to be capable of reaching a steady state is the impression creep test. Using a flat cylindrical punch, which yields a constant value of stress or indentation pressure, Chu and Li and Yu and Li show that a typical depth versus time curve under constant load resembles a conventional creep curve with both transient and steady state stages for LiF single crystals[7], succinonitrile crystals[6], and �-tin single crystals[8]. They state that the cause for the transient stage is believed to be due to the development of a steady state plastic zone under the punch, for example a sub grain or cell structure not very different from that developed during conventional steady state creep. This hypothesis is supported by the sub grain formation beneath the cleaved surface of a LiF single crystal after high temperature impression creep testing.
The notion of a steady-state stress or microstructure during indentation creep tests incorporating a sharpindenter has not been investigated to date due to the lack of a test technique capable of yielding a constant indentation strain-rate. While much of the previously discussed work using geometrically similar indenters has been conducted under the assumption that the fact that the stress is changing only marginally during the experiment does not greatly effect the results, no experimental evidence has been obtained that verifies this hypothesis.
2.2 Time Dependent Elasticity - Viscoelasticity
2.2.1 Definition and Terminology
Viscoelasticity refers to time-dependent elastic or recoverable deformation[61].
Consider for example the simple example give by McCrum[16] where a weight is suspended from a polymeric filament. The strain incurred as a result will not be
33 constant but will slowly increase with time at temperatures above 0 K. This effect is due to molecular rearrangements in the solid induced by and proportional to the stress. If the strains are small, on release of the stress, the molecules slowly return to their former spatial arrangement and the strain simultaneously returns to zero. The mechanisms of viscoelastic behavior differ greatly depending upon the material system, but are typically ascribed to a viscous flow process that results in the measured strain lagging behind the applied stress. The mechanical parameters defined to describe such a solid under stress must therefore depend on time (and/ortemperature which will not be discussed here).
In the consideration of viscoelastic behavior, two types of mechanical responses may be considered: a broad band quasi-static response function where the material behavior is observed over periods of several seconds and longer under conditions of constant applied stress or strain, or a frequency specific dynamic response function where the behavior of the material is studied over much shorter periods of time when a periodically varying stress or strain is imposed on the system. It is a variation of the dynamic response function that will be utilized in this work and will therefore be the focus of this review.
Before beginning it should be pointed out that the phenomenological description of viscoelastic behavior is well known and accepted in the polymer community and is the subject of many excellent texts. A large part of the phenomenological review contained herein is from McCrum[16], Nowick and Berry[17], or Zener[62], all of whom have done an excellent job in outlining the measurement of the dynamic mechanical response of materials.
34 2.2.2 Phenomenological Description
If a sinusoidal stress is applied to a viscoelastic solid, then the strain response generally lags behind the applied force by some phase angle. This phase lag results from the time necessary for molecular rearrangements in the material[16]. The dynamic response function of the material is most conveniently described in terms of complex notation[17]. Let the stress written be as
(3)
where 0'0 is the stress amplitude and OJ is the radial frequency of vibration (OJ=21tf, where f is the vibrational frequency). The resulting strain will also be periodic in nature and can be expressed in the form
(4)
where Yo is the strain amplitude and is the angle by which the strain lags behind the stress or conversely Y1 is the amplitude of the component of strain in phase with the stress and Y2 is the amplitude of the component of strain 90° out of phase with the applied stress. Consider first a perfectly elastic solid. A situation similar to that shown in Figure (3) will exist where the resulting strain is exactly in-phase with the applied stress, i.e., there is no damping and <1>=0. The ratio O'/Y then gives the shear modulus of the material G.
Now consider a linear viscoelastic solid. Here the strain is time-dependent and lags behind the stress by a phase angle as shown schematically in Figure (4). For this
35 Elastic Material
..0'- « -
-- Stress
.. -----Strain CJ) CJ) Q) ' +-' en
Time
Figure 3. A schematic of the strain response to applied stress for a purely an elastic material. The strain is exactly in phase with the applied stress.
36 Viscoelastic Material
... , , , • , , , , , I , , , , �t , , ,
c . - • , , � • -- , ...... Stress , , (J) • - I en , I • I en -----Strain • (]) I " • " :to... (J)...... =�tro
Time
Filjure 4. A schematic of the strain response to an applied stress for a viscoelastic material. The strain lags behind the applied stress by a phase angle <\>.
37 case the ratio a/y is a complex quantity. Nowick and Berry[l7] and McCrum[l6] define this complex modulus, G*(ro), as
* * * * = = G (G I -iG .. ) (5) a r r where
I ) = ao/ = ao/ G (m Ir1 Iro cosl/J (6)
and
) = ao / = ao/ G"(m Ir2 Iro sinl/J (7)
where G' and Gil are defined as the storage and loss modulus respectively and are related to the relaxed modulus, Gr, and unrelaxed modulus, Gu, measured in a creep response experiment according to
(26)
and
(27)
where is the relaxation time under conditions of constant strain. At sufficiently low 't£
38 frequencies, the strain will be proportional to the stress with the relaxed modulus as the proportionality constant such that G * (0) = G I = Gr . At very high frequencies,
G * ( 00)= G '= Gu . The function Gil is a Debye peak that resides at the frequency where
= 1 holds, i.e., at the resonant frequency of the mechanism. roT The importance of the phase angle
(28)
Several indirect methods have frequently been employed for specific loss measurements. These methods have been described by Zener[62], McCrum[16] and
N owick and Berry [17] and typically involve methods employing resonant systems vibrating at a natural frequency either in forced vibration or in free decay.
One indirect method of measurement involves measuring the amplitude of vibration under conditions of forced oscillation, the frequency of the excitation force being slowly varied while the amplitude is held constant. The amplitude of the resultant vibration will be a maximum when the excitation frequency is equal to the resonant frequency of a specific defect within the solid. In this type of experiment, if is the Llm change in the excitation frequency to change the amplitude from half-maximum on side of the resonant peak to half-maximum on the other side, then
(29)
39 where Q-l is analogous to the definition of Q found in resonant electrical circuits and corresponds to the sharpness of the resonance peak, COr is the resonant frequency of the defect and
A second, indirect, method of measuring mechanical losses involves the measurement of the energy per unit volume, t:,.W, dissipated per cycle of vibration.
Here, the energy of vibration, W, is defined as the strain energy per unit volume when the strain is at a maximum. At this point the energy is entirely potential in nature and is directly proportional to the square of the amplitude of the oscillation. A measure of the decay of the oscillation in free vibration therefore gives an indirect measure of the internallosses and is given by the logarithmic decrement, 0, where
(30)
w here A is the amplitude of the free vibration of the system and n is the number of the vibrational cycle. The log decrement is related to the internal friction through the relation
0=nt an l/J (3 1) where
To summarize the relationships between the previously defined quantities, the following relationships may be written:
t:,.W l =Q- =tan l/J (32) 2n W
40 where is the loss or phase angle, Q- l represents the sharpness of theresonant peak, and
AW IW is a measure of the energy lost per cycle of vibration as defined by equation (29). The relationships between the specific values of the damping coefficient, C, and stiffness, k, that are normally referred to in mechanical models and the parameters G' and Gil as defined in the dynamic response function for the system can be better understood by considering once again the energy losses that occur in the system during a complete cycle of vibration. The power absorbed by a damper responding harmonically at a given frequency is proportional to the square of the amplitude[63], i.e.,
(33)
where C is a damping coefficient of a dashpot element in standard units of N s / m. The total energy of the system can be calculated from the potential energy at the maximum amplitude, i.e.,
W = �kX2 (34)
where k is the stiffness of a spring element in units of N / m. From comparison with equation (28) it can be noted that
G" CmX2 (35) 1C Y6 oc 1C and
41 (36)
or simply stated, the loss modulus, Gil, is proportional to the damping coefficient of the material multiplied by the radial frequency and the storage or elastic modulus, G', is proportional to the stiffness of the materiaL These relationships will be fu rther expanded in the following section on measuring time-dependent elasticity from indentation experiments.
2.2.3 Mechanisms of Viscoelastic Relaxation
While not a specific topic of this study, the mechanisms of viscoelastic relaxation in polymers are important in aiding the understanding of the phenomenological measurement approach taken herein. A brief general survey of the different types of relaxation regions and relaxation mechanisms observed for solid polymers is therefore in order.
In order to identify and compare the locations of loss peaks for different polymers, the peaks are often labeled with the Greek letters �, etc.[16] According a., 'Y, to this nomenclature, the loss peak corresponds to the relaxation observed at the a. highest temperature (at a given frequency) or the lowest frequency (at a given temperature)[64]. The most pronounced mechanical relaxation region for polymers in the amorphous state is associated with the glass transition and occurs at temperatures at or around the glass transition temperature, a region where the effective viscosity increases rapidly for non-crystalline materials[61]. This transition is usually labeled and is a. referred to as the glass-rubber relaxation[16]. From a molecular point of view, it has been widely accepted for many years that the glass-rubber relaxation results from the
42 large scale rearrangements of the polymer chains occurring by rotations around main chain bonds[16]. Long chain polymers in the vicinity of their glass transition temperature are capable of marked viscoelastic behavior, the viscous component of their flow being involved with stress-assisted interchain sliding[61].
In addition to the glass-rubber relaxation, amorphous polymers usually exhibit at least one secondary relaxation region. These secondary loss regions result from motions within the polymer in the glass-like state. In this state the main chains are effectively
"frozen-in" so that these relaxations cannot be due to large scale rearrangements of the main polymer chain. Since the molecules of most amorphous polymers contain side groups capable of undergoing hindered rotations independently of the chain backbone, these secondary relaxations are often ascribed to such rotations.
43 Chapter 3 Experimental Equipment
3. Introduction
The use of depth-sensing indentation techniques to measure a variety of mechanical properties of small volumes of materials has seen vast expansion during the past decade. This has been made possible in large part due to advances in instrumentation and techniques capable of accentuating specific material properties by precisely controlling the load-time or displacement-time history of the experiment. Two distinct instruments were utilized in this study. The firsta Nano Indenter®n, is a state of the art commercial depth-sensing indentation system that is widely used in both industry and research. The second instrument is a prototype high temperature depth-sensing indentation system developed at the OakRidge National Laboratory. A brief description of the Nano Indenter®II system is followed by a more complete description of the new prototype high temperature instrument.
3.1 Description of Nano Indenter®ll Figure (5) is a schematic of the Nano Indenter® II (Nano Instruments, Inc., Oak
Ridge, TN, 37830) used for the majority of the room temperature experiments. The intricacies of the system have been detailed elsewhere [ 1] and will not be discussed here. The indenter is supported inside the indentation head by two leaf springs designed to limit the motion of the indenter column to only the vertical direction. The load on the indenter is generated through the use of a current carrying coil and permanent magnet assembly. The resolution of the loading system is 50 nN. The displacement of the indenter is measured with a three plate capacitive system with a resolution of 0.04 nm.
44 c
,------r--� CURRENT SOURCE
OSCILLATOR o
J===:i1rlii==:r-...... LOCK-IN AMPLIFIER
DISPLACEMENT SENSOR o
Figure 5. A schematic of the Nano Indenter® II. A-Sample; B-Indenter Column; C-Load Application Coil; D-Indenter Support Springs;
E-Capacitive Displacement Gauge; andF-Load Frame.
45 The operation of the system and the acquisition of data is controlled through a personal computer. Data is typically acquired at a rate of 3 load, displacement and time data points per second.
The Nano Indenter® II is typically operated in one of two modes; a constant loading rate mode achieved by direct control over the load ramp or a constant displacement rate mode achieved by means of a feedback loop between the calculated displacement rate and the loading rate. In addition to these two modes of operation, the system has the ability to apply step loads by a direct set of the digital to analog converter
(DAC) controlling the current to the load application coil.
3.2 Description of the High Temperature Mechanical Properties
Microprobe
3.2.1 Introduction
The High Temperature Mechanical Properties Microprobe (HTMPM) is an ultra low load, ultra-high vacuum (UHV), high temperature, depth sensing indentation system. A schematic of the system is shown in Figure (6). It combines the well developed technology found in the current ambient temperature mechanical properties microprobes[3] with the ability to perform similar experiments at temperatures ranging from -100°C to approximately 300°C. All functions of the system with the exception of the temperature control of the indenter and specimen are controlled from a personal computer and a Hewlett Packard (HP) 3852 Data Acquisition and Control Unit.
46 Zygo System ----� CURRENT SOURCE
Power Supply
Power Supply
A ~ Laser Pattern
Figure 6. A schematic of the High Temperature Mechanical Properties
Microprobe. A-Sample; B-Indenter; C-Load Application Coil; D-Indenter Support
Springs; E-Indenter Thermocouple; F-Indenter Temperature Controller; G-Indenter
Heater; H-Sample Thermocouple; I-Sample Temperature Controller; J-Sample Heating
Stage; K-Measurement Mirror;and L-Reference Mirror.
47 3.2.2 Loading System
The load on the indenter is generated via a coil and permanent magnet assembly as in the Nano Indenter® II. The load coil is rigidly attached to the indenter column and resides in the gap of a permanent magnet. The load on the indenter is varied by changing the current through the coil. The voltage (current) is supplied via a I8-bit digital to analog converter (DAC) and a I6-bit offset DAC. Four independent load ranges are available which differ in maximum loading capability and load resolution.
The load resolution of the system is 1.0 JlN. The voltage output of the load card is monitored continuously during experiments, thus giving a constant measure of the load being applied to the indenter.
3.2.3 Displacement Sensing System
The displacement measurement for the HTMPM is achieved through the use of an Axiom 2120™ He-Ne laser interferometric displacement sensing system manufactured by the Zygo Corporation. The complete interferometric system is mounted above the indentation head outside of the vacuum chamber. The beams are passed into the vacuum chamber through a strain-free sapphire viewport that is coated for zero reflectance. Two of the beams from the interferometer strike a mirror that is clamped between the indenter column and the indenter while the other two beams strike a mirror that is rigidly coupled to the specimen. This is shown schematically in Figure
(6 K&L). This configuration allows for a direct measure of the displacement of the indenter into the specimen with very little possibility for other contributions to the displacement measurement. Laser measurement techniques are extremely well suited for this type of application due to the inherent accuracy provided by interferometry and the very high data acquisition rate available from the system.
48 In a typical interferometer, a monochromatic light source is passed through a beam-splitter which transmits half of the beam to a moveable mirror and reflects the remainder 90° to a fixed mirror. The two beams are then recombined and their interference is observed. When one of the mirror is displaced the intensity of the recombined beams will vary as the light waves from the two beams interfere by adding and canceling. The cycle of intensity change of the interference of the recombined beams represents a half-wavelength displacement of travel of the measurement mirror.
If the wavelength of the light is known, the displacement of the measurement mirror can be determined. The accuracy of such a system can be further enhanced by utilizing two frequency laser interfe rometric techniques which are found in the Axiom 2/20 system.
In a two frequency heterodyne system, the laser beam contains two frequency components which differ in frequency by a fixed amount. In the Axiom 2120 system, this is achieved by first polarizing the beam so that only one frequency is present and then passing this single frequency beam through an acousto-optic frequency shifter which transmits one-half of the beam unaltered while the other is diffracted at a small angle and shifted up 20 MHz. One of these frequency components is used as the measurement beam and is reflectedfr om the moving indenter while the other frequency component is used as a reference signal and is reflected back from a stationary mirror that is rigidly coupled to the front of the specimen. These two beams are then recombined and the interference between the 20 MHz beat frequency and the unaltered beam are used to determine the change in displacement. Utilizing a double pass interferometer thus yields a displacement resolution of 1.25 nm. Since the measurement update rate of the laser interferometer is determined by the frequency of the measurement signal (20 MHz), the Axiom 2120 system is capable of producing an updated displacement reading 2 million times per second. Due to
49 hardware limitations, the rate at which data can be acquired from the system is limited and displacement data are normally acquired at a rate of approximately 3 points per second. However, it is possible to conduct dedicated measurements from the laser system for very short periods of time in order to obtain displacement data at exceedingly high rates. By passing data directly into an integer array into the data acquisition control unit's mainframe memory, displacement data can be acquired at rates up to 190,000 data points per second. A similar type of displacement reading into a real array yields a data acquisition rate of 3700 data points per second. These "bursts" of displacement readings can be performed at any time during the experiment, for example following a step-load, without interrupting normal experimental flow and the data retrieved on experiment completion. This very exciting fe ature of the measurement system gives the HTMPM very unique capabilities not found in other mechanical properties microprobes.
3.2.4 Vacuum / Manipulation System
The HTMPM indentation system is completely contained within a DHV manipulation chamber. DHV conditions are achieved through the use of four liquid nitrogen sorption pumps for rough pumping of the system from atmospheric pressure and two ion pumps and one titanium sublimation pump for further pressure reduction.
Typical operating pressure is =10-8 Torr.
The manipulation system of the chamber allows for five independent axes of motion of the specimen stage for laser alignment and specimen positioning. Visual imaging of the specimen is done by rotating the specimen 90° about the x-axis to a long working distance Questar™ telescope located outside the vacuum chamber. Positioning of the specimen is achieved by a calibrated move from the focal point of the telescope to the indenter.
50 3.2.5 Temperature Monitoring and Control
Temperature monitoring in the HTMPM is achieved through the use of two type
K thermocouples, one placed directly on the tip and one on the backside of the specimen as shown in Figure (6). These thermocouples are monitored by two Eurotherm type 801 temperature controllers that output a PID (ProportionallIntegrallDerivative) control signal to two independent programmable power supplies which in tum supply power for tip and specimen heating. Heating is achieved through the use of a tungsten resistance heater located just above the indenter and tantalum resistance heater located in the specimen stage. Temperatures of less than ambient are achieved by flowing liquid nitrogen or another cooling medium through cooling channels in the copper shroud that contains the indentation head and the copper stage which supports the specimen.
51 Chapter 4 Developing a Set of Indentation Creep Constitutive Equations and Defining Indentation Creep Terminology
4. Introduction
As previously discussed in the literature review, the phenomenological description of uniaxial creep at intermediate and elevated temperatures is quite complete. While very distinct differences exist between uniaxial and indentation testing, analogies can be made between the two types of tests in order to aid in describing the phenomenon of indentation creep. It is also necessary to define the terminology that will be used to describe the indentation creep process. These topics are discussed in the following sections.
4.1 Differences Between Uniaxial and Indentation Creep
4.1.1 Introduction
In attempting to compare the mechanical properties of materials measured by differenttechniques, it is important to realize that quite often the measurement technique itself may have a bearing on the measured properties. A few of the important questions are: "Is the test a measure of steady state or transient properties?"; "What is the stress distribution?"; "What volume of material is determining the measured response?";
"What is the geometry of the 'specimen' ?" A brief look at the differences between uniaxialtesting and indentation testing in light of these questions is in order.
52 4.1.2 Steady State versus Transient Behavior
The majority of uniaxial creep data presented and analyzed in the literature deal with steady state creep, i.e., the relationship between stress and strain rate under the conditions where the control variable is held constant and the dependent variable has arrived at a constant value indicative of the imposed state variables. Uniaxial testing typically imposes a constant stress (creep test) or a constant strain rate and then looks for the other to become constant at large strains, i.e., the onset of steady state creep.
In most types of indentation creep tests, the indentation strain rate, and therefore hardness, continuously change during the experiment. For instance, in a constant load creep test the indenter is loaded at a specified rate and the load is then held constant for a period of time while the displacement is monitored. The indentation strain rate changes both during loading due to the imposed loading rate and also in the subsequent hold segment under constant load due to the decreasing pressure (hardness) as the contact area increases. While these changes in the indentation strain rate and hardness may be small, the actual response is being governed by the material properties that are of interest. As the strain rate and hardness are constantly changing, the microstructure is continually evolving. While the deformation under a pyramid shaped indenter has typically been viewed as being geometrically similar from a time-independent point of view, this type of test does not yield geometrical similarity from a rate-dependent point of view. That is to say, as the deformation proceeds under the indenter, the strain-rates experienced by geometrically similar portions of material change as the deformation proceeds. It is therefore desirable to perform an indentation experiment during which the indentation strain rate and therefore hardness remain constant.
53 4.1.3 Homogeneous Stress Distribution versus Radial Stress
Distribution
While the uniaxial test yields a uniform distribution of stress on a macroscopic
scale, the stress available for creep under the indenter is known to scale as some radial
function away from the surface[6]. The strain rate in any given volume of material will
therefore depend on the local state of stress. As continuity must be preserved, it
therefore seems likely that there is some volume of material beneath the indenter that is responsible for controlling the rate of deformation.
4.1.4 Constant Volume versus Expanding Volume
During plastic deformation in a uniaxial test, volume is always conserved[65].
As previously discussed, in a uniaxial test, the microstructure evolves to a steady state during the course of primary creep due to the non-uniform stress distribution on a microscopic scale[14]. The onset of steady state creep is believed to be associated with the point at which the further hardening of the material is balanced by the rate of recovery. Any perturbations in the control parameter will typically result in a transient period during which the microstructure will evolve to a new state representative of the new set of conditions. Given enough time, steady state conditions can again be reached.
The arrival at a new steady state is believed to be representative of a new microstructural
state in the deforming volume.
In an indentation test, the deformed volume of material is continually expanding to encompass previously undeformed material. The material underneath the indenter is very often likened to an expanding cavity with a hydrostatic core where no deformation
is occurring and an expanding elastic/plastic boundary. The creep process is believed to depend upon the rate at which the elastic/plastic boundary can proceed into the material.
54 In contrast to the uniaxial test, the indentation creep process is therefore seen to entail an ever increasing volume of material as deformation proceeds.
4.1.5 User defined Geometry versus Material Defined Geometry
Another central difference between uniaxial testing and indentation testing lies in the geometry of the two tests. In any mechanical tests, forces and displacements are controlled and/or measured depending upon the type of apparatus being used. However, the displacement response alone of a sample to an applied force tells nothing of the properties of the material without incorporatingthe geometry of the specimen.
In uniaxial testing, the geometry of the test is definedby the user. Specimens are typically geometrically simple, e.g., a cylindrical compression specimen or a dog-bone tensile specimen.
In an indentation test, the geometry of the test is actually being controlled by the very properties of the material that are of interest, e.g., the hardness, Young's modulus, or strain rate sensitivity. The test itself is typically designed to examine the properties that are of interest.
4.2 Analogy Between Indentation and Uniaxial Geometries
Figure (7a) is a schematic of a simple compression specimen along with a short list of previously discussed equations to describe the mechanical response. The uniaxial strain rate is definedas some characteristic change in size of the specimen as a function of time, e.g., the displacement rate, divided by a characteristic length scale, e.g., the length of the specimen. The stress that the samplewill support is defined as the applied force divided by the cross-sectional area of the specimen. Finally, the uniaxial strain rate can expressed as a functionof the microstructure, fuCA), the activation energy for
55 p p �
I
T 1
dl e. = -I e'. -I dh U dt = I I h dt u f = H = P;A P;A T(H n Bi = h(J,.)e -% )
(a) (b)
Figure 7. Geometric comparison of uniaxial compression specimen (a) and
indentation creep 'specimen' (b) and equations for describing each.
56 creep, Q, and the stress exponent for creep, n. Figure (7b) is the corresponding geometrical representation for an indentation creep 'specimen.' It is possible to definean indentation strain rate as some characteristic change in size of the specimen as a function of time, e.g., the displacement rate, divided by some characteristic length scale, e.g., the depth of the indentation[50, 51, 53]. The mean stress or pressure that the 'specimen' will support can be defined as the load divided by the contact area between the indenter and the specimen, a quantity typically defined as the hardness of the material. Finally, the indentation strain rate can be expressed as a function of the microstructure, fiO,,) , the activation energy for indentation creep, Qi, and the stress exponent for indentation creep, ni.
4.3 Definition of Hardness as Indentation Stress
The most general definition of hardness is that it is the equivalent of the average pressure under the indenter, calculated as the applied load divided by the projected area of contact between the indenter and the sample[I]. In a conventional microhardness test, hardness is calculated by dividing the applied load by the projected contact area calculated by measuring the diagonals of the hardness impression once the load has been removed. As the scale of indentation testing samples has become increasingly smaller, the need for the ability to obtain hardness data without the need for imaging the indentations has become more prevalent. Depth-sensing indentation techniques can be used to measure the hardness of a material without the need for post-imaging of the indentation[l, 3]. The hardness can be calculated as
= Pmax H (37) A
57 where H is the hardness defined as the mean pressure the material will support under the peak load, Pmax, and A is the projected area of contact calculated from the depth of the
indent (either the total depth or more typically the plastic or contact depth, see [1, 3])
anda precise knowledge of the shape function of the indenter (the function relating the depth of penetration to the projected cross-sectional areaof contact).
In its simplest definition, hardness is typically thought of as describing an athermal strength of a material. However, as the flow stress of a material is a function of strain rate and temperature, the hardness of a material should be expected to vary in an analogous way[lO]. For this work, hardness will be definedas the instantaneous mean pressure that the surface will support as calculated from the load divided by the projected contact area. The contact area is calculated fromthe total depth of the indent and the area function for the perfect Berkovich indenter except for the case of the amorphous aluminafi lm and sapphire where the contact area was calculated based on an elastically corrected contact depth. The accuracy of defining the hardness based on the total depth of the indent rather than an elastically corrected depth is discussed in
Appendix B. The hardness will be considered to be a function of the indentation strain rate and temperature in a manner analogous to the flowstress in a uniaxial creep test.
4.3 "Steady State Indentation Strain Rate and Hardnesstt
The term "steady-state" is seen to appear throughout the review of uniaxial creep in Chapter 2. It was pointed out that steady-state implies that at a given temperature, a balance has been reached between hardening and recovery effects such that for a given applied stress or strain rate, the resulting strain rate or stress is constant. Steady-state is also typically thought to imply a constant microstructural state in the material.
58 In the literature review of indentation creep using depth-sensing indentation techniques it was pointed out that a possible problem associated with the currentstate of indentation creep testing with sharp indenters is that in none of the currently used test techniques is the indentation strain rate, iz/h, or hardness constant. While it has been pointed out by a number of authors that phenomenological descriptions developed for constant stress can be used since the hardness is changing very slowly, this fact has never been born out experimentally due to the lack of a test technique with a geometrically similar indenter capable of performing a constant strain rate or constant stress experiment.
59 Chapter 5 Indentation Creep - Materials/Sample Preparation
5.1 Lead-65 at % Indium
Pb-65 at% In has a melting point of 468 K (195°C). Its most common use is as a corrosion resistant plating material and solder for glass and silicate materials[66]. The
alloy was chosen for study due to its low melting point and well characterized creep characteristics[66] .
The Pb-65 at% In alloy used for both indentation and compression testing was prepared from commercially obtained Pb and In of greater than 99.999% purity. The indentation specimens were prepared by rolling the homogenized alloy to approximately
0.75 mm and then annealing for 1.5 hours at 170° C. The resulting grain size was
approximately 1 mm. The compression specimens were prepared by swaging an as-cast
3/8" rod to 1/4", homogenizing for 24 hours at 1700 C, swaging to 118" and then machining to approximately a 3:2 height to diameter ratio. The machined specimens were then annealed 40 minutes at 100° C; the resulting average grain size was 50 ).Lm.
The surface was prepared for indentation testing by microtome polishing at room temperature with a diamond knife.
5.2 Indium
Pure In has a melting point of 429.8 K (156.6°C). Because of its bright color, light reflectance and corrosion resistance, it is a valued plating metal, especially for reflectors[67] . Its creep properties are also well known over a wide range of homologous temperature[39].
60 The indium specimens used for all indentation tests were prepared from commercially available In of greater than 99.999% purity. Specimens were cast using
standard techniques to a thickness of 0.3885". The as cast material was then rolled to a final thickness of 0.0825" before being cut to the final dimensions and subsequently annealed at lOO°C for 1 hour. The surface was prepared for indentation by electropolishing in a 3: 1 solution of methanol and nitric acid maintained at 20°C. This process resulted in a mirror finish surface with a final grain size of approximately 2-3 rnrn.
5.3 Aluminum
Pure AI has a melting point of 933.5 K (660.4°C). It is perhaps the most widely studied of all materials in terms of its creep properties having been investigated by numerous authors. It was chosen for this study for this reason as well as its interesting creep behavior in the intermediate temperature range accessible with the HTMPM.
The aluminum specimens used for all of the indentation experiments were prepared from commercially available AI of greater than 99.999% purity. Specimens were cast using standard vacuum-arc techniques to a thickness of 0.2885", The as cast material was then rolled to a final thickness of 0.0905" before being cut to the final dimensions and subsequently annealed at 350°C for 1 hour. The surface was prepared for indentation using the method outlined in Oliver and Pharr[3]. Specifically, the specimen was ground with successively finergrits of SiC abrasive taking care to remove the damage layer at each step. The specimen was then polished for 8-12 hours on a vibratory polisher using 3 J.lm diamond paste and water on a TEXMETTM cloth followed by 8-12 hours using 0.5J.lmdiamond paste and water on a MASTERTEXTM cloth. Final
61 polishing was performed using colloidal silica and a MASTERTEXTM cloth for 15-20 minutes.
5.4 A1zD3
Single crystal Ah03 has a melting temperature of 2030°C (2303K). Alumina ceramics are the most widely used oxide-type ceramic, chiefly because alumina is plentiful, relatively low in cost, and equal to or better than most oxides in mechanical properties[67]. The amorphous alumina film and sapphire were both chosen as examples of materials whose time dependent properties could only be characterized
using indentation creep techniques.
The amorphous alumina used for the indentation study was a 1.9 Jlm thick electron beam evaporated filmdeposited on a single crystal Al203 (sapphire) substrate having its c-axis normal to the surface. Indentations were made in both the filmand the bare substrate in order to compare the time dependent properties of the two materials.
62 Chapter 6 Indentation Creep - Experimental Techniques and Details
6. Introduction
A variety of experimental load-time histories are used at various times throughout this study in an attempt to determine the most appropriate method for measuring indentation creep properties. The experiment types are presented essentially in chronological order. The reasons for the transition from one experiment type to another will become clear as the results obtained from each are presented. The presentation of the data for any given material on which a number of different types of experiments have been performed will follow this chronological order unless for the sake of continuity it becomes cumbersome to do so. All of the load-time histories used in this work will be presented in this chapter and referred to according to the experiment type located in parentheses after each section heading.
Two other topics important to the results are the indenter geometry and the problem of thermal drift in the experiment. These issues are discussed subsequent to the presentation of the load time histories.
6.1 Load Time Histories
6.1.1 Step LoadIHold (Step/Hold)
Step loads are achieved in the Nano Indenter® by multiplying the magnitude of the step load required, �P, by the reciprocal of the load calibration (e.g., V/JlN) to determine the magnitude of the current change required in the load coil. The appropriate
63 information is then written to the digital to analog converter controlling the actual load current.
A load-time history of a typical step load experiment utilized in this study is plotted in Figure (8). In this experiment, the indenter was step loaded from the point of contact to the desired maximum load after which the load was held constant and the displacement monitored as a function of time. This type of loading enabled very high displacement rates to be achievedduring the firstportion of the constant load segment.
Data was initially acquired at a rate of 3 data points per second during the initial rapid descent of the indenter. The data acquisition rate was slowed to approximately one data point every five seconds after the first fifteen seconds of the experiment. The hold segment was continued until a specified minimum displacement rate was reached. The indenter was then step unloaded to a load of 20 JlN, held at constant load, and then ramp reloaded and ramp unloaded to obtain the elastic and plastic properties of the material.
This experiment was carried out at room temperature on three materials: the Pb-65 at%
In alloy, the 1.9 Jlmamorphous Al203 film andits sapphire substrate.
6.1.2 %Step/Hold
A variation of the step load experiment was utilized in the HTMPM to take advantage of the high data acquisition rates in determining the effect of the percentage of the maximum load that was applied in a step fashion. Figure (9) is a plot of the load versus time history used for this experiment. All of the experiments were conducted to a maximum load of 10 mN. As shown in Figure (9), the load was applied by first ramping the load to a percentage of Pmax in 10 seconds and then immediately applying the remainder of the load in a step fashion. The load was then held constant for a period of time such that the displacement versus time response under constant load could be
64 Constant Load 80
..- 60 z Ramp Load -E "C ca 0 Step ..J 40 Unload
20 Hold @ Constant Load
OULJ...L.L..L.L.u...JL...W.r.I...... L.L ...L.L.:LLJ o 50 100 150 200 250 Time (8)
Fi&ure 8. A plot of the load time history for step load experiments conducted on
Pb-65 at% In, amorphous Ab03, and sapphire.
65 10
8 Percent of - P Ramped On z max g 6 -- 1 00/0 "'0 as o --- - - 200/0 ....J 4
•••••••- 70%
2
�=o for Hold Segments Following Step Load
o ������ ..���� .. � o 5 10 15 20 25 30 Time (8)
Figure 9. A plot of the load time history for the study of step loading effects on In. Following a load ramp to 10, 20, 30, and 70% percent of the maximum load, the remaining percentage of Pmax was applied in a step fashion.
66 monitored. Experiments were conducted in which 90, 80, 70, and 30% of the maximum load was applied in a step fashion. Note the definition of t=O that will be used for the displacement versus time response to this experiment type. These experiments were carried out on pure In at room temperature in the HTMPM.
6.1.3 Constant Rate of LoadingIHold (CRLlHold)
The CRUHold type of experiment was first utilized by Mayo et al[55] in their study of nanophase Ti02. This type of experiment utilizes the ability of the Nano Indenter® and the HTMPM to incrementally apply the load on the indenter at a rate specified by theuse r.
Figure (10) is a plot of the load time history for a variety of constant rate of loading experiments. As shown in the figure, the load on the indenter was ramped to a specified maximum at a constant loading rate and then held constant for a period of time.
Elevated temperature experiments on In and Al were conducted using a 1 second ramp to maximum load, 10 mN for In and 100 mN for Al respectively. The load ramp was immediately followed by a hold period under constant load to monitor the displacement versus time (indentation creep). Typical hold time under constant load was 180 seconds. Room temperature experiments were conducted on In at four different loading rates; 0. 1, 0.3, 1 , and 10 mN/s to a maximum load of 10 mN. This type of experiment allowed the indentation strain rate and hardness to be determined during a segment when both were expected to be changing.
67 15
• P,(mN/s) --0-10 1 0.3 -f:r-O.1 10
- z
-E "'0 ctS 0 --.J
5
o ��--��--� o 20 40 60 80 100 Time (8) Figure 10. A plot of load versus time for the constant rate of loading experiments. A 1 second ramp (10 roNls) was used in the elevated temperature experiments on In and AI. The remaining ramp rates were used in the study of the effects of loading rate on the indentation strain rate and hardness of In at room temperature. The maximum applied load for all experiments was 10 roN.
68 6.1.4 Constant pip
6.1.4.1 Development of a Constant "Indentation Strain Rate" Experiment
The technique for conducting constant indentation strain rate experimentscan be developed from the equation for the hardness of a material. Recall from equation (37),
p P H=-= (38) A ch2
where H is the hardness, P is the load, A is the projected contact area, h is the contact depth, and c is a constant that depends upon the geometry of the indenter (24.56 for the perfect Berkovich geometry). Equation (38) can be rewritten as
(39)
Differentiating equation (39) with respect to time leads to
2 2chhH + ch iI== P (40) which can be simplified as follows:
2 2chhH = P - ch iI (41)
2 . P ch iI h= - -- (42) 2chH 2chH
69 h P ch2if = - (43) h 2ch2H 2ch2H
Substituting in from equation (38) yields
h p if -h = -2P --2H (44)
or
(45)
where ei is the indentation strain rate. Equation (45) suggests an indentation experiment perfonned with a pyramid shaped indenter during which the loading rate is controlled so that the loading rate divided by the load is constant can result in a constant value of the indentation strain rate if a steady state value of the hardness can be reached and if= O. This type of test is very appealing as the loading rate is a directly ® controllable parameter in the N ano Indenter rr.
6.1.4.2 Experimental
To test the hypothesis that controlling the loading rate in such a manner as to maintain Pip constant would result in a constant indentation strain rate, a series of experiments at a variety of Pip values were conducted on pure In at room temperature. Figure (1 1) is a plot of the load time histories used for these experiments. As shown in the figure, the loading rate was controlledin such a way to maintain the instantaneous
70 15
O��.Iriiii:.I...... a-""""""""'a.....&.. ""'" a 200 400 600 800 1000 Time (8)
Figure 11. A plot of load versus time for the experiments, conducted on In, during which Pip was held constant. Four constant Pip experiments were conducted to a maximum load of 10 mN to determine if a constant Pip loading scheme would result in a constant indentation strain rate.
71 value of Pip constant up to a maximum load of 10 mN. The load was then held constant for a period of time and the displacement versus time was monitored. Values of P'1 P of 0.2, 0.1, 0.02, 0.01 and 0.005 s-1 were used.
6.1.5 it/p Change In order to investigate the response of pure In to an abrupt change in the indentation strain rate, a series of experiments were conducted in which the value of Pip was changed during the experiment at some specified depth. This type of experiment allowed the investigation of the existence of a path independent indentation steady state. Figure (12) is a plot of the control variable, Pip, during these experiments. Experiments wereconducted in which Pip was abruptly changed from a value of 0.02 to 0.1 as well as from 0. 1 to 0.02 at a depth of 3000 nm.
6.2 Indenter Geometry
All of the experiments conducted in this work were performedusing a Berkovich indenter; a three sided pyramidal indenter with the same depth to arearatio as a Vickers indenter. The projected area of contact for a perfect Berkovich geometry is given by
A = 24. 56h; (46)
where he is the contact depth of the indentation. The Berkovich geometry is typically chosen over a Vickers indenter as it is much easier to produce near perfect tips due to the natural intersection of three planes at a point. Unless otherwise specified, the area function used for calculating the hardness will be that of equation (46).
72 Indium
0.1
. • �,(S-l) 0.1 - 0.02 �,(S-l) • > 0.02 0.1 -0- ->
0.01 2000 3000 4000 5000 Displacement (nm)
Figure 12. A plot of Pip versus displacement for Pip change experiments conducted on In. The initial value of Pip was maintained up to a displacement of 3000 nm at which time Pip was either increased or decreased according to the figure. The new value of Pip was maintained up to a depth of 5000 nm.
73 Associated with a given geometry of indenter is the concept of a characteristic or effective strain[68]. An empiricalrelation was fo und by Tabor[68] which suggested that the hardness of a material as measured with a Vickers indenter was approximately equal to the flow stress of the material at a strain of 8-10%, i.e., the deformation due to the indentation itself produces an additional strain of 8-10%. Tabor proposed an effective strain of 0.2 cot� for a cone or pyramid with an included angle of 2�. The concept of a characteristic strain will be discussed further in comparing the indentation creep results to the results for uniaxial testing of the same material.
6.3 Thermal Drift
Three types of displacements are measured by a depth-sensing indentation system: elastic displacements of the sample surface and the load frame, plastic displacements into the sample, and displacements due to experimental (typically thermal) drift[69]. In order to interpret the data, it is necessary to accurately separate the three components of displacement, or perform the experiments in such a way that certain components of the displacement can be neglected. The elastic displacement of the sample surface is an issue that will be dealt with subsequently, it is the issue of thermal drift that will be addressed here. As the issue pertains differently to the Nano Indenter®
II and the HTMPM, the two instruments will be dealt with separately.
6.3.1 Nano Indenter® II Experimental drift in the Nano Indenter® II is caused by thermal expansions which are detected by the capacitance gauge. Two sources of drift are typically identified: (I) thermal drift that occurs due to fluctuations in the temperature of the laboratory, and (2) thermal drift that occurs due to Joule heating in the load application
74 coil(69]. Specific steps were taken to reducethe effects of these two possible sources of error.
First, the Nano Indenter® is isolated in a laboratory where the temperature is controlled to within ±O.5°Cat all times. In addition to the tight temperature control of the environment, the instrument is located within another isolation cabinet and has a very large thermal mass preventing rapid changes in the temperature of the system.
Finally, the specimens were typically placed within the laboratory for at least 24 hours before the experimental run to ensure thermal stability.
The effects of Joule heating were dealt with in two ways. In the step loadlhold experiments conducted on Pb - 65 at% In, amorphous A1203, and sapphire, which were performed in an earlier version of the Nano Indenter®, the magnitude of the step load applied was such that the current through the load coil under load was exactly equal to and opposite in sign of the current required to support the weight of the indenter (it should be pointed out for clarity that the typical mass of the indenter is around 7 grams, therefore the forces generated in the load application coil are typically supporting a large fraction of the weight of the indenter rather than actually pushing on the indenter). This minimized the effects of differential thermal expansion due to changes in the power dissipated in the load coil. The remaining room temperature experiments were conducted in a version of the Nano Indenter® with an Invar indenter tube. This material which is known for its extremely low coefficient of thermal expansion around room temperature is expected to minimize the effects of heating due to the load application coil. With these steps to minimize the effects of experimental drifttaken, the effects of thermal drifton the experimental results from the Nano Indenter® II were assumed to be negligible and no correction for this contribution to the displacement was made.
75 6.3.2 HTMPM
As previously discussed, the design of the displacement sensing system of the
HTMPM is very different than that of the Nano Indenter®. The interferometric displacement measurement is made strictly on a differential basis with a measurement mirror mounted directly above the indenter and a reference mirror rigidly coupled to the upper surface of the sample. This configuration leaves little material to contribute to experimental drift due to thermal expansion or contraction of components of the indentation system. As the temperature of the indenter tip and sample were directly controlled by independent systems throughout the experiment, no effects due to changes in the environment were expected. These two facts also led to the effects of thermal drift on the displacement measurement in the HTMPM to be neglected.
One note, as will be pointed out in the results and discussion section, a long term thermal stability problem was encountered at temperatures of less than RT in the
HTMPM. These problems did not stem from the heating system, rather the cooling system. In order to achieve temperatures of less than RT, liquid nitrogen was passed through cooling channels in the indentation head and specimen stage. Due to the violent boiling of the liquid nitrogen in the cooling channels, the supply had to be valved off during an experiment to reduce vibrational noise to an acceptable level. This resulted in the thermal expansion of an elastic hinge that was responsible for one axis of alignment of the laser measurement system. The problem was not directly one of a thermal instability in the displacement measurement, rather one of system stability as the temperature of vital components changed. This problem was avoided at temperatures greater than room temperature by using nitrogen gas for cooling rather than liquid nitrogen.
76 Chapter 7 Indentation Creep - Results and Discussion
7.1 Effects of Stress
7.1.1 Introduction
The central theme of this section of the dissertation is to investigate fully the relationship between indentation strain rate and hardness. Data from a variety of experimental techniques as well as from a variety of materials will be presented. The results obtained from the different experimental techniques will be compared and evaluated.
7.1.2 Results and Discussion
7.1.2.1 Pb-65 at% In
Figure (13) is a typical plot of displacement versus time obtained for the Pb-65 at% In alloy during the hold segment under constant load using the Step LoadIHold load time history. The displacement rate during the hold period was typically observed to vary from approximately 70 nmls during the initial stages of the hold period under constant load to approximately 0.1 nmls at the end of the 1 hour hold under constant load. By differentiating the displacement versus time data, the displacement rate of the indenter, it,can be obtained continuously during the constant load hold period. The indentation strain rate can thenbe obtained by dividing the instantaneous descent rate of the indenter by the displacement at that point in time, it/h. By defining hardness as the load divided by the instantaneous contact area as previously outlined, plots of
77 (dh/dt) 0.08 nm/s = --" 5300
5200
- § 5100 --
+-'C Q) 5000 Q)E () Ctj � 4900 o 4800
(dh/dt) 67 nm/s 4700 Pb - 65 at % In 4600 o 1000 2000 3000 4000 Time (s) Figure 13. A typical plot of displacement versus time during 3600 second hold period under constant load following an 80 mN step load for Pb-65 at% In. The displacement rate of the indenter was observed to vary nearly four orders of magnitude during the hold period under constant load. 78 indentation strain rate versus hardness analogous to strain-rate versus stress curves obtained using conventional bulk testing techniques can be obtained. Figure (14) shows the results of the analysis of the indentation creep properties of the Pb-65 at% In alloy. The importance of obtaining all of the data from a single indentation is evident fromthe fact that indent-to-indent variations in the hardness of the material have little effect on the overall shape of the curve, just its placement on the hardness axis. The slope of this plot of log hardness versus log indentation strain rate yields the stress exponent for indentation creep. A stress exponent of 90 was obtained at the high strain-rate portion of the curve with the stress exponent approaching 6 as the strain rate decreased. 7.1.2.2 Indium % Step/Hold Results Figure (15) is a semi-log plot of the displacement versus time response of the indenter obtained for indium for the %Step/Hold loading conditions to a maximum load of 10 mN. The upper four curves show the displacement response to an applied step load with varying amounts of the total load being applied during the ramp prior to the step to maximum load. The percentage of Pmax stepped on at t=O is indicated for each curve. There are several important fe atures to note from these four curves. First, note the wide range of descent rates experienced by the indenter as indicated by the span of eight orders of magnitude on the time axis. This reveals one of the useful aspects of the high data acquisition rates available with the interferometric displacement sensing technique being used in the HTMPM. Secondly, the depth achieved immediately following the step load is largest for the largest percentage Pmax stepped on. This is believed to be due to the large inertial or 79 - ..- I en - CD 3 ...... 10. ctS a:: c .- ctS ...... 10... en c 0 ...... ctS ...... 4 C 10- CD "'C C Pb - 65 at % In 10.5 0.1 0.12 0.14 0.16 0.18 0.2 Hardness (GPa) Figure 14. A log-log plot of indentation strain rate versus hardness for step load experiments conducted on Pb-65at% In. The slope of the line is the stress exponent for creep. A stress exponent or approximately 6 was observed at the lower strain rates. 80 Indium, P =10mN max 0/0 of P Ramped 4000 max on During Pre-load I - A cE 700/0 �+--;?""'7;?; -- ...... 3000 c Q) EQ) Cd() a. 2000 en o 1000 1s Ramp to P max O Io..- 6 ...... -.a..-...... - ...... 2 - ----...... ° ....----oIIo- 2 10- 10-4 10- 10 10 Time (s) Figure 15. A semi-log plot of displacement versus time for In showing the response of the material to a variety of step loads. The oscillatory response to large step loads is enclosed in "A" and is enlarged in Figure 16. 81 dynamic forces that are being experienced by the specimen as are evident from the oscillatory response of the indenter following the applied step as shown in "A" in Figure (15) and enlarged in Figure (16). Finally, immediately following the step load, an incubation period is observed where the displacement rate of the indenter is observed to be smallest for the largest step-fraction and the four curves are seen to reverse their order in terms of overall displacement. Since under normal data acquisition conditions the first data point on these curves is obtained at approximately 1 second, the region on the time axis where these four curves intersect is not normally observed. This initial reduced creep rate is believed to be due to (1) the stronger structure resulting from the higher strain rate experienced by the material when a larger fraction of the maximum load is applied in a step fashion, and; (2) the larger contact area and therefore the lower stress that results from the dynamic overload. Finally, afterthe initial incubation period, the displacement rates for the four curves become similar. While not directly related to the creep properties of the material, the oscillatory response to the dynamic overload warrants discussion. Figure (16) is an enlargement of the area marked in flAil in Figure (15). It is a plot of displacement versus time showing the oscillatory response of the indenter as it comes to rest during the 80 ms immediately following an applied step load of 9 mN (90% of Pmax). Inertia calculations using the 2 2 maximum acceleration, d xl dt , experienced by the indenter reveal that a dynamic overload of as much as 40% of the applied load can be experienced by the specimen. This phenomenon is seen to decrease as the step-fraction is reduced and is completely absent when the step-fraction is reduced below 30% as shown in Figure (15). A number of interesting observations can be made about the observed oscillation following the step-load. The natural frequency of the oscillation, which is determined 82 3170 1\ i, {\ I : \ \ ," - i i \ 1('\' ,"\ i l I j \ E , c ;. !\\i\/\I \, \ t"j --- \'" " � I �/ � 3165 ! ; ; v ...... V C ! V CD , E V CD ! (.) � - 3 160 c.. .-en o ro =21t{975c/s }=6126.1/s n t: � j S=mro 2 =2.82x1 05 N/m 3155 \J n 3150 � 0.002 0.004 0.006 0.008 0.01 Time (s) Fi�ure 16. A plot of displacement versus time showing the oscillatory response of the indenter during the 80 ms immediately following a step load of 9 mN on In. The frequency of the oscillation is directly related to the stiffness of the contact. The exponential decay of the oscillation is representative of the losses in the material. 83 predominantly by the stiffness of the contact, S, and the mass of the indenter, m, is observed to be 975 cyclesls (6126 rad/s). Using an equation for a simple harmonic oscillator, = ..J together with the known mass of the indenter yields a contact OJn Slm , stiffness of 2.82x 1 05 N/m. If the contact area of the indentation is calculated from the mean displacement of the oscillation, the storage modulus of the material can be calculated from[70] E= -JH � 2{A (47) where S is the stiffness of the contact. This analysis results in a modulus of 13.8± 0.2 GPa, a bit higher than the literature value of 12.7 GPa. This slightly higher value of the modulus is reasonable as the high frequency or unrelaxed modulus of a material is expected to be somewhat higher than a low frequency or relaxed modulus. It is also possible from the exponential decay of the oscillation to calculate an effective damping coefficient of 7 N s 1m. After carefulcharacterization of the dynamics of the indentation system as very accurately detailed in Appendix A, it was determined that this number most likely represents the damping or energy being absorbed by the sample material in the vicinity of the indentation and is not energy absorbed elsewhere in the system. Using the fact that the loss modulus is known to also scale as the square root of the contact area (see Chapter 12), it is possible to calculate a loss modulus according to (48) This calculation results in a loss modulus of 2.4 GPa. This value is seen to be roughly 20% of the storage modulus. 84 CRLlHold Results Room Temperature Figure (17) is a plot of the indentation strain rate versus displacement obtained using the CRLlHold loading schemes. The strain rates were calculated by taking the time derivative of the displacement during both the load rampand during the subsequent hold period under constant load and dividing the instantaneous rates by the displacement at that point in time. The indentation strain rate is observed to decrease continuously through the constant loading rate loading period and, even more rapidly, during the constant load hold period. Figure (18) is a plot of the calculated hardness versus displacement over the same displacement range. (The accuracy of the calculated hardness is discussed in Appendix B.) The hardness is observed to decrease as the experiment proceeds and the indentation strain rate decreases. The change in the hardness is gradual during the loading segment when the indentation strain rate is not changing rapidly and then becomes much more pronounced once the loading stops and the strain rate drops rapidly. Figure (19) is plot of the indentation strain rate versus the hardness from the CRLlHoldexperiment s. Each curve represents an average of five indentation performed under each loading condition. It is observed that the data immediately following a constant loading rate segment undergo a transient period immediately upon completion of the load ramp. These data are then observed to converge to a common curve over a period of time which is similar to the loading time. This transition region which appears to have a higher stress exponent is believed to be due to the transient microstructural evolution occurring immediately after the completion of the load ramp and is not 85 101 . Indium P,(mN/s) -0-10 - T""" -0-1 I en - 0.3 Q) 10° -0- ...... ct'S -----fr- O. 1 II: c Oct'S ...... I- CJ) c 0 1 -:;:;ct'S 10- ...... c Q) -c c 2000 3000 4000 5000 Displacement (nm) Fi�ure 17. A plot of indentation strain rate versus displacement for CRLfHold experiments conducted on In. The indentation strain rate during a constant loading rate segment is observed to decrease as a functionof displacement. 86 0.04 • P,(mN/s) 0.035 1 0 -<>- 1 - as 0.. 0.3 (!J 0.03 - en ----fr- 0.1 en Q) c "'C '- � 0.025 0.02 Indium 0.015 ...... &..&..,jL..L.&..&..L.L ...... �L..L.I. � 2000 3000 4000 5000 Displacement (nm) Figure 18. A plot of hardness versus displacement for CRLlHoid experiments conducted on In. The hardness of a rate sensitive material is seen to decrease during a constant loading rate segment due to the decreasing indentation strain rate. 87 0 10 �------�------p---� Indium - ,- , . en '-'" P,(mN/s) D 1 0 <> 1 o 0.3 6 0.01 6.1 1 4 10------0.01 0.02 0.03 0.04 Average Hardness (GPa) Figure 19. A log-log plot of the average indentation strain rate versus the average hardness for series of constant loading ratelhold experiments on In. The material is observed to undergo a transient response immediately following the completion of the load ramp. The data are then seen to relax to single curve. A stress exponent of 6.1 is obtained from the lower portion of the curve. 88 believed to be associated with power law breakdown. This point will be further discussed in light of the results from the constant Pip analysis. As previously pointed out, the indentation strain rate-hardness data during the hold segment converge to a single curve afteran initial transient period. A power law fit to the linear portion of the data in this region yields an average slope of 6.1 which is the stress exponent for indentation creep in the material. This value is in good agreement with the constant stress tensile data of Weertman[39]. Elevated Temperature Figure (20) is a plot of the displacement versus time results for indium obtained during the constant load hold period immediately following the load ramp as a function of the test temperature. All of the data were acquired using the CRLlHold technique with a 1 sec ramp to 10 mN. The data acquired for the temperatures less than room temperature are terminated after 40 s due to longer term system stability problems at these low temperatures. Some important features to note from these curves are 1.) the displacement achieved by the indenter is less as the temperature is reduced indicating the material is capable of supporting greater stresses and 2.) the sharpnessof the breakin the curves at temperatures less than room temperature suggests a very high stress exponent, i.e., the stress is becoming nearly constant very quickly. Figure (21) is a log-log plot of indentation strain rate versus hardness for experiments conducted at room temperature (28 °C), 50 °C and 75 °C. The lower temperature data is not included due to the previously mentioned experimental stability problems. 89 Indium 7000 6000 - E c --- C 5000 Q) E Q) � 4000 - c. .-CI) Cl 3000 2000 1000 � o 50 100 150 200 Time (5) Fiiure 20. A plot of displacement versus time during the hold period under constant load following a 1 second ramp to 10 mN for In at temperatures ranging from 90 102 28°C. , Indent 28°C, Indent 50°C, Indent � 1 !:::. 10 50°C, Indent 75°C, Indent ",-.... 75 • °C, Indent ,...I (f) ---Q) ° 10 Termination of +-'CO Load Ramp 0: 0 • s:::: 0 CO 1 1 • � 10- en+-' s:::: 0 � 2 5 +-'s:::: 10- Q) "'C s:::: 1 0-3 Indium 10-4 10-2 10-1 Hardness (GPa) Figure 21. A log-log plot of indentation strain rate versus hardness for indentation creep experiments on In at temperatures of 28, 50, and 75°C. The data are observed to undergo a transient period immediately fo llowing the completion of the load ramp yielding a higher apparent stress exponent for creep. 91 All of the data exhibit similar trends again showing a transient region in progressing from the load ramp to the hold period under constant load. An interesting note is that the length of the transientperiod seemsto be reduced as the temperature is raised perhaps due to the greater thermal energy available for microstructural rearrangement. All of the data are seen to be linearnear the end of the hold period with a stress exponent of approximately 5 at the higher temperatures. Constant P/p Figure (22) is a plot of the indentation strain rates versus displacement obtained using the constant Pip loading schemes as well as the data from the 1 mN/s CRLlHold experiment for comparison. The strain rates were again calculated by taking the time "------...,,.------��� derivative of the displacement, both during the loading segment and during the " �"",-"" '-"'--"''' -'''' '-- "'' .>'<-'''''-----", ,,.,•• _, ...,_� ,,,,,--r;-,�.•. ''''''''''''''''''''''' ''',-. _'''''_''''''i.. .'�_r_"rl'.t"",''�-,1'__ __ ''�'''_,,'h< subsequent hold segment, and dividing the instantaneous rates by the displacement at .. ,' .-" � that t:?int in !ime. As suggested by equation (45), the indentation strain rates are ,-�"." (,,, ," constantduring the loading segment and equal to 0.5 Pip suggesting that an indentation steady state has in fact been reached and if= o. The indentation strain rate is again observed to decrease once the load ramp terminates. Figure (23) is a plot of the calculated hardness versus displacement for the constant Pip experiments as well as for the 1 mN/s CRLlHold experiment again fo r comparative purposes. The calculated hardness is observed to remain constant during the constant Pip loading segment and to systematically decrease as Pip , and thus izlh, is decreased. Since for each Pip experiment a constant izlh and H is achieved, it is possible to tabulate "indentation steady state" strain rate-stress pairs analogous to uniaxial data. Figure (24) is a log-log plot of indentation strain rate versus hardness for the data from 92 . . ) P,(mNI s) �,(S-1 • 1.0 0.2 .... - ,,-- 0.1 '00 - -0- 0.02 Q) Cti 0.1 a: c ro � en+-' c Indium o . - Cti C 0.0 Q) 5 "C C o 2000 3000 4000 5000 Displacement (nm) Figure 22. A plot of indentation strain rate versus displacement for three different constant Pip experimentsconducted on In showing constant indentation strain rates equal to 0.5 Pip. Data from the constant P=l mN/s experiment are included for comparison. 93 0.03 . P,. (mN/s) �,(S-1) • 1.0 0.2 -0- --0-- 0.1 0.02 --0- 0.0 1 ---t:r-- 0.025 ill 0.005 ..-. ctS c.. -{!J en en (]) c "C Indium :I.- ctS I 0.02 3000 4000 5000 Displacement (nm) Figure 23. A plot of hardness versus displacement for five diffe rent constant Pip experiments conducted on In showing indentation steady state behavior in response to constant Pip load ramp. Data from constant P=lmN/s experiment are included for comparison. 94 - ,....I en -(]) +-'ro a: c ro � en+-' c 0 10-2 +-'ro +-'c (]) '"0 C (]) 0') ro (])� > « · Indium, Constant -3 10 �------��----�--�� 0.01 0.02 0.03 0.04 Average Hardness (GPa) Fi�ure 24. A log-log plot of average indentation strain rate versus average hardness for constant Pip experiments conducted on In. A constant power law relationship is observed over the entire range of indentation strain rates investigated. 95 the constant Pip experiments. The data were generated by averaging the constant hardness/constant izlh data between 2000 nm and the termination of the loading segment. Each data point represents the average of five experiments at each Pip . A power law fitto the data yields a slope of 7.3 (R=0.995) which is the stress exponent for creep in the material. Over the range of strain rates investigated the power law relationship was observed to remain constant. The stress exponent for indentation creep of 7.3 is very near the value of 7.6 obtained for indium tested uniaxially by Weertman[39] near room temperature. Hardness versus Time (Sargent and Ashby Analysis) As discussed in the literature review, one of the earliest techniques fo r gaining information into the stress exponent for creep from an indentation test was to monitor the optically measured hardness as a function of dwell time under load, the slope of a plot of log hardness versus log time being equal to -(lIn) . This same analysis can be performed with a depth-sensing indentation system. The centra!diff erenc�Jsthat r�ther than removing the load to optically measure the size of the indentation, the hardness can be calculated continuously as a function of time since the load on the indenter is known and the contact area of the indentation can be obtained from the depth of the indentation and the functional relationship between the contact area and the depth. Figure (25) is a log-log plot of hardness versus time showing typical curves obtained for In at room temperature for three different loading times that most closely approximate the probable loading times used in a rnicrohardness system. These data are plotted along with the data for In from Atkins et alL 10] obtained for a Vickers indenter and conical indenters with apical angles of 105, 120, 136, and 150°. 96 I 1 sec loading I _ •••• 10sec loading \\ ········· I· 30 loading " . sec "\". --0- Atkins et. al. Cone :\ • Atkins et. al. Vickers -':rc.. -";.�'-. I .e • .-.... • co • c.. I. (!J - en en -2 Q.) 10 c:: '"'C CO"- :c Indium 3 1...-_ __ "'-_...... _ ...... 10- ...... _---1. 1 ° 1 2 3 4 1 0- 10 10 10 10 10 Time (5) Figure 25. A log-log plot of hardness versus time under load for In. The data shown are from Atkins et al[lO] and from CRLlHoldexperiments to 10 mN from this study. The slope of the line is -lin. Source: A. G. Atkins, A. Silverio, and D. Tabor, J. lnst. Metals, 94, 369 (1966). 97 According to Atkins et aI, this analysis should result in a linear relationship, between log hardness and log time. The data from their work is seen to be linear with slopes of -0.095 and -0.086 for the data from the conical indenters and the Vickers indenter respectively. These slopes result in stress exponents of 10.5 for the conical indenters and 11.6 for the Vickers indenter as reported by Atkins et al in the cited work. Note that due to some ambiguity in the paper by Atkins et al, some question as to the absolute magnitude of the hardness values shown in this plot exists. This confusion does not effect the stress exponents obtained from the data, rather the placement of the data on the hardness axis. Comparison of the data from the current study to that of Atkins et al raises three points. First, as observed by Atkins et aI, the data for short loading times , e.g., 1 second, deviates significantly from linearity. This deviation is observed not only during the initial portion of...!ht'? data where a transient state probably exists, but in the latter stages of the data as well for reasons that are not completely understood. Second, neglecting the data from the 1 second loadingexperiment, the slope of the log hardness- log time data does seem to become constant once the initial transient period on going from the load ramp to the hold period is surpassed. Finally, a lower stress exponent is obtained for the data from this study than was obtained by Atkins et al. The average slope of -0.17 results in a stress exponent of 5.9 as compared to values of roughly twice that from the data of Atkins et al. The value of 5.9 obtained from this study agrees well with both the stress exponent obtained from the log indentation strain rate-log hardness data and the data from Weertman[39]. One of the potential problems of the technique of Atkins et al is that it requires a measure of an absolute variable, the time, rather than being based on a differential variable such as the indentation strain rate. This is also a problem from a fundamental 98 standpoint in that the hardness as a function of time has no physical meaning, it is simply a representation of the data based on the fact that the strain rate is decreasing as a functionof time under load. An excellent example where this can introduce problems is observed in comparing the data from a number of experiments conducted with a 10 second loading time. Figure (26) is a log-log plot of hardness versus time for five experiments conducted with a 10 second loading ramp. The scatter in the initial portion of these data is obvious. A log-log plot of indentation strainrate versus hardness for the same data does not however show the same scatter as shown in Figure (27). Rather, it is seen that the data all lie very nicely on a single curve even during the initial portion of the loading segment. Strain Rate versus Time One final issue that warrants discussion at this time is the analysis by Sherby[12] suggesting that the average indentation strain rate scales with the characteristic strain of the indenter divided by the time under load. Figure (28) is a plot of the indentation strain rate, hlh , versus time for the 1, 10 and 30 second load ramp data on indium at room temperature plotted along with the indentation strainrate predicted by dividing the characteristic strain of a Berkovich indenter (0.08) by the time of the experiment. Once the transient period on proceeding from the load ramp to the constant load hold period is surpassed, good agreement is observed between the experimental indentation strain rate and the indentation strain rate predicted by Sherby. Comparison of Indentation Techniques for Obtaining Stress Exponent Figure (29) is a log-log plot of indentation strain rateversus hardness for the data from both the constant Pip experiments as well as the constant rate of loading/hold 99 ,• I I \. '" " I ' � I I , ' . " 'I . \ .. " �,\ . . ... ' . I ' \. , t\ 4 a. \ CJ) a. \. •. CJ) . . , CD • c "E � J: 1 6.5 Indium ° 2 10 101 10 Time (s) Figure 26. A log-log plot of hardness versus time for In for experiments conducted using 10 second constant rate of loading ramp to 10 mN followed by a hold period under constant load. Note the scatter in the initial portion of the data due to definition of absolute time. The slope of the line is equal to -lin and gives a reasonable determination of the stress exponent at long times. 100 1 01 �------�----�--�� Indium - ,- I -en Q) +-' ctS a: 0-1 C 1 10-ctS +-' en c .-0 +-' +-'ctS C Q) "'C C 6.3 1 -4 1 �.0�1------0-.�0 2---0-.0�3--0 .-0 4-- Hardness (GPa) Fi�ure 27. A log-log plot of indentation strain rate versus hardness for experiments conducted using 10 second loading ramp to 10 mN followed by a hold period under constant load. The indentation strain rate versus hardness relationship is observed to follow a single curve throughout the entire experiment. 101 1 01 1 sec loading ------10sec loading - 30sec loading 0 '.\ 0.08/Time, Sherby - 0 T""" 1 Prediction I CIJ --- Q) -\ ...... ',\ ro .� a::: ,t 0- 1 '.� C 1 1;' . ,- ro � '- ...... \/'\ : � I en : "\, , \ C ': "'\ 0 ...... 0-2 \ ro 1 ...... \ c Q) "'C C 3 1 0- Indium 1 0-4 L...... I...... L...... I ...... &..I. .LI.IIIL...... I .LI.IIIL...... I...&..I..&.I.IIII 0 2 3 1 0- 1 1 0 1 01 1 0 1 0 Time (5) Figure 28. A plot of indentation strain rate versus time for 1, 10, and 30 second load ramps to 10 mN followed by hold period under constant load for In. According to the model of Sherby[12], the indentation strain rate should scale as the characteristic strain of the indenter (0.08 for Berkovich) divided by the time of the experiment. 102 0 10 �------�----�--� Indium ,.... I en - 1 7.3 . - ..... CO ..... c • a> "'C C P,(mN/s) a> 10 C) CO o l0- 1 a> o 0.3 � 6.3 6 0.1 1 10-4 ...... ------...... -- --.....&...------1 0.01 0.02 0.03 0.04 Average Hardness (GPa) Figure 29. A log-log plot of indentation strain rate versus hardness comparing the results of the CRLlHoldexperiments to the results of the constant Pip experiments. The stress exponent obtained from the two techniques is in reasonably good agreement. 103 experiments. Strain rate-stress pairs were calculated for both the loading segment and hold segment from the CRLlHold experiments. Each set of data represents the average of five separate indentation experiments performed for each loading rate. Several interesting observations can be made abouHhese data. It was previously hypothesized that the transition region on going from a load ramp to a constant load was due to transient microstructural evolution occurring immediately afterthe completion of the load ramp and was not associated with power law breakdown. This is supported by the fact that even at higher strain rates the constant Pip data obey a constantpower law relationship. The similarityof the stress exponent obtained from the two types of experiments and the offset in the hardness leads to two conclusions. One, the dominant factor in both types of experiments seems to be the stress exponent for creep. While the two sets of data were acquired under very different conditions, the appropriate stress exponent seems to come out of both sets of data. Secondly, the data from the CRL/Hold experiments appears to have a stronger microstructure as indicated by the higher hardness determined at the same strain rate. This is reasonable since the material has undergone a wide range of strain rates beginning at very high values and decreasing throughout the experiment. The material is therefore evolving from a strong microstructure to a weaker microstructure as the experiment proceeds. This is in direct contrast to the constant Pip experiments where the indentation steady state is always expected to be approached from a hardening direction. Comparison of the stress exponents obtained from the hardnessversus time data, Figure (25), and the indentation strain rate versus hardness data, Figure (29), shows that a similar value for the stress exponent is obtained from the two analyses, except at short 104 loading times. The values of the stress exponent obtained for each technique are summarized in Table (1). Table (1 ) . Summaryof Stress Exponents Determined for Indium. Technique Analysis Stress Exponent, n CRLIHold £i versus H 6.1 CRLIHold (High T) £i versus H 5 Pip £i versus H 7.3 CRL/Hold H versus time 5.9 7.1.2.3 Aluminum CRLlHold Results Room and ElevatedTemp erature Figure (30) is a plot of the typical displacement versus time results for Al obtained during the constant load hold period using the CRL/Hold loading technique as a function of the test temperature. Again note the sharpnessof the break in the curves at lower temperatures suggesting a very high stress exponent at the low homologous temperatures. Figures (31) through (35) are log-log plots of indentation strain rate versus hardness for experiments conducted at temperatures ranging from 21°C to 250°C. (The accuracy of the calculated hardness data is discussed in Appendix B.) The average stress exponent or the range of stress exponents from each set of data is shown on each plot. The stress exponent is observed to range from extremely high values near room 105 Aluminum 4500 -... c:E - ...... c: Q) 4000 Q)E ctS(.) 0- CIJ o 3500 3000 o��--���--�� 50 100 150 200 250 300 350 Time (5) Figure 30. A plot of displacement versus time during hold period under constant load following 1 second ramp to 10 mN for Al at temperatures ranging from 100°C to 106 1 10- ______Aluminum, 21°C - .,.- 2 I - 10 0 � -CIJ Q) ....,as a: c: as ,... -3 CI)...., 10 c: 0 +=i 77-130 as ....,c: Q) "0 c: 4 10- 1 -5 1 0 ...... ,I...,j ...... L-I...L- ..I...Ir....&..L...L..&...L..:...... 0.4 0.5 Hardness (GPa) Figure 31. A log-log plot of indentation strain rate versus hardness for indentation creep experiments on Al conducted at 21°e. 107 1 10- ______Aluminum, 1 aaoe .-.. .,- 0 I Cf) f::,. 0 -- 0 Q) ...... «S cr: c . «S- 29 � C/)...... c 0 +=i «S ...... C Q) "C C 0-5 1 ""----'-...a.....II...I.....a.....IL...I....a...... L.--&....a...... IL..&..&...L..1..&..I 0.3 0.4 Hardness (GPa) Fi�ure 32. A log-log plot of indentation strain rate versus hardness for indentation creep experiments on Al conducted at 100°C. 108 2 10- ...---___--- ..__-_ o - ,.- I en --- Q) 10-3 .....ctS 36 a::: c "ffi J.... CJ)..... c 0 +=i ctS 4 .....C 10- Q) '"C C Aluminum, 150°C 1 0-5 1..----_...&.-___'--- _----' 0.2 0.3 0.4 Hardness (GPa) Fi�ure 33. A log-log plot of indentation strain rate versus hardness for indentation creep experiments on AI conducted at 150°c' 109 Aluminum, 200°C - 2 ,.....I 10- rn -Q) +""a:s +""c Q) -c 4 C 10- 5 10- �------�----�----� 0.2 0.3 0.4 Hardness (GPa) Figure 34. A log-log plot of indentation strain rate versus hardness for indentation creep experiments on Al conducted at 200°C. 110 Aluminum, 250°C 8� 0 8-1 0 1 -4 1.--_--'-_---'-_ 0 ...... -'- ....&-...... 0.1 1 Hardness (GPa) Fi/iure 35. A log-log plot of indentation strain rate versus hardness for indentation creep experiments on Al conducted at 250°C. 111 temperature to values of 8-10 at 250°C. Figure (36) is a log-log plot of indentation strainrate versus hardness for the data obtained at all temperatures. A progression from high stress exponents at high stresses to lower stress exponents at lower stresses is observed over the complete range of temperatures investigated, however, no constant power law region is observed. This range of stress exponents is not unexpected given that at low homologous temperatures , see Table (2), and higher stresses the data are expected to obey power law breakdown type behavior. Several key issues must be kept in mind when considering these results. First, it is important to remember that the indentation test is representative of a constant strain of approximately 10%. At low homologous temperatures or high stresses, the dominant feature of plastic flow in Al is work hardening. It is therefore more likely that these data are representative of the transientcreep properties of the material rather than large strain steady state properties. Secondly, all of these data were acquired from a hold segment following a 1 second ramp to maximum load. From the results on In, it is now known that a transient period will exist at the beginning of these data that may effect the stress exponent determination. At low homologous temperatures it is conceivable that this transient period may persist well into the hold segment due to the absence of thermal energy for microstructural rearrangement. Table (2). Range of temperatures for indentation creep of Aluminum. Temperature (Oe) Temperature (K) TfTm 21 294 0.32 100 373 0.4 150 423 0.45 200 473 0.51 250 523 0.56 112 ° 21°C • 100°C '\, .7 150°C • 200°C .- .... m 250°C I en - Q) Cij a:: c .- ca ' CI)+-' C o 1a +-'C Q) "'C C Aluminum 0.3 0.5 0.7 Hardness (GPa) Figure 36. A log-log plot of indentation strain rate versus hardness for indentation creep experiments on Al conducted at 21, 100, 150, 200, and 250°C. 113 Figure (37) is a plot of the displacement versus time response from the hold period under constant load for the amorphous alumina film obtained using the Step LoadIHold loading technique. Note the wide range of descent rates experienced by the indenter during the60 s hold period under constant load. Figure (38) is log-log plot of indentation strain rate versus hardness for the amorphous alumina film. T��.1�lQtted here, ha�alculated from a� elastically corrected contact rather than the total depth of the indentation as was done for the soft metals. A linear relationship between the indentation strain rate and _ .. �_ ..__ __ ,_ _ """"-",,__ n,,,,">4""'''''''¥-''''''"- '''�_ '''''''-_ -"""'V, '" hardness on the log-log plot was observed over the range of strain rates obtained. A stress exponent of 55.6±O.72was obtained fromthe slope of the log-log plot. Figure (39) is a plot of the displacement versus time response from the hold period under constant load for the single crystal Al203 substrate also obtained using the Step LoadlHold loading technique. Again, note the wide range of descent rates experienced by the indenter during the 60 second hold period under constant load as well as the overall decrease in the absolute displacement as compared to the amorphous �"'�"",. ___ �c'eo��_- - alumina filmdenoting the ability of the sapphire sample of supporting higher stresses. A linear relationship between the indentation strain rate and hardness on a log log plot was also observed for the sapphire over the range of strain rates obtained as shown in Figure (40). The hardness of the sapphire specimen was also calculated using ------""" ,.. --, *��" ---'-'��"" "'�'"""""" "-> " "",,,,,,,,,,,,...... -. �,--,�,.- ,, ���-->" ...- '>" '"" .' .-" ...,,.-..------"'-. � an elastically corrected contact depth rather than the total depth of the indent. A stress i�"")('� �--�."" ..,�...,.....1'1"""' ... -."-" '-;_..,,,""" ,,' - ,___ ,------...' ... -� ...... " _,�_ ...... _""", ...... ,tT_> __ ....,w,..�_"'.,�-� ...... -..... -- .... � exponent of 113±4.7 was obtained from the slope of the log-lo� elot. . __ "_S " __ ,� � ", _ "" < �, ." � ....�At ...,.:.·"",/w,t -- 11'- �"'" ---... ""'__ .... __ ....."' '''''.... _.''', ''''... '''',, ''< ''' --.,,_-�,->�,"w,'_>''__ n ....-_"- " � _ . ::*,Oi" :, �� The wide range of indentation strain rates shown in Figures (38) and (40) demonstrates the utility of the indentation experiment. ���l���!� :-����� �! ��e . _ . " ,, stress exponent are indicative of high strainrate tests performed on materials at small ," _� ___-- ..-'-�-"" "'-" -."'.. ",' , " �"-" '-�- ' , � -,,�.--,,-"'-�' '.-�'� .. '''«,� _ • ,- �_iI' '" I I 114 ..-.. E c: - ...... c: Q) E Q) (.) -co 0.. .-(f) o Amorphous AI 0 on Sapphire 2 3 10 20 30 40 50 60 Time (8) Figure 37. A plot of the displacement versus time response to 80 mN step load for a 1.9 Ilm amorphous alumina film on sapphire. Data shown are for the 60 second hold period under constant load immediately following the step load. 115 - 1 1 0 __------__------__ Slope = Stress Exponent, n Standard Deviation = 0.72 2 - 1 - 0r- 0 o en - Q) ..... ctS a: c -3 55.6 ctS 1 10... 0 ..... en c 0 ..... ctS ..... c Q) -4 "'C 1 0 C Amorphous AI 0 on Sapphire 2 3 -5 1 0 ------6 7 8 Hardness (GPa) Figure 38. A log-log plot of indentation strain rate versus hardness for 1.9 J.lm amorphous alumina film on sapphire. The high stress exponent for creep is typical of materials at low homologous temperatures. 116 - 350 E c -- +-' C Q) E Q) () co c.. en 0 345 Sapphire 10 20 30 40 50 60 Time (s) Figure 39. A plot of displacement versus time response to 80 mN step load for sapphire. The data shown are for the 60 second hold period under constant load immediately following the step load. 117 1 10- ...----.-_ _ __ _------. Sapphire Slope = Stress Exponent, n Standard Deviation = 4.7 - ..- len -- 2 Q) 10- � ctl a: + c: ctl " � C/) c: o Cii �c: Q) "'C c: 4 10. ____...... ____ ...... ___ ---1 1 8 19 Hardness (GPa) Figure 40. A log-log plot of indentation strain rate versus hardness for sapphire. The stress exponent of the crystalline Al203 is observed to be nearly twice that of its amorphous counterpart. 118 fractions of their melting temperature. The ability to obtain information into the time dependence of flow from these materials alone attests to the value of the indentation creep technique. No other test technique is known of that is capable of yielding comparative data for two materials such as these. 7.2 Effects of Temperature 7.2.1 Introduction The central theme of this section of the dissertation is to investigate the effect of temperature on the relationship between indentation strain rate and hardness. Data from both In at high homologous temperatures and Al at intermediate homologous temperatures will be presented and discussed. 7.2.2 Results and Discussion 7.2.2.1 Indium Hardness versus Tff m Figure (4 1) is a plot of the hardness versus homologous temperature calculated from the applied load and the displacement at 15 s. It is observed that over the homologous temperature range of 0.4 to 0.8 the hardness decreases by over an order of magnitude. The change in hardness versus test temperature at a constant time under load will be discussed again in an analysis for determining the apparent activation energy for indentation creep. 119 o p =3 mN max A P =10 mN max .-... ct:S 0... ---C!J en 1 en 10- Q) c "C'- ct:S I Indium -2 10 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Homologous Ternperature, T / T m Figure 41. A plot of hardness versus homologous temperature for In over the homologous temperature range of 0.4 to 0.8. The hardness was calculated from the load divided by the contact area calculated from the total displacement 15 seconds into the hold period. 120 Determination of Apparent Activation Energyfor Indentation Creep As previously discussed, at temperatures above 0.5 T creep in pure metals in m, which dislocation glide is easy is thought to be controlled by the climbing of dislocations out of their glide planes to overcome obstacles to motion. The climb velocities are limited by the rate at which vacancies can diffuse in the material. As such, creep at elevated temperatures is expected to follow an Arrhenius type of behavior of the form s = so exp(-Q I RT) (12) where Q is the apparent activation energy for creep, R is the gas constant and T is the absolute temperature. The slope of a semi-log plot of strain-rate at constant values of hardness versus lIT will then yield the activation energy for the rate controlling mechanism. Figure (42) is a plot of the natural log of the indentation strain rate at constant hardness versus lIT for experiments conducted between 28°C and 75°C. The apparent activation energy for indentation creep obtained by a linear fit to the data is 77.9 kJ/mol. This is in excellent agreement with the . activation energy for self-diffusion in pure indium which is reported to be in the range 75-78 kJ/mol[71, 72] and in accord with the observation that the activation energy for creep in pure metals is nearly equal to the activation energy for self diffusion[5]. By compensating the indentation strain rate for the temperature of the experiment, a normalized plot of indentation strain rate versus hardness can be obtained. Figure (43) is a log-log plot of temperature compensated indentation strain rate versus 121 0 3 Slope = -Q / Rx1 0 Q) = ...... , -1 Q 77.9 kJ/mol co a: c 0 co '- -2 ...... , en c 0 +=i -3 co ...... , c Q) "0 c -4 ..- 0 0> 0 ....J -5 - co '- ::::J ca z -6 -7 � 2.8 2.9 3 3.1 3.2 3.3 3.4 3 (1 / T)x1 0 (K·1) Fiiure 42. A plot of natural log of indentation strain rate at constant values of hardness versus the reciprocal absolute temperature for In. The slope of the line is equal to -QlRx103 where Q is the apparent activation energy for creep. An activation energy of 77.9 kllmol was determined for the high temperature indentation creep of In. 122 -o E ----� o o (J) - f l"- I" II o I 5 c: o � .....c: Q) Indium -0 8 c: 10 ���--�--������ 10-2 10-1 Hardness (GPa) Fi&ure 43. A log-log plot of temperature compensated indentation strain rate versus hardness for 28, 50 and 75°C experiments on In. The indentation strain rates at the three temperatures were normalized using an activation energy for creep of 77.9 kJ/mol. 123 hardness for the three sets of data shown in Figure (21). The fact that all three sets of data, when compensated for the test temperature, lie on a single curve supports the premise that the deformation is being controlled by a single thermally active process. The slope of five agrees with observed values for stress exponents in pure metals obtained from conventional means where the rate controlling process is dislocation climb [4]. As previously discussed in Chapter 2, in calculating the temperature compensated creep rate it is often necessary to take into account the temperature dependence of the shear modulus, G, in order arrive at an appropriate activation energy[37, 38]. It has been shown for a number of materials that taking values at a constant stress rather than at a constant value of a/G(T) can lead to overestimation of the activation energy for creep[39]. Figure (44) is a plot of both the Young's modulus, E, and the shear modulus, G, as a function of temperature[66] for pure In. By normalizing the hardness for the temperature dependence of the modulus and determining indentation strain rates at constant values of HJG and HiE, the activation energy for the modulus compensated data can be determined. Figure (45) is a plot of the natural log of the indentation strain rate at constant values of HJG andHiE versus Iff. A linear fitto the two groups of data yields values for the apparent activation energy for indentation creep of 59.3 kJ/moland 58.0 kJ/mol respectively. One possible explanation for the lower activation energy is that the mechanism is not self diffusion but perhaps some coupled diffusion mechanism such as a combination of lattice diffusion and dislocation core diffusion as has been observed in Al at intermediate temperatures. However, at such high homologous temperatures, this explanation seems suspect. A more likely explanation is that the differences in the 124 11 10 9 -o- Young's Modulus E = 11.676 - O.0338T - ro 8 c.. (!J "'-'" C/) 7 :::J :::J "'C 0 6 ::E * Shear Modulus 5 G = 4.67 - O.0135T ...... 4 ...... do ... 3 20 30 40 50 60 70 80 Temperature (oC) Fi�ure 44. A plot of the shear modulus and Young's modulus of In as a function of temperature [ 66]. Source: John C. Wei, "High Temperature Creep of Pb-In Alloys." Ph. D. Dissertation in Materials Science, Stanford University, December 1980. 125 -3 Q) +-'ca -3.5 a: c .-ca !o.,.. -4 (j)+-' C 0 .- +-'ca -4.5 +-'C Q) "'C c -5 '+-0 C> 0 --l -5.5 ca ::J!o.,.. • H/E(T) +-'ca z -6 0=58.0 kJ/mol -6.5 2.8 2.9 3 3.1 3.2 3.3 3.4 1 /Tx1 03 (K-1) Figure 45. A plot of the natural log of the indentation strain rate at constant values of RIG and HIE versus the reciprocal absolute temperature for In. The slope of the line is equal to -QlRx l03 where Q is the apparent activation energy for creep. This analysis yields lower activation energies than were determined using values of indentation strain rate at constant hardness. 126 activation energies determined from the two analyses, which are roughly 20% different, are simply within the scatter of the data being used for determining the apparent activation energy for creep. A second technique for determining the activation energy for indentation creep is that of Sargent and Ashby[13] which involves plotting the natural log of the hardness determined at a constant time under load versus the reciprocal homologous temperature, the slope of such a line being equal to Q/nRT Figure (46) is a plot of In H versus m. TmIT for Inover the homologous temperature range of 0.7 to 0.81. An activation energy of 74 kJ/mol is found from the slope of this plot. This value is also in good agreement with the activation energy for selfdiffusion in the material. 7.2.2.2 Aluminum Hardness versus Tffm Figure (47) is a plot of hardness versus homologous temperature for Al calculated from the applied load and the displacement at 15 s. It is observed that over the homologous temperature range of 0.31 to 0.56 the hardness decreases from 0.43 GPa to 0.17 GPa. Apparent Activation Energy for Indentation Creep Due to the limited range of indentation strain rates and hardnesses covered at each temperature and the fact that the activation energy exhibits a stress dependence due to the dislocation core diffusion taking place in the intermediate temperature range studied, an accurate determination of the apparent activation energy for indentation creep of Al based on the results of this study was unattainable. It was therefore decided 127 -3.4 Slope 4.1634 Q/nRT = = en m "C C Q=74 kJ/mol o -3.6 (])(.) en LO � @) -3.8 en en (]) c "Eco I -4 -4.2 Indium -4.4 ...... - 1.2 1.25.....- ...... 1.3 1.35 ...... 1. 4 -.....1...... 45 T IT m Figure 46. A plot of natural log hardness at constant time under load versus the reciprocal homologous temperature for In. The slope of the line is equal to Q/nRT m where Q is the apparent activation energy for creep. An activation energy of 74 kJlmol was determined from the slope. 128 0.5 ...... 1""'1""1"' ...... 1""'1""1"' .....-...... 0.45 .-. ctS 0.4 a... CJ-- en 0.35 en Q) c "'0 0.3 ctS� T I Q) C) ctS 0.25 �Q) � 0.2 0.15 Aluminum 0.1 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Homologous Temperature, T / T m Figure 47. A plot of average hardness versus homologous temperature for Al over the homologous temperature range of 0.3 1 to 0.56. 129 to apply the effective diffusion coefficient model of Luthy et al[40] to the indentation data to determine qualitatively the accuracy of the method for indentation creep data in the intermediate homologoustemperature range. The effective diffusion coefficient theory of Luthy et al is defined in equation (14) and is based on all published diffusion data for AI, namely two different lattice self diffusioncoef ficients and one dislocation pipe diffusion coefficient and the fraction of atoms participating in each type of diffusion. Two different lattice diffusion coefficients exist in the literature; one for high temperature diffusion as determined by tracer diffusion, and one for intermediate temperature diffusion as determined by void shrinkage methods. Luthy et al use a bilinear relation for the latticedif fusion coefficientof the form ( ) ( ) -4 -142 7 -115 2 DL = 1. ?xlO exp + 6xlO- exp (m Is) (49) RT RT and assume that the fraction of atoms participating in lattice diffusion is approximately 1. Luthy et al[40] definethe core diffusion contribution as (50) where the coefficientfor dislocation pipe diffusion was previously determined by Volin et al[?3]. The problem then becomes one of defining fe , the fraction of atoms participating in dislocation core diffusion. According to Luthy et al[40], the value of fe is equal to nplN where n is the number of atoms contributing to core diffusion, N is the density of atoms per m2, and p 130 is the dislocation density. n is assumed to be 4 from the work of Volin et al[73] while the atomic density of AI is approximately lx1019 m·2. The density of dislocations is found by Luthy et al from the Taylor relation[74] which is given by (51) where a is the steady state stress and E is the Young's modulus. Two major problems are associated with this analysis concerningthe indentation creep test. First, the stress in a constant load indentation creep test is not constant but rather changes throughout the course of the hold period under constant load. Secondly, at these low homologous temperatures, the measured hardness is certainly not indicative of a steady state flow stress considering the fact that a characteristic strain of roughly 10% is associated with the indentation. With these factors in mind, the most logical choice for replacing (alE) in the Taylor relation seems to be the instantaneous value of (Hl3E). Using this parameter allows for the dislocation density to change as the hardness decreases during the experiment. It seems only logical that as the hardness and strain rate decrease during the experiment, the dislocation density in a volume of material at a normalized distance rlh from the indenter, where r is the radial distance from the tip of the indenter and h is the depth of the indent, would decrease as well. Figure (48) is a plot of the temperature dependence of the modulus used in normalizing the hardness data and for calculating the dislocation density from the modified Taylor relation. Figure (49) is a plot of the dislocation density obtained for the range of (HIE) encountered in the experiments on Al obtained using equation (51) to calculate the dislocation density replacing (alE) by (HI3E). The soundness of this range 131 -- E = 68.691-0.039904T R= 0.99996 66 - Cd Q.. C!J--- en ::::J 64 ::::J "'C o � en C) 62 c ::::J o >- 60 Aluminum 58 � o 50 100 150 200 250 300 Temperature (OC) Fi�ure 48. A plot of Young's modulus versus temperature for Al[40]. Source: H. Luthy, A. K. Miller, andO. D. Sherby, Acta Met., 28, 169 (1980). 132 Modi'fied Taylor Relation • • p=2x102DX(H/3E)2 • I .-. (\J I I • E "-" C I en c Q) o I c o +=i as (.) o ,. --en • o I' • " • • o � 0.002 0.004 0.006 0.008 HIE Figure 49. Plot of dislocation density calculated from the Taylor relation by replacing the normalized stress dependence, crlE, of the Taylor relation with (H/3E). '. - '. ", 133 of dislocation densities can be examined by considering a similar analysis put forward by Taylor[75] for the work hardening of metalcrystals such as AI. Taylor assumed that the average distance a dislocation moved before it was stopped was L. If the density of dislocations after a given deformation is p. then the strain is given by e=pxLxb (52) where b is the burgers vector. Rearranging this equation and solving for the dislocation density, p, using a value of L=lx1O-4 cm from Taylor[75], a burgers vector for Al of 10 nm, and the characteristic strain of the Berkovich indenter of 0.1, an average dislocation density of 1x1013 m-2 is calculated. This value is in fair agreement with that obtained using the modifiedTaylor relation above. The contribution of the core diffusion term to the effective diffusion coefficient can then be written as (53) where the fraction of atoms participating in core diffusion is represented by the second bracketed termwhich depends on the ratio of Hl3Efrom equation (5 1). Figure (50) is a plot of the indentation strain rate normalized by the effective diffusion coefficient calculated from equations (14), (48), and (52) versus the hardness normalized for the temperature dependence of the modulus. Examination of the plot shows that the effective diffusion coefficient approach seems to bring the data together on to a single curve in an acceptable manner. Attempts to fit these data using the hyperbolic-sine function of Gara(al� �ere �nsuccessful. ' 134 °00 2 1 Aluminum 10 �!�g C9§ ::e �8 Q) ° 21 °C 0 ...... (J) • 100°C +""" as .Fg � a: • A ,•• 150°C • c tit· f II as • • • • J.- 200°C • • +""" •• C/) ,"1t c 0 0 250°C I. +"""as � .. .-+"""C (J) '"0 c HIE Figure 50. A log-log plot of temperature compensated indentation strain rate (indentation strain rate divided by the effective diffusion coefficient) versus HIE for AI. The ratio of the hardness divided by three times the modulus (Hl3E) was used in the effective diffusion coefficient equation in calculating the contribution from the dislocation core diffusion term. 135 An examination of the indentation strain rate hardness data shows that no constant power law region of behavior is observed to exist for Al over the temperature range investigated. An examination of the values of if Deff obtained using the effective diffusion coefficient shows temperature compensated indentation strain rates ranging from 1013 to 102 1. Based on the observation from Sherby and Armstrong[12] that power law breakdown is observed to occur in most metals at values of i I Deff greater than 109, the entire range of data covered in these experiments is expected to be in the power law breakdown regime . 7.3 Steady State Path Independent Hardness 7.3.1 Introduction The central theme of this section of this dissertation is to investigate the existence of a path independent steady state hardness in an indentation test utilizing a geometrically similar indenter. It has been shown in uniaxial creep tests that the substructural changes that occur during the creep process are insensitive to temperature and depend only on time and stress[22] according to equation (15). This constancy of microstructure for a given level of stress has been shown by incorporating stress-change tests where the applied stress is abruptly changed once the steady state has been reached as discussed in Chapter 2 and shown in Figure (2). Analogous data from Pip change experiments will be presented. The results will be analyzed and discussed based on comparison to the stress change tests from bulk uniaxial creep testing. 136 7.3.2 pip change Tests on Indium In order to test the hypothesis that a steady state value of hardness could be reached that depends only on the imposed strain rate and temperature, a set of experiments was conducted using the constant Pip loading technique that was previously discussed. However, rather than maintaining the value of Pip constant throughout the entire loading segment, Pip change experiments were conducted in which the value of Pip was abruptly changed from an initial value to a new value at a predetermined displacement. This type of experiment allowed the response of the material to changes in the imposed indentation strain rate to be determined. Figure (51) is a plot of the resulting indentation strain rate, calculated as the instantaneous displacement rate divided by the instantaneous displacement, versus displacement for the Pip change experiments. The initial indentation strain rate is again observed to be constant andequal to 0.5 Pip. Upon a change in Pip , the indentation strain rate is observed to go through a transient period and then approach a new constant izlh after a delta displacement of approximately 500 nm. Figure (52) is the resulting hardness versus displacement for the Pip change experiments. The measured hardness resulting from the Pip change is alsoobserved to undergo a transient period and then approach the indentation steady state value from the constant Pip experiments. The displacement required seems to be somewhat a function of whether Pip was increased or decreased. The delta displacement required to reach the indentation steady state value at a Pip of 0.1 is approximately 1000 nm after the Pip increase from 0.02 while the additional displacement required on going from a Pip of 0.1 to 0.02 requires nearly 2000 nm of displacement. The requirement of additional strain rather than a characteristic time to reach a new indentation steady state value of hardness can be understood by reexamining the 137 Indium - � I -(f) Q) ...... ro a: c .- l) ro -<>-�,(0.1S- ...... '- CJ) -0-- 0.02 c 0.02->0.1 0 • '+=i 0.1->0.02 ro * ...... c Q) '1J C 10-2 2000 3000 4000 5000 Displacement (nm) Figure 51. A plot of indentation strain rate versus displacement fo r Pip change experiments on In. The indentation strain rate is observed to go through a transient period and then reach a new indentation steady state value after the Pip change. 138 0.03 �,(S-1) 0.028 --e- 1.0 -0-0.1 0.02 -0- 0.026 • 0.02 0.1 -> - ctS c.. • CJ - 0.024 en en (]) c: "C 0.022 ctS� I 0.02 0.018 Indium 0 1 6 O. 1oooI..I..I..L.L..L.L.L...I..I-I..&..L.II...L.I..II..I..L.&..&..L..I..L.L..L.L.L.... 2000 3000 4000 5000 Displacement (nm) Figure 52. A plot of hardness versus displacement for Pip change tests on In. In a manner analogous to the indentation strain rate, the hardness is observed to go through a transient period after the Pip change and then to reach a new steady state valuewith furtherdispla cement. 139 indentation process. As previously discussed, the plastically deformed volume in the indentation test is continually expanding into previously undeformed material. The indentation steady state is therefore always being approached from a hardening direction rather than a softening direction, even when pip has been decreased. It is interesting to compare the pip change indentation experiments to the uniaxial stress change experiments shown in Figure (2). While the transient response as well as the progression to a new steady state is very different in the indentation and uniaxial tests, comparison of the indentation pip change experiments to the uniaxial stress change experiments reveals remarkably similar behavior between the two. 7.4 Comparison to Uniaxial Data 7.4.1 Introduction While indentation creep testing is a viable technique for comparing the time dependent properties of materials that cannot be characterized by standard bulk techniques, how the indentation creep results can be compared to uniaxial creep data is a subject of major importance. This section presents and discusses on an empirical level how the indentation creep data and the uniaxial data compare. The specific ideas of relating the indentation creep data to its uniaxial counterpartbased on the ideas of (1) a constraint factor relating the hardness and the uniaxial flow stress, (2) a characteristic strain associated with the Berkovich indenter, and (3) a constant relating the indentation strain rate to the uniaxial strain rate will be explored. 140 7.4.2 Pb-65 at % In In order to directly compare the indentation creep data obtained for Pb-65 at% In to uniaxial data, a series of compression tests were performed on the bulk alloy. Figure (53) shows the stress-strain results for a series of uniaxial compression tests performed at a variety of constant cross head speeds. Figure (54) shows the corresponding stress strain rate results as a function of strain. Note that the stress exponent that one determines from the compression data is highly dependent on the percent strain at which the stress-strain rate pairs are tabulated. Figure (55) is a plot of the indentation creepfo r Pb-65 at% In data along with the uniaxial compression data at strains of 10 and 80%. According to Tabor[68], the indentation hardness of a material should be approximately 3.3 times its strength at a characteristic strain of 8-10%. When the indentation hardness data for Pb-65 at% In are divided by 3.3 and plotted versus the indentation strain rate along with the compression stress-strain rate data at 10 and 80% strain, the plot shown in Figure (56) results. The similarity of the indentation creep data to the uniaxial data at 10% strain implies that indentation creep strength measured with this type of experiment can be represented by the flow strength of the material when strained approximately 10%. This implies that the indentation creep experiments as conducted on Pb-65 at% In, i.e., Step Load/Hold experiments, are most likely measures of transient, and not steady state behavior. These data suggest that the constraint factor and the concept of a characteristic strain seem to be reasonably valid for Pb-65 at% In. 7.4.3 Indium Since the constant Pip indentation experiments yield an indentation steady state hardness, this type of experiment is believed to most closely approximate the results 141 50 - co c.. 40 -:E en en (]) '- ...... 30 en co 'x co c 20 :::J 10 o o 0.2 0.4 0.6 0.8 1 Strain Figure 53. A plot of stress versus strain from uniaxial compression experiments run on Pb-65 at% In alloy at a variety of crosshead speeds. The decrease in the stress with increasing strain is believed to be due to dynamic recrystallization at the lower strain rates due to the high homologous temperature. 142 10-1 Pb - 65 at% In % 80 Strain -0- 0% Strain 4 % 2 • 20 Strain - 10- % ...... 1 0 Strain I CJ) -0- ---- Q) +-'CO a: c 3 °ffi 1 0- s... CJ)+-' CO X CO 0-c ::::> 1 0-4 Uniaxial Stress (GPa) Figure 54. A log-log plot of stress versus strain rate as a function of strain for uniaxial compression tests on Pb-65 at% In. The stress exponent is seen to be highly dependent upon the level of strain at which it is detennined. 143 Pb - 65 atO/o In Uniaxial Data @ 0% Strain --0- 1 -0- Uniaxial Data @ 80% Strain Indentation Creep Data • - .... I Q) ca a:: c . - � (J)+-' -5 ______10 ...-- ___ ...- 2 10- 1 0-1 Uniaxial Stress or Hardness (GPa) Figure 55. A log-log plot of indentation strain rate versus hardness and uniaxial strain rate versus stress at 10 and 80% strain for Pb-65 at% In. The indentation creep data are seen to display similar behavior to the uniaxial data at 10% strain. 144 Pb - 65 at% In Uniaxial Data @ 10% Strain -0- Uniaxial Data @ 80% Strain ---0-- Indentation Creep Data • - T"" • en 2 '-' 1 0- Q) ...... a:s a:: c: - . 3 a:s 10- � ...... (/) 0.1 Uniaxial Stress or Hardness/3.3 (GPa) Fi�ure 56. A log-log plot of indentation strain rate versus hardness/3.3 and uniaxial strain rate versus stress at 10 and 80% strain for Pb-65 at% In. The constraint factor of Tabor is seen to bring the two curves closer to coincidence but does not fully account for the observed difference. 145 from the constant stress tensile experiments and will therefore be compared to the steady-state uniaxial data. Figure (57) is a log-log plot of both the indentation pip data and the constant stress tensile data obtained by Weertman[39] at room temperature. Both the uniaxial data and the indentation data are seen to exhibit constant power law behavior. As previously discussed, the stress exponents obtained from the two types of data are for all practical purposes identical, however, a finite offset is observed to exist between the uniaxial and indentation data. Figure (58) is a log-log plot of the temperature compensated uniaxial strain rate versus uniaxial stress plotted along with the temperature compensated indentation strain rate versus the hardness. The data have been compensated for temperature using the activation energy for self diffusion, 75 kJ/mol[39]. A slightly lower value for the stress exponent is observed for the indentation data as compared to the tensile data at room temperature when compared on a temperature compensated strain rate basis. Figure (59) is a log-log plot of the steady state uniaxial strain rate versus uniaxial stress and indentation strain rate versus hardness/3.3 according to the analysis of Tabor. The constraint factor is again observed to bring the two sets of data closer together but does not completely account for the observed difference. The steady state like behavior of the constant pip data can be better understood by considering the observations of Weertman who found that the creep rates of In were typically found to be constant over a wide range of strain, typically 5 to 25%. Recalling that the characteristic strain of a Berkovich indenter is approximately 8 to 10%, the strains experiencedin the indentation creep test were perhaps of sufficient magnitude to reach steady state conditions in the material. This is in direct contrast to the Pb-65 at% 146 --GJ- Weertman, Constant Stress --C-This Study, Constant pip ..- ,.... I en -- 7.3 (J) ...... ct:I a: c 1 "ct:I ...... So... (f.) Indium .5 oo._--..1...... 10 ...... 10.3 10.2 Stress or Hardness (GPa) Figure 57. A log-log plot of indentation strain rate versus hardness for constant Pip experiments on In plotted with tensile strain rate versus stress data from Weertman[39]. Source: 1. Weertman, Trans. of AIME, 218, 207 (1960). 147 1015 - .,..... I EEl Weertman, Constant Stress CIJ - <) This Study, Constant pip Q) ...... ro EE a:: <> OC <> ro <> .... CJ)...... EEl () <) "'C Q) ...... ro CIJ JW c EElEEl Q) a. Ei±l EEl E EBEE 0 ffitJEEl () Q) .... t! ::J EE ...... ro EEl ....Q) a. EEl E Q) Indium I- 3 EEl 1 Q -4 1 0 Stress or Hardness (GPa) Fi�ure 58. A log-log plot of temperature compensated indentation strain rate versus hardness for constant Pip experiments on In plotted with temperature compensated tensile strain rate versus stress data on In fromWeertman [39]. Source: 1. Weertman, Trans. of AIME, 218, 207-218 (1960). 148 --ITl- Weertman , Constant Stress �This Study, Constant pip �.-.. , en - 10-2 (J.) +oJctS a: c: - . ctS -3 +oJ"- 10 C/) Indium 5 1 0- i.---a..--a...... &....Or....&..&...... _ ....i-....i-...... 10-3 10-2 10- 1 Stress or Hardness/3.3 (GPa) Figure 59. A log-log plot of indentation strain rate versus hardness/3.3 for constant Pip experiments on In plotted with tensile strain rate versus stress data from Weertman[39]. Source: J. Weertman, Trans. of AIME,218, 207-218 (1960). 149 In data where 10% strain is observed to be still in the transient region of the stress strain curve. Another method for relating the indentation data of a power law creeping solid to its uniaxial counterpart is that of Bower et al[76] who used fi nite element modeling to arrive at an expression for the 'effective stress' and 'effective strain rate' under the indenter. Their effective stress is taken as the contact pressure or hardness, and the effective strain rate is determined by the shape of the indenter and the displacement rate. The effective stress is assumed to be related to the effective strain rate by the uniaxial stress-strain response. Figure (60) is a plot of the Bower factor for relating indentation and uniaxial strain rates a function of the stress exponent. The data are observed to as follow an exponential increase as the stress exponent decreases. Assuming that the hardness is equal to the uniaxial stress, the indentation strain rate divided by the Bower factor should yield the uniaxial strain rate. Figure (61) is a plot of the indentation data modified according to the appropriate Bower factor determined from an exponential fi t to the line in Figure (60). The Bower factor serves to shift the indentation strain rate nearly four orders of magnitude on the strain rate axis. While this modification factor also serves to bring the two sets of data closer together, it also fails to completely reconcile the observed differences. While the offset between the two sets of data cannot be completely accounted for by either constraint factors[68] or by predictions from time dependent finite element calculations[76], the utility of the constant Pip indentation data is evident. The most often sought after parameter in these typesof experiments is how the rate of deformation of the material changes in response to a change in the applied stress, Le., the stress exponent for creep. From the comparison of the two sets of data it is obvious that the stress exponent obtained from the two types of experiments is very similar. 150 (-1 .2093x) 6.2975 e y = o� '+- o� t) ctJ u.. Q)� o� al Stress Exponent, n Fi�ure 60. A plot of the Bower factor[76] for relating indentation strain rate to uniaxial strain rate versus stress exponent for creep. Source: A. F. Bower, N. A. Fleck, A. Needleman, and N. Ogbonna, Proc. R. Soc. Lond. A, 441, 97 (1993). 151 .. I en --- o10- ..... --rn- Weertman , Constant Stress ctS() LL � Modified by Bower Factor Q) «i Indium 0: � 1 0-5 1-- 3 ...... _ 1---10...&..&...... 2_ "'--""'--...... � 10- 10- 10-1 Uniaxial Stress or Hardness (GPa) Figure 61. Log-log plot of indentation strain ratelBowerfactor versus hardness for constant Pip experiments on In plotted with tensile strain rate versus stress data from Weertman[39]. Source: J. Weertman, Trans. of AIME, 218, 207-218 (1960). 152 7.4.4 Aluminum Figure (62) is a log-log plot of temperature compensated indentation strain rate versus hardness for Al plotted along with the steady state torsion and tension data of Luthy et a1[40]. Figure (63) is a plot of the indentation creep data for Al where the hardness has been modified by the constraint factor of Tabor[68] again plotted along with the steady state data from Luthy et al as well as uniaxial stress strain rate data from Trozera et al[77] tabulated at a strain of 10%. The constraint factor serves to bring the two sets of data closer together, especially at the higher strain rates, but cannot account completely for the offset. Two things must be kept in mind regarding this set of indentation data. First, it must again be kept in mind that the indentation data represent a constant strain of approximately 10%. While at high homologous temperatures this level of strain may be sufficient to reach steady-state conditions, this is not the case for AI. From 0.2 to 0.6 Tffm, strain hardening is the dominant feature of plastic flow in AI. This fact leads to experimental difficulties in obtaining steady state flow stresses since extremely large strains are required to reach steady state[40]. The data from Luthy et al were all obtained in torsion requiring large strains to reach steady-state, typically on the order of 100 to 200%. It is therefore unrealistic to expect that the indentation data in the same homologous temperature range should reflect the steady-state properties obtained from the bulk testing at all, rather, the data should more likely represent the properties of the material having undergone approximately 10% strain which is likely far from steady state conditions. At small strains, such as 10%, the stress for a given strain rate is expected to be significantly lower than the steady state flow stress. This would result in apparently higher stress exponents for creep than would be obtained at higher strains as exemplified by the data of Trozera et al[77]. As the temperature of the material is 153 .-.... C\II Aluminum --E .....,C <> This Study. CRUHold CJ) "I::J "l- . - o t5 Fi�ure 62. A log-log plot of indentation strain rate divided by the effective diffusion coefficient versus HIE for Al plotted with bulk data obtained in torsion and tension from Luthy et a1[40]. Source: H. Luthy, A. K. Miller, andO. D. Sherby, Acta MeL, 28, 169 (1980). 154 <> This Study, CRUHold, H/3.3 • Luthy et ai, Steady State • Trozera et ai, 10% Strain • • • • • • I I· 1 3 10 __��� __�� __ 3 �.-��� 10-4 10- 10-2 Stress/E or Hardness/3.3E Figure 63. A log-log plot of indentation strain rate divided by the effective diffusion coefficient versus Hl3.3E for AI plotted with bulk data obtained in torsion and tension from Luthy et al[40] and Trozera et a1 [77]. Source: H. Luthy, A. K. Miller, and O. D. Sherby, Acta Met., 28, 169 (1980). T. A. Trozera, O. D. Sherby, and J. E. Dom, Trans ASM, 49, 173 (1957). 155 raised, the required strain to reach steady state behavior is expected to decrease. This would result in a stress closer to the steady state value being obtained at lower strains. The comparison of the indentation creep data on Al seems to follow this explanation. At low homologous temperatures, the stress exponent obtained from the indentation data is observed to be similar to that obtained from the 10% data of Trozera et al. As the temperature is raised, the stress exponents are observed to approach those obtained from the steady state data of Luthy et al. Why the constraint factor of Tabor does not appropriately account for the observed differences between the indentation and uniaxial data at low temperatures can perhaps be explained by considering that the surface of the Al was not in a well annealed state as was expected but had residual work hardening due to the mechanical polishing. This could result in hardness values corrected by the constraint factor being closer to the steady state values of stress as is observed since the flow stress would notchange nearly as dramatically as a function of strain. Secondly, these data were acquired using the CRLfHold technique. This technique has been seen to result in a transient period on completion of the load ramp yielding higher apparent stress exponents during the initial stages of the hold period under constant load. At smaller fractions of the melting temperature, these transients could be expected to persist longer into the hold period resulting in abnormally high stress exponents being observed well into the hold period, i.e., the hardened microstructure due to the higher strain rates simply does not have sufficient thermal energy to recover. Examination of the transient region for the elevated temperature data on In does seem to suggest that the temperature is increased, the harness-strain rate as behavior comes to a constantpower law relation more quickly. 156 Chapter 8 Indentation Creep - Conclusions and Suggestion for Future Research 8.1 Conclusions U sing a variety of depth-sensing indentation techniques at both room and elevated temperatures, the dependency of the indentation hardness on the variables of indentation strain rate (stress exponent for creep, n) and temperature (apparent activation energy for creep, Q), and the existence of a steady state behavior in an indentation creep test with a Berkovich indenter were investigated. The indentation creep response of five materials, Pb-65 at% In (at RT), high purity indium (from RT to 75°C), high purity aluminum (from RT to 250°C), a vapor deposited amorphous alumina film (at RT), and single crystal alumina (sapphire) (at RT), was measured. It was shown for the first time that the indentation strain rate, defined as iz/h, could be held constant during an indentation experiment using a Berkovich indenter by controlling the loading rate such that the loading rate divided by the load, Pip, remained constant. The constant Pip technique yields the most unambiguous determination of the stress exponent for indentation creep and seems to most closely approximate the steady state results fromuniaxial testing. Comparison of the results from the constant Pip experiments to the results from conventional CRLlHold indentation creep experiments in Figure (30) shows that the transition from the load ramp to the hold period under constant load shows a brief period with an apparent higher stress exponent for creep. This feature is now believed to be due a transient stage on going from the load ramp to the hold under constant load and not to be associated with power law breakdown behavior. This is supported by the fact 157 that even at higher strain rates the Pip data obey a constant power law relationship. The experiments conducted at different constant loading rates to the same maximum load show that even though the indentation strain rates and hardnesses experienced during the load ramps are very different, the indentation strain rate-stress data during the subsequent hold period under constant load converge to a single curve after an initial transient period. The similarity of the stress exponent obtained from the constant Pip and CRLlHoldexperiments and the offset in the hardness leads to two conclusions. One, the dominant factor in both types of experiments seems to be the stress exponent for creep. Secondly, the data from the CRL/Hold experiments appears to have a stronger microstructure as indicated by the higher hardness determined at the same strain rate. This result is expected due to the higher strain ratesexperienced by thematerial during the experiment. Comparison of the stress exponents obtained from the hardness versus time data, Figure (26), and the indentation strain rate versus hardness data, Figure (30), for In shows that a similar value for the stress exponent is obtained from the two analyses, except at short loading times. While both techniques seem to yield similar results, the relationship between the indentation strain rate and the hardness is based on a valid material constitutive law. The measurements of the stress exponents for creep on amorphous alumina and sapphire demonstrate the indentation test's unique ability to measure differences in the time dependent response of materials that cannot be tested with other techniques. The ability to obtain information into the time dependence of flow from these materials alone attests to the value of the indentation creep technique. 158 The indentation strain rate - hardness data for In at high homologous temperatures obtained using the CRLlHold technique display a stress exponent of 5 in excellent agreement with that obtained for high temperature uniaxial creep where dislocation climb is the rate controlling mechanism[4]. The apparent activation energy for indentation creep in indium was found to be approximately 78 kJ/mol, in excellent agreement with the activation energy for self diffusion in the material. Temperature compensating the indentation strain rate using the experimentally determined activation energy is seen to bring all of the data at the various temperatures together onto single master curve supporting the hypothesis of a single thermally activated rate controlling mechanism. The technique for determining the activation energy for indentation creep of Sargent and Ashby[13], which involves plotting the natural log of the hardness determined at a constant time under load versus the reciprocal homologous temperature, yields an activation energy for indentation creep of 74 kJ/mol for In. This value is also in good agreement with the activation energy for self diffusion in the material. In light of the transient period that exists on going from the load ramp to the hold period, caution must be exercised to ensure that the hardness versus time data are obtained at times greater than the relaxation time. As the relaxation time scales with the loading time, the time required to apply the load would seem to be a logical lower limit on data obtained using this technique. The indentation strain rate - hardness data for Al show an appropriate decreasing trend in the stress exponent for indentation creep as the test temperature is raised. The temperature dependence of the indentation creep process was found to be reasonably well described by an effective diffusion coefficientas described in the literature[ 40] for bulk aluminum at intermediate temperatures. 159 By perfonning pip change experiments it was shown that a steady state path independent hardness could be reached in an indentation test with a geometrically similar indenter. The arrival at a new steady state value of hardness seems to depend on the accumulation of strain rather than a relaxation time. Comparison of the pip change indentation experiments to the uniaxial stress change experiments show that while the transient response as well as the progression to a new steady state are very different, the two types of experiments show remarkably similar behavior. A final of discussion is how specifically the simple ideas of a characteristic strain, constraint factor, and constant relating the indentation strain rate to the uniaxial strain rates are either upheld or disputed. To begin, it is important to point out that no clear cut method still exists for comparing indentation creep data to uniaxial creep data. However, comparing the indentation data from this study to both the experimental and literature data does reveal some important insights. Certainly in the case of the Pb - 65 at% In data at room temperature, a direct comparison with the uniaxial data at 10% strain shows the most promising correlation suggesting that those data are representative of small strain plasticity. If the hardness is divided by 3.3 and plotted along with the uniaxial data, the two sets of data observed to come remarkably close to one another. are The issue simply becomes with these two sets of data, "How close is close enough?" The comparison of the indium data to the uniaxial data from the literature also shows that the constraint factor serves to bring the two sets of data closer together, again justifying the concept. The steady state like behavior of the constant pip data can be better understood by considering the observations of Weertman who found that the creep rates of In were found to be constant over a wide range of strain, typically 5 to 25%. Recalling that the characteristic strain of a Berkovich indenter is approximately 8 to 10%, the strains experienced in the indentation creep test were perhaps of sufficient 160 magnitude to reach steady state conditions in the material. This is in direct contrast to the Pb-65 at% In data where 10% strain is observed to be still in the transient region of the stress strain curve. The correlation of the indentation creep Al data to the uniaxial data can also be explained by considering the characteristic strain concept. As previously discussed, over the temperature range studied, the dominant feature of the stress-strain behavior of Al is work hardening[77]. Therefore, at small strains, such as 10%, the stress for a given strain rate is expected to be significantly lower than the steady state flow stress. This would result in apparently higher stress exponents for creep than would be obtained at higher strains. As the temperature of the material is raised, the required strain to reach steady state behavior is expected to decrease. This would result in a stress closer to the steady state value being obtained at lower strains. The comparison of the indentation creep data on AI seems to follow this explanation. At low homologous temperatures, the stress exponent obtained from the indentation data is observed to yield higher stress exponents at similar strain rates similar to that of the small strain data of Trozera et al[77]. As the temperature is raised, the stress exponents are observed to approach those obtained from the steady state data of Luthy et al [40]. Why the constraint factor does not serve to more appropriately account for the observed differences is can perhaps be explained by the fact that the surface of the Al sample was still in a work hardened state from the mechanical polishing. This would result in less work hardening and hardnesses closer to the uniaxial steady state data. An interesting note is that even though the indentation data appears to go from a transient like to steady state like behavior over the range of temperatures studied, temperature compensating the data based on the effective diffusion coefficient concept is observed to bring the data together onto a single master curve suggesting that the mechanisms responsible for the deformation are the same in the indentation test. 161 8.2 Suggestions for Future Research In light of the results from this work, several opportunities exist for extending the study of indentation creep. First, it would be most interesting to repeat the experiments at the elevated temperatures using the constant Pip loading technique. This would allow a more thorough study of the temperature dependence of indentation creep at these temperatures without the ambiguities associated with the changing indentation strain rate and hardness. An important additional amount of work, now that the experimental details have been worked out, would be to perform experiments that specifically test the ideas of characteristic strain, constraint factor, and constant relating indentation strain rate to uniaxial strain rate. An example would be a series of indentation experiments on specimens that have been pre-strained various amounts, for instance, Al at room temperature tested at a variety of strain rates. By performing tests out to 10% strain increments, then performing indentation creep tests on the pre-strained samples, the concept of a characteristic strain could perhaps be more fully investigated. An even more challenging study would be to attempt to perform constant strain rate indentation experiments with a spherical indenter where the strain is changing as a function of depth rather than remaining constant as is the case for the geometrically similar indenters. Information on how the indentation data can then be related to uniaxial stress-strain data could then perhaps be obtained. While the indentation creep technique has been proven to yield appropriate stress exponents for both viscous flow (n=l) and dislocation climb controlled creep (n=5), no experiments have been performed on an alloywhere dislocation glide (n=3) is expected to be the rate controlling mechanism. An exciting extension of this would be to perform 162 experiments on a material that is known to undergo a glide to climb controlled transition where the stress exponent is expected to change. One aspect of the constant Pip experiment that was not addressed in this study is the transient associated with reaching an "indentation steady state." In a constant strain rate uniaxial test, the stress gradually builds up with time until such a point when the rate of hardening is balanced by the rate of recovery. How the dynamics of the indentation experiment manifest themselves in the transient response of the material is certainly a point of interest. Another exciting aspect in the continuation of this work would be an upgrade of the HTMPM to allow indentation creep properties to be investigated at higher absolute temperatures. Certainly in the case of AI, the temperature regime accessible with the currentca pabilities of the HTMPMis in the most complicated of all temperature regimes for creep. The ability to achieve higher operating temperatures would allow a wider range of relevant engineering materials to be investigated at high homologous temperatures. Obviously, these experimental techniques are aimed at the characterization of the creep properties of thin films andsmall volumes of material that cannot be characterized by existing bulk techniques. The creep properties of thin films may be very different from their bulk counterparts due to the different diffusion paths and microstructures. The understanding of the elevated temperature creep properties of these films would yield vital information to the microelectronics industry that cannot perhaps be obtained from any other technique. 163 Chapter 9 Developing Indentation Viscoelasticity Constitutive Equations 9. Introduction As discussed in Chapter 2, one phenomenological description for analyzing data from bulk viscoelastic testing techniques is given by a set of equations describing the dynamic response of the material as a function of frequency. In order to relate the response measured in a frequency specific indentation test to that measured in the bulk, an analogous set of equations for describing the response measuredwith the indentation test is required. 9.1 Analysis of Indentation Data The technique for measuring the viscoelastic properties of materials with depth sensing indentation techniques involves an extension of the dynamic indentation technique developed by Pethica and Oliver[78] for continuously monitoring the stiffness of the contact between the indenter and the materiaL The technique involves applying an oscillatory force to the indenter and measuring the amplitude and phase angle of the displacement response at that frequency and is more completely described in Appendix A dealing with dynamic modeling. However, in order to more completely describe the properties of a viscoelastic material, it is desirable to appropriately account for the mechanical losses that are occurring in the material as well. This is accomplished by modeling the contact between the indenter and the specimen in terms of a frequency dependent parallel spring and dashpot rather than a simple elastic element as was done 164 by Pethica and Oliver. This allows for the damping properties of the specimen to be considered in the analysis. The dynamic analysis of the indentation test can proceed in a manner similar to that outlined in the formal theory of viscoelasticity. The difference in the analysis is that in the indentation experiment the geometry of the test is dealt with according to contact mechanics in order to arrive at values for moduli. The analysis begins by writing the equation for Hooke's law in terms of complex notation, i.e., * * F =(S+iCm)h (54) where F* is the complex force, S is the storage or in-phase portion of the stiffness, Ceois the viscous or out-of-phase portion of the stiffness and h* is the complex form of the displacement response. The result that is of interest is II G* = G I ( m ) + iG ( m ) (55) where G* is the complex modulus given by the sum of the storage part of the modulus G' due to the stiffness S and the loss part of the modulus Gil due to the damping Ceo. If an oscillatory force of the form F=Foe(icot) is applied to the system, a displacement response of h=hoe(iol+q,) is obtained. A simple direct analysis can be constructed by considering the response equations for the dynamic system as given by Pethica and Oliver[78] but considering the more complex model for the material that includes both spring and dashpot elements. The analysis begins by writing equations (63) and (64) fromAppe ndix A for the amplitude and phase angle as 165 (56) and roCeq tan¢ = 2 (57) Keq -mro where Keq is the equivalent stiffness of the spring elements in the model (contact, indenter support springs, and load frame) and Ceq is the equivalent damping coefficient of the viscous elements in the model (contact and air in capacitive gap). By combining these two equations it is possible to solve explicitly for Keq and Ceq in terms of the amplitude and phase response. This analysis results in the expressions (58) and Fo . - = ro (59) S Ill At'f' Ce q flo These two equations can be expanded if the equivalent stiffness and damping coefficients are replaced by the appropriate quantities. Equations (58) and (59) then become 166 (60) and Fo sin= (Cro)-( Ciro) (6 1) ho Equations (60) and (61) can be solved independently for the stiffness of the contact, S, and the damping coefficient of the contact, Coo,by separating the displacement response of the indenter to the oscillating force into the in-phase and out-of-phase components respectively. 167 Chapter 10 Indentation Viscoelasticity Materials/Sample Preparation The material used in this study was an unvulcanized natural rubber or Hevea Rubber. It is obtained from the latex (or milk) of the tree Hevea brasiliensis. It has the chemical structure poly-cis l,4-isoprene as shown below Natural rubber is amorphous and rubbery at room temperature. It flows above 60°C and crystallizes on cooling below O°C or upon stretching. It has a density of approximately 1200 kglm3 and a glass transition temperature of -72 °C. Polyisoprene was chosen for a variety of reasons; 1.) due to its tack, it was possible to have stable contact areas under conditions of nearly zero load, and therefore zero creep; 2.) due to its relatively high loss modulus, Gil, at room temperature, 0.029 MPa, as compared to a storage modulus, 0', of 0.41 MPa[79], and; 3.) its dynamic behavior has been well documented over a wide range of frequencies [80]. The polyisoprene specimen was obtained in bulk form. The specimen was prepared for indentation by mechanically shearing the surface in a manner similar to microtome preparation. The actual rms roughness of the surface is not known but is not expected to effect the results due to the large depths of indentation used for the study. 168 Chapter 11 Indentation Viscoelasticity - Experimental Techniques 11.1 Introduction Two distinct types of indentation experiments were conducted to investigate the response of polyisoprene both as a function of the excitation frequency (Constant Contact ArealVariable Frequency) and as a function of the size of the contact area between the indenter and the material (Constant FrequencyNariable Contact Area). All of the experiments were conducted at room temperature (T=23.9°C) on a Nano Indenter® II operating in a frequency specific indentation mode. The fundamentals of the operation of the instrument in this mode may be found in Appendix A, in Pethica and Oliver[78], or in Oliver and Pharr[3]. 11.2 Constant Contact Area/Variable Frequency Figure (64) is a typical load versus time history for a constant contact area/variable frequency experiment. This experiment was designed to investigate the frequency response of polyisoprene at a constant contact area. The experiment consisted of the following: 1. The indenter approached the surface at a constant velocity of 10 nmls until contact was detected. 2. The load on the indenter was then increased at a constant loading rate to a predetermined maximum loads of either 150, 300,lor 450 I..LN. 169 Constant Contact ArealVariable Frequency - z ---E "'C <0 o .....I 0.001 -0.001 50 100 150 200 3600 Time (s) Figure 64. A plot of a typical load time history for constant contact area/variable frequency experiment conducted on polyisoprene. 170 3. The load on the indenter was then held constant for a 15 s period during which the displacement versus time response was monitored. 4. The load on the indenter was then decreased at a constant unloading rate to a predetermined minimum load. 5. The load on the indenter was again held constant for a 15 s period during which the displacement versus time response was monitored. 6. The load on the indenter was then increased back to approximately zero load, i.e., the initial load at the point of contact. 7. The load on the indenter was then held constant for a period of 1 hr during which the dynamic scan through the spectrum of frequencies was performed. 8. The indenter was unloaded at a constant unloading rate until the contact between the indenter and the material was broken .. The dynamic scan through the range of frequencies during Segment 7 of the experiment was conducted in a single frequency mode using an EG&G 5302 digital signal processing CDSP) lock-in amplifier. The range of frequencies investigated was from 50 mHz to 500 Hz. At the outset of the experiment, the phase angle between the output of the oscillator of the lock-in amplifier and the measured force oscillation actually being sent to the load application coil was set to zero degrees. All of the data were then acquired by first measuring the actual force excitation at the excitation frequency and maintaining the phase angle between the output of the oscillator and the measured force 171 at zero degrees In actuality, the stability of the oscillator never required the phase angle between the output of the oscillator and the resulting force oscillation to be reset after an initial set at the beginning of the experiment. The displacement response to the force excitation was subsequently measured as well as the phase angle between the two. While a measurement of the amplitude of both the in-phase and out-of-phase components was available from the lock-in amplifier, the difference in magnitude of the two components and the inability to set the sensitivity of the measurement independently prevented this fe ature from being used. Measurements were made at a frequency of 0.5 Hz at the beginning of the 1 hour hold period, at the 30 minute mark, and at the end of the hold period to ensure that the size of the contact was in fact remaining stable. 11.3 Constant FrequencyNariable Contact Area In order to more clearly definethe relationship between the damping coefficient, Coo,and the area of contact between the indenter and the material, a second type of experiment was performed in which the frequency of the excitation was held constant as the indenter was loaded into the material. This experiment consisted of loading the indenter at a rate of approximately 10 nmlswhile applying an oscillatory force at a fixed frequency of 1.1 Hz and a fixed force excitation of 1.5 J.!N. This type of experiment allowed the dynamic response of the material to be characterized as a function of depth at a constant frequency. 172 Chapter 12 Indentation Viscoelasticity - Results and Discussion 12.1 Introduction The central theme of this chapter of this dissertation is to present the results of the frequency specific indentation measurements made on bulk poly isoprene. Experiments from both constant contact area/variable frequency and constant frequency/variable contact area experiments will be presented in an attempt to quantitatively measure the viscoelastic properties of polyisoprene. 12.2 Constant Contact AreaIVariable Frequency Figure (65) is a plot of load versus displacement for the load-time history shown in Figure (64). Note that while significant amounts of both forward and reverse creep were observed at the upper and lower holds under constant load during the 15 s periods, only a very small amount of creep was observed during the 1 hr hold at nearly zero load and the majority of this displacement was observed to occur during thefirst few seconds of the hold. The fact that the area of contact remained nearly constant throughout this entire period was confirmed by performing reference measurements at a frequency of 0.5 Hz at the beginning, halfway through, and at the end of the hold period. Figure (66) is a plot of the dynamic compliance, the ratio of the displacement response to the force excitation, as a function of frequency showing the response of the free-hanging indenter as well as the composite response of the system with the indenter in contact with the material under conditions of constant contact area for three different peak loads. Note that not only has the compliance of the composite system decreased, 173 15s hold� . • . • . • 200 • • • • • • • • •• • • • Loading •• • •• • •• • 100 Segment • • •• • •• • - • • • • Z •• 1hr hold : ::::1. •• -- :, : "'C 0 : Cd • •. •: • • • 0 • • • .....I • • • • • • • • • • • • F orce •. •. •: 100 • • - • • • • • • • • • • • • ,- : : . .. 200 • . - • • �15 s hold • • • •• 300 - � o 5 10 15 20 25 Displacement (Jlm) Figure 65. A typical plot of load versus displacement for polyisoprene during constant contact area/variable frequency experiment. Note the stability of the contact during the 1 hour hold at near zero load. 174 Polyisoprene ...- z ...... ""-"E Ll...... c:: - Q) ()c ca 10-2 C. E0 () () caE c >. Free Hanging Indenter 0 • - - 6. - - 450 IlN Indent - -0 - - 300 JlN Indent - - 0 - -150 JlN Indent -3 ��������--��� 10 1 1 1 0- 1 00 1 0 1 02 Radial Frequency, (0 (rad/s) Fiiure 66. A log-log plot of dynamic compliance, hIF, versus radial frequency showing response of free hanging indenter as well as the response of the indenter in contact with polyisoprene under conditions of three different constant contact areas. 175 i.e., the dynamic stiffness has increased, the response of the system as a function of frequency appears to have taken on a frequency dependence, i.e., dynamic compliance is no longer approaching a constant value as was observed for the free hanging indenter but is continuing to increase as the frequency is lowered. Figure (67) is a plot of the phase angle, 4>, between the applied force oscillation and the resulting displacement oscillation for the same set of experiments. By using these two sets of data and equations (60) and (6 1) it is possible to calculate both the contact stiffness, S, and contact damping coefficient, Ceo. The results of this analysis are shown in Figure (68) which is a plot of both S andCeo versus the radial frequency, eo. The storage part of the stiffness is related to the storage modulus, 0', by equation (47) (G=E/2(1+v), with v=0.5 for natural rubber) which is a well known result from classical contact mechanics. It is proposed that the loss modulus could be related to the contact damping coefficient in the same fashion, i.e., (62) Figure (69) is a plot of both the storage and loss components of the modulus calculated according to equations (47) and (62) as a function of the radial frequency. It is seen that both expressions reasonably bring all three sets of data together onto a single curve as a function of frequency. While the general trend of the data suggests that the hypothesis in equation (62) was correct, a more definitive experiment to test the conclusion was desired. 176 Polyisoprene 160 • Free Hanging Indenter - 450 flN Indent - -6 - -300 flN Indent - -0- - - -150 flN Indent ---0 120 -e- .. CD - C) " c: I'" " ..... Q' 0 2 3 10- 10 Radial Frequency, co (rad/s) Figure 67. A semi-log plot of phase angle, <\>,versus radial frequency showing response of free hanging indenter as well as the response of the indenter in contact with polyisoprene under conditions of three different constantcontact areas. 177 S 450 • /IN Indent S 300 /I • N Indent • S 150 /IN Indent • • E 2 • 2 t • 10 • 10 ...... • Z • • • CJ --- • • $l) • • • • • en • • 3 .. • -C _. (J) • • :::l (J) (Q Q) ... c: ::= () +=i 8 l:::. l:::. en - 0 l:::. � 0 0 l:::. l:::...... 0 0 Z (.) 0 ...... CU l:::. 0 0 0 0 3 .....c: 0 0 --- 1 0 0 8 10 101 0 l:::. Cm 450 /IN Indent 0 Cm 300 /IN Indent Cm 150 /IN Indent 0 1 ° 2 10- 10 101 10 Radial Frequency, OJ (rad/s) Figure 68. A log-log plot of stiffness, S, and damping, Cm, versus radial frequency for polyisoprene under conditions of constant contact area. The data have been corrected for the contributions of the indentation system and represent only the contributions from the material. 178 1 1 (L=450 A G' �N ) I (L=300 • G �N ) I G (L=150 �N ) • • • - i I CO a.. I • 0r � • '-' A en en (!) s: - CJ) 0 ::J a. - c: ::J Polyisoprene -C c: 0 ...en � Ci) Q) C) 0.1 <> <> 0.1 � I::. � 0 <> <> 0 I::. <> "1J ..... I::. I::. � Q) (j) I::. '-' il (L=450 I::. G �N ) il (L=300 <> G �N ) il 0 G (L=150 JlN ) 1 10 100 Radial Frequency, ill (rad/s) Filjure 69. A plot of the storage modulus, 0', andloss modulus, Gil, as a function of radial frequency for polyisoprene under conditions of constantcontact area. 179 12.3 Constant Frequency/Variable Contact Area In order to more clearly define the relationship between the damping coefficient, Coo, and the area of contact between the indenter and the material, a second type of experiment was performed in which the frequency of the excitation was held constant as the indenter was loaded into the material. Each experiment consisted of loading the indenter at a rate of approximately 10 mnlswhile applying an oscillatory force of 1.5 JlN at a fixed frequency of 1.1 Hz. If the hypothesis is correct that the damping coefficient scales as the square root of the contact area then a plot of the measured damping coefficient versus indenter displacement should yield a straight line with the loss modulus then being given by I I =.J1iSlop e G OJ (63) 6 ...)24 .56 where the slope of the line is dC/dh where h is the contact depth which can be related to the area of contact for a perfectBe rkovich indenter by A=24.5h2. Figures (70) and (71)show the measured dynamic compliance and phase angle during the loading segment as a function of position within the capacitive plates as well as the response of the indentation head with no contact over the same absolute voltage range. While the voltage is linearly related to the displacement through a calibration factor, a knowledge of the absolute position of the center plate in the gap was crucialfor appropriately correcting the measured response for the contributions of the indentation system. By using the calibration data for both the damping coefficient of the indentation head and the stiffness of the support springs as a function of position (as detailed in Appendix A) it was possible to subtract out the contributions to the measured response due to the machine leaving only the response due to the contact between the indenter 180 .-. z ...... E 0.015 - u...... c: - Q) ()c: ctS .- -0- ·0.01 0E 0 .-() ctSE c: � 0.005 £:) -- Indenter Polyisoprene Response + --fr-- Free Hanging Indenter o --�� - 4 -3 -2 - 1 0 1 2 3 4 Indenter Position in Capacitive Gap (V) Figure 70. A plot of dynamic compliance versus absolute displacement of capacitive plate in gap for conditions of no contact between the indenter and sample and conditions of increasing contact area between indenter and polyisoprene. The contribution of the indentation system was removed based on the dynamic characterization discussed in Appendix A. 181 -- Indenter + Polyisoprene Response --!:r- Free Hanging Indenter - 0 - 60 -e- - Q) en c:: « Q) C/) co ..c:: D- 30 0 ...... -4 -3 -2 -1 0 1 2 3 4 Indenter Position in Capacitive Gap (V) Figure 71. A plot of the phase angle versus absolute displacement of capacitive plate in gap for conditions of no contact between the indenter and sample and conditions of increasing contact area between indenter and polyisoprene. The contribution of the indentation system was removed based on the dynamic characterization discussed in Appendix A. 182 and the material. Figure (72) is a plot of the calculatedstif fnessof the contact versus the depth of the indenter. Note that the data are linear as is expected from theory. Figure (73) is a similar plot for the calculated damping coefficient of the material versus the depth of the indenter. Note that these data are also linear supporting the earlier hypothesis. A storage modulus of 0.37 MPa can be calculated from the slope of the plot of stiffness versus indenter depth and a loss modulus of 0.054 MPa from the plot of the damping coefficient versus the indenter depth. Both of these values agree well with the data reported in the literature for polyisoprene[16]. One final check as to the validity of the results can be made by adding the data from the fixedfr equency/variable contact area experiments to the plot obtained for the variable frequency/constant contact area experiments. Figure (74) is the data from Figure (69) replotted with the results of the fixed frequency experiments. Considering the completely different dynamics of the experiments, good agreement is observed between the two types of measurements. ),: ". '," , 183 AS Slope = Mt -Iii Slope . '-- E' - - G - • 5 200 V - 0 3 6 �24.5' - E G' =0.37 MPa ...... z - '" 150 en en OJ c :::: ':.j:j (j) 100 t5 Cd ...... c o () 50 -- s = 1.1097 + 6.2101h; R= 0.99957 o o 5 1 0 15 20 25 30 35 40 Displacement (J.lm) Figure 72. A plot of contact stiffness, S, versus displacement for polyisoprene. The slope of the line is related to the storage modulus of the material. Good agreement is found between these results and literature values for polyisoprene. 184 de Slope = M 5 - E ""'- Gtf= 0.054 MPa en Z - 4 ...., c . Q)- () '+ '+- Q) 3 o o 0') c . - 0. 2 E en o 1 -- c = 0.37045 + 0.14006h; R= 0.99678 o o 5 10 15 20 25 30 35 40 Displacment (Jim) Figure 73. A plot of the contact damping coefficient, C, versus displacement for polyisoprene. The data are observed to be reasonably linear over the range of displacement measured. The slope of the line is related to the loss modulus of the material. 185 1 1 • G' Loading • G' , L=450 G' L=300 • , • ' ... G , L=1 50 • -ctS i • c.. r * • 0 • CJ) � • CJ) ---- • (!J s: '" 0 en a. :::J c: :::J c: "'0 CJ) 0 ... � G> Q) - 0> 0 0.1 0 & 0.1 s: � � 0 0 ""'0 0 f::. 0 § 0 0> +-' 0 � C/) ---- l G i Loading il 0 G , L=450 il G , L=300 l Gi L=1 50 f::. , 0.1 1 10 100 Radial Frequency, ro (rad/s) Figure 74. A log-log plot of the storage modulus, G', and loss modulus, G", of polyisoprene as a function of radial frequency showing data from constant contact area/variable frequency experiments and constant frequency/variable contact area experiments. 186 Chapter 13 Indentation Viscoelasticity - Conclusions and Suggestions for Future Research 13.1 Conclusions Using a depth-sensing indentation system operating in a frequency specific measurement mode, it was possible to measure both the storage and loss modulus of polyisoprene as a function of frequency. The relationship between the measured damping coefficient andthe loss modulus was confirmed to scale as the square root of the contact area between the indenter and the material as was previously known for the relationship between the stiffness of the contact and the storage modulus. The results obtained from the frequency specific indentation test agree well with the bulk values reported in the literature. 13.2 Suggestions for Future Research The directions for future research in this area are virtually limitless as this work presents the first such study of viscoelastic behavior using frequency specific depth sensing indentation techniques. The application of the technique to other materials to completely validate the results contained herein is therefore high on the priority list of future work. Obviously, the goal of this work is aimed specifically at the characterization of the viscoelastic properties of thin films and small volumes of material that cannot be characterized by existing bulk techniques. As such, specific instrumentation needs are required due to the scaling down of the depth of the indentation as required for the testing of these small volumes. While the dynamics of the indentation head are able to 187 be modeled quite accurately, the losses occurring in the indentation head itself due to the capacitive displacement gauge are expected to be large compared to the response from the material as the size of the indentation is reduced. Hardware is therefore needed to reduce the losses in the system to a level such that the measured response is sensitive to the damping coefficient of the contact. The characterization and utilization of this new hardware is therefore ofgreat interest. 188 Bibliography 189 Bibliography 1. M. F. Doerner and W. D. Nix, J. Maters. Res., 1, 601 (1986). 2. S. I. Bulychev, V. P. Alekhin, M. K. Shorshorev, A. P. Ternovskii, and G. D. Shnyrev, Zavod. Lab., 41, 1137 (1975). 3. W. C. Oliver and G. M. Pharr, J. Mater. Res., 7, 1564 (1992). 4. O. D. Sherby, Acta Met., 10, 135 (1962). 5. O. D. Sherby and A. K. Miller, Trans. ASME,J. Engr. Mat. Tech., 101, 387 (1979). 6. S. N. G. Chu and J. C. M. Li, 1. Mater. 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Tabor, Proc. Phys. Soc. London, 78, 169 (196 1). 194 Appendices 195 Appendix A. Dynamic Modeling To determine mechanical properties with a depth-sensing indentation test it is necessary to know four quantities: the displacement of the indenter into the material, the load on the indenter, the time of the measurement, and the "stiffness" of the contact between the indenter and the material. Stiffness has been enclosed in quotes here to denote that there can be more than one component of the stiffness that is of interest. In a standard depth-sensing indentation test, the load on the indenter is varied while the displacement and time of the experiment are continuously monitored yielding three of the four quantities. The stiffness of the contact can only be obtained during a partial unload of the contact when the relationship between the load and the displacement is being determined by the elasticity of the problem. A standard depth-sensing indentation test must therefore involve at least a loading and unloading segment to completely characterize both the elastic and plastic properties of a material. A technique to avert the necessity of unloading the contact in order to measure the stiffness has been developed. This technique involves treating the contact problem dynamically and modeling the problem as a system of masses, springs and dashpots as normally seen in vibrational analysis[78]. In the simplest sense, the contact between the indenter and the material can be thought of as an elastic element or a spring with a stiffness related to the Young's modulus of the material through a geometric factor. In a more complex sense, the contact can be thought of as possessing both the capacity for storing energy (stiffness) and the ability to dissipate energy (damping coefficient). By appropriately modeling the dynamics of the system and the contact it is therefore possible to completely model the indentation problem on a differential basis. 196 A.1 Calibration and Characterization of Nano Indenter®II Before delving into the properties of any material it is first necessary to characterize the experimental apparatus such that the role that it plays in the measured experimental response can be accurately accounted for. Any dynamic system in general can be thought of as including a means for storing potential energy (springs or elastic elements), a means for storing kinetic energy (mass or inertia elements), and a means for dissipating energy (damper or dashpot)[63]. The first step in developing the dynamic model for the indentation system is to examine the physical makeup of the system and develop a model that represents all the important components of the system for the purpose of deriving analytical equations governing thebehavior of the system. Figure (5) shows a schematic of the components of the Nanoindenter II necessary for consideration of a dynamic model. The indentation column of mass m is supported by two leaf springs of stiffness Ks which are designed to have very high stiffness in the plane of the springs and to have relatively low stiffness in the vertical direction, effectively limiting the motion of the indenter to a single degree of freedom. The indenter is driven by a coil and permanent magnet assembly, the coil being rigidly attached to the top of the indentation column. The displacement of the indenter column is measured with a parallel plate capacitive displacement gauge. This introduces a damping coefficient Ci into the system which is expected to vary as the fourth order of the position in the gap. When the indenter is not in contact with the specimen, the system can be modeled as a single degree of freedom simple harmonic oscillator with viscous damping. As briefly described above, the force on the indenter is generated by a coil and permanent magnet assembly. The magnitude of the force is controlled by varying the current through the coil residing inside the magnetic field. It is also possible to add a 197 harmonic component to the current that is driving the coil. This achieved by using an oscillator from a lock-in amplifier or similar device. By applying a harmonic force of known amplitude at a specific frequency, and measuring the displacement response of the system at that frequency and the phase angle between the two, it is possible to treat the indentation system dynamically. The system previously described is quite easy to analyze as it is simply a as single-degree of freedom, damped system undergoing forced vibration. The displacement response of this system, h=hoe(irot+$), is easily found for a driving force of the form F=Foe(irot) to be (64) and the phase angle, $, between the applied force andthe resulting displacement is Cm tan l/J= _�l --;:- (65) K -mm2 s One technique to investigate the dynamic response of the experimental apparatus is to apply a harmonic force at different frequencies and measure the displacement response at that frequency. Figure (75) shows a plot of the dynamic ® compliance, hIF, as a function of the radial frequency for the Nano Indenter II for the case when the indenter is not in contact with the sample. Note the absence of a resonant peak in the curve suggesting that the system is over-critically damped. By fitting this curve with equation (64)it is possible to determinethe values for the dynamic constants 198 1 1 0. .:------.�--...---.----..--...... -""'!I 11K s 1/C co ---\. i Free Hanging Indenter 6 10· __� ______� __ � __ � 0.01 1 100 104 Radial Frequency, co (rad/s) Figure 75. Log-log plot of dynamic compliance versus radial frequency for the Nano Indenter® II free hanging indenter. The data show no resonance peak: indicating that the system is over-critically damped. The data are fit extremely well by the equation for a simple harmonic oscillator. 199 of the system. The results of such a fit are given in Table (3). Note the near perfect fi t of the data by the simple harmonic oscillator function. Table (3), Summary of dynamic constants for the NanoIndenter ® II obtained from fit of equation (64) to the dynamic complianceversus radialfr equency. Support spring stiffness, ks 27 N/m Damping coefficient, Cj 6.4 Ns/m Indenter mass, mj 0.006 kg Correlation coefficient,R 0.999999 While this analysis precisely determines the mass of the indenter, the damping coefficientand the stiffness of the leaf springs are known to vary over the range of travel and must therefore be characterized as a function of position. This is accomplished by stepping the indenter through its range of motion within the capacitive gap and measuring the dynamic response at each position at a fixedfreq uency. These results can be analyzed by simultaneously solving equations (64) and (65) for the stiffness and damping coefficients. By solving these two equations it is possible to obtain a functionalfit for both ks and Cj as a function of absolute position. Figures (76) and(77) show the results of this analysis conducted at a frequency of 1.1 Hz. This specific frequency was chosen as Ks and Cjffiare of nearly equal magnitude at this frequency. A polynomial fit to these data was used to obtain a functional relationship for both Ks and Cj as a function of position. The results of these curve fits are given in Tables (4) and (5). 200 28 - E ...... "-"z � 26 "" (/J +""C ctS +""en c o () 24 C) c " ;:: C en 22 o --()- K from Amplitude s and Phase Angle 20 -4 -3 -2 -1 0 1 234 Indenter Position in Capacitive Gap (V) Fi�ure 76. A plot of Nano Indenter® II support spring stiffness as a function of the absolute position of the indenter with respect to the capacitive displacement gauge. The data were fit by a third order polynomial to determine a functional relationship for the data. 201 -o- c. from Amplitude 12 I - and Phase Angle E en z 11 o ...... c 10 .Q)- (.) � ...... Q) o 9 o C> c a.. 8 E ctS o 7 6 -4 -3 -2 -1 0 1 234 Indenter Position in Capacitive Gap (V) Fi&ure 77. A plot of Nano Indenter® II damping coefficient as a function of the absolute position of the indenter with respect to the capacitive displacement gauge. The data were fit by a fourth order polynomial to determine a functional fit for the data. 202 Table (4). Coefficients of third order polynomial fi t to the Nano Indenter® II support spring stiffness as a function of absolute position within the capacitive gap. Co 27.312 ct 1.2173 C2 -0.13229 C3 0.0034337 Correlation Coefficient, R 0.996 Table (5). Coefficients of fourth order polynomial fit toind enter damping coefficient as a function of position within the capacitive gap. Co 6.4435 ct -0.015829 C2 0.39336 C3 0.0026807 C4 0.015915 Correlation Coefficient,R 0.9999 A.2 Modeling the Elastic Contact Once the indenter is brought into contact with the sample the analysis then involves appropriately modeling the system and the contact between the indenter and the sample. This has been done for the case of the simple elastic response of the material by Pethica and Oliver[78]. The result is the introduction of two additional springs into the dynamic model; one representing the contact (S). the other representing the finite stiffness of the load frame (Kf). The amplitude response of the system is then found to be 203 (66) and the phase difference between the force and displacement signals is given by (67) The stiffness of the load frame can be determined from a method outlined by Oliver and Pharr[3]. This model can be extended to include damping or energy losses in the material as discussed in the section on viscoelastic measurements with the Nano Indenter® II. A.3 Calibration and Characterization of HTMPM Figure (78) shows a schematic of the components of the HTMPM necessary fo r consideration in a dynamic modeL The indenter is supported inside the indentation head by two leaf springs designed to have very low stiffness in the z direction (vertical indentation direction) and to be very rigid in plane, thereby limiting motion of the indenter to only the z-direction. The indenter can therefore be thought of as a mass, mi, supported by a spring, Ks, and dashpot, Ci, in parallel as shown schematically in Figure (78). The specimen stage shown schematically in Figure (78) is mounted to a long cantilever tube that is rigidly mounted to a fi ve axis manipulation system. In the simplest sense, it can be modeled as a mass mst supported by a spring Kst (cantilever tube) and dashpot Cst in parallel as shown in the lower portion of Figure (78). When the two are not in contact they may each be considered as a single degree of freedom 204 Dynamic Machine Response, No Contact Ks c h D tv o VI Mass of Stage F Figure 80. A schematic ofthe components of the HTMPM for consideration in the dynamic model. Shown are: A -Sample; B-Indenter column assembly with mass mi; C-Load application coil; D-Indenter support springs with stiffness Ks; E-Reference mirror; andF -Specimen support stage with mass mst. Damping coefficient Cst> and Stiffness, Kst .. system, i.e., the motion of each mass may be completely described by one coordinate axis. The dynamic properties of the indenter and stage can be determined by analyzing the vibratory response of the separate mass-spring-dashpot systems when excited by an external force. First, consider the free oscillation of the indenter when excited away fr om its equilibrium position. Figure (79) is a plot of the indenter displacement as a function of time in response to an impulse load. By analyzing this data it is possible to obtain all of the dynamic properties of the indenter mass-spring-dashpot system. Figure (80) is a plot of the cycle number of the oscillation versus the time of oscillation. The slope of this line yields the damped natural frequency of the system. Figure (81) is a plot of the absolute value of the peak indenter displacement as a function of time showing the exponential decay of the free oscillation. The exponential term in the curve fit is determined by the damping coefficientof the indentation head. The dynamic constants determined for the HTMPM indentation head are s arized in Table (6). umm Table (6). Summaryof dynamicconstants for HTMPM indentation head. Measured Quantities Applied Load, F 10.2 JlN Resulting Displacement, X 161 nm 1 Decay Constant, 'Y 11.132 s- Freguency of Free Oscillation, f 14.48 Hz Calculated Quantities Support Spring Stiffness, Ks 63.2 N/m Damped Natural Frequency, rod 91.0 s-1 Natural Frequency, ron 91.7 s-1 Mass of Indenter, mj 7.518 g Damping Coefficient, Cj 0.167 Ns/m 206 Free Vibration of Indenter 100 50 ..-. E • c: • --- • • • +-'c: 0 • • • Q) • • • • • • E • • Q) • • • • () • • ctS • -50 • a. en . V - 0 • -100 -150 0.1 0.2 0.3 0.4 0.5 Time (s) Figure 79. A plot of displacement versus time showing free vibration of HTMPM indenter in response to step load. The frequency of the oscillation and the exponential decay of the oscillation are representative of the dynamics of the indentation head. 207 4 1 Free Hanging Indenter -- y = 0.48771 14.483x R= 0.99997 + o --�--�----�--�----�� o 0.05 0.1 0.1 5 0.2 0.25 0.3 Time (5) Fiiure 80. A plot of cycle number versus time yielding the damped natural frequency of the HTMPM indenter mass, spring, dashpot system. 208 - Decay of amplitude of free c:E - oscillation of indenter ...... c: Q) E 150 Q) (,) ctj a. en .- o .::s!.ctj Q) 100 CL ..... o Q) :J ctj > Q) 50 ...... -:J o en ..c Fi�ure 81. A plot of the absolute peak displacement of the HTMPM indenter versus time during free vibration showing exponential decay of the vibrational amplitude. The decay constant is representative of the damping coefficient of the indentation head. 209 Next consider a similar analysis for the specimen stage. In order to study only the motion of the stage, the two interferometer beams that normally strike the indenter mirror were blocked off by a stationary mirror placed on the sapphire viewport. The stage was then excited into vibration by first stepping the indenter onto the surface in order to move the stage away from its eqUilibrium position, and then stepping the indenter off of the surface to observe the free vibration ofthe specimen stage. Figure (82) is a plot of the specimen stage displacement during free vibration as a function of time. By analyzing these data it is possible to obtain all of the dynamic properties of the stage mass-spring-dashpot system. Figure (83) is a plot of the cycle number of the oscillation versus the time of oscillation. The slope of this line yields the damped natural frequency of the stage system. Figure (84) is a plot of the absolute value of the peak stage displacement as a function of time showing the exponential decay of the free oscillation. The dynamic constants determined for the stage are summarized in Table (7). Table (7). Summary of dynamic constants for HTMPM specimen stage. Measured Quantities Stage Stiffness, Kst 6.67x105 N/m Decay Constant, 'Y 5.76 s-l Frequency of Free Oscillation, f 23.05 Hz Calculated Quantities Damped Natural Frequency, rod 144.81 s-l Natural Frequency, COn 144.92 s-l Mass of Stage, mst 31.76 Kg Damping Coefficient, Cst 365.5 Ns/m 210 Free Vibration of 600 Specimen Stage 400 ..-... § -- 200 ...... c CD CDE 0 () a:s �-200 o -400 -600 -800 o� 0.1 0.2 0.3 0.4 0.5 Time (s) Figure 82. A plot of displacement versus time showing free vibration of the HTMPM specimen stage in response to a step load. The frequency of the oscillation and the exponential decay of the oscillation are representative of the dynamics of the specimen stage. 211 14------ y = 0.50791 + 23.047x R= 0.99999 -- 12 10 '-Q) ..c 8 E ::J Z .::t::.C\1 Q) 6 c.. 4 2 Free Vibration of Specimen Stage o o� 0.1 0.2 0.3 0.4 0.5 0.6 Time (5) Figure 83. A plot of cycle number versus time yielding the damped natural frequency of the HTMPM stage mass, spring, dashpot system. 212 -... Decay of amplitude of free E c oscillation of specimen stage -- +-'C 600 Q) EQ) (.) as 500 C en o .::s:. m 400 c...... o Q) � 300 > .$::J o 200 en .c « 100 o�--��--�� 0.1 0.2 0.3 0.4 0.5 Time (s) Figure 84. A plot of the absolute peak displacement of the HTMPM stage versus time during free vibration showing exponential decay of the vibrational amplitude. The decay constant is representative of the damping coefficient of the specimen stage. 213 A second method for investigating the frequency dependence of a system is to drive the system at a specific frequency and force and monitor the amplitude of the resultant oscillation at the excitation frequency. This is extremely important for the HTMPM since the specimen stage must be considered as a sink for vibrational energy during dynamic experiments. By plotting the amplitude of the motion of the stage as a function of the driven frequency it is possible to determine the frequency regime over which the stage has little or no effect in the measured damping. Figure (85) is a plot of the resultant amplitude of the stage as a function of the driven frequency. The resonant peak for the system occurs at · a frequency of approximately 22 Hz which corresponds nicely with the natural frequency obtained fromthe free vibration experiments. In order to measure material properties with the HTMPM it becomes necessary to extend the dynamic model that was developed for the system in terms of its parts to include the contact between the indenter and the material. In the simplest sense, the contact canbe modeled as an additional spring and dashpot. The contact between the indenter and the specimen couples the motion of the two single degree of freedom systems. The analysis is no longer a simple one-degree of freedom system, but a more complex two degree of freedom system requiring two independent axes to completely describe the motion of the two masses. For the dynamic analysis it is convenient to consider the indenter, of mass mI, damping coefficient CI and support spring stiffness k}, being driven by a force of the form F=Foe(iwt) andbeing displaced in the x-direction about the axis (x is used rather Xl than z fo r the vertical axis to avoid ambiguity with the impedance matrix). The specimen stage, of mass m2, damping coefficient C3, and stiffness k3, is then considered as being driven by the motion of the indenter and is displaced in the x-direction about 214 2500 Forced Vibration of Specimen Stage - Ec --- Q) '"C :::J ...... - c.. E « co c 0 ..... � .0 > 500 o o 10 20 30 40 Frequency (Hz) Fi!:ure 85. A plot of amplitude versus frequency for forced vibration of the HTMPM specimen stage. The stage was excited using the indenter in contact with the reference mirror of the stage. A resonant peak was observed at a frequency of approximately 22 Hz. 215 the X2 axis. The two masses are coupled by the stiffness of the contact, k2, and the damping coefficient of the contact, C2. The solution to this mathematical model is a set of equations of the fonn [ m ].i+ [ C ]i + [k ]x = F (68) where [m] is the mass matrix, [c] is the damping matrix and [k] is the stiffness matrix. From the force balance, the equations of motion for the two coupled masses can be written as mIX] +{Cl +C2)XI -C2X2 +(k1 +k2)xr -k2X2 = Fo (t) (69) m2x2 + (C2 + C3)X2 - c2Xl + (k2 + k3)X2 - k2Xr = 0 The general problem of a two degree of freedom system under external forces is described in Rao[63] on page 245 as well as in Seto[81] Chapter 2. Using the Mechanical Impedance Method, by inspection from Rao, equation (69) can be rewritten as Definingthe impedance matrix as (71) 216 where Zi= F. X is then given by Z-I F and by inspection from Rao, it is seen that (72) and (73) where 2 Zl 1 = [-m ml +im(cl +c2 )+(k1 + k2 )] 2 222 = [-m m2 +im(c2 +c3)+(k2 + k3 )] (74a, b, c) Z12 = [-im c2 - k2 ] For simplicity, let (kl+k2)=kA, (Cl+C2)=CA, (k2+k3)=kB , and (C2+C3)=CB . Equations (74a, b, c) then become ZI I = [kA - m2ml +imcA ] 2 Z22 = [kB - m m2 + imcB] (75a, b, c) ZI2 = [-im c2 -Is] The denominator of the equations (72) and (73), ZlIZ22 - Z;2' can be simplified to the form C+iD where 217 and Therefore equation (72) for X 1 becomes where A=(kB-mw2) and B=( (79) Similarlyfor X2, F __ + __ G + iH -Z k2 iOJC2 x 1 2 O F F (80) 2 -- 2 - - -- Z Z22 - 7 C + I'D C + I'D 11 "42 0 0 where G=k2 and H=C2W. The magnitude of X2 is then (81) 218 It is the ratio (XI-X21X2) that is of importance in this analysis as that ratio defines the re lative motion of the two masses, however, a complete description of this dynamic problem involves completely defining the motion of the two masses in terms of the amplitude and phase angle. Recalling from equations (79) and (81) that Xl and X2 can be written as the ratio of two complex numbers, fromcomplex number identities, X I and X2 can be rewritten as A+iB i Xl = . Fo = Fo (K + zL) = Xl I ¢I (82) C+ID I e . where G +iH i¢2 X2 = . Fo = Fo(M + iN) = X2 I (83) I e C+ID . I where (84) and (85) The ratio (XI-X2)1X2 can easily be found from equations (82) and(83) to be 219 It is seen that the relative motion of the two masses is controlled by the elements that couple the indenter and stage well as by the properties of the stage. Figure (86) is a as plot of equation (86) a function of frequency for the stiffness, 3.3x105 N/m, and as damping coefficient,7 N slm, calculated from the oscillatory response for In to the 9 mN step load. While the plot says nothing about the absolute magnitude of the displacements of either the indenter or the specimen stage, it can lend valuable information. It was previously determined that the resonant frequency of both the indenter and the stage in the uncoupled mode were 14.5 and 23.0 Hz respectively. The logical interpretation of the increase in the ratio for increasing frequency is that the amplitude of the very massive stage is going to zero. This plot in conjunction with Figure (85) showing the response of the stage to forced vibration lends strong support to the hypothesis that the damping coefficient determined from the step load experiments is representative of the losses in the sample. 220 4 5 10 Contact Stiffness=3.3x10 N/m Damping Coefficient= 7 Ns/m 102