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Creep in pure single and polycrystalline aluminum at very low stresses and high temperatures: an evaluation of Harper-Dorn creep
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Creep in pure single and polycrystalline aluminum at very low stresses and high temperatures: an evaluation of Harper-Dorn creep
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CREEP IN PURE SINGLE AND POLYCRYSTALLINE ALUMINUM AT VERY LOW STRESSES AND HIGH TEMPERATURES: AN EVALUATION OF HARPER – DORN CREEP by Praveen Kumar A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) August 2007 Copyright 2007 Praveen Kumar ii DEDICATION To my mother Late (Smt.) Deomati Devi (Rai) iii ACKNOWLEDGEMENTS It gives me great pleasure to express my gratitude towards my supervisors Professors T G Langdon and M E Kassner for their continued guidance, encouragement and financial support during the entire period of my doctorate study. Continuous discussions with them have always given me the right perspective for the experiments and analysis of the data. I am grateful to them for granting me long and repetitive vacations during this period which allowed me to visit India and show my love and service to my ailing mother. I wish their laid foundation will always show me the way in my future research endeavors as well as in academic life. I am grateful to my dissertation committee member, Professor Edward Goo for his evaluation of my Ph.D study. I would like to take this opportunity to show my sincere thanks to Professor W Blum who not only allowed me to use the creep facilities in his laboratory at the University of Erlangen – Nuremberg (Germany) but also educated me through many scientific discussions. Also, his social interactions with me made my stay in Germany relatively much easier. I want to thank my colleagues at University of Erlangen - Germany, Drs. Philip Eisenlohr and P Sadarbadi for discussions related to creep and its microstructural aspects. I want to show my gratitude to Drs. H Chilukuru, S Mekala and Y Li for taking care of my experiments during my absence. I also want to thank technicians especially Mr. R Kosemala for their help in conducting experiments. I want to pass my sincere thanks to Professors Mughrabi and Gokhen for their encouragement during my stay in Germany. I would like to take this opportunity to thank my colleagues at USC for iv maintaining a healthy research environment in the lab. I want to say emotional thanks to my friends for their support and encouragement during this period. Without them, it would have been difficult for me to maintain enthusiasm and focus on research. Last but not the least, I would like to thank my father and other family members for their support, encouragement and inspiration which let me remain focused on the current academic assignment. v TABLE OF CONTENTS DEDICATION ii ACKNOWLEDGEMENTS iii TABLE OF CONTENTS v LIST OF TABLES ix LIST OF FIGURES x ABSTRACT xxiv CHAPTER 1. INTRODUCTION 1 1.1. Creep Processes 1 1.2. Debate over Harper – Dorn Creep 6 1.3. Significance of Creep at Very High Temperatures and Low 8 Stresses 1.3.1. Geological Applications 8 1.3.2. Commercial Application 9 1.3.3. Scientific Endeavor 9 CHAPTER 2. REVIEW OF LITERATURE 11 2.1. Major Experimental Features of Harper – Dorn Creep 11 2.2. Other Materials Systems where Harper – Dorn Creep has 29 been Reported 2.2.1. Cu 29 2.2.2. α - Zr 32 2.2.3. α - Fe 34 2.3. Harper – Dorn Creep in Ceramics and Geological Materials 36 2.4. Experimental Observations that are Inconsistent with Harper – 51 - Dorn Creep 2.4.1. Burton on pure Al 51 2.4.2. Muehleisen et al on pure Cu and Cu alloys 51 2.4.3. Blum and co-workers 52 2.4.4. Greenwood and Co-workers 54 2.5. Recent Observations by Mohamed and Co-Workers 56 vi 2.5.1. Effect of Impurity 56 2.5.2. Restoration Mechanism 59 3.5.3. Effect of Strain 62 2.6. Proposed Theories for Harper – Dorn Creep 62 2.6.1. Harper and Dorn 62 2.6.2. Friedel 63 2.6.3. Barrett, Muehleisen and Nix 63 2.6.4. Langdon and Yavari 65 2.6.5. Wu and Sherby 66 2.6.6. Nabarro 67 2.6.7. Wang and Co-workers 67 2.6.8. Ardell and Co-workers 68 2.7. Additional Observations on Harper – Dorn Creep 71 2.7.1. Weertman and Blacic 71 2.7.2. Raj and Co-workers 72 2.7.3. Nes and Co-workers 73 2.8. Compensation for The Friction 75 CHAPTER 3. MATERIALS AND EXPERIMENTAL PROCEDURE 80 3.1. Materials 80 3.1.1. Single Crystals 80 3.1.2. Polycrystals 82 3.2. Sample Preparation: General Procedure 86 3.2.1. Cylindrical Specimens 86 3.2.2. Square Cross – Sectioned Specimens 88 3.3. Testing Conditions 90 3.4. Creep Machines 91 3.4.1. University of Erlangen – Nuremberg, Germany 91 3.4.2. Institute of Physics of Materials, Brno 98 3.5. Grain Boundary Sliding Measurements 102 3.6. Microstructure Study 105 3.7. Features of Present Work 106 CHAPTER 4. EXPERIMENTAL RESULTS 109 4.1. Creep Testing: Single Crystals 109 vii 4.1.1. Sample 1 109 4.1.2. Sample 2 121 4.1.3. Sample 3 127 4.1.4. Sample 4 133 4.1.5. Sample 5 138 4.1.6. Sample 6 142 4.2. Creep Test: Polycrystals /Oligocrystals 146 4.2.1. Sample 7 146 4.2.2. Sample 8 146 4.2.3. Eisenlohr’s Data 154 4.3. Microstructural Study 160 4.3.1. Grain Boundary Sliding 160 4.3.1.1. Stress = 0.07 MPa 160 4.3.1.2. Stress = 0.12 MPa 163 4.3.1.3. Stress = 0.3 MPa 167 4.3.2. Etch Pit study 171 CHAPTER 5. DISCUSSION 180 5.1. Replotting the figures on Normalized Strain Rate and Stress Co- 180 Ordinates for Single Crystals 5.2. Comparison of the Present Study on Single Crystals with 188 Earlier Studies 5.2.1. Harper and Dorn 188 5.2.1.1. Comparison with Present Study 189 5.2.1.2. On the Threshold Stress 196 5.2.2. Barrett, Muehleisen and Nix 197 5.2.2.1. Comparison with Present Study 199 5.2.3. Mohamed and Ginter 206 5.2.3.1. Comparison with Present Study: 106 5.2.4. Ginter, Chaudhury and Mohamed 211 5.2.4.1. Comparison with Present Study 215 5.2.5. Overall Comparison 216 5.3. Dislocation Density Variation in Single Crystals 219 5.4. Possibility of a Size Effect in Single Crystals 221 5.5. Natural Power Law for Single Crystals? 222 viii 5.6. Polycrystal / Oligocrystal Results 225 5.7. Grain Boundary Sliding 230 5.8. Composite Plot 234 CONCLUSIONS 236 REFERENCES 238 ALPHABETIZED BIBLIOGRAPHY 247 APPENDICES 256 Appendix I 256 Appendix II 257 Appendix III 259 Appendix IV 261 Appendix V 268 ix LIST OF TABLES Table 1: Characteristics of creep mechanisms [13] 5 Table 2: Experimental Reports of Harper – Dorn creep 12 Table 3: Tensile creep tests on pure Al at very high temperatures and low stresses 13 Table 4: Compressive creep tests on pure Al at very high temperatures and low stresses 13 Table 5: Creep of Single Crystals 45 Table 6: Creep of Polycrystals 47 Table7: Values for D and G 50 Table 8: The Mean Free Path of the Dislocations in the Low Stress Range 74 Table 9: Experimental data and corresponding correction for κ 0 = 0.93 and an initial stress of 0.1 MPa 79 Table 10: Impurities associated with the single crystal Al with purity of 99.999 % 85 Table 11: Impurities associated with the polycrystal Al with purity of 99.99 % 85 Table 12: Features of the single crystal specimens used in the present study 110 Table 13: The stress exponent at various stress changes for Sample 2 124 Table 14 (a): Dimensions of Sample 3 prior to the start of the creep test 130 Table 14 (b): Dimensions of Sample 3 after the creep test 130 Table 15: The oligocrystalline specimens used in the present study and the earlier study of which the present study is a continuation. 147 Table AI: The shear modulus and the diffusion coefficient of pure Al at various temperatures. 260 x LIST OF FIGURES Fig 1: A schematic showing the flow of vacancies in Nabarro - Herring creep and Coble creep (modified and reproduced from [12, 13]). 4 Fig 2: A schematic showing various possibilities at the low stresses indicating the ambiguity of around an order in the deformation rate [15, 18, 23, 24, 35, 36]. 10 Fig 3: Temperature compensated strain rate vs. normalized stress for pure aluminum, showing data reported in the Harper – Dorn regime by Harper and Dorn [15], Barrett et al [18] and Mohamed et al [19]. The behavior predicted theoretically for the Nabarro – Herring creep is also indicated. The plot shows consistency in the creep behavior between the single crystals and polycrystals in the Harper – Dorn regime. The figure is adapted and reproduced from [43]. 15 Fig 4: Correlation between the creep behavior of bulk specimens and thin foils of Al. d c represents the critical grain size for the transition from Nabarro – Herring creep to Harper – Dorn creep. The plot is adapted and reproduced from [71]. 16 Fig 5: TEM micrograph of Al after creep testing in the Harper – Dorn regime at (a) 0.019 MPa and (b) 0.025 MPa [35]. The pictures show extensive step formation, indicating the possibility of cross – slip as the dislocation multiplying process. 18 Fig 6: (a) A random distribution of etch pits in the specimen of Al – 5% Mg tested at 0.14 MPa to within steady state flow in Harper – Dorn creep. (b) Evidence of the formation of subgrain boundaries in a specimen tested at 0.74 MPa to within steady state flow in the dislocation climb region [43]. (a) is tested in Harper – Dorn regime whereas (b) is tested in conventional 5 – power law regime. 19 Fig 7: TEM micrograph of pure Al (99.9995 % purity) in the Harper – Dorn regime after very large strain (~0.18) showing the wavy nature of the grain boundaries [26]. Wavy nature of the grain boundaries indicates the possibility of dynamic recrystallization as the restoration or recovery process during the creep deformation. 19 Fig 8: A reproduction of the log (strain rate) vs. log (applied stress) for Al tested at 923 K from [23]. Al (a) and Al (b) are prepared by cold work with the purity level of 99.999 % and 99.9995%, respectively whereas Al (c) is produced by hot work and has a purity of 99.99 % [35]. All of these samples had higher initial dislocation densities. 22 xi Fig 9: Variation of the dislocation density with stress for (a) Al – 5 % Mg which is mechanically similar to pure Al (reproduced from Table 2 of [43, 75]) and (b) pure Al [18, 35, 76]. 23 Fig 10: Typical creep curve for Al pre-strained at room temperature prior to testing at high temperature and low stress. The plot is reproduced from [18]. 25 Fig 11: Example of crystal recovery and creep recovery at 920° K in pure Al [15]. After removing the load, the sample undergoes a small recovery, hence leading to a net positive strain. In the case of vacancy diffusion creep, the recovery is complete and hence no net positive strain after unloading is observed. 26 Fig 12: Frequency of occurrence of the angle, Φ, between the dislocation line and the Burgers vector, in angular increments of 22.5°, for a specimen of Al – 5% Mg tested at 0.14 MPa to within steady state flow in Harper – Dorn regime [43]. The histogram indicates that most of the dislocations produced during deformation in Harper – Dorn regime are of edge type in nature. 27 Fig 13: Grain boundary shearing contribution to the total creep strain as function of stress [16]. 29 Fig 14: The steady-state creep behavior of copper [21, 31, 82-93]. The low stress regime shows a change in the stress exponent. 31 Fig 15: Compilation of various studies on α – Zr [52, 54, 96 - 102]. If δD gb is used for low stress data, they show good consistency [39]. 33 Fig 16: Compilation of various studies on α –Fe [50, 104 - 108]. The data from Cadek and Milicka [102] at a grain size of 216 μm were interpreted as Harper-Dorn creep by Fiala et al. [52]. The steady- state α-Fe data appear best normalized with D gb at low stress and D sd at high stress to best fit the data. 35 Fig 17: Normalized creep rate versus normalized stress for polycrystalline [57] and single crystal [114] CaO. 37 Fig 18: Normalized creep rate versus normalized stress for polycrystalline [115] and single crystal [116-120] LiF. 38 Fig 19: Normalized creep rate versus normalized stress for polycrystalline [121, 122] and single crystal [60, 123- 126] MgO. 39 xii Fig 20: Normalized creep rate versus normalized stress for polycrystalline [127] and single crystal [63, 128, 129] NaCl. 40 Fig 21: Normalized creep rate versus normalized stress for polycrystalline [130] and single crystal [68, 131] dry olivine ((Mg 4 Fe) 2 SiO 2 ). 41 Fig 22: Normalized creep rate versus normalized stress for polycrystalline [132] and single crystal [69] forsterite (Mg 2 SiO 4 ). The notation given for the single crystals denotes the loading direction with respect to the largest lattice parameter, c of the orthorhombic crystal. 42 Fig 23: Creep rate and stress relation at steady state as reported in Fig 3 of Blum and Maier [23]. Here Machine A and Machine B are the two sets of the creep machines used in the experiment. Machine A is without and Machine B is with the measurement of the load by load cell. The load acting on the specimen in machine A was determined from the external load neglecting friction in the load train [23]. 53 Fig 24: A reproduction of the results reported in [22, 23, 18, 15]. The plot compares the low stress creep behavior of aluminum at 873 to 913 K. Results have been normalized to 920 K using activation energy for creep of 149 kJmol -1 [15]. The results show scatter and very low strain rates. 55 Fig 25: (a) Strain rate vs. applied stress on a logarithmic scale for 99.99 Al and 99.9995 Al tested at 923 K. The figure has been adapted from [27] (b) Strain rate vs. applied stress on a logarithmic scale for 99.999 Pb and 99.95 Pb tested at 587 K. The figure has been adapted from [95]. 57 Fig 26: Creep curve in tension showing accelerated creep suggested to be associated with dynamic recrystallization (a) High purity Al (99.9995%) [26] (b) Highly pure lead [149]. The plots have been adapted from [26] 60 Fig 27: TEM micrograph showing new grains on the surface [26]. 61 Fig 28: Dislocation spacings (1/√ρ) versus normalized stress for pure Al, Al-5% Mg and NaCl in the H–D regime [18, 40, 169, 43, 170, 64, 116, 149]. Data for Al, LiF and Al-5% Zn in 5-power regime and lines proportional to bG/σ are shown for comparison. The transition to Harper – Dorn is suggested to occur at σ/G = 10 -5 for NaCl and 10 -6 for Al. Note:- The values for physical constants (G and b) for Al-5% Mg and Al-5% Zn used for plot is taken to be same as of pure Al. 70 xiii Fig 29: A schematic showing the friction cone for two samples with aspect ratios of 4 and 1.5, respectively [176]. The dotted line shows the axis of compression, l is the length of the sample and w is the width of the sample. 76 Fig 30: The relation between the nominal (σ 0 ) and the actual stress as experienced by the sample (σ). 78 Fig 31: The compensated steady state strain rate versus the modulus compensated steady state stress for pure Al (reproduced from [36, 182]). PLB stands for power law breakdown. 82 Fig 32: Variation of the Schmidt factor for the 12 most preferred slip systems of the FCC system with the deviation angle from the [100] orientation. The numbers 0 to 90˚ (FCC crystals have rotational symmetry of 90˚ with respect to [100]-axis) on the x – axis shows the azimuthal angle (the angle of sweep on the surface of the cone formed by the deviation vector when rotating it with respect to the [100]-direction). 83 Fig 33: The brass jacket – sample assembly used to avoid deformation in the sample while cutting it using the slow – speed saw. 87 Fig 34: The schematic of the grinding process to achieve homogeneous cross – section. 89 Fig 35: A digital photograph of the creep machine at University of Erlangen – Nuremberg. 92 Fig 36: The schematic of the optimized furnace coiling. This type of coil is very effective in reducing the slacking of the coils. 93 Fig 37: A schematic showing the TiO 2 strips over the sample surface. 96 Fig 38: A digital photograph of the creep machine at IPM, Brno. The pictures were taken from different angles to show all salient features of the equipment. 99 Fig 39: A schematic of the cage – type of holder for the specimen. (a) shows the schematic at IMP Brno, whereas (b) shows a possible set – up with proper alignment using a guide on 6lower holder as well as supporting rods for the upper holder. 100 xiv Fig 40: Appearance of three grains in a polycrystalline matrix (a) before tensile creep with longitudinal marker line AA’ and BB’ (b) after Harper – Dorn creep, and (c) after diffusion creep; the tensile axis is vertical [12]. 104 Fig 41: A digital photograph of Sample 1 after the creep test. The first row pictures show the side view whereas the second row of the pictures show primarily the top and bottom view of the specimen. 111 Fig 42: The stereographic projection of the active slip planes with respect to the [100] – direction. The red dots indicate the active slip planes. The most active slip plane seems to be (111), followed by ( 1 11). 112 Fig 43: The strain vs. time plot of Sample 1 (Al1SX1.0L E Cy). The long and almost flat part in the plot is due to the long term test at the lowest stress (0.06 MPa). 114 Fig 44: The strain rate vs. strain behavior of Sample 1. These strain – rate data are the “as measured” values from the experiment without any correction for friction or Taylor factor. The lines with slope of 4.5 (stress exponent for pure Al at very high temperature) are drawn. If the test condition is ideal then these lines should interpolate the region between the steady – state datum points between the same engineering stress, as the present test is a constant load test. Since, there is a missing strain (~10 %) in the measurement by laser system and also steady – state has not been reached in all cases, the lines with slope 4.5 do not interpolate the datum points between the same engineering stresses. 115 Fig 45: The reproduction of Fig 44 after compensating for the temperature (from the test temperature of 913 K to the plotting temperature of 923 K) based on the lattice diffusion coefficient ratios. The black dotted lines are connected as interpolating “parallel” lines between the same engineering stresses. The dark large circle at the stress of 0.36 MPa at the strain of 0.16 is taken as the reference point. This point is chosen as reference point as it seems to be in steady–state and also this strain is thought to be free from any kind of bulging effect (low strain). 116 Fig 46: The stress exponent calculated at different stress changes following the interpolation technique. The stress exponent is close to ~3 for the stress change to the lowest stress (0.06 MPa). 117 xv Fig 47: The stress dependence of the strain rate for Sample 1 for a temperature of 923 K. The large solid square represents the reference point whereas other points are derived based on the stress exponent and the reference point. The low stress behavior shows a deviation from both 5 – power law and Harper – Dorn creep. 120 Fig 48: The strain vs. time plot of the Sample 2. Engineering stresses are in MPa. Vertical lines show a stress change. 122 Fig 49: The strain rate vs. strain behavior of Sample 2. The points are joined by spline curves and an interpolation technique is used to join the data points of same engineering stress (as shown by dotted lines). The sudden jump in strain rate at ~1.2 % is attributed to the inhomogeneity in deformation with time and space. The larger circle shows the reference point, which was chosen according to the guidelines explained in the previous subsection. 123 Fig 50: Representation of the present work on stress - strain rate axes. The data for Sample 1 is also shown in order to see the consistency between the results on two different samples. The comparison shows excellent reproducibility of the data from sample to sample. 126 Fig 51: The strain – time plot for Sample 3, as recorded by the miscalibrated LVDT system. Based on the final and initial length measurement, strain accumulated in Sample 3 is ~3.8 % in the time period of 3.2 × 10 6 s (instead of 0.046 % as shown in the plot). 128 Fig 52: The strain rate – strain behavior for the Sample 3, as calculated from the LVDT data. It should be noted that the strain shown in the above plot is only 0.046 % whereas 3.8% of strain was accumulated in the sample. 129 Fig 53: The stress – strain rate behavior of the samples. The triangle with an error bar shows the 2/3 rd of the average strain rate for the sample at the stress of 0.04 MPa. The error bar is a factor of ±2 of the strain rate used in the plot. 132 Fig 54: The strain – time behavior of Sample 4. The cooling water circuit problem started after the vertical dotted line. A distinct change in the strain – time plot is seen before and after the occurrence of the cooling water problem. The circled regions show the time period in which strain was measured in the central 1/3 rd region of the specimen. 135 xvi Fig 55: (a) The strain rate - strain behavior of Sample 4. The accumulated strains are very small and hence, a conclusive prediction about the steady – state strain rate is not possible. Based on prior experience for the test under similar conditions, it can be speculated that the steady-state is very close to the final strain-rate shown in the above figure. (b) The strain rate - strain behavior of Sample 4b. It shows steady – state behavior. 136 Fig 56: The last strain rate result of Sample 4 shown by the inverted dark triangle. Also, the strain rate data for Sample 4b is shown. These datum points are substantially below the Harper - Dorn transition region but at the same time are faster than the corresponding 5 – power law prediction. 137 Fig 57: The strain – time behavior of Sample 5. The stress of 0.68 MPa (which is underlined) has a larger uncertainty associated with it. 139 Fig 58: The strain rate variation with strain for sample 5. The oscillation in the strain rate is attributed to the oscillation in the loading which occurs due to the evaporation of water. Due to the uncertain rate of evaporation towards the end of the test, the last load was uncertain. It is indicated by an underlined font in the above plot. The representative steady – state strain rate values are shown by the dotted horizontal lines. The stress exponents are shown at each stress jump. 140 Fig 59: The comparison of the results of Sample 5 with other observed results. The data point shows excellent consistency with the other set of the tests conducted during the present study. 141 Fig 60: The strain – time behavior of sample 6 shown along with sample 4. 143 Fig 61: The strain rate – strain behavior of Sample 6. The points corresponding to the same engineering stress are joined by an interpolating curve and the stress exponents were derived. The large dark circle shows the reference data for the test. 144 Fig 62: The stress – strain rate behavior of sample 6 with respect to other results in the present study. Lines with slopes 4.5 and 3 are shown in the plot. The results show very good consistency. 145 Fig 63: The strain – time behavior of Sample 7. 148 xvii Fig 64: The strain rate variation with strain for sample 7. The oscillation in the strain rate is attributed to the oscillation in the loading which occurs due to the evaporation of water. A stress exponent equal to ~5.4 is calculated based on the interpolation technique. The large circle shows the reference point for the test. 149 Fig 65: The comparison of the results of Sample 7 with other observed results. The data points of single and oligocrystals show a good match at moderately high stresses. The dotted lines represent the results from the single crystals and they show a good agreement at moderately high stresses if the Taylor factor is considered. 150 Fig 66: The strain – time behavior of Sample 8. The dotted lines show the interruption in the test in order to measure grain boundary sliding. 151 Fig 67: The strain-rate variation with the strain for sample 8. The strain rate shows a large reduction during the transient phase of the creep due to the interruption of the test after every stress. The stress exponents remain close to ~5. 152 Fig 68: A comparison of the results of Sample 8 with Sample 7. Sample 8 does not show any change in the stress exponent at the lower stresses as was shown by single crystals. 153 Fig 69: The strain rate – stress behavior of the single crystals and the oligocrystals deformed under compression in the present study. The Taylor factor is not included and the single crystals seem to be more susceptible to creep. The grain size for the oligocrystal samples was ~10 mm. 155 Fig 70: Reproduction of Fig 69 after compensating for the Taylor factor. The stress values for the single crystals were multiplied by a factor equal to 1.25. The test results show remarkable consistency for oligocrystals and single crystals at moderate stresses. Grain size for the oligocrystal samples was ~10 mm. 156 Fig 71: Strain rate – strain plot for various oligocrystals tested under compression by Dr. Eisenlohr. The segments with underlined engineering stress values are in steady – state and the shaded circles represent the corresponding strain rate. 157 Fig 72: The strain rate – stress variation of the oligocrystals tested by Eisenlohr. The strain rate values are based on Fig 71. The stresses have been modified for strain (i.e. true stress) and friction. 158 xviii Fig 73: A comparison of all results from the present work. At low stresses, the datum points from Eisenlohr are within a factor of 2 with respect to the single crystal data whereas the oligocrystal tested in the present work differs by a factor of 4 with the single crystal data. 159 Fig 74: A digital image of the sample after deformation at the stress of 0.07 MPa. The sample was deformed to a strain of 0.7 %. 161 Fig 75: Several features of GBS at a stress of 0.07 MPa after a strain of 0.7 %. The sample did not show any deviation from the 5-power law even though the stress belongs to the Harper – Dorn regime. The compression axis is horizontal in the plane of the paper. 162 Fig 76: A digital picture of the oligocrystal after deformation to a strain of 1.6 % at 0.18 MPa. The grains are fairly large in size (~ 10 mm) and the boundaries are clearly visible. 164 Fig 77: Some of the salient features related to grain boundary under a stress of 0.18 MPa after a strain of 1.6 %. 165 Fig 78: Some of the features related to grain boundary under a stress of 0.30 MPa after a strain of 1.6 %. 168 Fig 79: A comparison to the values of the GBS for the present study and the study of Harper et al. [20]. The values of GBS contribution are similar even though the trend in the strain rate – stress plot is different. 170 Fig 80: A schematic showing the shapes of the etch pits corresponding to various planes in a FCC material. The guideline for the shape of the pits are helpful in distinguishing the actual pits on the <100> plane and other surface features and irregularities. 172 Fig 81: Etch – pits micrographs of annealed specimen (just prior to loading) as observed under optical microscope at a magnification of 50X. The pits show a non-uniform distribution in the annealed specimens after etching for 8 seconds. (a) shows more pits relative to (b). 173 Fig 82: Etch – pits micrographs of Sample 1 as observed under optical microscope at a magnification of 200X. The last true stress used to deform the specimen was 0.3 MPa. Due to substantial bulging and slip activity observed at the surface of the sample, it was difficult to get the entire specimen focused at the same time. The surface was oxidized heavily and it is possible the electro-polishing and etching agent was not sufficient to show all dislocations (as the oxide layer might not been etched away) 174 xix Fig 83: Etch – pits micrographs of Sample 2 as observed under optical microscope at a magnification of 100X. The pits show a non- uniform distribution. The last true stress used in order to deform the sample was 0.052 MPa. (a) shows more pits relative to (b). 175 Fig 84: Etch – pits micrographs of Sample 5 as observed under optical microscope at a magnification of 100X. The last true stress used to deform the specimen was 0.9 MPa. The sample shows a very high dislocation density relative to last two samples tested in the low stress regime. Although the dislocation density was higher than earlier samples, no evidence of any sub-structure formation was observed. 176 Fig 85: Etch – pits micrographs of Sample 6 as observed under optical microscope at a magnification of 100X. The pits show a non- uniform distribution. The last true stress used in order to deform the sample was 0.018 MPa. (a) shows more pits relative to (b). 177 Fig 86: The stress dependence of the dislocation density as measured by etch – pits. The dislocation density does not remain constant in the low stress regime. The dislocation density increases with stress and it is consistent with the projected values based on 5 – power law observations. The references for this figure are [18, 19, 35, 169, 76]. 178 Fig 87: A comparative observation of the three early studies suggesting Harper - Dorn creep in pure Al [15, 18, 27]. All tests were conducted in tension. 181 Fig 88: Comparison of the present study with respect to the three earlier studies suggesting Harper – Dorn creep in pure Al [15, 18, 27]. The dark points represent the present study and they are not consistent with the Harper - Dorn prediction 182 Fig 89: A comparative representation of the three important studies prior to the present study not confirming Harper - Dorn creep in pure Al [23, 24, 36]. All the tests were conducted in compression. 184 Fig 90: Comparison of the present study with the three main studies [23, 24, 36] not confirming Harper - Dorn creep in pure Al. The dark points are from the present study and they are far away from the n = 4.5 line at the low stresses. 185 xx Fig 91: A comparative representation of the main studies on pure Al at very low stresses and at a temperature close of 0.99 T m . Dark symbols are from the present study. Sample 3 data are ambiguous as they are based on average strain rate. The first three studies [15, 18, 35] used tensile specimens whereas the remaining studies [23, 24, 36] used compression specimens. 186 Fig 92: A reproduction of Fig 91 with additional data by McNee et al [22]. The tensile data of McNee et al [22] shows an extensive scatter in the low stress regime. 187 Fig 93: The strain – time behavior of the Harper and Dorn samples (polycrystals) [15]. Only the samples showing positive strain rates are shown. Dark points show the samples which showed 5 – power law creep whereas open symbols lie in the Harper – Dorn regime. The study reported a negative strain (~ -0.00025 in 3.0 × 10 5 s) for a stress of 0.019 MPa. 190 Fig 94: The strain rate – strain plot for the Harper and Dorn study [15]. 191 Fig 95: The strain rate – strain plot showing a comparison of the present study with the Harper and Dorn report [15]. Actual stresses for Harper and Dorn are shown in the plot. Only (actual) stresses lower than 0.09 MPa are shown for which there was corresponding data in the present study. The “capped” lines with underlined numbers show the factor by which the values differ. 192 Fig 96: The strain rate – stress plot showing a comparison between the Harper – Dorn report and the present study. The term “present study” refers only to the single crystals, Harper - Dorn (true) corresponds to the true applied stress whereas Harper - Dorn (net) corresponds to the stress value after subtracting the threshold stress (~0.02 MPa.) from the actual stress (i.e. σ net = σ true - σ threshold ). 194 Fig 97: The extrapolation of strain rate – stress relationship to zero on linear – linear axes. It indicates a zero strain rate for zero applied stress and hence a threshold stress was not observed in the present study. 195 Fig 98: Variation of stress exponent with stress in the Harper – Dorn material [15]. The stress exponent increases at the lower stress with a decrease in the stress value but it is not drastic and hence does not clearly indicate the existence of a negative threshold stress. 198 Fig 99: A typical strain – time behavior of the polycrystalline Al samples tested by Barrett et al [18]. The sample deformed at 0.035 MPa was already pre-strained to a strain of 2 %. Unlike the Harper and Dorn study, the other strain – time plots were not shown in the report, hence the total strains achieved in several tests are not known. 200 xxi Fig 100: The strain rate – strain behavior of the creep data of Barrett et al [18]. The data points do not show the attainment of steady – state creep and the strains observed in the tests were less than 1 % which decreases with a decrease in the stress. 201 Fig 101: Comparison of the results obtained in the present study with the date of Barrett et al [18] data on strain rate – strain axes. A possible smooth extrapolation of the Barrett et al [18] data is drawn showing a consistency with the present test. 202 Fig 102: Strain rate – stress plot showing a comparison between Barrett et al [18] datum points and the present study. Only the datum points of Barrett et al [18] which might have achieved higher strains are shown here. Both studies show good agreement. 203 Fig 103: Strain rate – stress plot showing a comparison between Barrett et al [18] datum points and the results obtained in the present study. All the datum points reported by Barrett et al [18] for the polycrystalline samples are shown here. The datum points corresponding to the lowest two stress points are different from the present study. 204 Fig 104: Schematic of the double shear specimen used by Mohamed and Ginter [35]. All dimensions are in mm. 207 Fig 105: Typical strain – time behavior for the Mohamed - Ginter samples [35]. The remaining strain – time curves were not shown in the report, hence the total strains achieved in several tests are not known. 207 Fig 106: Strain rate – strain behavior of the creep data of Mohamed and Ginter [35]. The data points show the attainment of steady – state creep. The oscillation in the strain rate – strain curve may be due to grain growth. The authors did not mention a possible cause for the oscillations. 208 Fig 107: Comparison of the present study with the data of Mohamed and Ginter [35] on strain rate – strain axes. The “capped” lines with underlined numbers show the factor by which the values differ. Ginter and Mohamed [35] performed tests using double shear specimens and the shear stress was constant throughout the deformation process. Hence the true stress value at 0.45 % of strain is calculated for the present sample in order to compare it with the Mohamed and Ginter datum points. 209 Fig 108: The strain rate – stress plot showing a comparison between the Mohamed and Ginter report [35] and the present study. In general, the datum points show good agreement with the present study. 210 xxii Fig 109: Strain – time behavior of the creep data of Ginter et al [26]. The other strain – time curves were not shown in the report, hence the total strains achieved in several tests are not known. 212 Fig 110: Strain rate – strain behavior of the creep data of Ginter et al [26]. The data points show substantial oscillation. The authors suggested dynamic recrystallization as a possible cause for the oscillations but it may also be due to grain growth [12]. 213 Fig 111: The strain rate – stress plot showing a comparison between the Mohamed and Ginter report [26] and the present study. In general, the datum points show good agreement with the present study. 214 Fig 112: Comparison of the present study on single crystals with datum points from the earlier studies suggesting Harper – Dorn creep and which are accompanied with strain – time curves [15, 18, 26, 35]. 217 Fig 113: Comparison of the present study on single crystal data points from the earlier studies showing a deviation in the stress exponent at the lower stresses [15, 18, 35, 26]. 218 Fig 114: Analysis of the results of the present study with respect to natural third power – law creep for A 3 of Eq. (30) equal to (a) 1.0 and (b) 0.8 224 Fig 115: A possible interpretation of the oligocrystalline / polycrystalline data based on a higher stress exponent value in the 5 – power law regime. The underlined number (4) shows the factor by which the datum point at the lowest stress is higher than the extrapolation based on 5 – power law. The vertical dashed line shows the transition stress which is consistent with the transition stress in the case of single crystal samples. 226 Fig 116: Comparison of stress exponent values in 5 – power law regime for various purity level Al samples [191, 192, 193]. Dotted lines show lines with slope of 4.5. The leftwards arrow show that a lower purity material is more creep resistant. 228 Fig 117: A comparison of the results obtained in the present study using polycrystalline samples with the Harper and Dorn results [15]. The dashed line shows the consistency between the theory and the experiment of GBS at the lowest stress. 231 Fig 118: A composite plot reproduced to show various studies targeting the Harper – Dorn regime [15, 18, 19, 22 – 24, 26, 35, 40]. 235 xxiii Fig A1: A schematic showing the upward movement in the strips drawn over the surface of the specimen (used for the strain measurement using the Laser technique). The continuous lines show the actual profile of the sample after deformation. Dotted lines are drawn to show various features, such as ideal deformation profile, the central volume which is deforming ideally as it is free from the friction cone effect, etc. 258 Fig A2: A comparative plot showing the prediction based on dislocation network theory. The slope of the theoretical line is ~ 3.2 and the strain rate values are within a factor of 1.5 from the best fit line. 269 Fig A3 Variation of Hall – Patch constant, k HP , with temperature. The continuous line is a best – fit curve which has been extrapolated to the test temperature (650˚C) of the present investigation. 271 Fig A4 A comparison of the single crystal data and the polycrystal data observed during the present investigation. Single crystal data have been compensated for Taylor factor (1.25) and Hall – Patch factor (1.01) by multiplying the stress values with these factors and for impurity effects (1.36 in 5 – power law regime and 1.23 for 3 – power law regime) by dividing strain rate values by these factors. 273 xxiv ABSTRACT Harper – Dorn creep was proposed for materials with large grain size deforming at very low stresses (σ / G ~ 10 -6 where σ is the applied stress and G is the shear modulus) and high temperatures (~ 0.95 – 0.99 T m , where T m is the absolute melting temperature). Recently, this creep mechanism has become controversial and several other creep mechanisms, such as 5 – power law and Nabarro – Herring creep, have been proposed as governing the creep mechanism in the Harper – Dorn regime. An extensive study was conducted to evaluate several features of creep in the Harper – Dorn regime in order to determine an unambiguous creep mechanism. Compressive creep tests were conducted in the Harper – Dorn regime using single and polycrystalline samples of high purity aluminum. A stress exponent of ~ 3 was observed for single crystal samples whereas polycrystalline samples did not show any transition in the stress exponent value from the 5 – power law regime to the Harper – Dorn regime. An etch – pits study for single crystals showed that the dislocation density varies as the square of the applied stress in the Harper – Dorn regime and is in accordance with 5 – power law creep. The marker line method was applied to study grain boundary sliding in the polycrystalline samples and it showed an abrupt decrease in the grain boundary sliding contribution to the total strain at the transition stress for the 5 – power law regime to the Harper – Dorn regime with sliding remaining negligible in the Harper – Dorn regime. A review was conducted to compare the results obtained in the present investigation with the earlier studies supporting the conventional Harper – Dorn creep and in most of the cases a good consistency was observed. In the Harper – Dorn regime, xxv natural 3 – power law is proposed as the governing creep mechanism for the single crystals and a qualitative analysis showing grain boundary effects on the dislocation motion is presented for the polycrystals. 1 CHAPTER 1. INTRODUCTION 1.1. Creep Processes Creep in a material is defined as a plastic deformation process which is continuous with respect to time and occurs at a stress lower than the yield strength of the material under standard conditions (room temperature and at a strain rate of ~10 -3 s -1 ). Creep is observed at any temperature but it becomes more pronounced at higher temperatures (≥0.5 T m , where T m is the absolute melting temperature) where diffusion- controlled processes are reasonably rapid. The creep rate in any crystalline solid is dependent upon the testing temperature, the applied stress and the microstructural characteristics of the material. Generally, the variation of strain with time exhibits three distinct regions: (i) primary region where the rate of strain decreases with increasing strain, (ii) a secondary or steady-state region where the strain rate remains essentially constant and (iii) a tertiary region where the strain rate accelerates to final fracture. It should be noted that some investigators distinguish steady – state creep and secondary creep. Steady – state generally refers to a fixed creep rate under a constant stress due to a steady (i.e., strain independent) substructure and where hardening is balanced by dynamic recovery. Often this substructure is defined in terms of the dislocation features such as subgrain size, (network) dislocation density, and subgrain misorientation. Secondary creep can appear independent of steady-state. For example, a minimum creep rate as an inflection between tertiary and primary creep may be treated as secondary creep rate. Many of the theoretical creep mechanisms developed to date are concerned with predicting the rate of flow within the secondary or steady-state region. 2 Under the conditions of steady – state deformation, the dependence of the normalized creep rate on the normalized applied stress is represented as: n p n G σ d b A DGb kT ε = & (1) with − = RT Q exp D D 0 (2) where ε & is the creep rate, k is Boltzmann’s constant, T is the absolute temperature, D is the diffusion coefficient that characterizes the creep process, G is the shear modulus, b is the Burgers vector, A n is a dimensionless constant, d is the grain size, p is the inverse grain size sensitivity, σ is the applied stress, n is the stress exponent, Q is the activation energy for the diffusion process that controls the creep behavior and D 0 is the frequency factor for diffusion. By 1950, two creep mechanisms 1 taking place in pure metals were proposed: i. Dislocation Climb Based Mechanisms: Creep at intermediate temperatures (~0.5 – 0.7 T m ) and at intermediate stresses is supposed to follow a five power – law mechanism with the following modifications in Eq. (1): n = 4 – 7, p = 0, D equal to the lattice diffusion coefficient. This mechanism is governed by the dislocation climb and is an intragranular process [2]. In practice, essentially similar power-law creep is 1 Later on, creep mechanisms based on grain boundary sliding were also proposed and developed [1]. Grain boundary sliding mechanisms are dominant only in the temperature range above 0.5 T m . In this report, these three mechanisms are later summarized with their special features. None of the studies so far have reported any conflict between the GBS mechanisms and Harper – Dorn creep which occurs usually at very high temperatures. Therefore it will not be discussed in detail in the present report of the study of Harper – Dorn creep. 3 observed in a very wide range of crystalline materials including metals [3, 4], ceramics [5, 6], geological minerals [7] and ice [8]. ii. Stress Directed Diffusion Models: Creep at higher temperatures (≥0.7 T m ) was proposed to follow a mechanism based on the stress – directed diffusion of vacancies. In this mechanism, migration of vacancies occurs from grain boundaries aligned transverse to the stress axis to grain boundaries parallel to the stress axis. The creep rate is directly proportional to the applied stress (i.e. n = 1 in Eq. (1)) and varies with the grain size. Nabarro [9] proposed a model which suggested the motion of vacancies through the lattice/grain and later Herring [10] developed the mechanism and deduced an inverse grain size sensitivity of p = 2 and a stress exponent of n = 1 together with a diffusion coefficient, D equal to lattice diffusion. Later, for materials with smaller grain sizes and deforming at slightly lower temperatures than for Nabarro-Herring creep (<0.7 T m ), Coble [11] 2 proposed a mechanism which described the flow of vacancies through the grain boundaries, instead of through the lattice and suggested values for the constants in Eq. (1) as: p = 3, n = 1 and D is given by the grain boundary diffusion coefficient. Fig 1 shows a schematic of the flow of vacancies under the conditions of Nabarro – Herring creep and Coble creep [12, 13] and Table 1 summarizes the features of the various creep mechanisms. 2 Coble creep is dominant only at temperatures in the range of 0.7 T m or lower and in materials with smaller grain sizes. For the study of creep at very high temperatures (~0.95 – 0.99 T m ) and in materials with large grain sizes (as in the proposed study of Harper – Dorn creep), Coble creep has minimal effect. Hence, this creep mechanism will not be discussed in great detail in this report. 4 Fig 1: A schematic showing the flow of vacancies in Nabarro - Herring creep and Coble creep (modified and reproduced from [12, 13]). 5 Table 1: Characteristics of creep mechanisms [13] Mechanism Q p n GE GBO Ref Intragranular dislocation mechanisms Dislocation Climb Q l 0 4.5 Yes No [2] Dislocation Glide Q SD 0 3 Yes No [14] Harper – Dorn Creep Q l 0 1 Yes No [15] Grain Boundary Sliding (GBS) GBS (d>λ) § Q l 1 3 No Yes (Rachinger Sliding) [1] GBS in Superplasticity (d<λ) Q gb 2 2 No Yes (Rachinger Sliding) [1] Stress Directed Diffusion Creep and Vacancy Flow Nabarro – Herring Creep Q l 2 1 Yes Yes (Lifshitz Sliding) [9, 10] Coble Creep Q l 3 1 Yes Yes (Lifshitz Sliding) [11] § λ is the stable subgrain size GE: Grain Elongation GBO: Grain Boundary Offset 6 1.2. Debate over Harper – Dorn Creep As described above, creep at very high temperatures in materials with large grain sizes was considered to be Nabarro – Herring creep until 1957 when Harper and Dorn [15] reported their creep results for pure Al (99.99% purity) at 923 K (0.99 T m ) and at very low stresses (≤ 0.1 MPa). For a grain size of 3.3 mm, Harper and Dorn reported strain - rates which were ~ 1400 larger than the Nabarro – Herring prediction for the same test conditions. They argued that the observed stress dependence of the creep rate was equal to 1 and hence it was a Newtonian type of viscous flow but governed by a dislocation – based process (motion of jogged screw dislocations) instead of the stress directed diffusion of vacancies as in Nabarro – Herring creep. Harper and Dorn proposed the following changes in the form of Eq. (1): n = 1, p = 0, D given by the lattice diffusion coefficient. Following the original experiment by Harper and Dorn, several consistent studies [16 - 19] on different material systems and with samples of various geometrical shapes and crystalline features (single or polycrystals) supported Harper – Dorn creep as a unique and an independent creep mechanism. Burton [20] reported an absence of Harper – Dorn creep in experiments on pure Al at very low stresses and Muehleisen et al [21] were not able to reproduce Harper – Dorn creep in pure copper in the Harper – Dorn regime. Later, McNee et al [22] conducted experiments using pure Al and were also not able to reproduce the data of the original experiment of Harper and Dorn [15, 16] above a stress of 0.04 MPa for unknown reasons. On bulk samples of pure Al (99.99% purity), Blum and co-workers [23-25] performed creep tests in compression and reported a stress sensitivity of ~ 6.6 instead of 1 in the Harper – Dorn regime; hence, they argued that Harper – Dorn creep is not a 7 genuine creep mechanism but can be described by the conventional five power – law mechanism in the Harper – Dorn regime. Later, Mohamed and co-workers [26 - 28] pointed out that most of the studies performed showing the existence of Harper – Dorn creep (Newtonian viscous flow) might not have attained the steady – state and hence a higher value of the stress exponent (~2.5) is expected when samples are deformed for a larger strain. Their argument also refers to the conclusion of Weertman [29] that the Newtonian behavior noted in samples crept for small strains under very low stresses may not represent the true behavior for large strains and at least a strain of 0.1 is needed to achieve steady – state [30]. In a recent study on OFHC Cu by Srivastava et al [31], a stress exponent of 2 was observed in the low stress and high temperature regime. At present, there is a lack of studies which can deal with the above conflicts. Also, numerous conflicting theories are proposed to explain the phenomenon of Harper – Dorn creep. An elaborate study investigating the effects of higher strain, sample size, purity level of the sample, crystalline structure and dislocation sub-structure on the metallic as well as on the non-metallic material systems is needed in order to resolve the above conflicts. In order to analyze the existence of Harper – Dorn creep, it is required not only to show that the Harper - Dorn regime cannot be explained by the conventional Nabarro – Herring creep mechanism but it is also necessary to show that Harper – Dorn creep is an independent mechanism as in conventional five power – law creep. The experiments conducted as a part of the proposed study are planned in order to collect conclusive data to resolve the debate regarding the existence of Harper – Dorn creep. If the results provide proof of the existence of Harper – Dorn creep, an effort will be made to develop a model explaining the phenomenon. If the experimental results 8 contradict the basic principles of Harper – Dorn creep, an effort to explain the error in the earlier experiments will be put forward, along with mechanistic explanation of the observed features. To place this study in perspective, it is important to justify the significance of performing creep at low stresses and very high temperatures. 1.3. Significance of Creep at Very High Temperatures and Low Stresses It is important to understand the need for a study of deformation process at very low stresses (σ/G ≤ 10 -5 ) and at exceptionally high temperatures (≥ 0.9 T m ), which leads to very low strain rates (≤ 10 -8 s -1 ). At present, this deformation process seems to be commercially less important as it does not have any obvious application in day – to – day industrial life. But at the same time, research should not be limited to the present applications and only for commercial benefits and thus it is important to note some reasons which confirm the feasibility of studying of creep at very high temperatures and low stresses: 1.3.1. Geological Applications There are several suggestions that creep at very low stresses and very high temperatures may be important in the flow of planetary and lunar interiors [32 – 34]. The core and the mantle of the earth have exceptionally high temperature and comparatively low shear stress [34]. Earthquakes may take place by the creep failure of locks in the geodesic plates. Also, ice at the shore of Antarctica or in any glacier is very close to its melting temperature and deforms under its own weight (i.e. equivalent to small stresses). There may be other applications, where the change is occurring at 9 very low rates (~10 -16 - 10 -14 s -1 ) and it is very important to understand them in order to predict the future course of geological activities. Fig 2 shows the several different possibilities for the strain rate or deformation rate at such low stresses. If the deformation rate in pure Al is governed by the classic 5– power law instead of Harper – Dorn, it will be 20 times slower even at a stress of 0.05 MPa. At the same stress, the difference in strain rate will be a factor of 5 between the 3– power law and 5–power law mechanisms. Hence, it is important to study this deformation mechanism to reduce the uncertainties related to such phenomenon by an order of years. 1.3.2. Commercial Applications At present, this creep process may have some importance in the sintering of powders [15]. As the demand for higher efficiency increases, the operating temperature of heat sources may increase and creep in such heat sources at low stresses will become important, as in nuclear reactors where the applied stress in the enclosure is relatively low and the design limit is 1 % of strain in about 2 × 10 5 hrs (>20 years). Also, the possibility of achieving superplasticity and commercially interesting strain rates by manipulating the temperature and stress in the high temperature and low stress region is challenging but not impossible. 1.3.3. Scientific Endeavor The nature of creep at very high temperatures and low stresses is ambiguous. Due to its nature of extreme deformation conditions, an understanding of this phenomenon will certainly assist with the endeavor of developing a unified and consistent theory for creep. 10 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 n = 4.5 factor of ~5 n = 3 n = 1 factor of ~20 Pure Al T = 923 K Harper and Dorn, 1957 Barrett et al., 1972 Mohamed and Ginter, 1982 Straub and Blum, 1990 Blum and Maier, 1999 Blum et al., 2002 ε (s -1 ) σ/G 0.05 MPa σ (MPa) at 0.99 T m Fig 2: A schematic showing various possibilities at the low stresses indicating the ambiguity of around an order in the deformation rate even at a stress of 0.05 MPa [15, 18, 23, 24, 35, 36]. 11 CHAPTER 2. REVIEW OF LITERATURE 2.1. Major Experimental Features of Harper – Dorn Creep Over the last five decades, several research groups have conducted creep experiments on various material systems at very low stresses and very high temperatures. Several reviews of Harper – Dorn creep studies are available in the literature [37 - 39]. Table 2 shows some of the features of the materials where Harper – Dorn creep has been reported. Some of the additional features about the experimental details of these important studies dealing with pure aluminum at very high temperatures are shown in Tables 3 and 4. As shown in Table 2, most of the experiments on Harper – Dorn creep were conducted above 0.95 T m on materials with very large grain size (of the order of ~ 1 – 4 mm) and at the stresses of several hundred kilo-Pascal. Based on these studies, the following important features about Harper – Dorn creep were observed (mainly on aluminum 5 ): i. The creep rate is directly proportional to the applied stress [15, 18, 43] and hence, it is referred as Newtonian type or viscous flow with a strain rate sensitivity of 1. Fig 3 shows typical stress and creep rate behavior. ii. The creep rate is independent of the grain size as shown in Fig 4, even in the limit of the single crystals [15, 18]. iii. The activation energy for creep in the Harper – Dorn regime is equal to self – diffusion [15, 18]. 5 In the subsequent subsections, results on other material systems will also be presented. 12 Table 2: Experimental Reports of Harper – Dorn creep Material Loading Conditions Homologous Temperature Stress (MPa) Grain Size (mm) Ref. Metals at High Temperatures Al T 0.99 <0.1 3.3 § [15] Al T 0.98 – 0.99 <0.1 10 § [18] Al DS 0.99 <0.1 9 § [19] Al DS; T 0.99 <0.1 6;8 [35] Al C 0.99 <0.1 S [40] Al – 2% Mg C 0.97 <0.2 4 [41] Al – 3% Mg DS 0.98 – 0.99 <0.3 3 [17] Al – 5% Mg DS 0.96 <0.4 0.5, 0.9 [42] Al – 5% Mg T 0.96 <0.5 1.1 [43] Cu T 0.97 0.25 2.7 § [44] Pb DS 0.92 – 0.98 <0.2 1.5 [19] Sn DS 0.98 <0.4 2 [19] Pb – 9% Sn DS 0.84 – 0.93 <0.2 2.5 [45] Metals at Low Temperatures β - Co T ξ 0.61 <8 × 10 -5 G ψ 0.4 – 0.7 [46, 47] α-Ti T ξ 0.51 – 0.54 <1.0 0.1 – 0.8 [46, 48] α-Fe T ξ 0.47 – 0.51 <5 × 10 -5 G ψ 0.120 [49–51] α-Zr T ξ 0.35 – 0.48 0.08 – 1.8 0.13–0.34 [52-54] Ni – 15Cr T ξ 0.56 – 0.65 <1 0.1 [55] Al T ξ 0.78 < 1 > 2.1 [56] Ceramics CaO C 0.52 < 30 0.02 § [57- 58] UO 2 C 0.6 – 0.74 <2 × 10 -4 E ψ 0.0 –0.055 [59] ж MgO C 0.62 – 0.64 <3 × 10 -5 G ψ S [60, 61] TiO 2 C 0.51 – 0.66 - S [62] 13 Table 2: Contd. BeO C 0.53 7.85 × 10 -5 G ψ > 0.028 [63] Al 2 O 3 C 0.74 2.4 × 10 -4 G ψ 0.025 [63] NaCl C 0.85 – 0.95 <1 × 10 -5 G ψ S [64] KZnFe 3 C 0.87 - 0.98 < 9 S [65] KTaO 3 C 0.87 – 0.99 <9 S [66] CaTiO 3 C 0.61 – 0.66 2.08 × 10 -4 G ψ 0.008 [62] CaCO 3 C 0.66 7.1 × 10 -4 G ψ 0.01 [63] SiO 2 C 0.45 – 0.55 1.48 × 10 -3 G ψ 0.2 [63] Complex Compounds Mn 0.5 Zn 0.5 Fe 2 O 4 C 0.88 1.11 × 10 -4 G ψ 0.1 [63] Co 0.5 Mg 0.5 O C 0.66 – 0.71 2.14 × 10 -4 G ψ 0.01 [67] MgCl 2 .6H 2 O C 0.75 – 0.96 < 4 0.1 - 1 [62] (Mg,Fe) 2 SiO 4 C 0.74 3.48 × 10 -4 G ψ 0.015 (Dry) 0.03 (Wet) [68, 69] Feldspar C 0.79 – 0.96 <2 S [70] Others Ice C 0.99 9.13 × 10 -4 G ψ S [62] § Tests were conducted also with a single crystal; ξ Tests were conducted with helical spring specimens; ψ The exact value for the stress was not reported ж The article reinterprets the earlier published data T: Tension, DS: Double Shear, C: Compression, S: Single Crystal 14 Table 3: Tensile creep 6 tests on pure Al at very high temperatures and low stresses Polycrystals: Temperature (K) Purity Grain Size (mm) Thickness / Diameter (mm) Ref 923 99.99 3.3 2.54 (flat) [15] 914 - 925 99.99 10 10 [18] 920 - 923 99.999 – 99.9995 6 - 8 95.25 × 15.88 [35] Single Crystals Table 4: Compressive creep tests on pure Al at very high temperatures and low stresses Polycrystals: Temp. (K) Purity Grain Size (mm) Length (mm) Diameter /side (mm) Aspect Ratio (l/d) Lub. Ψ Ref 923 99.99 3.5 ± 0.7 10 10 × 10 1 No 923 99.999 0.5 10 10 1 No [36] 923 99.99 ~ 4 12 12 × 12 1 No [23] 923 99.99 Vary 35 29 × 29 1.2 No [24] Ψ Lub. : Lubrication 6 Mohamed and Ginter [35] also used double shear specimens for the testing. Table 3 contains the data for the tensile specimens. Temperature (K) Purity Thickness / Diameter (mm) Ref 920 99.99 [16] 914 - 918 99.99 12.5 × 3 [18] 15 10 -7 10 -6 10 -5 10 -4 10 -20 10 -19 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 4.5 1 1 σ/G εkT/DGb 3.3 10 9 Al Polycrystal d (mm) Single Crystal 1 Harper and Dorn (1957) Barrett et al (1972) Mohamed et al (1973) Nabarro - Herring Creep d (mm) 3.3 9 Fig 3: Temperature compensated strain rate vs. normalized stress for pure aluminum, showing data reported in the Harper – Dorn regime by Harper and Dorn [15], Barrett et al [18] and Mohamed et al [19]. The behavior predicted theoretically for the Nabarro – Herring creep is also indicated. The plot shows consistency in the creep behavior between the single crystals and polycrystals in the Harper – Dorn regime. The figure is adapted and reproduced from [43]. 16 10 5 10 6 10 7 10 8 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 2 10 3 10 4 d (μm) Burton (1972) [thin films] Harper and Dorn (1957) Barrett et al (1972) Mohamed et al (1973) γ 1 ~ γ HD γ NH < γ HD γ 1 ~γ HD + γ NH γ 1 ~ γ NH τ normalized /γ normalized d/b d c γ NH > γ HD Al single crystals predicted curve Fig 4: Correlation between the creep behavior of bulk specimens and thin foils of Al. d c represents the critical grain size for the transition from Nabarro – Herring creep to Harper – Dorn creep. The plot is adapted and reproduced from [71]. 17 iv. The creep rate is almost 3 orders higher than the values given by Nabarro – Herring creep. This is also shown in Fig 3 where the Nabarro – Herring creep projections for the various grain sizes used in earlier studies are indicated by broken lines. v. The strain observed in the material is almost uniform throughout the sample [15]. vi. Sub-grain formation was not observed in the original study of Harper and Dorn [15]. Later, Barrett at al [18] reported the formation of subgrains in material in the Harper – Dorn regime and they noted that Harper and Dorn could not observe any subgrains in their study due to the following three reasons: (a) the grain size (3.3 mm) was of a comparable size to the stable sub-grain size (2.3 mm) formed under the given stresses, (b) there was an annealing effect due to slow cooling of the crept sample after the test (instead of quenching) and (c) the small strain for creep. Mohamed and Ginter [35] addressed the above three points in their experiments but could not observe any distinct formation of sub-grains. Instead, they reported the formation of a dislocation sub-structure which showed long dislocations with step formation. The step formation in the long dislocations shows the presence of cross slip in a dislocation – multiplying process instead of conventional Frank – Read sources (which cannot operate here due to the very low applied stress) [35]. Fig 5 shows the TEM micrograph of the sample crept within the Harper – Dorn regime, whereas Fig 6 shows an etch – pits micrograph reported by Yavari et al [43] illustrating the formation of subgrains only at a higher stress (outside the Harper – Dorn regime). Yavari et al [43] concluded that subgrains can form in Harper – Dorn creep only if the stable subgrain size is significantly smaller than the grain size of the specimen. Later, Ginter et al [26] reported the formation of new grains in the Harper – Dorn regime at very large strains, as shown in Fig 7. 18 Fig 5: TEM micrograph of Al after creep testing in the Harper – Dorn regime at (a) 0.019 MPa and (b) 0.025 MPa [35]. The pictures show extensive step formation, indicating the possibility of cross – slip as the dislocation multiplying process. (a) (b) 19 Fig 6: (a) A random distribution of etch pits in the specimen of Al – 5% Mg tested at 0.14 MPa to within steady state flow in Harper – Dorn creep. (b) Evidence of the formation of subgrain boundaries in a specimen tested at 0.74 MPa to within steady state flow in the dislocation climb region [43]. (a) is tested in Harper – Dorn regime whereas (b) is tested in conventional 5 – power law regime. Fig 7: TEM micrograph of pure Al (99.9995 % purity) in the Harper – Dorn regime after very large strain (~0.18) showing the wavy nature of the grain boundaries [26]. According to the authors, wavy nature of the grain boundaries indicates the possibility of dynamic recrystallization as the restoration or recovery process during the creep deformation. 20 vii. In the original study, Harper – Dorn reported the surface tension between the surfaces of the Al – Al 2 O 3 and Al 2 O 3 - air and interpreted it as an acting threshold stress [15, 16] whereas Mohamed and co-workers [19, 35] did not report any threshold stress and an extrapolation on the stress vs. strain rate plot showed zero strain rate at zero stress. Langdon and co -workers [43, 72] also did report a threshold stress in their model for the Harper – Dorn creep. Nabarro [73] developed a mechanism for Harper – Dorn creep based on the internal stresses produced by an equilibrium concentration of dislocations exerting a stress equal to the Peierls stress on its neighbors and hence argued the presence of the threshold stress in this low stress creep deformation. viii. The value for A HD (the dimensionless constant in Eq. 1 for the Harper - Dorn regime), and hence the value of ε & /σ, is almost the same for Al, Al -2% Mg, Al – 3% Mg and Al – 5% Mg [43], but as shown in Table 2 the critical stress at which the transformation to power – law creep occurs from Harper – Dorn creep increases with an increase in the alloy content. Also, the experimental value of A HD (and hence, ε & /σ) is similar for almost all materials tested at very high temperature (>0.8 T m ) in the Harper – Dorn regime [74]. ix. A very large strain (~0.18) under the Harper – Dorn regime is reported [35]. x. In the presence of precipitates, Harper – Dorn creep is suppressed and the activation energy becomes much larger than for self diffusion [18] xi. Mohamed and Ginter [35] showed in their study on pure Al (purity: 99.99 – 99.9995 %) having different thermo-mechanical history (and hence different dislocation densities) that only a material with low dislocation density can undergo Harper – Dorn creep. In the same study on pure Al, the authors calculated ~2 × 10 9 m -2 as the critical 21 dislocation density above which Harper – Dorn creep is not observed. Fig 8 shows the creep curve observed in the above study. xii. There is a random but reasonably uniform distribution of dislocations throughout the crept sample [43]. The dislocation density is independent of the steady – state stress value in the Harper – Dorn regime [43, 72]. Figure 9 (a) shows the variation of the dislocation density with the stress in the Harper – Dorn regime and it gives an average value of (4.7 ± 0.7) × 10 9 m -2 for Al – 5 % Mg [75] 7 . Figure 9 (b) shows the variation in the dislocation density values for pure Al deformed at 923 K. The data of Barrett et al [18] varies in the low stress regime but does not show any trend and the authors inferred a constant dislocation density in the low stress regime. Lin et al [76] showed a constant dislocation density in the Harper - Dorn regime which increases rapidly in the 5 – power law regime. Weertman and Blacic [77] argued that a cyclic variation in temperature with a moderate amplitude (±1 K) and time period (~ 40 – 50 minutes) can produce significantly more dislocation density than as generated by the applied load. Hence, a constant and higher than expected dislocation density (based on 5 – power law) can be observed during the Harper - Dorn experiments. Later, using X – Ray topography Nes [78] showed that the dislocation density was dependent on stress to a power of 1.3 (ρ = Cσ 1.3 ), however, the test temperature ≤ 693 K was rather low so that creep was presumably negligible at the low stresses applied. Nes [78] also argued that the conventional TEM studies cannot predict the exact value of the dislocation density due to inherent problems with TEM sample preparation and the lower dislocation density associated with Harper – Dorn creep. 7 The value for dislocation density reported in [43] was corrected in [75] 22 10 -2 10 -1 10 0 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 n = 4.5 ε (s -1 ) σ (MPa) Mohamed et al (1973) Al (a), 99.999 %, cold working Al (b), 99.9995 %, cold working Al (c), 99.99 %, hot working Single Crystal Al n = 1 Mohamed and Ginter (1982) Fig 8: A reproduction of the log (strain rate) vs. log (applied stress) for Al tested at 923 K from [35]. Al (a) and Al (b) are prepared by cold work with the purity level of 99.999 % and 99.9995%, respectively whereas Al (c) is produced by hot work and has a purity of 99.99 % [35]. All of these samples had higher initial dislocation densities. 23 0.01 0.1 1 10 8 10 9 10 10 10 11 10 12 Yavari and Langdon (1981) Corrected by Owen and Langdon (1996) Dislocation Density (m -2 ) Stress (MPa) 5-Power Law Al - 5 % Mg 0.01 0.1 1 10 6 10 7 10 8 10 9 10 10 Barrett et. al. (1972) Mohamed and Ginter (1982) Lin, Lee and Ardell (1986) Dislocation Density (m -2 ) Stress (MPa) 5-Power Law Pure Al Lin et al Barrett et al 3 Fig 9: Variation of the dislocation density with stress for (a) Al – 5 % Mg which is mechanically similar to pure Al (reproduced from Table 2 of [43, 75]) and (b) pure Al [18, 35, 76]. (a) (b) 24 xiii. Almost all of the studies have shown a small but distinct primary stage in Harper – Dorn creep. An inverted transient creep is recorded in materials with pre-straining. Barrett et al [18] showed a lower initial creep rate, which accelerated and subsequently decelerated in the material (pure Al) with pre-straining at room temperature. Figure 10 shows an inverted transient feature. There is a reverse creep strain of about 0.0002 on unloading and a similar transient on reloading, which is contrary to the stress directed vacancy diffusion based mechanisms (e.g. in Nabarro – Herring and Coble creep) [15] as shown in Fig 11. This observation is not consistent with any known vacancy diffusion mechanisms as the equilibrium concentration of vacancies should not be higher at the start of the deformation than its value during the steady-state. As the earlier studies always addressed creep at high temperatures for large grain size material as in Nabarro – Herring creep, this argument was proposed in favor of Harper – Dorn creep to establish it as a new creep mechanism. xiv. The dislocations in the Harper – Dorn creep are predominantly close to the edge orientation (>50%); only a small proportion (~10 %) are close to the screw orientation [43], as shown by the histogram in Fig 12. 25 0 1x10 5 2x10 5 3x10 5 4x10 5 5x10 5 6x10 5 0.000 0.002 0.004 0.006 0.008 Barrett et al. (1972) ε ss = 7.8 x 10 -9 s -1 σ = 0.035 MPa T = 923 K Pre-strain, ε = 0.0002 Purity: 99.99 % ε Time (s) Al Fig 10: Typical creep curve for Al pre-strained at room temperature prior to testing at high temperature and low stress. The plot is reproduced from [18]. 26 . 0.0 5.0x10 4 1.0x10 5 1.5x10 5 0.0000 0.0004 0.0008 0.0012 0.0016 0.0020 Loaded Load Removed Loading Unloading ε,True Strain Time (s) Load Removed Al Harper and Dorn (1957) Fig 11: Example of crystal recovery and creep recovery at 920° K in pure Al [15]. After removing the load, the sample undergoes a small recovery, hence leading to a net positive strain. In the case of Nabarro – Herring creep, the recovery is complete and hence no net positive strain after unloading is observed. 27 0.0 22.5 45.0 67.5 90.0 0 20 40 60 80 Edge Mixed Mixed Frequency (%) Φ (°) Al - 5 % Mg Yavari et al (1982) T = 823 K σ = 0.14 MPa d = 1.1 mm Screw Fig 12: Frequency of occurrence of the angle, Φ, between the dislocation line and the Burgers vector, in angular increments of 22.5°, for a specimen of Al – 5% Mg tested at 0.14 MPa to within steady state flow in Harper – Dorn regime [43]. The histogram indicates that most of the dislocations produced during deformation in Harper – Dorn regime are of edge type in nature. 28 xv. In the classic work of Harper et al [16], careful study was conducted to observe the variation in the nature of grain boundary sliding from the 5 – power law to the proposed Harper - Dorn regime. The results showed an increasing contribution of the grain boundary sliding at the low stresses, and then a decrease during the transition stress regime and finally a reduction to a very low value (~ 5 – 10 %) at very low stresses in the Harper - Dorn regime. In the Nabarro – Herring region, the sliding contribution remains constant with a decrease in the stress. Also, the contribution of grain boundary sliding in Harper – Dorn creep was significantly lower than that predicted or observed in Nabarro – Herring creep where it has been shown to be consistently high and invariably close to ~ 60 % [79 - 81]. Harper et al [16] argued that sudden decrease in grain boundary sliding in the Harper - Dorn regime, still leading to an increased strain rate, indicated the existence of a new creep mechanism. Figure 13 shows their results. The increase in grain boundary sliding at the moderate stress may be attributed to the occurrence of grain boundary sliding (Rachinger sliding) which varies with a stress exponent of 2 or 3. 29 0.01 0.1 1 0 10 20 30 40 50 60 70 Dislocation Creep (n = 4.5) Actual Stress (σ) Net Stress (σ - σ 0 ) Calculated (Net Stress) Calculated (Actual Stress) Contribution of Sliding, ξ (%) Stress, σ (MPa) Pure Al Harper et al. (1958) T = 920 K d = 3.3 mm Harper - Dorn Creep (n = 1) Fig 13: Grain boundary shearing contribution to the total creep strain as function of stress [16]. 30 2.2. Other Materials Systems where Harper – Dorn Creep has been Reported Following the earlier studies on Al, numerous studies suggested Harper- Dorn creep in different material systems. Table 2 lists materials for which Harper-Dorn has been suggested to occur, either by the original experiments or by the subsequent reinterpretation of original data by other investigators. However, the Harper-Dorn conclusions are ambiguous in several instances. In the following subsections, some of those studies dealing with metals and simple ceramics will be examined. 2.2.1. Cu Figure 14 shows classic 5-power-law behavior in copper. Recently, Srivastava et al. [31] tested OFHC Cu in tension at temperatures close to the melting temperature and obtained creep rates that were approximately two orders of magnitude faster than the creep rates predicted for Nabarro – Herring creep. These very rapid rates are similar to the earlier creep data reported for Cu by Pines and Sirenko [44] which were subsequently interpreted by Mohamed [94] as possibly indicative of the occurrence of Harper – Dorn creep. The Cu was polycrystalline in the above study [31] and some grain boundary sliding was also reported which is contrary to the usual Harper – Dorn behavior. The strain rates were consistent with those anticipated for Harper-Dorn creep although the measured stress exponent was closer to n ≈ 2 rather than n = 1. The authors concluded that the interior dislocations caused slip but they did not identify it as Harper-Dorn creep [31]. However, this latter result is consistent both with the recent results of Ginter et al. [26] on aluminum tested to strains of >0.1 and with results obtained from experiments currently in progress on specimens of Pb of 99.999% purity [95]. 31 10 -5 10 -4 10 -3 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 x x Barrett and Sherby (1964) Barrett et al (1967) Lloyd and Embury (1969) Muehleisen et al (1970) Pahutova et al (1970) Raj and Langdon (1991) Wilshire and Palmer (2002) x Wilshire and Palmer (2002) Kassner et al(2002) Srivastava et al (2005) Okrainets and Pishchak (1977) Feltham and Meakin (1959) G σ DGb kT ε & Copper Single Crystal 4.5 2 1 x (99.96 - 99.999 % Purity) Fig 14: The steady-state creep behavior of copper [21, 31, 82-93]. The low stress regime shows a possible change in the stress exponent. 32 2.2.2. α-Zr Figure 15 illustrates the steady-state creep behavior of α-Zr. Harper-Dorn creep was suggested for helical specimens of α-Ti, α-Fe, and α-Zr and β-Co [46 - 54]. Although Hayes and Kassner [103] found that the creep rate of zirconium at low values of σ/G varied approximately proportional to the applied stress, the rate controlling mechanism for creep within this regime was not clear. At low stresses, a grain size dependency may exist below the grain size of 90 μm, suggesting a diffusional or grain boundary sliding mechanism for creep. A grain size independence at larger grain sizes supports Harper-Dorn but the low observed activation energy (~ 90 kJ/mol) is not consistent with those observed at similar temperatures at higher stresses in the 5 – power law regime (270 kJ/mol). None of the studies have plotted stress – strain behavior of the helical samples at the higher stresses; hence, it is difficult to have a direct comparison of the results in the case of the helical and the flat samples. Helical samples are of special interest at the lower stress levels as various strain rate data can be achieved simultaneously using the same sample and loading condition. Hence, spring samples save time, especially as the probable strain rates are very low at these low stress levels. Furthermore, the main problem with the helical samples arises due to the fact that the strains achieved in these tests are very low (of the order of 10 -4 – 10 -5 ) and, hence, the data do not give information on steady-state creep. The low values of strain may only account for the elastic and anelastic strains, instead of plastic strains. Helical samples may not sustain higher strains (roughly less than < 10 -3 ) due to the instability in the sample and, hence, these strains are always insufficient for non-ambiguous steady – state data. 33 10 -6 10 -5 10 -4 10 -3 10 -2 10 -17 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Ardell and Ashby (1967) Bernstein (1967) Fiala et al (1991) MacEwen et al (1981) Novotry et al (1985) Pahutova and Cadek (1973) Prasad et al (1992) Perez- Prado et al (2005) Ardell (PhD, 1964) DGb kT ε & G σ Open: Helical Samples Closed: Flat Samples α−Zirconium 6.4 1 Fig 15: Compilation of various studies on α – Zr [52, 54, 96 - 102]. If δD gb is used for low stress data, they show good consistency [39]. 34 2.2.3. α-Fe Figure 16 illustrates the steady-state creep behavior of α-Fe. As in α-Zr, at the lower stresses a low stress exponent was interpreted to be due to Harper-Dorn for samples with grain size larger than 123 µm and Coble creep for the smaller grain size samples [50]. However, an examination of all of the data does not reveal a systematic grain size trend. Most of the studies conducted at lower stresses used helical samples and hence had the limitations explained earlier. Lack of a sufficient number of studies showing the Harper – Dorn regime and very low levels of strain achieved in the studies proposing Harper – Dorn creep are the reasons for an unambiguous conclusion about the behavior of α-iron at the lower stress values. At low temperatures, Harper – Dorn behavior was reported to be controlled by pipe diffusion of dislocations or dislocation core diffusion for both α – Zr and α – Fe [46 - 54]. Contrary to studies at very high temperatures, these studies used samples with very high initial dislocation densities (~ 10 12 m -2 for α – Zr [52]) and hence reported the existence of threshold stresses. Unlike the studies at higher temperatures, where A HD (the dimensionless constant for creep in Eq. (1)) is constant with temperature as well as constant through different metallic systems [73], a temperature dependence of A HD was reported for low temperature Harper – Dorn creep [52]. In conclusion, the behavior of the “low temperature Harper – Dorn creep” is not only significantly different than the behavior of the “high temperature Harper – Dorn creep” but also the reported data are not reliable due to the use of helical samples which acquires only very limited strains. 35 σ / G 10 -6 10 -5 10 -4 10 -3 10 -2 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Cadek et al (1969) Longdale and Flewitt (1978) Fiala et al (1983) Davies and Williams (1969) Cadek and Milicka (1968) Cadek and Milicka (1968) Towle and Jones (1976) α - Iron DGb kT ε & Closed: Flat Specimens Open: Spiral Speciemens 7 1 High Stress Low Stress Fig 16: Compilation of various studies on α –Fe [50, 104 - 108]. The data from Cadek and Milicka [102] at a grain size of 216 μm were interpreted as Harper-Dorn creep by Fiala et al. [52]. The steady-state α-Fe data appear best normalized with D gb at low stress and D sd at high stress provide a good fit to the data. 36 2.3. Harper – Dorn Creep in Ceramics and Geological Materials Most of the reports suggesting the possibility of the occurrence of Harper-Dorn creep in non-metallic materials are based on re-interpretations of published data. For example, the low stress creep data on polycrystalline and single crystal CaO [57] were subsequently interpreted as indicative of Harper-Dorn creep because the results were generally in agreement with this mechanism including a similarity in creep rates for the single crystal and polycrystalline samples [58]. Several reports are now available suggesting that Harper-Dorn creep may occur in a wide range of ceramic and geological materials [62, 66, 67, 70, 109 - 111]. However, these reports must be interpreted with caution because subsequent inspection has shown that at least some of the proposals (as for MgCl 2 .6H 2 O (CO 0.5 Mg 0.5 )O and CaTiO 3 ) are probably erroneous because the available data do not fulfill the requirements for unambiguous establishment of the advent of the Harper-Dorn flow mechanism [112, 113]. Figures 16 – 19 show various experimental creep data plotted logarithmically in the form of the normalized strain rate, b DG / kT ε & , versus the normalized stress, σ/G, for CaO, LiF, MgO and NaCl, respectively. Similar plots are shown in Figs 20 and 21 for dry olivine ((Mg 4 Fe)2SiO 2 ) and forsterite (Mg 2 SiO 4 ). 37 10 -5 10 -4 10 -3 10 -2 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 Dixon-Stubbs and Wilshire (1982) Single Crystal Dixon-Stubbs and Wilshire (1982) Duong and Wolfenstine (1991) CaO 1.6 5.4 Polycrystal DGb kT ε & σ/G 1 1 Fig 17: Normalized creep rate versus normalized stress for polycrystalline [57] and single crystal [114] CaO. 38 10 -5 10 -4 10 -3 10 -2 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 Streb and Reppich (1972) Cropper and Pask (1973) Ruoff and Rao (1975) Yu and Li (1977) Biberger and Blum (1989) Polycrystal Cropper and Langdon (1968) G σ DGb kT ε & LiF 3.5 Single Crystal 1 Fig 18: Normalized creep rate versus normalized stress for polycrystalline [115] and single crystal [116-120] LiF. 39 10 -5 10 -4 10 -3 10 -2 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 Hensler and Cullen (1968) Langdon and Pask (1970) Single Crystal Cummerow (1963) Rothwell and Neiman (1965) Routbort (1979) Ramesh et al (1986) Wolfenstine and Kohlstedt (1988) MgO 3.2 1.0 3.2 Polycrystal DGb kT ε & σ/G 1 1 1 Fig 19: Normalized creep rate versus normalized stress for polycrystalline [121, 122] and single crystal [60, 123- 126] MgO. 40 10 -6 10 -5 10 -4 10 -3 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 Burke (1968) Single Crystal Blum and Ilschner (1967) Poirier (1972) Banerdt and Sammis (1985) 1.3 3.5 σ/G DGb kT ε & NaCl Polycrystal 1 1 Fig 20: Normalized creep rate versus normalized stress for polycrystalline [127] and single crystal [64, 128, 129] NaCl. 41 10 -5 10 -4 10 -3 10 -2 10 -7 10 -6 10 -5 10 -4 10 -3 Schwenn and Goetze (1978) Single Crystal Kohlstedt and Goetze (1974) Justice et al (1982) Dry Olivine ((Mg 4 Fe)2SiO 2 ) 1.1 3.5 Polycrystal DGb kT ε & σ/G 1 1 Fig 21: Normalized creep rate versus normalized stress for polycrystalline [130] and single crystal [68, 131] dry olivine ((Mg 4 Fe) 2 SiO 2 ). 42 10 -5 10 -4 10 -3 10 -2 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 Relandeau (1981) Single Crystal Darot and Gueguen (1981), [110] c Darot and Gueguen (1981) [101] c Darot and Gueguen (1981) [011] c Forsterite(Mg 2 SiO 4 ) 1.2 3.3 Polycrystal DGb kT ε & σ/G 1 1 Fig 22: Normalized creep rate versus normalized stress for polycrystalline [132] and single crystal [69] forsterite (Mg 2 SiO 4 ). The notation given for the single crystals denotes the loading direction with respect to the largest lattice parameter, c of the orthorhombic crystal. 43 With the exception of LiF, all of the above materials show evidence for a transition to a creep regime having a low value of n at the lowest stresses. Furthermore, where the data for polycrystalline specimens and single crystals are in reasonable agreement at the lowest stresses, as in CaO in Fig. 17 and olivine in Fig. 22, the evidence is strong that the creep behavior is not associated with Nabarro-Herring diffusion creep and may instead represent the occurrence of Harper-Dorn creep mechanism. Figures 17, 19 and 21 show a transition in the slope to a lower value in the low stress regime. Surprisingly, there is no such transition in the data for LiF despite the very large number of individual datum points extending over a range of nine orders of magnitude in normalized strain – rate. For NaCl as shown in Fig. 20, the results of Banerdt and Sammis [64] at the lowest stresses have been interpreted as evidence for Harper-Dorn creep. LiF and NaCl belong to the same family of ceramics (halides with FCC crystal structure). NaCl shows a change in the stress exponent at σ/G ≈ 10 -5 , whereas the lowest stress achieved in any of the studies for LiF is only σ/G ≈ 2 ×10 -5 which is slightly higher than the transition stress level observed in NaCl. This probable similarity can also be seen in terms of normalized strain rate, b DG / kT ε & , which is equal to ~ 3 × 10 -15 for both NaCl and LiF at a stress level of σ/G ≈ 10 -5 . Dry olivine and forsterite also show a transition in the stress exponent which becomes very close to 1 at the lowest stresses. The data for MgO appear to divide into two separate sections within the power- law region with both regions having a stress exponent of n ≈ 3.2. The reason for this division is not known but it may reflect differences in impurity levels for different 44 batches of the single crystals. The purer material, being soft generally, shows higher strain rates for a given stress in the 5 - power law regime as compared to the less pure material. As shown in Fig 19, only the softer material shows a transition to the lower stress exponent. This is consistent with recent work by Mohamed on Al and Pb [26, 95]. Also, there is a clear division in Fig. 22 within the power-law region for forsterite (an orthorhombic crystal) but this is simply a reflection of the different orientations of [110], [101] and [011] used for the single crystals. The relevant sources for all of the creep data are summarized in Table 5 for single crystals and in Table 6 for polycrystalline materials. In constructing these plots (metals and ceramics), the values used for the diffusion coefficient D and the shear modulus G are summarized in Table 7. The value of D is expressed as D o exp (-Q/RT) where D o is the frequency factor, Q is the activation energy for diffusion (for the active ion in the case of ceramics) and R is the gas constant. The value of the shear modulus of elasticity G is expressed as (G o − ΔG) where G o is the value of the shear modulus extrapolated linearly to absolute zero and ΔG is the rate of decrease of the shear modulus with increasing temperature. For CaO and forsterite, the values for the above materials properties were taken from an earlier report [38] where a detailed reasoning is given for the selection of these values. It should be noted that in practice the diffusion of Si (which is the slowest moving ion in forsterite as well as olivine) appears to be ~30 times faster in olivine than forsterite within the temperature range of 1130-1530°C [147] but nevertheless this only introduces a factor which has no influence on the normalization and the subsequent relative agreements between the various sets of datum points shown in Fig. 22. 45 Table 5: Creep of Single Crystals Material Temperature (K) Homologous Temperature Orientation Purity / Impurity Ref. Cu 830 0.61 Unknown OFHC (unknown) Wilshire and Palmer [91] 1473 0.52 <111>, <100> 99.997 % CaO, 1 ppm Fe , 2 ppm Si, 1 ppm Na, 10 ppm Mg and 300 ppm Sr Dixon-Stubbs and Wilshire [57] CaO 1623 and 1673 0.57 and 0.58 <100> 99.997 % CaO, 80 ppm Fe , 80 ppm Al, 60 ppm Si Duong and Wolfenstine [114] 523 - 1023 0.47 - 0.97 <100> Divalent impurity < 0.7 mol ppm of Mg 2+ Streb and Reppich [116] 923 – 1023 0.81 – 0.88 <100> Divalent impurity < 17 ppm Cropper and Pask [117] 923 – 1023 0.81 – 0.90 <100> Ruoff and Rao [118] 923 0.81 <100> Divalent impurity < 50 ppm Yu and Li [119] LiF 673 – 1113 0.59 – 0.98 <100> Divalent impurity < 1 ppm Biberger and Blum [120] 1929 – 1973 0.62 – 0.63 Possibly same as Ramesh et al. [75] - using same supplier Cummerow [123] 1673 0.54 Possibly same as Ramesh et al. [75] - using same supplier Rothwell and Neiman [124] MgO 1678 – 2073 0.54 – 0.66 <100> 99.97 – 99.99 % (in ppm: Al 100, Ca 100, Fe 50, Si 20 and Ti 10 && Al 10 – 100, Ca <10, Fe 10 and Ti 30) Routbort [125] 46 Table 5: Contd. 1948 – 2008 0.62 – 0.64 <100> 99.93 % (in ppm. (Na + K) 10, Si 300, Ca 330, Fe 70 and (Ti + Ni) < 10) Ramesh et al. [60] MgO 1573 – 1773 0.50 – 0.57 <100> 99.962 % (in ppm: Al 45, Ca 280, Cd <1, Co 3, Cu 1, K 10, Mn 22, Mo <1, Ni 15, Pb <1 and Zn 2) Wolfenstine and Kohlstedt [126] 1010 0.94 ~ 60 ppm of Ca 2+ Blum and Ilschner [128] 750 –1060 0.7 – 0.9 <001> Purest grade Harshaw NaCl (impurity ~ 10 ppm of Ca 2+ ) Poirier [129] NaCl 920 – 1010 0.85 - 0.95 <100> Optical quality (impurity < 100 ppm) Banerdt and Sammis [64] 1701 – 1923 0.8 – 0.9 San Carlos (Fo 92 ) Kohlstedt and Goetze [68] Dry Olivine 1843 0.86 San Carlos (Fo 92 ) Justice et al. [131] Forsterite 1673 -1923 0.78 – 0.90 [110] c , [101] c , [011] c Fo 100 Darot and Gueguen [69] 47 Table 6: Creep of Polycrystals Material Temperature (K) Homologous Temperature Grain Size (µm) Purity / Impurity Ref. 680 - 980 0.50 – 0.72 30 99.99 Feltham and Meakin [82] 686 - 800 0.50 – 0.59 30, 400, 1000 > 99.995 Barrett and Sherby [83] 679 - 899 0.50 – 0.66 30 - 700 99.995 Barrett et al [85] 694 - 908 0.51 – 0.67 70 99.99 Lloyd and Embry [86] 1297 - 1331 0.95 – 0.98 500 99.9 Muehleisen et al [21] 550 - 1025 0.40 – 0.75 450 99.99 Pahutova et al [87] 612 - 925 0.45 – 0.68 500 99.99 Okrainets and Pishchak [88] 623 – 973 0.46 – 0.72 250 ± 15 99.98 Raj and Langdon [89] 720 - 760 0.53 - 0.56 30 - 450 OFHC Wilshire and Palmer [91] 823 0.61 420 99.9985 Kassner et al [92] Cu 1123 - 1347 0.83 – 0.99 ~1000 OFHC Srivastava et al [31] 48 Table 6: Contd. 840 0.45 300 99.84 Fe Ardell [96] 973 – 1083 0.51 – 0.58 300 99.84 Fe Ardell and Sherby [97] 840 0.45 51 99.95 % Fe Bernstein [98] 673 – 1023 0.25 – 0.54 150 99.78 Fe Pahutova and Cadek [99] 673 – 1023 0.25 – 0.54 30 ± 5 99.84 Fe MacEwen et al [100] 873 0.46 16.8 99.879 Prasad et al [101] 673, 1073 0.36, 0.57 - 99.89 Perez-Padro et al [102] 743 – 973 0.40 – 0.54 158 - 243 99.78 % Fe Novotny et al [49] Zr 750 – 980 0.40 – 0.52 48 - 87 99.78 % Fe Fiala et al [50] 843 – 918 0.47 – 0.51 218 - Cadek and Milicka [104] 818 – 1118 0.45 – 0.62 140 98.57 Cadek et al [105] 815 – 978 0.45 – 0.54 100 99.95 Davies and Williams [106] Fe 873 0.48 80 99.8 Towle and Jones [107] 49 Table 6: Contd. 873 0.48 60 99.93 Longdale and Flewitt [108] Fe 823 0.45 145 99.97 Fiala et al [54] CaO 1473 0.52 20 98.86 % CaO, 0.53% SiO 2 , 0.50 % Al 2 O 3 , 0.04 % Fe 2 O 3 and 0.07 % other oxides Dixon-Stubbs and Wilshire [57] LiF 673 – 823 0.59 – 0.72 160, 3000 Divalent impurity < 20 ppm Cropper and Langdon [115] 1573 – 1773 0.50 – 0.57 13 – 68 Hensler and Cullen [121] MgO 1473 0.47 12 – 62 99.98 % (Impurities: Fe < 20, Si: 30, Al: 10, Cu: 5, Ni < 10, Sr < 50, Ca: 40 and Li: 75) Langdon and Pask [122] NaCl 638 –1015 0.60 – 0.95 200 – 300 Burke [126] Dry Olivine 1253 – 1873 0.59 – 0.88 25 – 2000 Schwenn and Goetze [130] Forsterite 1718 – 1873 0.80 – 0.88 60, 98 and 132 Fo 100 Relandeau [132] 50 Table7: Values for D and G Material Active Ion D 0 (m 2 s -1 ) Q (kJ mol -1 ) Ref. G o (MPa) Δ Δ Δ ΔG (MPa K -1 ) Ref. Cu - 2.5 × 10 -5 197 [133, 134] 4.71× 10 4 16.72 [135] Zr - 3 × 10 -5 190 [136] 4.18× 10 4 20.3 [137] Fe - 2 × 10 -4 251 [138] 7.258×10 4 28.6 [138] CaO O 2- 2.0 × 10 -10 345 [114, 38] 91.46 0.021 [140] LiF F − 6.4 × 10 -3 214 [141] 5.52 × 10 4 33.2 [142] MgO O 2- 2.5 × 10 -10 261 [143] 1.39 × 10 5 26.2 [144] NaCl Cl − 1.2 × 10 -2 214 [145] 1.79 × 10 4 9.6 [142] Olivine Si 4+ 1.5 × 10 -10 376 [146] 8.49 × 10 4 13.30 [146] Forsterite Si 4+ 1.5 × 10 -10 376 [147,146, 38] 8.49 × 10 4 13.30 [147, 38] 51 2.4. Experimental Observations that are Inconsistent with Harper – Dorn Creep In the previous subsections, details of the basic features for Harper – Dorn creep and some results for a wide range of materials were outlined. This section contains a compilation of some of the studies where experimental observations contrary to the formulation of Harper – Dorn creep were reported: 2.4.1. Burton [20] on pure Al In the study on thin samples of pure Al (99.99% purity), Burton failed to observe Harper – Dorn creep in large grain samples in the Harper – Dorn regime. Rather, he reported the dominance of Nabarro – Herring creep for specimens tested in vacuum. 2.4.2. Muehleisen et al [21] on pure Cu and Cu alloys Bulk samples of polycrystalline copper with a purity of 99.999 % were tested in compression in order to observe the effect of a dislocation – based mechanism as well as to compare the data on bulk samples versus thin (foil like) samples where Nabarro – Herring creep usually occurs. The tests were conducted at 1323 K (0.98 T m ). The experimental results showed no evidence of Newtonian viscosity but rather a stress exponent of ~5 was observed. In addition, an activation energy of creep (197 kcal/mol) equal to the activation energy of self diffusion led Muehleisen et al [21] to conclude that the material behaves similar in the high temperature – low stress range and in the intermediate temperature – high stress range (i.e. according to 5 - power law mechanism). Similar conclusions were proposed for some copper alloys (Cu -10% Zn and Cu – 3% Si) in the same study. 52 2.4.3. Blum and co-workers [23 – 25, 36] By conducting creep test in compression on bulk sized samples of Al, Straub and Blum [36] showed a stress exponent of >3 instead of 1 as predicted for the Harper – Dorn regime. They concluded that Harper – Dorn creep is only an extension of 5 - power law and is not a genuine creep model. Nabarro [148] suggested that the stresses used by Blum and Maier [23] were more than the transition stress (0.093 MPa) from Harper – Dorn to the 5 - power law regime. Following this suggestion, Blum et al [24] repeated their experiment at much lower stress (0.06 MPa) but still observed a stress exponent of 5 instead of 1. Blum and co-workers suggested that the above discrepancy in the results was due to the lack of a genuine steady – state condition in the earlier studies reporting Harper – Dorn creep as smaller strains were achieved and hence Harper – Dorn creep is simply a transient feature of the genuine 5 - power law. Fig 23 shows the creep behavior as reported by Blum and co – workers in their studies. Based on the dislocation network model proposed by Ardell et al [40, 149], Nes et al [25] argued that the slip distance for the dislocations will be comparable to the dimension of the creep samples for which Harper – Dorn creep was observed earlier and hence Harper – Dorn creep is an outcome of a size effect. Hence, they concluded that the legendary Harper – Dorn creep is not a unique creep mechanism but a special case of the 5 - power law creep which is observed when the dimension of the creep sample is smaller than the dislocation slip distance and hence it is not observed in the larger samples. 53 10 -2 10 -1 10 0 10 1 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 920 Harper and Dorn (1957) 920 ± 5 Barrett et al (1972) 902 ± 1 Mecking et al (1988) 923 Straub and Blum (1990) 923 Machine A 923 Machine B ε (s -1 ) σ (MPa) Al T (K) Blum and Maier (1999) Fig 23: Creep rate and stress relation at steady state as reported in Fig 3 of Blum and Maier [23]. Here Machine A and Machine B are the two sets of the creep machines used in the experiment. Machine A is without and Machine B is with the measurement of the load by load cell. The load acting on the specimen in machine A was determined from the external load neglecting friction in the load train [23]. 54 Kumar et al [38] argued that the slower strain rates observed by Blum and co- workers [23, 24] were due to the fact that they used an aspect ratio of ~ 1 during the compression test. The flat samples have a larger volume fraction of the friction cone and this will reduce the freely deforming volume leading to much slower strain rates if the strains are calculated based on the entire sample volume (length). 2.4.4. Greenwood and Co-workers [22, 31] McNee et al [22] performed creep test on Al samples with purity of 99.999%, although some samples may have been of lower purity than that used by Dorn and co- workers [15, 16]. They reported results consistent with Blum and co-workers [22, 24] who suggested that the so called Harper – Dorn creep is an extension of the 5 - power law creep. Fig 24 shows results as reported by McNee et al [22]. McNee et al [22] used very thin polycrystalline samples where grains can be thought of as stacks along the axis of the applied load. The samples failed to deform to larger strains, but rather showed quick failure even at a strain as low as 0.1 %. The results show significant scatter and some of the samples did not show even a recordable strain. In their recent study on thin samples of OFHC Cu, Srivastava et al [31] observed a stress exponent of 2. They reported independence of the creep rate with the grain size and their experimental points were closer to the Harper – Dorn prediction but the stress exponent was different than 1. They reported grain boundary shearing but did not conclude if it was Harper – Dorn creep. 55 10 -2 10 -1 10 0 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 n = 4.5 McNee et al (2001) Blum and Maier (1999) Barrett et al (1972) Harper and Dorn (1957) ε (s -1 ) σ (MPa) Al n ~ 1 Fig 24: A reproduction of the results reported in [15, 18, 22, 23]. The plot compares the low stress creep behavior of aluminum at 873 to 913 K. Results have been normalized to 920 K using activation energy for creep of 149 kJmol -1 [15]. The results show scatter and very low strain rates. 56 2.5. Recent Observations by Mohamed and Co-Workers Recently, Mohamed and co – workers [26 - 28, 95] conducted several studies on polycrystalline samples of pure Al (99.99 % and 99.9995 % purity) and Pb (99.999%) and reached various conclusions. 2.5.1. Effect of Impurity After conducting experiments on double shear creep specimens of pure aluminum of various levels of purity, Mohamed and co- workers [26 – 28, 95] showed that Harper – Dorn creep can occur only in specimens with very high purity. Their reports showed the existence of “Harper – Dorn creep” 8 in samples of 99.9995 % purity Al and 99.999 % purity Pb, but in specimens of relatively lower purities (99.99 % purity Al and 99.95 % purity Pb) Harper – Dorn creep was not observed. Also, they did not observe cyclic acceleration and deceleration in the strain – time curves of the lower purities samples which is a prominent feature in the higher purity samples. On the other hand, 99.99 % purity Al showed a higher stress exponent (~4.5) as shown in Fig 25(a). A similar trend was shown in preliminary results of 99.999 % purity Pb [95] which is shown in Fig 25(b). As shown in Figs 25, a lower stress-exponent (~ 3.8) even in the conventional 5 – power law regime for the high purity samples may indicate a transition regime between the two independent creep mechanisms controlling the low and high stress exponents, respectively [150, 151]. 8 Mohamed [95] also uses Harper – Dorn creep terminology for the conditions where a relatively higher stress exponent (~ 2.5) has been reported. He marks the deviation from the conventional power law as Harper – Dorn creep. 57 10 -2 10 -1 10 0 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 99.9995 Al 99.99 Al Transient ε after 200 hr Earlier Data on H-D creep ε (s -1 ) σ (MPa) Al Ginter et al (2001) Fig 25 (a): Strain rate vs. applied stress on a logarithmic scale for 99.99 Al and 99.9995 Al tested at 923 K. The figure has been adapted from [26]. (b): Strain rate vs. applied stress on a logarithmic scale for 99.999 Pb and 99.95 Pb tested at 587 K. The figure has been adapted from [95]. (a) 58 Fig 25 Contd. 10 -2 10 -1 10 0 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 n = 1 n = 2.2 Mohamed et al (1973) 99.999 % 99.95 % ε (s -1 ) σ (MPa) Pb n = 4.9 Mohamed (2007) Fig 25 (a): Strain rate vs. applied stress on a logarithmic scale for 99.99 Al and 99.9995 Al tested at 923 K. The figure has been adapted from [26]. (b): Strain rate vs. applied stress on a logarithmic scale for 99.999 Pb and 99.95 Pb tested at 587 K. The figure has been adapted from [95]. (b) 59 It should be noted that evidence for purity effects tend to be negated by the earlier work of Lee and Ardell [40] where excellent agreement was obtained with the behavior anticipated for Harper – Dorn creep. Lee and Ardell [40] conducted their experiments in compression using single crystals of Al of 99.99 % purity. 2.5.2. Restoration Mechanism Based on the observations on 99.9995% purity Al creep samples, Mohamed and co-workers [27, 28] argued that the restoration process taking place in Harper - Dorn creep is dynamic recrystallization, instead of static recovery as argued by Nes et al [25]. In support to the above observation, they gave an example of the experiment by Gifkins [152] (Fig 26) and conclusions of Yagamata [153 - 155] that dynamic recrystallization is the only possible restoration mechanism if a material of very high purity is deforming within the Harper – Dorn regime. Langdon [12] argued that the acceleration in creep curves of Ginter and Mohamed [27] were not at the regular intervals (as it is usually observed in the case of dynamic recrystallization in samples of very high purity) and hence it was more reasonable to argue it was due to grain growth instead of dynamic recrystallization. Later, Mohamed [28] suggested dynamic recrystallization based on: (i) nucleation of new grains near the surface which grow in the bulk due to the high purity of the sample, as shown in Fig 27 (ii) the absence of a regular array of subgrains (iii) the presence of wavy grain boundaries (Fig 7) (iv) the high gradient of the dislocation density across the grain boundaries, and (v) the occurrence of periodic accelerations over the entire strain – time curve. Mohamed and co-workers [26] found well defined sub-grain boundaries in the sample of Al of purity 99.99%. 60 0.0 5.0x10 6 1.0x10 7 1.5x10 7 2.0x10 7 0 2 4 6 8 10 ε × 10 2 Time (s) 99.9995 Al T = 923 K σ = 0.02 MPa Ginter et al (2001) 0.0 5.0x10 6 1.0x10 7 1.5x10 7 2.0x10 7 0 2 4 6 8 10 ε × 10 2 Time (s) High Purity Pb T = 373 K σ = 2.76 MPa Giffkins (1958) Fig 26: Creep curve in tension showing accelerated creep suggested to be associated with dynamic recrystallization (a) High purity Al (99.9995%) [26] (b) Highly pure lead [152]. The plots have been adapted from [26] (a) (b) 61 Fig 27: TEM micrograph showing new grains on the surface [26]. 62 2.5.3. Effect of Strain Mohamed and co-workers [26 – 28, 95] deformed samples of pure Al and Pb to very high strains and suggested a non-linear relationship between creep rate and applied stress. They showed that only at lower creep strains a stress exponent of 1 is found in the Harper – Dorn regime but at the higher strains the stress exponent can be higher (~2.5). This feature is also shown in Figs 25 and 26. 2.6. Proposed Theories for Harper – Dorn Creep Numerous theories have been proposed to explain Harper – Dorn creep. All theories use a dislocation – based mechanism to explain a stress exponent of 1 and a constant dislocation density throughout the deformation process. In the following subsections, a brief summary of various theories will be given. 2.6.1 Harper and Dorn [15, 16, 156] As explained in earlier subsections, after elaborate experiments to disqualify stress directed diffusion of vacancies as a possible creep mechanism, Harper and Dorn [15] speculated a dislocation climb creep mechanism for creep in the low stress – high temperature regime. Following the initial postulation [15], Dorn and co-workers [16, 156] proposed a model based on the production of vacancies by jogs in moving screw dislocations, analogous to that described by Mott [157] and later by Barrett and Nix [158]. This leads to a steady – state strain rate, 0 1 12 . sd js G kT Gb D b = τ ρ π γ l & (3) 63 where ρ is the dislocation density, ℓ js is the jog spacing in screw dislocations. Langdon and Yavari [72] criticized this model on the basis that it requires unrealistically small jog spacings (~0.3b). Also, it appeared that screw dislocations are not present substantially in the samples deforming in the Harper – Dorn regime [43]. 2.6.2. Friedel [159] Friedel suggested that the so-called “Harper – Dorn creep” is actually a diffusion creep (e.g., Nabarro-Herring) where vacancies diffuse between the relatively close subgrain boundaries: 0 1 2 . sd NH G b kT Gb D A = τ λ γ & (4) where λ is the subgrain size. As Langdon and Yavari [72] pointed out, subgrains are not always observed in the Harper-Dorn region. When subgrains form, the size, according to Barrett et al [18] varies inversely with the applied stress and this would increase the stress dependence to 3 in Eq. (4), which is contrary to the observed value of 1. 2.6.3. Barrett, Muehleisen and Nix [18] Barrett et al [18] assumed that the dislocation generation occurs by a diffusion controlled climb process which is driven by the applied stress. This occurs at a fixed number of sources per unit volume, which is independent of the applied stress [158]. The rate of dislocation generation is x v C 0 ρ ρ = + & (5) 64 where ρ 0 is the fixed dislocation length per unit volume, v C is the climb velocity, and x is the distance over which climb must occur to create glide dislocations. The climb velocity under a climb stress, σ, is [160] kT Db v C σ 2 = (6) Barrett et al [22] assume σ Gb x = (7) which appears to be Taylor hardening, according to Kassner et al [39]. Dislocation annihilation is assumed to occur only at subgrain boundaries. The annihilation rate is λ ρ ρ g v = − & (8) where v g is the glide velocity. The subgrain size phenomenologically varies with stress as σ λ λ 0 = . (9) It was assumed that σ 0 v v g = (10) At steady – state, 0 = − = − + ρ ρ ρ & & & (11) Combining Eqs. (5) to (11), one finds that the steady – state dislocation density, 0 0 0 v G kT Dbλ ρ ρ = (12) 65 is independent of σ. Combining with the Orowan equation, the steady – state creep rate is G kT λ σ Db ρ v b ρ 0 2 0 g ss = = ε & (13) Again, critical to this derivation is a stress exponent of 1 for v g and the assumption (6) which appears questionable as subgrain boundaries may not always exist. If subgrains are not formed then, as shown by Langdon and Yavari [72], the stress exponent in Eq (13) will become 2 and the strain rate will vary inversely with the grain size, which is not compatible with the conventional principles of Harper – Dorn creep. Also, the assumptions of fixed density of dislocation sources and a diffusion controlled generation rate are not well defended in the model and this seems to fail with a varying dislocation density. 2.6.4. Langdon and Yavari [72] Dislocation jogs may become saturated with vacancies at high temperatures and the dislocation velocity is then controlled by the rate of diffusion of vacancies to and from the dislocation line [159]. Later, Mohamed et al [161] considered a possible Harper – Dorn mechanism based on the climb of jogged edge dislocations. Following this approach, Langdon and Yavari [72] elaborated the idea of Hirth and Lothe [160] and suggested the following equation: ( ) 2 1 2 1 2 1 6 . sd G kT Gb D b n b = τ ρ ρ π γ l & (14) This model was able to explain almost all the requirements of Harper – Dorn creep but 66 failed to give the correct value of A HD . 2.6.5. Wu and Sherby [162] The fact that both Harper – Dorn creep and 5 – power law creep are controlled by diffusion controlled dislocation activities led Wu and Sherby [162] to propose a unified relation that describes the creep behavior over both ranges. This model incorporates an internal stress which arises from the presence of random stationary dislocations within subgrains. They assumed that during steady – state flow half of the dislocations moving under an applied stress are aided by the internal stress field, whereas the motion of the other half is inhibited by the internal stress. The internal stress is calculated from the dislocation density by the dislocation hardening equation (τ = αGb ρ , where α ≅ 0.5). The unified equation is, ( ) − − − + + = n i i i n i eff WS ss E E b D A σ σ σ σ σ σ σ σ ε 2 2 1 & (15) where A WS is a constant and σ i is the internal stress. At high stresses, where σ >> σ i , σ i is negligible compared to σ, Eq. (15) reduces to the (5 - power-law) relation, n eff ss E b D A = σ ε 2 10 & (with n = 4-7) (16) At low stresses, where σ << σ i (Harper-Dorn regime), Eq. (15) reduces to a form where n = 1. A reasonable agreement has been suggested between the predictions from this model and experimental data [161, 37] for pure aluminum, γ-Fe and β-Co. The internal stress model was criticized by Nabarro [73], who claimed that a unified approach to 67 both 5 – power law and Harper – Dorn creep is not possible since none of these processes are well understood and unexplained dimensionless constants were introduced in order to match theoretical predictions with experimental data. Also, the dislocation density in Harper – Dorn creep is constant, whereas it increases with the square of the stress in the power – law regime. Thus, the physical processes occurring in both regimes must be different. Additionally, Nabarro [73] argued the required dislocations are not in thermal equilibrium, and in order to produce a strain of 0.1 (which has been observed in Harper – Dorn creep), the dislocations must move by ~ 10 m which is unrealistic due to the sample size limitations used in the creep tests. 2.6.6. Nabarro [73] According to Nabarro [73], an equilibrium concentration of dislocations is established during steady-state creep which exerts a stress on its neighbors equal to the Peierls stress. The mechanism of plastic flow would be the motion of these dislocations that is controlled by climb. Nabarro [73] hence concluded that the materials with higher stacking fault energy will be preferred for Harper – Dorn creep, as they would probably have higher Peierls stress, therefore higher dislocation density (proportional to the square of the Peierls stress) and strain rate. 2.6.7. Wang and Co-workers [163 - 166] The internal stress model of Sherby and Wu [161] was also criticized by Wang [161], who proposed that the transition between power – law creep and Harper – Dorn creep takes place at a stress equal to the “Peierls stress, σ p ,” [164, 165]. Wang et al 68 [166] suggested that the steady-state dislocation density is related to the Peierls stress which is consistent with the idea of Nabarro [73]. In equilibrium, the stress due to the mutual interaction of moving dislocations is in balance not only with the applied stress but also with lattice friction which fluctuates with an amplitude of the Peierls stress. As a result, the steady – state dislocation density ρ in dislocation creep can be written as: 2 1 2 2 2 1 3 1 τ + τ = ρ G G . b p (17) where τ is the applied shear stress. When τ >> τ p , the dislocation density is proportional to the square of the applied stress and 5 - (or 3 -) power – law creep is observed. This is consistent with the recent work of Kassner [167]. Conversely, when τ << τ p , the dislocation density is independent of the applied stress and Harper-Dorn occurs which is consistent with Nabarro [73]. 2.6.8. Ardell and co-workers [40,149, 168] A different and fairly extensive approach to Harper – Dorn is based on the dislocation network theory by Ardell and coworkers [40, 149, 168]. The dislocation link length distribution contains no segments that are long enough to glide freely. Thus, the longest links of length L m are smaller than the critical link length to activate a dislocation source (for example, Frank – Read source). Therefore Harper – Dorn creep is a phenomenon in which all the plastic strain in the crystal is a consequence of dislocation network coarsening. The recovery of the dislocation density during Harper – Dorn creep is comparable to static recovery in the absence of an applied stress; climb of 69 nodes is driven by the line – tension of dislocation links. The stress – dependence of the strain rate arises because the applied stress biases the collisions, since the lengths of all the links must increase as σ increases, thereby increasing the collision possibilities. The climb velocity of the nodes is mostly affected by the resolved force arising from the line tensions of the dislocations at the nodes. Accidental collisions between these links can refine the network and stimulate further coarsening so that ρσ π ε kT D Cb SD 2 3 = & (18) where ( ) ( ) ∫ − > < = − c u m du u u u u C 0 2 2 1 Φ α (19) and Φ u is the scaled-link-length (u) distribution function, α ≈ 0.5 and m is a phenomenological exponent. The independence of ρ with σ is a consequence of the frustration of the dislocation network coarsening which arises because of the exhaustion of Burgers vectors that can satisfy Frank’s rule at the nodes. Figure 28 displays the average dislocation spacings ρ -0.5 as function of normalized stress. Allowing for an experimental uncertainty of a factor of 2 for determination of the dislocation spacings, the majority of the data points in the Harper – Dorn regime lies in the bG/σ band. Ardell’s proposal of frustration of network coarsening implies that the dislocation spacing would approximately follow bG/σ in the 5 – power regime until frustration sets in and the data level off. However, for Al the frustration level given by Ardell is not supported by the Al data to a significant extent. From the results of Barrett et al [18] for stress dependent subgrain size, it is also clear 70 10 -6 10 -5 10 -4 10 -3 10 -1 10 0 10 1 10 2 10 3 Pure Al Barrett et al. (1972) " Ardell and Lee (1986) " Kassner and McMahon (1987) Al - 5 % Mg Yavari et al. (1982) Al - 11 % Zn Blum (1991) NaCl Banderdt and Sammis (1985) LiF Streb and Reppich (1972) Network Frustration Ardell (1997) G σ ) m ( ρ 1 μ σ bG Open : etch pits Closed: TEM σ bG 2 σ bG 5 . 0 Fig 28. Dislocation spacings (1/√ρ) versus normalized stress for pure Al, Al-5% Mg and NaCl in the H–D regime [18, 40, 43, 64, 116, 149, 169, 170]. Data for Al, LiF and Al-5% Zn in 5-power regime and lines proportional to bG/σ are shown for comparison. The transition to Harper–Dorn is suggested to occur at σ/G = 10 -5 for NaCl and 10 -6 for Al. Note. The values for physical constants (G and b) for Al-5% Mg and Al-5% Zn used for the plot are taken to be the same as for pure Al. 71 that the total dislocation density (i.e. the sum of lengths of free dislocations and subgrain boundary dislocations per crystal volume) will depend markedly on stress. Rather, the majority of data lie above bG/σ. Best support for network frustration seems to come form the data for Al – Mg. However, it depends on a single data point and the frustration level is one order of magnitude in dislocation spacing (two orders of magnitude in dislocation density) below the level for Al. It seems that at the present state of microstructural knowledge there is no conclusive evidence for constancy of dislocation spacings in the Harper – Dorn regime. 2.7. Additional Observations on Harper – Dorn Creep 2.7.1. Weertman and Blacic [77] Weertman and Blacic [77] attempted to explain the constant dislocation density behavior of Harper – Dorn creep. A cyclic change in temperature results in a non- equilibrium concentration of vacancies, leading to a chemical potential or an internal stress which is cyclic in nature. Due to this cyclic nature, the stress does not produce any net strain in the sample, but depending on the amplitude and the time period of the temperature oscillation it can be very large compared to the applied stress and hence it produces a dislocation density which can be significantly larger than those produced solely by the applied stress. For the case of pure Al at 0.99 T m , a temperature fluctuation of ± 1 K in a time period of 5 and 50 minutes will produce a saturation dislocation density of 2.3 × 10 8 and 5 × 10 7 m -2 , respectively which is an order of magnitude higher than the expected values based on 5 – power law extrapolation. These values are the same as those reported in the Harper – Dorn regime [40, 43]. Hence, a temperature 72 fluctuation with larger amplitude and moderate time period can produce a significantly large number of dislocations, leading to a higher strain rate and probably Harper – Dorn creep [73]. Based on the above arguments, Weertman and Blacic [77] based on above arguments, concluded that Harper – Dorn creep cannot be a genuine creep mechanism. No later studies made any attempt to test the validity of the above argument, experimentally. Nabarro [73] calculated the activation energy, Q, based on the above formulation and found a value of Q ≈ 0.69 Q SD , where Q SD is the activation energy of the self diffusion. Nabarro [73] believed this value to be consistent with the observed activation energy in the Harper – Dorn regime, but it is not consistent with the original work of Harper and Dorn [15] where it was shown to be equal to Q SD . 2.7.2. Raj and Co-workers [171 - 173] Raj and co-workers [171 - 173] suggested that the sample size can have an active influence on the nature of Harper – Dorn creep. Raj [171] observed that the grain sizes in the studies showing Harper – Dorn creep were comparable to the sample thickness. Further, based on the observation of a critical stress, σ/G, for the transition from 5 – power law creep to Harper – Dorn creep and the ratio of the sample thickness to the grain size, Raj [171] proposed the following relationship: 8 0 6 10 3 7 . d t . G − × = σ (20) where t is the thickness of the sample and d is the grain size. Later, Raman and Raj [172] proposed a mechanism using the surface dislocation sources to account for the observed higher strain rate as well as constant dislocation 73 density in Harper – Dorn creep. Surface sources might be activated by a climb controlled mechanism involving the flow of vacancies from the surface to the dislocation line and the stresses required to activate such surface sources are within the Harper – Dorn regime. Elaborating the idea of the t/d – ratio dependence of Harper – Dorn creep, Raj [173] proposed a grain size dependence ( ε & ∞ (b/d) p , where 1 ≤ p ≤2) in the Harper – Dorn regime for samples with a t/d – ratio larger than 4 whereas this dependence disappears for samples with smaller t/d – ratios. Hence, Raj and co- workers [171 - 173] concluded that Harper – Dorn creep is not a bulk sample phenomenon and can be operated only in a thin sample. 2.7.3. Nes and Co-workers [25] Nes et al [25] explained the occurrence of Harper – Dorn creep as a sample size effect. The subgrain size, λ, at a stress σ is equal to k 1 bG/σ, where k 1 is a constant. For pure Al, the value of “k 1 ” is ~ 20. Based on similar relationship, the mean free path of the dislocations, λ 1 is calculated using the following equation [174]: σ ψ λ Gb 1 1 = (21) where 1 ψ is a constant. It is assumed that the dislocation would move across ~ 5 subgrains before they are stored and hence Ψ 1 is roughly equal to 100 [175]. Table 8 shows the mean free path of the dislocations at low stresses (in the Harper - Dorn regime). Table 8 shows that λ 1 is comparable to or even larger than the sample sizes used in various experiments in the proposed Harper – Dorn regime. Hence, it is expected that dislocations can travel to the surface of the specimens and will leave the 74 Table 8: The Mean Free Path of the Dislocations in the Low Stress Range Stress (MPa) Mean Free Path for Dislocations (mm) 0.02 24.20 0.03 16.13 0.04 12.10 0.05 9.68 0.06 8.07 0.07 6.91 0.08 6.05 0.09 5.38 0.1 4.84 75 specimen, which in turn may increase the deformation rate. Based on the model proposed by Nes et al [25], it is expected that the strain rate will vary inversely with the square of the characteristic size of the specimen in the region where dislocation characteristic length (~mean free path) is larger than the sample size: G D kT Gb S C sd σ ε 3 2 1 1 = & (22) where C 1 is an empirical constant, S 1 is a geometric parameter directly proportional to the sample size and D sd is the diffusion coefficient for self diffusion. Nes et al [25] argue that the so called Harper – Dorn creep can be reproduced only in small sized samples and hence it is not a genuine creep mechanism. 2.8. Compensation for Friction Due to the compressive nature of loading, a friction cone is formed near the end of the sample [176]. A friction cone is defined as the volume of the sample which does not (or only partially) deform under the applied stress. This is also a region where the actual flow stress is lower than the applied stress. Friction between the sample and the sample holder constrains the free movement of the sample end surfaces and this gives rise to the formation of the friction cone. Fig 29 shows a schematic of the friction cone formed in compression tests [177]. Both FEM and experimental studies have suggested this volume as a cone with roughly coinciding base with the sample ends [178, 179]. The volume fraction of the friction cone depends on the aspect ratio and hence it changes during the course of the deformation (as the aspect ratio decreases with strain 76 Fig 29: A schematic showing the friction cone for two samples with aspect ratios of 4 and 1.5, respectively [177]. The dotted line shows the axis of compression, l is the length of the sample and w is the width of the sample. l/w = 4 l/w = 1.5 77 during the compression test). Widinger et al [180], after careful experiments on samples of Al and NaCl, proposed an empirical equation to correlate the compression tests at various ratios values of κ (in this report this is addressed as “κ-ratio”) and the tensile test gives: q c exp − = κ σ σ 1 0 (23) where σ 0 is the nominal true stress which is calculated from the applied load and the instantaneous cross section area. The constant “κ” is defined as length/square root of area and hence is a strain dependent quantity and σ is the actual stress (or flow stress) in the sample under compression. The constants “q” and “c” are the empirical constants fixed for a set of experiments. The value of “κ” is calculated using the equation ) 5 . 1 exp( 0 ε − κ = κ (24) where κ is the instantaneous κ-ratio, κ 0 is the κ – ratio at zero – strain defined by Eq 25 and ε is the true strain: 0 0 0 A l = κ (25) where l 0 is the initial length of the sample and A 0 is the initial cross section area. Based on the above set of equations, Fig 30 is drawn showing the variation of σ/σ 0 with strain for various κ 0 . Table 9 shows the σ/σ 0 ratio for a sample with κ 0 equal to 0.93 in tabular form. A κ 0 ratio of 1.9 corresponds to an aspect ratio of 1.6 for cylindrical specimen. Fig 31 does not show any significant effect of friction at lower strains. Even for a k 0 ratio of 0.93 (aspect ratio 0.82), Table 9 shows a decrease of only 10 % in the σ/σ o value at a strain of 5 %. 78 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00 0.25 0.50 0.75 1.00 30 % k 0 =1.9 σ/σ 0 Strain p = 0.95, c = 0.47 σ/σ 0 = (1 - exp(-k/c)) q k = k 0 exp(-1.5ε) k 0 =1.06 5 % Fig 30: The relation between the nominal (σ 0 ) and the actual stress as experienced by the sample (σ). 79 Table 9: Experimental data and corresponding correction for κ 0 = 0.93 and an initial stress of 0.1 MPa Strain σ 0 (MPa) σ 0 / G k σ (MPa) σ / G σ/σ 0 0.005 0.10 5.82 × 10 -06 0.93 0.10 5.82 × 10 -06 1.00 0.048 0.22 1.43 ×10 -05 0.87 0.20 1.31 × 10 -05 0.91 0.138 0.38 2.49 ×10 -05 0.76 0.34 2.20 × 10 -05 0.88 0.102 0.14 9.35 ×10 -06 0.80 0.13 8.38 × 10 -06 0.90 0.150 0.38 2.48 ×10 -05 0.75 0.33 2.18 × 10 -05 0.88 0.169 0.30 1.98 ×10 -05 0.72 0.26 1.73 × 10 -05 0.87 0.189 0.20 1.29 ×10 -05 0.70 0.17 1.12 × 10 -05 0.86 0.083 0.21 1.38 ×10 -05 0.82 0.19 1.24 × 10 -05 0.90 0.206 0.13 8.77 ×10 -06 0.69 0.12 7.52 × 10 -05 0.86 0.209 0.05 3.24×10 -06 0.68 0.04 2.78 × 10 -06 0.86 0.218 0.14 8.85×10 -06 0.67 0.12 7.54 × 10 -06 0.85 0.229 0.18 1.2 ×10 -05 0.66 0.16 1.02 × 10 -05 0.85 0.245 0.28 1.83 ×10 -05 0.65 0.24 1.54 × 10 -05 0.84 0.270 0.33 2.19 ×10 -05 0.62 0.28 1.82 × 10 -05 0.83 80 CHAPTER 3. MATERIALS AND EXPERIMENTAL PROCEDURE 3.1. Materials Dorn and co-workers [15, 16] conducted their experiments on pure Al (both single and polycrystals). After that, many other works were conducted in the Harper – Dorn regime using Al with varying purity levels [17 – 28]. Also, several reports on Al – Mg alloys are available with the observation of Harper – Dorn creep [17, 41 – 43]. Pure Al has a low melting temperature (~933 K) and hence the temperature control for the high temperature creep testing for an extensive period of time is comparatively easy. Also, pure Al has a higher stacking fault energy, γ (γ/Gb ~ 18.9 × 10 -3 at melting temperature as compared to 3.2 × 10 -3 for Cu at its melting temperature, 1356 K [181]); which might lead to the onset of probable Harper – Dorn regime at relatively higher stresses [73]. In creep experiments, a better control for loading is achieved if the applied loads are higher. There has been numerous works on Al in other creep regimes and there is a large data base on creep for pure Al. Figure 31 shows some of the data over a wide range of temperatures and stresses for pure Al. 3.1.1. Single Crystals Most of the tests during the present work were conducted on 99.999 % pure aluminum single crystals. The crystallographic orientation of the samples was [100]. This orientation of a FCC crystal has 12 most favored slip systems out of which 8 have the same Schmidt factor, ~ 0.41 (with respect to the compression axis along the [100] direction), hence leading to a more uniform deformation compared to other possible orientations. A quantitative study was conducted to observe the effect on the Schmidt factor by a deviation in the orientation [100] by an arbitrary angle. 81 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -20 10 -18 10 -16 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 PLB 4.5 εkT/DGb σ/G 1 Pure Al ? 5 - Power Law Fig 31: The compensated steady state strain rate versus the modulus compensated steady state stress for pure Al (reproduced from [36, 182]). PLB stands for power law breakdown. 82 Based on the results of the study, it was concluded that a deviation of up to < 4˚ should allow multiple slip, but any deviation of more than 5˚ should lead to a non-uniform slip over different slip systems. Figure 32 shows the variation of the Schmidt factor with deviation in azimuthal angle (as defined in the spherical coordinate system). Based on this analysis, a deviation within ± 1 degree (the specified error for the present sample) in the orientation was ignored. A code was written in Mat Lab for the above analysis and is attached as Appendix I. The material was provided by MaTeck GmbH, Im Langenbroich, Germany. The impurities of the material are given in Table 10. The as – received material was in the form of a cylinder with a diameter of 25 mm and a length of 125 mm. The larger diameter of the samples avoids size – effects of samples as well as the surface oxide effects [20]. A single crystal is good for the experiments as: (a) there is no presence of Coble or Nabarro – Herring creep due to an absence of grain boundaries and (b) the other effects of grain boundaries can be avoided. 3.1.2. Polycrystals The material was provided by University of Erlangen-Nuremberg, Germany. The purity of available cast Al was 99.97 %. The impurities are listed in Table 11. One sample had 5-6 randomly oriented grains with an average grain size of ~ 10 mm. The material had a square cross – section and the outer surfaces were not smooth. Conducting experiments on the polycrystalline Al provides important information, such as (a) the effect of the grain boundaries in the Harper – Dorn regime, (b) the independence of strain rate from the grain size and (c) the contribution of other creep mechanisms (e.g. Nabarro – Herring creep, etc) in the Harper – Dorn regime. 83 0 20 40 60 80 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Schmidt Factor Deviation in Azimuthal Angle ( 0 ) Deviation of Crystal Orientation with Respect to [100] by an Angle of 0 0 0 20 40 60 80 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Schmidt Factor Deviation in Azimuthal Angle ( 0 ) Deviation of Crystal Orientation with Respect to [100] by an Angle of 1 0 Fig 32: Variation of the Schmidt factor for the 12 most preferred slip systems of the FCC system with the deviation angle from the [100] orientation. The numbers 0 to 90˚ (FCC crystals have rotational symmetry of 90˚ with respect to [100]-axis) on the x – axis shows the azimuthal angle (the angle of sweep on the surface of the cone formed by the deviation vector when rotating it with respect to the [100]-direction). 84 Fig 32 Contd. 0 20 40 60 80 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Schmidt Factor Deviation in Azimuthal Angle ( 0 ) Deviation of Crystal Orientation with Respect to [100] by an Angle of 4 0 0 20 40 60 80 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Schmidt Factor Deviation in Azimuthal Angle ( 0 ) Deviation of Crystal Orientation with Respect to [100] by an Angle of 15 0 Fig 32: Variation of the Schmidt factor for the 12 most preferred slip systems of the FCC system with the deviation angle from the [100] orientation. The numbers 0 to 90˚ (FCC crystals have rotational symmetry of 90˚ with respect to [100]-axis) on the x – axis shows the azimuthal angle (the angle of sweep on the surface of the cone formed by the deviation vector when rotating it with respect to the [100]-direction). 85 Table 10: Impurities associated with the single crystal Al with purity of 99.999 % Table 11: Major Impurities associated with the polycrystal Al with purity of 99.97% 9 Total Mass Index (TMI) is the aggregate total of all the impurities. Element Na Zn Mn Si Cu Ti Mg Fe Cr TMI 9 Presence (ppm) 0.5 0.5 0.2 1.0 1.0 0.2 1.0 2.0 0.2 10 Element C Zn Na Mn Co Si Cr Cu Ti Mg Fe Pb Ca Presence (ppm) 60 10 <5 <5 <5 37 <5 37 <5 <5 110 10 <5 86 3.2. Sample Preparation: General Procedure 3.2.1. Cylindrical Specimens Specimens were cut from as – received material using a slow – speed saw of thickness ~ 0.5 mm. Since pure Al is a very soft material and fixing it directly in the sample holder of a cutting machine could impart large stresses, a sample holder of brass was prepared and the sample was glued inside it using a non-reactive polymer. This assembly is shown in Fig 33. This method was effective in avoiding any kind of unintentional stresses in the material during the cutting. After the specimen was cut, both flat ends were ground on grit papers of 800, 1200 and 2500, respectively. This was conducted to ensure the ends were plane parallel. The samples were plane parallel within 30 µm. Samples were then electro-polished using a solution of 10 % percholeric acid with 90 % of acetone. Electro – polishing was conducted at room temperature. Electro-polishing was performed to provide a better observation of the surface features, such as slip lines, etc. Prior to applying the load, specimens were annealed at the test temperature (913 K) for ~ 50 hrs. Annealing was performed to achieve (a) the removal of any unintentional stress effects during sample preparation and (b) a very low dislocation density which is a necessary condition for Harper – Dorn creep. During the annealing process, a very low load (~0.002 MPa) was applied to maintain contact between the sample, platens and the connecting / loading rods. 87 Fig 33: The brass jacket – sample assembly used to avoid deformation in the sample while cutting it using the slow – speed saw. Brass Jacket Al Sample Glued by Polymer 88 3.2.2. Square Cross – Sectioned Specimens Square cross – section specimens were required for the following two reasons: a. Compared to the cross - sectional area of the as received single crystal, some of the creep machines used in the present study required smaller cross - sectional areas for the samples. Based on the availability of the machining tools for the present study, producing a square specimen induces less machining damage compared to the preparation of a cylindrical specimen. b. The as – received polycrystalline specimens were of square cross – section. For the single crystal, a cylindrical piece of the sample was cut with the desired length of a square cross – section sample. Using a low – speed diamond coated wire saw, a square cross – section was cut out of the circular cross – section. Due to the long length of the specimens, the wire cutting could not give a smooth surface along the length and also the cross – section was not completely orthogonal or perpendicular at either end. To achieve flat, smooth and orthogonal surfaces, the specimens (both single crystal, prepared from the cylindrical as – received material and polycrystals received in square cross – section) were ground using 800, 1200 and 2500 grit papers. This was performed using a special die made of hardened steel and by repetitive and consecutive polishing over the various faces leading to the reduction of the error in each consecutive pass. This process is shown in Fig 34. Mechanical polishing was conducted over a rotating table. A similar attempt was made to make the end surfaces plane parallel. The criterion for plane parallelism was 30 µm, as in the case of cylindrical specimens. As with the cylindrical specimens, the side – surfaces of the square cross – sectioned specimens were electro-polished. 89 Fig 34: The schematic of the grinding process to achieve homogeneous cross - section 90 The sample was annealed at the test temperature (913 K) for ~ 45 hrs in a separate furnace. Following the anneal, the sample was kept in the test furnace at the test temperature for 5 hrs before applying the desired load. 3.3. Testing Conditions The average cross – section was calculated based on the mass and length measurements. In order to calculate the average cross – section of the sample, its mass was divided by the product of the density of Al (2702 kg/m 3 ) and the measured length. In all cases, the average cross – section was equal (under the uncertainty associated with measurements) to the cross – section calculated from the measured dimensions of the sides or the diameter. This type of method allowed a quick and accurate measurement of the cross – section. Consistency in the measured value of the cross – sectional area and the calculated value based on direct measured values of diameter (or sides) indicated an absence of pores in the sample. All tests were conducted in compression. The temperature of the test was 913 K and the temperature was maintained within a limit of ± 1 K. Since the testing temperature is very high, the direct contact of the sample with the connecting / loading rods will lead to the formation of the eutectic point and hence the sample will melt. In order to avoid the formation of the eutectic point, ceramic (Al 2 O 3 ) platens were used between the sample and the loading rods. In most of the tests, proper lubrication (Al based liquid lubricant or boron nitride) was used between the sample and the platens in order to reduce the effect of the friction. Higher friction constrains otherwise free movement of the end surfaces leading to the bulging of the sample during the 91 deformation. Due to the long wait prior to application of test load and due to the very high heat conductivity of pure Al, a homogeneous temperature distribution across the sample was achieved. The temperature variation along the length was within ±2 K for the creep machine at University of Erlangen – Nuremberg and it was within ±1 K for the creep machine at the IPM. Brno. At the University of Erlangen, the load was applied directly on sample using the simple connecting/loading rods while at IPM, Brno, a lever-arm was used to apply the desired load on the sample. The details of these two creep machines are outlined in the following sections. 3.4. Creep Machines 3.4.1. University of Erlangen – Nuremberg, Germany Figure 35 shows the creep machine. It is composed of the following parts: a. Furnace: It is necessary to have a furnace which provides adequate temperature distribution as well as a much longer operation life. A two – zone furnace was used to heat the sample. The small size of the furnace ensures a better temperature distribution and hence, a smaller temperature gradient across the sample. The creep tests in the Harper – Dorn regime usually run for months. Due to the nature of the tests, the furnace should be engineered for a long – time use. The following ensures a longer life i. The windings were filled with molten ceramic (Al 2 O 3 ) which was placed in an inside – outside manner through the coils. As shown in Fig 36, it can be understood as winding the heating wire over the ceramic rods and stacking them side by side. This led to the solidification of the ceramic powder in such a way that it supports the heating coil and hence prevents a slacking of the coil due to its own weight at very 92 Fig 35: A digital photograph of the creep machine at University of Erlangen – Nuremberg. 93 Fig 36: The schematic of the optimized furnace coiling. This type of coil is very effective in reducing the slacking of the coils. Base Ceramic Coils over Supporting Ceramic Rods Supporting Ceramic Rods after Solidification 94 high temperatures. In this way, melting of the coils due to the heat concentration from slacking could be prevented, resulting in a very long life for the small furnace. ii. The length of the heating wire was increased by reducing the pitch of the heating coil helix. Due to this, the surface load (heat energy produced per unit surface area) over the heating wire can be reduced and hence the life of the furnace is increased. This step was easier due to Step i, as slacking in the coils was reduced and hence more wire length may be accommodated. b. Cooling Water Circuit: Radiation from the furnace – jacket can damage some delicate equipment (e.g. the laser equipment, electronic controllers, etc) which are kept nearby. Therefore two cooling water circuits to cool the exterior of the creep machine and eliminate any foreseen damage: i. The cooling water jacket over the furnace took water from the common cooling water reservoir of the university. There were copper tubes over the furnace outer jacket through which water passed. This cooling water was susceptible to clogging and a better reservoir may be needed for further tests. ii. For the cooling water used to keep the upper and lower supporting rod cool, a separate water tank – pump system was used. This contained demineralized water as the parts through which the water passes were made of expensive materials. The load cell was also kept cool using this circuit. c. Loading and Load Cell: The sample was supported using a lower and upper metallic rod (Fig 35). Since the creep tests were conducted at temperatures very close to the melting temperature of the sample, a direct touch with another metal may lead to the melting of the sample by the formation of the eutectic point. To avoid such melting 95 of the sample, ceramic (Al 2 O 3 ) platens were used between the metallic supporting rod and the sample. The load was applied directly to the upper rod whereas the load cell was kept at the end of the fixed lower supporting rod. Dead weights were balanced by a pulley-weight mechanism which was attached to the upper rod. d. Laser System: A laser system was used to measure the displacement and hence to measure the strain in the creep. A rotating source produced the red laser which fell on the creep sample. The inbuilt system received the reflections from the sample and measured the interference pattern. The device measured the distance between the interference patterns and hence measured the distance between the reflecting surfaces. The laser box transferred the data to a computer where it was analyzed to provide the instantaneous height of the specimen and hence the total strain accumulated. The data was then transferred to the main computer for plotting and storage. The following describes the steps for the measurement of the strain using this laser extensometer system: I. Two strips of TiO 2 were put on the surface of the sample or on the platens to act as the non-reflective parts on the sample. These points were treated as the length of the sample by the laser system. The error in the laser measurement decreased as the separation between the strips increased. Figure 37 shows a schematic of the strips on the sample surface. II. The laser source could be turned on by the rotating a key. A glass window made of boron was fitted in the furnace so that the laser beam was able to pass through the furnace wall. Boron is a good insulator for the heat; hence the heat loss through this window was minimized. 96 Fig 37: A schematic showing the TiO 2 strips over the sample surface. TiO 2 Strips (Sensed by Laser) 97 III. The laser system needed to be trained for the incoming signals. The computer attached to it showed the interference pattern created by the laser rays. Using a few buttons, a band pass filter was calibrated to eliminate noises. The controller also showed the reflected signal along with its first and the second derivative.The morphological changes (e.g. the ceramic platen ends, surface strips / markers etc) occur at the point where the first derivate is either maximum or minimum and the second derivative is zero showing a mathematical inflection point. At this point, a burst in the signal was shown by the system. Separate bursts were registered for all non – reflective entities. The computer was capable of measuring the distance between the burst points. In order to work as an extensometer for the creep tests, the computer needed to recognize the exact number of these bursts. This information is fed into the system following a sequence in which the bursts are shown in the scope. Hence, the bursts become numbered and the computer measures the length between any given two numbered bursts which corresponds to the length of the desired section on the sample. e. The Main Computer: The data was recorded in a computer by the software “CatMan”. This software not only recorded the data but also provided a visual platform where the stress, strain, strain rate, temperature and force were plotted with respect to time or strain. The software was able to export all data to an Excel file and hence it was then used for any further analysis. The main computer interacted with its devices using an electronic platform called spider and received all required data from the load cell, laser system computer and temperature thermocouple. 98 3.4.2. Institute of Physics of Materials, Brno Unlike the testing facility available at the University of Erlangen, the creep machine at IPM, Brno used the lever arm mechanism for loading the sample. The lever – arm ratio was 1:2. The creep machine was designed for providing constant stress. The creep machine at IPM, Brno, is illustrated in Fig 38. There was no provision in the machine to measure the load on the specimen in real time but the measurements of the actual load on the specimen were performed when the test was not running. Due to several mechanical contacts in the load transfer – chain, a friction shielding of the applied load was expected. Nevertheless, it was characterized beforehand. The creep machine was designed and manufactured at the machine shop of IPM, Brno. The machine used a cage type of the sample holder that compressed the sample even though a tensile stimulus was applied in the connecting rods. A schematic of the sample holder is given in Fig 39 (a). The schematic shows three numbered parts, 1, 2 and 3, respectively. Sample is shown by red color and is denoted with a symbol “S”. The parts “1” and “2” are joined together by a rod (shown in Fig 39(a) by a heavy line) and hence it constrains these two parts to move in the same direction. Part “1” is attached to the upper rod of Fig 38 which in turn is connected with the lever arm and it is pulled in the upward direction. Part “3” is fixed with the lower supporting rod and it does not move in the course of the creep test. The transducer measuring the force on the sample is placed below part “3”. As part “1” (and hence part “2”) move up and part “3” stays fixed, the sample “S” undergoes a compressive load. 99 Fig. 38: A digital photograph of the creep machine at IPM, Brno. The pictures were taken from different angles to show all salient features of the equipment. 100 Fig 39: A schematic of the cage – type of holder for the specimen. (a) shows the schematic at IMP Brno, whereas (b) shows a possible set – up with proper alignment using a guide on lower holder as well as supporting rods for the upper holder. Connecting/ Loading Rod Supporting Rod (Upper Holder) Fixed Base with Guide for Lower Holder Fasteners Lower Holder (Moving) Sample Upper Holder (Stationary) Guide on Lower Holder (a) (b) 101 Due to these simple but well machined parts, the basic tensile motion in the connecting rods was transmitted as compressive load over the sample. As shown in Fig 39, the design of the cage was not friction – free and hence dry powder of molybdenum - sulphide was used as a lubricant to reduce the friction. The thick red lines across the sample “S” are the ceramic plates. Fig 39 (b) shows a 3 – D view of the sample holder and its position in the creep machine. Other features of the creep machine are outlined below: i. An LVDT was used to measure the strain in the sample. The strain was measured between the sample ends instead of the central 1/3 rd region 10 . This is consistent with the later tests conducted at University of Erlangen – Nuremberg. ii. A temperature gradient was maintained within ±1 K over a length of 50 mm; hence, for the length of the present sample (24 mm), a gradient of ±0.5 K was expected. During the course of long term tests, a maximum fluctuation of ±1 K was expected. iii. A computer – based system was used to receive and store the data. Electronics were attached to the creep machine to monitor various parameters, such as the temperature, displacement, etc, and for a feedback response. iv. An alignment rod was attached to the upper rod (shown in Fig 38) to ensure the upper rod remains vertical during the creep test. v. The load is placed on the “load pan” (shown in Fig 38). The pan was balanced with some additional weight before the real weights were applied. 10 The central 1/3 rd region is assumed to be free from friction effects and hence represents the actual stress [176]. 102 vi. The furnace was seamless. The sample was mounted in the cage (Fig 39) and it was attached to the lower rod, then fixed to the upper rod (attached to the lever arm) and then the whole upper rod was pulled upward. Later, the hook on the upper rod was attached with the lever arm. The mounting of the sample was therefore complicated and must be performed carefully. vii. Lubricants were used between all mechanical joints to reduce the friction. A solid molybdenum sulphide – based lubricant was used between different parts of the sample holder while an Al – based liquid lubricant was used between the sample and the ceramic platens. 3.5. Grain Boundary Sliding Measurements Lifshitz sliding is an inherent consequence of the vacancy flow from one grain boundary (which is transverse to the stress axis) to a grain boundary which is parallel to the stress axis and it takes place as an accommodation process and results in elongated grains [183]. Hence, Lifshitz sliding occurs in Nabarro – Herring and Coble creep. Lifshitz sliding is mechanistically different from Rachinger sliding which occurs in conventional superplasticity [1] when the grains of a polycrystal move over each other in direct response to an external stress. By contrast, Harper-Dorn creep appears to take place through an intragranular dislocation process, the measured strain rate is independent of the grain size, there is no mass flow of vacancies and accordingly no offsets are produced in any surface markers. Langdon [12, 13, 184] suggested a measurement of Lifshitz sliding as one of the ways to analyze the importance of stress – directed diffusion of vacancies in Harper – 103 Dorn creep. Figure 40 gives a schematic showing Lifshitz sliding and hence provides an experiment which can be conducted on highly pure Al polycrystals. Based on this information, the method of marker lines is suggested here to measure the contribution of grain boundary sliding in Harper – Dorn creep. For the marker lines method, thin marker lines were drawn over the polycrystalline samples in the longitudinal direction with respect to the stress axis. Before drawing the marker lines, all surfaces of the samples were polished to achieve a mirror – like surface finish using 800, 1200 and 2400 grit papers. Following the mechanical polishing, the sample surfaces were electropolished to produce scratch – free mirror – like surfaces. Then marker lines were scribed by rubbing a diamond paste of size 3 µm soaked in acetone on a lens paper over the surface very gently and only once. The sample was tested in the creep machine for a strain ~ 1 - 2 % at a selected stress. The sample was then cooled under load and taken out of the furnace. The sample was examined under an optical microscope using magnifications of 100X to 200X. Optical micrographs were taken and the transverse offsets in the marker lines with respect to the compression axis were recorded. Several offset measurements were performed at various grain boundaries to obtain a statistically – consistent value of the average grain boundary sliding. The following equations gave a value for the grain boundary sliding [185]: t gb L w Φ = ε (26) total gb gb ε ε = ξ (27) 104 Fig 40: Appearance of three grains in a polycrystalline matrix (a) before tensile creep with longitudinal marker line AA’ and BB’ (b) after Harper – Dorn creep, and (c) after diffusion creep; the tensile axis is vertical [12]. (a) (b) (c) 105 where ε gb is the strain due to the grain boundary sliding, Φ is an empirical constant usually equal to 1.5 (compared to π/2 used by Harper et al [16]), w is the average offset in the transverse direction, L t is the mean intercept length of grain in the longitudinal direction with respect to the applied stress, ξ gb is the contribution of the grain boundary sliding in the total strain, ε total . Grain boundary shearing was measured at three different stresses (0.07, 0.18 and 0.3 MPa) using the same sample. These three stresses are in the Harper - Dorn regime, the transition region and the five power – law region, respectively. After the first set of experiments at one stress, the sample was repolished using 4000 grit paper and was electro-polished. New marker lines were scribed and the sample was deformed following the previously described procedure. 3.6. Microstructure Study It is an important aspect for any experiment to analyze the micro-structural details of the material. Experiments in the Harper – Dorn creep have shown a lower value of the dislocation density using the etch – pits technique for a measurement of the dislocation density. In order to observe the dislocation microstructures, an etch – pits study was conducted over the surface of the deformed as well as annealed specimens. Only the <100> plane was observed. Samples were electro-polished at room temperature using a mixture of 90% acetone and 10% perchloric acid. Samples were not polished mechanically and hence any mechanical damage was excluded. Due to the low melting temperature of Al, samples showed some oscillations over the surface after 106 electropolishing. This was reduced considerably by cooling the sample in liquid nitrogen prior to the electro-polishing. Proper care was taken in deciding the total time for the electro-polishing in order to avoid any possible etching or oxidation of the sample due to over electro-polishing. Next, samples were etched for 6 – 8 seconds using a solution prepared with 50% HCl, 47% HNO 3 and 3% HF. The etched parts of the samples were later observed under an optical microscope at various magnifications. The time spent between the electro-polishing and the etching was less than 1 to 2 minutes. Several photo-micrographs were taken, both following a line and at random locations. Etch – pits were counted per picture and the dislocation density was calculated using the following equation [186]: A 2N = ρ (28) where, ρ is the dislocation density, N is the total number of the etch-pits found in the area, A. Special attention was taken to examine for any possible dislocation substructure formation. Photo-micrographs were taken at optimum magnification in order to quantify the sub-structure. 3.7. Features of Present Work Tests were conducted on both relatively smaller – sized specimens (cross – sectional area of ~50 mm 2 ) and bulk – sized specimens (cross – sectional area of ~ 500 mm 2 ). • Bulk – Sized Samples: Bulk - sized samples of highly pure Al single crystal were tested in compression. The diameter of these specimens was 24 mm. Creep tests in compression on the bulk – sized Al samples with higher purity of 99.999 % can infer if 107 the high stress exponent observed by Blum and co-workers [23 - 25] was due to the effect of impurity or is inherent to tests on the bulk material. • Smaller – Sized Samples: The smaller – sized samples were tested in compression and this can be used to critically examine the hypothesis of the size – effect as proposed by Nes et al [30]. The minimum cross – sectional area of the specimen used in the present test was 50 mm 2 (7 × 7 mm). Since the nature of the experiments is very delicate and very slow, and they were conducted at very high temperatures and very low stresses, the following additional care was taken: i. Steady – State: The tests were conducted to achieve very high strains to make sure that the steady-state was reached [30]. Additionally, in order to ensure steady – state, plots of the creep rate against creep strain were obtained. A steady – state was represented as zero rate of the change of the strain rate and hence the strain corresponding to the minimum point in the strain rate vs. strain plot will clearly explain the ambiguity of steady state which has been considerably highlighted in the recent literature on Harper – Dorn creep [23 - 28]. None of the prominent studies in the Harper – Dorn literature seems to employ this technique to confirm steady – state. ii. Measurement Technique: Due to the use of very low stresses in these experiments, the use of conventional extensometers and mechanical sensors for measuring the strains in the specimen will induce some additional effects, such as shielding of the applied load by friction, etc. A laser extensometer was used in most of the tests to accurately measure the deformation without making any contact with the sample. iii. Load Cycling: Tests were conducted with and without changing the stress. 108 Changing weights may lead to discontinuous dynamic recrystallization [23] and hence a few tests were conducted at a constant stress also. Stress change tests give accurate results for the stress exponent and less time is required to attain higher strains and hence steady – state. 109 CHAPTER 4. EXPERIMENTAL RESULTS 4.1. Creep Testing: Single Crystals Table 12 gives a short summary of all the tests conducted during the present study using single crystals. In the following subsections, results for each sample will be explained in detail. 4.1.1. Sample 1 Special Note: Due to the lack of sufficient space between the connecting / loading rods, there was a pre-strain of 5.9 % in the specimen which was caused by the thermal expansion of the sample during heating. Hence, the κ- ratio and the aspect ratio prior to “actual” loading were reduced to 1.1 and ~1.0, respectively. Sample 1 was deformed up to a strain of ~40 % and it underwent significant bulging. Bulging is attributed to the large friction present between the sample ends and the platens as no lubricant was used. A digital image of Sample 1 after the deformation is shown in Fig 41. It showed dominance of only one slip plane out of the four slip planes. From a calculation based on a stereographic projection, it was concluded that the most active slip plane is (111) followed by ( 1 11). The stereographic projection shown in Fig 42 shows active planes by red dots. Fig 42 is developed based on Fig 41 which shows that the active slip planes, when projected on a plane, lie diametrically opposite. Other slip planes (which should lie perpendicular to the above diameter) were not visible on the surface. It is consistent with the fact that a lower aspect ratio constrains the slip. 110 Table 12: Features of the single crystal specimens used in the present study. Sample Number Temp (K) Cross – Section Length (mm) Diameter /side (mm) Aspect Ratio (l/d) κ- Ratio (l/√A) Lub. ξ Stress Cycling Strain Measurement, Methodology 1 913 Cylindrical 23.54 24.1 1.0 1.2 N Y 90 % of length, Laser 2 911 Cylindrical 24.35 24.1 1.0 1.14 Y Y Central 1/3 rd zone, Laser 3 * 913 Square 24.01 12.5×12.7 1.9 1.9 Y N Whole Sample, LVDT 4 913 Cylindrical 40.44 24.1 1.7 1.9 Y N Whole & 1/3 rd zone, Laser 5 913 Square 15.16 7.1 × 7.2 2.2 2.1 N Y Whole Sample, LVDT 6 913 Cylindrical 41.56 24.1 1.7 1.95 Y Y Whole Sample, Laser Y: Yes, N: No ξ Lub : Lubrication * This test was conducted at IPM, Brno (Czech Republic). Other tests were conducted at WW I, University of Erlangen – Nuremberg (Germany) 111 Fig 41: A digital photograph of Sample 1 after the creep test. The first row pictures show the side view whereas the second row of the pictures show primarily the top and bottom view of the specimen. 112 Fig 42: The stereographic projection of the active slip planes with respect to the [100] – direction. The red dots indicate the active slip planes. The most active slip plane seems to be (111), followed by ( 1 11). 113 There were problems in the measurement of the displacement due to the bulging of the specimen. The strips are drawn on the surface of the specimen and hence they are susceptible to any upward movement of the surface material due to the bulging. Due to significant bulging at later stages of loading as observed in Sample 1, the strain rate measured was considerably lower (by a factor of ~2) than the actual strain rates (based on the measurement of the length, before and after deformation). This effect is termed as roll – over and is explained in Appendix II. Figure 43 shows the strain – time plot for Sample 1 whereas Fig 44 shows the strain rate vs. strain behavior. The test temperature was 913 K, but it is essential to represent all strain rates at 923 K (0.99 T m ), so that a comparison with the earlier studies become easier. Figure 45 is a reproduction of Fig 44 showing the corresponding strain rates at 923 K. The strain rates shown in Fig 44 were multiplied by a factor of 1.22 (i.e. D 923 K / D 913 K ) 19 to calculate the corresponding strain rates at 923 K. Figures 43, 44 and 45 show only a strain of ~30 % which is almost ~10 % less than the total strain calculated from the length measurements. This discrepancy is attributed to the upward movement of the surface strips due to barreling and is observed for all specimens tested using strips over the surface. 19 The temperature compensation factor, which appears in various plots, is based on the ratio of the lattice diffusion coefficients at the plotting temperature (923 K) and the test temperature (913 or 911 K). Strain rates have been normalized for the temperature of 923 K as most of the earlier studies, like Harper and Dorn [4], Barrett et al [22], Mohamed and Ginter [37] and Blum and co- workers [28 - 30] conducted the tests at 923 K (0.99 T m ); hence, it facilitates a direct comparison of the data collected in the present study with the earlier reports. Diffusion coefficients are given in Appendix III. 114 0 1x10 6 2x10 6 3x10 6 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.44 0.36 0.23 0.16 0.06 0.16 0.23 0.36 0.44 0.05 0.44 0.16 0.23 Strain Time (s) Al1SX1.0L E Cy 0.1 σ (MPa) = T = 913 K Fig 43: The strain vs. time plot of Sample 1 (Al1SX1.0L E Cy) 20 . The long and almost flat part in the plot is due to the long term test at the lowest stress (0.06 MPa). 20 Al1SX1.0L E Cy stands for : Aluminum, Sample number 1, Single Crystal (OX: Oligocrystal), Aspect Ratio was 1.0, laser system was used to measure strain (T instead of L denotes LVDT), E shows that the strain was measured between the ends of the sample , (C for central 1/3 rd region, P for platens) and it was a cylindrical specimen (Sq stands for square cross section) 115 0.00 0.05 0.10 0.15 0.20 0.25 0.30 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 0.06 T : 913 K 0.36 0.36 0.44 0.44 0.44 0.16 0.16 0.16 0.23 0.23 0.23 0.10 ε (s -1 ) ε unloading & reloading 0.23 4.5 Al1SX1.0L C Cy σ (MPa) = Fig 44: The strain rate vs. strain behavior of Sample 1. These strain – rate data are the “as measured” values from the experiment without any correction for friction or Taylor factor. The lines with slope of 4.5 (stress exponent for pure Al at very high temperature) are drawn. If the test condition is ideal then these lines should interpolate the region between the steady – state datum points between the same engineering stress, as the present test is a constant load test. Since, there is a missing strain (~10 %) in the measurement by laser system and also steady – state has not been reached in all cases, the lines with slope 4.5 do not interpolate the datum points between the same engineering stresses. 116 0.00 0.05 0.10 0.15 0.20 0.25 0.30 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 σ 0 (MPa) = 0.06 T : 923 K 0.36 0.36 0.44 0.44 0.16 0.16 0.16 0.23 0.23 0.23 0.10 ε (s -1 ) ε Al1SX1.0L C Cy Fig 45: The reproduction of Fig 44 after compensating for the temperature (from the test temperature of 913 K to the plotting temperature of 923 K) based on the lattice diffusion coefficient ratios. The black dotted lines are connected as interpolating “parallel” lines between the same engineering stresses. The dark large circle at the stress of 0.36 MPa at the strain of 0.16 is taken as the reference point. This point is chosen as reference point as it seems to be in steady–state and also this strain is thought to be free from any kind of bulging effect (low strain). 117 0.00 0.05 0.10 0.15 0.20 0.25 0.30 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 3.1 4.1 4.5 σ 0 (MPa) = 0.06 T : 923 K 0.36 0.36 0.44 0.44 0.16 0.16 0.16 0.23 0.23 0.23 0.10 ε (s -1 ) ε n =3.4 Al1SX1.0L C Cy Fig 46: The stress exponent calculated at different stress changes following the interpolation technique. The stress exponent is close to ~3 for the stress change to the lowest stress (0.06 MPa). 118 In order to prepare Fig 45 21 , the points at the same engineering stress were interpolated by parallel lines (across several engineering stresses). As is shown in Fig 45, the interpolating lines are neither parallel nor do they have a slope of 4.5. This shows that the strain rates measured at the higher strains are less than the expected values by a factor of ~2 in most cases. This missing strain is attributed to a fault in the measurement system which used strips on the bulged surface of the sample as its reference points and also steady–state was not achieved at the lower strains. Drawing parallel lines was therefore the best way to minimize the error. Also, the interpolation technique also minimizes the errors related to poorly defined steady – state during the test, especially at lower strains. This is due to the fact that instead of showing several strain rate values corresponding to one engineering stress, only one “average” strain rate is calculated for one stress. A very critical issue in the above analysis is the selection of the reference point, which is usually performed as follows: • Only one reference point is chosen for the entire test. • The point should be as close / near to the steady–state as possible, hence a higher strain value is preferred over a low strain value for the same stress. • The point should lie close to the median of the strain achieved in the test, as it then avoids the effect of bulging, etc. With respect to the reference point chosen in above way, strain – rates corresponding to the other stresses are calculated based on: • The strain rate at the reference point 21 For all other samples, analogous analysis was performed to get the reference point, stress exponent and the resulting strain rate – stress plots. 119 • The stress exponent for various stress changes Nevertheless, the results of Fig 45 are of considerable importance. Even if the measured strain-rates are lower that the expected values the strain rate corresponding to low stress (0.06 MPa) is higher than the 5 – power law and it lies between the Harper – Dorn and the 5 – power law regimes. As shown in Fig 46, the stress exponent for the lowest stress is ~3 based on the present strain rate measurements. Since the stress exponent calculation is based on the ratio of the measured values rather than absolute values, the stress exponent calculation may be free from any systematic error. Also, a trend in stress exponent values can be observed in Fig 46. The stress exponent decreases with the decrease in the stress (from n = 4.5 to n = 3) and this may indicate a possible transition between two independent creep mechanisms, one dominant at high stresses and the other dominant at lower stresses [148]. Figure 47 represents results on the strain rate – stress axes. The large dark point represents the reference point whereas the smaller points are the strain rates calculated based on the stress exponent shown in Fig 46 and the reference point. This type of analysis used the interpolation technique between the same stress levels and hence reduced the scatter as well as the error in the data representation. Figure 47 clearly shows a deviation from the conventional 5 – power law at the lower stresses but the slope at the low stresses does not indicate the onset of the Harper – Dorn behavior in the present sample. It should be noted that the stress values are compensated for the friction according to Eq. (23). 120 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 H-D Line Pure Al Al1SX1.0L E Cy ε (s -1 ) σ/G n = 4.5 σ (MPa) at 0.99 T m Fig 47: The stress dependence of the strain rate for Sample 1 for a temperature of 923 K. The large solid square represents the reference point whereas other points are derived based on the stress exponent and the reference point. The low stress behavior shows a deviation from both 5 – power law and Harper – Dorn creep. 121 4.1.2. Sample 2 This test was conducted in a series of stress cycling between two stresses (0.07 and 0.056 MPa) only. Both of these stresses belong to the probable Harper –Dorn regime. Sample 2 was deformed up to a strain of 6.8 % and it also showed bulging in the specimen even at such a low strain. Bulging indicates a malfunction or inefficiency of the lubricant. An Al - based liquid lubricant was used for Sample 2 which was changed in the subsequent tests. Contrary to Sample 1, it did not show the dominance of one slip plane under the microscope which means that the slip activities were uniform up to a strain of ~ 7 %. This may also indicate that the non – uniformity in the stress distribution due to a low aspect ratio is not as severe at small strains as in the case of higher strains (for example, Sample 1 which accumulated ~40 % of the strain). Figure 48 shows the strain – time plot for Sample 2 and Fig 49 shows the strain rate vs. strain behavior. Fig 49 also indicates the single stress exponent calculated for the test based on the interpolation technique between the same engineering stresses. The strain measured by the laser was 4.6 % whereas it was 6.8 % by the length measurement (before and after the test). This discrepancy is also attributed to roll over or bulging in the specimen. The absence of any considerable transient between the stress changes indicates the absence of strong substructure formation. Table 13 shows the stress exponent measured at various stress changes during the creep test. The stress exponent value remains close to 3.3; instead of 4.5 or 1. The problem of the smaller measured strain than the actual strain adds into this confusion. 122 0 1x10 6 2x10 6 3x10 6 4x10 6 5x10 6 6x10 6 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.056 0.071 0.056 0.056 0.071 Strain Time (s) T : 911 K 0.071 Al2SX1.0L C Cy σ (MPa) = Fig 48: The strain vs. time plot of the Sample 2. Engineering stresses are in MPa. Vertical lines show a stress change. 123 0.00 0.01 0.02 0.03 0.04 0.05 10 -10 10 -9 10 -8 10 -7 10 -6 n = 4.1 σ(MPa) = 0.056 0.071 0.071 0.071 0.056 T : 923 K 0.056 ε (s -1 ) ε Al1SX1.0L C Cy Fig 49: The strain rate vs. strain behavior of Sample 2. The points are joined by spline curves and an interpolation technique is used to join the data points of same engineering stress (as shown by dotted lines). The sudden jump in strain rate at ~1.2 % is attributed to the inhomogeneity in deformation with time and space. The larger circle shows the reference point, which was chosen according to the guidelines explained in the previous subsection. 124 Table 13: The stress exponent at various stress changes for Sample 2 22 Strain ε & (s -1 ) Engg. Stress (MPa) True Stress (MPa) n Avg. σ (MPa) Δσ (MPa) 0.012 4.19 × 10 -9 0.056 0.055 0.013 6.36 × 10 -9 0.071 0.070 1.80 0.063 0.007 0.031 2.03 × 10 -8 0.071 0.069 0.036 7.80 × 10 -9 0.056 0.054 3.96 0.061 0.007 0.036 7.80 × 10 -9 0.056 0.054 0.040 2.23 × 10 -8 0.071 0.068 4.52 0.061 0.007 0.040 2.23 × 10 -8 0.071 0.068 0.046 1.10 × 10 -8 0.056 0.053 2.90 0.061 -0.007 Average 3.3 22 The stress jumps are shown by vertical lines in Fig 49. The points are in the sequence of strain. 125 As shown in Fig 49, there is a sudden rise in the strain rate values for the same stress (a kink in the strain – time curve at 0.071 MPa, as shown in Fig 48). The strain was measured between the strips drawn in the central 1/3 rd region of the sample. It may be possible that the deformation is not homogeneous throughout the sample. Hence, it is possible that at the beginning of the test the outer region undergoes most of the deformation. Only after a certain strain, deformation becomes more homogeneous leading to higher strain rates in the central region also. This may be a possible reason for this behavior. It should be noted that an increase of the strain rate by a factor of 1.5 (the ratio of actual strain and the measured strain by laser) for the lower stress will push the results towards the Harper – Dorn regime. In any case, it is expected that the strain rates shown in Fig 49 are smaller than the actual strain rates and this factor can be 1.5, especially closer to the end of the experiment (i.e. at the higher strains). Figure 50 shows the behavior of sample 2 on the strain rate – stress axes. It also compares the results of Sample 2 with the results of Sample 1. Again, the dark large point represents the reference point and the smaller points show the strain rate derived based on the stress exponent and the reference point. In this case, the reference point strain-rate is well defined and is well inside the steady state. Both samples show good agreement in the strain rate – stress behavior. Although the stress values are compensated for friction using Eq. (23), the values do not change significantly due to the low strain achieved in the test. 126 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 H-D Line Pure Al Al1SX1.0L E Cy Al2SX1.0L C Cy ε (s -1 ) σ/G n = 4.5 σ (MPa) at 0.99 T m Fig 50: Representation of the present work on stress - strain rate axes. The data for Sample 1 is also shown in order to see the consistency between the results on two different samples. The comparison shows excellent reproducibility of the data from sample to sample. 127 4.1.3. Sample 3 This sample was tested at IPM, Brno. Like Sample 2, an Al- based liquid lubricant was used to reduce the friction between platens and sample. A Molybdenum sulphide – based lubricant was used between other contacts, for example cage elements. During the test, a new LVDT system was used which was not correctly configured and hence an error occurred in the strain measurements. Figure 51 shows the strain – time plot which was recorded by the same (i.e. incorrect) LVDT system. Figure 52 is based on Fig 51 and shows the strain rate variation with respect to the measured strain. The specimen deformed inhomogeneously. Barreling was observed on one side - face and inhomogeneous broadening was observed on the perpendicular face. Table 14 lists the dimensions of the specimen before and after the creep test. These two readings show an excellent match of the volume and hence the measurements are consistent. There are two ways to calculate a meaningful strain-rate from this test: i. An attempt to calculate the average strain rate may be useful. The total change in the length of the specimen was ~0.9 mm in a time period of ~4 x 10 6 seconds. The initial length of the specimen was 24.01 mm. This results in a strain of ~3.8 % and an average strain rate of 9.5 × 10 -9 s -1 . It should be noted that the average strain rate is always larger than the steady – state strain rate for the creep mechanisms showing a distinct primary stage. To achieve a more realistic value for the strain rate, it is assumed that the transient is not strong and to be conservative a strain rate equal to the 2/3 rd of the average (6.3 × 10 -9 s -1 ) is taken for the strain rate – stress plot 23 . 23 The steady – state strain rate for Sample 4 is ~1.8 times smaller than the average strain rate. A factor of 1.5 is close to this value. 128 0 1x10 6 2x10 6 3x10 6 4x10 6 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 Strain Time (s) T : 913 K σ (MPa) = 0.04 Al3SX1.9T P Sq Fig 51: The strain – time plot for Sample 3, as recorded by the miscalibrated LVDT system. Based on the final and initial length measurement, strain accumulated in Sample 3 is ~3.8 % in the time period of 3.2 × 10 6 s (instead of 0.046 % as shown in the plot). 129 0 2x10 -3 4x10 -3 6x10 -3 8x10 -3 1x10 -2 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 0.04 T : 923 K ε (s -1 ) ε Al3SX1.9T P Sq σ (MPa) = Fig 52: The strain rate – strain behavior for the Sample 3, as calculated from the LVDT data. It should be noted that the strain shown in the above plot is only 0.046 % whereas 3.8% of strain was accumulated in the sample. 130 Table 14 (a): Dimensions of Sample 3 prior to the start of the creep test Length (mm) Width (mm) Height (mm) 12.699 12.536 24.021 12.699 12.537 24.034 12.703 12.527 24.014 12.703 12.513 23.991 12.696 12.493 24.024 12.70 ± 0.003 12.52 ± 0.02 24.01 ± 0.02 Table 14 (b): Dimensions of Sample 3 after the creep test 24 Length (mm) Width (mm) Height (mm) Average 13.02 ± 0.03 12.75 ± 0.07 23.11 ± 0.02 Remark “barrel“ ?? “wedge” ?? 24 These reading were taken by Dr Dobes at IPM, Brno. Based on these readings, the average strain rate was calculated. 131 ii. The other way is to scale up the measured strain – time plot to fit into the actual strain. This step is reasonable as the miscalibrated LVDT shifts the actual displacement signals by a constant factor 25 . The strain – time plot (Fig 51) shows a strain of 0.045 % and hence it is ~85 times smaller than the actual strain (3.8 %) accumulated in the sample. The minimum strain rate as shown in Fig 52 is 6.9×10 -11 s -1 ; hence by the rule of proportionality, 5.9 × 10 -9 s -1 is expected as the actual minimum strain rate. It is interesting to observe that the strain rate calculated in steps 1 and 2 are equal within an error of 7%. Based on the above analysis, a value of 6 × 10 -9 s -1 for the steady – state strain rate was calculated which is reasonable. Following this, the strain rate is multiplied by a factor of 1.25 to compensate for the temperature of 923 K (from the test temperature of 913 K). An error bar equal to a factor of ±2 is shown in Fig 53 and in the following similar figures. Due to a high κ-ratio (1.9) of Sample 3, correction in the stress value for friction is insignificant. 25 It is possible that the linear zone of the LVDT is small and an elaborate relationship is required to define proportionality for different intervals on the length scale. 132 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 H-D Line Pure Al Al1SX1.0L E Cy Al2SX1.0L C Cy Al2SX1.9T P Sq ε (s -1 ) σ/G n = 4.5 σ (MPa) at 0.99 T m Fig 53: The stress – strain rate behavior of the samples. The triangle with an error bar shows the 2/3 rd of the average strain rate for the sample at the stress of 0.04 MPa. The error bar is a factor of ±2 of the strain rate used in the plot. 133 4.1.4. Sample 4 A paste of boron nitride powder with acetone was used as the lubricant between the platens. This specimen was subjected to various interruptions during the testing process. Due to the irregularities in the cooling water flow, the furnace could not run for several hours continuously and that made this specimen undergo several annealing cycles. In some of these cycles of a few hours, the specimen was loaded also but could not show any recordable strain in the short periods due to the very low applied stress (0.03 MPa). Cooling was always performed with load (to preserve the microstructure) and the sample was heated without the load. The sample was then kept under no load and at room temperature for 3 months while tests on the other samples were conducted. This specimen was then deformed for an additional 0.4 % at 922 K 26 and the result for this part or the second run of the test is reported as Sample 4b. Unlike Samples 1 and 2, the strips were put not only in the central region of the specimen but also on the platens so that the total deformation in the sample was also measured. The objective behind putting these two additional pairs of strips (i.e. over platens) was to observe the difference in the deformation in the central region with respect to the deformation taking place in the entire specimen. This could be helpful in understanding the reason for the sudden jump in the strain rate which was observed in Sample 2 (Fig 49). In addition to this advantage, the strain measurement between the platens was not affected by the bulging of the specimen and hence the undesired 26 Dr. Y Li took care of the experiment. The result for this part of the test is shown as Sample 4b in the present report 134 decrease in the reported strain by the laser – based deformation extensometer will not occur (i.e. the roll-over effect can be eliminated). Figure 54 shows the strain – time plot for Samples 4 and 4(b). The strain – time plot for Sample 4 shows a sudden change in the slope immediately after the start of the problems with the cooling water. It also shows a relatively higher strain rate if strain is measured over the entire sample compared to the case when strain is measured only in the central 1/3 rd region. The large circles in Fig 54(a) show the region where the strain was measured in the central 1/3 rd region of the sample. Figure 55(a) shows the strain rate vs. strain plot for Sample 4. This plot does not show any sign of steady – state in the given strain range, but it can be assumed to be very close to the steady-state based on the transient behavior of the earlier tests. Figure 55(b) shows the strain rate vs. strain plot for Sample 4b and it shows a steady – state behavior. In Fig 55(a), a vertical dashed line is drawn corresponding to a displacement of 30 µm (i.e. equal to the limit of plane parallelity of the specimen’s end surfaces). Any strain rate prior to this displacement is due to the initial machine adjustments and should be discarded while observing the initial transient behavior of the sample. Thus, this line can be treated as the zero strain line. The end strain rates from these tests are shown in Fig 56 on a strain rate vs. normalized stress plot. The end strain rates are substantially lower than the Harper – Dorn creep rates and are also far away from the 5 – power law. The correction for friction is negligible due to a high initial κ-ratio. 135 0 1x10 6 2x10 6 3x10 6 4x10 6 5x10 6 6x10 6 0.000 0.005 0.010 0.015 0.03 Strain Time (s) T : 910 K Al4SX1.7L C,P Cy σ (MPa) = 0 1x10 6 2x10 6 3x10 6 4x10 6 5x10 6 6x10 6 0.000 0.005 0.010 0.015 0.032 Strain Time (s) T : 922 K Al4bSX1.7L C,P Cy σ (MPa) = Fig 54: The strain – time behavior of Sample 4. The cooling water circuit problem started after the vertical dotted line. A distinct change in the strain – time plot is seen before and after the occurrence of the cooling water problem. The circled regions show the time period in which strain was measured in the central 1/3 rd region of the specimen. 136 0.00 0.01 0.02 0.03 0.04 0.05 10 -10 10 -9 10 -8 10 -7 10 -6 0.03 T : 923 K ε (s -1 ) ε Al4SX1.7L C,P Cy σ (MPa) = 0.00 0.01 0.02 0.03 0.04 0.05 10 -10 10 -9 10 -8 10 -7 10 -6 0.03 T : 923 K ε (s -1 ) ε Al4bSX1.7L C,P Cy σ (MPa) = Fig 55: (a) The strain rate - strain behavior of Sample 4. The accumulated strains are very small and hence, a conclusive prediction about the steady – state strain rate is not possible. Based on prior experience for the test under similar conditions, it can be speculated that the steady-state is very close to the final strain-rate shown in the above figure. (b) The strain rate - strain behavior of Sample 4b. It shows steady – state behavior. (a) (b) 137 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 Pure Al Al1SX1.0L E Cy Al2SX1.0L C Cy Al3SX1.9T P Sq Al4SX1.7L C,P Cy Al4bSX1.7L P Cy ε (s -1 ) σ/G H-D Line σ (MPa) at 0.99 T m n = 4.5 Fig 56: The last strain rate result of Sample 4 shown by the inverted dark triangle. Also, the strain rate data for Sample 4b is shown. These datum points are substantially below the Harper - Dorn transition region but at the same time are faster than the corresponding 5 – power law prediction. 138 4.1.5. Sample 5 Like Sample 4, a paste of boron nitride powder with acetone was used as the lubricant. This test was conducted using a second creep machine available at University of Erlangen – Nuremberg, Germany. The second creep machine used a simple loading system (similar to the first creep machine) where load was directly applied to the sample. A separate lever arm was also attached to the machine and this facility was used for applying larger stresses. Some fluctuation in the stress occurred due to the evaporation of water kept over the sample to cool the LVDT system. The fluctuation in loading due to this evaporation was ~ 20 grams in a span of ~ 12-15 hrs. A consistent effort was applied to maintain the water level as close to the ideal level as possible by keeping a close vigilance of the water level in the bowl. The temperature fluctuation over the time period of the test was within ±2 K. Figure 57 shows the variation of strain with time. The slight wavy nature of the strain – time plot is attributed to changes in the load over the time period of the test. The stresses written with underline fonts have larger uncertainty (also, this uncertainty cannot be quantified with available information), hence they were discarded from further plots. Figure 58 shows the strain rate – strain plot whereas Fig 59 compares the present study with the earlier samples. Figure 58 also shows the stress exponent for each stress jump and the stress exponent between the known stress values is ~ 3 instead of 4.5 or 1. The results on the strain rate – stress plot are consistent. The effect of friction is negligible in the test due to the higher value of the initial κ – ratio (or aspect ratio). 139 5.0x10 5 1.0x10 6 1.5x10 6 2.0x10 6 2.5x10 6 3.0x10 6 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.068 0.12 0.075 Strain Time (s) T : 913 K σ (MPa) = Al5SX2.2T P Sq Fig 57: The strain – time behavior of Sample 5. The stress of 0.68 MPa (which is underlined) has a larger uncertainty associated with it. 140 0.00 0.01 0.02 0.03 0.04 0.05 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 n = 3.5 0.12 0.068 0.075 T : 923 K ε (s -1 ) ε Al5SX2.2T P Sq σ (MPa) = n = 3 Fig 58: The strain rate variation with strain for sample 5. The oscillation in the strain rate is attributed to the oscillation in the loading which occurs due to the evaporation of water. Due to the uncertain rate of evaporation towards the end of the test, the last load was uncertain. It is indicated by an underlined font in the above plot. The representative steady – state strain rate values are shown by the dotted horizontal lines. The stress exponents are shown at each stress jump. 141 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 n = 4.5 Pure Al Al1SX1.0L E Cy Al2SX1.0L C Cy Al3SX1.9T P Sq Al4SX1.7L C,P Cy Al4bSX1.7L P Cy Al5SX2.2T P Sq ε (s -1 ) σ/G H-D Line σ (MPa) @ 0.99 T m Fig 59: The comparison of the results of Sample 5 with other observed results. The data point shows excellent consistency with the other set of the tests conducted during the present study. 142 4.1.6. Sample 6 A paste of boron nitride powder with acetone was used as the lubricant between the platens and the specimen. This specimen was prepared with the same dimension as Sample 4 in order to make a confirmatory test of the results of Sample 4. Sample 4 was tested at a very low stress of 0.03 MPa and it showed a much lower strain rate with respect to the Harper - Dorn prediction, but at the same time the test was interrupted several times due to cooling water circuit failure and hence it was necessary to confirm the results by conducting another identical test with no interruptions. Initially, the sample was loaded at 0.03 MPa but once the result confirmed the earlier results (i.e. from Sample 4), the sample underwent some small stress changes. In order to search for a possible threshold stress, the sample was loaded to stresses as low as 0.018 MPa. Figure 60 shows the strain – time plot for Sample 6. It also shows the strain – time plot for Sample 4. The end segment of Sample 4 curve shows a slightly lower strain rate than Sample 6. Figure 61 shows the strain rate vs. strain plot. The strain – time plot shows a tendency for steady – state behavior even at lower strains. It should be noted that even at the low stresses of 0.018 MPa there is no net negative strain in the sample and hence it rules out the possibility of a threshold stress in the present test. Figure 62 shows the very low stress behavior on a strain rate vs. stress plot. The strain rates are lower than the Harper – Dorn creep rates but they are substantially faster than the 5 – power law. Also, the results show good consistency with the earlier tests on single crystals in the present study. The correction for friction in the present case (Sample 6) is negligible due to a high initial κ-ratio. 143 0 1x10 6 2x10 6 3x10 6 4x10 6 5x10 6 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 T : 913 K 0.018 0.03 0.046 0.058 0.046 Al6SX1.7L C,P Cy Al4SX1.7L C,P Cy 0.03 0.03 Strain Time (s) Pure Al Except circular regions, strain was measured between the platens Fig 60: The strain – time behavior of sample 6 shown along with sample 4. 144 0.00 0.01 0.02 0.03 0.04 0.05 10 -10 10 -9 10 -8 10 -7 10 -6 0.018 0.03 3.9 n = 3.5 0.058 0.046 0.03 T : 923 K ε (s -1 ) ε Al6SX1.7L C,P Cy Fig 61: The strain rate – strain behavior of Sample 6. The points corresponding to the same engineering stress are joined by an interpolating curve and the stress exponents were derived. The large dark circle shows the reference data for the test. 145 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 Single Crystal Pure Al Al1SX1.0L E Cy Al2SX1.0L C Cy Al3SX1.9T P Sq Al4SX1.7L C,P Cy Al4bSX1.7L P Cy Al5SX2.2T P Sq Al6SX1.7L C,P Cy ε (s -1 ) σ/G H-D Line n = 3.1 σ (MPa) at 0.99 T m n = 4.5 Fig 62: The stress – strain rate behavior of sample 6 with respect to other results in the present study. Lines with slopes 4.5 and 3 are shown in the plot. The results show very good consistency. 146 4.2. Creep Test: Polycrystals /Oligocrystals Table 15 summarizes various features of the experimental details for the oligocrystalline samples. 4.2.1. Sample 7 Like Sample 5 (single crystal), this test was conducted using the second creep machine available at University of Erlangen – Nuremberg, Germany. Figure 63 shows the variation of strain with time. The slight wavy nature of the strain – time plot is attributed the changes in the applied load over the extended period of time which is due to the evaporation of cooling water used for the LVDT system. Figure 64 shows the strain rate – strain plot whereas Fig 65 shows the sample behavior on strain rate – stress axes. The effect of friction is negligible in the test due to the higher value of the initial κ – ratio. Figure 65 also uses the interpolation – based analysis (as explained earlier in subsection 4.1.1) for the stress exponent between these two stresses. 4.2.2. Sample 8 Sample 8 was tested at the usual creep facility using laser extensometer for the measurement of strain. This sample was deformed for an observation of the grain boundary sliding, hence the test was interrupted after each stress and the sample was studied for possible grain boundary sliding. Following optical microscopy, the sample was mechanically – polished using 4000 SiC paper and subsequently electro-polished. After the initial deformation, no further annealing was performed on the sample for the next set of loading in order to preserve the last microstructure and to give an overview of the stress-change tests. Fig 66 shows the variation of strain with time, Fig 67 shows the strain rate–strain plot whereas Fig 68 shows the results on the strain rate–stress axes. 147 Table 15: The oligocrystalline specimens used in the present study and the earlier study of which the present study is a continuation. Sample Number Temp (K) Cross Section Length (mm) Side (mm) Aspect Ratio (l/d) κ-Ratio (l/√A) Lub. Stress Cycling Strain Measurement, Methodology 7 913 Square 17.9 11 × 11 1.6 1.6 N Y Whole Sample, LVDT 8 913 Square 19.53 10.5×10.5 1.8 1.8 Y Y Whole Sample, Laser 9 923 Square 1.1 1.1 N Y 90% of length, Laser 10 923 Square 1.1 1.1 N Y 90% of length, Laser 11 923 Square 1.1 1.1 N Y 90% of length, Laser 12 923 Square 1.1 1.1 N Y 90% of length, Laser Y: Yes, N : No 148 5.0x10 4 1.0x10 5 1.5x10 5 2.0x10 5 0.00 0.05 0.10 0.15 0.20 0.25 0.26 0.52 0.26 Strain Time (s) T : 913 K Al7OX1.6T P Sq σ (MPa) = Fig 63: The strain – time behavior of Sample 7. 149 0.00 0.05 0.10 0.15 0.20 0.25 0.30 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 0.52 n = 5.4 0.26 0.26 T : 923 K ε (s -1 ) ε Al7OX1.6T P Sq Fig 64: The strain rate variation with strain for sample 7. The oscillation in the strain rate is attributed to the oscillation in the loading which occurs due to the evaporation of water. A stress exponent equal to ~5.4 is calculated based on the interpolation technique. The large circle shows the reference point for the test. 150 10 -7 10 -6 10 -5 10 -4 10 -10 10 -7 10 -4 10 -1 10 -2 10 -1 10 0 n = 3 Pure Al Al7OX1.6T P Sq ε (s -1 ) σ/G n = 4.5 Based on Single Crystal Data σ (MPa) at 0.99 T m Fig 65: The comparison of the results of Sample 7 with other observed results. The data points of single and oligocrystals show a good match at moderately high stresses. The dotted lines represent the results from the single crystals and they show a good agreement at moderately high stresses if the Taylor factor is considered. 151 0 1x10 6 2x10 6 3x10 6 4x10 6 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.3 0.07 Strain Time (s) T : 913 K 0.18 Al8OX1.8L P Sq Fig 66: The strain – time behavior of Sample 8. The dotted lines show the interruption in the test in order to measure grain boundary sliding. 152 0.00 0.01 0.02 0.03 0.04 0.05 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 n = 5.3 0.30 0.07 0.18 T : 923 K ε (s -1 ) ε σ (MPa) = n = 5.1 Al8OX1.8L P Sq Fig 67: The strain-rate variation with the strain for sample 8. The strain rate shows a large reduction during the transient phase of the creep due to the interruption of the test after every stress. The stress exponents remain close to ~5. 153 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 Oligocrystal Al7OX1.6T P Sq Al8OX1.8L P Sq Pure Al ε (s -1 ) σ/G n = 4.5 H-D Line σ (MPa) at 0.99 T m Fig 68: A comparison of the results of Sample 8 with Sample 7. Sample 8 does not show any change in the stress exponent at the lower stresses as was shown by single crystals. 154 As shown in Fig 68, the oligocrystalline samples do not show any change in the stress exponent value at the lower stresses. The best fit line passing through all the data points have a slope of 4.5 which indicates that the oligocrystals deform according to the classic 5 – power law creep. This behavior at the low stresses is in contrast with the single crystal behavior. The reasons for this difference are not known but the following two facts are possibilities: 1. Low purity level (uncertain purity) 2. Grain boundary effects which do not allow dislocations to move freely as was the case in the single crystals. The grain size for the oligocrystals was ~ 10 mm. Figure 69 compares the data of the single crystal and the oligocrystals which were observed in the present study. Figure 70 is reproduced from Fig 69 after compensating for the Taylor factor. The stresses for the single crystals are multiplied by a factor of 1.25 (= Taylor Factor for polycrystals divided by the Taylor factor for the single crystals = 3.06 / 2.45) to compensate for the Taylor factor. The data points in Fig 70 show good agreement at moderately higher stresses whereas they fail to agree at the lower stresses. 4.2.3. Eisenlohr’s Data Figure 71 shows the strain rate – strain plots for the various oligocrystals tested in compression by Dr. Philip Eisenlohr at the creep facility available at the WW I, University of Erlangen- Nuremberg Germany. In both of these studies (present and by Eisenlohr), samples prepared from the same as - received material were used. Figure 72 shows the strain rate – stress plot for the Eisenlohr samples whereas Fig 73 summarizes all the data collected during the present study and Eisenlohr’s study. 155 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 n = 3.1 Single Crystals 99.999 % Oligocrystals 99.99 % Pure Al ε (s -1 ) σ/G n = 4.5 H-D Line Purity σ (MPa) at 0.99 T m Fig 69: The strain rate – stress behavior of the single crystals and the oligocrystals deformed under compression in the present study. The Taylor factor is not included and the single crystals seem to be more susceptible to creep. The grain size for the oligocrystal samples was ~10 mm. 156 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 Single Crystals 99.999 % Oligocrystals 99.99 % Purity n = 3.1 Pure Al ε (s -1 ) σ/G n = 4.5 H-D Line Taylor Factor = 1.25 σ (MPa) at 0.99 T m Fig 70: Reproduction of Fig 69 after compensating for the Taylor factor. The stress values for the single crystals were multiplied by a factor equal to 1.25. The test results show remarkable consistency for oligocrystals and single crystals at moderate stresses. Grain size for the oligocrystal samples was ~10 mm. 157 0.00 0.01 0.02 0.03 0.04 0.05 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Oligocrystal 0.07 0.05 0.05 0.113 0.092 0.086 0.13 0.1 0.1 0.07 0.07 0.06 0.07 0.07 0.113 0.113 0.165 0.05 0.92 0.10 0.113 Al9OX1.1L E Sq Al10OX1.1L E Sq Al11OX1.1L E Sq Al12OX1.1L E Sq ε (s -1 ) ε n = 4.5 0.07 Pure Al σ (MPa) = Fig 71: Strain rate – strain plot for various oligocrystals tested under compression by Dr. Eisenlohr. The segments with underlined engineering stress values are in steady – state and the shaded circles represent the corresponding strain rate. 158 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 n = 3.5 (?) Eisenlohr Pure Al Oligocrystal ε (s -1 ) σ/G σ (MPa) at 0.99 T m Fig 72: The strain rate – stress variation of the oligocrystals tested by Eisenlohr. The strain rate values are based on Fig 71. The stresses have been modified for strain (i.e. true stress) and friction. 159 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 Single Crystals 99.999 % Oligocrystals 99.99 % Eisenlohr 99.99 % Purity n = 3.1 Pure Al ε (s -1 ) σ/G n = 4.5 H-D Line Taylor Factor = 1.25 4 σ (MPa) at 0.99 T m Fig 73: A comparison of all results from the present work. At low stresses, the datum points from Eisenlohr are within a factor of 2 with respect to the single crystal data whereas the oligocrystal tested in the present work differs by a factor of 4 with the single crystal data. 160 4.3. Microstructural Study Various microstructural studies were conducted during the course of the study. In the following subsections, the results will be explained in detail. 4.3.1. Grain Boundary Sliding Grain boundary sliding measurement were conducted at three different stresses, 0.07 MPa (Harper – Dorn regime), 0.18 MPa (transient region between Harper – Dorn regime and 5-power law) and 0.3 MPa (5 – power law regime). In the following subsections, various optical micrographs showing grain boundary sliding and other boundary features such as grain migration and rotation are examined. 4.3.1.1.Stress = 0.07 MPa Fig 74 shows a digital image of the sample after deformation to a strain of 0.7 % at the stress of 0.07 MPa. It is difficult to see the grain features clearly in the original picture; hence blue dotted lines are drawn free hand as an aid. As shown in Fig 74, it appears that the grain boundary sliding is very small for this case. Fig 75 shows various features related to GBS in the sample. Fig 75(a) shows a region / boundary where grain boundaries have slid over each other without producing any other feature, like grain boundary branching, rotation, etc. Fig 75(b) shows a boundary where negligible sliding took place. There were a few places over the sample where branching of the grain boundaries was observed. Sample surfaces also showed a wiggling feature and it was very unique to this test condition as other tests did not show such feature. Based on several micrographs, GBS was calculated for the above sample after a strain of ~0.7 % and its contribution to the total strain was equal to ~11 % using a value of L t = 10 mm. 161 Fig 74: A digital image of the sample after deformation at the stress of 0.07 MPa. The sample was deformed to a strain of 0.7 %. 162 Fig 75: Several features of GBS at a stress of 0.07 MPa after a strain of 0.7 %. The sample did not show any deviation from the 5-power law even though the stress belongs to the Harper – Dorn regime. The compression axis is horizontal in the plane of the paper. (a) (b) Negligible GBS 163 4.3.1.2. Stress = 0.12 MPa Fig 76 shows a digital image of the sample after deformation through a strain of 1.6 % at the stress of 0.18 MPa. As shown in Fig 76, the grain boundary features are clearly visible after deformation and the sample surface was no longer smooth due to the movement of the grains. Fig 77 shows various features related to GBS in the sample. Figure 77(a) shows a region / boundary where grain boundaries show sliding. Similar sliding features were observed at most of the boundaries. Figure 77(b) shows an area near the end of the sample surface and clearly shows the change in the surface level across grain boundaries. Figure 77(c) combines some features showing migration or rotation of the grain boundaries. There were very few places in the sample where such branching of the grain boundaries was observed (as shown in Fig 77 (c)) indicating only limited grain boundary migration or rotation under the deformation condition. An important feature of all these tests is shown by the top and bottom micrographs in Fig 77 (c), which show the division of a grain boundary in three branches (top picture) which combines to form one grain boundary again (bottom picture). Based on several micrographs, GBS was calculated for the above deformation condition and its contribution to the total strain was equal to ~30 % using a value of L t = 10 mm. 164 Fig 76: A digital picture of the oligocrystal after deformation to a strain of 1.6 % at 0.18 MPa. The grains are fairly large in size (~ 10 mm) and the boundaries are clearly visible. 165 Fig 77: Some of the salient features related to grain boundary under a stress of 0.18 MPa after a strain of 1.6 %. (a) (b) 166 Fig 77 Contd. Fig 77: Some of the salient features related to grain boundary under a stress of 0.18 MPa after a strain of 1.6 %. (c) 167 4.3.1.3. Stress = 0.3 MPa The sample was deformed through a strain of 1.6 % at a stress of 0.3 MPa. Figure 78 shows various features related to GBS in the sample. Figure 78(a) shows a region / boundary where grain boundaries show sliding. Figure 78(b) shows a region close to the upper edge of the sample and it also shows a large amount of GBS even though this area should be dominated by the friction cone. Figures 78(c) and 78(d) show some features of migration or rotation of the grain boundaries. This is a general feature of the deformed sample, showing extensive grain boundary migration and rotation. This is different from last two set of tests where it was limited to only a few places. Because of these additional features, apart from only GBS, the sliding measurement cannot be performed at each marker line. It is also expected that the additional grain boundary features will affect the overall sliding in the sample. Based on several micrographs, GBS was calculated for the above deformation condition and was equal to ~37 % using a value of L t = 10 mm. Figure 79 compares the GBS values with the data of Harper et al [16] on a similar purity polycrystalline sample. Even though the sample in the present study showed no change in slope or deviation from the 5-power law, the grain boundary sliding data show good consistency with the results of Harper et al [16]. Some additional test conditions for GBS would be helpful for a firm conclusion for the comparison. The GBS contribution to the total strain does not depend on the strain and it is supposed to be the same throughout the deformation process. So, even if the underlying deformation process is the same, the reported strain-rates by Harper et al [16] might be different than the present study if steady – state was not achieved in the earlier study. 168 Fig 78: Some of the features related to grain boundary under a stress of 0.30 MPa after a strain of 1.6 % (a) (b) 169 Fig 78 contd. Fig 78: Some of the features related to grain boundary under a stress of 0.30 MPa after a strain of 1.6 %. (c) (d) Grain Boundary Migration 170 0.01 0.1 1 0 10 20 30 40 50 60 70 10 Dislocation Creep (n = 4.5) Actual Stress (σ) Net Stress (σ - σ 0 ) Present Investigation Contribution of Sliding to total strain, ξ (%) Stress, σ (MPa) Pure Al Harper - Dorn Creep (n = 1) grain size (mm) 3.3 Fig 79: A comparison to the values of the GBS for the present study and the study of Harper et al. [16]. The values of GBS contribution are similar even though the trend in the strain rate – stress plot is different. 171 4.3.2. Etch Pit study An Etch pits study was conducted in order to measure the dislocation density and examine the substructure formation. Etch-pits do not induce any kind of mechanical damage. Most of the earlier Harper – Dorn studies used etch – pits to calculate the dislocation density and to observe the substructure formation in the sample mainly due to the fact that a very low density of dislocations is expected at such low stresses. Etch – pit studies were conducted only for single crystal samples. Figure 80 shows a schematic of the shapes of the etch pits for a FCC crystalline material. Since the orientation of the single crystal was <100>, a square / quadrilateral shape for the pits is expected. This is one of the ways by which the pits can be distinguished from other surface features such as an artifact. Etch pits study was conducted for four samples out of six single crystal samples deformed in the present study. Sample 4 (single crystal) is currently undergoing test whereas Sample 3, which was tested at IPM, Brno, does not have well-defined stress information for the last stress. Figures 81 to 85 show optical micrographs from the etch pit studies. Figure 86 shows the variation of the dislocation density with stress. The dislocation density increases with stress. The variation in the dislocation density in the low stress region shows a slope of 2 and it matches well with data at high stresses. The dislocation density at the highest stress of the present study (0.33 MPa) is smaller than the projected curve (with slope 2). This can be attributed to the heavy oxidation of the surface which was not completely etched in the applied time period (which works well for other less – oxidized samples). It is hence suggested to conduct confirmatory etch- 172 Fig 80: A schematic showing the shapes of the etch pits corresponding to various planes in a FCC material. The guideline for the shape of the pits are helpful in distinguishing the actual pits on the <100> plane and other surface features and irregularities. 173 Fig 81: Etch – pits micrographs of annealed specimen (just prior to loading) as observed under optical microscope at a magnification of 50X. The pits show a non-uniform distribution in the annealed specimens after etching for 8 seconds. (a) shows more pits relative to (b). (a) (b) 174 Fig 82: Etch – pits micrographs of Sample 1 as observed under optical microscope at a magnification of 200X. The last true stress used to deform the specimen was 0.3 MPa. Due to substantial bulging and slip activity observed at the surface of the sample, it was difficult to get the entire specimen focused at the same time. The surface was oxidized heavily and it is possible the electro-polishing and etching agent was not sufficient to show all dislocations (as the oxide layer might not been etched away) 175 Fig 83: Etch – pits micrographs of Sample 2 as observed under optical microscope at a magnification of 100X. The pits show a non-uniform distribution. The last true stress used in order to deform the sample was 0.052 MPa. (a) shows more pits relative to (b). (a) (b) 176 Fig 84: Etch – pits micrographs of Sample 5 as observed under optical microscope at a magnification of 100X. The last true stress used to deform the specimen was 0.9 MPa. The sample shows a very high dislocation density relative to last two samples tested in the low stress regime. Although the dislocation density was higher than earlier samples, no evidence of any sub-structure formation was observed. 177 Fig 85: Etch – pits micrographs of Sample 6 as observed under optical microscope at a magnification of 100X. The pits show a non-uniform distribution. The last true stress used in order to deform the sample was 0.018 MPa. (a) shows more pits relative to (b). (a) (b) 178 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 1 Dislocation Density (m -2 ) Al - 5 % Zn Present Investigation (All Pits) Present Investigation (Bigger Pits) Barrett et. al. (1972) Mohamed et al (1973) Mohamed and Ginter (1982) Kassner and McMahon (1987) Lin, Lee and Ardell (1989) 5-Power Law Pure Al σ/G 2 Blum (1993) Fig 86: The stress dependence of the dislocation density as measured by etch – pits. The dislocation density does not remain constant in the low stress regime. The dislocation density increases with stress and it is consistent with the projected values based on 5 – power law observations. The references for this figure are [18, 19, 35, 76, 169]. 179 pits study on this particular sample from the inner part of the specimen. This can be conducted after removing a few microns of the material from the surface using electro – polishing. As seen in Fig 85, there are a few small pits also along with the larger ones. Fig 86 also shows the dislocation density values based on only bigger pits. The nature of dislocation density does not change whether only big pits are counted or all pits are counted. The presence of small pits is attributed to the fact that etch – pits solution did not etch entire oxide layer from the surface instantaneously and hence nucleation of pits occurred at different time. The pits which were nucleated earlier grew to a larger size. Subgrain formation was not observed for the above samples. The expected subgrain size (~ 1.1 mm for 0.1 MPa) is within the limits of the window of the observation under the optical microscope. The dislocations seem to follow a random arrangement and hence suggest the weak nature of subgrain formation. This is also supported by the observation of the weak transients (in the strain rate – strain plots) for the stress change tests in the low stress regime. These two observations suggest that subgrains / substructures in the low stress regime are not well developed. 180 CHAPTER 5. DISCUSSION 5.1. Replotting the figures on Normalized Strain Rate and Stress Co-Ordinates for Single Crystals Figure 87 shows some of the earlier studies suggesting Harper - Dorn creep in pure Al. The applied stress values for the single crystals are not compensated for the Taylor factor due to the uncertainty associated with various types of polycrystalline samples used in earlier studies. The datum points of Barrett et al [18] show considerable scatter in the transition region from the 5-power law to the Harper - Dorn regime. In this region, the average slope is higher than 1 which is shown by an enclosing circle in Fig 87. The results of the previous studies also show scatter in the transition region. Figure 88 compares the above studies with the present study. The results obtained in the present study for Samples 1, 2 and 5 are close to the lower outer envelope of the scatter zone and the proposed value for the strain rate for Sample 3 is also consistent with the reported strain rates in the transition region. The results from Samples 4 and 6 differ from the Harper - Dorn datum points. For the stresses within the Harper - Dorn regime, the strain rates for all tested specimens are higher than the 5 – power law prediction but they are consistently lower than the Harper - Dorn prediction. Figures 41 and 42 show that slip in Sample 1 was more pronounced only in one slip – plane compared to the other three equally possible planes. Due to the smaller aspect ratio(s) of Samples 1 (and 2), it is possible that creep was suppressed as slip activities were confined to only a few slip systems but results from samples with larger aspect ratios (for example, Samples 4, 5 and 6) are also not consistent with the conventional Harper - Dorn behavior in the present study. 181 10 -7 10 -6 10 -5 10 -4 10 -19 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 0.01 0.1 1 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 σ (MPa) at 923 K ε (s -1 ) at 923 K n = 1 Pure Al Harper and Dorn (1957) Barrett et al. (1972) Mohamed and Ginter (1982) εkT/DGb σ/G n = 4.5 Fig 87: A comparative observation of the three early studies suggesting Harper - Dorn creep in pure Al [15, 18, 27]. All tests were conducted in tension. 182 10 -7 10 -6 10 -5 10 -4 10 -19 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 0.01 0.1 1 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 σ (MPa) at 923 K ε (s -1 ) at 923 K n = 1 Pure Al Harper and Dorn (1957) Barrett et al. (1972) Mohamed and Ginter (1982) Al1SX1.0L E Cy Al2SX1.0L C Cy Al3SX1.9T P Sq Al4SX1.7L C,P Cy Al4bSX1.7L P Cy Al5SX1.6T P Sq Al6SX1.7L C,P Cy εkT/DGb σ/G n = 4.5 Fig 88: Comparison of the present study with respect to the three earlier studies suggesting Harper – Dorn creep in pure Al [15, 18, 27]. The dark points represent the present study and they are not consistent with the Harper - Dorn prediction 183 Unlike the earlier studies confirming Harper - Dorn creep, the scatter in the strain rates observed in the present tests is not random, but it indicates a distinct slope which is close to ~ 3. This will be discussed in detail in subsequent sections. Figure 89 summarizes some of the studies [23, 24, 36] which did not observe Harper - Dorn creep in the low stress regime where Harper – Dorn creep was suggested earlier. The reported results from these studies lie on a line with a slope of 4.5 even up to a stress of 0.06 MPa. The datum points do not show any significant scatter and are consistent from one study to another. One of the main criticisms of these studies [38] is the use of test samples with low aspect ratios (~ 1) which inhibit creep in compression due to the formation of friction cones which cover a larger fraction of the test sample. Fig 90 compares these results with the results of the present study. In the present study, samples with small (~1.1) as well as large aspect ratios (~1.7 – 2.0) were deformed in the Harper – Dorn regime. At the lower stresses (Harper – Dorn regime), the strain rates observed in present study are consistently higher than the above studies. Figure 91 shows the overall representation of the various important studies conducted at very low stresses and very high temperatures using pure Al. The strain rates observed in the present tests lie consistently between the Harper – Dorn regime and the 5-power law and show a probable stress exponent of ~ 3. Figure 92 is a reproduction of Fig 91 with additional data from McNee et al [22]. McNee et al [22] used polycrystalline Al (d ≥ 5 mm) produced using cold work. Their datum points show considerable scatter and in some cases the authors [22] could not observe any strain at the low stresses. In the present study, higher strain rates were consistently observed in the region where they failed to show any deformation. 184 10 -7 10 -6 10 -5 10 -4 10 -19 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 0.01 0.1 1 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 σ (MPa) at 923 K ε (s -1 ) at 923 K n = 1 Pure Al Straub and Blum (1990) Blum and Maier (1999) Blum et al (2002) εkT/DGb σ/G n = 4.5 Fig 89: A comparative representation of the three important studies prior to the present study not confirming Harper - Dorn creep in pure Al [23, 24, 36]. All the tests were conducted in compression. 185 10 -7 10 -6 10 -5 10 -4 10 -19 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 0.01 0.1 1 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 σ (MPa) at 923 K ε (s -1 ) at 923 K n = 1 Pure Al Straub and Blum (1990) Blum and Maier (1999) Blum et al (2002) Al1SX1.0L E Cy Al2SX1.0L C Cy Al3SX1.9T P Sq Al4SX1.7L C,P Cy Al4bSX1.7L P Cy Al5SX1.6T P Sq Al6SX1.7L C,P Cy εkT/DGb σ/G n = 4.5 Fig 90: Comparison of the present study with the three main studies [23, 24, 36] not confirming Harper - Dorn creep in pure Al. The dark points are from the present study and they are far away from the n = 4.5 line at the low stresses. 186 10 -7 10 -6 10 -5 10 -4 10 -19 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 0.01 0.1 1 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 n = 3.1 σ (MPa) at 923 K ε (s -1 ) at 923 K n = 1 Pure Al Harper and Dorn (1957) Barrett et al(1999) Mohamed and Ginter (1982) Straub and Blum (1990) Blum and Maier(1999) Blum et al (2002) Al1SX1.0L E Cy Al2SX1.0L C Cy Al3SX1.9T P Sq Al4SX1.7L C,P Cy Al4bSX1.7L P Cy Al5SX1.6T P Sq Al6SX1.7L C,P Cy εkT/DGb σ/G n = 4.5 Fig 91: A comparative representation of the main studies on pure Al at very low stresses and at a temperature close of 0.99 T m . Dark symbols are from the present study. Sample 3 data are ambiguous as they are based on average strain rate. The first three studies [15, 18, 35] used tensile specimens whereas the remaining studies [23, 24, 36] used compression specimens. 187 10 -7 10 -6 10 -5 10 -4 10 -19 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 0.01 0.1 1 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 n = 3.1 σ (MPa) at 923 K ε (s -1 ) at 923 K n = 1 Pure Al Harper and Dorn (1957) Barrett et al(1999) Mohamed and Ginter (1982) Straub and Blum (1990) Blum and Maier(1999) Blum et al (2002) McNee et al (2002) Al1SX1.0L E Cy Al2SX1.0L C Cy Al3SX1.9T P Sq Al4SX1.7L C,P Cy Al4bSX1.7L P Cy Al5SX1.6T P Sq Al6SX1.7L C,P Cy εkT/DGb σ/G n = 4.5 Fig 92: A reproduction of Fig 91 with additional data by McNee et al [22]. The tensile data of McNee et al [22] shows an extensive scatter in the low stress regime. 188 The following reasons may be attributed to the above differences: 1. Prior to creep, the sample was annealed in the presence of a small load of the value of 0.025 and 0.035 MPa which belong to the Harper – Dorn regime and hence the sample might have pre-crept. In the present study, a very low stress of 0.002 MPa was applied while heating the sample to maintain the contacts. 2. A variation in the loading axis with respect to the cold working direction might lead to considerable scatter if proper annealing was not conducted. The annealing process is not clearly described in the report. 3. The purity is not known in some cases giving a threshold stress for lower purity. 4. The samples underwent very small strains during the test and in some cases fractured within a few hundredths of the strain. This behavior is not explained by the authors and it is contrary to the fact that very large strains are possible at such low stresses and high temperatures [35]. 5. For high purity polycrystals (99.999 % purity), grain growth might occur leading to higher strain rates but the contrary was observed in the above study [22]. Arguments 4 and 5 indicate possible errors in either the experimental set – up or in the material preparation. In spite of this fact, the results are important because even after extensive scatter none of the datum points show Harper – Dorn creep. 5.2. Comparison of the Present Study on Single Crystals with Earlier Studies 5.2.1. Harper and Dorn [15, 16] Harper and Dorn [15, 16] tested both single crystals and polycrystals in air at 920 – 923 K. The purity of samples was 99.99 % and the grain size was 3 mm for the 189 polycrystals. These tests were conducted in tension using a constant stress technique. The sample dimensions were: 95.25 × 12.7 × 2.54 mm. Figures 93 and 94 summarize the results of Harper and Dorn [15] on strain – time and strain rate – strain axes. Figure 94 shows the strain rate – strain plot for Harper and Dorn [15]. Most of the samples were crept to a strain less than 1 % and strains in the experiments decrease with the decreasing stress. For the minimum stress which produced a positive strain (0.025 MPa), the accumulated strain was only 0.1 %. This trend of a decrease in the strain may be attributed to the fact that the study was limited by a time – factor (~5000 hrs), instead of a strain factor. The following two trends are inferred from Fig 94: i. The steady – state might not have been achieved in the samples deformed under stresses equal to or lower than 0.073 MPa. ii. For the lower stresses, especially at 0.073 and 0.045 MPa, a cusp in the strain rate – strain plot is seen. The strain rate decreases with the strain and starts increasing after the minimum strain rate is achieved. This trend is also weakly observed in the sample deformed under a load of 0.114 MPa. After the cusp, the strain rate increases by a factor of ~2. These tests were conducted in tension and hence an initial necking may be attributed to the increase in the strain rates. But no such necking was mentioned by the authors [15], hence the reason for this feature of the strain rate – strain curve is not clear. 5.2.1.1. Comparison with Present Study Figure 95 compares the present study with the above study [15] on a strain rate – strain plot. The strain rate – strain plot shows a remarkable similarity between the present study and the Harper and Dorn [15] results. The behavior at a stress of 0.045 190 0 1x10 5 2x10 5 3x10 5 0.000 0.003 0.006 0.009 0.012 0.015 0.025 0.028 0.045 0.073 0.09 0.114 0.152 0.176 0.207 Strain Time (s) σ (MPa) = 0.252 Pure Al Harper and Dorn (1957) T = 920 K Fig 93: The strain – time behavior of the Harper and Dorn samples (polycrystals) [15]. Only the samples showing positive strain rates are shown. Dark points show the samples which showed 5 – power law creep whereas open symbols lie in the Harper – Dorn regime. The study reported a negative strain (~ -0.00025 in 3.0 × 10 5 s) for a stress of 0.019 MPa. 191 0.00 0.01 0.02 0.03 0.04 0.05 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 σ (MPa) = 0.025 0.028 0.045 0.073 0.09 0.114 0.152 0.176 0.207 Harper and Dorn (1957) ε (S -1 ) ε 0.252 Pure Al T = 920 K Fig 94: The strain rate – strain plot for the Harper and Dorn study [15]. 192 0.000 0.005 0.010 0.015 0.020 0.025 10 -10 10 -9 10 -8 10 -7 T = 923 K 2 2 0.046 0.071 0.030 σ (MPa) = 0.028 0.045 0.073 Open: Harper and Dorn (1957) Closed: Present Investigation ε (s -1 ) ε 2 Pure Al Fig 95: The strain rate – strain plot showing a comparison of the present study with the Harper and Dorn report [15]. Actual stresses for Harper and Dorn are shown in the plot. Only (actual) stresses lower than 0.09 MPa are shown for which there was corresponding data in the present study. The “capped” lines with underlined numbers show the factor by which the values differ. 193 MPa is consistent with the present work up to the cusp formation, beyond which the Harper and Dorn sample shows an increase in the creep rate whereas the sample in the present work shows a decrease in the strain rate which is a common observation. It is observed that the “continuous” decrease in the strain rate in the Harper and Dorn report at a stress of 0.073 MPa is much faster than expected for a constant stress test (it should be horizontal) and hence it indicates that the steady – state was not reached in the earlier study [15]. If the straight line passing through the last three stages is extended to the strain at which observation was made in the present study, a good agreement is found between the Harper-Dorn study and the present work. The behavior at the stress of 0.03 MPa is interesting. Both studies show a remarkable consistency, but it seems that the deformation in the earlier study [15] was stopped at a lower strain and hence the minimum strain rate in the earlier study is higher than the actual steady – state value. Based on the above arguments, the following conclusion is made: The strain rate – strain plot for both studies show a good agreement, but the minimum strain rates reported by Harper and Dorn [15] were higher than the actual steady – state values at the same true stresses. Due to this reason, the tendency of the points on the strain rate – stress plot is to move upwards from the corresponding steady – state values. Figure 96 shows the strain rate – stress behavior for both studies. Harper and Dorn observed negative strain at a stress of ~0.019 MPa and hence concluded that the samples have a threshold stress due to the presence of the surface tension (surface energy) between different layers, namely Al – Al 2 O 3 and Al 2 O 3 and air [15]. From the interpolation on the strain rate – stress (linear – linear) plot, a value of 0.02 MPa as the 194 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 n = 4.5 n = 1 Present Investigation Harper and Dorn (1957) True stress, σ Net stress, σ − σ 0 Pure Al ε (s -1 ) σ/G n = 3 σ (MPa) at 0.99 T m Fig 96: The strain rate – stress plot showing a comparison between the Harper – Dorn report and the present study. The term “present study” refers only to the single crystals, Harper - Dorn (true) corresponds to the true applied stress whereas Harper - Dorn (net) corresponds to the stress value after subtracting the threshold stress (~0.02 MPa.) from the actual stress (i.e. σ net = σ true - σ threshold ). 195 0.0 5.0x10 -6 1.0x10 -5 1.5x10 -5 2.0x10 -7 4.0x10 -7 6.0x10 -7 0.00 0.05 0.10 Present Investigation Best fit curve ε = 4 x 10 8 σ 3 ε = 4 x10 8 σ 3 Pure Al ε (s -1 ) σ/G σ (MPa) at 0.99 T m Fig 97: The extrapolation of strain rate – stress relationship to zero on linear – linear axes. It indicates a zero strain rate for zero applied stress and hence a threshold stress was not observed in the present study. 196 threshold (negative) stress was obtained and this was used by these authors. Use of such a threshold stress value shifts the curves towards the left by a significant amount, especially at the lower stresses. As shown in Fig 96, the datum points in the present study match well with the Harper and Dorn points if the threshold stress is not used. After introducing the threshold stress, the data points of the Harper – Dorn report move towards the left and result in a slope of ~ 1. Hence, for the slope of n = 1 in the Harper and Dorn study [15], the following two reasons are responsible: 1. The presence of a threshold stress of ~0.02 MPa which moves the datum points towards the left and thus exhibits a strong effect. 2. The minimum strain rate is higher than the actual steady – state strain rate which moves the datum points upwards although this effect is less severe than the previous one. 5.2.1.2. On the Threshold Stress None of the subsequent studies on Harper – Dorn creep observed this feature, even though the sample thickness was similar as well as the test environment was same (vs. the Barrett et al [18] single crystal specimens). Mohamed and Ginter [35] made an attempt to extrapolate the strain rate – stress plot to zero stress but could not find any threshold stress. As shown in Fig 97, a similar analysis was performed in the present study also and it gave zero strain rate for zero stress indicating an absence of any threshold stress. For tests in compression (as in the present study), the total surface area decreases with strain and hence it should have an enhancing effect (or positive threshold stress) on stress and should lead to a higher strain rate. By using a large sample in the 197 present study, this possibility of a threshold stress was eliminated. Usually, a threshold stress is recognized by a sudden rise in the stress exponent. Fig 98 shows the stress exponent variation for the Harper and Dorn study. The calculation of the stress exponent was based on the actual stresses and strain rate values was taken from Fig 96. At the low stresses, the stress exponent increases but the increase is very mild and it does not unambiguously indicate the existence of a threshold stress in the Harper and Dorn samples. It may be possible in the present study that inclusions or impurities can interact with dislocations giving rise to a threshold stress. Attempts were made to find a threshold stress in the present test, but at a stress value as low as 0.018 MPa, the deformation behavior did not show any anomaly indicating about the existence of a threshold stress. If there is a threshold stress in the samples, it is insignificant for the stress range used during the present study. 5.2.2. Barrett, Muehleisen and Nix [18] Barrett et al [18] conducted creep tests using both single crystals and polycrystals at 920 – 923 K in air. The purity of the samples was 99.99 % and the grain size was 1 cm for the polycrystals. These tests were conducted in tension using a constant load technique. The sample dimensions were: 1. Polycrystals:- length: 3.75 cm, diameter: 1 cm 2. Single Crystals:- 3.75 × 1.25 × 0.3 cm 3 198 0.0 0.5 1.0 0 1 2 3 4 5 6 7 8 9 10 Harper and Dorn (1957) Stress Exponent, n σ (MPa) Pure Al Fig 98: Variation of stress exponent with stress in the Harper – Dorn material [15]. The stress exponent increases at the lower stress with a decrease in the stress value but it is not drastic and hence does not clearly indicate the existence of a negative threshold stress. 199 Some of the tests were also conducted on the pre-strained samples. Also such pre-straining was conducted to a strain of 2%. Figures 99 and 100 summarize some of the results reported by Barrett et al [18]. Fig 100 shows the strain rate – strain behavior for the sample deformed at the stress of 0.059 and 0.035 MPa. Again, both these points do not show attainment of a steady – state to within a factor of 2 or 3. It is interesting to see the inverted primary creep behavior for the pre-strained samples which indicate a similarity between the creep behavior at low stresses and in the 5 – power law regime. 5.2.2.1. Comparison with Present Study Fig 101 compares the data of Barrett et al [18] with the results of the present study on a strain rate – strain plot. Because of the lack of attainment of an unambiguous steady – state by Barrett et al [18], a smooth curve is drawn using the last few points of Fig 99. The extrapolated curve shows good agreement with the present study. Since the pre-strained samples in the present study do not have any corresponding data, the above extrapolation was performed only for the datum point corresponding to a stress value of 0.059 MPa. Figures 102 and 103 compare the results obtained in the present study with the reported results of Barrett et al [18] on a strain rate – stress plot. Due to the lack of information about the strain achieved by Barrett et al [18], it is difficult to compare the various data points. An effort was made to identify data which might be associated with higher strains. The following was assumed before deriving such a plot (i.e. Fig 102 which represents datum points with higher strains): 200 0 1x10 5 2x10 5 3x10 5 4x10 5 5x10 5 6x10 5 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.035 Strain Time (s) 0.059 σ (MPa) = Pure Al T = 923 K Barrett et al (1972) Fig 99: A typical strain – time behavior of the polycrystalline Al samples tested by Barrett et al [18]. The sample deformed at 0.035 MPa was already pre-strained to a strain of 2 %. Unlike the Harper and Dorn study, the other strain – time plots were not shown in the report, hence the total strains achieved in several tests are not known. 201 0.00 0.01 0.02 0.03 0.04 0.05 10 -10 10 -9 10 -8 10 -7 10 -6 σ (MPa) = 0.035 Barrett et al (1972) ε (s -1 ) ε 0.059 Pure Al T = 923 K Fig 100: The strain rate – strain behavior of the creep data of Barrett et al [18]. The data points do not show the attainment of steady – state creep and the strains observed in the tests were less than 1 % which decreases with a decrease in the stress. 202 0.00 0.01 0.02 0.03 0.04 0.05 0.06 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 Present Investigation (Al2SX1.0L C Cy) Barrett et al (1972) 0.059 0.056 σ (MPa) = ε (s -1 ) ε Pure Al n =3 Fig 101: Comparison of the results obtained in the present study with the date of Barrett et al [18] data on strain rate – strain axes. A possible smooth extrapolation of the Barrett et al [18] data is drawn showing a consistency with the present test. 203 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 Present Investigation Pure Al ε (s -1 ) σ/G H-D Line n = 4.5 Barrett et al (1972) - High ε σ (MPa) at 0.99 T m n = 3 Fig 102: Strain rate – stress plot showing a comparison between Barrett et al [18] datum points and the present study. Only the datum points of Barrett et al [18] which might have achieved higher strains are shown here. Both studies show good agreement. 204 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 Present Investigation Pure Al ε (s -1 ) σ/G H-D Line n = 4.5 Barrett et al (1972) σ (MPa) at 0.99 T m n = 3 Fig 103: Strain rate – stress plot showing a comparison between Barrett et al [18] datum points and the results obtained in the present study. All the datum points reported by Barrett et al [18] for the polycrystalline samples are shown here. The datum points corresponding to the lowest two stress points are different from the present study. 205 • In the Barrett et al [18] tests where a stress change was conducted, the first reported “steady – state” strain rate was excluded since the strain might be too low. The following (i.e. after the first stress change) “steady – state” strain rates are shown in the plot assuming that a higher strain might have been attained. • The samples which were pre-strained are also shown in the plot, since such samples might have achieved relatively higher strains and hence they should be closer to the steady – state. As shown in Fig 99, it should be noted that the samples which were pre-strained also might not have achieved steady – state as they may lack enough strain; for example the sample tested at 0.035 MPa after pre-straining could accumulate only ≤ 0.5 % of strain. Following the above two guidelines, Fig 101 was prepared. As shown in Fig 101, the datum points show a remarkable consistency with the present test results and do not show n = 1 slope. Figure 102 shows all the datum points from the report of Barrett et al [18] and it has a tendency for a slope of n = 1 at fairly low stresses but the strain rates are lower than in the Harper and Dorn study [15] by a factor of 1.5. If the datum points corresponding to the two lowest stresses are discarded on the basis of attainment of very low strains, the results of the present study show a good agreement with Barrett et al [18]. Based on the previous arguments, the following conclusion can be made: The sample creep curves of Barrett et al [18] study does not show steady – state. But the datum points of the above study [18] which might have achieved relatively higher strains (e.g. in the stress cycling tests) show a good match with the present study. 206 5.2.3. Mohamed and Ginter [35] Mohamed and Ginter [35] tested polycrystalline samples of pure Al in air at 920 – 923 K to relatively large strains (γ ~ 0.1 – 0.2). The purity of the samples was 99.999 % and the grain sizes were 6 and ≥ 8 mm for the polycrystals. These tests were conducted using double shear specimens 27 , resulting in a constant shear stress throughout the deformation. The dimensions of the specimens are shown in Fig 104. Figure 105 summarizes the results on strain – time axes. Figure 106 shows the strain rate – strain behavior of the above two strain – time plots. The samples were deformed to large strains and there is no doubt steady – state was achieved in these samples. 5.2.3.1. Comparison with Present Study Fig 106 compares the Mohamed and Ginter data [35] with the present study. This shows a good agreement with the present study. The present work shows a small oscillation in the strain rate with strain and this is also consistent with Mohamed and Ginter [35]. The steady – state value is the same within a factor of 2 as shown in Fig 107. Taking into the account that the earlier study was conducted using double shear specimens, the consistency is excellent. As shown by Fig 108, the data points show a good consistency with the present test results and do not show any evidence of n = 1. It should be noted that the above study made no attempt to find strain rates at very low stresses. Based on the previous arguments, following conclusion can be made: The data points of the present study show a good match with the Mohamed – Ginter report [35]. 27 For conversion, ε = 2/3 γ and σ = 2 τ 207 Fig 104: Schematic of the double shear specimen used by Mohamed and Ginter [35]. All dimensions are in mm. 0.0 4.0x10 6 8.0x10 6 0.00 0.05 0.10 0.15 T = 923 K 0.050 0.058 Strain, ε Time (s) σ (MPa) = Pure Al (99.999 % ) Mohamed and Ginter (1982) Fig 105: Typical strain – time behavior for the Mohamed - Ginter samples [35]. The remaining strain – time curves were not shown in the report, hence the total strains achieved in several tests are not known. 208 0.00 0.03 0.06 0.09 0.12 0.15 10 -10 10 -9 10 -8 10 -7 10 -6 σ (MPa) = 0.05 Mohamed and Ginter (1982) ε (s -1 ) ε 0.058 Pure Al (99.999 %) T = 923 K Fig 106: Strain rate – strain behavior of the creep data of Mohamed and Ginter [35]. The data points show the attainment of steady – state creep. The oscillation in the strain rate – strain curve may be due to grain growth. The authors did not mention a possible cause for the oscillations. 209 0.00 0.02 0.04 0.06 0.08 0.10 0.12 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 Present Investigation (Al2SX1.0L C Cy) Mohamed and Ginter (1982) 3 2 0.050 0.053 σ (MPa) = ε (s -1 ) ε Pure Al Fig 107: Comparison of the present study with the data of Mohamed and Ginter [35] on strain rate – strain axes. The “capped” lines with underlined numbers show the factor by which the values differ. Mohamed and Ginter [35] performed tests using double shear specimens and the shear stress was constant throughout the deformation process. Hence the true stress value at 0.45 % of strain is calculated for the present sample in order to compare it with the Mohamed and Ginter datum points. 210 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 Present Investigation Pure Al ε (s -1 ) σ/G n = 4.5 Mohamed and Ginter (1982) σ (MPa) at 0.99 T m H-D Line n = 3 Fig 108: The strain rate – stress plot showing a comparison between the Mohamed and Ginter report [35] and the present study. In general, the datum points show good agreement with the present study. 211 5.2.4. Ginter, Chaudhury and Mohamed[26] Ginter et al [26] tested polycrystalline samples of pure Al in air at 920 – 923 K to relatively large strains (γ ~ 0.2). The purity of the samples was 99.9995 % and the grain sizes were 6 and ≥ 8 mm for the polycrystals. Like Mohamed and Ginter [35], these tests were also conducted using double shear specimens whose dimensions are shown in Fig 104. Figure 109 summarizes the results of two samples on strain – time axes. Figure 110 shows the strain rate – strain behavior of the above two strain – time plots. Fig 110 shows oscillations in the strain rate – strain curves. The authors suggested dynamic recrystallization as the restoration process for this feature, but Langdon [12] suggested it was as an outcome of (dynamic) grain growth (which may be inhibited at low purities). From the strain rate – strain plot (Fig 110), Langdon’s suggestion gets support as: 1. The oscillations in the strain rate – strain plot is irregular whereas in dynamic recrystallization (DRX) it is at regular strain intervals. 2. At a stress equal to 0.05 MPa, the oscillations are damped at the higher strains which should not be the case if DRX is the restoration process. Due to the damping of the oscillations, a conventional steady – state may be attained for this test. 3. The random nature of oscillations at a stress equal to 0.02 MPa suggests that some unknown processes are affecting the applied load which is likely to be the case as the applied load is very small (τ = 0.01 MPa). Small loads are susceptible to effects due to friction in the load chain, LVDT, etc. 212 0.0 5.0x10 6 1.0x10 7 1.5x10 7 2.0x10 7 0.00 0.05 0.10 0.15 0.20 σ (MPa) = 0.05 0.02 Strain Time (s) Ginter et al (2001) Pure Al T = 923 K Fig 109: Strain – time behavior of the creep data of Ginter et al [26]. The other strain – time curves were not shown in the report, hence the total strains achieved in several tests are not known. 213 0.00 0.02 0.04 0.06 0.08 0.10 0.12 10 -10 10 -9 10 -8 10 -7 10 -6 σ (MPa) = 0.02 Ginter et al (2001) ε (s -1 ) ε 0.05 Pure Al T = 923 K Fig 110: Strain rate – strain behavior of the creep data of Ginter et al [26]. The data points show substantial oscillation. The authors suggested dynamic recrystallization as a possible cause for the oscillations but it may also be due to grain growth [12]. 214 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 ε = 2/3 γ Present Investigation (99.999 SX Al) Ginter et al. (2001) (99.9995 Al) Pure Al ε (s -1 ) σ/G H-D Line σ = 2τ n = 4.5 σ (MPa) at 0.99 T m n = 3 Fig 111: The strain rate – stress plot showing a comparison between the Mohamed and Ginter report [26] and the present study. In general, the datum points show good agreement with the present study. 215 5.2.4.1. Comparison with Present Investigation Figure 111 shows good consistency in the nature of the stress exponent between the above study and the results of the present investigation. At the lower stresses, Ginter et al [26] derived a stress exponent of 2.5 whereas the present study suggested a stress exponent equal to ~ 3. It is important to note that Ginter et al [26] speculated uncertainty associated with their reported stress exponent value due to the periodic accelerations and due to the absence of a well defined steady – state. As can be seen in Fig 111, Ginter et al [26] consistently reported higher strain rates than the present study and this difference is more pronounced at the lower stresses. This difference may not be real and may be due to the following artifacts: 1. The calculation of steady – state strain rate by Ginter et al [26] is not based on the last segment of the strain – time curve. They calculated one strain rate for each segment (i.e. one period of oscillation) and assumed the average of all such strain rates as the steady – state strain rate for that stress. As shown in Fig 110, the strain rate value for the 0.05 MPa test converges to a well defined steady – state value at higher strains and since the above type of analysis includes the higher strain rates calculated at the beginning of the test also, it will hence always result in a higher strain rate value than the actual steady – state value. For example, the converged value for the strain rate in the case of 0.05 MPa is ~ 10 -8 s -1 (from Fig 110) and this value is consistent with the present study, but based on the above analysis Ginter et al [26] reported a value equal to ~2 × 10 -8 s -1 for the same conditions. 2. The tests conducted by Ginter et al [26] used double shear specimens and hence the conversion factors for converting shear stress and shear strain rate to normal stress 216 and normal strain rate, respectively, if not chosen correctly, may shift the datum points. From the above critical analysis, it is concluded that the present study is consistent with Ginter et al [26] in terms of the change in stress exponent and it is possible that the reported strain rates are also similar within experimental uncertainty. 5.2.5. Overall Comparison Figure 112 shows the strain rate – stress points at the low stresses. From the earlier studies suggesting Harper – Dorn creep, only the values which have been derived from the strain rate – strain plots (from the reported strain – time plots) are illustrated. The datum points of Ginter et al [26] corresponding to σ = 0.02 MPa are not shown in Fig 112 due to the ambiguity associated with the selection of the strain rate from Fig 110. It should be noted that some of these points (e.g. for Barrett et al [18] and Harper and Dorn [15]) may be away from the steady – state values. The Harper and Dorn [15] datum points are not corrected for the threshold stress. In this plot, the various datum points show good agreement within the uncertainty of the experimental observations (i.e. a factor of ± 2). Figure 113 shows all datum points from the various studies. These datum points are taken directly from the reported values. The Harper and Dorn [15] data are not compensated for the threshold stress. Again, this figure shows a good agreement between various studies except for the lowest two stress levels of Barrett et al [18]. The lowest two stress datum points of Barrett et al [18] are subject to uncertainties on the basis of the low strain accumulation. 217 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 n = 4.5 Present Investigation Pure Al ε (s -1 ) σ/G n = 3 Harper and Dorn (1957) Barrett et al (1972) Ginter and Mohamed (1982) Ginter et al (2001) σ (MPa) at 0.99 T m H-D line Fig 112: Comparison of the present study on single crystals with datum points from the earlier studies suggesting Harper – Dorn creep and which are accompanied with strain – time curves [15, 18, 26, 35]. 218 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 n = 4.5 Present Investigation Harper and Dorn (Actual) (1957) Ginter et al. (2001) Pure Al ε (s -1 ) σ/G n = 3 Barrett et al (1972) Ginter and Mohamed (1982) σ (MPa) at 0.99 T m H-D line Fig 113: Comparison of the present study on single crystal data points from the earlier studies showing a deviation in the stress exponent at the lower stresses [15, 18, 26, 35]. 219 After a critical examination of the studies suggesting the conventional Harper – Dorn creep, it can be concluded that the lack of steady – state attainment in most of the studies together with the ambiguity in interpretation of the experimental data might have confirmed the stress exponent of 1 in the Harper – Dorn regime. From the above analysis of the data reported in the same set of studies and the results obtained on single crystals during the present study, it is suggested that a stress exponent of ~ 3 is more appropriate in the Harper – Dorn regime rather than n ~ 1. 5.3. Dislocation Density Variation in Single Crystals The dislocation substructure formation is a key signature of the deformation mechanism governing creep. Figure 86 shows the dislocation density variation for the single crystals deformed during the present study and compares them with the published reports on pure Al polycrystals and a mechanistically – similar Al – 5 % Zn alloy. The following can be inferred from Fig 86: 1. Contrary to earlier reports on Harper – Dorn creep [40, 42], the dislocation density does not remain constant with stress. The dislocation density varies approximately as the square of the applied stress so that ρ = βσ 2 , where ρ is the dislocation density, β is a constant and σ is the applied stress. 2. The extrapolation from 5 – power law fits with the dislocation density value in the Harper – Dorn regime. This implies that the basic dislocation generation processes in the Harper – Dorn regime is the same as in the 5 – power law regime. It is important to analyze the reasons responsible for a constant dislocation density observed in the earlier studies. As explained in Section 2.7.1, Blacic and 220 Weertman [77] suggested a mechanism based on the chemical potential produced by a cyclic fluctuation in the temperature as the reason for the constant dislocation density observed in the Harper – Dorn regime. In the present study, the temperature fluctuations measured for a short term test had an amplitude of less than 0.25 K and the average time period of oscillation was ~ 4400 s. Using the following equation, the dislocation density produced due to the above temperature fluctuation (and hence the chemical potential of Blacic and Weertman) is 1.5 × 10 7 m -2 which is close but smaller than the minimum dislocation density measured during the present test: 3 2 1 f 0 t Tb T Q c β Ω μ δ β = ρ (29) where ρ is the dislocation density (as set by the fluctuation of chemical stress, σ cc, which is given by σ cc = Q f δT/µT), β is a constant (~ 1), c 0 is the equilibrium vacancy concentration ( 9.4 × 10 -4 for Al close to the melting temperature), Q f is the activation energy for the formation of a vacancy (0.66 eV for Al close to the melting temperature), δT is the temperature fluctuation, µ is the shear modulus (16.9 GPa for Al at 923 K), T is the temperature (923 K), b is the Burgers vector (2.84 × 10 -10 m), Ω is the activation volume and t 1 is the time period for the fluctuations. It is assumed here that the long term tests also have the same temperature fluctuation profile as the short term test, as most of the tests on single crystals were conducted on the same creep machine. For the long term tests, datum points were measured only after a time interval of 4000 s, which is comparable to the above time period (~4400 s) and hence such oscillations might have been lost during measurement. 221 Since the dislocation density produced by the chemical potential is smaller than the measured dislocation density, the present study is not affected by the temperature fluctuation and hence it was possible to measure the actual behavior of dislocation density as a function of applied stress. If earlier studies had temperature fluctuations with amplitudes of 1 K and time periods of 40 to 4000 s, they may have been affected by the chemical potential of Blacic and Weertman [77] and hence reported a constant dislocation density with respect to the applied stress. In conclusion, it can be stated that the dislocation density variation in the Harper – Dorn regime is similar to the 5 – power law regime and a study should be carried out to analyze the chemical potential hypothesis of Blacic and Weertman [77] which may explain the earlier reported constant dislocation density behavior in Harper – Dorn creep. 5.4. Possibility of a Size Effect in Single Crystals The diameter of the cylindrical specimens in these experiments was 25 mm whereas the widths of the square cross – section samples were either 12 mm or 7 mm. Table 8 shows the mean free path of the dislocations at low stresses within the Harper - Dorn regime. Table 8 shows that the mean free paths of the dislocations are comparable to the sample dimensions. Hence, it is expected that dislocations move to the surface of the specimens and leave the specimens which will then increase the deformation rate. Based on the model proposed by Nes et al [25] it is expected that the strain rate should vary inversely with the square of the width of the specimen as given by Eq. (22). 222 The similarity in the strain rate values of the samples with different cross sections leads to the contrary observation. The Sample 5 cross section area (~50 mm 2 ) is ~ 10 times smaller than the cross section of Sample 2 (~470 mm 2 ); hence the size effect model predicts that Sample 5 should creep ~ 100 times faster than Sample 2. No such difference was observed between the two strain rates, thereby giving an indication that this model may not fit the present study. In conclusion, it may be stated that the size effect model fails to explain the results in the present study. 5.5. Natural Power Law for Single Crystals? Weertman [189, 190] showed that, if all ad hoc assumptions are rejected, theoretical models lead to the natural third power law which is given by the following equation: 3 2 3 s G kT b DG A σ Ω = ε & (30) where Ω is the atomic volume and A 3 is a non – dimensional constant. The third-power law arises in a natural way because the strain rate is related to the dislocation density ρ and to the average dislocation velocity v through Orowan equation bv ψρ = ε & (31) where ψ is a constant. The dislocation density is proportional to the square of the applied stress so that 2 2 G b σ α = ρ (32) 223 where α is a constant. Substituting Eq. (32) in Eq. (31) for ρ and assuming that v is directly proportional the applied stress, one gets ( ) χσ σ α ψ = ε b G b 2 2 & (33) where χ is a constant relating dislocation velocity and the applied stress. Rearranging Eq. (33) gives the form of Eq. (30). Assuming Ω equal to ~b 3 for a cubic crystal structure and rearranging Eq. 26, one gets: 3 3 s G A DGb kT σ = ε & (34) In Section 5.3, it was shown that the dislocation density variation in the low stress regime for the single crystals tested during the present investigation varied approximately as the square of the applied stress. This observation is consistent with Eq. (32). Single crystals in the low stress regime also showed a stress exponent equal to ~ 3. The above two observations suggest a possibility of a natural power law creep (third power law creep) for very high purity Al single crystal samples in the Harper – Dorn regime. A value of A 3 equal to 1.0 is suggested for natural power law creep [36, 190]. Figures 114 (a) and (b) show the prediction of Eq. (34) assuming a value of A 3 equal to 1 and 0.8, respectively. Both values of A 3 shows good match with the experimental data but a value of 0.8 for A 3 shows a better match between the theory and the single crystal results at low stresses as shown in Fig 114(b). The test sample was of very high purity (99.999 %) which suggests that there were few impurity obstacles for dislocations. The 224 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 0.01 0.1 1 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 n = 3.1 Single Crystals 99.999 % Pure Al ε (s -1 ) σ/G n = 4.5 H-D Line Purity ε kT/DGb = (σ/G) 3 εkT/DGb σ (MPa) at 923 K 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 0.01 0.1 1 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 n = 3.1 Single Crystals 99.999 % Pure Al ε (s -1 ) σ/G n = 4.5 H-D Line Purity ε kT/DGb = 0.8 (σ/G) 3 εkT/DGb σ (MPa) at 923 K Fig 114: Analysis of the results of the present study with respect to natural third power – law creep for A 3 of Eq. (30) equal to (a) 1.0 and (b) 0.8 (a) (b) 225 test on a single crystal excludes the imperfections (grain boundaries and their effects) of polycrystals. A compression test along [100] leads to a uniform slip activity in the sample and the very large size of the specimens excludes the artifacts due to surface tension, surface oxidation and other size effects. The test temperature was very high (~ 0.98 T m ) leading to a very high diffusion rate. The above test conditions suggest they may be the ideal testing conditions and hence ideal or natural power law creep may take place. In conclusion, it may be stated that the natural third power – law creep seems to be a viable creep mechanism at low stresses, but a further critical analysis including elaborate creep tests and microstructural study is required to fully justify it. 5.6. Polycrystal / Oligocrystal Results Figures 69 – 70 show the strain rate – stress behavior of polycrystals (or oligocrystals) tested during the present study. Contrary to the single crystals, the polycrystals do not show any change in the stress exponent to a lower value in the Harper – Dorn regime. The datum points fit consistently on a line of slope 4.5 which indicates that the usual 5 – power law is the creep governing mechanism in the polycrystalline samples in the given stress range. Since the datum points acquired during the present study were limited in number, several interpretations of the results are possible. One of the possible interpretations is shown by Fig 115 where a change in the stress exponent is inferred on the basis of a higher value of stress exponent (5.5) in the 5 – power law region instead of the usual value of 4.5. The downward arrows show that a conclusive steady – state 226 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 n = 3 Sample 7 Sample 8 Pure Al Oligocrystal ε (s -1 ) σ/G n = 5.5 4 σ (MPa) at 0.99 T m Fig 115: A possible interpretation of the oligocrystalline / polycrystalline data based on a higher stress exponent value in the 5 – power law regime. The underlined number (4) shows the factor by which the datum point at the lowest stress is higher than the extrapolation based on 5 – power law. The vertical dashed line shows the transition stress which is consistent with the transition stress in the case of single crystal samples. 227 (from Fig 67) may have moved those points to a slightly lower value. In that case, oligocrystals show a change in the stress exponent at the lower stresses and it occurs at the same stress level as for the single crystals. Oligocrystals show lower strain rate than the single crystals and this is due to the lower purity (99.99%) compared to the single crystal samples (99.999%). In order to justify the above argument, the most critical point is to justify a higher stress exponent for the lower purity samples. Figure 116 shows the results on Al samples in 5 – power law creep. Samples with varying purity show the same stress exponent equal to 4.5, although samples with lower purity are creep resistant compared to the purer materials. Based on this observation, it is concluded that a stress exponent of 5.5, as argued in above paragraph, is doubtful and hence the oligocrystalline samples do not show any change in the stress exponent value. The difference in the trend of the stress exponent between the oligocrystal and the single crystal indicates that, apart from the impurity effect, the grain boundaries are playing an important role in the deformation process. It is contrary to earlier reports on the low stress regime where the deformation process was shown to be independent of the crystalline features (i.e. single crystals and polycrystals showed the same strain rate). At such low stresses the mean free path of dislocation is comparable to the grain size of the material, for example at a stress of 0.07 MPa the mean free path for dislocations is 6.9 mm (Table 8). The grain size of the sample was ~ 10 mm and hence it is probable that the dislocations are only stored at the grain boundaries and the cell interior is free from dislocations. Due to the lack of dislocations in the grain interior, deformation based on a dislocation mechanism will be inhibited and the overall 228 10 -7 10 -6 10 -5 10 -4 10 -3 10 -17 10 -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 T > 600 K 99.5 Al, Straub (1989) 99.99 Al, Straub (1989) 99.99 Al, Blum et al (1989) 99.999 Al, Straub (1989) 99.999 Al, Kassner et al (1991) εkT/DGb σ/G Pure Al 4.5 Fig 116: Comparison of stress exponent values in 5 – power law regime for various purity level Al samples [191 – 193]. Dotted lines show lines with slope of 4.5. The leftwards arrow show that a lower purity material is more creep resistant. With respect to 99.99 % purity samples, the best fit line for 99.999 % purity is shifted left on stress axis by a factor of ~ 1.07. 229 deformation will be lower if grain boundary sliding is also low. For a less pure sample, grain growth is inhibited as grain boundaries are not free to move due to impurity impingement. Hence, at the lower stresses a purer polycrystalline sample will show higher strain rate than the less pure sample. If the above argument is correct then at very low stresses the strain rates should be even slower than the 5 – power law predictions for a lower purity sample. This deviation will be more severe at lower stresses and for more impure material. Also, even a very high purity sample will show a decrease in the strain rate but at slightly lower stresses than the lower purity samples. The following observations support the above hypothesis on the active role of grain boundaries: 1. Ginter et al [26] observed a change in the stress exponent only for a higher purity sample where grain growth was possible. A change in the stress exponent was accompanied by an oscillation in the strain – time curve, which is the signature of the grain growth. Grain growth gives rise to the possibility that the dislocations may be stored in the grain interior leading to a faster creep rate. For the less pure samples, such oscillations were not observed and the stress exponent did not decrease to a lower value since grain growth was inhibited due to impurities. 2. The strain rate reported at the lowest stresses for Al and Pb by Ginter et al [26] and Mohamed [95] is much lower than the prediction of 5 – power creep. This is shown in Figs 25 and 26. In the case of 99.95 % pure Pb, this difference is more than one order whereas for 99.99 % pure Al it is more than by a factor of 5. 3. The datum points shown in Fig 25 for Al 99.9995 % also support the above hypothesis. At the lowest three stresses, the stress exponent increases and the overall 230 creep curve shows an “s- shape”. Below 0.02 MPa, a best fit line with higher slope can be drawn. At 0.02 MPa, the mean free path of dislocations is ~24.2 mm (Table 8) and this is comparable to the dimension of the test sample (Fig 104) and hence larger than the grain size (the grain size of the samples after the test was not reported by Ginter et al [26]). This suggests that for very pure samples, due to the absence of grain growth, the above effect of lowering the strain rate from the ideal 3 – power law occurs at the lower stresses compared to less pure samples. In conclusion, it may be stated that the creep process has an inherent threshold stress due to the lack of storage of dislocations in the grain interior. A model may br developed based on the concept that the high angle boundaries can act as a natural sink and represent the closest storage place for the dislocations at the lower stresses. This should be accompanied by a study on samples with varying impurity levels. 5.7. Grain Boundary Sliding Figure 79 shows the grain boundary sliding contribution to the deformation process of oligocrystalline sample tested in the present study. Figure 79 shows that grain boundary sliding is higher at the moderate stresses and it decreases to a very low value at the lowest stress. Also, the GBS contribution to the total strain is similar in the present study and in the work of Dorn and co-workers [15, 16]. It is important to understand the reasons which are responsible for a higher strain rate for the Harper and Dorn samples [15] as compared to the oligocrystalline samples tested in the present study. 231 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 0 20 40 60 Polycrystals Harper and Dorn (1957) 3.3 Present Investigation 10 Pure Al ε (s -1 ) σ/G n = 4.5 H-D Line Grain Size (mm) ξ (%) σ (MPa) Fig 117: A comparison of the results obtained in the present study using polycrystalline samples with the Harper and Dorn results [15]. The dashed line shows the consistency between the theory and the experiment of GBS at the lowest stress. 232 The average grain size of samples used by Harper and Dorn [15] was 3.3 mm whereas it was ~ 10 mm in the present study. Figure 117 shows that at the lowest stress (0.07 MPa) the grain boundary sliding contribution to the total strain, ξ, is equal for both the present study and Harper and Dorn [15]. According to the arguments given in the last section, the formation of subgrain was not feasible at this stress as the mean free path of dislocations and hence the stable subgrain size was larger than the grain size so that GBS will be accommodated by dislocations moving throughout the grain. This means that the inverse grain size sensitivity, p, is equal to 2 for this case. Since the grain size of the polycrystal tested in the present study was 3 times more than the Harper and Dorn sample, the following calculation shows that the strain rate for the present study should be ~ 1 order smaller than those of Harper and Dorn [15] at this stress: ξ H-D ≈ ξ Present – Study = ξ PS (35a) or (ε gb /ε t ) H-D ≈ (ε gb /ε total ) PS (35b) taking time derivative of both sides one gets: ( t gb ε ε & & ) H-D ≈ ( t gb ε ε & & ) PS (35c) Putting the condition that ( gb ε & ) H-D ≈ (10/3.3) 2 ( gb ε & ) PS (35d) the following relation is derived ( t ε & ) H-D = 10( t ε & ) PS (35e) This is consistent with the strain rates observed in both studies, as shown in Fig 117 by the dashed line. But such type of analysis shows that the strain rate observed by Harper and Dorn [15] at moderate stresses should be 2 to 3 times higher than the strain 233 rate observed in the present study. It should be noted that at the moderate stresses where subgrain sizes are smaller than the grain size, p will be 1 instead of 2 (as in the case of the smallest stress) [1, 194]. As shown in Fig 117, the strain rates reported by Harper and Dorn [15] are the same as that reported by the present study in the 5 – power law regime. Also, Barrett et al [18] also used a polycrystalline sample with a grain size equal to 10 mm but reported the same values for strain rates in the moderate stress regime. Similarity in the grain boundary sliding study and the strain rates indicate a lack of theoretical understanding of the deformation processes taking place in polycrystalline samples. One possible reason for the abrupt decrease in the GBS at the low stresses may be due to a threshold stress. A threshold stress has been analyzed in detail in the superplasticity literature and it was found that threshold stresses exist in superplasticity and are dependent on the impurity level of the material [195]. Since the main governing mechanism in superplastic deformation is GBS [1], it is possible that a threshold stress is required for grains to initiate sliding. In such case, GBS will decrease steeply at the lower stresses which is the case in the present study as well as in the Harper and Dorn [15] study. It is hence concluded that the GBS study shows consistency in the experiment and the theory at the lowest stress but it is not possible to explain the observations at the moderate stresses. It is necessary to conduct a detailed study on GBS at low stresses to clarify the various issues raised above. 234 5.8. Composite Plot Figure 118 shows a composite or summary plot of most of the studies conducted in the low stress regime on pure Al at a very high temperature (0.99 T m ). It is observed that most of the published data in the Harper – Dorn regime lie between the Harper – Dorn line and the 5 – power law line with a larger concentration over a line with slope of 3 if the test samples attained larger values of strain and the results are not modified for the presence of a threshold stress. 235 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -2 10 -1 10 0 Harper and Dorn (Net),PX, 99.99 (1957) Harper and Dorn (Actual),PX, 99.99 (1957) Harper and Dorn (Actual),SX, 99.99 (1957) Barrett et al,SX, 99.9994 (1972) Barrett et al,PX, 99.99 (1972) High ε Barrett et al,PX, 99.99 (1972) Remaining Barrett et al,PX, 99.99 (1972) Pre-Strain ~0.02 Mohamed et al,SX, 99.99 (1973) Mohamed et al,PX, 99.99 (1973) Mohamed and Ginter,PX, 99.9995 (1982) Mohamed and Ginter,PX, 99.99 (1982) Ardell and Lee,SX, 99.99 (1982) Blum and Maier,PX,99.99 (1999) Ginter et al,PX, 99.9995 (2001) Ginter et al,PX, 99.99 (2001) Blum et al,PX,99.99 (2002) McNee et al,99.95-99.999 (2004) Al1SX0.95L e Cy, 99.999 Al2SX1.01L c Cy, 99.999 Al3SX1.67T p Sq, 99.999 Al4SX1.92L c Cy, 99.999 Al4bSX1.92L c Cy, 99.999 Al5SX2.12T p Sq, 99.999 Al6SX1.72L p Cy, 99.999 Al7OX1.56T p Sq, 99.99 Al8OX1.8T p Sq, 99.99 Eisenlohr, OX, 99.99 Pure Al ε (s -1 ) σ/G n = 3 (ε = 5 x 10 8 σ 3 ) T = 923 K n = 4.5 σ (MPa) at 0.99 T m H-D line Fig 118: A composite plot reproduced to show various studies targeting the Harper – Dorn regime [15, 18, 19, 22 – 24, 26, 35, 40]. 236 CONCLUSIONS 1. Creep tests in the Harper – Dorn regime (very high temperatures and very low stresses) were conducted in compression using 99.999 % purity single crystalline and 99.99 % purity polycrystalline samples of Al. Consistent results were observed which did not show any scatter in the strain rate – stress plots. 2. The strain rates observed in the single crystal samples used in the present test lie consistently between the Harper – Dorn regime and the 5-power law and show a probable stress exponent of ~ 3. On the other hand, polycrystalline samples showed a stress exponent of 4.5 and are consistent with the 5 – power law regime. Both type of samples showed a stress exponent of 4.5 in the moderate and high stress regime. 3. After a critical examination of the studies suggesting conventional Harper – Dorn creep, it can be concluded that a lack of steady – state attainment in most of the studies as well as an ambiguity in interpretation of the experimental data led to the observation of the stress exponent of 1 in the Harper – Dorn regime. From the analysis of the data reported in the same set of studies and the results obtained on single crystals during the present study, it is suggested that a stress exponent of ~ 3 is more appropriate in the Harper – Dorn regime rather than n ~ 1. 4. Results obtained using small as well as large cross – section single crystal samples showed consistency and hence the size effect model of Nes et al [25] is not able to explain the results of the present study. 5. The natural third power – law seems to be a viable creep mechanism at low stresses for the single crystals, but a further critical analysis including elaborate 237 creep tests and microstructural study is required to fully justify it. 6. The creep process in polycrystalline samples may have an inherent threshold stress due a lack of storage of dislocations in the grain interior at very low stresses. A model can be developed based on the concept that the high angle boundaries may act as a natural sink and the closest storage place for the dislocations at the lower stresses. This should be accompanied by a study on samples with varying impurities. 7. Etch – pits measurements for single crystals do not show a saturation in dislocation density in the Harper – Dorn regime and their variation with stress is consistent with the 5 – power law prediction. 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Due to friction, there is no or only partial deformation in the friction cone and the deformation becomes inhomogeneous. Due to this fact, the central zone of the sample deforms as in ideal conditions whereas the entire volumes of the end regions are not free to expand. This results in bulge formation which pulls the strips away from each other. This motion is opposite to the compressive action and hence leads to an artifact in the measurement system due to which the measured strain is always less than the actual strain. Fig A1 shows the theory in the form of a schematic and in the context of the present study. 258 Fig A1: A schematic showing the upward movement in the strips drawn over the surface of the specimen (used for the strain measurement using the Laser technique). The continuous lines show the actual profile of the sample after deformation. Dotted lines are drawn to show various features, such as ideal deformation profile, the central volume which is deforming ideally as it is free from the friction cone effect, etc. 259 Appendix III: Value of Shear Modulus and Diffusion Coefficient for Pure Al The shear modulus, G, decreases linearly with temperature. The following equation is used to calculate the values of G at various temperatures: T dT dG G ) T ( G 0 − = (A1) where G 0 is the shear modulus at 0 K and dT dG is the variation of G with temperature. The temperature T is given in degrees K. For pure Al, G 0 is equal to 29.48 GPa and dT dG is equal to 1.361 × 10 7 MPa K -1 . These values have been calculated from Sutton [P M Sutton, Phy Rev 91 (1953) 816]. The values of the shear modulus from 900 to 923 K are listed in Table A1. The lattice diffusion coefficient, D l , increases exponentially with temperature. The following equation was used to calculate the values of D l at various temperatures: ) RT Q exp( D ) T ( D l 0 l − = (A2) where D 0 is the frequency factor, Q is the activation energy for the lattice diffusion and R is the universal gas constant. Again, T is given in degrees K. For pure Al, D 0 is equal to 1.74 × 10 -4 ms -2 , Q l is 142 kJ mol -1 and R is 8.314 J mol -1 K -1 . These values have been taken from Lundy and Murdock [T S Lundy and J F Murdock, J App Phy 33 (1962) 1671]. Values of the lattice diffusion coefficient for the temperatures from 900 to 923 K are listed in Table A1. Also in Table A1, the ratios of D l (923 K)/D l (T) are given which are used as the compensation factor for strain rates in various plots. 260 Table A1: The shear modulus and the diffusion coefficient of pure Al at various temperatures. Temperature (K) G (GPa) D l (ms -2 ) G(923 K)/G(T) D l (923 K)/ D l (T) 900 17.24 9.74 × 10 -13 0.98 1.60 901 17.22 9.95 × 10 -13 0.98 1.57 902 17.21 1.02 × 10 -12 0.98 1.54 903 17.19 1.04 × 10 -12 0.98 1.51 904 17.18 1.06 × 10 -12 0.98 1.48 905 17.17 1.08 × 10 -12 0.99 1.44 906 17.15 1.10 × 10 -12 0.99 1.42 907 17.14 1.13 × 10 -12 0.99 1.39 908 17.13 1.15 × 10 -12 0.99 1.36 909 17.11 1.18 × 10 -12 0.99 1.33 910 17.10 1.20 × 10 -12 0.99 1.30 911 17.09 1.23 × 10 -12 0.99 1.28 912 17.07 1.25 × 10 -12 0.99 1.25 913 17.06 1.28 × 10 -12 0.99 1.22 914 17.04 1.30 × 10 -12 0.99 1.20 915 17.03 1.33 × 10 -12 0.99 1.18 916 17.02 1.36 × 10 -12 0.99 1.15 917 17.00 1.39 × 10 -12 1.00 1.13 918 16.99 1.41 × 10 -12 1.00 1.11 919 16.98 1.44 × 10 -12 1.00 1.08 920 16.96 1.47 × 10 -12 1.00 1.06 921 16.95 1.50 × 10 -12 1.00 1.04 922 16.94 1.53 × 10 -12 1.00 1.02 923 16.92 1.56 × 10 -12 1.00 1.00 261 Appendix IV: Network Based Creep Model A three dimensional dislocation network has been reported in experiments rather than dislocation pile – ups, hence a creep model based on 3 – dimensional Frank network of dislocation is more appropriate 28 . It is assumed that dislocation glide is responsible for the most of the strain in the material but climb controls the creep rate. It is assumed that the distribution of the dislocation – links length is uniform with the smallest length equal to b and the largest link length equal to the critical link length for a Frank – Read source (i.e. l c = Gb/τ, where τ is the applied shear stress [J C Fisher, E W Hart and R H Pry, Phy Rev 87 (1952) 958, M E Kassner and M T Perez – Prado, “Fundamentals of Creep in Metals and Alloys”, Elsevier (2004)]). For the dislocation network in 3 –dimensions, both nodes and dislocation – links may climb to release dislocations and relax the network by coarsening it. The slowest of above two will govern the creep rate. Based on the concentration of vacancies and assuming dislocation – length as perfect sink and source of vacancies, the following climb velocities are calculated: kT Fb D v sd n π 4 = (A3) = b x ln kT Fb D v sd l 2 2π (A4) where v n is the climb velocity of nodes, F is the total force per unit length of the dislocations, v l is the climb velocity of dislocation links and x is the average mesh size 28 The model presented here borrows heavily from H E Evans and G Knowles, Acta Mater 25 (1977) 963 262 of the dislocation network. Eq. (A4) is originally from Weertman [J Weertman, In: Dorn Mem Symp, Ed: J C M Li and A K Mukherjee, Plenum, New York (1974) 315] who used the following form: − = 1 2 kT exp b R ln b Db v * l Ω σ π (A4b) where R is distance from the dislocation to the point at which the vacancy concentration is equal to the average vacancy concentration in the crystal and σ * is the stress acting on the dislocation that produces a climb force. Eq (A4b) reduces to the form of Eq (A4) if R = 2 x , Ω = b 3 , F = σb and σ * Ω/kT <1. x is given by the following equation: σ α ρ MGb ' x = = 1 (A5) where ρ is the dislocation density. Eq. (A5) is reasonable as 1/√ρ has been shown to be equal to the dislocation spacing in the samples deformed at high temperatures. Based on Fig. 86, value for α’M was calculated to be equal to 1.44. Also, Eq. (A5) is substantiated from the following equation which appears phenomenologically correct: a ss MGb ' ρ α σ = (A6) where a ~0.5. From Eq. (A3) and (A4), the ratio of v n and v l is given by = b x ln v v l n 2 2 (A7) 263 which is >1 (~ 20 for stresses used in the present study 29 ) and hence the climb of dislocation – links (length) will govern the creep or deformation. Vacancies can diffuse both through dislocations and through lattice. Since in a network, dislocations - links may provide the short circuit path for the vacancies from one point to any other (as all points are connected through dislocation – links), it may be possible that pipe diffusion becomes dominant over lattice diffusion. The following ratio decides which one will be governing: = σ α σ α 2 3 2 2 2 G ln D D G P p sd (A8) where D p is the pipe diffusion coefficient. For P >> 1, lattice diffusion will be the creep governing process. Based on the values of D p and D sd it can be concluded that at higher temperatures and low stresses, lattice diffusion will be the creep controlling processes. Now, let’s introduce a few more terms and their relationships with known quantities. Let φ be the swept area by a dislocation loop which can be given by s l = φ (A9) where l is the average dislocation link length and s is the slip distance of the dislocation loop. Eq (A9) used an average link length whereas Evans and Knowles used mesh size. It is assumed here that slip area is caused by the movement of individual links instead of all links inside one mesh. Now, let’s calculate the expression for F which was used in Eq (A4) for calculating climb velocity of dislocation – links. It is assumed here that only external 29 this number is also in accordance to the experimental values of dislocation densities at these stresses 264 forces acting on the climbing dislocation should affect the dislocation climb velocity. There may be two kinds of external forces on the dislocations: (a) the applied stress and (b) stresses due to other dislocations. It is assumed (contrary to Evans and Knowles) that the force due to coarsening of mesh or line – tension of dislocations will be internal in nature and should not be accounted in calculation of external forces 30 . Also, this force seems to affect the node (climb) velocity (which is an order of magnitude higher than the climb velocity of the dislocation – links) rather than the climb velocity of the climbing dislocation – link if no change in length of the climbing link is assumed. Hence the following will be the expression for force per unit length of the dislocations: x ) ( Gb b F n ν π σ − + = 1 2 2 (A10) where σ n is the force component along the direction of climb and υ is the Poisson’s ratio. The first term in above expression is due to applied load whereas the second term comes from the elastic interaction of dislocations. Assuming σ n = σ/√2 (since b = [110] for present case of single crystals whereas the loading direction is [100]) 31 the following expression 32 is found: x ) ( Gb b F ν π σ − + = 1 2 2 2 (A11) Using the value of x from Eq (A5) and taking υ equal to 0.3, the following expression for F is calculated: 30 This force was also neglected by H M Zbib, T D de la Rubia, M Rhee and J P Hirth, J Nucl Mater 276 (2000) 154 in their 3 – dimension dislocation network model 31 Evans and Knowles used: σ n = σ/2 which is Tresca criterion for polycrystals 32 The second term is ~20 % of the first term 265 F = γσb (A12) where γ is a constant which is equal to ~ 0.87 for the single crystals used in the present study. So substituting the value of F from Eq. (A12) in Eq. (A4), one gets b b x ln kT b D v sd l γσ π = 2 2 (A13) Another important quantity is the rate of release of the dislocation loops from the network, N & : − = 3 3 x ) l l ( v N c l & (A14) where l is the mean link length and 3 3 x gives the total number of dislocation – links per unit volume. Eq. (A14) uses ) l l ( c − whereas Evans and Knowles used onlyl . A multiplying factor equal to 2 was used by Evans and Knowles to show that the probability of the release of a dislocation loop from either a Frank – Read source (for which critical length is l c ) or due to the breakage of nodes (i.e. when the dislocation – link length is less than l c but the resultant concentrated force on a node is more than node strength). It was a very conservative assumption as the breaking of nodes is less likely than the release of a dislocation loop from Frank – Read source because node (or jog) strength is much larger than the critical stress required to activate a Frank – Read source. Hence, in the present work, the factor of 2 was discarded. So, after substituting the value of v l from Eq (A13) in Eq. (A14) the following equation is calculated for N & : 266 − = b x ln kT b b D x ) l l ( N sd c 2 2 3 1 3 γσ π & (A15) − = b x ln kT b D x ) l l ( sd c 2 6 2 3 σ πγ (A16) Now, substituting the value for x from Eq. (A5), assuming 3 x l = 33 and taking τ bG l c = , − = σ α σ σ α σ α τ πγ MbG ' b ln kT b D MbG ' MbG ' bG N sd 2 1 3 6 2 3 & (A17) For a single crystal, τ = Sσ where S is the Schmidt factor (= 0.41 for the single crystals used in the present investigation), hence the following expression: ( ) − = σ α σ σ α α πγ G M ' ln kT b D bG M ' M ' . N sd 2 3 25 2 6 2 4 3 & (A18) Strain rate is given by b Nφ β ε & & = (A19) 33 The link length is the distance between two dislocation – nodes in the network whereas x is the side length of the cube which can contain the paralleloid formed by above dislocations. For example, in a FCC crystal, three mutually independent Burgers vectors will be 0.5 [110], 0.5 [101] and 0.5 [011] (with length equal to 0.5 √2) and these will be the dislocation – links of the smallest possible network. A cube which has side equal to the base diagonal of the above paralleloid will contain the entire structure inside it, hence the side of the cube (equal to x ) will be 0.5 [2 1 1] (or length equal to 0.5 √6). From this small calculation one can see that 3 x l = . 267 where β is a constant to convert the shear strain into uniaxial strain (S for single crystals). Now substituting the values of parameters in Eq. (A19) from Eqs. (A19) and (AIV18), ( ) ( )b s l G M ' ln kT b D bG M ' M ' . sd − = σ α σ σ α α πγ β ε 2 3 25 2 6 2 4 3 & (A20) Taking x = s i.e. slip occurs for one mesh size (this is true if hardening is independent of the stress which holds true for the case of single crystals where subgrain formation was not observed and the sample size was much bigger than network mesh size) and 3 x l = , ( ) ( ) b x G M ' ln kT b D bG M ' M ' . sd 3 2 3 25 2 6 2 2 4 3 − = σ α σ σ α α πγβ ε & (A21) ( ) b MbG ' G M ' ln kT b D bG M ' M ' . sd 2 2 4 3 2 3 25 2 3 6 − = σ α σ α σ σ α α πγβ (A22) ( ) − = σ α σ σ α α πγβ G M ' ln kT b D bG M ' M ' . sd 2 3 25 2 3 6 3 2 (A23) ( ) − = σ α σ α α πγβ G M ' ln kT b D G M ' M ' . sd 2 3 25 2 3 6 3 2 (A24) ( ) 3 2 1 3 25 2 3 2 − = G kT bG D G M ' ln M ' M ' . sd σ σ α α α πγβ (A25) 268 Now, for a FCC single crystals oriented in [100] direction, β =0.41, γ = 0.87 and α’M = 1.44 (from experimental observation, Fig 86), 3 72 0 9 1 = G kT bG D G . ln . ss sd ss σ σ ε & (A26) Fig A2 compares the theory with the experimental results of the single crystals and it shows a good match between the theory and experimental data. Despite of the fact that values of several constants used in the derivation of Eq (A26) may not be accurate (but they are within the correct order), the experimental points are within a factor of 4 from the theory. The slope of the theoretical line is close to 3.2 which is also consistent with the stress exponent of the experiment. Based on this discussion, an alternative creep model for single crystals at the low stresses is proposed. Assumptions in the above model do not contradict natural 3- power law creep and it may be seen as a mechanistic explanation for natural 3- power law creep. Appendix V: Correction for Taylor Factor, Hall – Path Factor and Impurity In order to compare the results of the single crystals and the polycrystals, it is important to study the effects of grain boundaries in terms of Taylor factor and Hall – Patch factor. Since the purity level of single crystals and polycrystals used in the present study were different, it is also required to study the effect of the impurities in order to compare their results. In the following subsections, proper factors for the compensation due to above three factors will be derived and subsequently used. 269 Fig A2: A comparative plot showing the prediction based on dislocation network theory. The slope of the theoretical line is ~ 3.2 and the strain rate values are within a factor of 1.5 from the best fit line. 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 0.01 0.1 1 10 -18 10 -17 10 -16 10 -15 10 -14 10 -13 n = 3.1 Single Crystals 99.999 % Pure Al ε (s -1 ) σ/G n = 4.5 H-D Line Purity 3 72 0 9 1 = G kT bG D G . ln . sd σ σ ε & εkT/DGb σ (MPa) at 923 K 270 Correction for Taylor Factor: In a polycrystalline material, several single crystals are arranged in the form of grains. These grains are misoriented with respect to each other and hence give additional strength to the polycrystals compared to single crystals 34 . For an untextured FCC polycrystal, for example pure Al, Taylor factor was calculated to be equal to 3.06 [J F W Bishop and R Hill, Phil Mag 42 (1951) 1298, U F Kocks, Metall Trans 1 (1970) 1121]. For the single crystals, Taylor factor is the inverse of Schmidt factor. Schmidt factor is 0.41 for the single crystals tested in the present investigation and hence the Taylor factor was 2.25. A higher Taylor factor indicates higher strength and hence the polycrystals were 1.25 (i.e. 3.06 /2.25) stronger than the single crystals. In order to compensate for the Taylor factor, the stress values of the single crystals are multiplied by 1.25. Correction for Hall - Patch Factor: Grain boundaries attribute additional strength to the polycrystals and this is given by Hall – Patch relationship. Hall – Patch relationship is given by the following equation: 5 0 0 . HP y d k − + = σ σ (A27) where σ y is the yield strength of the material, σ 0 is the single crystal strength (in MPa), k HP is the Hall – Patch constant (in Nmm -1.5 ) and d is the grain size given in mm. σ 0 is taken as 2.95 MPa for pure Al. It was shown by Kassner and Li [M E Kassner and X Li, Scripta Metal Mater 25 (1991) 2833] that k HP is a function of temperature and it decreases with the temperature. Fig A3 shows the variation of k HP in pure Al with 34 This is due to the Von Mises criterion of having 5 independent slip systems in a crystal to have homogenous deformation and continuity in slip on the slip systems across the grain boundaries of a polycrystal is required in order to maintain geometric continuity / consistency between the grains. 271 0 100 200 300 400 500 600 700 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Kassner and Li (1991) Best Fit Curve (y = 2.363 e -0.005x ) Hall - Patch Constsnt, k Temp ( 0 C) Pure Al T = 650 0 C, k = 0.092 Fig A3: Variation of Hall – Patch constant, k HP , with temperature. The continuous line is a best – fit curve which has been extrapolated to the test temperature (650˚C) of the present investigation. 272 respect to temperature. At 923 K, the extrapolation curve gives a value of 0.092 for Hall – Patch constant. Using Eq (A27), σ y is calculated to be equal to 2.98 MPa for the polycrystalline sample with a grain size of 10 mm which is. the grain size of the polycrystalline samples used in present investigation. Hence, the polycrystalline samples are 1.01 (i.e. σ y / σ 0 = 2.98/2.95) times stronger than the single crystalline samples. In order to compensate for the Hall - Patch factor, the stress values of the single crystals are multiplied by 1.01. Correction for Impurity: In the present investigation, the purity of polycrystalline samples were 99.99 % whereas it was 99.999 % for the single crystals, hence it is required to calculate a factor by which the strain rates (/ stress) values of single crystals should be divided (/ multiplied) in order to compensate for the impurity effects. Based on Fig 116, it was calculated that the best fit line for the 99.999 % samples is shifted left on stress scale by a factor of ~1.07 in 5 – power law regime with respect to the 99.99 % purity samples and this factor corresponds to ~1.36 on the strain rate axis. Unlike the previous two factors (Taylor factor and Hall – Patch factor), this compensation factor is not fundamental and it does not change the stress field inside the sample. Its effect is observed in lowering the strain rates and hence the strain rate values corresponding to the single crystals lying in 5 – power law are divided by a factor of 1.36 whereas this factor was 1.23 in the 3 – power law regime. Fig A4 shows all corrections for the single crystals and compares the data with polycrystals. 273 10 -7 10 -6 10 -5 10 -4 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -2 10 -1 10 0 n = 3.1 Single Crystals (Before Corrections) Single Crystals (After Corrections) Oligocrystals n = 3.1 Pure Al ε (s -1 ) σ/G n = 4.5 H-D Line σ (MPa) at 0.99 T m Fig A4: A comparison of the single crystal data and the polycrystal data observed during the present investigation. Single crystal data have been compensated for Taylor factor (1.25) and Hall – Patch factor (1.01) by multiplying the stress values with these factors and for impurity effects (1.36 in 5 – power law regime and 1.23 for 3 – power law regime) by dividing strain rate values by these factors.
Abstract (if available)
Abstract
Harper-Dorn creep was proposed for materials with large grain size deforming at very low stresses (sigma / G ~ 10^-6 where sigma is the applied stress and G is the shear modulus) and high temperatures (~ 0.95 - 0.99 Tm, where Tm is the absolute melting temperature). Recently, this creep mechanism has become controversial and several other creep mechanisms, such as 5-power law and Nabarro-Herring creep, have been proposed as governing the creep mechanism in the Harper-Dorn regime. An extensive study was conducted to evaluate several features of creep in the Harper-Dorn regime in order to determine an unambiguous creep mechanism.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Kumar, Praveen
(author)
Core Title
Creep in pure single and polycrystalline aluminum at very low stresses and high temperatures: an evaluation of Harper-Dorn creep
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Mechanical Engineering
Publication Date
06/05/2007
Defense Date
04/17/2007
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Aluminum,creep,Harper-Dorn,natural 3-power law,OAI-PMH Harvest
Language
English
Advisor
Kassner, Michael E. (
committee chair
), Langdon, Terence G. (
committee chair
), Goo, Edward K. (
committee member
)
Creator Email
praveenk@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m506
Unique identifier
UC1417476
Identifier
etd-Kumar-20070605 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-502212 (legacy record id),usctheses-m506 (legacy record id)
Legacy Identifier
etd-Kumar-20070605.pdf
Dmrecord
502212
Document Type
Dissertation
Rights
Kumar, Praveen
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
creep
Harper-Dorn
natural 3-power law