Backstress from dislocation interactions quantified 1 by nanoindentation load-drop experiments 2

17 Recent work has identified the importance of strain hardening and backstresses among dislocations 18 in the deformation of geologic materials at both high and low temperatures, but very few experimental 19 measurements of such backstresses exist. Using a nanoindentation load drop method and a self-similar 20 Berkovich tip, we measure backstresses in single crystals of olivine, quartz, and plagioclase feldspar at a 21 range of indentation depths from 100–1750 nm, corresponding to densities of geometrically necessary 22 dislocations (GND) of order 10–10 m. Our results reveal a power-law relationship between backstress 23 Preprint submitted to Geophysical Research Letters and EarthArXiv and GND density, with an exponent ranging from 0.44 to 0.55 for each material, in close agreement with 24 the theoretical prediction (0.5) from Taylor hardening. This work provides experimental evidence of Taylor 25 hardening in geologic materials and supports the assertion that backstress must be considered in both high26 and low-temperature deformation. 27 Plain Language Summary 28 As a material is plastically deformed at room temperature, it often becomes stronger. In minerals, 29 this strengthening is typically caused by the accumulation of linear defects within the material. These 30 defects repel each other and push back with a strength predicted to be proportional to the square root of the 31 defect density, but this relationship has not typically been observed for geologic materials. We developed 32 a method to measure the strength of this pushback at very small scales for which the density of these defects 33 in the crystal is high. Our results for three common geologic materials agree with predictions from theory 34 and demonstrate that these defects must be considered when modeling deformation of rocks in Earth’s 35 interior. 36


Introduction
creep experiments is challenging because it may be caused by the interplay of multiple deformation Preprint submitted to Geophysical Research Letters and EarthArXiv dislocation interactions) occurs during nanoindentation. However, due to the significantly larger stresses in 75 nanoindentation, the dislocation density was much higher than in samples deformed in Hansen et al. (2019). 76 In the present paper, we quantify the relationship between GND density and backstress in three 77 common geologic materials (olivine, quartz, and plagioclase feldspar) using a novel nanoindentation 78 method. Because nanoindentation localizes deformation in a small volume of material, the sample is 79 essentially self-confined, and extremely high stresses can be applied without inducing fracture. 80 Additionally, nanoindentation using a Berkovich (3-sided pyramid) tip offers a significant advantage in that 81 it can be used to probe different microstructures (i.e., GND densities) at the same strain (~8%) due to its 82 self-similar geometry. We utilize this technique to demonstrate excellent quantitative agreement between 83 our experiments and theoretical predictions of Taylor hardening (Taylor, 1934), which suggests that 84 backstress should scale as the square root of GND density. 85

86
We have developed a method to measure the backstress from GNDs created during nanoindentation 87 experiments. This method is similar to a stress-reduction test, a common technique used on macroscopic 88 samples to measure anelasticity (e.g., Takeuchi & Argon, 1976;Blum & Weckert, 1987;Caswell et al., 89 2015; Hansen et al., 2020), with one key difference. Because the indentation stress is controlled by the 90 mechanical response of the sample and not its physical dimensions, this type of experiment is more 91 accurately described as a "load-drop" test. Only the applied load is prescribed in the experiment, and neither 92 stress nor strain rate are controlled. Syed Asif and Pethica (1997) presented the only previous study that 93 utilized load drops to measure changes in indentation creep behavior, but they did not quantify the 94 backstress systematically in their study of tungsten and gallium arsenide single crystals. 95 Each of our experiments consisted of four parts: 1) an initial loading phase, 2) a short hold at 96 constant load to measure indentation creep behavior, 3) a rapid load drop, and 4) another longer hold at a 97 reduced constant load to measure the mechanical response of the sample. Segment 1 can be completed 98 using any number of standard nanoindentation protocols, such as constant loading rate or constant nominal 99 Preprint submitted to Geophysical Research Letters and EarthArXiv strain rate, as the main function of this step is to set the initial microstructure (i.e., GND density) beneath 100 the indenter tip. The GND density, ρGND, below the indenter tip for a pyramidal geometry is a function of 101 the tip shape, the indentation depth, h, and the Burgers vector, b, of the material (e.g. Pharr et al., 2010) and 102 is given by 103 where is the angle formed between the surface and the indenter (19.7° for a Berkovich tip). Thus, deeper 105 indents formed by larger applied loads will result in a lower GND density. 106 In the results presented here, all experiments were performed in a load-controlled nanoindentation 107 apparatus with Ṗ/P = 0.2 for segment 1, where P is the applied load and Ṗ is its time derivative. The 108 indentation hardness, H, is the mean contact stress, defined as 109 where A is the projected contact area between the tip and the sample. The value of A is calibrated as a 111 function of depth using a standard of known Young's modulus (usually fused silica) and given by the 112 where C1, C2, C3...C7 are constants, and hc is the contact depth (i.e., the true depth at which the tip and 115 sample are in contact, with elastic deflection of the surface of the sample removed). The contact depth is 116 given by 117 where h is the measured indentation depth, ϵ is a constant associated with the geometry of the indenter (0.75 119 for Berkovich), and S is the contact stiffness. With known contact stiffness and contact area, the reduced 120 elastic modulus, Er , of the tip-sample contact can be determined using 121 (Eq. 5) 122

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Utilizing known values of the elastic constants of the diamond tip and an assumed Poisson's ratio of the 123 sample, we determined the sample's elastic modulus using 124 where E is the Young's modulus, v is Poisson's ratio, and the subscripts s and i refer to the sample and 126 indenter tip, respectively. Our experiments were performed using the continuous stiffness measurement 127 (CSM) method with a dynamic frequency of 110 Hz and a target dynamic displacement of 2 nm, which 128 allowed us to measure the contact stiffness (and therefore the contact depth, hardness, and elastic modulus 129 of the sample) continuously as a function of time (Li & Bhushan, 2002;Oliver & Pharr, 2004). 130 Segment 2 of our load-drop method is optional, but in these experiments we performed a 60-s hold 131 to measure the creep behavior. Due to possible thermal drift of the instrument, this portion of the test and 132 all subsequent measurements were obtained using the CSM method. In this portion of the test, and all 133 following steps, we used the measured elastic modulus from Segment 1 and rearrange Eq. 5 to solve for 134 contact area as 135 This approach is preferable to relying on the depth measurement to acquire contact area using Eqs. 3 and 4 137 because the depth measurement is highly sensitive to temperature fluctuations. Thus, our subsequent 138 measurements of hardness from Eq. 2 are calculated from the measured contact stiffness, the previously 139 derived elastic modulus, and the current applied load. 140 Segments 3 and 4 are the additions of our method and encompass a load drop and subsequent hold. 141 In Segment 3 of our experiments, we reduced the load linearly over 1 s by a prescribed amount, ranging 142 from 1% to 99% of the maximum applied load. A small amount of dynamic overshoot occurred for large 143 reductions in applied load, but these variations did not significantly influence any of our results. After the 144 load drop, the new applied load was held constant for the duration of Segment 4. In the results presented 145 here, we held the load at the reduced value for 3600 s before completely unloading the sample.
In summary, this method determines the hardness and elastic modulus as a function of indentation 147 depth and the creep behavior during a short hold at high stress. In addition, by testing a range of reductions 148 in load for a given peak load, we can determine the magnitude of the backstress in a material, as 149 demonstrated below. Repeating a series of experiments at different peak loads and thus different maximum 150 depths, corresponding to different GND densities, also allows us to explore the influence of microstructure 151 on backstress. 152

153
We performed a total of 155 load-drop experiments on single crystals of San Carlos olivine, 154 synthetic quartz, and natural plagioclase feldspar (labradorite). For each experiment, we recorded the 155 applied load, indentation depth, and contact stiffness at a rate of 100 Hz, from which we derived the elastic 156 modulus, hardness, and creep behavior at each point in the test. 157 For each material, Segments 1 and 2 were reproducible for a given maximum load (e.g., Figure 1a), test. Contact stiffness versus time for each experiment before and after the load drop are presented in Figure  165 1a, with each test segment labeled. An initial steep increase in contact stiffness occurs in Segment 1, and a 166 small increase over time occurs in Segment 2. The abrupt reduction in contact stiffness occurs as the applied 167 stress is reduced in Segment 3 and is associated with some elastic recovery of the material. 168 In all experiments, one of three behaviors was observed after the load drop: 1) the contact stiffness

(forward creep), increases (backwards creep), or remains the same (no creep). 185
We use the measured contact stiffness, the elastic modulus calculated from Segment 1, and Eqs. 2 186 and 7 to determine the hardness in the experiments in Figure 1a. These data are presented in Figure 1b for  for all data presented in Figure 2a.

266
We have performed nanoindentation load-drop experiments on single crystals of olivine, quartz, 267 and plagioclase feldspar to measure the backstress created by long-range elastic interactions among 268 dislocations. To vary the GND density, we applied a range of maximum loads using a self-similar 269

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Berkovich indenter tip to achieve a range of indentation depths. Our results demonstrate that the 270 backstress in all three materials scales approximately with the square root of GND density, as predicted 271 from the Taylor   All data used in this study are available at https://upenn.box.com/s/mo9txpz9n6dzvdup6dt5yq8t6ltd80tt. 282 Funding for this study was provided by NERC 1710DG008/JC4 to L.N.H. and C.A.T. and NSF EAR-283 1806791 to K.M.K. 284