Spatial-Temporal Heterogeneity of Rock Samples’ Deformation and Damage: Experimental Study and Digital Image Correlation Analysis

: In-situ observations and laboratory experiments showed that slow deformation waves widely exist 11 in geomedia under loading conditions. Slow deformation waves’ behavior exhibits some similarities in media 12 ranging from the scale as large as the Earth's crust to the scale as small as the laboratory test samples. However, 13 the mechanism underlying their generation has not been clarified yet. In this research an experimental study 14 was performed on small-scale red sandstone samples subjected to uniaxial compression at the displacement 15 rates of 0.1, 0.5, and 1 mm/min. Slow deformation waves under different loading rates were analyzed by 16 speckle photography for microscopic characterization combined with the digital image correlation (DIC) 17 technique. The Luders deformation bands were predominantly observed in the flow channels formed at the 18 stage of macro-elastic deformation. The spatial-temporal heterogeneity of the rock sample surface was 19 quantified, and the deformation waves' propagation velocities under different loading rates were obtained. The 20 linear relationship between the propagation velocities of slow deformation waves and the loading rates was 21 determined. The research findings shed some new lights on the evolutionary characteristics of the slow 22 deformation waves.


Introduction
In-situ observations showed that slow deformation waves with propagation velocities much lower than those 27 of transverse and longitudinal waves generated by earthquakes exist in Earth's crust. Slow deformation waves 28 have different names with less essential difference, for example, seismoactive waves (Mogi 1973), creeping 29 stress waves (Savage 1971), tectonic waves (Elsasser 1969; Kasahara 1981, plastic flow waves (Wang 1993; 30 Wang and Zhang 1994) etc. According to the concept of epicenter migration-related waves (Richter 1958), 31 their propagation velocities in the seismic belt may reach only several dozens of kilometers per year (Levina 32 and Ruzhich 2015; Elsasser 1969). The presence of creeping waves can be related to the instability caused by 33 frictional sliding (Ruina 1983). As to the occurrence mechanism of these waves, it is believed that they are 34 jointly triggered by external factors and the changes in the properties of the media in the focal area (Feng 1986). 35 Inside the Earth's crust, there is an interlayer composed of volatile media. Under certain conditions, the 36 propagation of slow deformation waves may be triggered in such volatile media, and they play a crucial role 37 in seismic development and occurrence (Geng and Xu 1990). Studies on the plastic-flow network and plastic 38 flow waves of the continental lithosphere (Wang and Zhang 1994;Wang 1993) have shown that the driving 39 force at the margin of the continental plate is transmitted over a long distance, primarily through the network- The rock samples under study correspond to the third category of the red sandstone. The latter exists in two 86 main structure forms, including granular-clastic and mud cement forms, and has high hygroscopicity, porosity 87 ranging approximately from 0.1 to 0.3, low bonding performance, poor overall performance, and is less prone 88 to disintegration than ordinary sandstone. Nine red sandstone samples with dimensions of 89 50mm×50mm×100mm were prepared using the method recommended by the International Society for Rock 90 Mechanics (ISRM)( Fairhurst and Hudson 1999). Every two opposite sides were parallel to each other, and all 91 of them were smooth. All samples were prepared from the same batch of blocks harvested on-site. Those with 92 defects and joints were eliminated. 93 The rock samples were subdivided into three groups, each containing three samples, and were subjected to 94 displacement-controlled compressive loading at three loading rates, using a DRTS-500 test bench composed 95 of the loading system (with the maximum axial load of 500KN) and the data collection system. The 96 displacement control mode was adopted since the test goal was to investigate slow deformation waves under 97 different loading rates. The rock sample geometry and the experimental equipment are shown in Fig. 2. 98

Fig.2 99
Given that speckle photography was adopted to capture the rock samples' images, the samples were first 100 sprayed with a layer of ordinary white paint and left to dry. Then, another layer of black particle paint was 101 sprayed to form random speckles (Munoz et al.2016;Heinz and Wiggins 2010;Yang et al.2015). 102 was 80 MPa. The loading was imposed under the displacement-controlled mode with three different rates: 0.1, 107 0.5, and 1 mm/min, respectively. In each group, speckle photography was used for microscopic characterization 108 from the start to the end of the loading process. Later, the collected images were subjected to the digital image 109 correlation (DIC) analysis. The evolution of the displacement vector r(x, y) field at the marked points on the 110 sample surface was tracked and recorded to provide data for the further analysis. 111 The stress-strain curves of the rock samples obtained from the uniaxial compression tests are shown in Fig.  112 3, where 1  is the axial strain; 3  is the transverse strain; and V  is the volumetric strain. The pressure in the 113 figure below is defined as positive. A typical stress-strain curve of an intact rock sample can be roughly 114 subdivided into the following four stages. The first stage is the compaction stage, where the curve bends 115 slightly upwards, and the initial microcracks are closed due to the compression. The curve is a straight line at 116 the second stage, so this is the linear elastic stage. At the third stage, the curve bends slightly downwards, and 117 unstable microcracks are generated in the direction parallel to the loading direction. The third stage is called a 118 non-elastic (plastic) stage, while the fourth one is referred to as the failure stage. At the plastic and failure 119 stages, cracks appeared on the sample surface, so the DIC technology would fail to achieve the desired effect. 120 Therefore, DIC is only applicable to the data at the first two (compaction and elastic) stages. In metals and 121 alloys, features of strain localization that resemble those of the deformation waves have been observed at the 122 linear elastic stage in the samples under compression or tension (Danilov et al.2005(Danilov et al. ,2009). In the present study, 123 the deformation waves' propagation velocities at the linear elastic stage (segment AB in Fig. 3) were of the 124 primary concern during the loading process. At this stage, the wave propagation velocity was closer to the slow 125 waves velocities observed in Earth's crust and exhibited similar evolution patterns. 126    According to the vertical views in Fig. 5a and 5b, as loading continued, the fluctuations became more 180 pronounced, and the degree of waviness increased. The peak values gradually increased and occurred more 181 frequently. Moreover, the waviness became increasingly apparent. 182 It is noteworthy that as the loading rate increased, the fluctuations on the left half side became more intensive. 183 The peaks were also greater and denser. At the loading rate of 1 mm/min, the sample surface was subjected to left corner, its value being approximately five times the strain values at other places on the same surface (Fig.  186   6). 187

Fig. 6 188
Taken together, whatever the loading rate is, the strain component was greater at the end to which the load 189 was applied but smaller at the other end. As the loading continued, the strain components at both ends became 190 more consistent. During this process, the slow deformation wave propagated through the media, accompanied 191 by energy accumulation and conversion. The slow deformation waves were generated by adjusting the 192 imbalance between the driving force and the viscous force inherent inside the solids. This process involved the 193 conversion between the elastic strain and kinetic energies.   strong tensile deformation zones from the beginning. For example, in the nephograms (Fig. 9), corresponding 232 to points 2 and 3 in Fig. 8, the strong tensile deformation, as indicated by the strain component , periodically 233 appeared at 20-60mm, with a magnitude of about 0.1% (as indicated by the arrowhead at the point 3 in Fig. 9). 234 These regions of strong tensile deformation corresponded to the flow channels observed in the nephogram of 235 . This phenomenon was observed throughout the compaction and elastic stages of loading. 236 As observed in the nephogram of , the nucleation of Luders bands (LB), i.e., shear bands formed in 237 regions of stress concentration, occurred during the loading process (point 2 in Fig. 9). At first, the nuclei had 238 small oval cross-sections and appeared in pairs. One of them was a region of compressive strain with the 239 maximum strain value being -0.5%; the other was a region of tensile strain with the maximum strain value 240 being 0.2%. The grain deformation and LB nucleations were more conspicuous in the nephogram of and 241 observed roughly at the same positions as those in the hephogram of . The maximum strain value was 242 0.15%. The observed nucleation pattern can be explained as follows. The displacements originated from the sample surface, whose deformation exceeded the bulk one, i.e., the average deformation of the sample. During 244 the sample's loading-induced deformation, the displacement was rapidly released, thereby resulting in the LB 245 nucleation. 246 After grain deformation and LB nucleation, the Luders bands were gradually formed. In the beginning, the 247 Luders bands were obscure. As the loading continued, the Luders bands became clearer and began to show 248 more conspicuous banding features. This trend started from both sides of the sample simultaneously, with slow 249 interconnection and penetration in the transverse direction. This phenomenon indicated that the Luders bands 250 were already formed (points 3 and 4 in the nephogram of in Fig. 9) before the plastic deformation stage. 251 The Luders bands were mainly concentrated on the half part of the sample near the end where the load was 252 applied; however, the strain was smaller on the other half part. This situation was not observed on the 253 As to the component , the flow channels were parallel to the Luders bands. However, the Luders bands 263 and the initial stress concentrators (i.e., nucleation sites) were located outside the flow channels. From this, we 264 inferred that they were of different origins. 265  Fig. 9). The temporal variations of the axial strain components 268 at these three points during the loading process were tracked and represented, as shown in Fig. 10. The curves 269 in Fig. 10 depict fluctuations of the vertical strain component during the loading process. Generally speaking, the deformation is greater near the end when the load was applied and smaller farther away from this end. At 271 30 mm, 0.3%<ε xx <0.35%; at 50mm, 0.35%<ε xx <0.4%; at 70mm, 0.4%<ε xx <0.48%. These three points were 272 close to the three flow channels generated during the loading process (Fig. 9). 273 Thus, the maximum deformation amplitudes of the three flow channels can be ranked as follows: 274 They are determined by their duration. The longer the time it appears, 275 the greater its deformation. As the width of the Luders bands increased, these three flow channels successively 276 entered active deformation regime. Therefore, it was presumed that the Luders bands' formation was related to 277 the flow channels' maximum strain value. As the deformation was intensified, the Luders bands interacted with 278 the flow channels nearby to generate the maximum strain values. 279 280

Propagation velocity analysis of the deformation waves at the elastic stage 281
Deformation waves are generated during the loading process of the red sandstone samples. Some researchers 282 have studied the propagation of deformation waves and estimated their propagation velocity range. However, 283 the relationships between the propagation velocities of the deformation waves and the loading rates have not 284 been fully clarified. In our study, such relationships were determined based on the experimental data. 285 Deformation waves has one unique feature: it propagates along the sample's axis. The loading diagram is 286 shown in Fig. 11. Since the samples were loaded at a steady loading rate, an approximate dependence ε~t was 287 assumed. The deformation waves' strain peaks (i.e., peaks and troughs of the deformation waves) were plotted 288 on the X-axis. The slope of the straight line representing the relationship of X vs. t was calculated. On this basis, 289 the motion velocities of points with the maximum and minimum values of strain, that is, the propagation 290 velocities of the deformation waves, were estimated. 291 Taking the case with a loading rate of 0.5 mm/min as an example. After the microscopic characterization of 293 the specimen, the deformation diagram of the strain component on the centerline at 8 time moments was 294 selected in the elastic stage (Fig.12). According to the treatment method of local strain maximization, the points 295 of local strain maximum are chosen for comparative analysis. The results are shown in Fig. 13. 296 After data processing, the deformation wave's propagation velocities were estimated as 5.4×10 -5 , 6.0×10 -5 , 299 and 6.3×10 -5 m/s in the three groups of experiments under the loading rate of 0.5 mm/min. After averaging, the 300 strain peak's propagation velocity (of deformation waves) was 5.9×10 -5 m/s in the samples. It exceeded the 301 axial boundary advance propagation velocity of 0.8×10 -6 m/s (equivalent to the loading rate of 0.5 mm/min) 302 by one order of magnitude. Therefore, the propagation velocities of the deformation waves was significantly 303 higher than that of the matter particles. 304 The propagation velocity of the slow deformation waves was calculated under the loading rate of 0.5mm/min. 305 Under this loading rate, stable regions of localized deformation were formed due to the slow deformation waves. 306 These regions did not move over time. It can be observed from the dashed box in Fig. 12 that at 60-100 mm on 307 the sample surface, the peaks and troughs' positions were relatively fixed, and the peak and the basic shape of 308 the wave only change slightly. However, the waveform and the peaks' positions changed significantly for the 309 regions formed at an earlier stage, indicating the normal propagation and fluctuations. This situation persisted 310 until the last diagram. The above region was stable and no longer experienced dramatic plastic deformation 311 over time. The propagation of slow deformation wave occurs at the initial stage of loading, that is, the elastic 312 stage and the compaction stage. At the initial stage, the damage of the sample develops uniformly, and the 313 slow deformation wave propagation is produced. As the loading process goes on, the plastic localization occurs 314 gradually, and the propagation of slow deformation wave no longer appears. At this stage, the plastic 315 deformation is concentrated to the adjacent weak points, thus showing the plastic localization, and then these 316 weak points are gradually connected, resulting in the fracture of a plane. 317 Using the same method, the propagation velocties of the deformation waves in the three groups of samples 318 under the loading rate of 0.1mm/min were estimated as 3.0×10 -5 , 3.3×10 -5 , and 3.5×10 -5 m/s, respectively. The 319 average wave propagation velocity was 3.27×10 -5 m/s. Under the loading rate of 1 mm/min, the propagation 320 velocities of the deformation waves in the three tested samples were 10.1×10 -5 , 10.5×10 -5 , and 10.7×10 -5 m/s, 321 respectively, and the average velocity was 10.4×10 -5 m/s. The wave propagation velocities under the three 322 loading rates were roughly equivalent to those of the plastic flow waves controlling earthquake migration in The deformation waves' propagation velocities obtained at the loading rates of 0.1, 0.5, and 1 mm/min by 325 testing nine samples are plotted in Fig.14. As the loading rate increases, the propagation velocities of the 326 deformation waves increases linearly. Thus, a positive linear correlation between these parameters with a high 327 correlation coefficient R 2 =0.9964 was revealed. The above dependence was best-fitted by a linear regression 328 formula presented in Fig.14. 329  the three loading rates, the deformation waves' propagation rates were 3.0×10 -5 , 5.9×10 -5 , and 10.4×10 -5 m/s, 335 respectively. They exceed by one order of magnitude the axial loading rates, i.e., the boundary advance speeds, 336 of 0.16×10 -6 , 0.8×0 -6 , and 1.6×10 -6 m/s, respectively, for the loading rates of 0.1, 0.5, and 1 mm/min. Thus, 337 the propagation rate of deformation waves is significantly higher than that of the mass particles. The local 338 thickening and uplift caused by boundary advancement continue to propagate with the help of elastic force and 339 inertial force. Thus, the matter particles' initial high displacement rate under the elastic force's action tended to 340 decrease due to viscous resistance. As the flow rate decreased, the subsequent substance deformation was 341 accumulated, leading to local bulging and increased elastic potential. This would further accelerate the matter 342 particles' displacement velocity. Such alternation of high and low velocities is manifested as wave propagation. 343 It can be considered that this is a kind of deformation wave that contains the migration of matter particles which 344 is essentially the particle momemntum waves proposed by Fitzgerald (1966), 345 The deformation wave propagation is a process of mutual conversion between the elastic and kinetic energies 346 under the boundary-driven condition and a process of adjustment for the imbalance between the driving force 347 and viscous force inherent in solids. The situation where the driving boundary induces the deformation wave 348 propagation is known as "boundary-driven wave generation" or "external-driven wave generation". Viscous (1) The deformation waves slowly propagated from one end of the sample, to which the load was applied. 360 The strain magnitude was much higher at one end to which the load was applied than that at the other one at 361 the beginning of loading. However, the strain values became comparable over time over the sample, thus 362 demonstrating deformation wave propagation in the laboratory tests of small-scale rock samples. 363 (2) In the samples under loading, the deformation waves are influenced by the elastic standing waves，which 364 could be inferred from the positions of the minimums and maximums of the elastic standing waves (regions of 365 stress concentration) and the flow channels' distributions. These flow channels were formed at the compaction 366 and elastic stages and were involved in the subsequent stages. As the stress increased, the LB nuclei interacted 367 with the flow channels nearby to generate the minimum and maximum strain values ( ). 368 (3) As the loading rate increased, the propagation velocity of the deformation waves increased linearly. Thus, 369 a positive linear correlation between these parameters with a high correlation coefficient R 2 =0.9964 was 370 revealed. 371

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Some or all data, models, or code that support the findings of this study are available from the corresponding 383 author upon reasonable request. 384 Figure 1 Propagation of slow waves originated from the Himalayan Arc in the East Asian continent (Wang 1987, Wang et al.1990) Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. This map has been provided by the authors.   The loading scheme of rock samples  Dependence of the propagation rate of the deformation waves on loading rate