Experimental Study on Rock Deformation Localization Using Digital Image Correlation and Acoustic Emission

: In this study, the digital image correlation (DIC) method and acoustic emission (AE) technology were combined to study the evolution of rock deformation localization in detail. The second-order spatial–temporal subset DIC (STS-DIC) algorithm was proposed and used for measuring strongly heterogeneous deformation ﬁ elds of red sandstone specimens under uniaxial compression. The evolution of the deformation ﬁ eld was analyzed with a focus on the deformation localization stage. The length and width of the deformation localization band (DLB) were measured, and the relationships between the relative sliding rate of the DLB, the relative opening rate of the DLB, and the AE counts were identi ﬁ ed. Deformation localization was found to result from the rapid evolution of the strain concentration before the peak stress. The complete development of the DLB is an inducing factor for catastrophic rock failure, and the failure modes of the rock specimens were consistent with the ﬁ nal state of the DLB. A good correlation was identi ﬁ ed between the AE counts and the relative displacement rate of the DLB, and the sliding rate was found to have a signi ﬁ cant in ﬂ u-ence on the AE counts.


Introduction
Rock is the most important construction medium of underground space engineering works such as mines, tunnels, and water conservancy.The deformation and failure mechanisms of rock affect the safety of such work [1,2].Rock is a natural and heterogeneous material that contains many primary micro-defects [3,4].When a rock is loaded, the defects in the rock expand, connect, and concentrate in one or several banded regions; this is called deformation localization [5].The evolution of the deformation localization has an important influence on the macroscopic mechanical behavior of rock [6,7].Therefore, studying the evolution of rock deformation localization in detail is important for understanding the mechanism of rock failure and assessing the safety of rock engineering.
The evolution of rock deformation localization has been considered in several studies [8][9][10][11].At present, the research work on the characteristics of rock deformation localization evolution has mainly been carried out by experimental means [12][13][14][15].Zuev et al. [16] showed that the deformation of some rocks under quasi-static compression remains localized throughout the loading process up to the sample's fracture.Munoz and Taheri [17] discussed the process of local damage and pointed out that large, irreversible deformation and deterioration of the material stiffness occur progressively, which lead to localized damage.Panteleev et al. [18] studied the spatiotemporal evolution of rock deformation localization under uniaxial tension.Dewers et al. [19] observed the deformation localization process of sandstone under uniaxial compression.However, most research on the characteristics of rock deformation localization has been focused on qualitative analysis.Few in-depth and quantitative studies have been conducted.Chajed and Singh [20] explored the possible application of the AE technique to identify the favorable or unfavorable state of rock salt and quantitatively analyzed the relationship between the AE signal and the fracture factors of rock salt.The results show that tensile microcracks release 85% of the AE energy.Hamadouche et al. [21] analyzed the degree of damage during the rock deformation process by using the deformation field value.
Digital image correlation (DIC) [22][23][24][25][26][27][28] and the acoustic emission (AE) technique [29][30][31][32][33][34][35] are widely used monitoring methods in rock mechanics experiments.In experimental research on rock deformation localization, accurate measurement of the deformation field in the deformation localization stage is required for in-depth study.DIC has the advantages of being a noncontact and full-field measurement method [36].However, the trade-offs between the subset size, shape function, and spatial resolution mean that the traditional subset-based DIC algorithm cannot measure the strongly heterogeneous deformation field of rock deformation localization with sufficient accuracy [37][38][39][40].In addition, the AE technique has been combined with DIC in rock mechanics experiments [41][42][43][44][45][46][47], but previous studies did not present a detailed analysis on the relationship between the deformation field evolution characteristics and the AE parameters.
In summary, experimental research on rock deformation localization has had the following shortcomings: for measurement, no suitable DIC algorithm is available for measuring the strongly heterogeneous deformation field that results from the deformation localization process; few quantitative analyses have been performed on the evolution of the rock deformation field during the deformation localization stage; and the correlations between the macro-mechanical behavior of rock, the evolution of the deformation field, and AE have not been studied in detail.Therefore, a quantitative and in-depth study on the evolution of rock deformation localization is needed.
In this study, the second-order STS-DIC algorithm was proposed for measuring strongly heterogeneous deformation fields.Compared to traditional DIC methods, the second-order STS-DIC algorithm considered the heterogeneousness of the deformation field and introduced time continuity constraints; its displacement function is more suitable for heterogeneous deformation field measurement.Uniaxial compression experiments were performed on red sandstone specimens, and the second-order DIC method and the AE technique were combined for measurements.The evolution characteristics of the deformation field during the rock deformation localization stage were analyzed quantitatively, and the correlations between the evolution characteristics of the deformation field and the AE parameters were identified.The purpose was to study the evolution characteristics of the rock deformation localization stage by analyzing the relationship between heterogeneous deformation field parameters and the AE parameters.

Brief Introduction to Second-Order STS-DIC
For the traditional subset-based DIC, as shown in Figure 1a, the reference image is first divided into a number of subsets.Each subset is tracked in the deformed image with the correlation-matching algorithm to determine its deformation [48].Let f represent the processing subset of the reference images and gt (t = t1, t2, …, tn) represent the subset of the deformed images captured at time t.Under the assumption of a constant intensity, there exists an equation for each subset pair f and gt as follows: where X is the coordinate of the pixels and U(X, t) is the shape function of subset gt relative to subset f.The shape function of the subset gt in the traditional, subset-based DIC can be written as follows: where u and v are the horizontal and vertical components of displacement U(X, t), Δx and Δy are the coordinate differences between the center point and any point in the subset, and a is the unknown matrix to be solved.
In the traditional subset-based DIC, the deformation mode of the subset (i.e., shape function) is generally assumed to be first-order [38,39,49].However, the complexity of the deformation field evolution in the deformation localization process of rock is mainly reflected by a heterogeneous distribution in space and a nonlinear change over time.The first-order shape function cannot accurately describe strongly heterogeneous deformation.In fact, the deformation of solid materials is continuous over time.If deformed speckle images are captured continuously within a short time interval during an experiment, then the characteristic of continuous deformation over time can be introduced into an image analysis [50].In this study, the subsets of the deformed image series captured in a short period were considered.According to the characteristics of rock deformation, a scheme termed second-order spatial-temporal subset (with the second-order shape function in spatial and a short period) was proposed, as shown in Figure 1b.The 2m + 1 (m = 1, 2, 3, …) deformed subsets of the image series captured in the period of [ti−m, ti+m] were averaged [50,51].This changes Equation (1) to where is the spatial-temporal subset, which can describe the displacement of the strongly heterogeneous deformation field well.The spatial-temporal subset in Equation ( 3) can improve the quality of the speckle images in certain circumstances as compared to the traditional spatial subset for the subset-based DIC [50].According to the characteristics of rock deformation, the displacement of the spatialtemporal subset can be suitably expressed by a second-order expression in spatial and a short period, as shown in Figure 1b.Thus, the shape function of the second-order spatialtemporal subset can be denoted as follows: where Δt = t − ti is the time interval between two images and b is the unknown matrix to be solved.
In practice, the number of pixels in the spatial-temporal subset is much larger than the number of unknowns b.Thus, Equation ( 3) is overdetermined and can be solved with an optimization method.The optimization objective function is given by: where C is the correlation coefficient and can be defined in different ways.The Newton-Raphson method was used to solve this optimisation problem iteratively: where b (l) is the solution to b in the l-th iteration.
Compared with the traditional subset-based DIC, the second-order STS-DIC algorithm is more suitable for measuring the strongly heterogeneous deformation field of rock during the deformation localization stage.In the following, the second-order STS-DIC was used to analyze the deformed images captured in the uniaxial compression experiments to obtain a more detailed deformation field.

Experiment Protocol
Red sandstone mainly presents two typical structural forms: granular debris structure and mud-like cementation structure.Due to differences in cementation materials and weathering degree, its strength varies greatly.The density of natural red sandstone is generally around 2.5 g/cm 3 , the Poisson's ratio is between 0.2 and 0.35, the elastic modulus is 5-20 GPa, and the compressive strength is around 15 MPa.According to the Chinese standard GB/T 23561.7-2009[52], the red sandstone was processed into five cuboid specimens with dimensions of 50 × 50 × 100 mm 3 .According to the standard, a specimen aspect ratio of 2.0 can effectively avoid the influence of end effects on the experimental results.Uniaxial compression tests were carried out on a test machine.The DIC and AE systems were combined to monitor the deformation information, as shown in Figure 2a.The experiments were carried out at a loading rate of 0.05 mm/min in order to load specimens under static.The specimen surfaces were painted with a random speckle pattern, and images of the speckle pattern were captured with a charge-coupled device (CCD) camera (Basler, Ahrensburg, Germany, acA1600-20gm) at a resolution of 1600 × 1200 pixels and a frame rate of 3 fps.The length-pixel ratio of the imaging system was 0.167 mm/pixel.Two lightemitting diodes (LEDs) were used as light sources to provide a stable light intensity field for the experiments.The AE system (PAC, Princeton Jct, NJ, USA, PCI-2) was used to monitor the AE signal of the specimens throughout the loading process.Six AE sensors (PAC, Princeton Jct, NJ, USA, R15A) were arranged on two sides of the specimen, as shown in Figure 2b, and Vaseline was used to enhance the coupling effect between the specimen and the AE sensors.The gain of the amplifier was set to 40 dB, and the sampling rate was set to 2.5 MHz.We used the pencil break method in the experiment and, based on the test results, the threshold was set to 50 dB.Speckle patterns were spray-painted on the specimen's surfaces, and the region of interest (ROI) was selected for the deformation field calculation, as shown in Figure 2b.After the AE sensors were arranged, the signal acquisition effect of each channel was tested to make sure that the AE system was operating normally.A preload of 0.5 kN was applied to clamp the specimen and compact the gap between the joints of the loading device.Then, the CCD camera was adjusted so that the camera-imaging plane was parallel to the specimen's surface, and the intensity of the light sources was adjusted to ensure that the image quality.After the experimental system was set up, the loading, image acquisition, and the AE systems were calibrated with regard to time to synchronize the three data acquisition systems.At the beginning of the experiment, the three experimental operators clicked the start button at the same time to trigger the loading system, DIC system, and the AE system.Each system collected data separately during the test until the specimen reached failure.Based on the above settings, a total of five groups of experiments were carried out.

Deformation Field Evolution
The image captured at the beginning of an experiment was taken as the reference image, and the other images captured during the experiment were the deformed images.The second-order STS-DIC algorithm was used to calculate the displacement fields, and then the strain fields (i.e., maximum shear strain fields) could be obtained.For the DIC process, the size of the spatial-temporal subset is 13 × 13 × 5 pixels.As shown in Figure 3, six points (A-F) were marked on the stress-time curve, and the corresponding strain fields were determined.Points A and B belonged to the initial loading stage, where the strain values were small, and the strain distribution was uniform.With increasing stress, some small strain concentration regions occurred in the strain field, as shown by point C, but the strain values were still small.At point D, an obvious deformation concentration region occurred in the lower part of the strain field, and this was defined as the starting point of the deformation localization.Point E corresponded to the first peak stress, and a distinct deformation concentration region with a band shape was formed that was called a deformation localization band (DLB).At the peak stress (point F), another DLB formed.After an abrupt drop in stress, the specimen failed because of the rapid growth of macro-cracks (point G).The deformation fields shown by the images at point F and point G indicate that the final failure of the rock specimen resulted from the evolution of the deformation localization.According to the above analysis, the evolution of the deformation localization plays a decisive role in rock deformation and failure.To analyze the evolution of the deformation field in the deformation localization stage in detail, the strain fields corresponding to points 1-6 in Figure 4 were selected, as shown in Figure 5. Point 1 had a stress of 18.44 MPa (84.9% of σc) and was after the start of the deformation localization.Several obvious deformation concentration areas were observed that were independent of each other (regions 1-3 in Figure 5a).Point 2 had a stress of 20.21 MPa (93.1% of σc); because of the further development of micro-cracks in region 2, some deformation concentration areas connected to form a DLB (Figure 5b).Point 3 had a stress of 20.94 MPa (96.5% of σc) and was after the first stress drop; because of the greater connection between regions 2 and 3, an obvious DLB formed throughout the specimen (Figure 5c) and was defined as DLB1.Meanwhile, region 1 did not continue to develop.This indicates that the development of the deformation concentration areas is unbalanced during the evolution of the deformation localization, and some areas may not develop into the final DLB.Point 4 was the minimum of the first stress drop; the strain in DLB1 continued to increase, and a new deformation concentration area (region 4) appeared in the upper right part of the specimen, as shown in Figure 5d.At Point 5, the stress began to increase again, and region 4 developed into a new DLB (Figure 5e) that was defined as DLB2.Point 6 had a stress of 21.28 MPa (98.1% of σc); the strain in DLB1 and length of DLB2 continued to increase, as shown in Figure 5f.In summary, when the DLB first started to form, its influence on the global mechanical properties of the rock specimen was small, and the rock specimen still bore the load as a whole.However, as the DLB evolved, the mechanical behavior of the rock specimen changed dramatically.The rock specimen showed obvious structural deformation characteristics, and instability failure occurred very quickly.

Evolution of the Deformation Localization Band
The length and width of the DLB were measured to quantitatively analyze its evolution.The correlation between the relative displacement rate on both sides of the DLB and the evolutions of its length and width was analyzed.Figure 6 shows the stress, length, and width of the DLB vs. time in the deformation localization stage.According to the analysis in Section 4.1, the deformation localization started when the loading time was 5.96 × 10 3 s.In stage I, the evolution of the DLB was mainly characterized by the growth of its length.
The length of the DLB at the end of stage I was about 45.46 mm, while the width was basically unchanged at about 2.53 mm.In stage II, both the width and length of the DLB did not increase.In stage III, the width of the DLB remained constant, but the length further increased to 100.12 mm.In stage IV, DLB1 was completely formed; the length of the DLB grew slowly to 108.93 mm, but the width increased rapidly to 5.22 mm.According to the evolution of the deformation field, the increases in the width and length of the DLB before stage V were mainly due to the formation of DLB1.In stage V, the appearance and evolution of DLB2 rapidly increased the length and width of the DLB to 169.43 mm and 7.16 mm, respectively.The substantial drop in stress indicates that the rapid development of the DLB reduced the load-bearing capacity of the specimen.After stage V, the length and width of the DLB continued to grow.At the peak stress, the length and width of the DLB reached 200.91 mm and 9.63 mm, respectively.Because of the damage accumulation, macro-cracks appeared at the position of the DLB, and the rock specimen rapidly failed after a short period of stress adjustment.
The rate of relative displacement on both sides of the DLB were calculated as follows [53,54]: First, the strain field and failure mode presented in Section 4.1 were used to determine the location of the DLB.Second, on both sides of the DLB, two spatial-temporal subsets with their connecting line perpendicular to the DLB were selected with a spatial subset size of 15 × 15 pixels and a temporal subset size of 5, as shown in Figure 7a.The displacements in the x and y directions of the center points P1 and P2 of the two spatialtemporal subsets were calculated, where P1 and P2 are a pair of measurement points.As shown in Figure 7a, five pairs of measurement points were arranged along DLB1 and DLB2, respectively.Third, the relative displacements ∆u and ∆v of each pair of measurement points were calculated.As shown in Figure 7b, both ∆u and ∆v were decomposed along the tangent and normal directions of the DLB, where the vector sums along the tangent and normal directions of the DLB were the relative sliding displacement and relative opening displacement, respectively.Finally, the average value of the five pairs of measurement points was taken as the relative displacement of each DLB, and the sliding rate and opening rate of each DLB were obtained by differentiating the relative displacements with respect to time.According to a preliminary analysis, DLB1 was dominated by sliding, while DLB2 was dominated by opening, as shown in Figure 7c.Therefore, only the sliding rate of DLB1 and the opening rate of DLB2 are presented in this paper.To be clear, the length value is the sum of the lengths of all the DLB, and the width value is the maximum width of the DLB.After the initiation of the deformation localization (5.96 × 10 3 s), although the length of the DLB increased, the relative displacement rate and width did not increase.In the short period before 6.10 × 10 3 s, the length, width, and relative displacement rate of the DLB remained almost constant.Afterward, the sliding rate of DLB1 began to increase, and the length of the DLB began to increase correspondingly, but the width did not increase.During the loading period from 6.10 × 10 3 s to 6.21 × 10 3 s (interval marked by arrows), the sliding rate increased rapidly.However, because DLB1 had been basically formed, the length of the DLB increased slowly, and the increase in the sliding rate caused the width of the DLB to rapidly increase.This indicates that the increase in the sliding rate contributed greatly to the increase in the width of the DLB, which was formed by shear deformation.During the loading period from 6.21 × 10 3 s to 6.24 × 10 3 s, the opening rate began to increase.DLB2 began to develop, and the length and width of the DLB increased rapidly.After the loading time of 6.24 × 10 3 s, the sliding rate became nearly 0, and the opening rate decreased gradually.However, the length and width of the DLB continued to increase.This indicates that the opening rate contributed greatly to the increases in the length and width of DLB2.

Correlation between the Acoustic Emission and Deformation Characteristics
AE is an important physical index that reflects the evolution of rock deformation.To study the characteristics of AE and rock deformation evolution during the deformation localization stage, the relationships between the relative displacement rate of the DLB and the AE counts were analyzed.It should be clarified that the AE data we used are based on the acoustic signals of sensor 2, as this sensor records a relatively rich amount of data throughout the entire loading process.For one sensor's data, one signal source corresponds to one event [55], and the number of events can be determined based on the density of the curve.The AE counts can reflect the intensity of the AE signal source, so we combined the AE counts with deformation field data for analysis to explore the evolution mechanism of rock deformation localization.
Figure 9 plots the curves of the AE counts, the relative displacement rate, and stress vs. time.At 5.96 × 10 3 s, the deformation localization was initiated, and the AE counts increased from 6.10 × 10 3 s to 6.21 × 10 3 s.The sliding rate of DLB1 increased gradually, and the AE counts increased with the sliding rate.From 6.21 × 10 3 s to 6.24 × 10 3 s, which corresponded to the first stress drop, the sliding rate was large.For each moment with a sudden increase in the sliding rate, a sharp increase in the AE counts was also observed during this period.At 6.24 × 10 3 s, both the sliding rate and opening rate increased sharply, and the AE counts were also high.After the loading time of 6.24 × 10 3 s, the sliding rate became 0, and the opening rate was about 1.5 × 10 −3 mm/s; the single AE signal had a high counts value.At 6.33 × 10 3 s, the sliding rate and opening rate increased suddenly, which corresponded to the post-peak stress drop.At this moment, the AE counts for a single AE signal reached its maximum.The above correlations among the different characteristics indicate that the sliding rate and opening rate greatly influenced the AE counts.

Conclusions
In this study, DIC and AE were combined to obtain the deformation information of red sandstone under uniaxial compression.The second-order STS-DIC algorithm was developed for measuring strongly heterogeneous deformation fields and was used to calculate the deformation field during the deformation localization stage of the rock specimens.The experimental results for the deformation field and AE were used to analyze the evolution of the rock deformation localization.
First, the evolution of the deformation field during rock deformation localization was analyzed in detail by using second-order STS-DIC and AE.In the deformation localization process, many small deformation concentration regions were observed; some eventually connected to form DLBs, but others stopped developing.The evolution of the DLB was crucial to the rock's mechanical behavior.The DLB fully developed before the peak stress was reached.After the peak stress, macro-cracks appeared at the DLB, which led to rock instability and failure.The failure mode of the rock specimen was consistent with the final state of the DLB.Second, the evolution of the DLB was analyzed.When the deformation localization was initiated, the length of the DLB increased continuously, but the width increased only when the length reached a certain value.The sliding rate had a significant effect on the length of the DLB formed by shear deformation.However, the opening rate had a greater influence on the length and width of the DLB formed by tensile deformation.The sliding rate and opening had a significant influence on the AE counts.
This study mainly focused on the deformation localization of rock and had several deficiencies.For the analysis on rock deformation characteristics, the relationship between the initiation of the deformation localization and the stress level (i.e., the competition mechanism among DLBs) needs to be studied in more detail.The characteristics of instability and failure also need to be studied in more detail.To realize instability failure analysis, highly accurate DIC algorithms need to be developed for measuring discontinuous deformation fields.Higher-performance measurement systems need to be developed, such as an adaptive image acquisition system for different deformation states (e.g., lowfrequency acquisition during early loading, medium-frequency acquisition during the deformation localization, and high-frequency acquisition during instability).

Figure 2 .
Figure 2. Experiment setup: (a) schematic of the experiment system and (b) specimen and AE sensors, (1-6) are the number of sensors.

Figure 3 .
Figure 3. Stress-time curve and deformation field for the entire loading process, (A-G) are the marker points on the stress-time curve.

Figure 4 .
Figure 4. Marked points at different stress levels during the deformation localization stage.

Figure 5 .
Figure 5. Strain fields corresponding to the marked points in Figure4: (a) corresponding to point 1, (b) corresponding to point 2, (c) corresponding to point 3, (d) corresponding to point 3, (e) corresponding to point 5, (f) corresponding to point 6, (1-4) are the marker of deformation concentration area.

Figure 6 .
Figure 6.Length and width of the DLB: (I-V) are the marker of deformation localization stages.

Figure 7 .
Figure 7. Principle of calculating the relative displacement: (a) layout of measurement points, (b) decomposition of the displacements, and (c) schematic of the dominate deformation forms.

Figure 8
Figure 8  plots the curves of the normalized length and width of the DLB vs. time and the relative displacement rate vs. time in the deformation localization stage for comparison.To be clear, the length value is the sum of the lengths of all the DLB, and the width value is the maximum width of the DLB.After the initiation of the deformation localization (5.96 × 10 3 s), although the length of the DLB increased, the relative displacement rate and width did not increase.In the short period before 6.10 × 10 3 s, the length, width, and relative displacement rate of the DLB remained almost constant.Afterward, the sliding rate of DLB1 began to increase, and the length of the DLB began to increase correspondingly, but the width did not increase.During the loading period from 6.10 × 10 3 s to 6.21 × 10 3 s (interval marked by arrows), the sliding rate increased rapidly.However, because DLB1 had been basically formed, the length of the DLB increased slowly, and the increase in the sliding rate caused the width of the DLB to rapidly increase.This indicates that the increase in the sliding rate contributed greatly to the increase in the width of the DLB, which was formed by shear deformation.During the loading period from 6.21 × 10 3 s to 6.24 × 10 3 s, the opening rate began to increase.DLB2 began to develop, and the length and width of the DLB increased rapidly.After the loading time of 6.24 × 10 3 s, the sliding rate became nearly 0, and the opening rate decreased gradually.However, the length and width of the DLB continued to increase.This indicates that the opening rate contributed greatly to the increases in the length and width of DLB2.

Figure 8 .
Figure 8. Correlation between the relative displacement rate and the length or width of the DLB.

Figure 9 .
Figure 9. Correlation between the relative displacement rate and the AE counts.