Method for identifying the position of crack tip by displacement distribution based on digital image correlation

Determining the position of crack tip is crucial for rock fracture characteristic analysis. This study aims to establish a quantitative standard for determining crack tip positions in crystalline rocks based on digital image correlation (DIC). Semi-circular bending (SCB) tests were conducted on four types of rocks under three-point bending. Optical microscopy and DIC were used to investigate characteristics of the horizontal displacement fields around the crack tips. A significant disparity in the displacement gradient was observed across the crack tips, with the internal gradient being approximately 3–5.1 times higher than the external gradient. A critical value ranging from 0.41%–0.64% was determined for the high internal displacement gradient (∂u/∂x) at a 20 pixel resolution equivalent to 1 mm. Furthermore, a novel DIC-based method was proposed that used three patterns of opening displacement gradients to classify SCB specimens into non-cohesive, cohesive, and elastic zones, providing a quantitative determination of Type I crack tip positions. This study offers valuable insights and a crack tip determination method based on DIC gradients, applicable for assessing parameters such as the crack extension length and fracture process zone length.


Introduction
Internal flaws, such as cracks and pores in rocks, are key factors that influence the deformation of rock masses.Under external loads, natural fractures develop, extend, and interconnect, eventually leading to instability and failure of engineering structures.Therefore, the issue of rock mass damage and fracturing has become a research topic in rock mechanics.
In recent years several researchers have investigated rock damage and fracturing issues using experimental mechanics methods [1][2].Among these methods, digital image correlation (DIC) has gained widespread application in rock fracture experiments owing to its full-field, noncontact nature, high automation, high precision, low environmental requirements [3][4], and adjustable scale and precision.By capturing the displacement field around a crack tip in rocks, DIC enables dynamic observations.Zhao et al. [5][6]  Accurate measurements of stress intensity factors, which are essential for assessing rock fracture toughness, tracking the expansion of structural planes, and evaluating the stability of engineering rock masses, have been conducted using DIC.Dai et al. [7][8][9] determined the stress intensity factors and crack tip positions for Type I and I-II composite cracks in metals and rock materials using DIC technology.
The measurement principle of the stress intensity factor is based on fitting the displacement field function at the crack tip.Therefore, accurately identifying the crack tip position is a key issue when measuring the stress intensity factor.Lin et al. [10] determined from the DIC displacement data that the internal gradient value in a small area near the crack tip in sandstone was approximately 4-5 times that of the external gradient because of the stress concentration at the notch.Ji et al. [11] discovered that the intersection point of the horizontal displacement field contour lines in marble and yellow sandstone analyzed using DIC technology is the tip of the fracture process zone.Zhang et al. [12] used the DIC method to analyze the variation of the displacement field at the crack tip of semi-circular bent sandstone specimens of different sizes, and discovered that the displacement gradient at the middle position of the horizontal displacement curve at the notch tip during peak times is 4.4-5.7 times the critical horizontal displacement gradient.However, despite the use of the DIC method to determine fracture mechanics parameters at the crack tip, research on methods for determining the crack tip is limited, and there is no established quantitative technique to determine the crack tip position.
Therefore, this study conducted semi-circular disc three-point bending tests on four types of crystalline rocks.The position of the crack tip was determined using an optical microscope, followed by a systematic analysis and summary of the displacement field distribution characteristics at this position.Subsequently, a quantitative standard for determining the rock crack tip position based on DIC was proposed.The new method used in this study can improve the accuracy of measurements such as the calculation of stress intensity factors, crack propagation length, and fracture process zone length in rock fracture indicators.The established indicators provide a reference for the quantitative determination of the crack tip position.

Specimen preparation and methods
This study selected four types of rocks with different crystal structures (three types of monzogranite and one type of diabase), numbered X, L, M, and S respectively Based on the observation results under an optical microscope and the X-ray diffraction (XRD) test results, it was found that all four samples contained four minerals: quartz (white), K-feldspar (reddish-brown), albite (gray), and biotite (black).Nevertheless, noticeable variations were identified among the specimens in terms of the color, morphology, content, and arrangement of the mineral grains.
The four types of crystalline rock samples were processed into two groups of semi-circular bending (SCB) specimens with preexisting central cracks in Groups A and B. Each specimen had a diameter (D) of 100 mm, thickness (B) of 20 mm, and support span (S) of 80 mm.The SCB specimens were cut using a wire cutting machine, and the crack inclination angle was set to 0 °for each group.The notch length (a) was 15 mm and the notch width (b) was 1 mm.The specimens were subjected to a three-point bending loading, as illustrated in figure 1.The loading device used in the test was an MTS815.04high-rigidity rock mechanics testing system at Shaoxing University, as shown in figure 2. A three-point bending loading method was used to load the SCB specimens.The crack mouth opening displacement (CMOD) was measured using an extensometer and servo-controlled to increase at a speed of 0.02 mm/min until the Load-CMOD curve approached a horizontal trend, at which point the loading was halted.For real-time monitoring of the surface speckle field changes during loading, the recording system utilized an i-speed 716 high-speed camera from Tongji University's Key Laboratory of Geotechnical and Underground Engineering.The camera had a resolution of 2048 × 1536 pixels and operated at a capture rate of 10 fps.Two 600 W red-headed halogen lamps were placed on either side of the camera to provide stable lighting during the experiment, facilitating the real-time capture of speckle patterns on the surface of the specimen.The analysis system used proprietary software developed by Professor Zhao Cheng's team at Tongji University.Digital images obtained from the recording system were processed using a computer analysis system to generate displacement-strain field evolution diagrams.In this study, the application steps of the analysis software were as follows: First set the (1) initial image, (2) calculation images, (3) region of interest (ROI), and (4) calculation parameters.Then perform the (5) DIC calculations, (6) displacement formatting, and (7) strain calculations.Based on the impact of parameters such as subregion size on the measurement accuracy and computational workload, the following settings were set during the post-processing of the DIC data analysis software, including parameter configuration and data output: • An image collected before loading was selected as the initial image, and an image collected at the end of the test was selected as the calculated image to reduce the computational workload.• Select a region of interest (ROI) covering the area from the crack tip to the upper loading point, ensuring a sufficient width to cover the entire crack propagation path, thus reducing unnecessary computational workload.• The coordinate system was established at the top-left corner of the images captured by the camera, with one calculation point for every 1 pixel, where the length of each pixel was set to 1 pixel = 0.05 mm.• In the calculation parameter settings, the subdomain radius was set to R = 15 pixels, the calculation point interval was 0 pixels, and the number of calculation threads was ten.Select subset segmentation for non-continuous analysis and crack identification, set the threshold G to 0.3, and keep the other parameters at their default values in most cases.

Observation of crack tip position of specimen based on DIC
In this study, optical microscopy was employed to observe the surfaces of rock specimens.Microscopic images of the mineral grain morphology and crack propagation paths on the specimen surface were captured after the completion of loading.The optical microscope used in this study was a TD-10 L-4KC-15.6,featuring an optical size of 1/1.7", a resolution of 3840 × 2160 pixels, and provides a clear visualization of the grain morphology.An optical microscope is equipped with practical functions such as measurement, crosshair, scale, and screenshot capabilities, enhancing operational convenience and efficiency.Owing to limitations in the shooting range, multiple photographs were captured along the crack propagation path to document the details of the crack propagation trajectory comprehensively.The captured photographs were imported into Photoshop software for post-processing.By leveraging the powerful photomerge functions of Photoshop, multiple photographs were automatically stitched together to form a complete image, thereby facilitating a comprehensive view of the crack propagation path.

Crack tip displacement distribution characteristics of specimen
Because the crack tip position is known, when combined with the DIC analysis results at the final loading stage, the distribution characteristics of the DIC displacement field at the crack tip can be identified.Considering the S-A-0 specimen as an example, a series of horizontally equidistant sampling lines were arranged above and below the crack tip coordinates of Y = 521 pixels (figure 5).The spacing between the sampling lines was 20 pixels with a length of 100 pixels.Horizontal displacement distribution maps were generated for different sampling lines near the crack tip at the peak load (figure 6) to analyze the displacement distribution characteristics near the crack tip of the specimen.Figure 6.Horizontal displacement curves along the horizontal reference lines.As shown in figure 6, as the sampling lines approached the pre-existing crack, the displacement jump points A and B at the ends of the fracture process zone (FPZ) became more distinguishable, and it became more challenging to identify the abrupt displacement change point near the elastic zone.From the graph, it is evident that lines L1 (Y = 501) and L2 (Y = 521) approximately overlap.The L1 line exhibits no significant displacement jumps and displays continuous displacement changes.Conversely, within the range 1020 < x < 1060, the L2 line shows noticeable discontinuity in displacement, while the displacement curves at both ends outside this range are relatively smooth, and the displacement gradient u / x tends toward 0. According to the fracture mechanics theory, the region within L2 represents the process zone of rock failure.It clearly distinguishes the continuous and discontinuous horizontal displacements.This finding is consistent with microscopic observations at the tip of the fracture process zone.Calculation shows that the critical value of the internal high-displacement gradient u / x within the fracture process zone near the crack tip is approximately 0.53%, whereas the external displacement gradient is approximately 0.12%.

Displacement gradient distribution at the crack tip
Based on the crack tip displacement distribution characteristics of S-A-0 specimen, the crack tip position can be quantitatively described by the gradients.Similarly, horizontal displacement distribution maps along the sampling lines near the crack tips were generated for the other specimens after the completion of loading (figure 7).The displacement gradient distribution relationships around the crack tip were then calculated, revealing the following patterns: • For the S-B-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.51%, while the external displacement gradient is 0.1%.• For the M-A-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.44%, while the external displacement gradient is 0.14%.For the M-B-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.64%, while the external displacement gradient is 0.18%.• For the L-A-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.58%, while the external displacement gradient is 0.12%.For the L-B-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.41%, while the external displacement gradient is 0.1%.• For the X-A-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.45%, while the external displacement gradient is 0.09%.For the X-B-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.42%, while the external displacement gradient is 0.14%.The findings can be summarizes as follows: the difference in the displacement gradient between the inside and outside of the fracture process zone near the crack tip for different specimens is approximately 3-5.1 times.The critical value of the high displacement gradient u/ x inside the tip ranges from 0.41% to 0.64%.The precision was 20 pixels, with 1 pixel representing 0.05 mm.

Discussion
This study investigated the quantified gradient relationships of the DIC horizontal displacement field distributions around the crack tips of eight SCB specimens fabricated from four different crystalline rock materials.The study discovered that by analyzing the variation in displacement gradients around the crack tip, it is feasible to detect whether the sampling line is located within the fracture process zone and to understand its boundary conditions.This study summarized three displacement distribution patterns to distinguish between specimens with no cohesive cracks, cohesive zones, and elastic zones (figure 8).Consequently, a new method based on DIC was proposed to quantify the crack position.Among them, the displacement distribution pattern corresponds to specimens with no cohesive crack.The criterion for judgment is that outside the high-displacement-gradient region, the material cannot transmit stress, and the displacement gradient is approximately 0, whereas the internal gradient is greater than 0.64%, leading to a jump in the displacement.Displacement distribution pattern corresponded to specimens with a cohesive zone.According to the cohesive crack model, the material outside the high-displacement-gradient region experiences a certain tensile stress, generating a positive displacement gradient.The judgment criteria are that the internal gradient is approximately 3-5.1 times the external gradient, and the critical value of the high internal displacement gradient u/ x at the tip is between 0.41% and 0.64%.The displacement distribution pattern corresponded to specimens with an elastic zone.The displacement change tended to be continuous, and jump points A and B were not easily identifiable.The internal gradient value is less than 3 times the external gradient value, and the critical value of the high internal displacement gradient u/ x at the tip is less than 0.41%.The crack tip position was the point where the displacement changed from a sudden jump (Type displacement distribution pattern) to a gradual change (Type displacement distribution pattern), and the top of the fracture process zone was the point where the displacement on both sides became smooth (Type displacement distribution pattern).It is noteworthy that the displacement distribution pattern is influenced by the DIC analysis parameters, such as the subset radius and calculation point spacing.The gradient judgment criteria's 0.41%-0.64%,applicability to other rock materials and DIC analysis parameter results requires further verification.Nevertheless, the research approach and method for determining the quantitative criteria presented in this study have a major reference value for locating crack tips based on DIC.

Conclusion
In this study, we used optical microscopy and DIC methods to investigate the distribution characteristics of the horizontal displacement field around the crack tips in SCB specimens of four different crystalline rock materials.Our findings led to two key conclusions: • A significant disparity was observed in the displacement gradient across the crack tip of the SCB specimens, with the gradient inside the tip being approximately 3-5.1 times higher than that outside.The critical value of this high internal displacement gradient ( u/ x) was determined to be in the range of 0.41%-0.64%measured with a precision of 20 pixels, which correspond to 1 mm.• A novel DIC-based method is proposed for the quantitative determination of Type I crack tip positions that use three distinct patterns of opening horizontal displacement gradients to effectively classify SCB specimens into non-cohesive, cohesive, and elastic crack zones.
This study offers valuable insights and presents a method for quantitatively determining crack tips using DIC, which can be applied to the quantitative assessment of parameters such as the crack extension length and fracture process zone length, providing both theoretical and practical implications.However, the primary focus of this study was heterogeneous crystalline rock materials, and its applicability to other rocks requires further verification.Expanding the investigation to include different material characteristics will contribute to the comprehensive understanding of the mechanisms of the horizontal displacement fields at the crack tips enabling the assessment of the applicability and accuracy of the DIC method.Furthermore, future investigations should delve into the impact of DIC analysis parameters, such as the subset radius and point spacing, on the distribution patterns of the horizontal displacement fields at the crack tips.This provides additional validation of the suitability of the proposed crack-tip judgment criteria, with gradients ranging from 0.41% to 0.64%.

Figure 2 .
Figure 2. Mechanical testing system and digital image recording system.In this study, the application steps of the analysis software were as follows: First set the (1) initial image, (2) calculation images, (3) region of interest (ROI), and (4) calculation parameters.Then perform the (5) DIC calculations, (6) displacement formatting, and (7) strain calculations.Based on the impact of parameters such as subregion size on the measurement accuracy and computational workload, the following settings were set during the post-processing of the DIC data analysis software, including parameter configuration and data output:• An image collected before loading was selected as the initial image, and an image collected at the end of the test was selected as the calculated image to reduce the computational workload.• Select a region of interest (ROI) covering the area from the crack tip to the upper loading point, ensuring a sufficient width to cover the entire crack propagation path, thus reducing unnecessary computational workload.• The coordinate system was established at the top-left corner of the images captured by the camera, with one calculation point for every 1 pixel, where the length of each pixel was set to 1 pixel = 0.05 mm.• In the calculation parameter settings, the subdomain radius was set to R = 15 pixels, the calculation point interval was 0 pixels, and the number of calculation threads was ten.Select subset segmentation for non-continuous analysis and crack identification, set the threshold G to 0.3, and keep the other parameters at their default values in most cases.

Figure 3 .
Figure3.The crack propagation path of the specimen (considering the S-A-0 specimen as an example) By observing the crack propagation path in the microscopic photographs, the position of the crack tip of the specimen can be determined, as indicated by the arrows in figure3.Considering the SA-0 specimen as an example, DIC was used to analyze the displacement field of the SCB specimen at the peak load (figure4).Based on the actual image of the microscopic observation and a comparison of the horizontal displacement field and horizontal strain field, the position of the crack tip was identified at Y = 521 pixels, representing the vertical distance from the top-left corner of the camera-captured image to the crack tip.Similarly, the Y-coordinates for the crack tips of the other specimens were determined as follows: X-A-0 (495), X-B-0 (474), L-A-0 (496), L-B-0 (513), M-A-0 (484), and M-B-0 (565), and S-B-0 (476) pixels.

Figure 4 .
Figure 4. Microscopic images of the crack tip analysis region and DIC horizontal displacement field cloud map.

Figure 5 .
Figure 5. Layout of the sampling lines.Figure6.Horizontal displacement curves along the horizontal reference lines.As shown in figure6, as the sampling lines approached the pre-existing crack, the displacement jump points A and B at the ends of the fracture process zone (FPZ) became more distinguishable, and it became more challenging to identify the abrupt displacement change point near the elastic zone.From the graph, it is evident that lines L1 (Y = 501) and L2 (Y = 521) approximately overlap.The L1 line exhibits no significant displacement jumps and displays continuous displacement changes.Conversely, within the range 1020 < x < 1060, the L2 line shows noticeable discontinuity in displacement, while the displacement curves at both ends outside this range are relatively smooth, and the displacement gradient u / x tends toward 0. According to the fracture mechanics theory, the region within L2 represents the process zone of rock failure.It clearly distinguishes the continuous and discontinuous horizontal displacements.This finding is consistent with microscopic observations at the tip of the fracture process zone.Calculation shows that the critical value of the internal high-displacement gradient u / x within the fracture process zone near the crack tip is approximately 0.53%, whereas the external displacement gradient is approximately 0.12%.

6 Figure 7 .
Figure 7. Horizontal displacement distribution maps along the sampling lines near the crack tips of each specimen.•For the S-B-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.51%, while the external displacement gradient is 0.1%.• For the M-A-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.44%, while the external displacement gradient is 0.14%.For the M-B-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.64%, while the external displacement gradient is 0.18%.• For the L-A-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.58%, while the external displacement gradient is 0.12%.For the L-B-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.41%, while the external displacement gradient is 0.1%.• For the X-A-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.45%, while the external displacement gradient is 0.09%.For the X-B-0 specimen, the critical value of the high displacement gradient u/ x inside the crack tip is 0.42%, while the external displacement gradient is 0.14%.The findings can be summarizes as follows: the difference in the displacement gradient between the inside and outside of the fracture process zone near the crack tip for different specimens is approximately 3-5.1 times.The critical value of the high displacement gradient u/ x inside the tip ranges from 0.41% to 0.64%.The precision was 20 pixels, with 1 pixel representing 0.05 mm.

Figure 8 .
Figure 8. Strain localization band and displacement distribution patterns.Among them, the displacement distribution pattern corresponds to specimens with no cohesive crack.The criterion for judgment is that outside the high-displacement-gradient region, the material cannot transmit stress, and the displacement gradient is approximately 0, whereas the internal gradient is greater than 0.64%, leading to a jump in the displacement.Displacement distribution pattern corresponded to specimens with a cohesive zone.According to the cohesive crack model, the material outside the high-displacement-gradient region experiences a certain tensile stress, generating a positive displacement gradient.The judgment criteria are that the internal gradient is approximately 3-5.1 times the external gradient, and the critical value of the high internal displacement gradient u/ x at the tip is between 0.41% and 0.64%.The displacement distribution pattern corresponded to specimens with an elastic zone.The displacement change tended to be continuous, and jump points A and B were not easily identifiable.The internal gradient value is less than 3 times the external gradient value, and the critical value of the high internal displacement gradient u/ x at the tip is less than 0.41%.The crack tip position was the point where the displacement changed from a sudden jump (Type displacement distribution pattern) to a gradual change (Type displacement distribution pattern), and the top of the fracture process zone was the point where the displacement on both sides became smooth (Type displacement distribution pattern).It is noteworthy that the displacement distribution pattern is influenced by the DIC analysis parameters, such as the subset radius and calculation point spacing.The gradient judgment criteria's 0.41%-0.64%,applicability to other rock materials and DIC analysis parameter results requires further verification.Nevertheless, the research approach and method for determining the quantitative criteria presented in this study have a major reference value for locating crack tips based on DIC.
used DIC technology to conduct macro-and micro-multiscale studies on the crack propagation process in quasi-rock brittle materials under uniaxial compression, obtaining