Deformation and Fracturing Behaviors of Sandstone Disc Containing a Circular Inclusion under Diametrical Compression: Insights from Integrated SG and DIC Experiments

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Introduction
Rock mass, as a typical geological material, exhibits signifcant nonlinearity and heterogeneity due to inherent defects or discontinuities in various forms, such as pores, fssures, joints, faults, and shear bands [1][2][3][4]. Tese discontinuities can lead to excessive stress concentrations and uncoordinated deformations in rock masses when subjected to external loads, resulting in crack initiation and propagation along with irreversible damage or even structural failure of the rock [5][6][7][8][9]. Up to date, numerous studies have been conducted to explore the cracking mechanism of rock or rock-like materials containing preexisting faws under various compression conditions [10][11][12][13][14][15]. However, most of these studies focused on artifcial faws with lengths of 15-20 mm and small widths of 1-2 mm, while the presence of macroporosity or open holes that also have detrimental efects on the mechanical properties of rock were rarely considered [16]. It has been observed that rocks such as lithophytic tuf, vesicular basalts [17], and sandstone [18] commonly contain holes with much larger sizes than the artifcially created faws. Te number, shape, size, and distribution of these open defects have been found to signifcantly infuence the mechanical responses of rocks, as demonstrated by experimental and numerical studies [19][20][21][22][23]. To this end, investigations have been carried out to study the fracturing behavior of rock specimens with one or multiple holes under tension or indirect tensile conditions [24][25][26][27][28]. Previous studies primarily focused on the mechanical properties and cracking characteristics of rocks with open faws or holes, which are crucial for in situ rock engineering applications.
In practical scenarios, however, the fssures or joints within rock masses are often flled with weak interlayers such as clay, sand, and gypsum [29][30][31][32]. Tese inclusions can occur naturally as a result of physical/chemical weathering or joint shearing actions [33] (see Figures 1(a) and 1(b)) or can be manually flled through grouting/shotcreting during the tunnel construction [29,[34][35][36] (see Figure 1(c)). Te fllings in the holes or fssures introduce more complex stress distributions within the rock mass, leading to diferent failure mechanisms compared to rocks with open faws [37]. Terefore, investigating the mechanical properties and fracturing evolution of rocks with flled faws holds great engineering signifcance.
In this regard, extensive investigations have been conducted on rock specimens with flling materials, as depicted in Figure 2 [38][39][40][41][42][43][44][45]. Under the direct shear conditions, Tian et al. [46] conducted experimental and numerical studies on the shear behavior of cemented concrete-rock joints. Lu et al. [47] investigated the shear deformation and strength behavior of cement-grout-flled joints. Salimian et al. [48] found that grouting positively afected the shear strength of cement-flled joints. In terms of specimens subjected to compression-shear conditions, Butron et al. [49] studied the mechanical behavior of silica sol-grouted hard rocks under unconfned compression and triaxial conditions. Miao et al. [50] performed unconfned compression tests on fawed sandstone specimens, considering factors such as inclination angles and inflling materials (e.g., gypsum, cement, and resin). Du et al. [51] investigated the strength properties and failure patterns of plate-shaped sandstones with two circular inclusions under uniaxial compression. Zhu et al. [39] studied the deformation and cracking processes of sandstones containing various inclusions using digital image correlation (DIC) and acoustic emission (AE) techniques, considering the infuences of hole shape on strength and deformation properties. It was observed that inclusions alter crack propagation behaviors and reduce stress concentration intensity at the crack tip zone. However, most existing studies on inflled rock specimens have focused on uniaxial/ biaxial compression or shear loading, with limited investigations into the mechanical responses and failure behaviors under tensile loading. To explore the fracturing evolution of specimens under tensile stress, Chang et al. [52] conducted numerical studies on the mechanical behaviors of cement mortar specimens with circular inclusions under diametrical compression. Sharafsafa et al. [33] performed diametrical compression on 3D printed rock-like discs to investigate the efect of inflling materials on the cracking behavior. Nevertheless, the understanding of the deformation and cracking processes of inflled disc specimens under diametrical compression is still not fully elucidated.
In this study, red sandstone and grey sandstone discs with circular inclusions were prepared and submitted to diametrical compression tests. Te inclusions had diferent strengths/stifness and sizes. Te deformation characteristics, mechanical properties, and cracking evolution were systematically explored using SG measurement and the DIC technique. Te infuences of the inclusion strength and diameter on local strain concentration and cracking behavior were deeply analyzed.

Materials and Methodology
2.1. Specimen Preparation. Two distinct types of sandstones, distinguished by their colors and geological textures, were selected for experimentation in this study. Te red sandstones were sourced from Liuyang City, Hunan Province, while the grey sandstones were extracted from an open-pit quarry in Miluo City, Hunan Province. To minimize artifcial damage to the rock specimens, both types of sandstones were obtained from the bedrock using mechanical cutting methods. Brazilian discs with dimensions of 50 mm in diameter and 25 mm in thickness were drilled from the rock blocks. A circular hole was then created at the center of each disc specimen. Diferent hole diameters, including 10 mm, 15 mm, and 20 mm, were considered to investigate the efect of hole size on cracking behaviors. Similar hole diameters were also adopted in previous studies, such as Chang et al. [52] and Wu et al. [22].
To fll the central holes, cement mortars with two different composite ratios were prepared. One composite ratio consisted of cement, sand, and water in a mass ratio of 1: 5.27: 1.16, generating a low-strength cement mortar material (Type I). Te other composite ratio consisted of cement, sand, and water in a mass ratio of 1: 3.47: 0.64, resulting in a high-strength cement mortar material (Type II). Te specimens were prepared in the same batch to ensure consistent interfacial bonding strength among them. In addition, the inflled specimens were cured under standard temperature and humidity conditions for 28 days. Following these curing conditions, the cement mortar exhibits optimal mechanical performance and forms satisfactory adhesions with the prefabricated hole walls.
To determine the basic mechanical properties of the materials, solid specimens of red sandstone, grey sandstone, Type I cement mortar, and Type II cement mortar were subjected to standard unconfned compression tests and Brazilian disc tests. Te stress-strain curves obtained from the unconfned compression tests and the load-displacement curves obtained from the Brazilian disc tests are presented in Figure 3. Table 1 summarizes the basic physical and mechanical parameters of the rocks and cement mortars.
Te red sandstone exhibits a P-wave velocity of 2799 m/s, which is 20.8% higher than that of the grey sandstone. Accordingly, the unconfned compressive strength (UCS), Young's modulus, and tensile strength of the red sandstone 2 Shock and Vibration   Shock and Vibration 3 are 2.5 times, 1.4 times, and 3.1 times larger than those of the grey sandstone, respectively, indicating distinct mechanical behaviors between the two types of sandstones. Regarding the cement mortars, CM-II (high-strength cement mortar) demonstrates signifcantly higher mechanical properties compared to CM-I (low-strength cement mortar). Specifcally, the UCS, Young's modulus, and tensile strength of CM-II are 2.4 times, 1.9 times, and 3.7 times larger than those of CM-I, respectively. Furthermore, the P-wave velocity, UCS, and Young's modulus of grey sandstone closely match those of CM-II, as evident from similar stress-strain responses in Figure 3(a), although noticeable discrepancies in Poisson's ratio and tensile strength are observed. Te basic physical and mechanical parameters of the four materials difer signifcantly, which is advantageous for investigating the efect of inflling on the combined specimens. Terefore, four groups of inflled specimens, such as ReS-I, ReS-II, GrS-I, and GrS-II, as shown in Figure 4, were generated. In addition, solid Brazilian disc specimens (BD) and ringshaped (RS) specimens without infllings were prepared for comparative analysis. Each confguration comprises three specimens, and the averaged parameters obtained from the experiments were used for subsequent analyses.

Experimental Methodology and Equipment.
A hydraulic servo-controlled loading system (Model RMT-150C) was used to conduct the diametrical compression tests. Te disc specimens were positioned on the loading platen and compressed diametrically until failure. Te loading speed was set to 0.002 mm/s using a displacement control module. Tis speed ensured that the specimen deformed at a quasistatic strain rate. Te applied load was monitored using a small force sensor with a range of 0-20 kN. Strain gauges were attached to the specimen's surface to record the local strains at points of interest during the diametrical compression.
Te fracturing process of the specimen under diametrical compression was captured and recorded using a high-speed camera (HS-camera) system. Since the disc specimens were tested under diametrical compression conditions, the total time required to fracture the specimen was approximately 6 minutes, as determined in the preexperiments. Consequently, a lower frame rate was selected to capture and record as many images as possible for subsequent analysis.
After calibration, a frame rate of 50 fps was chosen with a resolution of 320 × 360 pixel array size. Tese images were then used for digital image correlation (DIC) analysis. A speckle pattern was applied to the specimen's surface to enhance contrast. Tis was achieved by frst spraying white paint to create a white base and then spraying black paint to generate black speckles. Figure 5 illustrates the experimental setup, incorporating the strain gauge measurement and HScamera system. Figures 6 and 7 illustrate the typical vertical load-displacement curves of the red sandstone and grey sandstone specimens with diferent confgurations. It is observed from Figures 6(a) and 7(a) that the load-carrying capacities (i.e., maximum load in the curve) of the ring-shaped ReS specimens and GrS specimens decrease dramatically with increasing inner diameter. When the inner diameter is 10 mm, there is only one force drop observed on the load-displacement curve. Nevertheless, as the inner diameter increases to 15 mm or 20 mm, an additional force drop occurs after the maximum load, indicating that the specimen experiences two fracturing responses to the loading. Tis phenomenon has been well explained in previous studies [53,54]. Te frst fracture occurs at the roof and foor of the inner hole surface due to tensile stress concentration, while the subsequent fracture takes place at     On the other hand, when the specimen is inflled, the shape of the load-displacement curve difers signifcantly from that of the RS specimen. As shown in Figures 6(b), 6(c), 7(b), and 7(c), for the same hole diameter, the inflled specimen exhibits a signifcantly higher load-carrying capacity compared to the ring-shaped sandstone. Moreover, the inflled specimen displays multiple force drops in the postpeak phase, indicating the occurrence of multiple local failures during the loading process. As illustrated in Section 3.4, these failures were induced not only within the sandstone and cement mortar but also at the interface between the rock and cement mortar.

Load-Displacement Curves.
When comparing specimens inflled with diferent cement mortars (i.e., low-strength CM-I and high-strength CM-II), it is observed that the specimen inflled with CM-II has much larger vertical deformation and more complex postpeak mechanical behaviors (see Figures 6(c) and 7(c)). Tis can be attributed to the higher strength and elastic modulus of CM-II compared to CM-I, which provides a more pronounced support efect on the inflled specimen. In terms of the red sandstone and grey sandstone specimens having the same inclusion material and geometric confguration, it is evident that the red sandstone has a higher load-carrying capacity than the grey sandstone, while the vertical displacement of the red sandstone is smaller than that of the grey sandstone. Tis phenomenon is largely infuenced by the mechanical properties of the rocks, such as the greater strength and Young's modulus of the red sandstone compared to the grey sandstone (refer to Table 1).

Stifness and Maximum Peak Load.
In this section, the stifness index is introduced to examine the overall deformation characteristics of the inflled specimens. Te stifness is defned as the ratio of the force to the vertical displacement during the linear phase of the loaddisplacement curve before the frst peak load. Following this defnition, Figure 8 presents the relationship between stifness and hole diameter for the specimens. It is observed that the trends in the stifness variation with hole diameter are generally similar for both the red sandstone and grey sandstone specimens. Specifcally, for the ring-shaped specimens without inclusion, the stifness decreases monotonically with increasing the hole diameter, manifesting that the hole has a weakening efect on stifness. Nevertheless, when the specimens are inflled with CM-I or CM-II, the stifness exhibits a trend of initially increasing and then decreasing as the hole diameter changes from 10 mm to 15 mm and subsequently to 20 mm. Te stifness reaches the highest value when the hole diameter is 15 mm. Furthermore, the stifness of the ReS15-II and GrS15-II specimens are higher than that of the Brazilian disc specimen, indicating that the high-strength cement mortar provides strong support to the specimens.
As demonstrated in Section 3.1, the inflled specimens exhibit multiple peaks in the load-displacement curves, indicating a progressive cracking process during the loading. In this section, the maximum peak load (MPL) value is selected to further study the relationship between the peak load and hole diameter as well as inclusion material. Figure 9 presents the changes in MPL with respect to the hole diameter for the ReS and GrS specimens with various inclusions. For the RS specimen, the MPL value decreases with increasing hole diameter, similar to the trend observed in stifness. Under the same hole diameter, the MPL of the inflled specimen is higher than that of the unflled specimen, indicating an enhancement in the load-carrying capacity due to the inclusion. For instance, the MPL of the GrS20-II specimen is more than 22 times greater than that of the GrS20 specimen without inclusion and 1.5 times greater than that of the GrS20-I specimen. However, the degree of enhancement in load-carrying capacity varies signifcantly among diferent confgurations. In the case of the GrS specimen inflled with CM-II, the MPL is almost the same as the tensile strength of the solid Brazilian disc. Tis phenomenon occurs probably due to the similar mechanical properties (e.g., Young's modulus and UCS) of CM-II and the grey sandstone, indicating that the inclusion ofsets the weakening efect of the hole, as observed in the RS specimen.
For the ReS-II specimen, the MPL monotonically increases with increasing hole diameter. Nevertheless, for the ReS and GrS specimens with CM-I inclusion, the MPL frst increases and then decreases with increasing hole diameter, reaching its highest value when the hole diameter is 15 mm. Tis phenomenon clearly indicates that the CM-I inclusion material can enhance the load-carrying capacity of both the GrS and ReS specimens. However, the enhancement efect will be negatively compensated when the hole diameter increases. Alternatively, the efective load-carrying capacity of the inflled specimen results from a competitive interplay between the inclusion enhancement and the hole weakening efect.

Evolutions of Strain Readings.
As indicated in Section 3.1, the deformation of the inflled specimen was closely dependent on the inclusion material and hole diameter. In this section, to further study the deformation characteristics of the inflled specimens under diametrical compression, the strain evolution of several locations adjacent to the inclusion-rock interface was examined using strain gauge measurement. Te arrangement of the strain gauges is illustrated in Figure 10. SG5 was positioned at the center of the inclusion. SG2 was located at the lateral boundary of the inclusion-rock interface, while SG3 was placed at the roof boundary parallel to the compression line. SG1 was placed on the left side of SG3 and was 5 mm away from SG3. SG4 was located 10 mm away from SG3 along the compression diameter. SG1 was used to record the strain evolution of the point of interest that is not in the loading line. Figure 11 illustrates the strain evolutions recorded by the SGs on the inflled ReS specimens with various hole diameters and inclusions. Te time-to-fracture line indicates when the specimen loses partial or entire load-carrying capacity. Positive values in the SG readings represent tensile strain, while negative values represent compressive strain. It can be seen that irrespective of the inclusion material or hole diameter, the tensile strain in the SG3 reading is consistently higher than that in the SG4 reading, indicating a change in the tensile strain distribution in the specimen due to the presence of the inclusion. In addition, the tensile strain concentration occurs at SG3 due to the hole efect, even when the hole is inflled with cement mortar. Furthermore, when comparing Figure 11(a) with Figure 11(b) and Figure 11(c) with Figure 11(d), it is found that the SG3 reading increases signifcantly with increasing hole diameter, indicating a more pronounced intensity of tensile strain concentration in cases with a larger hole diameter. Te SG1 reading remains positive before failure in all test scenarios and increases with increasing loading time. Nevertheless, the SG1 reading is considerably lower than the SG3 reading, demonstrating that the highest tensile strain occurs at the inclusion boundary. Tis observation is further supported by the smaller SG5 reading compared to the SG3 reading. Furthermore, the SG2 reading remains consistently negative throughout the entire cracking process, demonstrating that the lateral boundary of the inclusion remains in a state of compression. Tese results indicate that although Shock and Vibration the central hole is flled with CM-I or CM-II, tensile strain concentration still occurs at the roof boundary of the inclusion along the compression line. Similar tensile regions have been observed in ring-shaped rock specimens, as shown in Li et al. [55]. Te mismatch in the deformation compatibility at the inclusion boundary may be a contributing factor to the initiation of interfacial cracking, which will be further explained in the subsequent sections. Figure 12 shows the strain evolutions recorded by the strain gauges on the inflled GrS specimens. It is seen that, the variations in the strain values obtained by the strain gauge for the inflled GrS specimens are similar to those of the inflled ReS specimens. Te main diference lies in their responses to load drops in the load-time curves. For example, in the case of the GrS20-II specimen (Figure 12(d)), multiple load peaks occur during the diametrical compression process, leading to fuctuations in the strain readings of SG3 and SG4 that correspond to the load drops.
To further investigate the efects of the inclusion and hole diameter on the deformation features, Figure 13 summarizes the changes in the SG1 and SG3 readings at the MPL point for the inflled ReS and GrS specimens. It can be seen that, for all the ReS and GrS specimens, the values of SG1 and SG3 increase with increasing hole diameter, irrespective of the inclusion material types. Tis phenomenon aligns with previous experimental and numerical fndings [30,52].    Shock and Vibration Furthermore, for cases with the same hole diameter, specimens with low-strength inclusion (CM-I) exhibit higher SG1 and SG3 readings compared to specimens with highstrength inclusion (CM-II). However, it is worth noting that the ring-shaped specimens without inclusion demonstrate the highest SG1 and SG3 readings, as shown in Figure 13.
Tese results show that a more intensive strain concentration area is generated at the inclusion boundary when the Shock and Vibration hole diameter is larger and the inclusion material has lower strength. Consequently, when the hole diameter is the same, the RS specimen without inclusion exhibits the lowest loadcarrying capacity, followed by the CM-I flled specimen and then the CM-II flled specimen (Figure 9).

Crack Evolution Process from the DIC Measurement.
Te images photographed by the HS-camera during the loading process were used to calculate the distribution of the maximum principal strain. Trough visual inspections of the specimen surface and statistical analyses of the strain distributions, it was observed that macrocracking occurs when the maximum principal strain exceeds 3.5%. Terefore, areas with a maximum principal strain exceeding 3.5% are considered as locations where macrofracture initiates. To better demonstrate the cracking process, the maximum principal strain contour was limited to the range of 0-3.5%. Figure 14 presents the cracking process of red sandstone specimens with varying hole diameters and inclusions. It can be observed that specimens with identical hole diameters but diferent inclusion materials exhibit similar crack modes. For example, regardless of whether the inclusion is CM-I or CM-II, the inclusions in the ReS10-I and ReS10-II specimens remain intact (Figures 14(a) and 14(b)), although debonding failure occurs at the inclusion-rock interface. Te specimens with a 20 mm hole diameter (Figures 14(c) and 14(d)) were split into two halves along the loaded diameter. Tis phenomenon demonstrates that the crack mode of red sandstone appears to be independent of the strength of the inclusion. Tis is likely due to the signifcant disparity between the strength of the red sandstone and the inclusion (i.e., the strength of the red sandstone is 2.3 times greater than that of the high-strength inclusion and 5.5 times for the low-strength inclusion), which minimizes the infuence of the inclusion's strength on the cracking behavior. However, as mentioned above, when the inclusion material is the same, signifcant diferences in crack mode are observed for different hole diameters. For example, with a larger hole diameter of 20 mm, the fracture plane occurs along the compression line. Nevertheless, specimens with a smaller hole diameter of 10 mm tend to lose their load-carrying capacity due to interfacial debonding. Figure 15 illustrates the cracking process of the grey sandstone specimens with varying inner diameters and inclusions. Te cracking behaviors of grey sandstone difer signifcantly from those of red sandstone. In the case of grey sandstone with a 10 mm hole diameter, the fnal fracture plane generally passes through the inclusion, but it does not align with the center compression line (Figures 15(a) and 15(b)). Comparing these results with Figures 14(a) and 14(b), it can be observed that when the hole diameter is small, the growth of the fracture plane is signifcantly infuenced by the hole size efect, making the fracture more prone to propagate towards the inclusion-rock interface. However, this crack mode is markedly diferent from that of the grey sandstone with a 20 mm inner diameter (Figures 15(c) and 15(d)), where failure is primarily induced by interfacial debonding. Te cracking process of inflled sandstone is closely related to the hole diameters and inclusions, as explained in Section 3.2. Based on the aforementioned illustrations, three types of failure can be observed in inflled specimens: interfacial debonding, tensile splitting in the sandstone, and tensile splitting in the inclusion. Interfacial debonding initially occurs at either the left or right side of the interface due to a deformation mismatch. Subsequently, tensile cracking is initiated at the crown and foor of the hole as a result of compression. With further increases in the diametrical load, splitting failures in the sandstone and inclusion become evident.

Discussion
From the above analyses, it is found that the presence of inclusion leads to nonuniform stress distribution and uncoordinated deformation at the inclusion-rock interface during diametrical compression. Tis ultimately results in various types of failures, as depicted in Figures 14 and 15. Te stress feld of a homogeneous disc under diametrical compression has been thoroughly discussed in previous studies [56,57]. Based on the linear elasticity theory, the analytical solution for a homogeneous Brazilian disc under diametrical compression can be expressed as follows [58,59]: where P is the applied load, R is the disc radius, and t is the disc thickness. Equation (1) suggests that along the compression line (x � 0), the horizontal stress σ x exhibits tensile behavior and equals a constant value of P/(πRt). Tis value can be used to determine the tensile strength of the homogeneous disc when P reaches its maximum. Along the transverse diameter line (y � 0), the vertical stress σ y is compressive, and the maximum value occurs in the center of the disc.
In this study, the Brazilian disc specimens are not homogeneous but are instead inflled with diferent inclusions. Deriving a closed-form solution for the stress distribution in the inflled disc is challenging. Terefore, fnite element (FE) analyses were conducted to explore the stress distribution in the sandstone, inclusion, and the rock-inclusion interface. Figure 16 presents the numerical model with the loading and boundary conditions. A quarter model was used to reduce computation time. Based on the experimental fndings in Section 3, it was observed that failures primarily occur along the vertical diameter or at the rock-inclusion interface. Hence, it was crucial to examine the stress distributions along the loaded diameter and the transverse diameter, particularly the rock-inclusion interface to illuminate the cracking mechanism. To facilitate comparison, the stresses derived from the numerical simulation were normalized by the equivalent tensile stress (P/(πRt)). Figure 17 presents the normalized Y-stress distribution along the X-axis and the normalized X-stress distribution along the Y-axis. Te analytical solution and FE solution for the homogeneous Brazilian disc are also included for comparison. Positive stress values denote tensile stress, while negative stress values denote compressive stress. It is evident from Figure 17(a) that the center of the inflled disc experiences a compression state, but the magnitude is considerably smaller than that of the homogeneous disc. At the rock-inclusion interface, the compressive stress dramatically increases, leading to a concentrated stress zone. However, the extent of stress concentration diminishes with increasing inclusion diameter. Since the numerical results in Figure 17 are based on the properties of red sandstone, the rockinclusion interface of the 10 mm inflled disc exhibits a higher stress concentration compared to that of the 20 mm inflled disc. Consequently, the 10 mm inflled disc is more prone to fracture at the interface, as observed in Figure 14, where debonding failure occurs. In contrast, as shown in Figure 18(b), the X-stress (in tensile behavior) exhibits a substantial increase at the interface between the inclusion and rock, with the magnitude escalating as the inclusion diameter increases. Te numerical results verify that higher tensile stress occurs along the compression line for discs with larger inclusion diameters, making them more susceptible to splitting along the loaded diameter. Tis mechanism also explains the splitting of red sandstone discs with a 20 mm inclusion diameter, as depicted in Figure 14.      It should be noted that the failure mechanism of the inflled disc under diametrical compression is complex and infuenced by various factors, such as the mechanical properties of the rock, the mechanical properties of the inclusion, and the inclusion diameter. To further explore these relationships, additional FE modeling studies were conducted. Te inclusion diameter varied from 5 mm to 25 mm, and the elastic modulus of the inclusion varied from 1 GPa to 45 GPa. Figure 18 illustrates the relationships between the normalized maximum stress and the normalized inclusion diameter (D inc /D disc ) and the normalized inclusion elastic modulus (E inc /E disc ). It is observed that as the inclusion diameter increases, the normalized maximum Xstress along the loaded diameter monotonically increases, while the compressive stress concentration at the transverse interface weakens. Tis phenomenon is consistent with the observations in Figure 17. However, as shown in Figure 18(b), with increasing the normalized elastic modulus, the normalized maximum stresses initially decrease and then increase. Specifcally, when E inc /E disc is less than 1.0, the maximum stress concentration intensity decreases with increasing the normalized elastic modulus. Nevertheless, when E inc /E disc is larger than 1.0, the maximum stress concentration intensity (either tensile or compressive) increases with increasing normalized elastic modulus. Moreover, the location of the maximum stress shifts from the inclusion-rock interface to the center of the inclusion when E inc /E disc exceeds 1.0 (see Figure 18(b)). It is important to highlight that the FE analyses presented here were based on the elastic deformation of the inflled disc. However, under diametrical compression, the deformation and cracking characteristics are also infuenced by the bonding strength at the interface. Te failure of the inflled disc is a competing result of tensile limit, compressive limit, and bonding strength, which should be further, investigated in future studies, including the elastoplastic and postpeak deformations of the inflled disc subjected to diametrical compression.

Conclusions
Red sandstone and grey sandstone disc specimens, flled with inclusions of diferent strengths and diameters, were prepared for diametrical compression tests. Te local strain concentration intensity and cracking process at the interface between the rock and inclusions were studied using the SG and DIC measurement techniques. Te stress distribution along the loaded diameter and transverse diameter was analyzed using the elastic fnite element method. Te fndings from the study lead to the following conclusions: (1) Te presence of the inclusion introduces more complex deformation and cracking behaviors in the inflled sandstones. Te sandstone inflled with CM-II exhibits signifcantly larger vertical deformation and more intricate postpeak mechanical behaviors in comparison to the sandstone with CM-I inclusion.
(2) Te MPL of the red sandstone inflled with CM-II increases continuously as the inclusion diameter decreases. However, the MPL of the grey sandstone inflled with CM-II remains approximately constant regardless of the hole diameter. Te efective loadcarrying capacity of the inflled sandstone was Normalized maximum X-stress along the Y-axis Normalized maximum Y-stress along the X-axis Normalized maximum X-stress along the Y-axis Normalized maximum Y-stress along the X-axis Normalized X-stress at the roof interface Normalized X-stress at the transverse interface (b) Figure 18: (a) Relationship between normalized maximum stress and D inc /D disc ; (b) relationship between normalized maximum stress and E inc /E disc . determined by the competing efects of inclusion enhancement and hole weakening.
(3) When the inclusion diameter is larger and the inclusion strength is lower, a more pronounced strain concentration occurs at the inclusion boundary. Te crack mode in the inflled red sandstone is independent of the inclusion strength. In cases where the hole diameter is larger, the fracture planes form along the compression line, while interfacial debonding occurs in the red sandstone with a smaller hole diameter.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.