The sandstone samples were taken from a mine in Sichuan, China. Thin section identification and scanning electron microscope analysis were conducted on the sandstone samples. The sandstone was mainly composed of clay, orthoclase, gypsum, plagioclase, quartz, calcite, and dolomite. The particles consist primarily of minerals like quartz and feldspar, exhibiting distinct cleavage development on the surface of the feldspar particles and a high degree of self-filling of chlorite between the particles. To ensure the reliability of the constitutive model in numerical analysis, strain gauges were affixed to the specimen. Additionally, the development process of surface cracks in rock was monitored using the corner correlation method. Cuboid sandstone samples with dimensions of 50 mm × 50 mm × 250 mm were prepared, adhering strictly to the test standards of rock mechanics. The parallelism and flatness of the sample’s cross section were carefully maintained, as depicted in Figure 2A. Grid marks in the form of spray paint points were arranged on the sample, with 4 rows and 25 columns of 5 mm × 5 mm grids placed on the shooting surface. Along the longitudinal symmetry axis, 25 strain gauges were applied to the rear surface of the rock, with each strain gauge corresponding to a specific grid. The strain gauge model employed was BE120-3AA, featuring dimensions of 6.4 mm × 3.5 mm and a resistance of 120 ± 0.1 Ω, as illustrated in Figure 2B.

The rock sample underwent a uniaxial impact compression test using the Split Hopkinson Pressure Bar (SHPB) test device with a rod diameter of 50 mm, located in the Impact Mechanics Laboratory of the University of Science and Technology of China. The incident and transmission bars of the experimental apparatus are both 100 cm in length and are made of steel. The impact speed was set at 8.08 m/s, resulting in a strain rate of 98 s⁻¹. The sample experienced complete damage during the test. The entire dynamic impact process was captured using a high-speed camera known as the Thousand-eye Wolf X213. The frame rate of the camera was configured at 50,000 fps, capturing an image every 11 μs. The experimental system, as shown in Figure 3. To ensure uniform loading and prevent damage to the ends of the rock sample caused by uneven force, a steel plate was inserted between the rock sample and the incident and transmission rods. This steel plate compensates for the difference in cross-sectional area between the rock sample and the rods. Importantly, the area of the steel plate exceeds that of the rock sample. To minimize interface friction, petroleum jelly was evenly applied to both sides of the steel plate, ensuring optimal contact between the rock sample, steel plate, and rods.

Throughout the experiment, a high-speed camera captured images of the sample every 11 μs. The experimental images selected depict several key times when the crack width changes, as illustrated in Figure 4. Furthermore, Figure 5 provides zoomed-ins of the crack localization zone.

Figure 4 illustrates that no significant changes occurred on the sample’s surface between 77 and 143 μs. During this period, the internal micro-cracks of the sample were extruded and opened, resulting in small macroscopic cracks on the surface, a process imperceptible to the naked eye. From 143 to 231 µs, with the action of stress waves, the micro-cracks inside the sample gradually extend and reach the elastic limit. As depicted in Figure 5A, a micro-crack becomes visible to the naked eye when it reaches the surface of the specimen. At this stage, the crack expansion is minimal, and the sample’s strength does not reach the limit value. Subsequently, from 242 to 341 µs, as shown in Figures 5B–J, the crack accelerated due to the repeated reflection of stress waves and the impact of the incident and transmission rods, leading to the imminent penetrate of the internal crack of the sample. Ultimately, when the stress value of the sample reached the compressive strength limit, the crack spread throughout the entire sample, resulting in its destruction.

The axial displacement field of the crack region was calculated by the corner correlation method, as shown in Figure 6.

When the crack initiates and starts to propagate, the axial displacement exhibits a consistently positive trend, indicating a continuous compression process within the sample. Figure 6 clearly illustrates the development of displacement from the incident end towards the transmitted end, with the displacement field continuously evolving. Eventually, a concentration phenomenon occurs at the crack site, as depicted in Figure 6. At 77 µs, the displacement on both sides of the initiation point begins to increase, reaching approximately 0.017 mm by 110 μs. Starting from 143 µs, the crack progressively expands and extends, leading to a gradual increase in displacement at the crack site. By 242 µs, the displacement on both sides of the crack reaches roughly 0.1 mm. At 341 µs, the displacement field becomes concentrated within the crack region, resulting in rapid crack width expansion and a displacement value of approximately 0.51 mm on both sides.

Corner points on both sides of the region where the crack is located are taken, and the distance between each two-corner points is 5 mm. The numbers 1–8 represent eight cross sections. The displacement difference between the left and right corner points of each section is represented by the crack width. The crack width calculation diagram, as shown in Figure 7, and the crack width time curve, as shown in Figure 8.

Figure 8 presents the crack width time curve, while Figure 9 illustrates the evolution process of the axial strain field. Throughout the loading process, the crack width variation curve with time was delineated into four stages. The first stage is the loading stage, during which no cracks manifest on the sample’s surface, and it undergoes elastic deformation, as depicted in Figure 8. The sample experiences compression, compacting tiny cracks within, yet without significant expansion, signifying the gradual input of external energy and internal stress readjustment, as shown in Figure 9. In the second stage, at 77 µs, external input energy accumulates within the sample in the form of elastic strain energy. All curves exhibit a slight increase, indicating the onset of internal micro-crack expansion, with a small crack emerging on the surface, imperceptible to the naked eye, as evident in Figure 8. At this time, it can be seen from the strain diagram that stress concentration has occurred at the failure site, and the elastic strain energy has begun to be used for crack expansion. The internal micro-crack deformation reaches the elastic limit value, followed by a drop to approximately 0 at 143 µs, signifying the accumulation of elastic strain energy once again due to stress wave reflection within the sample. As mentioned above, “the stress wave propagates in the sample once is about 60 µs,” indicating that cracks have appeared in the sample during the first propagation of stress waves. Subsequently, in the third stage, post-143 µs, all curves exhibit a steady upward trend, indicating the reappearance and development of surface cracks during stress wave reflection. At this juncture, internal micro-cracks once again reach the elastic limit value, commencing expansion and extension. The strain diagram reflects the elastic strain energy stored inside the sample is released. Stress concentration at the failure site becomes pronounced, with a portion of the stored elastic strain energy utilized for crack propagation and extension, while the remainder contributes to radiation energy and frictional heat generation. Finally, in the fourth stage, at 242 µs, all curves demonstrate a marked increase, signifying the commencement of rapid expansion of micro-cracks within the sample, culminating in the formation of through cracks. At this stage, the complete release of elastic strain occurs, leading to a rapid increase in dissipation energy, ultimately contributing to the sample’s fracture failure.

The axial strain field’s evolutionary process demonstrates that the strain in the failure zone exhibits compressive strain before 77 µs, as depicted in Figure 9, reflecting the entire compression stage. From 77 to 143 µs, the failure zone strain transforms into tensile strain, with a concentrated appearance towards the failure zone. By 242 µs, a concentrated zone has formed, and with the crack’s expansion, the strain color in the crack region deepens and expands, signifying the rapid macroscopic crack expansion. Stages 2 and 3 of Figure 8 are respectively magnified in Figure 10 and Figure 11.

The crack widths of section 1 and sections 6-8 exhibit significant increases from 77 μs onwards, while the crack widths of other sections show minimal change. This suggests that the crack initiation point of the observation plane is at the upper and lower ends, and the crack progresses from these points towards the middle region, as illustrated in Figure 10. Following 110 µs, the crack widths of all sections decrease, attributed to the accumulation of strain energy resulting from stress wave reflection. In addition, at the initial stage, the specimen contains a small number of micro-cracks and has a large elastic modulus, leading to a gradual reduction in the original crack widths. By 143 µs, the crack width values of all sections are negative, representing the local deformation of the specimen under the compressive force of stress waves.

As depicted in Figure 11, between 143 and 154 μs, the crack widths of sections 1–8 are observed to increase, suggesting the extension of the crack to the entire observation area during this period. The distance of section 1–8 is measured at 35 mm. Consequently, the calculated average crack propagation velocity is approximately 3,182 m/s. The crack width exhibits a gradual decrease in the order of sections 8-1, indicating that more cracks penetrated from front to back at the lower end of the observation area during this stage.

The crack width growth rate presents a fluctuating property, which is a nonlinear change of the crack growth rate caused by the reflection and transmission of stress waves. It can be seen from Figure 12 that the crack width growth rate is negative at 110–143 µs, indicating that the crack width gradually decreases at this stage. The crack width growth rate of section 7 is the highest at 319 µs (10.04 m/s), and the crack width growth rate of section 6 is the highest at 330 µs (9.22 m/s), indicating that the damage of the two sections is more serious and the damage progress is the fastest at the above two times.

The rock density measures approximately 2.2 g/cm^3, with a porosity ranging from 12% to 14%. The calculation model, as illustrated in Figure 13, is based on the ANSYS/LS-DYNA18.1 power display calculation method. A linear elastic constitutive model is employed for the bullet and the incident bar, with a density of 7.9 g/cm, an elastic modulus of 210GPa, and a Poisson’s ratio of 0.3. The bullet is in contact with the incident rod, and the incident rod is in contact with the specimen. The bullet speed is 8.08 m/s, with the specimen section size measuring 5 cm × 5 cm and the unit surface size at 0.5 cm × 0.5 cm, as depicted in Figure 13.

This paper adopts the RHT constitutive model (Riedel-Hiermaier-Thoma), a mechanical model extensively utilized to characterize the dynamic response of rock and concrete materials under high strain rate and substantial deformation conditions, such as explosion impact and projectile penetration. The model incorporates the failure surface equation, elastic limit surface equation, and residual stress surface equation to define the failure surface. While the parameter determination of the model is more intricate, it offers better simulation adaptability. Consequently, it has found widespread application in numerical simulation, with relevant material parameters detailed in Table 1.

The measured value and calculated value of the strain time history curve at the same position on the rock specimen at the impact velocity of 8.08 m/s are compared. The strain time history curve is analyzed at the measuring points 5, 15, and 30 mm away from the impact end, as depicted in Figure 14. The numerical simulation results exhibit a strong agreement with the measured results in terms of stress wave peak value and pulse width, indicating the reliability of the numerical simulation outcomes. This also underscores the capability of the RHT constitutive model of rock and the selected material parameters to effectively characterize the pertinent properties of the rock materials used in the experiment. Consequently, this model and set of parameters can be confidently employed for further numerical simulations involving this type of rock.

To minimize the superposition of trans-reflective waves, we modified the experimental conditions by increasing the length of the specimen to 100 cm. We then selected points every 10 cm along a straight line beginning 1.5 cm away from the impact end face, as illustrated in Figure 15. All other experimental parameters remained unchanged.