Strain-rate effects and energy dissipation of frozen rocks under dynamic impact loading

2.1. Specimen parameters and basic physical parameters

Following the standards of the International Society for Rock Mechanics (ISRM), yellow sandstone, white sandstone, and granite collected from an open-pit mine in western China were machined into cylindrical specimens of Ф50 mm×50 mm, with end-face unevenness ≤ 0.02 mm and non-parallelism ≤ 0.05 mm. To obtain parameters such as dry density, saturated density, and dry and saturated P-wave velocities, the specimens were first dried. Immediately after removal from the drying oven, their mass was measured using a high-precision electronic balance (accuracy 0.01 g), and their exact dimensions were measured using a high-precision electronic caliper. The dry P-wave velocities of the three rock types were measured using a wave velocity tester, and the wave impedance of the rock was calculated from the P-wave velocity and density of the rock [21]. Subsequently, the specimens were subjected to saturation treatment. After saturation, their saturated mass and saturated P-wave velocities were measured. Following the measurements, the specimens were wrapped with plastic wrap to prevent surface moisture evaporation during freezing, and then placed in a freezing chamber for 48 h to freeze. The specific operating procedure is shown in Fig. 1, and the relevant parameters are listed in Table 1.

Fig. 1Pre-test preparation flow. Photo by Tianxiang Deng and Qi An, 2025.12, Anhui University of Science and Technology, Huainan, Anhui 232001, China

2.2. Modification of the SHPB test system and experimental scheme

The SHPB system used in the test consists of four parts: a power system, a bar system, a damping system, and a data acquisition system. The power system includes a high-pressure nitrogen cylinder, a launch chamber, and an air pressure controller. The bar system comprises a striker bar, an incident bar, and a transmission bar. The damping system is a damper. The data acquisition system consists of strain gauges, an ultra-dynamic strain amplifier, and an oscilloscope, as shown in Fig. 2(a-b). To meet the testing requirements of frozen rock at low temperatures, a temperature control module was added to the system: a temperature-controllable (–40 °C to 60 °C) insulation chamber was installed between the incident bar and the transmission bar, with holes matching the bar diameters on both sides of the chamber, as shown in Fig. 2(c-d). The device consists of a temperature circulating bath and an insulation chamber. Industrial ethanol solution is added to the temperature circulating bath, and circulation and diversion are used to achieve temperature control. The ethanol circulates inside the pipelines of the insulation chamber, and a fan inside the chamber ensures uniform temperature distribution, with a temperature accuracy of ±1 °C.

In order to attenuate high-frequency oscillations and prolong the rising edge of the incident wave, a pulse shaper (a copper disc of 2 mm in thickness) was affixed to the impact end of the incident bar for all experiments, so that the specimen could reach stress equilibrium before fracture. The validity of pulse shaping was confirmed by the stress equilibrium verification presented in Fig. 3(b).

Fig. 2SHPB test and NMR processing. Photo by Tianxiang Deng and Qi An, 2025.12, Anhui University of Science and Technology, Huainan, Anhui 232001, China

The material parameters of the bars in the SHPB test system are listed in Table 2.

This experiment requires the assumption of one-dimensional elastic stress wave propagation in the bars and the assumption of uniform distribution of the stress and strain fields along the length of the specimen. To obtain the stress, strain, and strain rate of the specimen under impact loading, only the reflected and transmitted strain signals ${\epsilon }_i\left(t\right)$ and ${\epsilon }_t\left(t\right)$, respectively are used. This method is referred to as the “two-wave method” [22]. Provided that the two basic assumptions are satisfied, the results obtained by the two-wave method and the three-wave method are in good agreement, and the two-wave method is relatively simpler and faster for calculation. Therefore, after stress equilibrium has been verified, the two-wave method is used for processing and calculation in this study.

The corresponding governing equations are listed below:

where $L_S$ and $S_S$ are the length and cross-sectional area of the specimen under impact loading; for the bar, the longitudinal wave velocity, elastic modulus, and cross-sectional area are denoted as $c_0$, $E$, and $S_B$; the incident strain, transmitted strain, and reflected strain of the specimen are ${\epsilon }_i\left(t\right)$, ${\epsilon }_t\left(t\right)$, and ${\epsilon }_r\left(t\right)$; the strain rate of the specimen is ${\dot{\epsilon }}_s\left(t\right)$; the strain of the specimen is ${\epsilon }_s\left(t\right)$; and the stress of the specimen is ${\sigma }_s\left(t\right)$.

All test specimens were subjected to forced water saturation. The freezing temperatures were set at –10 °C, –20 °C, and –30 °C, with room temperature (20 °C) serving as the control group. The experimental apparatus was a self-developed split Hopkinson pressure bar (SHPB) system capable of controlling a low-temperature environment. Preliminary impact tests were conducted to determine four impact pressure levels: 0.35 MPa, 0.45 MPa, 0.55 MPa, and 0.65 MPa. Under the forced water-saturated condition, a factorial test program was conducted with temperature, impact pressure, and strain rate as variables. Each test group had three parallel specimens to ensure data reliability.

The typical waveform diagram and the stress equilibrium verification diagram of the specimen are shown in Fig. 3. It can be seen from the figure that the incident wave propagates in the form of a rectangular wave. Although the rectangular wave exhibits certain oscillations, these oscillations remain within a controllable range and the waveform is satisfactory. In addition, it can be observed from the strain time history curves that the strain curve formed by the superposition of the incident and reflected waves shows good agreement with the transmitted strain curve, which verifies that the specimen can achieve a state of uniform stress and satisfy dynamic stress equilibrium in this test. As shown in Fig. 3(b), the three curves “In”, “Re” and “Tr” represent the incident wave, the transmitted wave, and the reflected wave.

Fig. 3Typical waveform and stress equilibrium verification diagram

a) Typical waveform diagram of the specimen

b) Verification of dynamic stress equilibrium for a typical specimen

2.3. Changes in porosity

Under dynamic loading, rock specimens may not necessarily develop macroscopic cracks or obvious structural failure; however, their internal structure is inevitably affected by dynamic disturbance. Nuclear Magnetic Resonance (NMR) is a technique capable of analyzing the pore distribution within a specimen. Therefore, NMR testing was employed to quantitatively characterize the internal pore distribution of the specimens before and after a single impact load. The NMR analyzer is shown in Fig. 2(e) [23].

Fig. 4Yellow sandstone porosity

a) 20 ℃

b) –10 ℃

c) –20 ℃

d) –30 ℃

The porosities of the rocks under different test conditions were obtained by nuclear magnetic resonance (NMR) measurements, and scatter plots with fitted curves for the three rock types are presented in Figs. 4-6. In Fig. 7, $x=$ 0 MPa represents the initial porosity of each specimen before impact, which is approximated here by the average porosity of three randomly selected specimens prior to testing. Only specimens that remained macroscopically intact after impact were subjected to NMR testing; specimens with visible cracks or fragmentation were excluded, because their pore structures would represent the post-impact failure state rather than the damage caused by a single impact. For example, in the relationship between porosity and impact pressure for yellow sandstone shown in Fig. 7(a), at room temperature (20 °C) the specimen maintained structural integrity only after the lowest impact pressure (0.35 MPa), while the specimens under other pressure conditions had already failed; therefore, porosity data for the other pressure conditions were not measured.

Fig. 5White sandstone porosity

a) 20 ℃

b) –10 ℃

c) –20 ℃

d) –30 ℃

As shown in Fig. 7, the initial porosity of the three rocks in descending order is yellow sandstone, white sandstone, and granite, indicating that granite is the most compact, followed by white sandstone, while yellow sandstone is the loosest. Among them, the porosity variations of white sandstone and yellow sandstone are relatively large, with the most significant increase observed in yellow sandstone, reflecting that its internal structure is more susceptible to impact loading.

The influence of temperature on porosity evolution demonstrates that both sandstone varieties exhibit greater structural deterioration under room-temperature impact conditions, accompanied by a more pronounced increase in porosity. This phenomenon suggests that moderate cooling promotes the formation of a frozen internal structure, thereby improving the impact resistance of yellow sandstone and white sandstone to a certain degree. Nevertheless, further cooling to −30 °C reverses this trend, with both sandstone types showing higher porosity increments than those observed at −10 °C and −20 °C. This behavior may be attributed to the increased brittleness induced by extremely low temperatures, where thermal contraction and stress concentration facilitate the initiation and propagation of internal cracks, ultimately resulting in a more developed pore network. A comparable porosity evolution pattern is also observed in granite specimens subjected to the combined effects of severe cooling and relatively high impact pressure.

Fig. 6Granite porosity

a) 20 ℃

b) –10 ℃

c) –20 ℃

d) –30 ℃

Fig. 7Porosity changes of three pure rock specimens before and after impact

a) Yellow sandstone porosity

b) White sandstone porosity

c) Granite porosity

3.1. Characteristics of stress-strain curves

Based on Eq. (1), the valid data obtained from the SHPB test system can be used to compute the dynamic stress-strain curves of various rocks under different conditions. According to the post-impact state of the specimens and the variation characteristics of the curves, the stress-strain curves of the three lithologies (yellow sandstone, white sandstone, and granite) are plotted as shown in the Figs. 8-10.

Fig. 8Stress-strain curve of yellow sandstone specimen

a) 20 ℃

b) –10 ℃

c) –20 ℃

d) –30 ℃

The peak stress point marks the critical transition of rock from steady-state fracturing to unstable failure. The stress and strain at this point best reflect the material’s bearing capacity and deformation characteristics under specific temperature and strain rate conditions. As shown in Fig. 8, when the temperature ranges from 20 °C to –20 °C, the peak stress and the corresponding strain range under different impact pressures increase continuously; however, at –30 °C, this range narrows rapidly. Within the investigated temperature interval from 20 °C to –20 °C, yellow sandstone exhibits an increasing dependence on strain rate with decreasing temperature, indicating that its dynamic response becomes progressively more rate-sensitive under cooling conditions. However, a further reduction in temperature to –30 °C leads to a decline in the strain-rate strengthening effect, resulting in a reduced sensitivity of yellow sandstone to loading-rate variations.

Fig. 9Stress-strain curve of white sandstone specimen

a) 20 ℃

b) –10 ℃

c) –20 ℃

d) –30 ℃

As shown in Fig. 9, the variation characteristics of peak stress and the corresponding strain of white sandstone are strongly affected by freezing temperature. Compared with specimens tested at room temperature, those subjected to –10 °C and –20 °C exhibit a more concentrated peak region with an extended post-peak deformation stage. In addition, both the peak stress and the associated strain increase approximately linearly as the impact pressure increases. These observations indicate that moderate freezing conditions modify the dynamic deformation behavior of white sandstone and increase its dependence on loading rate.

Fig. 10 demonstrates that granite exhibits a relatively stable dynamic response at room temperature (20 °C), with variations in peak stress and the associated strain caused by different impact pressures being less pronounced than those observed in the two sandstone types. This indicates that the mechanical behavior of granite is less sensitive to temperature-related effects compared with sandstone under the investigated loading conditions. Under low-temperature conditions, the fissure structure in water-saturated sandstone specimens is markedly improved by the pore water-ice phase transition, leading to a substantial increase in mechanical strength. In contrast, owing to its inherently low porosity and low water absorption, although granite exhibits certain variations in its peak stress parameters, the magnitude of the temperature effect is notably smaller than that observed in the sandstone specimens.

Fig. 10Stress-strain curve of granite specimen

a) 20 ℃

b) –10 ℃

c) –20 ℃

d) –30 ℃

3.2. Dynamic mechanical properties

The effect of impact pressure on the peak stress response of the three rock types was evaluated under various temperature conditions. Fig. 11 summarizes the evolution of peak stress with increasing impact pressure. Within the range of 0.35-0.65 MPa, yellow sandstone and granite exhibited a progressive increase in peak stress, indicating a positive correlation between impact pressure and dynamic strength. White sandstone showed a similar tendency under most testing conditions; however, an exception occurred at –30 °C combined with an impact pressure of 0.65 MPa, where the peak stress deviated from the overall increasing trend.

In terms of the growth trend, the fitted curves for yellow sandstone and granite are well described by linear relationships, and the $R^2$ value of the linear fit for granite is closer to 1, indicating that granite most closely follows the linear variation trend. For white sandstone, although it does not satisfy linear growth well, its trend is can be well described by a quadratic relationship.

For yellow sandstone and white sandstone under the four different temperature conditions, as shown in Fig. 11(a-b), the data points and their corresponding four curves are clearly distinguished by peak stress, indicating a relatively strong influence of temperature. In contrast, although the data points and fitted curves of granite also exhibit certain variations with temperature, the degree of influence is considerably smaller.

Fig. 11Peak stress-impact pressure relationship diagram of three types of rock specimens

a) Yellow sandstone

b) White sandstone

c) Granite

To quantify the temperature-dependent variation in peak stress among rocks with different lithologies, the results obtained at an impact pressure of 0.55 MPa were selected for comparative analysis. The peak stress values under different temperature conditions were normalized against the corresponding values at room temperature (20 °C), and the temperature-induced increments and growth rates for the three rock types are summarized in Fig. 12.

Fig. 12Peak stress growth of three types of rock specimens under 0.55 MPa impact pressure with temperature influence

With progressive cooling, both sandstone types exhibit a gradual enhancement in peak stress, whereas the magnitude of this increase varies significantly among different lithologies. The temperature-induced strengthening effect follows the order of yellow sandstone, white sandstone, and granite, with yellow sandstone showing the most pronounced response and granite exhibiting the weakest variation. This is because under freezing conditions, the internal water in the specimens begins to freeze into solid ice due to the low-temperature environment. Although the strength of ice is lower than that of the rock matrix, the frozen state can fill the pre-existing pores inside the specimen, thereby better inhibiting the initiation of internal damage and the development of fissures under loading. Macroscopically, this enables the specimen to withstand higher stress without failure and significantly increases its peak stress. The different lithologies exhibit distinct water-storage capacities. Compared with granite, the sandstone specimens contain a larger amount of pore water under saturated conditions due to their higher initial water-holding capacity. Consequently, a greater volume of pore ice can be generated during freezing, which contributes to the filling of internal voids and enhances the bonding effect within the rock skeleton. Similarly, the porosity of the two sandstones is also much higher than that of granite. Under freezing conditions, the sandstone structure is “strengthened” to a greater extent, and the macroscopic manifestation is that sandstone is influenced by temperature factors much more significantly than granite.

4.1. Method of energy dissipation

According to the composition of the SHPB test system, the total energy (loading energy) during the entire test is the kinetic energy of the striker acting on the striker bar (projectile) when released under high-pressure nitrogen, denoted as $W_B$. The energies carried by the incident stress wave, reflected stress wave, and transmitted stress wave collected in the test are $W_I$, $W_R$, and $W_T$. In addition, the energy absorbed by the specimen under impact loading is $W_A$. The above energies are generally calculated using the following equations, as expressed by Eqs. (4), (5), (6), and (7) [24]:

where $C_B$, $A_B$, and $E_B$ denote the longitudinal wave velocity, cross-sectional area, and elastic modulus of the elastic bar and ${\sigma }_I\left(t\right)$ , ${\sigma }_R\left(t\right)$, and ${\sigma }_T\left(t\right)$ are the specific values of the incident stress wave, reflected stress wave, and transmitted stress wave collected and processed from the test at time $t$ [25].

According to the energy partition characteristics, the absorbed energy ($W_A$) is predominantly consumed by internal deformation and fracture processes of the specimen, accounting for approximately 95 % of the total input energy. In comparison, the energy associated with fragment separation and material ejection is relatively limited (5 %). Thus, only $W_A$ is considered in the subsequent fracture energy evaluation, while the minor energy components are neglected. The data calculated based on the above formulas are presented in the following Tables 3-5.

For the convenience of comparison and analysis of the relationships among the reflected energy $W_R$, transmitted energy $W_T$, and dissipated energy $W_A$ of the specimen, the corresponding proportions relative to the incident energy are analyzed [26].

The formulas for the proportions of the three energies to the incident energy are as follows:

where $P_R$ is the reflected energy proportion, $P_T$ is the transmitted energy proportion, and $P_A$ is the dissipated energy proportion of the specimen.

4.2. Comparison of energy evolution of three frozen rocks

According to Eqs. (8-10), the images of $W_R$, $W_T$, and $W_A$ as proportions of the incident energy, denoted as $P_R$, $P_T$, and $P_A$ respectively, for yellow sandstone, white sandstone, and granite are plotted.

For yellow sandstone under different impact pressures, the variation of the proportions of the three energies (reflected, dissipated, and transmitted) in the incident energy with temperature is as follows. Overall, $P_R$ accounts for the highest proportion, followed by $P_A$ of the specimen, while $P_T$ accounts for the lowest proportion.

Fig. 13Energy characteristic value relationship diagram of yellow sandstone specimen

a) 0.35 MPa

b) 0.45 MPa

c) 0.55 MPa

d) 0.65 MPa

Under different impact pressures, the energy distribution of yellow sandstone exhibits similar but slightly varying trends with decreasing temperature: $P_R$ generally decreases, $P_A$ generally increases, and $P_T$ increases modestly. At 0.35 MPa, $P_R$ continuously decreases from approximately 60 %, $P_T$ increases from about 6 % to about 10 %, and $P_A$ steadily rises as temperature decreases. At 0.45 MPa, $P_R$ decreases from 64.81 % to 50.72 %, $P_T$ increases from 5.61 % to 13.97 %, and $P_A$ increases from 29.58 % to 35.30 %. At 0.55 MPa, the $P_R$ first decreases from 69.20 % to 53.67 % at –20 °C and then rises to 58.16 % at –30 °C; $P_T$ increases from 4.57 % to 14.35 % and then decreases to 10.52 %; $P_A$ rises to 34.25 % and then remains generally stable at approximately 31.5 %, fluctuating between 26.24 % and 34.25 %. At 0.65 MPa, $P_R$ decreases from 70.79 % to 48.10 %; $P_T$ continuously increases from 4.61 % to 12.88 %; $P_A$ increases from 24.61 % to 39.01 %. As impact pressure increases, the initial value of $P_R$ becomes larger but its decrease becomes more pronounced, and the increase in $P_A$ is more significant. Moreover, at 0.55 MPa, there is a turning point where $P_R$ first decreases and then increases, while $P_A$ first increases and then decreases, indicating the existence of an optimal freezing temperature. At a high impact pressure of 0.65 MPa, $P_R$ continuously decreases, and both transmitted and dissipated energy proportions continuously increase, suggesting that low temperature and high strain rate synergistically enhance energy absorption and internal damage.

Fig. 14Energy characteristic value relationship diagram of white sandstone specimen

a) 0.35 MPa

b) 0.45 MPa

c) 0.55 MPa

d) 0.65 MPa

The proportion of reflected energy to incident energy in yellow sandstone shows a gradually decreasing trend with increasing impact pressure, and this decreasing trend is more pronounced under freezing conditions. The proportion of transmitted energy is the lowest under all conditions, but it exhibits a slight increasing trend with decreasing temperature, indicating that the low-temperature environment has a certain but limited effect on the energy transmission capacity. The proportion of dissipated energy in the specimen increases with decreasing temperature, suggesting that the freezing environment enhances the brittle failure of yellow sandstone, causing the specimen to absorb more energy for internal damage; this trend is particularly evident under high impact pressures.

Under different impact pressures, the energy distribution of white sandstone exhibits a consistent trend with decreasing temperature: $P_R$ generally decreases, $P_T$ continuously increases, and $P_A$ gradually increases under moderately low temperature conditions. A detailed comparison shows that at 0.35 MPa, P_R decreases from 43.55 % to 25.93 %, $P_T$ increases from 14.73 % to 29.59 %, and $P_A$ increases slightly from 41.72 % to 44.48 %. At 0.45 MPa, $P_R$ first increases slightly from 41.11 % at 20 °C to 42.32 % at −10 °C and then decreases to 23.65 %; $P_T$ increases from 16.01 % to 31.66 %; $P_A$ first decreases from 29.58 % at 20 °C to 17.65 % at −10 °C and then rises to 31.66 %. At 0.55 MPa, $P_R$ decreases from 43.68 % to 26.19 %; $P_T$ increases from 14.92 % to 28.88 %; $P_A$ first decreases slightly from 41.40 % at 20 °C to 40.69 % at −10 °C and then increases to 44.94 %. At 0.65 MPa, $P_R$ decreases from 48.43 % to 36.56 %; $P_T$ increases from 11.00 % to 20.57 %; $P_A$ increases from 40.56 % to 42.87 %. With increasing impact pressure, the initial value of $P_R$ becomes larger but its decrease becomes more significant, while the increase in the transmitted energy proportion is relatively smaller. Although the dissipated energy ratio shows only a slight increase under the coupled action of elevated impact pressure and freezing conditions, its absolute value remains relatively high. The interaction between intense dynamic loading and low-temperature effects promotes energy dissipation within the specimen, resulting in enhanced damage accumulation and more severe failure.

Fig. 15Energy characteristic value relationship diagram of granite specimen

a) 0.35 MPa

b) 0.45 MPa

c) 0.55 MPa

d) 0.65 MPa

For white sandstone, $P_R$ decreases gradually with decreasing temperature, indicating that low temperature enhances the energy absorption capacity of white sandstone, allowing more energy to be utilized by the specimen. $P_T$ increases under all working conditions, and the improvement is relatively significant, suggesting that the elastic wave transmission capacity of white sandstone is enhanced at low temperatures, rather than being solely used for fragmentation and energy absorption. The proportion of dissipated energy reaches its maximum at the lowest freezing temperature under all impact pressures, indicating that the brittleness of white sandstone increases at low temperatures and more energy is converted into internal damage.

For granite specimens under different impact pressures, the variation of the proportions of the three energies with temperature is as follows. Unlike the two sandstones, granite exhibits the lowest $P_R$, followed by $P_A$ of the specimen, and the highest proportion is $P_T$. This is mainly attributed to the high strength of granite itself and its good wave impedance matching with the bars.

Within the impact pressure range of 0.35-0.65 MPa, the energy distribution of granite shows a clear contrast: transmitted energy always dominates, generally in the range of 45 %-60 %; $P_A$ remains stable at 30 %-35 %; $P_R$ is maintained at a low level of 10 %-20 % overall, but exhibits different temperature responses with varying impact pressure. At 0.35 MPa, P_R surges to about 20 % at –30 °C, indicating that extremely low temperature enhances local reflection. At 0.45 MPa, $P_R$ fluctuates between 10 % and 20 %, while $P_A$ first increases and then slightly decreases at –30 °C. At 0.55 MPa, $P_R$ briefly rises to 15 %-20 % at –20  °C, with a corresponding decrease in transmitted energy; $P_A$ remains around 35 % at different temperatures. At 0.65 MPa, $P_R$ continuously decreases with decreasing temperature to about 10 %, and $P_T$ remains stable at 45 %-55 %. Except for slight fluctuations in reflected and transmitted energies at 0.55 MPa, the proportions of each energy vary only slightly under other conditions, indicating that the energy distribution of granite is only slightly affected by temperature, the dominant role of transmission remains unchanged, and the failure mode is relatively stable. Granite has the highest proportion of transmitted energy, indicating that its compact structure allows more wave energy to be transmitted rather than absorbed or reflected. Under the coupled effects of low temperature and high impact pressure, the dissipated energy of granite exhibits certain fluctuations; however, the variation amplitude is much smaller than that observed in sandstone, suggesting a stronger resistance of granite to impact-induced failure. Moreover, under high impact pressure, transmitted energy still dominates, demonstrating that even at extremely low temperatures, granite still exhibits good energy transmission capability rather than entering a complete failure mode.

Comparing the energy variations of the three rock types under low-temperature conditions reveals that the proportion of dissipated energy in yellow sandstone and white sandstone increases significantly, reflecting that low temperature enhances the brittleness characteristics of the two sandstones. This enables them to absorb more incident energy and convert it into the fracture energy required for internal damage and crack propagation. The observed tendency is further amplified at higher impact pressures, indicating the coupled strengthening contribution of strain-rate effects and low-temperature effects. In contrast, owing to its dense structure, extremely low porosity and water absorption, as well as favorable wave impedance matching with the steel bars, the energy partitioning of granite is dominated by transmitted energy. Since most of the incident energy is carried away through elastic wave propagation rather than being converted into fracture-related dissipation, the energy dissipation response of the specimen shows limited sensitivity to temperature variation. By integrating the energy characteristics of the three rock types, it can be concluded that lithology is the key factor determining the energy partitioning pattern – for rocks with lower wave impedance and higher porosity, the increase of $P_A$ at low temperatures is more significant and the brittleness enhancement effect is more pronounced; whereas for dense rocks with high wave impedance, energy propagates primarily through transmission, and the low-temperature influence is weak.