Experimental Investigations of Dynamic Response and Fatigue Damage Characteristics of Granite Under Multi-Level Cyclic Impacts

Abstract

Dynamic fatigue of rocks under repeated cyclic impact is a nonconservative property, as surrounding rocks in real environments subjects them to variable impact disturbances, and the degree of damage varies under different energy level loads. To evaluate the dynamic response and fatigue damage characteristics of rocks under multi-level cyclic impacts, uniaxial cyclic impact tests were carried out on granite with various stress paths and energy levels using a modified split Hopkinson pressure bar. Dynamic deformation characteristics of specimens under different loading modes were investigated by introducing the deformation modulus of the loading stage. Evolution of macroscopic cracks during the impact process was investigated based on high-speed camera images, and the microscopic structure of damaged specimens was examined using SEM. In addition, cumulative energy dissipation was used to assess the damage of rocks. Results show that the deformation modulus of the loading stage, dynamic peak stress and strain of specimens increase with the impact energy, and the deformation modulus of the loading stage decreases as the damage level increases. Propagation rate of tensile cracks in rock was correlated with participation time of the higher energy level, which observed the following sequence: linearly decreasing > same > linearly increasing energy level, and cyclic loading of nonlinear energy level produced more tensile cracks and rock spalling than the same energy level. Compared with cyclic impacts of the same energy level, multi-level impacts form more microcracks and fatigue striations. The cumulative rate of specimen damage under the same energy change rate is as follows: linear decreasing > same > linear increasing loading. This provides a new case study for evaluating the dynamic damage, crushing efficiency and load-bearing capacity of rocks in real engineering environments.

1. Introduction

Many engineering activities of human societies are based on natural rock mass, utilizing their solidity to create many large-scale, structurally complex and long-lasting geotechnical engineering environments such as mines, energy storage reservoirs and rocky slopes [1]. In fact, during construction and service, surrounding rock mass is subjected to different levels of dynamic disturbances (earthquakes, blasting and rock drilling, etc.) at different times [2]. Figure 1 illustrates the effects of multi-level cyclic dynamic loading on surrounding rock during multiple blasting of an open-pit slope. Dynamic strength and damage type of rock materials are closely related to strain rate, in situ stress and environmental conditions [3]. Therefore, it is unreasonable for safety procedures to still use dynamic properties of rocks under single or repeated cyclic dynamic loading as an engineering reference, ignoring the actual conditions in which surrounding rocks are exposed to multi-frequency and multi-level dynamic loading.

Currently, many researchers have conducted extensive research on dynamic mechanical properties of rock materials at medium and high strain rates (10¹s⁻¹~10³s⁻¹), and investigated dynamic strength, deformation, and fracture properties of rocks under single loading at varying loading rates [4,5,6,7,8,9]. In addition, considering in situ stress level and environmental factors, some scholars have simulated field conditions and prepared damaged rock by applying static loads [10,11], dynamic loads [12,13], coupled static-dynamic loads [14,15,16], heating treatment [17,18,19], freeze–thaw cycles [20,21,22] and wet–dry cycles [23,24,25] to investigate dynamic properties of rocks under different working conditions. However, frequent dynamic disturbances can cause a continuous effect on surrounding rock, that is, damage accumulation effect, and researchers have started to investigate crack propagation process, mechanical property degradation and energy dissipation characteristics of rock under repeated cyclic dynamic loading [26,27,28,29,30,31,32,33,34,35]. For example, Zhu et al. [27] conducted uniaxial cyclic impact compression tests on granite specimens using a modified large-diameter SHPB device. They found that the rate of increase in cumulative damage varied from small to large with the increase in number of impacts, and main increment in cumulative damage of rock before failure was generated by the last impact. Dai et al. [30] and Wang et al. [31] conducted cyclic impact tests with dynamic and static coupled loading on pre-fabricating cracks on granite using SHPB systems that can apply axial and radial static stresses, respectively, to study dynamic characteristics of granite in a complex stress environment. Wang et al. [32] carried out uniaxial cyclic impact tests on granite specimens heat-treated at different temperatures: 20 °C, 200 °C, etc. Results showed that there were different damage thresholds for heat-treated specimens at different temperatures, and when impact load was less than the threshold, damage caused by repeated impacts was not significant. Liu et al. [34] investigated microstructural characteristics of rocks after cyclic dynamic impacts using nuclear magnetic resonance (NMR). Results showed that the cyclic dynamic impact process favors the formation of small and medium-sized pores in rock and reduces the number and size of large pores.

These creative efforts have advanced our knowledge of the dynamic response of rock under cyclic impact. However, whether it is multiple blasting or repeated rock drilling, impact loads applied to surrounding rock are not the same each time. Therefore, it is necessary to study the dynamic response of rocks under complex cyclic dynamic loading to accurately assess the degree of dynamic disturbance, damage, and failure of rocks in real engineering environments. For example, Xie et al. [36] have developed a device capable of coupling triaxial dynamic impact and triaxial prestress with a true triaxial electromagnetic Hopkinson rod system, which provides a new test platform to carry out research on three-dimensional rock dynamics. Wu et al. [37] utilized an SHPB system to investigate damage anisotropy characteristics of marble under different impact directions and loads and explored how loading direction affects the damage mechanism. Yan et al. [38] and Zhang et al. [39] attempted cyclic dynamic impacts with progressive loading on weathered granite and red sandstone, respectively, in order to study permeability characteristics and energy dissipation features of rocks under complex dynamic loading environments. Although some studies have been conducted on the dynamic mechanical behavior of rocks under complex dynamic loading conditions, only a few studies have examined dynamic response and fatigue damage characteristics of rock under cyclic impacts with multiple energy levels. To address this research gap, this study conducted a test to explore dynamic response and fatigue damage characteristics of rock under cyclic impacts with multiple energy levels, which can provide important guidance to understand crushing efficiency, and damage and load-bearing capacity of rocks in real engineering environments.

In this work, granite specimens were first subjected to single-impact pre-testing at different energy levels using a modified SHPB system to obtain failure threshold intervals. Subsequently, cyclic impact tests were conducted on specimens at different energy change rates for the same, linear increasing, linear decreasing, increasing–decreasing and decreasing–increasing energy levels. Effects of cyclic paths and energy change rates on dynamic mechanical properties and deformation behaviors of granite were investigated. Based on results obtained from high-speed cameras and SEM, fracture characteristics of macro- and micro-cracks in specimens under different impact modes were studied. In addition, evolution of energy dissipation and average strain rate during multi-level impact was analyzed, and cumulative energy dissipation was used to evaluate the degree of rock damage under different impact modes.

2. Experimental Procedures

2.1. Specimen Preparation

The fine-grained granite used in this study was collected from Zhangzhou (Fujian, China), and its basic parameters are provided in

Table 1. Basic physical and mechanical parameters.

Properties	Values
Density $\rho$ (kg/m3)	2801.64
Longitudinal wave velocity $V_P$ (m/s)	5543.28
Young modulus $E_S$ (GPa)	71.33
Poisson ratio $v$	0.23
Uniaxial compressive strength ${\sigma }_{\mathrm{c}}$ (MPa)	183.95
Brazilian tensile strength ${\sigma }_{\mathrm{t}}$ (MPa)	12.11

. Specimens were processed to 50 mm in diameter with a length-to-diameter ratio of 1:1, and the two end surfaces had a parallelism error of less than 0.02 mm. Specimens with similar P-wave velocities (varying less than 5%) were selected using an ultrasonic tester (HS-YS4A) for the tests.

2.2. Experimental Methods

2.2.1. Dynamic Test System and Principle

The main instrumentation used for the test is presented in Figure 2. This setup consisted of a modified SHPB system, the specific details of which are provided in the previous work [40]. Moreover, a high-speed camera (Vision Research Inc., Wayne, NJ, USA—Phantom V711) was connected to the SHPB test system via an oscilloscope (Yokogawa Electric Corporation, Tokyo, Japan—DL850E) to synchronize the recording of the specimen failure process under high strain rate, and a waveform storage system was used to receive and store incident, reflected and transmitted wave signals from the oscilloscope in real time.

According to one-dimensional stress wave theory, when displacement velocities of particles on both sides of the elastic rod–specimen interface are continuous and internal forces of the rods on both sides of the interface are equal, stress, strain, and strain rate of the specimen in the system at moment t can be expressed as follows:

$${\sigma }_s(t)={\displaystyle \frac{A_{\mathrm{e}}}{2A_s}}[{\sigma }_I(t)-{\sigma }_R(t)+{\sigma }_T(t)],$$

(1)

$$\epsilon (t)={\displaystyle \frac{1}{{\rho }_e{C}_e{L}_s}}{\displaystyle {\int }_0^{\tau }[{\sigma }_I(t)-{\sigma }_R(t)+{\sigma }_T(t)]}dt,$$

(2)

$$\dot{\epsilon }(t)={\displaystyle \frac{1}{{\rho }_e{C}_e{L}_s}}[{\sigma }_I(t)-{\sigma }_R(t)+{\sigma }_T(t)]$$

(3)

where ${\sigma }_I(t)$, ${\sigma }_R(t)$ and ${\sigma }_T(t)$ correspond to incident, reflected and transmission stress at time instant t, respectively. ${\rho }_{\mathrm{e}}$, $E_e$, $A_e$ and $C_e$ denote density, elastic modulus, cross-sectional area and longitudinal wave velocity of the elastic rod, respectively. $\tau$ is duration of the stress wave, and $A_s$ and $L_s$ are cross-sectional area and length of specimen, respectively.

2.2.2. Multi Energy Level Cyclic Impact Test

For analysis of cyclic impact tests performed at different energy levels and cyclic paths, it should be ensured that cumulative incident energy is as constant as possible. In the SHPB system, kinetic energy from the shaped striker by pneumatic acceleration is the source of incident energy, and the striker impacts and transfers energy to the incident bar in a relatively short period of time, where the kinetic energy loss in this process can be neglected [14]. Theoretically, when the position of the striker in the cavity is fixed, incident energy is proportional to magnitude of impact air pressure through same cumulative impact air pressure maintenance; total incident energy was approximated controlled, so as to apply multi energy level cyclic (MELC) impact load to rock. Air pressure thresholds were determined by pre-testing of a single one-dimensional impact compression. Test design is shown in

Table 2. Design of multi-level cyclic impact test.

Cyclic Path		Specimen No.	Pressure Level Design
Linear	SEL	S1~S3
	IEL-0.05	I1~I3
	IEL-0.1	I4~I6
	DEL-0.05	D1~D3
	DEL-0.1	D4~D6
Nonlinear	IDEL-0.05	ID1~ID3
	IDEL-0.1	ID4~ID6
	DIEL-0.05	DI1~DI3
	DIEL-0.1	DI4~DI6

, and all specimens are categorized into five groups based on loading paths, with same energy level (SEL), increasing energy level (IEL), decreasing energy level (DEL), increasing–decreasing energy level (IDEL) and decreasing–increasing energy level (DIEL) cyclic loading. Additionally, two different energy change rates are set up under the same path, and three specimens are utilized for each rate to carry out repetitive tests; cyclic loading was stopped once the designed loading path was completed.

3. Results and Discussion

3.1. Dynamic Stress Balance Verification

Figure 3 demonstrates the dynamic stress equilibrium at both ends of typical specimen S1. Superposition of incident and reflected wave signals shows stresses at the specimen’s incident bar interface, and transmitted wave signal shows stresses at the specimen’s transmitted bar interface. Signals of stresses at two ends of the specimen during the impact test reflect the following relationship: ${\sigma }_I+{\sigma }_R\approx {\sigma }_T$. Therefore, we can consider that axial inertia effect is reduced to a negligible level during the impact test and deformation of the specimen is approximately in stress equilibrium [41].

3.2. Stress Waveform Characteristics of MELC Impact

Figure 4 shows the stress waveforms obtained by MELC impact loading. All waveform curves are smooth and appear bell-like, amplitude of incident wave is close between same energy level loading, and incident wave amplitude between different energy level loading shows significant variations, which indicates that it is feasible to control the magnitude of incident energy for MELC dynamic loading through impact air pressure. Ju et al. [42] simulated the same triangular incident wave for cyclic impact, and results showed that transmitted wave amplitude gradually became smaller while reflected wave amplitude gradually became larger. Variation in trends of stress wave amplitudes of specimen S1 in Figure 4a supported this finding.

With varying incident energy levels, changes in patterns of transmitted and reflected wave amplitudes in MELC impacts tend to be complicated. IEL cyclic impact, transmitted and reflected wave amplitudes of specimen I1 increased continuously (Figure 4b), and incident energy emerged as the dominant factor influencing changes in transmitted and reflected wave amplitudes. The difference is that in IDEL and DIEL cyclic impacts, amplitude changes in transmitted and reflected waves affected by both the magnitude of incident energy and the number of impacts, which manifested by amplitude of transmitted and reflected waves generated by the impact of a high energy level, are generally larger than that of a lower energy level. However, in the process of two impacts before and after the same energy level, the amplitude of the transmitted wave generated by the latter impact is smaller and the amplitude of the reflected wave is larger.

3.3. Dynamic Mechanical Response of Granite Under MELC Impact

3.3.1. Dynamic Stress–Strain Relationship

Dynamic stress ${\sigma }_s(t)$ and strain $\dot{\epsilon }(t)$ of the specimen at each impact can be obtained using Equations (1) and (2), respectively. Figure 5 illustrates stress–strain curves in the MELC compression test. During each impact, specimens experienced five distinct stages: initial compaction, linear elastic deformation, stable crack propagation, stress collapse and hysteretic rebound. Among them, the initial compaction stage was generally short, and duration decreased with the increase in impact air pressure, indicating that granite specimens would be compacted at a faster rate during high-energy impact. In addition, curves showed different degrees of rebound after reaching peak strain, and elastic strain energy was released during this period. The rebound loop shape depends on both the number of impacts and the cyclic loading path, as shown in Figure 5i, which shows that rebound loops do not have same morphology even though specimen S1 was impacted with the same air pressure at each time. The slope of the rebound segment after peak strain reflects the local elastic modulus, indicating that cyclic impacts caused different deformations to the specimen and the cumulative rate of damage within the specimen was variable. At the same impact air pressure and number of impacts, Figure 5c,d used different loading paths with different morphologies of rebound loops. This suggests that the cyclic path can influence the cumulative velocity of damage in granite.

Although there are two identical energy levels during the nonlinear loading process, their strain peaks differ. For example, strain peaks of the fifth impact in Figure 5e,f are all larger than the first one, strain peaks of the fourth impact are all larger than the second one, and strain peaks of the latter impact of the same energy level in Figure 5g,h are all larger than the previous one. From the perspective of damage accumulation, the accumulation of cyclic impacts progressively compromises the rock’s deformation resistance capacity, thus, same energy level stress loads can cause more strains.

3.3.2. Deformation Characteristics of Granite

As shown in Figure 6, peak stress in each of the cyclic impacts of same energy level is close, and the overall trend is an initial increase followed by a decrease. The first impact compacts the specimen and improves its resistance to external impact load. However, with an increase in impact number, peak stress value began to decline; the 4th impact of specimen S4 decreased by 3.6% compared with the 1st impact, where the specimen showed a deterioration of dynamic strength. Peak stress in MELC impacts shows an obvious strain rate effect, and increase or decrease in stress peak corresponds to the trend of the energy level. Results in Figure 6b show that in two impacts of same energy level before and after the same specimen, the peak stress of the latter one shows a decrease. For example, the 5th impact of specimen ID4 decreased by 25.7% compared to the 1st one, and the 6th impact decreased by 10.9% compared to the 2nd one, which indicates a progressive degradation in rock’s resistance to external cyclic impact loading.

At high strain rates, elastic deformation contributes minimally to the behavior of brittle hard rocks during loading, and the linear elastic response phase is too short to represent dynamic deformation behavior of rock. The study examined stress and strain values of specimens during the loading stage; the stress–strain curve of the rock is relatively flat near the 50% peak stress point and has a long period of stable deformation. Defining the modulus at this point helps minimize the influence of initial nonlinearities. Therefore, the tangent slope at 50% peak stress was defined as the overall deformation modulus of the loading stage [43], which can be used to evaluate deformation behavior under dynamic loading.

Figure 7a shows that the deformation modulus of the loading stage of specimen S1 decreases gradually with increasing impact numbers, but an increase occurs at the last impact. Analyzed from a microstructural perspective of rock, the stress concentration of the tensile wave at tips of pre-existing or newly formed microcracks exceeded the rock strength, promoting crack initiation and extension. Damage accumulation reduced the dynamic strength of rock, which was manifested by a gradual decrease in the deformation modulus of the loading stage. During the final macroscopic failure of the rock, coalescence of microcracks led to an increase in the deformation modulus.

Research also indicated that the degree of rock damage and strain rate jointly affect the deformation modulus of the loading stage, where the effect of damage degree was mainly in the early and middle stages of cyclic impact. For instance, peak stresses of specimens I1 and I4 increased with the increase in impact energy level and strain rate. But compared to initial value, the deformation modulus of the loading stage of specimen I1 only increased by 0.08 GPa at the third impact, and the deformation modulus of specimen I4 even decreased, which indicates that there is a slowing down process of propagation of internal cracks in rock. The specimens in Figure 7b exhibited crack propagation deceleration during the third impact, which means some of the microcracks caused by the first two impacts were closed during the third impact. However, when impact number increases, the degree of rock damage gradually increases; microcracks have been partially coalesced or newly nucleated, closure of old cracks is slower than generation and propagation of new cracks, and dynamic strain rate or energy level becomes the main factor affecting the deformation modulus of the loading stage. In addition, statistical analysis was conducted using three replicate samples from each group, and Figure 8 and Figure 9 provide the mean value and standard deviation of peak stress and the deformation modulus, respectively. The results show that the standard deviation of the sample data is small and concentrated, and the standard deviation is generally large during high-energy loading. As the number of cyclic impacts increases, the trend of peak stress and the deformation modulus of the loading stage remain consistent with the above experimental results.

3.4. Cyclic Fracturing Behavior and Failure Modes

3.4.1. Macroscopic Crack Propagation During Impact Process

To study crack evolution of granite under MELC impact, the camera’s frame rate and resolution were set at 79,161 fps and 256 × 256 pixels, respectively, and the specimen was constrained from rotational movement along the circumference to capture the fixed strain region during entire cyclic impact.

Table 3. Cyclic impact process under linear energy levels.

Impact No.	Specimen Type
	S3	I5	D5	I2	D2
1st impact
2nd impact
3rd impact
4th impact		End of impact test	End of impact test		Complete damaged
5th impact		End of impact test	End of impact test		Complete damaged
6th impact		End of impact test	End of impact test	End of impact test	End of impact test

Note: Dynamic loading direction: horizontally from right to left.

demonstrates the cyclic impact process of rocks under linear energy level paths, where specimen S3 was cyclically impacted at the same energy level. Applying linearly increasing and decreasing energy levels of impact loading to specimens presents opposite results. Specimens I5 and I2 withstood all impact levels without complete failure and developed fewer tensile cracks. However, specimen D5 developed a through-going macroscopic crack at the last impact and specimen D2 encountered failure before completing five cycles of impact testing. This suggests that dynamic loading with linearly decreasing energy levels contributes to the crack initiation and propagation in rock, which is significantly faster than linearly increasing energy level loading, and crack propagation in rocks subjected to the same energy level loading falls in between these two. In addition, the rate of energy change has a limited effect on crack propagation rate. I5 formed longer tensile cracks in early impacts, but the length of the cracks was close to that of the final cracks formed in I2, and an early appearance of long cracks may have been related to the premature application of high-energy impacts in cyclic impacts.

Unlike linear energy level cyclic loading, damage and failure behavior of rocks under nonlinear energy level cyclic impact is more complicated. As shown in

Table 4. Cyclic impact process under fluctuating energy levels.

Impact No.	Specimen Type
	S3	DI5	ID5	DI2	ID2
1st impact
2nd impact
3rd impact
4th impact
5th impact
6th impact				End of impact test	End of impact test

Note: Dynamic loading direction: horizontally from right to left.

, nonlinear energy level cyclic loading resulted in more cracks, and most cracks propagated axially parallel to the specimen’s height direction. As the stress wave propagated, tensile stress was concentrated at the free surface, inducing radial tensile cracks. In addition, nonlinear energy level cyclic impact exacerbated damage at the end of the specimen, and coalescence of microcracks at the end caused increased rock spalling. The accumulation of spalling gradually reduced the overall strength of the specimen, eventually causing complete failure. Based on results of the first two impacts in Table 4, specimins DI5 and DI2 demonstrate that high energy impact contributes to the propagation and coalescence of microcracks within rocks. However, tensile cracks of rock after experiencing multiple nonlinear energy level impacts exhibited consistent results; the number of cracks is higher than that of specimen S3 under same energy level loading, which indicates that energy alternation in loading can accelerate deterioration of the dynamic strength of rock compared to constant energy level loading.

3.4.2. Microscopic Fatigue Failure in Granite

To investigate microscopic fatigue characteristics of specimens under different loading modes, scanning electron microscopy was performed on fractured specimens [44]. Typical SEM results for a selected magnification of 500 are shown in Figure 10, where Figure 10a,b shows specimens fractured by a single impact in pre-testing. The fracture surface is flat and dominated by brittle fracture; most of the cracks propagate along boundaries between mineral grains to form intergranular fracture, and localized stress concentration leads to transgranular fracture, pores, and rock debris. Figure 10c,d demonstrates microstructures of specimens S1 and S2 after fracture, respectively, and results show that specimens under SEL cyclic impacts are progressive-failure dominated by one primary crack and secondary cracks are scattered in each stress concentration region. In addition, fatigue striation is concentrated in the ductile fracture region, which is attributed to fatigue of rock by cyclic dynamic impacts, and brittle fracture region is dominated by transgranular fracture. However, as shown in Figure 10e–h, specimens under MELC impact developed multiple major cracks to form a fracture network, with secondary cracks generating and intersecting. Some intersections of cracks appeared to be damaged by macro-pores, which was probably due to varying stress concentration zones from different energy level impacts inside the rock and induced fatigue damage in multiple areas. This conclusion is confirmed by the increase in regions showing fatigue striations.

3.5. Energy Evolution Characteristics of Granite Under MELC Impact

3.5.1. Evolution Process of Energy Dissipation

Granite absorbs, transforms and transfers energy during MELC impact tests, leading to macro-to-micro scale failure and fragment ejection; thus, understanding energy evolution in rock is essential for analyzing rock damage. Incident energy $E_I$, reflected energy $E_R$ and transmitted energy $E_T$ of an arbitrary impact stress wave are denoted as follows:

$$E_i={\displaystyle \frac{A_e}{{\rho }_e{C}_e}}{\displaystyle {\int }_0^{\tau }{\sigma }_i^2(t)}dt,i=I,R,T$$

(4)

where ${\rho }_e$ is the density of the elastic rod. ${\sigma }_I$, ${\sigma }_R$ and ${\sigma }_T$ are incident stress, reflection stress and transmission stress, respectively. The values of $A_e$ and $C_e$ are the same as in Equations (1) and (2).

According to conservation of energy, total absorbed energy $E_S$ by the specimen during impact can be expressed as follows:

$$E_s=E_I-E_R-E_T$$

(5)

where $E_I$, $E_R$ and $E_T$ are incident, projected and reflected energy, respectively.

In impact tests, total dissipated energy $E_D$ is converted into various forms of energy, including acoustic and thermal energy, in addition to causing rock damage and fracture and fragment ejection [45]. It is generally accepted that damage and fracture behavior contribute 95% of total absorbed energy, with the other categories totaling less than 5%. By estimating kinetic energy of major fragments, it was found that they account for less than 5% of total energy dissipation [46]. Therefore, total dissipated energy $E_D$ can be obtained from an approximation of total absorbed energy $E_S$ [47]. To evaluate energy dissipation characteristics during cyclic impact, energy dissipation density is defined as the dissipated energy per unit volume of specimen $E_V$ and is expressed as follows:

$$E_V={\displaystyle \frac{E_D}{V_S}}.$$

(6)

Equations (4)–(6) were used to calculate energy dissipation density $E_V$. Figure 11 statistically shows the relationship between average strain rate and energy dissipation density under MELC impact. Results show that strain rates vary significantly across different cyclic paths and energy levels, with a maximum extreme difference of 68.05 s⁻¹. $E_V$ increases linearly with average strain rate, and the adjusted R² of this linear regression model is 0.7456, indicating a strong positive correlation which is similar to that of results for granite under a single impact [40].

Furthermore, the evolution patterns of energy dissipation density and average strain rate across different loading processes were investigated. As shown in Figure 12a,b, the established linear regression models apply to both IEL and DEL cycles, with increasing impact number. Average strain rate and energy dissipation density in the IEL cycle both increase linearly, while they decrease linearly in the DEL cycle. In addition, slope magnitude of the energy dissipation density curve reflects the energy dissipation rate, and this dissipation rate increases with energy level, e.g., the energy dissipation rates of IEL-0.05 and DEL-0.05 are half of IEL-0.10 and DEL-0.10, respectively, and the rates of IEL-0.05 and IEL-0.10 are smaller than DEL-0.05 and DEL-0.10, respectively. This phenomenon suggests that the order of participation of energy levels affects the energy dissipation rate during MELC impacts, and premature application of high energy impacts accelerates energy dissipation rate in the specimens. Under the cyclic impact of nonlinear energy levels, there is no significant linear relationship between the average strain rate, energy dissipation density and the number of impacts. However, they showed synchronous trends and a certain positive correlation. This is attributed to the fact that with increasing average strain rate, cumulative damage level of rock increases and more internal microcracks appear, and these stress-concentrated zones will dissipate more energy during further damage, thus, energy dissipation density increases.

3.5.2. Cumulative Damage Characteristics

Long-term dynamic responses of geotechnical structures cause progressive damage accumulation. Under cyclic impact, rock undergoes progressive strength degradation through fatigue damage, and is eventually destroyed due to its inability to resist external loads. Accumulation of incident energy under different cyclic paths and impact energy levels is statistically presented in Figure 13. Results show that when average air pressure is the same, the mean accumulated incident energies across groups show <5% variation, and standard deviations are less than 20 J. For example, mean values of total incident energy of the first two impacts in the SEL, IDEL and DIEL cycles are 142.63, 154.27 and 155.53 J. Differences in total incident energy of MELC impacts are insignificant, which ensures comparability in studying damage degradation of the specimens under different energy level allocations.

Damage to a rock during impact will consume a certain amount of energy, and its damage evolution is a progressive accumulation of dissipated energy with increasing cycles, which is usually irreversible. Therefore, when controlling for the same incident energy, the energy dissipation serves as a quantitative indicator for assessing rock damage under cyclic loading [33]. The expression for the damage variable defined by cumulative energy dissipation is given below:

$$D_E={\displaystyle \frac{{\displaystyle \sum _{i=1}^N{E}_{vi}}}{E_{total}}}$$

(7)

where $E_{\mathrm{v}}$ is the energy dissipation per unit volume, N is the number of cyclic impacts and $E_{total}$ is the total sum of dissipated energy per unit volume of rock during the entire cycle test.

Figure 14 illustrates cumulative damage levels of rocks obtained using Equation (7) for different loading modes. Results show that there is a significant difference in trend in damage accumulation when the initially intact specimen undergoes MELC impact. As shown in Figure 14a, the third cyclic impact resulted in complete failure for all three groups of specimens, DEL-0.10, DEL-0.05 and IIEL+0.10, despite having equal cumulative incident energy. However, accumulated damage in IEL-0.05 and SEL groups of specimens had not yet exceeded 50%, indicating that early involvement of high energy levels can significantly accelerate accumulation of damage and overall failure. Under same energy change rate, the cumulative rate of damage in granite showed a DEL > IEL > SEL cycle, and under the same cyclic path, the cumulative rate of damage was faster at higher energy change rates than that at lower energy rates. In addition, contributions of IDEL, DIEL and SEL cyclic loading paths to rock damage are close to each other, and the energy change rate becomes a key factor affecting damage accumulation rate. As shown in Figure 14b, c, during the first four cyclic impacts, accumulated incident energies of the specimens were the same; however, the specimens accumulated damage faster at lower energy change rates than at higher ones. This phenomenon occurs because the rate of change decreases and more high-energy level impacts are involved in early loading, which accelerates fatigue damage accumulation.

4. Conclusions

(1)

Under MELC impacts, different paths significantly influence accumulated plastic strain rate in rock. Peak dynamic stress increases with impact energy level, and for two equal-energy impacts on the same specimen, the latter showed a decrease in peak stress, which indicates the ability of rock to resist decreasing external impact loading. In addition, the deformation modulus during the loading stage was introduced to analyze dynamic strain characteristics of granite. Results showed that deformation during the loading stage increased with the increase in impact energy level but decelerated during intermediate cycles, and the degree of damage and strain rate together affected the magnitude of the deformation modulus during the loading stage.

(2)

The crack evolution of the specimens during MELC impact was studied based on simultaneous recording with a high-speed camera. Under a linear energy level path, the rate of tensile crack initiation and propagation was influenced by early involvement of high energy levels, with the following ranking: DEL > SEL > IEL cycle. Under a nonlinear energy level path, each group developed more tensile cracks and rock spalling than the SEL cycle. SEM results show that the specimens exhibit fatigue damage dominated by one primary crack under SEL cyclic impacts, while under MELC impacts the damage manifested as a fracture network with multiple primary cracks and more regions showing fatigue striations. The rocks showed less resistance to dynamic fatigue under MELC impacts than SEL cycle impacts.

(3)

In the MELC impact test, a clear linear correlation exists between energy dissipation density and average strain rate. Damage coefficients were calculated based on the cumulative energy dissipation density of granite before and after impact. For linear energy level loading, it was found that the cumulative rate of damage in rock at the same energy change rate followed a DEL > IEL > SEL cycle. For the same cycling path, damage accumulation rate is faster at higher energy change rates than at lower energy change rates. In addition, under nonlinear energy level loading, the contribution of IDEL and DIEL cyclic loading paths to rock damage are similar, and the cumulative damage rate is greater at lower energy change rates than at higher ones.

(4)

This study covers the dynamic response of rocks under multi-level cyclic impact and compares the fatigue damage and failure of rocks under four different energy level change paths, providing a reference for the dynamic research of rocks under complex stress states. However, experiments under controlled conditions cannot fully replicate the complex conditions in which natural rocks are situated. This variability may be caused by various factors, such as simplified stress paths, material anisotropy and complex engineering geological environments, leading to conclusions exhibiting non-applicability. It is necessary to use methods such as multi-physics coupling and multi-scale damage observation to provide more case studies.