Abstract
To clarify the influence of pulverized coal gradation on the cyclic damage behavior of grouted consolidated bodies, this study used cement-pulverized coal composites as the research object. Three pulverized coal contents (10%, 20%, and 30%) and three pulverized coal particle radii (300 μm, 75 μm, and 28 μm) were designed, and cyclic loading-unloading tests, damping-energy separation analysis, and PFC numerical simulations were performed. Based on an equivalent single-degree-of-freedom model, a damping-energy calculation method was established, and a modified damage factor considering the damping effect was proposed to distinguish damping energy consumption from damage energy consumption within the hysteretic dissipated energy. The results show that, when the pulverized coal particle radius decreases from 300 μm to 28 μm, the specimen density increases by 9.94%–12.24%, whereas slurry fluidity decreases by 28.26%–82.93%. This indicates that fine pulverized coal can improve the compactness of the consolidated body but reduces slurry workability. The uniaxial compressive strengths of specimens with different gradations range from 10.17 to 46.04 MPa, suggesting that 28 μm pulverized coal is beneficial for enhancing the load-bearing capacity of the specimen, whereas a pulverized coal content of 30% weakens the continuous cemented structure of the cement matrix. During cyclic loading, the total mechanical energy density, elastic energy density, and damping energy density all increase nonlinearly with the number of cycles. The modified damage factor is lower than the conventional energy-based damage factor, indicating that ignoring the damping effect overestimates the actual damage degree of the material. PFC moment-tensor analysis shows that specimen failure is dominated by tensile cracks, with tensile cracks accounting for 28.00%–39.00%, whereas shear cracks and compression-closure cracks account for 21.00%–30.00% and 21.00%–28.00%, respectively. The results indicate that pulverized coal gradation controls the cyclic damage evolution of grouted consolidated bodies by altering particle packing, interfacial friction, damping dissipation, and crack propagation. These findings provide a quantitative basis for evaluating the vibration resistance of coal-pillar grouted reinforcement under cyclic mining-induced loading.
1 Introduction
During coal mining, the stability of coal pillars is a key factor in ensuring safe production (). Coal pillars must not only bear the static load of the overlying rock mass but also withstand cyclic vibrations generated during mining activities (; ; ). These cyclic vibrations further reduce coal pillar stability and thereby affect coal mine safety (; ). Grouting is a key measure for improving coal pillar stability. However, in a cyclic vibration environment, grouted reinforcement bodies may still undergo residual deformation accumulation, enhanced energy dissipation, and continuous damage evolution (; ; ). Therefore, conducting cyclic loading-unloading tests to investigate the mechanical properties of grouted reinforcement bodies under cyclic vibration has important engineering significance.
In coal-pillar grouting reinforcement systems, the macroscopic mechanical properties of the reinforced body depend not only on the slurry material itself but are also closely related to the particle size and content of pulverized coal (; ; ). Fine-grained cementitious materials can improve slurry penetration into pores and fractures in fractured coal-rock masses and enhance the compactness and reinforcement effect of the consolidated body (). revealed the grouting modification mechanism of fractured coal from the perspectives of interfacial porosity and microstructure, showing that interfacial pore structure and cementation morphology directly affect the overall performance of grouted coal. In addition, the pulverized coal content of cement-based fractured coal consolidated bodies can significantly alter peak strength, cohesion, internal friction angle, energy dissipation, and damage-deformation behavior (; ). However, existing studies have mostly focused on grouting material proportioning, slurry diffusion, consolidation strength, and interfacial microstructure. A systematic understanding is still lacking regarding how the particle size and content of pulverized coal affect the residual deformation, energy dissipation, and damage evolution of grouted reinforcement bodies under vibration.
Cyclic loading-unloading tests are an important method for studying the mechanical behavior and damage evolution of coal-rock, rock, and cement-based composites under cyclic vibration. Under cyclic loading, part of the externally input energy is stored inside the specimen as elastic strain energy, while the remainder is dissipated through pore compaction, particle frictional sliding, plastic deformation, crack initiation, and crack growth. Therefore, damage characterization methods based on energy evolution have been widely used to describe the progressive failure process of materials. Earlier studies established rock damage constitutive models under cyclic loading based on dissipated energy and revealed the seepage, acoustic emission, and energy dissipation characteristics of coal-rock under stepwise cyclic loading (; ; ; ). In recent years, research on coal-rock damage evolution under cyclic loading-unloading conditions has been further developed. investigated the mechanical characteristics and energy-damage evolution of coal specimens during cyclic loading and unloading, while analyzed failure and acoustic emission characteristics of coal under stepwise cyclic loading and unloading stress paths. These studies demonstrate that dissipated energy and residual deformation can effectively reflect the accumulation of internal material damage (; ; ). However, for cement-based composites containing pulverized coal particles, the hysteresis-loop area is not fully equivalent to damage energy; it also includes damping energy caused by the viscous effect of the material. If the entire hysteretic dissipated energy is directly used to calculate the damage factor, the actual damage degree of the specimen may be overestimated, thereby affecting the assessment of the load-bearing capacity of grouted reinforcement bodies.
The damping effect is an important component of energy dissipation in composites under cyclic vibration. Conventional cyclic constitutive models generally decompose the material response into elastic, viscous, and plastic components to describe hysteretic behavior under cyclic loading (; ; ). However, existing cyclic constitutive models usually involve many parameters and complex formulations, making them difficult to apply directly to the separation of damping energy and damage energy in grouted reinforcement bodies with different pulverized coal gradations. Although acoustic emission monitoring and particle discrete-element numerical simulation provide effective means for identifying mesoscopic cracks and analyzing damage mechanisms (; ; ), few studies have integrated the damping-corrected damage factor, residual deformation, energy evolution, and mesoscopic crack types to explain the mechanisms by which the particle size and content of pulverized coal affect the vibration resistance of grouted reinforcement bodies.
Based on the above issues, this study takes cement-based pulverized coal composite specimens with different pulverized coal particle sizes and contents as the research object and systematically conducts cyclic loading-unloading tests, energy evolution analysis, damping-corrected damage factor calculations, and PFC mesoscopic crack simulations. First, a damping-energy calculation method is developed based on a single-degree-of-freedom forced vibration model to separate damping energy from total dissipated energy during the cyclic hysteresis process, and a modified damage factor considering the damping effect is established. Second, three pulverized coal particle sizes and three pulverized coal contents are designed to analyze the evolution of specimen strength, residual strain, energy density, and damage factor under different pulverized coal gradations. Finally, PFC numerical simulations and moment-tensor acoustic emission analysis are combined to identify the proportions of tensile cracks, shear cracks, and mixed cracks in different models. This study aims to clarify the mechanisms by which pulverized coal particle size and content affect the vibration resistance and damage evolution of grouted reinforcement bodies, thereby providing theoretical reference for coal-pillar grouting reinforcement design and stability evaluation under cyclic mining-induced loading.
2 Materials and methods
2.1 Damping energy calculation model
At present, there is a relatively large amount of research on the cyclic loading and unloading model (; ; ). In general, these models are an effective combination of elasticity, viscosity and plasticity. However, plasticity can interfere with the calculation of damping energy. Therefore, in this paper, when constructing the damping energy calculation model, the material is simplified into a single degree of freedom forced vibration model. That is, the whole system is simplified into equivalent mass units, equivalent elastic units and equivalent damping units. The simplified model is shown in Figure 1.
FIGURE 1
It should be noted that the single-degree-of-freedom system adopted in this study is an equivalent mechanical model rather than a complete three-dimensional constitutive model for cement-pulverized coal composites. The purpose of this simplification is not to describe the local stress distribution inside the specimen but to separate damping-related energy from the macroscopic hysteresis curves obtained from cyclic loading-unloading tests and further distinguish it from damage-related energy. In the cyclic loading-unloading tests, the specimens are mainly subjected to axial compression, and the macroscopic response recorded by the testing system is primarily represented by the stress-strain relationship. Therefore, from the perspective of overall energy balance, the heterogeneous specimen can be equivalently represented as a single-degree-of-freedom system consisting of an equivalent mass, equivalent stiffness, and equivalent damping. In this equivalent framework, the stiffness term represents the overall ability of the specimen to resist axial deformation, whereas the damping term represents non-damage energy dissipation caused by viscous effects, internal friction, and deformation lag during cyclic loading. Therefore, the simplified model has a clearly defined scope of applicability. The proposed model is suitable for evaluating the overall damping energy and modified damage factor of the specimen based on macroscopic cyclic stress-strain curves, but it cannot explicitly consider material heterogeneity caused by pulverized coal particles, pores, and slurry-coal interfaces, nor can it directly reflect the three-dimensional stress state, lateral deformation, or local stress concentration.
On the simplified mechanical model, combined with the D’Alembert’s principle, inertial forces are applied to the equivalent mass units to obtain the mechanical calculation model as shown in Figure 2.
FIGURE 2
The formula for force balance is expressed as Formula 1.
Using a Fourier series expansion for the cyclic loading and unloading excitation, the excitation force formula is expressed as Formula 2.
Studying a single term of the expansion, Formula 1 can be expressed as follows.
Introduce new variables as shown below.where is the circular frequency, 1/s; is the damping ratio; and are the force per unit mass, N/kg.
The Formula 3 can be further simplified as shown below.
The roots of Formula 5 are shown below.
Using the characteristic root form presented by Formula 6, the hysteresis curve is constructed as shown.where , and are the coefficient to be determined.
Substituting Formula 7 into the differential Formula 5 yields an expression for the coefficients as shown below.
The area of the hysteresis loop formed by Formula 7 is the damping energy.
2.2 Specimens’ preparation
In order to reduce the variability of the experimental process, P.O 42.5 cement with a particle radius of 33 um was selected for the experiment. The pulverized coal used in this study was obtained from a coal mine in Tongchuan, Shaanxi Province, China. According to existing mechanical tests on the coal samples, its density is 1.38 g/cm3, its uniaxial compressive strength is 20.16 MPa, and its elastic modulus is 2.75 GPa.
The specimens in this study were prepared as standard cylinders with a diameter of 50 mm and a height of 100 mm, and the water-cement ratio was 0.4:0.6. This geometry is commonly used in uniaxial compression and cyclic loading-unloading tests because it helps reduce end-constraint effects and ensures a relatively uniform axial stress state in the middle of the specimen. During specimen preparation, cement, pulverized coal, and water were weighed according to the designed proportions, thoroughly mixed to form a uniform slurry, poured into cylindrical molds in layers, and properly compacted to reduce internal bubbles and improve specimen uniformity. After demolding, all specimens were cured under the same environmental conditions for 28 days and then dried to reduce the influence of differences in curing conditions on the test results. Before testing, the upper and lower end faces of each specimen were ground and polished to ensure full contact between the specimen and the loading platens; the non-parallelism of the two end faces was controlled within 30 μm. Specimens with obvious eccentricity, surface damage, uneven end faces, or other visible defects were excluded from subsequent tests to reduce local stress concentration caused by poor end-face contact and to improve the reliability and repeatability of the uniaxial compression and cyclic loading-unloading test results.
Based on the gradation of the pulverized coal, the specimens were numbered as follows C1R1, C2R1, C3R1, C1R2, C2R2, C3R2, C1R3, C2R3, C3R3. Here, C1, C2 and C3 represent pulverized coal contents of 10%, 20% and 30%, respectively, and R1, R2 and R3 represent pulverized coal particle radii of 300 μm, 75 μm and 28 μm, respectively. The selection of pulverized coal particle size was also based on variable control and engineering representativeness. In this study, particle radii of 300 μm, 75 μm, and 28 μm were selected to represent coarse, medium, and fine pulverized coal, respectively. The 300 μm pulverized coal represents the relatively coarse fine-particle fraction in fractured coal; the 75 μm pulverized coal corresponds to a commonly encountered fine pulverized coal scale; and the 28 μm pulverized coal is close to the particle size of P.O 42.5 cement, allowing the role of fine pulverized coal and cement particles in pore filling, particle packing, and interfacial cementation to be analyzed. The three particle sizes gradually transition from coarse pulverized coal to a scale close to that of cement particles, covering typical ranges of slurry workability, consolidated-body compactness, and interfacial structure changes during pulverized coal refinement. Specifically, coarse pulverized coal particles are more likely to form pores and local interfacial mismatches; medium-sized pulverized coal reflects the role of conventional fine pulverized coal in cement-based consolidation; and fine pulverized coal facilitates the filling effect and optimization of particle gradation. By combining three contents (10%, 20%, and 30%) with three particle radii (300 μm, 75 μm, and 28 μm), this study systematically analyzes the coupled effects of pulverized coal content and particle size on slurry fluidity, specimen compactness, cyclic load-bearing capacity, residual deformation, energy evolution, and crack development. Therefore, this parameter combination is not a simple scaled representation of the overall composition of an in-situ coal pillar; instead, it is designed to reveal, under controlled laboratory conditions, the mechanism by which pulverized coal gradation affects the vibration resistance of grouted consolidated bodies.
The end faces of the specimens are shown in Figure 3. The flowability of different gradation slurries and the density of the specimens after 28 days of curing and drying were tested, and the physical properties of the specimens are shown in Figure 4.
FIGURE 3
FIGURE 4
When the pulverized coal is coarse, the coal particles can be clearly seen on the specimen surface. This phenomenon becomes more pronounced as the amount of pulverized coal increases. If the pulverized coal is fine, it will fully integrate with the cement, resulting in a smoother and more polished surface.
When the pulverized coal particle size is the same, both slurry fluidity and specimen density decrease as the pulverized coal content increases from 10% to 30%. For the R1 group, fluidity decreases from 230 mm to 205 mm, a reduction of 10.87%, and density decreases from 1.61 g/cm3 to 1.38 g/cm3, a reduction of 14.29%. For the R2 group, fluidity decreases from 225 mm to 190 mm, a reduction of 15.56%, and density decreases from 1.67 g/cm3 to 1.52 g/cm3, a reduction of 8.98%. For the R3 group, fluidity decreases from 165 mm to 35 mm, a reduction of 78.79%, and density decreases from 1.77 g/cm3 to 1.54 g/cm3, a reduction of 12.99%. These results indicate that increasing the pulverized coal content simultaneously weakens slurry fluidity and the compactness of the hardened specimens. Among them, the fine-particle pulverized coal group shows the most pronounced decrease in fluidity, indicating that fine pulverized coal has a more sensitive influence on slurry workability under high-content conditions.
When the pulverized coal content is the same, slurry fluidity decreases markedly and specimen density generally increases as the pulverized coal particle size decreases from 300 μm (R1) to 28 μm (R3). In the C1 group, fluidity decreases from 230 mm to 165 mm, a reduction of 28.26%, while density increases from 1.61 g/cm3 to 1.77 g/cm3, an increase of 9.94%. In the C2 group, fluidity decreases from 220 mm to 130 mm, a reduction of 40.91%, while density increases from 1.47 g/cm3 to 1.65 g/cm3, an increase of 12.24%. In the C3 group, fluidity decreases from 205 mm to 35 mm, a reduction of 82.93%, while density increases from 1.38 g/cm3 to 1.54 g/cm3, an increase of 11.59%. This indicates that pulverized coal refinement has a dual effect on the slurry and hardened body. On the one hand, fine pulverized coal has a larger specific surface area, adsorbs more free water, and increases interparticle frictional resistance, thereby reducing slurry fluidity. On the other hand, fine pulverized coal can more fully fill pores in the cement matrix, improve particle packing density, and make the hardened specimen structure more compact.
The uniaxial compression tests and cyclic loading-unloading tests in this study were both performed using a DDL-200 electronic universal testing machine. The device has a maximum test force of 200 kN and supports force control, displacement control, and cyclic loading, satisfying the loading requirements for uniaxial compression and stepwise cyclic loading-unloading tests of the cement-pulverized coal composite specimens. During testing, the data acquisition system of the testing machine recorded load, displacement, and time data in real time, which were then used to calculate stress-strain curves, cyclic hysteresis curves, residual strain, and energy density. The test-force indication accuracy is ±1%, and the displacement measurement accuracy is ±0.5%, which meet the accuracy requirements for analyzing load-displacement response and energy evolution during cyclic loading and unloading. The uniaxial compression tests were conducted under force control at a loading rate of 500 N/s. The uniaxial compressive stress-strain curves of the specimens are shown in Figure 5.
FIGURE 5
Under the same pulverized coal mixing condition, as the radius of the pulverized coal particles decreases, the peak strength and ultimate strain of the specimen gradually increase. When the radius of the pulverized coal particles is the same, the peak strength of the specimen decreases and the ultimate strain gradually increases with the increase of the pulverized coal blending amount. This indicates that blending pulverized coal with smaller particle radius and higher blending amount can effectively improve the toughness of the specimen.
In order to analyze the cyclic loading and unloading mechanical properties and damage evolution of specimens under different stress amplitudes, a cyclic loading and unloading experimental design was developed as shown in Figure 6.
FIGURE 6
Design of test for cyclic loading and unloading: set the stress upper limits at 15 kN, 30 kN, 45 kN, 60 kN and 75 kN for 5 cyclic loading and unloading cycles. Each step consists of 5 cycles. After the 25th cycle, continue loading until the specimen is com-pletely destroyed. To ensure adequate contact between the specimen and the test ma-chine, the minimum contact force of the press machine shall be set at 1 kN. These cyclic stress settings have clear experimental and engineering significance. After conversion, the five cyclic stress levels increase stepwise from 7.64 MPa to 38.20 MPa, covering the main loading stages of the specimen from low-stress compaction, through intermediate-stress damage development, to high-stress conditions approaching failure. The low-stress stage mainly corresponds to pore compaction and initial crack closure, the intermediate-stress stage corresponds to microcrack initiation, interfacial sliding, and damage accumulation, and the high-stress stage corresponds to rapid crack propagation and unstable failure. Combined with the uniaxial compressive strength of each specimen, the above stress levels generally cover 20%–80% of the uniaxial strength of most specimens, which is broadly consistent with the stepwise stress levels commonly used in coal-rock cyclic loading-unloading tests.
3 Results
3.1 Mechanical performance
The uniaxial strength of each specimen was obtained as shown in Table 1. The cyclic loading and unloading test curves were obtained as shown in Figure 7.
TABLE 1
| Specimen no. | C1R1 | C2R1 | C3R1 | C1R2 | C2R2 | C3R2 | C1R3 | C2R3 | C3R3 |
|---|---|---|---|---|---|---|---|---|---|
| Uniaxial compression/MPa | 31.97 | 17.74 | 10.17 | 36.37 | 28.88 | 16.69 | 46.04 | 36.97 | 36.77 |
| Cyclic loading and unloading/MPa | 30.50 | 20.70 | 13.20 | 41.12 | 29.86 | 22.30 | 55.34 | 44.00 | 36.49 |
Peak strengths of specimens.
FIGURE 7
The strength of the specimen in the cyclic loading and unloading test is generally higher than that in the uniaxial compression test. There are three reasons for this. First, due to the closure of internal microcracks and micropores in the specimen under lower cyclic stress, the specimen has a larger effective load-bearing area under subsequent loading, thereby improving the ultimate strength. Secondly, the damping energy gradually increases during cyclic loading and unloading, consuming the energy storage within the specimen, allowing the specimen to withstand greater external energy input, thereby improving the ultimate strength. Thirdly, due to the longer cycle of the cyclic loading and unloading test, the internal cracks of the specimen have sufficient time to develop, reducing the probability of initiation of long-field cracks, thereby improving the ultimate strength of the specimen.
From the final cyclic stage before failure, the total mechanical energy density of specimens with different gradations ranges from 0.0494 to 1.6446 MJ/m3, the elastic energy density ranges from 0.0422 to 1.4408 MJ/m3, the dissipated energy density ranges from 0.0072 to 0.2037 MJ/m3, and the damping energy density ranges from 0.0023 to 0.0609 MJ/m3. Specimens with low pulverized coal content and fine particle size can withstand more cycles and accumulate higher energy, indicating stronger cyclic load-bearing capacity and energy storage capacity. In contrast, specimens with high pulverized coal content or coarse particles enter the failure stage after fewer cycles, showing weaker vibration resistance.
Under the same number of cycles, increasing the pulverized coal content significantly amplifies hysteretic dissipation. Taking the fifth cycle of the R1 group as an example, when the pulverized coal content increases from 10% to 30%, the dissipated energy density increases from 0.0038 MJ/m3 to 0.0072 MJ/m3. In the 10th cycle of the R2 group, the dissipated energy density increases from 0.0123 MJ/m3 to 0.0354 MJ/m3. In the 20th cycle of the R3 group, the dissipated energy density increases from 0.0601 MJ/m3 to 0.1473 MJ/m3. This indicates that, after the pulverized coal content increases, the number of coal-cement interfaces and weakly bonded zones increases, making interfacial sliding, particle dislocation, and deformation lag more likely during cyclic loading, thereby enlarging the hysteresis-loop area in Figure 7.
The influence of pulverized coal particle size on hysteretic dissipation is mainly reflected in differences in structural compactness and local stress concentration. Under the same content and at the same cycle step, reducing the pulverized coal particle size usually decreases early- or middle-stage hysteretic dissipation. For example, in the 10th cycle at a content of 20%, as the pulverized coal particle size decreases from R1 to R3, the dissipated energy density decreases from 0.0235 MJ/m3 to 0.0127 MJ/m3. In the fifth cycle at a content of 30%, the dissipated energy density decreases from 0.0072 MJ/m3 to 0.0056 MJ/m3. This indicates that fine pulverized coal can improve pore filling and particle packing, reduce interfacial mismatch and local stress concentration, and thereby decrease hysteretic energy consumption during the early stage of cyclic loading. However, because fine-particle specimens can withstand more cycles, their final accumulated energy may still be higher, which is consistent with their stronger cyclic stability and energy storage capacity.
3.2 Energy evolution
Cement is a viscous-elastic-plastic material and during cyclic loading and unloading tests the energy can be divided into total mechanical energy, elastic recovery energy, damage energy and damping energy. The total mechanical energy density is the area of the loading curve and the horizontal coordinate, the elastic energy density is the area of the unloading curve and the coordinate, and the sum of the damping energy density and the damage energy density is the area of the hysteresis loop. The energy distribution is shown in Figure 8.where is the total mechanical energy density, ; is the elastic energy density, ; is the dissipative energy density, ; is the damage energy density, ; is the damping energy density, .
FIGURE 8
During the test, the total mechanical energy of the specimen is the sum of elastic energy and dissipated energy (Equation 9). The dissipated energy consists of damping energy and damage energy (Equation 10).
Figure 9 shows the evolution of total mechanical energy density, elastic energy density, and damping energy density of specimens with different pulverized coal gradations during cyclic loading and unloading. Overall, all three types of energy density increase continuously with the number of cycles and show obvious nonlinear accumulation characteristics. In the low-stress cyclic stage, the energy density increases slowly, indicating that this stage is dominated by pore compaction, initial crack closure, and elastic energy storage. As the cyclic stress level increases stepwise, the energy growth rate increases markedly, suggesting that interfacial sliding, microcrack initiation, and microcrack propagation gradually develop inside the specimen. The total mechanical energy density is always higher than the elastic energy density, and the difference between them reflects the accumulation of irrecoverable energy dissipation during cyclic loading and unloading. Although the damping energy density is lower than the total mechanical energy density and elastic energy density, it increases synchronously with the number of cycles, indicating that viscous damping, interfacial friction, and deformation lag continuously participate in cyclic energy consumption of the specimen. Therefore, damping energy should not be simply incorporated into damage energy when calculating the damage factor.
FIGURE 9
Pulverized coal content has a significant influence on energy evolution. When the pulverized coal particle size is the same, as the pulverized coal content increases from 10% to 30%, the number of cycles that the specimen can withstand generally decreases, and the accumulated energy before failure also decreases markedly. For the R1 group, the 10%, 20%, and 30% specimens withstand 20, 10, and 5 cycles, respectively; their total mechanical energy densities before failure are 0.8834, 0.1875, and 0.0494 MJ/m3, their elastic energy densities are 0.7375, 0.1640, and 0.0422 MJ/m3, and their dissipated energy densities are 0.1459, 0.0235, and 0.0072 MJ/m3, respectively. For the R2 group, the three specimens withstand 25, 15, and 10 cycles, respectively; the total mechanical energy density before failure decreases from 1.4311 MJ/m3 to 0.2259 MJ/m3, the elastic energy density decreases from 1.2558 MJ/m3 to 0.1905 MJ/m3, and the dissipated energy density decreases from 0.1753 MJ/m3 to 0.0354 MJ/m3. In the R3 group, the 10% and 20% specimens both complete 25 cycles, whereas the 30% specimen fails after 20 cycles; their total mechanical energy densities before failure are 1.3893, 1.6446, and 1.0689 MJ/m3, respectively. These results indicate that excessive pulverized coal content weakens the cyclic load-bearing capacity and energy accumulation capacity of the specimen.
Pulverized coal particle size mainly affects energy evolution through pore filling, particle packing, and local stress concentration. When the pulverized coal content is the same, fine-particle pulverized coal specimens usually exhibit stronger energy storage capacity and cyclic stability. For example, C1R2 and C1R3 can complete more cycles, and their total mechanical energy density and elastic energy density are higher than those of C1R1. C2R3 also shows stronger cyclic load-bearing capacity than C2R1 and C2R2. Fine pulverized coal has a larger specific surface area and a better filling effect; it can fill pores in the cement matrix, improve particle packing, and enhance the compactness and structural uniformity of the hardened specimen. A denser structure can reduce local stress concentration, delay microcrack coalescence, and allow more external input energy to be stored inside the specimen as elastic energy. By contrast, coarse pulverized coal particles tend to form larger pores, interfacial mismatches, and local stress concentration zones. During cyclic loading, they are more likely to induce crack growth, particle frictional sliding, and interfacial debonding, thereby reducing energy accumulation capacity and accelerating failure.
The variation in damping energy further reflects the influence of pulverized coal gradation on the cyclic energy-consumption mechanism. Before failure, the damping energy density ranges from 0.0023 to 0.0609 MJ/m3. In the R1 group, the damping energy densities of the 10%, 20%, and 30% specimens are 0.0287, 0.0090, and 0.0023 MJ/m3, respectively; in the R2 group, they are 0.0609, 0.0187, and 0.0088 MJ/m3, respectively; and in the R3 group, they are 0.0601, 0.0574, and 0.0450 MJ/m3, respectively. Because damping energy before failure is a cumulative quantity, its magnitude depends not only on hysteretic energy consumption in a single cycle but also on the number of cycles that the specimen can withstand. Fine-particle specimens can undergo more cycles, and therefore their cumulative damping energy is usually higher. In contrast, high-content coarse-particle specimens may show stronger early-stage hysteretic energy consumption, but because they fail earlier, their final accumulated damping energy is relatively low. Thus, damping energy is not controlled solely by the magnitude of cyclic loading; rather, it is jointly determined by relative stress level, pore structure, interfacial friction, deformation lag, and cyclic life.
In summary, the influence of pulverized coal gradation on energy evolution is not controlled by a single factor. Although a higher pulverized coal content increases interfacial frictional energy consumption, it also weakens the overall cementation and compactness of the specimen, causing damage to develop earlier. A smaller pulverized coal particle size helps improve pore filling and particle packing, thereby increasing elastic energy storage capacity and cyclic stability. Therefore, fine-particle pulverized coal helps improve the vibration resistance of grouted reinforcement bodies, whereas excessive pulverized coal content may increase weak interfaces and accelerate damage accumulation. In practical coal-pillar grouting reinforcement, pulverized coal particle size, content, slurry fluidity, and consolidated-body compactness should be comprehensively considered to achieve better vibration resistance.
3.3 Residual strains
Figure 10 shows the evolution of residual strain in specimens with different pulverized coal particle sizes and contents during cyclic loading and unloading. Overall, the residual strain of all specimens gradually accumulates with increasing cycles and exhibits stepwise growth after each increase in cyclic stress level. Before failure, the final residual strain of each specimen ranges from 0.96 × 10−3 to 3.89 × 10−3. In the R1 group, the 10%, 20%, and 30% specimens experience 19, 10, and 5 cycles, respectively, with final residual strains of 3.09 × 10−3, 1.34 × 10−3, and 0.96 × 10−3. In the R2 group, they experience 25, 14, and 10 cycles, respectively, with final residual strains of 3.05 × 10−3, 1.43 × 10−3, and 1.80 × 10−3. In the R3 group, they experience 25, 25, and 20 cycles, respectively, with final residual strains of 2.17 × 10−3, 3.07 × 10−3, and 3.89 × 10−3. These results indicate that fine-particle pulverized coal specimens can withstand more cycles and accumulate larger residual deformation over a longer loading process, reflecting stronger cyclic deformation-bearing capacity. In contrast, coarse-particle or high-content specimens often enter the failure stage within fewer cycles.
FIGURE 10
From the perspective of pulverized coal content, increasing the content under the same particle-size condition increases the early-stage accumulation rate of residual strain. Taking the fifth cycle as an example, the residual strain in the R1 group increases from 0.41 × 10−3 to 0.96 × 10−3, an increase of 134.15%; in the R2 group, it increases from 0.42 × 10−3 to 0.56 × 10−3, an increase of 33.33%; and in the R3 group, it increases from 0.39 × 10−3 to 0.90 × 10−3, an increase of 130.77%. At the 10th cycle, the residual strains of the 10% and 20% specimens in the R1 group are 0.69 × 10−3 and 1.34 × 10−3, respectively, with the 20% specimen being 94.20% higher than the 10% specimen. In the R2 group, the residual strains of the 10%, 20%, and 30% specimens are 0.63 × 10−3, 0.98 × 10−3, and 1.80 × 10−3, respectively. In the R3 group, they are 0.60 × 10−3, 0.50 × 10−3, and 1.40 × 10−3, respectively. These results show that a high pulverized coal content increases the number of coal-cement interfaces and the proportion of weakly bonded zones, making interfacial sliding, particle dislocation, and deformation lag more pronounced during cyclic loading and thereby accelerating residual strain accumulation.
From the perspective of pulverized coal particle size, pulverized coal refinement has a dual effect on residual strain. Under the same content and at the same cycle step, fine-particle pulverized coal usually reduces early-stage residual strain. Taking the 10th cycle as an example, in the 10% content group, residual strain decreases from 0.69 × 10−3 to 0.60 × 10−3 as the particle size decreases from R1 to R3. In the 20% content group, it decreases from 1.34 × 10−3 to 0.50 × 10−3, a reduction of 62.69%. In the 30% content group, the residual strains of the R2 and R3 specimens are 1.80 × 10−3 and 1.40 × 10−3, respectively, with R3 being 22.22% lower than R2. This indicates that fine pulverized coal can fill pores in the cement matrix, improve particle packing, and enhance specimen compactness and structural uniformity, thereby reducing early-stage irrecoverable deformation. However, fine-particle specimens can withstand more cycles and thus may still develop large cumulative residual strain after high-stress cycling. For example, in the R3 group, the final residual strains of the 10%, 20%, and 30% specimens reach 2.17 × 10−3, 3.07 × 10−3, and 3.89 × 10−3, respectively. Therefore, residual strain is not controlled by pulverized coal particle size or content alone, but by the combined effects of pore filling, interfacial bonding strength, particle frictional sliding, number of cycles, and relative stress level. Overall, appropriate refinement of pulverized coal helps reduce early-stage residual deformation and improve cyclic stability, whereas excessive pulverized coal content increases weak interfaces and residual deformation accumulation, which is unfavorable for long-term vibration resistance.
4 Moment tensor acoustic emission simulation
A two-dimensional mesoscopic model of the cement-pulverized coal composite specimen was established using the PFC discrete-element method. The model dimensions were consistent with the axial section of the laboratory cylindrical specimen, with a width of 50 mm and a height of 100 mm. The particles in the model were divided into cement particles and pulverized coal particles. The cement particles were used to represent the cemented skeleton formed after hydration of P.O 42.5 cement, whereas the pulverized coal particles were used to represent pulverized coal components with different particle sizes and contents. In the experiment, the average particle radius of P.O 42.5 cement was 33 μm, while the pulverized coal particle radii were 300 μm, 75 μm, and 28 μm, corresponding to coarse, medium, and fine pulverized coal, respectively. Because the real particle scale is much smaller than the specimen scale, explicit modeling using the actual particle sizes in a 50 mm × 100 mm model would result in an excessively large number of particles and significantly increase computational cost. Therefore, this study adopted a proportional coarse-graining approach for PFC modeling, in which the particle radii were uniformly enlarged while maintaining the relative particle-size relationship between cement particles and pulverized coal particles. Specifically, the equivalent radius of cement particles was set to 0.165 mm, and the equivalent radii of R1, R2, and R3 pulverized coal particles were set to 1.50 mm, 0.375 mm, and 0.14 mm, respectively. This treatment preserves the gradation characteristics that R1 pulverized coal is significantly coarser than cement particles, R2 pulverized coal is slightly coarser than cement particles, and R3 pulverized coal is close to the cement particle scale. It can therefore be used to analyze the effects of pulverized coal refinement on pore filling, particle packing, and interfacial contact state.
Different pulverized coal contents were implemented by controlling the proportion of pulverized coal particles in the total particle mass of the model. C1, C2, and C3 correspond to pulverized coal mass fractions of 10%, 20%, and 30%, respectively. During particle generation, the area ratio and number ratio of the two particle types were first determined according to the densities of pulverized coal and cement and the target mass fraction. Cement particles and pulverized coal particles were then randomly generated within the model domain, and the model was brought to a stable initial state through servo compaction. With an initial porosity of 0.08 as the control condition, the generated models contained 4.56 × 104 to 5.92 × 104 particles. In the coarse pulverized coal group, the number of pulverized coal particles was smaller because each pulverized coal particle was larger; in the fine pulverized coal group, the number of pulverized coal particles increased markedly because each pulverized coal particle was smaller. These differences in particle number and particle-size distribution can reflect, at the mesoscopic scale, the influence of different pulverized coal gradations on the structure of the cement-pulverized coal composite.
The parallel bond model (PB model) was used to describe the cementation between particles, and different contact parameters were assigned according to particle type. Cement-cement contacts were assigned relatively high parallel-bond stiffness and bond strength to represent the continuous cemented skeleton formed by cement hydration products. Pulverized coal-pulverized coal contacts were assigned lower bond strength and a higher friction coefficient to represent the weaker cementation and frictional sliding behavior between pulverized coal particles. Cement-pulverized coal contacts were assigned intermediate bond strength to represent the coal-cement interfacial bonding zone.
By adjusting the normal stiffness, shear stiffness, parallel-bond tensile strength, cohesion, and friction coefficient of the three contact types--cement-cement, pulverized coal-pulverized coal, and cement-pulverized coal--the simulated uniaxial compressive stress-strain curves were matched with the laboratory test results (Table 2). Figure 11 compares the uniaxial compression test curves and PFC simulation curves of the nine specimen groups. The simulation curves can reasonably reproduce the elastic deformation stage and peak strength level of specimens with different pulverized coal gradations, indicating that the two-particle modeling method and the calibration results of the PB parameters are reasonable.
TABLE 2
| Contact type | Normal stiffness /GPa | Shear stiffness /GPa | Tensile strength /MPa | Cohesion /MPa | Friction coefficient |
|---|---|---|---|---|---|
| Cement-cement | 6.0 | 4.0 | 18.0 | 36.0 | 0.45 |
| Cement-pulverized coal | 4.0 | 2.7 | 10.0 | 20.0 | 0.55 |
| Pulverized coal-pulverized coal | 2.5 | 1.7 | 5.0 | 10.0 | 0.65 |
PB model parameters.
FIGURE 11
Acoustic emission techniques are effective in providing continuous, non-destructive monitoring of specimens throughout the experiment. Indoor experiments are affected by factors such as electronically controlled press noise and specimen flatness, reducing the comparability of results. However, numerical simulation can effectively overcome this shortcoming, while particle flow discrete element software (PFC) offers significant advantages in dealing with rock fracture and damage evolution by monitoring the kinetic energy of particles around the bond fracture and using it to directly quantify the acoustic emission energy emitted by the source. Finally, by clustering the fracture of multiple bonds in space and time, the acoustic emission characteristics of the specimens can be obtained. The crack cloud diagrams obtained for the specimens at different pulverized coal levels are shown in Figure 12.
FIGURE 12
The crack distributions in the figure reflect the mesoscopic fracture characteristics of models with different pulverized coal gradations after cyclic loading. Far-field fine cracks mainly refer to small-scale, non-through cracks that are distributed away from the dominant macroscopic crack band and loading boundaries. They can be used to characterize local damage and distributed fracture states before the formation of through-going failure in the model. For the R1 coarse pulverized coal group, as the pulverized coal content increases from C1R1 to C3R1, the number of distributed fine cracks in the model gradually decreases, while local macroscopic cracks become more obvious. This phenomenon indicates that coarse pulverized coal particles are more likely to form large pores, interfacial mismatches, and local stress concentration zones inside the cement-pulverized coal composite. Under cyclic loading, these weak regions are more likely to develop into localized crack bands rather than uniformly distributed microcracks. Therefore, coarse-particle specimens exhibit stronger brittle failure characteristics and a more concentrated crack propagation pattern. By contrast, R3 fine pulverized coal can more fully fill pores and improve the particle packing structure, making the internal structure of the model denser and more uniform. Therefore, before the formation of macroscopic cracks, the model can develop more distributed local deformation and damage, showing relatively better deformation coordination capacity.
To further identify the fracture mechanisms of different models, this study used the moment-tensor acoustic emission method to classify crack types. The theoretical basis of the moment-tensor acoustic emission method is derived from the acoustic emission source mechanism theory in elastodynamics. When microcracks initiate, propagate, or when interfaces become unstable within a material, transient elastic waves are released. Such an acoustic emission event can be regarded as an equivalent microseismic source, and its fracture mechanism can be characterized by a second-order moment tensor. Moment-tensor analysis can not only locate acoustic emission sources but also identify crack types and crack orientations. It has been widely used for analyzing mesoscopic fracture mechanisms in concrete, rock, and rock-like materials (). In acoustic emission moment-tensor theory, the elastic-wave displacement received at an observation point can be expressed as Formula 11 ():where u is the displacement response at the observation point x; G is the spatial derivative of the Green’s function; ξ is the acoustic emission source location; and M is the acoustic emission source moment tensor; t is the time. For a microcrack event, the moment tensor can also be written as Formula 12:where C is the elastic stiffness tensor, bk is the relative displacement vector of the crack surface, and nl is the normal vector of the crack surface. This expression indicates that the moment tensor contains information on both the normal opening displacement and tangential sliding displacement of the crack surface, and can therefore be used to determine the fracture type of microcracks.
In the PFC numerical simulations, parallel-bond breakage or contact instability between particles can be automatically recorded as a mesoscopic crack event. PFC can automatically track the particle contact state at each calculation step. When the normal or shear stress of a parallel bond exceeds its tensile strength or cohesive strength, the program identifies the bond as failed and records the crack location, time, fracture mode, and corresponding changes in contact force. Previous studies have shown that each particle bond breakage event in PFC can be regarded as an equivalent acoustic emission event, and the particle kinetic energy or contact-force change released during bond breakage can be used to characterize acoustic emission source intensity and further analyze the crack source mechanism using the moment-tensor method (). Therefore, PFC can provide not only the spatial distribution of cracks but also automatic identification and statistics of crack types based on the local mechanical information during bond breakage (Figure 13).
FIGURE 13
When the particle radius is R1, the percentage of tensile cracks first increases and then decreases; the percentage of shear cracks first decreases and then increases; the percentage of mixed cracks gradually decreases. When the particle radius is R2, the percentage of tensile cracks gradually increases; the percentage of shear cracks gradually decreases; and the percentage of mixed cracks gradually increases. When the particle radius is R3, the percentage of tensile cracks first decreases and then increases; the percentage of shear cracks first increases and then decreases; the percentage of mixed cracks first decreases and then increases.
From the perspective of pulverized coal particle size, when the pulverized coal content is the same, the proportion of tensile cracks generally increases as the particle size decreases from R1 to R3. In the 10% content group, the proportion of tensile cracks increases from 30% in C1R1 to 36% in C1R3; in the 20% content group, it increases from 31% in C2R1 to 34% in C2R3; and in the 30% content group, it increases from 28% in C3R1 to 39% in C3R3. This indicates that fine pulverized coal can improve specimen compactness and elastic energy storage capacity. However, during failure, the higher local cementation strength and particle contact stiffness make cracks more likely to release energy rapidly in a tensile mode. Therefore, fine-particle pulverized coal improves load-bearing capacity and structural integrity, but final failure remains dominated by tensile crack propagation.
According to the above change law, it can be seen that the crack type of the model reverses as the particle radius decreases. In order to prevent the model from being damaged by stretching, the proportion of pulverized coal mixture should be suitably reduced when the pulverized coal is coarse, and the proportion of pulverized coal mixture can be suitably increased when the pulverized coal mixture is fine.
5 Discussions
5.1 Factors influencing damping energy
From Formula 8 it can be seen that the damping energy is related to the circular frequency and damping ratio of the material. The hysteresis curves at different circular frequencies and damping ratios are obtained using the control variable method as shown in Figure 14.
FIGURE 14
In the influence diagram of circular frequency on hysteresis loop, as the circular frequency increases, the hysteresis loop rotates counterclockwise, and the strain de-creases continuously, indicating that the stiffness of the material is continuously enhanced, which conforms to Formula 4. Therefore, in actual engineering, when the load borne by the structure remains unchanged, the structure with greater stiffness has less deformation and less damping energy.
As the damping ratio increases in Figure 14b, the area of the hysteresis loop in-creases, appearing fatter in the figure; meanwhile, the stress and strain at the peak point of the curve both decrease with the increase of the damping ratio. This indicates that when the material stiffness and external load remain constant, the increase in the damping ratio makes the damping effect more obvious, and the proportion of damping force increases continuously.
5.2 Damage factor
Damage factors are important physical parameters of damage mechanics, which can objectively reflect the degree of damage to materials in different cyclic stages. After a long period of development, scholars have proposed different damage factor expressions based on residual strain and tangent modulus, but these expressions cannot well reflect the damage of strain-hardening materials. Therefore, a damage factor expression based on dissipated energy is proposed, the expression of which is shown below Formula 13.where is the damage factor of the cycle, ; is the dissipative energy density of the cycle, .
Since Formula 11 does not take into account damping effects, this paper proposed a revised damage factor calculation method that considers damping effects as follows (Formula 14).where is the modified cycle damage factor, ; is the damage energy density of the cycle, ; is the damping energy density of the cycle, .
Obtaining the damage factors under two calculation methods as shown in Figure 15.
FIGURE 15
From the figure can be seen, the damage factor before and after the correction of the development of the same law, all show with the number of cycles and increase, in the same stage of the growth rate gradually reduced the law of change. However, the damage factor after the correction is basically lower than the damage factor before the correction, indicating that the damage factor calculation method before the correction underestimates the load capacity of the specimen, which will cause some economic waste.
Comparing C1R1, C1R2 and C1R3, the damage factor values show C1R1>C1R2>C1R3 for the same number of cycles, again indicating that finer pulverized coal can effectively increase the ultimate load bearing capacity of the specimens and reduce the degree of damage to the specimens. Comparing C1R3, C2R3 and C3R3 for the same number of cycles, the damage factor values show C3R3>C2R3>C1R3, further indicating that higher pulverization reduces the ultimate load bearing capacity of the specimens and increases the damage to the specimens.
5.3 Correlation mechanism among pulverized coal gradation, energy evolution, damage factor, and crack development
The influence of pulverized coal gradation on cyclic damage is not reflected solely through strength or energy indices, but through the coordinated variations among damping energy, the modified damage factor, residual strain, and crack type. For specimens with coarse particles or high pulverized coal content, the cyclic hysteresis-loop area is larger, damping energy and residual strain accumulate more readily, and the modified damage factor increases more markedly. Meanwhile, the PFC moment-tensor acoustic emission results show that this type of specimen contains a higher proportion of tensile cracks, and cracks are more likely to evolve from local initiation to through-going failure. This phenomenon indicates that pulverized coal gradation first changes the initial pore structure and interfacial bonding state of the specimen, and then further controls energy dissipation and damage evolution through interfacial friction, particle dislocation, and crack opening-closing behavior during cyclic loading. In other words, the increase in damping energy is the macroscopic manifestation of interfacial sliding and deformation lag; the increase in the modified damage factor reflects the accumulation of irreversible structural deterioration; and the change in crack type reveals the mesoscopic mechanism by which damage transforms from energy dissipation to fracture failure.
After the pulverized coal content increases, the number of contact interfaces between coal particles and cement hydration products inside the specimen increases, the continuous cementation of the cement matrix is weakened, and weakly bonded zones and potential slip surfaces increase accordingly. During cyclic loading and unloading, these interfaces first undergo frictional sliding and local deformation lag, manifested as increased damping energy and residual strain accumulation. As the cyclic stress level increases, interfacial debonding, pore expansion, and microcrack coalescence are further intensified, causing more input energy to be converted into damage energy and ultimately leading to rapid growth of the modified damage factor. This process is consistent with existing understanding of energy evolution and damage deformation in cemented fractured coal, namely, that an increase in coal particle content changes the strength, interfacial structure, and energy dissipation mode of the cemented body (). Therefore, a high pulverized coal content does not necessarily imply better energy-consumption capacity. When it disrupts the continuous cemented structure, the increased energy consumption is mainly manifested as weak-interface sliding and damage propagation rather than effective vibration-resistant energy storage.
The influence of pulverized coal particle size is mainly reflected in pore filling and local stress concentration. Fine-particle pulverized coal can fill pores in the cement matrix, improve particle packing density and structural uniformity, enable the specimen to store more elastic energy during cyclic loading, and delay rapid microcrack coalescence. Therefore, fine-particle specimens usually show better cyclic stability and a lower damage development rate. By contrast, coarse pulverized coal particles are prone to forming larger pores and obvious interfacial mismatches, making local stress concentration more prominent. During cyclic compression, these heterogeneous regions induce particle dislocation, interfacial debonding, and local tensile stress concentration, thereby promoting tensile crack initiation. The dominance of tensile cracks in the PFC simulation results indicates that, although the specimen is under macroscopic compression, pore compaction, particle rearrangement, and interfacial debonding at the mesoscopic scale can still induce local tensile failure. This also explains why coarse-particle pulverized coal specimens are more likely to exhibit a higher damage factor and more pronounced crack propagation.
Therefore, a progressive relationship exists among damping energy, the damage factor, and crack development. In the early stage of cyclic loading, damping energy mainly originates from pore compaction, interfacial friction, and particle sliding. As the number of cycles and stress level increase, interfacial sliding gradually transforms into interfacial debonding and microcrack propagation, the proportion of damage energy increases, and the modified damage factor rises accordingly. When damage accumulates to a certain degree, local cracks coalesce and ultimately form a failure mode dominated by tensile cracks. Existing cyclic loading-unloading studies have also shown that dissipated energy, residual deformation, and acoustic emission activity can jointly characterize the progressive damage process of coal-rock materials (; ; ). Based on the results of this study, a reasonable pulverized coal gradation should strike a balance between improving filling compactness and avoiding an excessive increase in weak interfaces. Fine-particle pulverized coal helps improve structural compactness and cyclic stability, but excessive pulverized coal content increases interfacial sliding and irreversible damage, which is unfavorable for the long-term vibration resistance of grouted reinforcement bodies.
The prepared cement-pulverized coal composite specimens have uniaxial compressive strengths of 10.17–46.04 MPa, and some specimens have strengths close to or higher than representative raw coal strength. This indicates that these specimens can represent the improved load-bearing capacity of fractured coal pillars after grouting reinforcement. Based on the comparison between representative coal parameters and the strength levels of the specimens in this study, the conclusions are mainly applicable to cement-based grouting reinforcement of medium- and low-strength fractured coal in the Tongchuan mining area of Shaanxi Province and in mining areas with similar mechanical properties. They are particularly applicable to evaluating residual deformation, energy dissipation, damping-corrected damage, and crack evolution of grouted consolidated bodies under cyclic mining-induced vibration. For high-strength coal, coal with limited fracture development, or engineering conditions with significant differences in coal-rock composite structure, the relevant parameters and conclusions still need to be further modified based on field coal samples and grouted consolidated body tests.
6 Conclusion
A damping-energy separation method and a damping-corrected damage factor were proposed for cyclic loading–unloading tests. The corrected damage factor was lower than the conventional energy-based damage factor after excluding damping energy, indicating that ignoring damping effects may overestimate the actual damage degree of cement-pulverized coal composites.
Pulverized coal gradation significantly affected specimen compactness, strength and energy evolution. When the pulverized coal radius decreased from 300 μm to 28 μm, specimen density increased by 9.94%–12.24%, while slurry fluidity decreased by 28.26%–82.93%. The uniaxial compressive strength ranged from 10.17 MPa for C3R1 to 46.04 MPa for C1R3. Finer pulverized coal improved compactness and cyclic bearing capacity, whereas excessive pulverized coal content weakened the cementitious skeleton and accelerated damage accumulation.
Residual strain, damage factor and PFC moment tensor results indicate that cyclic failure is controlled by the coupling of particle packing, interfacial friction and crack propagation. Higher pulverized coal content increased weak interfaces and residual deformation, while finer pulverized coal improved pore filling and delayed crack coalescence. Tensile cracks dominated the failure process; therefore, pulverized coal content should be reduced for coarse particles, whereas an appropriate increase can be considered for fine particles.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
YX: Conceptualization, Funding acquisition, Writing – original draft. WZ: Methodology, Writing – review and editing. QL: Software, Visualization, Writing – original draft. GH: Validation, Writing – original draft. SP: Funding acquisition, Writing – review and editing.
Funding
The author(s) declared that financial support was received for this work and/or its publication. This research was funded by the Project of National Natural Science Foundation of China (grant number: 51904012), Science and Technology Research Program of Chongqing Municipal Education Commission (grant number: KJQN202304402).
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
cyclic loading and unloading, damage factor, damping energy, energy evolution, pulverized coal grading
Citation
Xiong Y, Zhong W, Liang Q, Huang G and Peng S (2026) Study of cyclic loading and unloading energy and damage evolution of different pulverized coal gradations considering damping effect. Front. Mater. 13:1858118. doi: 10.3389/fmats.2026.1858118
Received
17 April 2026
Revised
22 May 2026
Accepted
26 June 2026
Published
22 July 2026
Volume
13 - 2026
Edited by
Zhiwei Ma, Ansteel Beijing Research Institute, China
Reviewed by
Jun Wang, China University of Mining and Technology, China
Wei Yang, Chang’an University, China
Updates
Copyright
© 2026 Xiong, Zhong, Liang, Huang and Peng.
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*Correspondence: Yangtao Xiong, 26724905@alu.cqu.edu.cn
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