Mechanical properties and energy evolution of cemented tailings-rock powder backfill under uniaxial compression: effect of rock powder type and content

Abstract

Economic investment and mechanical characteristics represent primary limitations to the widespread adoption of cemented tailings backfill. This study introduces an approach of using quarry waste to generate rock powder, which is subsequently mixed with tailings to produce cemented tailings-rock powder backfill (CTRPB). This method achieves cost efficiency while facilitating synergistic waste management. Through uniaxial compression testing, the influences of rock powder type and content on the mechanical behavior, failure modes, and energy transformation of CTRPB were examined. A piecewise damage constitutive model was developed to explore damage progression mechanisms. Findings demonstrate that the incorporation of rock powder markedly improves the deformation resistance of CTRPB. Both uniaxial compressive strength (UCS) and elastic modulus exhibit an initial increase followed by a decline as content rises, with optimal levels and enhancement effects differing across various rock powders. The failure process of CTRPB encompassed four distinct phases, where rock powder amplified post-peak energy release and fragmentation behaviors. Energy assessment indicated that rock powder significantly boosted energy storage and dissipation capacities, elevating total energy density, elastic energy density, and dissipated energy density by 252.17%, 213.87%, and 478.99%, respectively. The formulated piecewise damage constitutive model correlated well with experimental data in the pre-peak regime. Damage evolution may be categorized into four stages, and damage values can act as indicators for assessing failure conditions in the resource utilization of mining waste and rock powder-tailings backfill technology. This research offers a theoretical foundation for the resource-oriented use of mine waste and rock powder-tailings backfill technology.

Introduction

The sustainable advancement of the mining industry is severely constrained by the management of tailings, a primary solid by-product generated during mineral processing. A staggering accumulation of over 25 billion tons of such solid waste has been stockpiled in China, a figure that continues to escalate at an estimated rate of 600 million tons annually¹. The considerable accumulation of tailings gives rise to critical environmental and safety concerns. These include placing a strain on land resources and posing severe risks, particularly the potential for dam failures and the extensive pollution of surrounding ecological systems^2–5. Within this framework, the backfill mining method has gained prominence as an effective approach to tailings management, combining safety with economic gains. It addresses the issue of ground pressure control in mined-out zones, averts surface subsidence, and facilitates the conversion of mining solid waste into a resource^6–11.

As a key structural element for maintaining the stability of goaf areas, the long-term stability of cemented backfill directly impacts mine safety production. Extensive research indicates that the mechanical properties of backfill are influenced by multiple internal and external factors^9,12–16. These include intrinsic structure^17–20, cementitious material type^15,21–25, and chemical additives^14,26–28. Among these, uniaxial compressive strength serves as a core indicator for evaluating backfill performance, and its enhancement has become a primary focus of scholarly research^29–37. Extensive experiments reveal that incorporating chemical additives or fiber materials into mixtures significantly enhances strength, toughness, and ductility while effectively inhibiting crack propagation and limiting post-failure strength degradation^38–44. For instance, Xue et al^21,41. systematically investigated the influence of fiber parameters on filling strength and established a strength prediction model; Chen et al²². revealed the reinforcement mechanism of polypropylene fibers through microstructural analysis; Cao et al⁴². compared the improvement effects of different fibers on the early-age properties of fillers. These studies confirm that the reinforcement effect of fibers primarily depends on factors such as tensile strength, interfacial bonding properties with the matrix, and distribution characteristics.

Although chemical additives and fiber materials can significantly enhance the performance of cemented backfill, their high costs and potential environmental impacts have restricted large-scale application in mines. In contrast, improving backfill performance by utilizing locally generated mine waste is a more economical and environmentally friendly strategy. Notably, for mines equipped with both stopes and quarries, the preparation of cemented backfill by blending rock powder processed from quarry waste (e.g., waste rock) with tailings exhibits distinct advantages. First, in terms of material properties, unlike fine tailings, which have complex chemical compositions and may interfere with hydration reactions or trigger environmental issues, rock powder is relatively chemically inert and stable, with its primary mechanism of action being physical filling rather than chemical activity. Second, in terms of economic and environmental benefits, producing rock powder from locally sourced quarry waste results in nearly zero raw material costs, thus achieving the goal of “treating waste with waste”. Although energy is consumed during the grinding process, compared with long-term reliance on expensive chemical additives or specialized fibers, this waste resource utilization scheme based on local materials may be more competitive in terms of life-cycle costs. Furthermore, it can simultaneously consume two types of solid waste, which is more in line with the development concept of green mines.

However, existing research lacks a systematic elucidation of the intrinsic mechanisms through which rock powder, as a physical filler, affects the mechanical behavior of cemented backfill. In particular, it remains unclear how rock powder incorporation modulates the energy storage, dissipation, and release behaviors of the backfill by altering its internal structure, as well as the differential laws governing the strengthening effects and mechanisms among different types of rock powder. Therefore, based on uniaxial compression tests, this paper systematically investigates the effects of rock powder types and dosages on the mechanical properties, failure modes, and energy evolution characteristics of cemented tailings backfill. It aims to reveal the strengthening mechanisms from a novel perspective of physical filling and energy regulation. The research results can provide a more in-depth theoretical basis and direct technical reference for the synergistic resource utilization of mine solid waste.

Materials and tests

Materials

The raw materials used in the experiment included tailings, rock powder, cement, and water. Among them, the tailings were sourced from a tungsten mine in Ganzhou City, Jiangxi Province. The rock powder types consisted of granite powder, basalt powder, and marble powder, these three types of rock are the most common waste materials in quarries and the construction industry. Their selection can meet the practical requirements of synergistic waste management, and provide direct references for the design of filling ratios in mining areas with different geological backgrounds (which generate waste rocks of varying lithologies).

The physicochemical properties of rock powder and tailings were analyzed and tested using a laser particle size analyzer and an X-ray spectrometer. The cumulative distribution results of particle sizes and chemical compositions of granite rock powder, basalt rock powder, marble rock powder, and tailings are shown in Fig. 1. It can be seen from Fig. 1 that the three types of rock powder have similar particle size distributions. Among them, the main chemical components of granite rock powder and basalt rock powder are SiO₂ and Al₂O₃, while those of marble powder are CaO and MgO. For the tailings, the effective particle size (D₁₀) is 5.57 μm, the median particle size (D₅₀) is 64.52 μm, the maximum particle size (D₉₀) is 381.39 μm, the uniformity coefficient is 11.58, and the curvature coefficient is 1.96. The contents of sulfur (S) and chlorine (Cl) in the tailings are relatively low, resulting in stable physicochemical properties and minimal influence on the strength of backfill materials.The cementitious material used in the study is P·O 42.5 ordinary Portland cement.

Fig. 1.

Particle size and chemical composition of rock powder and tailings.

Experimental design and methodology

This study aims to investigate the influence laws of rock powder types and dosages on the strength of cemented tailings backfill. Based on previous tests and the filling industrial practices of typical mines in this region, a baseline mix ratio was designed, with the cement-to-tailings ratio of the slurry being 1:4 and the mass concentration 70%. A control group without rock powder was set up for comparison, and multiple experimental groups were established, in which granite powder, basalt powder, marble powder and other rock powders were used to replace 3%, 6%, 9%, 12% and 15% of the aggregate mass, respectively. The detailed experimental scheme is presented in Table 1.

Table 1.

Experimental design.

Condition Number	Cement-to-sand Ratio	Mass Concentration	Rock Powder Type	Rock Powder Content
Unadded	1:4	70%	-	-
G3	1:4	70%	Granite Rock Powder	3%
G6	1:4	70%	Granite Rock Powder	6%
G9	1:4	70%	Granite Rock Powder	9%
G12	1:4	70%	Granite Rock Powder	12%
G15	1:4	70%	Granite Rock Powder	15%
B3	1:4	70%	Basalt Rock Powder	3%
B6	1:4	70%	Basalt Rock Powder	6%
B9	1:4	70%	Basalt Rock Powder	9%
B12	1:4	70%	Basalt Rock Powder	12%
B15	1:4	70%	Basalt Rock Powder	15%
M3	1:4	70%	Marble Rock Powder	3%
M6	1:4	70%	Marble Rock Powder	6%
M9	1:4	70%	Marble Rock Powder	9%
M12	1:4	70%	Marble Rock Powder	12%
M15	1:4	70%	Marble Rock Powder	15%

Following the experimental procedure illustrated in Fig. 2, slurries were first prepared by mixing according to the mix proportions specified in Table 1, and then poured into cylindrical molds with a height of 100 mm and a diameter of 50 mm. After 24 h, the specimens were demolded, grouped, and labeled sequentially before being transferred to a standard curing chamber for a 28-day curing period. Upon completion of curing, both ends of all specimens were ground to ensure flat and smooth surfaces, after which uniaxial compressive strength tests were conducted. The tests were performed on a WDW-20 electronic universal testing machine, which has a maximum load capacity of 20 kN, an accuracy of ± 1%, and a resolution of 0.1 kN. The loading process was carried out at a constant rate of 0.5 mm/min. To control accidental experimental errors, three parallel tests were set up for each test condition. After the tests, the results were initially screened; if the data dispersion within any individual group was excessively high, supplementary tests were conducted for that specific condition. Finally, the average value of all valid test results was calculated and adopted as the representative value of the mechanical property parameters for the specimen group, which was then used for subsequent analysis.

Fig. 2.

Experimental flowchart.

Experimental results

Effect of rock powder type and content on the mechanical properties of CTRPB

Characteristics of stress-strain curves

Uniaxial compression tests were performed on all mixtures outlined in Table 1, generating the stress-strain relationships plotted in Fig. 4. The curves in subfigures (a), (b), and (c) correspond to CTRPB specimens containing varying amounts of granite, basalt, and marble powder, respectively, while subfigure (d) depicts the characteristic curve of the plain backfill (control group). A consistent four-stage deformation pattern was observed across all specimens: (1) Pore Compaction (OA), where initial compression of internal voids results in an upward-concave curve; (2) Linear Elastic Deformation (AB), characterized by a nearly straight-line relationship between stress and strain, accompanied by microcrack initiation; (3) Plastic Deformation (BC), during which unstable crack propagation leads to peak stress and a subsequent downward-curving trajectory; and (4) Post-Fracture (CD), where macroscopic failure occurs, yet the curve does not drop to zero, indicating residual load-bearing capacity^45,46.

Fig. 4.

Effect of rock powder content on UCS of of CTRPB: (a) Granite rock powder; (b) Basalt rock powder; (c) Marble rock powder; (d) Comparison among rock powder types.

By comparing the stress-strain curves of the cemented tailings-rock powder backfill (CTRPB) specimens and the control group specimens in subfigures (a), (b), and (c), it can be found that the CTRPB specimens yield higher peak stress. This indicates that the incorporation of rock powder can optimize particle gradation, effectively fill the pores between the skeleton particles, and modify the structure and properties of the interfacial transition zone between cement paste and aggregates, thereby enhancing the compactness and homogeneity of CTRPB and further influencing its stiffness, peak strain, and post-failure mechanical behavior. In addition, the slopes of the CD-stage curves of the CTRPB specimens are generally steeper than those of the control group. This also demonstrates that the incorporation of rock powder not only significantly improves the uniaxial compressive strength of the backfill, but also results in a transition of the material properties from ductility to brittleness in the post-peak stage.

Fig. 3.

Stress-strain curves for each condition: (a) Granite rock powder; (b) Basalt rock powder; (c) Marble rock powder; (d) Classic stress-strain curve

Strength and deformation characteristics

Effect of rock powder type and content on the strength of CTRPB

Figure 4 depicts the impact of rock powder variety and content on the strength performance of CTRPB specimens. Data in Fig. 4(d) indicate that the inclusion of each type of rock powder markedly improved the unconfined compressive strength (UCS) of the backfill material. As illustrated in Fig. 4(a), (b), and (c), for a given rock powder type, the average UCS of the CTRPB specimens showed a consistent trend: it first increased and then decreased with rising rock powder content. Specifically, Fig. 4(a) reveals that compared to the control group, the average UCS increased by 26.92%, 29.81%, 44.77%, 23.48%, and 12.25% corresponding to the incremental granite rock powder content. Figure 4(b) demonstrates that the average UCS was enhanced by 37.24%, 45.82%, 69.38%, 22.38%, and 9.73% with higher basalt rock powder content. Similarly, Fig. 4(c) shows increments of 35.64%, 49.23%, 55.87%, 66.58%, and 33.90% for increasing marble powder content.

Analyzed from the microstructure point of view, at the initial stage of rock flour incorporation (3%−9% content) the rock flour particles effectively filled the voids between the tailings skeleton and reduced the initial defects. However, when the rock flour content exceeded the optimal value (9%−15%), the relative dilution of cementitious materials led to the weakening of the interfacial transition zone, which triggered the expansion of microcracks, and this dilution ultimately resulted in the reduction of strength.

In summary, all three rock powders contributed to an increased load-bearing capacity of the backfill. The optimal incorporation levels were identified as 9% for both granite and basalt powder, and 12% for marble powder. The enhancement efficacy decreased in the order: basalt rock powder (9%) > marble rock powder (12%) > granite rock powder (9%).

effect of rock powder type and content on the elastic modulus of CTRPB

The elastic modulus in this study refers to the secant modulus, whose value is obtained by calculating the slope of the stress-strain curve within the linear elastic deformation stage. The relationship between rock powder characteristics (type and content) and the elastic modulus of CTRPB specimens is presented in Fig. 5. Analysis of Fig. 5(a)-(c) shows that, for a given rock powder, the variation in elastic modulus mirrors that of the UCS, characterized by an initial increase succeeded by a decrease at higher contents. A key observation from Fig. 5(d) is the general reduction in elastic modulus for the CTRPB specimens relative to the control. This observed inverse relationship between enhanced UCS and reduced modulus suggests that rock powder addition homogenizes stress distribution, boosting deformation resistance. This property grants CTRPB specimens improved energy absorption and dissipation capabilities under dynamic or cyclic loading conditions.

Fig. 5.

Effect of rock powder type and content on the elastic modulus of CTRPB: (a) Granite rock powder; (b) Basalt rock powder; (c) Marble rock powder; (d) Comparison among rock powder types.

Effect of rock powder type and content on peak strain characteristics of CTRPB

Figure 6 depicts the impact of rock powder type and content on the peak strain characteristics of CTRPB specimens. According to Fig. 6(d), the inclusion of rock powder led to a consistent increase in peak strain. For a given rock powder type, the peak strain typically rose initially and then declined as the content increased. Specifically, Fig. 6(a) shows that the peak strain for granite powder increased by 25.93%, 45.31%, 23.55%, 19.74%, and 30.33% at the respective content levels compared to the control. Similarly, Fig. 6(b) reports increases of 59.10%, 69.92%, 11.66%, 2.26%, and 2.38% for basalt powder, while Fig. 6(c) shows enhancements of 41.14%, 78.84%, 46.74%, 24.98%, and 46.61% for marble powder. Therefore, the addition of rock powder enables the CTRPB specimens to withstand greater deformation and accumulate higher strain energy before failure. This also indicates that the material has greater energy absorption potential when subjected to static or dynamic loads, thereby potentially enhancing its adaptability to the deformation of surrounding rock in goafs and reducing the risk of support failure.

Fig. 6.

Effect of rock powder type and content on peak strain characteristics of CTRPB: (a) Granite rock powder; (b) Basalt rock powder; (c) Marble rock powder; (d) Comparison among rock powder types.

The comparative analysis reveals a distinct mechanical trade-off in CTRPB specimens relative to the control group: a general reduction in elastic modulus is observed alongside enhanced ultimate compressive strength (UCS) and peak strain. This behavior can be attributed to the denser internal microstructure resulting from the incorporation of rock powder. The increased compactness promotes a more uniform stress distribution under compressive loading, which improves the material’s overall deformation resistance. As a consequence, CTRPB specimens possess a greater capacity for energy absorption prior to attaining peak strength, directly contributing to the higher UCS. Nevertheless, two critical limitations must be noted: The improvement in strength-deformation characteristics achieved through rock powder addition has a finite threshold. The significant energy accumulation before peak stress is followed by an abrupt release during the post-peak stage, leading to a pronounced brittle failure mode.

Failure mode analysis

Figure 7 outlines the image analysis procedure for evaluating specimen failure. The right panel demonstrates how exposure and contrast adjustments were applied to the failure image to clarify the primary crack path. In the left panel, solid and dashed lines annotate the main crack propagation and surface peeling features, respectively. Figure 8 displays the typical failure mode of CTRPB specimens, which is predominantly characterized by tensile damage under uniaxial compression. With increasing axial stress, cracks initiate at the specimen ends and propagate steadily, leading to a complete loss of load-bearing capacity. Unlike the minor flaking seen in control specimens, CTRPB specimens underwent severe disintegration, reflecting lower integrity. This difference is mainly due to the greater energy accumulated in CTRPB specimens before failure, which causes a more violent energy release and rock-like fragmentation upon fracture.

Fig. 7.

Image processing method.

Fig. 8.

Failure mode of CTRPB specimen.

Effect of rock powder type and content on the energy evolution characteristics of CTRPB

Energy evolution behavior

Figure 9 depicts the energy calculation model. Specimen failure under uniaxial compression can be regarded as the result of energy accumulation and release. According to the law of energy conservation, external energy is transferred to the CTRPB specimen in the form of mechanical energy, accumulating internally as elastic strain energy, which is subsequently converted into plastic potential energy. Upon reaching peak stress, the accumulated strain energy within the specimen is abruptly released, leading to specimen failure^47–52.

Fig. 9.

Energy calculation model.

Assuming no external heat exchange occurs during the test process, the total strain energy of the specimen should be the sum of elastic strain energy and dissipated energy. This is expressed by the formula:

1

In the formula, U represents the total strain energy, represents the elastic strain energy, and represents the dissipated energy.

The total strain energy accumulated by the specimen is the area enclosed by the stress-strain curve and the horizontal axis^53–56, calculated using the formula:

2

In the formula, represents the axial compressive stress, represents the axial strain, and represent the axial compressive stresses at any two points on the stress-strain curve, and and represent the axial strains at any two points on the stress-strain curve.

The shaded area in Fig. 9 represents the elastic strain energy accumulated by the specimen before reaching peak stress. According to Hooke’s law, the formula for calculating elastic strain energy is as follows^57,58. For computational convenience, the initial elastic modulus can be substituted for the unloading elastic modulus⁵⁹. This is expressed by the formula:

3

In the formula, represents the unloading elastic modulus, and represents the initial elastic modulus.

By combining Eqs. (1), (2), and (3), the formula for calculating dissipated energy is:

4

Energy dissipation characteristics

Following Eq. (1) to (4), the total strain energy, elastic strain energy, and dissipated energy were determined for the complete failure process of all mixtures specified in Table 1. The progression of axial compressive stress and these energy metrics with increasing axial strain is illustrated in Fig. 10. Based on the distinctive features observed in these curves, the energy evolution of the CTRPB specimen can be categorized into distinct stages^52,60.:

Fig. 10.

Evolutionary curve of energy characteristics: (a) Unadded; (b) G3; (c) G6; (d) G9; (e) G12; (f) G15; (g) B3; (h) B6; (i) B9; (j) B12; (k) B15; (l) M3; (m) M6; (n) M9; (o) M12; (p) M15.

(1) Initial energy accumulation stage: Corresponding to segment OA in the figure, elastic strain energy exhibits a concave-up relationship with total energy input, while dissipated energy exceeds elastic strain energy. This primarily results from energy consumption due to pore and crack closure within the specimen. As axial strain increases, elastic strain energy gradually surpasses dissipated energy. This stage corresponds to the pore compaction phase in the stress-strain curve.

(2) Rapid Energy Accumulation Stage: This corresponds to segment AB in the figure. After the elastic strain energy curve crosses the dissipated energy curve, the specimen enters a linear energy storage phase. Within this stage, the growth rate of elastic strain energy gradually increases, and its accumulation exceeds dissipated energy. The slopes of the elastic strain energy curve and the total energy curve become similar. Dissipated energy shows no significant fluctuations during this stage, with its slope tending toward stability. At this point, most of the externally input energy is converted into elastic strain energy. This stage corresponds to the linear elastic deformation stage of the stress-strain curve.

(3) Energy Stabilization Stage: This stage corresponds to segment BC in the figure. As axial strain increases, both elastic strain energy and total energy continue to grow. However, the rate of increase in elastic strain energy gradually slows down, reaching its maximum at peak strength. The externally input energy has reached the specimen’s storage limit. During this stage, the slope of the dissipated energy curve increases, and the accumulation of dissipated energy surges abruptly. This phenomenon occurs because the specimen transitions from elastic to plastic deformation, with most of the externally applied energy being utilized for crack propagation. This stage corresponds to the plastic deformation phase of the stress-strain curve.

(4) Energy Release Stage: This corresponds to segment CD in the diagram. As the primary crack continues to propagate, the specimen’s overall structure becomes unstable, causing the accumulated elastic strain energy to be released as dissipated energy. The curve characteristics in the figure show a rapid decrease in the elastic strain energy curve and a rapid increase in the dissipated energy curve. Ultimately, the dissipated energy curve intersects the elastic strain energy curve again and overtakes it. This stage corresponds to the post-fracture phase of the stress-strain curve.

During uniaxial compression of CTRPB specimens, energy is initially stored as elastic strain energy and subsequently released as dissipated energy. The energy characteristics exhibit strong correlation with the UCS properties of CTRPB specimens, reflecting the enhancement of energy accumulation capacity achieved by incorporating rock powder. Therefore, we define the difference between the peak strain energy density of the CTRPB specimen and the control group as the energy density difference. The ratio of this energy density difference to the control group’s energy density is defined as the energy density optimization rate to investigate the extent to which rock powder optimizes the energy characteristics of CTRPB specimens. The calculation formula is as follows:

5

In the formula, T represents the total energy density optimization rate, represents the elastic energy density optimization rate, represents the dissipated energy density optimization rate, represents the total energy density of the CTRPB specimen, represents the total energy density of the control group, represents the elastic strain energy density of the CTRPB specimen, represents the elastic strain energy density of the control group, represents the dissipated energy density of the CTRPB specimen, and represents the dissipated energy density of the control group. All parameters are measured at peak strain conditions.

Table 2 presents the calculated energy characteristics of CTRPB specimens at peak strain. The table reveals that the incorporation of all three rock powders significantly increased the energy density of the grout specimens. When granite rock powder was used, the CTRPB specimens exhibited increases of 56.51% to 145.66%, 62.84% to 92.58%, and 0.65% to 323.79% in total energy density, elastic energy density, and dissipated energy density, respectively, compared to the control group. For basalt rock powder, the CTRPB specimens exhibited increases of 32.89%–252.17%, 47.55%–213.87%, and − 6.22%–353.89% in total energy density, elastic energy density, and dissipated energy density, respectively, compared to the control group. For marble rock powder, the CTRPB specimens exhibited increases of 130.72%–218.92%, 96.23%–153.13%, and 115.18%–478.99% in total energy density, elastic energy density, and dissipated energy density, respectively, compared to the control group.

Table 2.

Energy characteristics of CTRPB specimens at peak strain.

Group number	U/(kJ·m− 3)	/(kJ·m− 3)	/(kJ·m− 3)	Peak axial strain	T/(100%)	/(100%)	/(100%)
Unadded	8.3	5.665728	2.634272	1.0512	-	-	-
G3	15.22	9.544631	5.675369	1.3238	0.833735	0.684626	1.154435
G6	20.39	9.226214	11.16379	1.5275	1.456627	0.628425	3.237902
G9	19.11	10.34011	8.76989	1.2988	1.30241	0.825028	2.329151
G12	12.99	10.33852	2.65148	1.2587	0.56506	0.824747	0.006532
G15	16.15	10.91131	5.238688	1.35	0.945783	0.925845	0.988666
B3	24.06	17.78294	6.277063	1.6725	1.898795	2.138685	1.382845
B6	29.23	17.27336	11.95664	1.7862	2.521687	2.048745	3.538879
B9	18.88	14.45762	4.422379	1.1738	1.274699	1.551768	0.678786
B12	13.87	9.842974	4.027026	1.075	0.671084	0.737283	0.528705
B15	11.03	8.359537	2.470463	1.0537	0.328916	0.475457	−0.06218
M3	22.87	13.8158	9.054195	1.4837	1.755422	1.438487	2.437077
M6	26.47	14.34177	12.12823	1.88	2.189157	1.53132	3.604016
M9	21.38	14.06996	7.31004	1.5425	1.575904	1.483346	1.774975
M12	19.15	13.48169	5.668309	1.3138	1.307229	1.379516	1.151755
M15	26.37	11.11786	15.25214	1.5412	2.177108	0.9623	4.789888

Overall, the incorporation of all three rock powders enhances the energy storage capacity of the filling material. When dissipated energy approaches equilibrium while elastic strain energy increases, it indicates enhanced elastic recovery capacity, greater load-bearing capability, and superior crack resistance. This enables specimens to maintain stable performance under complex environmental conditions and elevated stresses, making them suitable for engineering applications requiring long-term stability. Conversely, when elastic strain energy approaches equilibrium while dissipated energy increases, it signifies heightened toughness during loading, effectively delaying specimen failure.

CTRPB damage constitutive model

Clearly defining the damage constitutive equation can effectively describe the damage evolution process of the specimen under external force and help predict the failure modes of CTRPB under different load conditions^61–63. For this purpose, this study established damage constitutive models for CTRPB specimens under different rock dust types and content conditions, using the peak stress point as a boundary, to describe the damage evolution process of CTRPB specimens under uniaxial compression conditions.

Damage constitutive model

The distribution of internal pores and microcracks in CTRPB specimens is random. Thus, CTRPB can be regarded as an isotropic continuous medium. The test process is assumed to be adiabatic or approximately adiabatic, with energy losses caused by heat exchange neglected. Energy calculation is based on macroscopic continuum mechanics and the law of conservation. According to the Lemaitre strain equivalence principle^64,65, the stress of the mesoscopic unit of the CTRPB specimen is expressed as follows:

6

In the formula, represents the effective stress of CTRPB, denotes the nominal stress of the CTRPB specimen, and D indicates the damage value.

Gao et al⁶⁶.proposed that when the strength of a specimen’s micro-unit follows a Weibull distribution, the probability density function of the micro-unit strength can be expressed as:

7

In the formula, represents the strength of the microscopic unit, whileand k denote the parameters of the Weibull distribution.

Figure 11 illustrates the failure process of CTRPB under external force. The shaded areas represent damaged and failed sections, while the unshaded regions indicate undamaged sections. The damage value N is defined as the ratio of damaged micro-elements to the total number of micro-elements, expressed as:

8

Fig. 11.

Schematic diagram of failure process.

In the formula, represents the number of damaged microscopic units, and N represents the total number of microscopic units. When, the number of damaged microscopic units is 0; when, the calculation formula for the number of damaged microscopic unitscan be expressed as:

9

Combining Equ. (8) and Equ. (9), the damage value can be defined by the following formula:

10

Assuming the strength of the micronutrient unit satisfies the yield criterion for maximum tensile strain, the maximum tensile strain of the micronutrient unit can be expressed as:

11

In the formula, represents the maximum tensile strain of the microscopic element.

During the initial consolidation phase of the stress-strain curve, no failure occurred in the mesoscopic elements. Therefore, based on the test data from the compaction stage, the constitutive model for CTRPB during the initial consolidation phase can be fitted as follows:

12

In the formula, represents the projection of the stress-strain curve onto the strain axis during the initial compaction stage.

At this point, the strength of the CTRPB microscopic element can be expressed as:

13

is the intercept of the elastic stage extension line with the strain axis. Combining Eqs. (10) and (13), we obtain:

14

Since the elastic modulus represents the slope during the linear elastic stage, the stress-strain relationship in the post-compaction stage can be expressed as:

15

In the formula, represents the elastic modulus of CTRPB, defined as:

16

In summary, the damage constitutive model for the stress-strain curve following the initial consolidation stage can be expressed as:

17

According to the research findings of Liu et al⁶¹., the parameters and k of the Weibull distribution are determined by the peak point of stress. Since the slope of the tangent line at the peak point of the stress-strain curve is zero, it follows that:

18

Where and represent peak stress and peak strain, respectively.

Combining Eqs. (17) and (18), the values of the Weibull distribution parameters k and are solved as follows:

19

20

The constitutive model for CTRPB damage is ultimately defined as follows:

21

In the equation, represents the critical point between the initial consolidation stage and the linear elastic stage.

Model validation and discussion

Table 3 presents the damage constitutive model established based on stress-strain curves, elastic modulus, and peak stress data obtained from uniaxial compression tests.

Table 3.

Damage constitutive model for CTRPB.

Condition Number	Damage constitutive model equation
Unadded
G3
G6
G9
G12
G15
B3
B6
B9
B12
B15
M3
M6
M9
M12
M15

Based on the established damage constitutive model, theoretical stress-strain curves for CTRPB specimens under various operating conditions were obtained, revealing the relationship between damage values and stress-strain values. As shown in Fig. 12, the theoretical stress-strain curve of CTRPB exhibits high agreement with the experimentally obtained stress-strain curve during the pre-peak stage. Furthermore, the damage model can predict the compressive strength of CTRPB under different rock powder types and content conditions. This indicates that the established damage constitutive model provides a theoretical foundation for analyzing the stability of CTRPB and offers guidance for engineering applications.

Fig. 12.

Relationship between stress-strain curve and damage-strain curve: (a) Unadded; (b) G3; (c) G6; (d) G9; (e) G12; (f) G15; (g) B3; (h) B6; (i) B9; (j) B12; (k) B15; (l) M3; (m) M6; (n) M9; (o) M12; (p) M15.

Figure 12 illustrates four damage stages of CTRPB: ① Undamaged stage: The specimen undergoes compaction and partial linear elastic deformation. Internal voids are compacted, and the damage-strain curve remains parallel to the strain axis. ② Initial damage stage: Damage values increase gradually from zero, forming a concave damage-strain curve. ③ Accelerated damage stage: Damage values increase linearly. ④ Stable damage stage: Damage value growth slows until reaching 1. The damage evolution trend of CTRPB under uniaxial compression is characterized as follows: As strain increases, the damage value rises slowly from 0, increases rapidly after reaching peak stress, and finally reaches 1 when CTRPB fails completely.

Test conditions G9 (9% granite powder content), B9 (9% basalt powder content), and M12 (12% marble powder content) exhibited damage values of 0.3123, 0.1322, and 0.1623, respectively, under peak stress. This correlates with their compressive strengths. Specifically, Test Case G9 underwent only three damage stages: no damage, initial damage, and accelerated damage. This indicates that when loading ceased upon failure in G9, the specimen was not completely damaged. In contrast, B9 and M12 fully traversed all four damage stages, demonstrating that these two groups reached a state of complete failure after loading cessation.

Discussion

Figure 13 shows the fitted curves of strength characteristics for three CTRPB specimens. As depicted, the unconfined compressive strength (UCS) of CTRPB exhibits an inverted U-shaped trend after rock powder addition, with significant variations in optimal dosage and strengthening effects among different rock powders. Compared to chemical additives or fiber materials^38–44, rock powder modification avoids the potential environmental impacts associated with chemicals and is more green and environmentally friendly. However, its post-peak failure behavior exhibits more pronounced brittle characteristics. This is because rock powder enhances the energy accumulation capacity of the specimens during loading, but the stored elastic strain energy is suddenly released upon failure, leading to rapid crack propagation and sudden failure. Additionally, rock powder particles may alter the crack propagation path, making the failure process more localized rather than following a progressive development. This phenomenon requires attention in engineering applications, especially in scenarios where backfill materials need to possess a certain degree of toughness. The post-peak performance can be improved by adding fibers or optimizing the cementitious material mix ratio.

Fig. 13.

Strength characteristic fitting curve for CTRPB.

The influence of rock powder type and content on CTRPB specimens is reflected not only in variations of unconfined compressive strength (UCS) but also in energy accumulation and dissipation characteristics. From an energy perspective, the incorporation of all three rock powders significantly enhances the energy density characteristics of the fill material. Data in Table 2 indicate that rock powder addition markedly increases the specimens’ energy accumulation capacity during loading, a phenomenon intrinsically consistent with the observed UCS improvement. A deeper analysis of the energy characteristics reveals that when dissipated energy approaches equilibrium while elastic strain energy increases, it indicates the specimen possesses stronger elastic recovery capacity and greater load-bearing capability. This property enables the grout to maintain excellent performance under more complex environmental conditions and higher stress levels, making it particularly suitable for engineering applications requiring long-term stability. Conversely, when elastic strain energy approaches saturation while dissipated energy increases, it indicates the specimen exhibits enhanced toughness during loading, effectively delaying the failure process.

Notably, although the established piecewise damage constitutive model can well describe the test data in the pre-peak stage, its predictive capability is limited in the post-peak stage. This is mainly because the rapid propagation of cracks in the specimen after the peak leads to deformation localization, while the current model is based on the assumption of a homogeneous continuous medium and thus cannot accurately characterize the nonlinear behavior during this stage. In addition, the constitutive model developed in this study is only applicable to uniaxial compression conditions. For practical engineering design, it must be calibrated and modified by means of triaxial tests or constitutive models that take confining pressure into account. Therefore, more in-depth research should be conducted in the future on the universality of the damage constitutive model for CTRPB.

Conclusion

This study investigates the influence of rock powder type and content on the mechanical behavior, failure characteristics, and energy dissipation properties of CTRPB based on uniaxial compression tests. A segmented damage constitutive model for CTRPB was established to analyze its damage evolution under different rock powder types and contents. The following conclusions were drawn:

(1) The incorporation of rock powder enhances the deformation resistance of CTRPB. Both uniaxial compressive strength and elastic modulus exhibit a pattern of initial increase followed by decrease as the rock powder content rises. Among these, basalt rock powder exhibited the most significant improvement in axial compressive strength at a 9% content, followed by marble rock powder at 12%, while granite rock powder showed a relatively smaller increase at 9%.

(2) The deformation failure of CTRPB undergoes four stages: pore compaction, linear elastic deformation, plastic deformation, and post-fracture stage. Compared to the control group, CTRPB exhibits a generally steeper slope in the post-fracture stage curve. Furthermore, the incorporation of rock powder causes CTRPB to release energy more violently during fracture, ultimately resulting in a rock-like disintegration phenomenon.

(3) The loading process of CTRPB exhibits four energy evolution stages: the elastic strain energy curve rises then falls, while the dissipated energy curve falls then rises. During compaction, energy dissipation density dominates. In the elastic stage, elastic strain energy outweighs dissipated energy. During plastic deformation, elastic strain energy decreases while dissipated strain energy increases. In the post-peak failure stage, dissipated energy becomes dominant.

(4) Rock powder incorporation significantly enhances the energy density characteristics of the grout. Compared to the control group, the total energy density, elastic energy density, and dissipated energy density of CTRPB increased by 252.17%, 213.87%, and 478.99%, respectively, indicating CTRPB possesses stronger energy accumulation capacity during loading.

(5) A segmented damage constitutive model for CTRPB has been established, with the calculated theoretical curve exhibiting high agreement with experimental results during the pre-peak stage. The damage evolution of CTRPB is divided into four stages: undamaged stage, initial damage stage, accelerated damage stage, and stable damage stage. The occurrence of complete failure in the specimen can be determined by the damage value.

Author contributions

Jiyong Zhang: Conceptualization, Writing - original draft, Methodology, Investigation, Data curation. Qianjin Zou: Writing - original draft, Methodology, Investigation, Formal analysis, Conceptualization. Wentao Cai: Investigation, Data curation. Longyue Zhang: Investigation, Data curation. Haoran Ge: Investigation, Data curation. Haigang Li: Supervision, Validation, Methodology, Funding acquisition.

Funding

This study was supported by the National Key Research and Development Program of China under the 14th Five-Year Plan (2023YFC3012200).

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.