1 INTRODUCTION
In tunneling, the surrounding rock and lining are generally considered to form an integrated structure, known as the surrounding rock–lining system, which together bear the geological load (Fan et al., 2023; Li et al., 2023; Yang, Li, et al., 2023). The lining is typically the first to sustain significant damage during tunnel fires, with the mechanical properties of the concrete being weakened (Casey, 2020; Ren et al., 2019). Although the surrounding rock is shielded by the lining and not directly exposed to the fire, the high temperatures induced by the fire can be transmitted through the lining to the surrounding rock (Caner et al., 2005; Chen, Liu, et al., 2023; Chen, You, et al., 2023). Additionally, since the surrounding rock and the lining concrete are different materials, the interface between them is considered the interface transition zone (ITZ), characterized by high porosity, low density and high permeability (Ansell, 2010; Korouzhdeh et al., 2022; Xie et al., 2015). Consequently, the interface is regarded as the weak surface of the surrounding rock–lining system (Mohammadi et al., 2024a, b; Sarfarazi et al., 2024; Zhou et al., 2023). Therefore, the mechanical behavior of the rock–concrete (surrounding rock–lining) interface plays a crucial role in ensuring tunnel stability (Bryne et al., 2014; Chen et al., 2024; Malmgren et al., 2005).
The rock–concrete (surrounding rock–lining) interface is generally under compression and prone to shear failure. Extensive research has been conducted on the mechanical behavior of the rock–concrete interface. Shear tests on rock–concrete interfaces with regular and irregular triangular surfaces, fractal surfaces and natural surfaces have been carried out and have shown that the interface geometry affects the shear behavior of the rock–concrete interface (Campos et al., 2024; Chen, Yang, et al., 2023; Gu et al., 2003; Haque & Kodikara, 2012; Jiang et al., 2021; Mouzannar et al., 2017). The influence of the boundary conditions of constant normal load and constant normal stiffness on the shear behavior of the rock–concrete interface was discussed by Gutiérrez-Ch et al. (2018), Tian et al. (2015) and Yang, Hu, et al. (2023). It was found that the controlling effect of constant normal stiffness on the shear behavior of rock concrete was more obvious than that of constant normal load. Atapour and Moosavi (2014), Li et al. (2022) and Zhang et al. (2024) investigated the effect of the shear rate on the shear behavior of rock concrete specimens with different rock types and concrete strengths, moisture contents and normal stress levels and found that shear strength increased and shear stiffness decreased with the increasing shear rate. These existing achievements focused on variables such as interface geometry, boundary conditions and stress states, with limited consideration given to temperature effects.
Relevant scholars have documented the effect of temperature on the mechanical behavior of the rock–concrete interface. Hu et al. (2020) investigated the mechanical behavior of the shotcrete–rock interface in geothermal tunnels, focusing on different curing temperatures (60, 80, and 100°C), which are well below the temperature of the fire. Nie et al. (2023) and Yu et al. (2023) performed the three-point bending test and the Brazilian test for the sandstone–concrete interface to investigate the bond failure and deformation behavior at elevated temperatures. However, in tunnels, the rock–concrete interface is generally under compression, and studying only the bond effect is insufficient to fully understand the interaction between the surrounding rock and the lining at elevated temperatures. Zhang et al. (2023) performed cyclic shear tests on rock–concrete specimens and found the degradation law for the mechanical behavior of the rock–concrete interface at elevated temperatures. However, no further analysis was carried out to propose a valid predictive model to guide similar situations beyond the experimental conditions.
In this study, shear tests of the rock–concrete interface were conducted at elevated temperatures of 20, 150, 300, and 450°C. The evolution of mechanical properties, including compaction shear displacement, shear stiffness, peak shear strength and residual strength, as well as interface damage and failure modes, was analyzed to quantify the degradation of mechanical properties of the rock–concrete interface during shear at elevated temperatures. Subsequently, a temperature-dependent shear constitutive model, based on damage mechanics and statistical theory, was developed to eliminate reliance on specific experimental variables and serve as a reference for similar situations beyond the experimental conditions.
2 EXPERIMENTAL OVERVIEW
2.1 Experiment scheme
In tunnel fires, the temperature of the free surface (fired surface) of the lining can exceed 1000°C. However, concrete, being a thermally inert material, exhibits a delayed temperature rise compared to tunnel fires. According to the statistics of tunnel fire accidents (Smith and Pells, 2008; Wasantha et al., 2021), the temperature at the rock–lining interface does not exceed 500°C. Research on normal concrete (Georgali & Tsakiridis 2005; Wróblewska & Kowalski 2020) indicates that free water is lost at temperatures around 100–150°C, and bound water is lost at temperatures around 200–350°C. Additionally, the cement paste shrinks due to dehydration, while the sand and gravel aggregate expands at temperatures around 300°C, leading to extensive cracking. Calcium hydroxide crystals disintegrate at temperatures above 400°C. Based on the above analysis, the heating temperatures of rock–concrete specimens were set at 20, 150, 300, and 450°C, with a heating duration of 120 min, as shown in Table 1.
| No. |
Temperature (°C) |
Normal stress (MPa) |
| 1 |
20 |
1 |
| 2 |
150 |
2 |
| 3 |
300 |
3 |
| 4 |
450 |
|
Urban rail transit tunnels with heavy traffic are typically shallow tunnels with minimal geological structural stress. The load of the overlying rock mass disturbed by tunnel excavation is the primary load borne by the lining. The stress state is constant and can be regarded as a constant normal load (Shrivastava & Rao, 2015; Yang et al., 2022). Therefore, the boundary conditions in the shear test were set to constant normal load (CNL) with initial normal stresses of 1.0, 2.0, and 3.0 MPa, as illustrated in Table 1. The rock–concrete interface roughness is 6–8 to represent the uneven contour surface of the surrounding rock. The shear rate and shear displacement are based on existing shear tests (Dang et al., 2018, 2022; Muralha et al., 2014), with a shear rate of 1 mm/min and a shear displacement of 10 mm.
The rock–concrete interface shear test was performed on the servo-controlled shear system of the Geotechnical Laboratory of Nagasaki University (Jiang et al., 2004; Wang et al., 2023; Wang et al., 2024), as shown in Figure 1 The shear test platform applies normal and shear force through the vertical and horizontal cylinders with a maximum output shear and normal force of 200 kN. The upper part of the shear box is fixed, while the lower part moves in the shear direction. The shear and normal displacements are monitored in real time by the LVDT displacement sensor. The accuracy of the displacement sensor is 0.001 mm, and the measurement range is 0–20 mm. Mechanical properties, including shear stress, normal stress, shear displacement and normal displacement, can be recorded during the test. The specimen size allowed by the equipment is 200 mm × 100 mm × 100 mm, with rock and concrete making up half the volume at 200 mm × 100 mm × 50 mm. The mixing proportion of concrete is listed in Table 2.
Rock–concrete interface shear test.
Table 2. Mixing proportion of concrete. (kg/m
3)
| Cement |
Fly ash |
Water |
Sand |
Gravel |
Admixture |
| 415 |
85 |
195 |
528 |
1176 |
16.5 |
2.2 Mechanical properties of rock and concrete at elevated temperatures
The multiphase system (solid, liquid and gas) of rock and concrete underwent a series of complex physical and chemical reactions at elevated temperatures. The mass loss of rock and concrete is illustrated in Figure 2. The mass loss rate of both rock and concrete increased with rising temperature, with concrete exhibiting a more pronounced mass loss rate than rock. The temperature rose from 20 to 150°C, and the mass loss rates of rock and concrete were 0.59% and 4.91%, respectively, due to the evaporation of free water. The temperature rose from 150 to 300°C, and the mass loss rates of rock and concrete were 0.24% and 2.11%, respectively, attributed to the evaporation of bound water. The temperature rose from 300 to 450°C, and the mass loss rates were 0.45% for rock and 0.77% for concrete, owing to the decomposition and water loss of clay minerals in the rock and the partial dehydration of hydrates, such as hydrated calcium silicate and calcium hydroxide crystals, in concrete.
Mass loss of rock and concrete after elevated temperatures.
The color and failure modes of rock and concrete after elevated temperatures are shown in Figure 3. As the temperature rose from 20 to 450°C, the color of the rock changed from blue-gray to dark red, while the color of the concrete changed from gray to light pink. The failure modes of rock and concrete also gradually changed from tensile failure to tension-shear mixed failure.
Color and failure modes of rock and concrete after elevated temperatures.
The uniaxial compressive strength of rock and concrete after elevated temperatures is shown in Figure 4. The uniaxial compressive strength of rock and concrete decreased with increasing temperature. At the temperature of 20°C, the uniaxial compressive strengths of rock and concrete were 86.83 and 45.36 MPa, respectively. As temperatures rose from 20 to 150°C, the uniaxial compressive strength of rock and concrete decreased by 2.05 and 1.51 MPa, with decrease rates of 2.36% and 3.33%, respectively. As temperatures rose from 150 to 300°C, the uniaxial compressive strength of rock and concrete decreased by 5.39 and 6.27 MPa, with decrease rates of 6.21% and 13.82%, respectively. As temperatures rose from 300 to 450°C, the uniaxial compressive strength of rock and concrete decreased by 12.33 and 9.31 MPa, with decrease rates of 14.20% and 20.52%, respectively. The adverse effect of temperature on the mechanical properties of concrete was much greater than that of rock.
Uniaxial compressive strength of rock and concrete after elevated temperatures.
3 INTERFACE DAMAGE AND FAILURE CHARACTERISTICS
Rock–concrete specimens after elevated temperatures are shown in Figure 5. The rock–concrete specimens exhibited no significant visual changes at 20 or 150°C (Figure 5a,b). However, distinct white traces appeared parallel to the rock–concrete interface on the concrete side at 300°C, as shown in Figure 5c. At 450°C, cracks formed at the interface and extended towards the concrete (Figure 5d), indicating that concrete is more sensitive to elevated temperatures than rock.
Rock–concrete specimens after elevated temperatures: (a) 20°C, (b) 150°C, (c) 300°C, and (d) 450°C.
The damage and failure characteristics of the rock–concrete interface are shown in Figure 6. Shear failure occurred mainly on the concrete side or at the rock–concrete interface. In the experimental conditions, failure modes of the rock–concrete interface are classified into three types: Mode I, where the failure occurred entirely in the concrete (Figure 6a), the morphology of the shear fracture surface fluctuated greatly and the rock surface was coated with a substantial quantity of concrete, suggesting that the strength of the concrete controls the failure of the rock–concrete interface. Mode II, where the failure occurred partly in the concrete and partly at the rock–concrete interface (Figure 6b), the shape of the shear fracture surface was undulated to one side, and part of the rock surface was covered with concrete, indicating that the strength of the concrete and the bond strength of the interface have an equal effect on the failure of the rock–concrete interface. Mode III, where the failure occurred exclusively at the rock–concrete interface (Figure 6c), the shear fracture surface extended completely along the rock–concrete interface and there were obvious scratches on the rock surface and the concrete surface, indicating that the interface had fractured after being subjected to elevated temperatures.
Failure modes of the rock–concrete interface: (a) Mode I, (b) Mode II, and (c) Mode III.
The failure modes of the rock–concrete interface under different temperatures and normal stresses are shown in Table 3. The shear fracture surface of the rock–concrete specimen gradually transitioned from within the concrete to the interface with increasing temperature and normal stress. It can be concluded that the rock–concrete interface was more sensitive to external effects (temperature and load) than rock and concrete, and the temperature not only weakened the bond between the rock and concrete but also reduced the strength of the rock and concrete.
Table 3. Failure modes of the rock–concrete interface.
| Load (MPa) |
Temperature, T (°C) |
| 20 |
150 |
300 |
450 |
| 1 |
I |
II |
II |
III |
| 2 |
I |
II |
III |
III |
| 3 |
II |
II |
III |
III |
The failure mechanism of the rock–concrete interface depended on the temperature. At the temperature of 20°C, the ridges and depressions distributed on the rough rock surface were conducive to the accumulation and embedding of hydration products in the concrete, which improved the density of the rock–concrete interface and enhanced the bonding and biting ability of the rock–concrete interface. Therefore, the strength of the interface was higher than that of the concrete, the shear fracture surface was within the concrete and the debris generated during the shear process was the block concrete, as shown in Figure 7a. At temperatures of 150 and 300°C, the free and bound water in the rock and concrete were lost, and microcracking was initiated in the rock–concrete specimen, particularly in the concrete. The shear fracture surface showed a mixture of features, some within the concrete and others at the interface. The debris produced during the shear process was gravel-like concrete and rock particles, as shown in Figure 7b. At the temperature of 450°C, due to the difference in thermal expansion coefficients between rock and concrete, the stress caused by the uneven expansion deformation between rock and concrete gradually exceeded the bondability of the rock–concrete interface, causing the interface to break completely and the rock and concrete to separate. Therefore, the shear fracture surface was at the interface, and the debris created during the shear process was crushed concrete and rock powder, as shown in Figure 7c.
Debris from the rock–concrete interface during the shear process: (a) at the temperature of 20°C, (b) at the temperatures of 150 and 300°C, and (c) at the temperature of 450°C.
4 EVOLUTION OF SHEAR PROPERTIES
4.1 Shear stress–shear displacement
The evolution of the shear stress–shear displacement of the rock–concrete interface was essentially similar. As the shear displacement increased, the shear stress initially increased until it reached the peak strength, after which it decreased and eventually stabilized at the residual shear stress, as shown in Figure
8. Notably, the shear displacement increased initially, while the shear stress remained constant. This phenomenon became more pronounced with rising temperature, resembling the compaction process observed in uniaxial compression tests. Therefore, the shear process can be divided into three stages: Stage I, the compaction stage; Stage II, the pre-peak stage; and Stage III, the post-peak stage, as shown in the enlarged curve in Figure
8.
1.
Stage I, the compaction stage: the shear displacement increased with rising temperature, while the shear stress remained low or even negligible, indicating that cracks and voids were generated in the rock–concrete specimen at elevated temperatures, thus requiring greater shear displacement to achieve a dense state during the shear process.
2.
Stage II, the pre-peak stage: The shear stress increased with the shear displacement. The increased rate of the shear stress tended to slow down as it approached the peak strength, showing strain-hardening characteristics. As the temperature rose, the peak shear displacement increased, while the peak shear stress decreased, and the pre-peak slope of the shear stress–shear displacement curve (also known as shear stiffness) also decreased, indicating that temperature degraded the shear properties of the rock–concrete interface.
3.
Stage III, the post-peak stage: The shear stress decreased with increasing shear displacement until it stabilized at the residual shear stress. The decrease in shear stress did not show a strong linear relationship with temperature due to the difference in normal stress and shear fracture surface. The rock–concrete interface was also completely fractured after the shear stress exceeded peak strength, and the depressions and bulges on the shear fracture surface gradually wore away during the subsequent shear process, eventually developing into stable frictional shear so that the shear stress reached the residual shear stress. Although the residual shear stress decreased with rising temperature, it was mainly controlled by the normal stress.
As was evident from the previous analysis, the effect of temperature on the shear behavior of the rock–concrete interface was primarily reflected in key parameters such as compaction shear displacement, shear stiffness, peak shear displacement, peak shear stress and residual shear stress. These parameters were the focus of the subsequent analysis.
Shear stress–shear displacement at the rock–concrete interface under normal stress: (a) 1 MPa, (b) 2 MPa, and (c) 3 MPa.
4.2 Shear strength
Shear strength is an important parameter that measures the shear properties of materials. In this test, the shear strength characteristics were mainly represented by the peak shear stress and residual shear stress.
1.
Peak shear stress
The variation of the peak shear stress is shown in Figure 9. The peak shear stress had a linear negative correlation with the temperature and a linear positive correlation with the normal stress. The maximum peak shear stress was 6.796 MPa at the temperature of 20°C and the normal stress of 3 MPa, and the minimum was 1.655 MPa at the temperature of 450°C and the normal stress of 1 MPa. The difference between the maximum and minimum was 5.141 MPa, and the rate of difference was 410.63%. The findings suggest that both temperature and normal stress have significant effects on the peak shear stress.
At the same temperature, as the normal stress increased from 1 to 3 MPa, the decrease in peak shear stress was 20°C: 3.066 MPa, 150°C: 2.266 MPa, 300°C: 1.868 MPa, and 450°C: 1.456 MPa, and the rate of decrease was 82.20%, 74.64%, 81.32%, and 87.98%. At the same normal stress, as the temperature rose from 20 to 450°C, the increase in peak shear stress was 1 MPa: 2.075 MPa, 2 MPa: 2.988 MPa, and 3 MPa: 3.685 MPa, and the rate of increase was 55.63%, 55.78%, and 54.22%. It is evident that the temperature is detrimental to the peak shear stress, while the normal stress is beneficial to the peak shear stress. The effect of temperature on the peak shear stress was greater than that of normal stress, but normal stress can inhibit the decrease in peak shear stress.
2.
Residual shear stress
The evolution of the residual shear stress is shown in Figure 10. The residual shear stress decreased with rising temperature and increased with increasing normal stress. At the normal stress of 1 MPa, the residual shear stress was 20°C: 1.178 MPa, 150°C: 1.111 MPa, 300°C: 0.853 MPa, and 450°C: 0.655 MPa. Except for 450°C, the remaining residual shear stresses are not different from the normal stress of 1 MPa. Therefore, the normal stress has a more significant effect on the residual shear stress than the temperature.
Variation of peak shear stress with: (a) temperature and (b) normal stress.
Evolution of residual shear stress with: (a) temperature and (b) normal stress.
At the same temperature, as the normal stress increased from 1 to 3 MPa, the increase in residual shear stress was 20°C: 1.722 MPa, 100°C: 1.525 MPa, 300°C: 1.368 MPa, and 450°C: 1.260 MPa; the rate of increase was 159.64%, 149.95%, 158.52%, and 162.37%. At the same normal stress, as the temperature rose from 20 to 450°C, the decrease in residual shear stress was 1 MPa: 0.334 MPa, 2 MPa: 0.610 MPa, and 3 MPa: 0.846 MPa; the rate of decrease was 30.09%, 32.34%, and 29.35%. The results indicate that the effect of the normal stress on residual shear stress is greater than that of the temperature.
As analyzed in interface damage and failure characteristics, before peak strength was reached, the shear stress was provided by the cohesion of the rock–concrete interface. When the shear stress reached peak strength, the shear fracture surface was completely penetrated, and the cohesion of the rock–concrete interface was completely lost. The shear stress was then only provided by the friction of the rough shear fracture surface. As the shear displacement increased, the asperities on the rough shear fracture surface were gradually worn and smoothed, resulting in a decrease in the shear stress to the residual strength. Therefore, the main controller of shear stress was temperature before peak strength was reached, and it changed to normal stress after peak strength was exceeded.
4.3 Shear displacement
The shear displacement characteristics mainly include compaction shear displacement and peak shear displacement.
1.
Compaction shear displacement
The compaction shear displacement had a linear positive correlation with the temperature and the normal stress (Figure 11). The maximum compaction shear displacement was 0.295 mm, and the minimum was 0.084 mm. The difference between the maximum and minimum was 0.211 mm, and the rate of difference was 251.90%. The findings suggest that the compaction shear displacement is significantly affected by temperature and normal stress. At the same temperature, as the normal stress increased from 1 to 3 MPa, the increase in the compaction shear displacement was 20°C: 0.067 mm, 100°C: 0.069 mm, 300°C: 0.082 mm, and 450°C: 0.093 mm; the rate of increase was 79.76%, 58.97%, 52.56%, and 46.04%. At the same normal stress, as the temperature rose from 20°C to 450°C, the increase in compaction shear displacement was 1 MPa: 0.118 mm, 2 MPa: 0.154 mm, and 3 MPa: 0.144 mm; the rate of decrease was 140.48%, 149.52%, and 95.36%. The analysis shows that the effect of the normal stress on the compaction shear displacement is greater than that of the temperature. It is assumed that although more microcracking was generated at the rock–concrete interface with rising temperature, these new microcracking and primary microcavities only closed under normal stress, so that the compaction effect became more significant with increasing normal stress.
2.
Peak shear displacement
The peak shear displacement also had a linear positive correlation with temperature and the normal stress (Figure 12). The maximum peak shear displacement was 0.705 mm, and the minimum was 1.738 mm. The difference between the maximum and minimum was 1.033 mm, and the rate of difference was 146.52%. It was consistent with the evolution of compaction shear displacement. At the same temperature, as the normal stress increased from 1 to 3 MPa, the increase in peak shear displacement was 20°C: 0.287 mm, 100°C: 0.354 mm, 300°C: 0.536 mm, and 450°C: 0.666 mm; the rate of increase was 40.71%, 44.58%, 57.08%, and 62.13%. At the same normal stress, as the temperature rose from 20°C to 450°C, the increase in the peak shear displacement was 1 MPa: 0.367 mm, 2 MPa: 0.570 mm, and 3 MPa: 0.746 mm; the rate of increase was 52.06%, 66.36%, and 75.20%. Thus, temperature and normal stress promoted each other in increasing the peak shear displacement, and the effect of normal stress was greater than that of temperature. Although the evolution of peak shear displacement and compression shear displacement was consistent, their causal mechanisms were different.
Development of compaction shear displacement with: (a) temperature and (b) normal stress.
Evolution of peak shear displacement with: (a) temperature and (b) normal stress.
The findings on shear strength in Section 4.2 indicated that the decrease in shear strength of the rock–concrete interface was due to the loss of cohesion at elevated temperatures. Therefore, based on the analysis of shear strength and shear displacement, the loss of cohesion can be attributed to two factors: First, the difference in thermal expansion coefficients between the rock and concrete causes microcracking at the interface, which is considered physical damage. Second, the hydrates at the rock–concrete interface gradually decompose at elevated temperatures, which is considered chemical damage. The physical damage caused by microcracking at the rock–concrete interface is re-closed during the compaction stage. The increase in peak shear displacement is a manifestation of the chemical damage due to the decomposition of hydrates at the rock–concrete interface.
The minimum ratio of compaction shear displacement to peak shear displacement was 12.28%, and the maximum ratio was 14.33%, indicating that compaction shear displacement cannot be ignored compared to peak shear displacement. In engineering practice, if the compaction shear displacement can be ignored, the shear stiffness of the rock–concrete interface will be underestimated, which may lead to safety but also to uneconomical results. In the experimental conditions, the error resulting from neglecting the compaction shear displacement increased with increasing temperature and normal stress, which is not conducive to a correct understanding of the temperature effect on the shear behavior of the rock–concrete interface. Therefore, the compaction shear displacement must be considered when studying the shear behavior of the rock–concrete interface at elevated temperatures.
4.4 Shear stiffness
Shear stiffness is an important parameter that reflects the ability of the rock–concrete interface to resist shear deformation. Its value is the slope of the approximate straight line before the peak strength of the shear stress–shear displacement curve. The evolution of shear stiffness is shown in Figure 13. Shear stiffness decreased rapidly with rising temperature and increased with increasing normal stress. At the temperature of 20°C, the shear stiffness was 1 MPa: 6.006 GPa/m, 2 MPa: 7.086 GPa/m, and 3 MPa: 8.081 GPa/m, the increase was 1.080 and 0.995 GPa/m and the rate of increase was 17.98% and 14.04%. At the temperature of 150°C, the shear stiffness was 1 MPa: 4.484 GPa/m, 2 MPa: 5.080 GPa/m, and 3 MPa: 5.511 GPa/m, the increase was 0.596 and 0.431 GPa/m and the rate of increase was 13.29% and 8.48%. The shear stiffness was approximately 3.0 GPa/m at the temperature of 300°C and 2.0 GPa/m at the temperature of 450°C. The difference in shear stiffness between different normal stresses was only 0.1 GPa/m, which is almost unaffected by the normal stress.
Variation of shear stiffness with (a) temperature and (b) normal stress.
Based on the analysis in Sections 4.1 and 4.2, the evolution of the shear stiffness can be explained as follows: before the peak shear strength was reached, the shear resistance of the rock–concrete interface was mainly attributed to the cohesion of the interface. Furthermore, the increase in normal stress did not alleviate the degradation of shear stiffness. This indicates that shear stiffness is more influenced by temperature than by normal stress.
4.5 Interface fracture energy
From the shear stress–shear displacement curve, it is evident that the shear stress decreased immediately after reaching peak strength, indicating that the rock–concrete interface experienced complete fracture. Therefore, the interface fracture energy can be defined as the area enclosed by the peak shear stress and the peak shear displacement, and the unit is kN/m instead of J/m2 to be consistent with the shear displacement–shear stress curve (Figure 14a). The interface fracture energy decreased with rising temperature and increased with increasing normal stress, as shown in Figure 14. The maximum interface fracture energy reached 2.882 kN/m at the temperature of 20°C and the normal stress of 3 MPa, the minimum reached 0.776 kN/m at the temperature of 450°C and the normal stress of 1 MPa. The difference between the maximum and minimum was 2.215 kN/m, and the rate of difference was 285.44%.
Interface fracture energy: (a) schematic diagram illustrating the calculation, (b) variation with temperature, and (c) variation with normal stress.
At the same temperature, as the normal stress increased from 1 to 3 MPa, the increase in interface fracture energy was 20°C: 1.881 kN/m, 100°C: 1.525 kN/m, 300°C: 1.368 kN/m, and 450°C: 1.260 kN/m; the rate of increase was 169.46%, 149.95%, 158.52%, and 162.37%. At the same normal stress, as the temperature rose from 20 to 450°C, the decrease in interface fracture energy was 1 MPa: 0.334 kN/m, 2 MPa: 0.610 kN/m, and 3 MPa: 0.995 kN/m; the rate of decrease was 30.09%, 32.34%, and 31.93%. At the same temperature, as the normal stress increased from 1 to 3 MPa, the increase in interface fracture energy gradually decreased, and the rate of increase was almost unchanged. At the same normal stress, as the temperature rose from 20 to 450°C, the decrease in interface fracture energy gradually increased and the rate of decrease was almost unchanged. The effect of the normal stress at the same temperature on the interface fracture energy was greater than the effect of temperature at the same normal stress, i.e. the interface fracture energy was more sensitive to the normal stress than to the temperature.
5 TEMPERATURE-DEPENDENT SHEAR CONSTITUTIVE MODEL
The results of the above shear tests indicate that the shear behavior of the rock–concrete interface is significantly dependent on temperature. However, existing shear constitutive models have not included temperature. Therefore, a novel shear constitutive model of the rock–concrete interface was established by considering the temperature effect using damage mechanics and statistical theories.
5.1 Establishment of the shear constitutive model
Damage theory suggests that materials or structures undergo damage when subjected to external factors such as loading, temperature and humidity. This damage is considered random and can be described using statistical methods (Cao et al.,
2019; Yang & Chen,
2011; Zhao et al.,
2017). Therefore, the temperature damage for the rock–concrete interface follows the statistical distribution. The ratio of damaged microelements
to total microelements
of the rock–concrete interface is defined as the damage variable
D.
(1)
Assume that the probability density function of the microelement strength is
. Microelements are damaged when the stress level
exceeds a certain strength. In the interval (
,
), the number of newly damaged microelements is
. Then, the damage variable
D can be calculated as
(2)
The temperature damage of the rock–concrete interface can be abstracted into two parts: the damaged area and the undamaged area, as shown in Figure 15.
Schematic diagram of temperature damage to the rock–concrete interface.
It is assumed that the total shear area of the rock–concrete interface is
, the undamaged area is
and the damaged area is
. The specific values of
and
are difficult to determine accurately. Therefore, the damage variable
D also follows Equation (
3).
(3)
Referring to the strain equivalence theory and the effective stress principle, the normal stress carried by the damaged area and the undamaged area is the same and equal to the nominal normal stress. The nominal shear stress
is equal to the sum of the shear stress
in the undamaged area and the shear stress
in the damaged area. Shear stress follows Equation (
4).
(4)
Equations (
3) and (
4) give
(5)
Based on the shear stress–shear displacement curve in Section
4, the shear stress
and shear displacement
in the undamaged area follow a linear relationship before reaching the peak shear strength. The shear stress
in the damaged area is equal to the residual shear stress, which can be expressed as
(6)
where
is the shear stiffness,
is the residual shear stress.
The experimental results in Section
4 showed that the shear stress remained at a negligibly low level, independent of the shear displacement, during the compaction stage, indicating that the undamaged area had not yet begun to bear the load. Therefore, the shear displacement
in the undamaged area can be expressed as
(7)
where
is the nominal shear displacement,
is the compaction shear displacement.
Equations (
5-7) give:
(8)
The experimental results in Section
4 also indicated that the compaction stage was only related to temperature and normal stress. Therefore, based on the phenomenological theory, the shear process of the rock–concrete interface can be divided into two stages, which gives
(9)
5.2 Statistical analysis of temperature damage
Temperature damage to heterogeneous rock and concrete is discrete and difficult to quantify directly. Therefore, functions reflecting statistical characteristics are introduced to organically combine the continuous damage theory with the statistical characteristics of mechanical parameters. Existing studies have also shown that the Weibull probability distribution function satisfies the statistical characteristics of the load-bearing failure of rock or concrete (Cao et al.,
2010; Xiao et al.,
2022,
2023; Xie et al.,
2020; Zhao et al.,
2018). Therefore, in this study, the microelements at the rock–concrete interface, after being subjected to temperature, are assumed to conform to the Weibull distribution function, shown in Equation (
10).
(10)
where
F stands for the elemental strength parameter or stress level;
is the scaling parameter, which is called characteristic stress and
m stands for the shape parameter, which determines the shape of the distribution.
Substituting Equation (
10) into Equation (
2) and then integrating gives
(11)
The classic Mohr–Coulomb shear strength criterion is used to characterize the strength
F of the microelements, as shown in Equation (
12).
(12)
where
is the shear stress carried by the microelement that is not damaged by temperature,
and
are the cohesion and the internal friction angle of the shear fracture surface of the rock–concrete interface, respectively. Substituting Equations (
6) and (
7) into Equation (
12) gives
(13)
where
is the cohesion at peak shear stress,
is the internal friction angle at peak shear stress.
5.3 Model establishment and parameter determination
Substituting Equation (
11) into Equation (
9) gives the temperature-dependent shear constitutive model, as shown in Equation (
14).
(14)
Obviously, the shear displacement reaches the peak shear displacement
, and the corresponding shear stress reaches the peak shear stress
. Substituting these into Equation (
14) gives
(15)
Based on the characteristics of extreme values, differentiating Equation (
15) yields
(16)
Combining Equations (
15) and (
16) produces
(17)
(18)
(19)
The model parameters
m and
F
0 in Equations (
18) and (
19) were obtained at specific temperatures and normal stresses that are not applicable to scenarios outside this experiment. Therefore, the model parameters
m and
F
0 need to be transformed into universal parameters that do not depend on the specific temperature and normal stress. Existing studies indicated that the shear test of rock or concrete was conducted under a specific normal stress, and the shear behavior should also be related to the normal stress. The model parameters were determined as follows.
1.
Determination of
The compaction shear displacement is not accounted for in existing shear studies that do not involve temperature. However, this experiment found that compaction shear displacement cannot be ignored compared to the peak shear displacement. Therefore, the relationship between the compaction shear displacement and the normal stress is shown in Figure 16a, which can be expressed as
(20)
where
and
are the temperature functions and their fitting curves with temperature are shown in Figure 16b, with the corresponding expression given as
(21)
(22)
2.
Determination of
As mentioned above, the shear stiffness a has a good linear relationship with the normal stress, as shown in Figure 17a, which can be expressed as
(23)
where
and
are the temperature functions, and their fitting curves with temperature are shown in Figure 17b. The two parameters can be fitted with linear and exponential functions as follows:
(24)
(25)
3.
Determination of
It is assumed that complete failure occurs at the rock–concrete interface when the shear stress reaches the peak value. The peak shear stress
and the normal stress
follow the Mohr–Coulomb failure criterion (Figure 18a).
(26)
where
are the internal friction angle when the shear stress reaches the peak shear stress. The cohesion
and the internal friction angle
decreased with rising temperature (Figure 18b), with the corresponding expression given as
(27)
(28)
4.
Determination of
As mentioned above, the peak shear displacement and the normal stress are shown in Figure 19a, which can be expressed as
(29)
where
and
are the temperature functions, and their fitting curves with temperature are shown in Figure 19b, with the corresponding expression given as
(30)
(31)
Determination of
in model parameters: (a) relationship between compaction shear displacement and normal stress and (b) fitting curves
and
with temperature.
Determination of
in model parameters: (a) relationship between shear stiffness and normal stress and (b) fitting curves of
and
with temperature.
Determination of
in model parameters: (a) relationship between peak shear stress and normal stress at test temperature and (b) fitting curves of
and
with temperature.
Determination of
in model parameters: (a) relationship between compaction shear displacement and normal stress and (b) fitting curves of
and
with temperature.
Finally, inserting the above expressions (20)–(31) into Equations (17) and (18) yields the model parameters m and F0. Substituting these model parameters into Equation (14) gives the temperature-dependent shear constitutive model for the rock–concrete interface.
5.4 Verification and discussion
The comparison between the shear constitutive model and the experimental results is shown in Figure
20.
1.
The shear constitutive model is in good agreement with the characteristic point of the experimental results, including compaction shear displacement, shear stiffness, shear stiffness and residual shear stress, reflecting the temperature-dependent properties.
2.
The shear constitutive model reflects the compaction shear displacement induced by the elevated temperatures well, but the curve fits the experimental results slightly worse in the linear elastic stage due to the heterogeneity of rock and concrete.
3.
The shear constitutive model does not account for the roughness abrasion of the shear fracture surface in the post-peak stage, leading to a slower decrease in shear stress compared to the experimental results.
Comparisons between shear constitutive model curves and experimental results at the normal stress of 3 MPa: (a) 20°C, (b) 150°C, (c) 300°C, and (d) 450°C.
6 CONCLUSIONS
In this study, shear tests of the rock–concrete interface at elevated temperatures were conducted, the evolution of the shear properties, interface damage and failure mode were analyzed and a temperature-dependent shear constitutive model of the rock–concrete interface was established.
1.
The mechanical properties of the rock and concrete gradually deteriorated with rising temperature. For the rock, the mass loss rate was 1.28%, and the decrease rate of the uniaxial compressive strength was 22.77%. For the concrete, the mass loss rate was 7.79%, and the decrease rate of the uniaxial compressive strength was 33.27%. The mechanical properties of concrete are more sensitive to temperature than those of rock under the experimental conditions.
2.
The shear failure modes can be classified as failure within the concrete (Mode I), failure partially within the concrete and partially at the interface (Mode II) and failure at the interface (Mode III). The shear fracture surface gradually shifted from within the concrete to the rock–concrete interface with increasing temperature.
3.
The shear process can be divided into three stages: the compaction stage, the pre-peak stage and the post-peak stage. As the temperature rose, the compaction shear displacement and peak shear displacement increased, and the peak shear stress decreased. It is worth noting that the compaction stage cannot be ignored, and the maximum ratio of compaction shear displacement to peak shear displacement was 16.97%.
4.
The cohesion and internal friction angle of the shear fracture surface decreased with rising temperature, resulting in a decrease in interface shear stiffness and interface fracture energy. The maximum rates of decrease in cohesion, internal friction angle, shear stiffness and interface fracture energy were 58.61%, 36.63%, 73.32%, and 32.34%.
5.
A temperature-dependent shear constitutive model of the rock–concrete interface was established based on damage mechanics and statistical theory, which can better simulate the shear behavior of the rock–concrete interface at elevated temperatures, in particular, to reflect the increase in compaction shear displacement and the decrease in shear stiffness during the shear process.
CONFLICT OF INTEREST STATEMENT
The authors declare no conflict of interest.
Biography
Yujing Jiang is a professor in the School of Engineering at Nagasaki University and a Member of the Engineering Academy of Japan. He obtained his Bachelor's and Master's degrees from Shandong University of Science and Technology, China, in 1982 and 1985, respectively, and his PhD from Kyushu University, Japan, in 1993. He serves as Director of the State Key Laboratory of Intelligent Strata Control and Green Mining Co-founded by Shandong Province and the Ministry of Science and Technology, China, and as Vice Chairman of the Sub-society for Soft Rock and Deep Disaster Control of the Chinese Society for Rock Mechanics and Engineering. Additionally, he was a recipient of the National Science Fund for Distinguished Young Scholars (Overseas), China. His research interests include environmental geotechnical engineering; rock mechanics and rock engineering; development and utilization of underground space; remote monitoring, forecast, and control of geological disasters; mining of deep-sea resources (natural gas hydrate) and control of seabed strata environments; and mine pressure and strata control. He has published 15 books in multiple languages and over 300 research papers.
