Introduction

The purpose of rock strength criteria is to determine the failure of rocks under certain stress states. Whether rock is damaged in the natural environment is related to many factors. In cold regions, rock is subjected to freeze-thaw effects throughout the year and are subjected to certain load. Therefore, studying the damage degradation mechanism and damage model of rock under the combined action of freeze-thaw cycles and load, and further developing into a rock strength theory considering freeze-thaw effects, has important theoretical significance and application value, whether in enriching and improving the knowledge system of rock mechanics basic research, or in making more reasonable use of material strength, scientific analysis and design of geotechnical structures.

The development of rock strength criteria has been over a hundred years, and scholars at home and abroad have established many valuable strength criteria based on various theories and methods, greatly promoting the development of rock mechanics. The M-C strength criterion was established based on the relationship between the rock shear strength parameters and the fitting parameters in the linear Mogi empirical strength criterion1. In addition, the M-C criterion2,3, D-P criterion4,5,6 and Hoek–Brown (H-B) criterion7 is commonly used as strength criteria for rock materials. Xie et al. analyzed the failure mechanism of rocks based on the principle of rock failure energy and established a rock energy failure strength criterion8. Li et al. considered the influence of deep high stress and established rock strength criteria under high stress conditions based on a new stress state combination9. Li et al. combined the nonlinear unified strength theory with the H-B strength criterion to establish a new three-dimensional rock strength criterion10. Niu et al. improved the H-B standard by considering the effects of critical confining pressure and intermediate principal stress11. Huang et al. modified the H-B strength criterion using the double shear mechanics model and double shear function12. Li et al. proposed a nonlinear strength criterion based on the variation law of rock strength in conventional triaxial tests and the extreme value of deviatoric stress13. Zhao et al. derived the spatial expressions for shear stress and normal stress on the failure surface of the MSDP criterion by establishing the geometric relationship between the strength envelope of the MSDP criterion and the Mohr stress circle14. Zhang et al. established a rock strength criterion considering freeze-thaw effects based on the geometric characteristics of extreme points in the stress-strain relationship curve of rock15. The current research mainly summarizes and outlines the development process of rock strength criteria, and derives their strength criteria through regression analysis of test data. There is relatively little theoretical derivation of rock strength criteria. In cold regions, the impact of freeze-thaw cycles on the strength failure of rocks cannot be ignored. However, there is currently little research on the theoretical derivation of rock strength criteria under freeze-thaw conditions. Therefore, the study of rock strength criteria under freeze-thaw conditions has significant theoretical guidance and high practical value.

The core of using damage theory to study damage problems of rock lies in selecting damage variables, determining the damage evolution equation, and finally establishing damage constitutive model. Research on damage model of rock can be roughly divided into two categories: one is based on experiments, assuming the relationship between damage variables and test parameters, and then determining parameter expressions through test simulations16. Another type is to use a certain random distribution to describe the randomness of elemental failure, derive the expression of damage variable, and establish a statistical damage model for rock based on this. At this point, the criteria for determining whether a micro element has been damaged are crucial. Existing research has established damage constitutive models considering different discrimination criteria, and the statistical results are shown in Table 1.

Table 1 Rock damage constitutive model based on different criteria for judging micro element damage.
Full size table

In addition, scholars have conducted theoretical research on rock damage models under freeze-thaw load based on existing strength criteria. Zhang et al. equated the damage of rock under freeze-thaw and load to level 2 damage, and proposed a calculation method for the total damage variable based on the extended strain equivalence principle, and a damage statistical model for rock under freeze-thaw and load was established33. Zhou et al.34, Huang et al.35, Fang et al.36, Zhang et al.37, and Jiang et al.38 based on Weibull distribution function, and Gao et al.39 based on Maxwell distribution theory, the damage constitutive models for rock under freeze-thaw cycles were established. Zhou et al.40 Gao et al.41 considered the characteristics of post peak softening and residual deformation, and established a damage constitutive model for rock under freeze-thaw load.

The research results have greatly enriched and developed the theory of rock mechanics. But there are the following issues: (1) Factors such as environment, temperature, humidity, and stress gradient can all affect the failure characteristics of rock. The freeze-thaw cycle in cold regions has a serious impact on the stability of engineering structures. Therefore, the damage strength criterion of rock considering freeze-thaw effects has important theoretical and practical significance for the safety and stability evaluation of engineering construction in cold regions. (2) At present, the criteria mainly describe the failure criteria of rock in various limit states. As a natural damage material, it ignores the essential characteristics of a large number of micro defects inside rock, that is, they fail to consider the damage characteristics of rock, and the calculated results do not fully match the actual failure form of rock, and cannot reveal the physical mechanism of rock failure. (3) There is a significant gap between the existing strength criteria and actual results, and the key issue is that the entire process of rock deformation and failure is not yet fully understood. In this process, not only the pre peak behavior of the stress-strain curve should be considered, but also the post peak characteristics should be accurately described, especially the residual characteristics of rock after complete failure. However, the influence of the existing strength criteria on the residual effect has not been studied, so it can not accurately reflect the essence of damage mechanics.

Based on this, the article takes the safety evaluation of geotechnical engineering in cold regions as the background, considering the characteristic of bearing strength even after complete failure of rock, the damage constitutive model of rock under freeze-thaw cycles by considering residual strength is established. Starting from the characteristics of the entire process of rock deformation and failure, the Revised M-C strength criterion of rock is established by considering freeze-thaw cyccles and residual effects. After comparison with test results, it is found that the strength criterion established in this paper better reflects the actual engineering situation.

Damage constitutive model for frozen-thawed rock considering residual stress

Establishment of damage constitutive model

In cold regions, the freeze-thaw cycles and load are the main factors causing damage and deterioration of rock. Under freeze-thaw cycles, internal cracks continue to expand and connect, ultimately leading to freeze-thaw failure. According to the strain equivalence principle, the load on rock is borne by undamaged part, so when rock is completely damaged, the residual strength must be 0 MPa. In fact, during the process of damage and failure, a macro fracture surface is formed inside rock, and the bearing capacity gradually decreases. However, it is not only fully borne by undamaged part, but also jointly borne by damage part. When rock is completely destroyed, the bearing capacity is the frictional force between fracture surfaces, which no longer changes with the increase of deformation, and is called residual strength. Therefore, when establishing a constitutive model, it is necessary to fully consider the freeze-thaw effect, and deformation characteristics of rock at various stages during load process. Firstly, make the following assumptions:

The total area of rock micro elements under freeze-thaw and load is abstracted into two parts: undamaged and damage (freeze-thaw damage, load damage, freeze-thaw and load coupling damage). The damage part bears residual stress, while the undamaged part bears effective stress. By analyzing the micro forces and geometric conditions of various parts, it can be concluded that

$$\sigma_{i} A = \sigma^{\prime}_{i} A_{1} + \sigma_{{\text{r}}} A_{2}$$
(1)
$$A = A_{1} + A_{2}$$
(2)
$$A_{2} = A_{{\text{n}}} + A_{{\text{d}}} - A_{{\text{s}}}$$
(3)

where A, A1, A2 are the total area of rock micro elements, the area of undamaged part, the area of damage part. An, Ad are the area of freeze-thaw damage part and load damage part. As is the area of freeze-thaw and load coupled damage part. \(\sigma^{\prime}_{i}\) is the effective stress. \(\sigma_{i}\) is the nominal stress. \(\sigma_{{\text{r}}}\) is the residual stress.

Considering the rock area alone, the freeze-thaw damage variable is defined as the ratio of the effective freeze-thaw damage area to total area after load action. The load damage variable can be defined as the ratio of the load damage area to undamaged area in frozen-thawed rock. Therefore, it can be obtained that

$$D_{{\text{n}}} = \frac{{A_{{\text{n}}} - A_{{\text{s}}} }}{A}$$
(4)
$$D_{{\text{d}}} = \frac{{A_{{\text{d}}} }}{{A - A_{{\text{n}}} + A_{{\text{s}}} }}$$
(5)

where Dn is freeze-thaw damage variable. Dd is load damage variable.

The ultimate damage and failure of rock is caused by the combined action of freeze-thaw and load. Therefore, the total damage variable of rock under freeze-thaw and load can be defined as

$$D_{{\text{z}}} = \frac{{A_{{\text{n}}} + A_{{\text{d}}} - A_{{\text{s}}} }}{A}$$
(6)

where Dz is total damage variable.

From Eqs. (1) to (6), the damage constitutive model for rock under freeze-thaw and load can be obtained

$$\sigma_{i} - \sigma_{{\text{r}}} = \left( {\sigma^{\prime}_{i} - \sigma_{{\text{r}}} } \right)\left( {1 - D_{{\text{d}}} } \right)\left( {1 - D_{{\text{n}}} } \right)$$
(7)

According to Eq. (7), the load damage of rock under freeze-thaw action can be considered as two progressive damage states. the first is the freeze-thaw damage state, and the second is the load damage state after freeze-thaw. The freeze-thaw and load, weaken the progress of material cohesion through different mechanical mechanisms, inducing coupling and mutual influence between the freeze-thaw damage and load damage.

According to the heterogeneity of rock in micro-structure, the distribution of mechanical properties of its internal micro-elements is random. When frozen-thawed rock is continuously loaded, the transformation of undamaged parts into damage is also a continuous process. When assuming that the strength of micro elements follows Weibull random distribution33, the load damage variable can be expressed as

$$D_{{\text{d}}} = 1 - \exp \left[ { - \left( {\frac{{F^{ * } }}{{F_{0} }}} \right)^{m} } \right]$$
(8)

where \(F^{ * }\) is the random distribution variable of rock micro element strength; F0 and m are model parameters, respectively.

By combining Eq. (8) with Eq. (7), the expression of the total damage variable of rock under freeze-thaw and load can be obtained

$$D_{{\text{z}}} = 1 - \left( {1 - D_{{\text{n}}} } \right)\exp \left[ { - \left( {\frac{{F^{ * } }}{{F_{0} }}} \right)^{m} } \right]$$
(9)

Assuming that the undamaged part follows the generalized Hooke’s law, and based on the significance of physical parameter Poisson’s ratio and the deformation coordination relationship of each part of frozen-thawed rock, and substituting Eq. (9) into Eqs. (7) can obtain

$$\sigma_{{1}} - \sigma_{{\text{r}}} = \left( {1 - D_{{\text{n}}} } \right)\left( {E_{{\text{n}}} \varepsilon_{{1}} + \mu_{{\text{n}}} \sigma_{{2}} + \mu_{{\text{n}}} \sigma_{{3}} - \sigma_{{\text{r}}} } \right)\exp \left[ { - \left( {\frac{{F_{1}^{ * } }}{{F_{0} }}} \right)^{m} } \right]$$
(10)
$$\sigma_{{2}} - \sigma_{{\text{r}}} = \left( {1 - D_{{\text{n}}} } \right)\left( {E_{{\text{n}}} \varepsilon_{{2}} + \mu_{{\text{n}}} \sigma_{{1}} + \mu_{{\text{n}}} \sigma_{{3}} - \sigma_{{\text{r}}} } \right)\exp \left[ { - \left( {\frac{{F_{2}^{ * } }}{{F_{0} }}} \right)^{m} } \right]$$
(11)
$$\sigma_{{3}} - \sigma_{{\text{r}}} = \left( {1 - D_{{\text{n}}} } \right)\left( {E_{{\text{n}}} \varepsilon_{{3}} + \mu_{{\text{n}}} \sigma_{{1}} + \mu_{{\text{n}}} \sigma_{{2}} - \sigma_{{\text{r}}} } \right)\exp \left[ { - \left( {\frac{{F_{3}^{ * } }}{{F_{0} }}} \right)^{m} } \right]$$
(12)

where \(\sigma_{{1}}\) is the nominal stress in axial direction, and \(\sigma_{{2}}\), \(\sigma_{{3}}\) are the nominal stress in lateral direction; \(\varepsilon_{{1}}\) is the strain in axial direction, and \(\varepsilon_{{2}}\), \(\varepsilon_{{3}}\) are the strain in lateral direction; En and \(\mu_{{\text{n}}}\) are the elastic modulus and Poisson’s ratio of rock under freeze-thaw cycles; \(F_{1}^{ * }\) is the random distribution variable of rock micro element strength in the axial direction; \(F_{2}^{ * }\), \(F_{3}^{ * }\) are the random distribution variable of rock micro element strength in the lateral direction.

Determination of distribution variable and model parameters

Before determining the model parameters, it is necessary to select appropriate strength criteria to reasonably measure the strength of rock micro-elements. Assuming that the failure of rock elements follows the M-C criterion42, in axial direction, the distributed variable can be expressed as

$$F_{1}^{ * } = \sigma^{\prime}_{1} - \alpha \sigma^{\prime}_{3}$$
(13)

where \(\alpha = {{\left( {{1} + {\text{sin}}\phi } \right)} \mathord{\left/ {\vphantom {{\left( {{1} + {\text{sin}}\phi } \right)} {\left( {{1} - {\text{sin}}\phi } \right)}}} \right. \kern-0pt} {\left( {{1} - {\text{sin}}\phi } \right)}}\).

Under conventional triaxial compression, substituting Eq. (7) into Eq. (13) can obtain

$$F_{1}^{ * } = \frac{{\sigma_{1} - \alpha \sigma_{3} - \left( {1 - \alpha } \right)\sigma_{{\text{r}}} }}{{\left( {1 - D_{{\text{d}}} } \right)\left( {1 - D_{{\text{n}}} } \right)}} + \left( {1 - \alpha } \right)\sigma_{{\text{r}}}$$
(14)

Substituting Eq. (10) into Eq. (14) can obtain

$$F_{1}^{ * } = \frac{{E_{{\text{n}}} \varepsilon_{{1}} + 2\mu_{{\text{n}}} \sigma_{{3}} - \sigma_{{\text{r}}} }}{{\sigma_{{1}} - \sigma_{{\text{r}}} }}\left[ {\sigma_{1} - \alpha \sigma_{3} - \left( {1 - \alpha } \right)\sigma_{{\text{r}}} } \right] + \left( {1 - \alpha } \right)\sigma_{{\text{r}}}$$
(15)

One of the relationships at the peak point of stress-strain curve is

$$\varepsilon_{1} = \varepsilon_{{{\text{sc}}}} ,\sigma_{1} = \sigma_{{{\text{sc}}}}$$
(16)

where \(\sigma_{{{\text{sc}}}}\) and \(\varepsilon_{{{\text{sc}}}}\) are the stress and strain corresponding to the peak point, respectively.

Substituting Eq. (16) into Eq. (10) can obtain

$$\exp \left[ { - \left( {\frac{{F_{{1{\text{c}}}}^{ * } }}{{F_{0} }}} \right)^{m} } \right] = \frac{{\left( {\sigma_{{1}} - \sigma_{{\text{r}}} } \right)}}{{\left( {1 - D_{{\text{n}}} } \right)\left( {E_{{\text{n}}} \varepsilon_{{1}} + 2\mu_{{\text{n}}} \sigma_{{3}} - \sigma_{{\text{r}}} } \right)}}$$
(17)

where \(F_{{1{\text{c}}}}^{ * }\) is the value in Eq. (10) when \(\varepsilon_{1} = \varepsilon_{{{\text{sc}}}}\).

Taking the derivative of \(\sigma_{1}\) for \(\varepsilon_{1}\) in Eq. (10) can obtain

$$\frac{{{\text{d}}\sigma_{{1}} }}{{{\text{d}}\varepsilon_{{1}} }} = \left( {1 - D_{{\text{n}}} } \right)\exp \left[ { - \left( {\frac{{F_{1}^{ * } }}{{F_{0} }}} \right)^{m} } \right]\left[ {E_{{\text{n}}} - m\left( {E_{{\text{n}}} \varepsilon_{{1}} + 2\mu_{{\text{n}}} \sigma_{3} - \sigma_{{\text{r}}} } \right)\left( {\frac{{F_{1}^{ * } }}{{F_{0} }}} \right)^{m} \frac{{\left( {F_{1}^{ * } } \right)^{\prime } }}{{F_{0} }}} \right]$$
(18)

Another condition should be met at the peak point of the stress-strain curve is

$$\frac{{{\text{d}}\sigma_{1} }}{{{\text{d}}\varepsilon_{1} }}\left| {_{{\varepsilon_{1} = \varepsilon_{{{\text{sc}}}} }} } \right. = 0$$
(19)

Substituting Eq. (19) into Eq. (18) can obtain

$$\left[ {E_{{\text{n}}} - m\left( {E_{{\text{n}}} \varepsilon_{{1}} + 2\mu_{{\text{n}}} \sigma_{3} - \sigma_{{\text{r}}} } \right)\left( {\frac{{F_{1}^{ * } }}{{F_{0} }}} \right)^{m} \frac{{\left( {F_{1}^{ * } } \right)^{\prime } }}{{F_{1}^{ * } }}} \right]\left| {_{{\varepsilon_{1} = \varepsilon_{{{\text{sc}}}} }} } \right. = 0$$
(20)

Combining Eqs. (15) and (20) can obtain

$$m = \frac{{F_{{1{\text{c}}}}^{ * } \left( {\sigma_{{{\text{sc}}}} - \sigma_{{\text{r}}} } \right)}}{{\left[ {\sigma_{{{\text{sc}}}} - \alpha \sigma_{3} - \left( {1 - \alpha } \right)\sigma_{{\text{r}}} } \right]\left( {E_{{\text{n}}} \varepsilon_{{{\text{sc}}}} + 2\mu_{{\text{n}}} \sigma_{{3}} - \sigma_{{\text{r}}} } \right){\text{ln}}\frac{{\left( {1 - D_{{\text{n}}} } \right)\left( {E_{{\text{n}}} \varepsilon_{{{\text{sc}}}} + 2\mu_{{\text{n}}} \sigma_{{3}} - \sigma_{{\text{r}}} } \right)}}{{\sigma_{{{\text{sc}}}} - \sigma_{{\text{r}}} }}}}$$
(21)
$$F_{0} = {{F_{{1{\text{c}}}}^{ * } } \mathord{\left/ {\vphantom {{F_{{1{\text{c}}}}^{ * } } {\left[ {{\text{ln}}\frac{{\left( {1 - D_{{\text{n}}} } \right)\left( {E_{{\text{n}}} \varepsilon_{{{\text{sc}}}} + 2\mu_{{\text{n}}} \sigma_{{3}} - \sigma_{{\text{r}}} } \right)}}{{\sigma_{{{\text{sc}}}} - \sigma_{{\text{r}}} }}} \right]^{\frac{1}{m}} }}} \right. \kern-0pt} {\left[ {{\text{ln}}\frac{{\left( {1 - D_{{\text{n}}} } \right)\left( {E_{{\text{n}}} \varepsilon_{{{\text{sc}}}} + 2\mu_{{\text{n}}} \sigma_{{3}} - \sigma_{{\text{r}}} } \right)}}{{\sigma_{{{\text{sc}}}} - \sigma_{{\text{r}}} }}} \right]^{\frac{1}{m}} }}$$
(22)

According to the theory of macro phenomenological Damage mechanics, the deterioration degree inside the material can be characterized by the response of the macro mechanical properties of rock, so the elastic modulus can be used to measure freeze-thaw damage, and can be defined as

$$D_{{\text{n}}} = 1 - \frac{{E_{{\text{n}}} }}{{E_{{0}} }}$$
(23)

where E0 is the elastic modulus of undamaged rock.

Verification of damage constitutive model

Analysis of mechanical property test results

According to the ISRM testing procedures, it is prepared into standard samples (ϕ50 mm × 100 mm), as shown in Fig. 1a, and the error of prepared samples is required to be no more than 0.3 mm, and the non parallelism of both ends is not more than 0.05 mm. After processing, the samples with good integrity are selected for acoustic wave measurement, and further select samples with good homogeneity. Based on different F-T cycles and confining pressures, 72 samples are made in the test, as shown in Fig. 1b, with every 3 samples as a group and the average value as the test result to minimize the randomness. According to the range of F-T temperature difference of geological bodies in cold regions, the freezing and thawing temperatures selected are -20 °C and 20 °C. Put the samples into the automatic F-T test box. Freezing for 6 h, thawing for 6 h, that is, every 12 h as a cycles. The freeze-thaw cycles are 0, 5, 10, 20, 30 and 40 times, respectively (The result is shown in Fig. 1c).

Fig. 1
Fig. 1The alternative text for this image may have been generated using AI.
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Red sandstone samples and freeze-thaw results.

During the freeze-thaw process of rock samples, it can be observed that in the initial stage of freeze-thaw cycles, the free particles appear on the surface and edges of sample. As the number of freeze-thaw cycles increases, bulging phenomena appear on the upper surface of sample. As the number of freeze-thaw cycles continues to increase, a large amount of surface layer of samples falls off. By the time of 40 freeze-thaw cycles, the surface layer of sample had completely peeled off. The reason can be explained as follows: when the temperature decreases and reaches below 0℃, the pore water gradually freezes from surface to inside, expands in volume, causing inconsistent expansion and contraction of different types of particle boundaries, and generating large frost heave forces between grains and pores. When the tensile stress generated by this force is greater than the corresponding ultimate tensile strength, it leads to an increase in pores and the extension of original cracks. When the temperature exceeds 0℃, the ice gradually melts into water, and the frost heave force generated by freezing is relieved. The melted water flows along the through cracks and pores, resulting in a decrease in the bonding ability between particles. As the number of freeze-thaw cycles continues to increase, the sample undergoes repeated water ice phase transitions, resulting in a continuous increase and connection of pores, rapid expansion and connection of cracks, namely the emergence of new cracks, the extension of original cracks, and ultimately the development of macro cracks with surface peeling.

The selected large red sandstone strata have a burial depth ranging from 100 to 300 m, and the Eq. σ3 = 0.023H is used to select confining pressures values, where H is burial depth. Finally, conduct triaxial compression test with confining pressuress of 0, 2, 4 and 6 MPa on samples under different freeze-thaw cycles. Therefore, the test data on mechanical parameters such as compressive strength and elastic modulus of rock samples under different freeze-thaw cycles and confining pressures are obtained, as shown in Fig. 2. When confining pressure was 0 MPa, the failure mode of red sandstone is brittle failure, the residual strength value is very small and difficult to obtain. Therefore, the residual strength value at a confining pressure of 0 MPa is set to 0 MPa.

Fig. 2
Fig. 2The alternative text for this image may have been generated using AI.
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Mechanical parameters of red sandstone under different freeze-thaw cycles and confining pressures.

As shown in Fig. 2, When the confining pressure is constant, with the increase of freeze-thaw cycles, the deterioration degree of internal structure of red sandstone gradually intensifies, leading to a decrease in its ability to resist deformation and damage, manifested as a gradual decrease in compressive strength and elastic modulus. Taking a confining pressure of 2 MPa as an example, during the freeze-thaw process from 0 to 40 times, the compressive strength decreases by nearly 15 MPa, and the elastic modulus decreases by more than 1.5 GPa. After being completely destroyed by load, the sample particles that have undergone more freeze-thaw cycles are relatively loose, so the residual strength provided by the frictional force between particles decreases. Taking a confining pressure of 4 MPa as an example, the residual strength is 34.43 MPa at 0 time freeze-thaw cycles and decreases by more than 40% at 40 times freeze-thaw cycles. The freeze-thaw cycle leads to an increase in internal pores and voids, resulting in increased deformation and peak strain and Poisson’s ratio under load. Taking a confining pressure of 6 MPa as an example, the peak strain and Poisson’s ratio increased by 0.0132 and 0.2997 at 0 time freeze-thaw cycles, and 7.12% and 5.38% at 40 freeze-thaw cycles, respectively.

The confining pressure suppresses the internal damage and improves internal structure of red sandstone. Therefore, when the number of freeze-thaw cycles is constant, the ability to resist deformation and damage increases with the increase of confining pressure. The compressive strength and elastic modulus gradually increase. When the confining pressure increases from 0 to 6 MPa, the compressive strength after 0 time freeze-thaw cycles increases by 23.50 MPa and the elastic modulus increases by 0.797 GPa. The confining pressure causes the internal pores and voids to be compacted, resulting in an increase in strain during load process, and an enhancement of plastic characteristics. At the same time, the confining pressure limits the lateral deformation, leading to a gradual decrease in Poisson’s ratio. Taking 20 times freeze-thaw cycles as an example, the peak strain is 0.0094 at confining pressure of 0 MPa, and increased to 0.0138 at confining pressure of 6 MPa, an increase of 31.88%. The Poisson’s ratio is 0.3275 at confining pressure of 0 MPa, and increased to 0.3163 at confining pressure of 6 MPa, an increase of 3.42%. Due to the presence of confining pressure, after the complete failure of red sandstone, the particles become more tightly packed, resulting in an increase in residual strength provided by the frictional force between particles. Taking 40 freeze-thaw cycles as an example, the residual strength is 13.62 MPa at confining pressure of 2 MPa, and increased to 20.06 MPa at confining pressure of 6 MPa, an increase of 32.10%.

Substituting the elastic modulus of Fig. 2 into Eqs. (23) to obtain freeze-thaw damage variable of red sandstone under different freeze-thaw cycles and confining pressures, the result is shown in Fig. 3.

Fig. 3
Fig. 3The alternative text for this image may have been generated using AI.
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Freeze-thaw damage variable of red sandstone under freeze-thaw cycles and confining pressures.

It can be seen from Fig. 3 that after freeze-thaw, new pores and cracks appear inside red sandstone, causing freeze-thaw damage. Under uniaxial compression, with the increase of freeze-thaw cycles, the damage to red sandstone becomes more significant, reaching nearly 70% after 40 freeze-thaw cycles. When the freeze-thaw cycles is constant, the increase in confining pressure under high freeze-thaw cycles has a more significant effect on red sandstone. After 40 freeze-thaw cycles, the damage caused by confining pressure of 6 MPa is reduced by nearly 25% compared to 0 MPa. As the number of freeze-thaw cycles increases, red sandstone damage intensifies, leading to deterioration of macro mechanical properties and a decrease in its bearing capacity. Under confining pressure, the pores of red sandstone are compacted, the bonding strength between particles is enhanced, the internal structure is improved, the freeze-thaw damage is weakened, and the macroscopic mechanical properties are improved. With the increase of confining pressure, the degree of freeze-thaw damage weakening is greater, and the red sandstone has stronger resistance to deformation and failure.

Verification of constitutive model for frozen-thawed rock considering residual effects

After introducing Eqs. (15), (21), (22) and (23) into Eq. (10), the data from Figs. 2 and 3 is introduced to obtain the theoretical curves of damage constitutive model for rock under freeze-thaw cycles considering residual effects. The result are compared with test curves, as shown in Fig. 4.

Fig. 4
Fig. 4The alternative text for this image may have been generated using AI.
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Verification of damage constitutive model.

It can be seen from Fig. 4 that the theoretical curves and the test curves are not completely consistent. In order to better verify the rationality of constitutive model, the variance R2 is selected to reflect the degree of dispersion between theoretical results and test data. By comparing the corresponding points of the test curves and the theoretical curves, the variances in Fig. 4(a)-(f) are 0.1, 0.15, 0.55, 0.53, 0.41, and 0.31, respectively. The variances are small, indicating that the theoretical curves of model are in good agreement with test curves, thus verifying the rationality of the constitutive equation and damage variable description, and can better reflect the mechanical behavior of rock under various confining pressures and freeze-thaw cycles.

When the confining pressure is constant, with the increase of freeze-thaw cycles, the stress-strain compressibility increases, the elastic stage shortens, and the elastic modulus continuously decreases, which is shown in the graph as a decrease in the slope of the stress-strain curve in elastic stage. After the peak, red sandstone exhibits brittle failure under uniaxial compression, at this stage, with minimal deformation, the stress drops sharply. When the maximum bearing capacity is reached, the bearing capacity decreases rapidly as the load continues to be applied. The stress-strain curve shows a sharp and abrupt decrease trend, and the macro surface of sample experiences rapid cracking, resulting in brittle failure. While under triaxial compression, red sandstone exhibits plastic. when the peak stress is reached, the curve gradually decreases with the increase of strain, and the ability to bear load gradually decreases until complete failure, indicating that the micro elements undergo irreversible plastic deformation and the crack propagation rate accelerates. As the deformation continues to increase, a macro fracture runs through the interior of red sandstone, and the cohesive force between the fracture surfaces is basically lost. The bearing capacity is completely provided by the frictional force between the fracture surfaces and maintained at a stable value, which no longer changes with the increase of deformation. When the number of freeze-thaw cycles is constant, with the increase of confining pressure, the elastic stage of the stress-strain curve is prolonged, the elastic modulus continues to increase, and the peak point moves to the upper right, indicating that the bearing capacity and resistance to deformation and failure of red sandstone are enhanced. The failure mode changes from brittle uniaxial to ductile failure, and this characteristic becomes more obvious with the increase of confining pressure. Considering that there was no damage during the initial compaction stage, the author equated this stage with the elastic stage during the establishment of constitutive model33,42, resulting in a linear distribution of the theoretical curve obtained.

This model has the following advantages: (1) The model fully reflects the entire process of rock deformation and failure, especially the post peak softening and residual deformation characteristics of rock. (2) This fully reflects the influence of freeze-thaw and stress state on rock strength. As the number of freeze-thaw cycles increases, the rock strength decreases and deformation increases; As the confining pressure increases, the strength and deformation of rock also increase, which is consistent with the actual situation. (3) This model has obvious advantages over existing research results, and the calculation results can better reflect engineering practice, laying a solid foundation for the establishment of rock damage strength theory.

Revised Mohr-Coulomb strength criterion of rock based on freeze-thaw and residual effects

Establishment of Revised Mohr-Coulomb strength criterion

In order to determine the strength criterion of this article more simply and conveniently, the distribution variables of Eq. (15) are transformed and introduced into the constitutive model of Eq. (10). After transformation, Eq. (15) can be represented as

$$F_{11}^{ * } - \left( {1 - \alpha } \right)\sigma_{{\text{r}}} = \frac{{E_{{\text{n}}} \varepsilon_{{1}} + 2\mu_{{\text{n}}} \sigma_{{3}} - \sigma_{{\text{r}}} }}{{\sigma_{{1}} - \sigma_{{\text{r}}} }}\left[ {\sigma_{1} - \alpha \sigma_{3} - \left( {1 - \alpha } \right)\sigma_{{\text{r}}} } \right]$$
(24)

Under the conventional triaxial stress state, according to the law of total differentiation of multivariate functions, treating \(\sigma_{1}\) and \(\sigma_{3}\) as functions of \(\varepsilon_{1}\),\(\varepsilon_{3}\) and n, differentiation of \(\sigma_{1}\) and \(\sigma_{3}\) can be obtained

$$\begin{aligned} & d\sigma_{1} = \frac{{\partial \sigma_{1} }}{{\partial \varepsilon_{1} }}d\varepsilon_{1} + \frac{{\partial \sigma_{1} }}{{\partial \varepsilon_{3} }}d\varepsilon_{3} + \frac{{\partial \sigma_{1} }}{\partial n}dn \\ & d\sigma_{3} = \frac{{\partial \sigma_{3} }}{{\partial \varepsilon_{1} }}d\varepsilon_{1} + \frac{{\partial \sigma_{1} }}{{\partial \varepsilon_{3} }}d\varepsilon_{3} + \frac{{\partial \sigma_{3} }}{\partial n}dn \\ \end{aligned}$$

Under the conventional triaxial stress state, taking differentiation separately from Eqs. (13) and (15) can obtain

$$\left\{ {\begin{array}{*{20}l} {d\sigma_{1} = \alpha_{1} d\varepsilon_{1} + \alpha_{2} dF_{11}^{ * } + \alpha_{3} dm + \alpha_{4} dF_{0} + \alpha_{5} dE_{n} + 2\mu_{n} d\sigma_{3} + 2\sigma_{3} d\mu_{n} } \hfill \\ {d\sigma_{3} = \beta_{1} d\varepsilon_{3} + \beta_{2} dF_{3}^{ * } + \beta_{3} dm + \beta_{4} dF_{0} + \beta_{5} dE_{n} + \left( {\sigma_{3} + \sigma_{1} } \right)d\mu_{n} + \mu_{n} d\sigma_{1} + \mu_{n} d\sigma_{3} } \hfill \\ \end{array} } \right.$$
(25)

where \(\alpha_{1} = \frac{{\partial \sigma_{1} }}{{\partial \varepsilon_{1} }}\),\(\alpha_{2} = \frac{{\partial \sigma_{1} }}{{\partial F_{11}^{ * } }}\), \(\alpha_{3} = \frac{{\partial \sigma_{1} }}{\partial m}\), \(\alpha_{4} = \frac{{\partial \sigma_{1} }}{{\partial F_{0} }}\), \(\alpha_{5} = \frac{{\partial \sigma_{1} }}{{\partial E_{n} }}\), \(\beta_{1} = \frac{{\partial \sigma_{3} }}{{\partial \varepsilon_{3} }}\), \(\beta_{2} = \frac{{\partial \sigma_{3} }}{{\partial F_{3}^{ * } }}\), \(\beta_{3} = \frac{{\partial \sigma_{3} }}{\partial m}\), \(\beta_{3} = \frac{{\partial \sigma_{3} }}{\partial m}\) , \(\beta_{4} = \frac{{\partial \sigma_{3} }}{{\partial F_{0} }}\), \(\beta_{5} = \frac{{\partial \sigma_{3} }}{{\partial E_{n} }}\).

In Eqs. (10) and (12), \(dF_{11}^{ * }\) and \(dF_{3}^{ * }\) can be represented as

$$\left\{ {\begin{array}{*{20}l} {dF_{11}^{ * } = F_{11} d\varepsilon_{1} + F_{12} d\sigma_{1} + F_{13} d\sigma_{3} + F_{14} dn} \hfill \\ {dF_{3}^{ * } = F_{21} d\varepsilon_{3} + F_{22} d\sigma_{1} + F_{23} d\sigma_{3} + F_{24} dn} \hfill \\ \end{array} } \right.$$
(26)

where \(F_{11} = \frac{{\partial F_{11}^{ * } }}{{\partial \varepsilon_{1} }}\), \(F_{12} = \frac{{\partial F_{11}^{ * } }}{{\partial \sigma_{1} }}\), \(F_{13} = \frac{{\partial F_{11}^{ * } }}{{\partial \sigma_{1} }}\), \(F_{14} = \frac{{\partial F_{11}^{ * } }}{\partial n}\), \(F_{21} = \frac{{\partial F_{3}^{ * } }}{{\partial \varepsilon_{3} }}\), \(F_{21} = \frac{{\partial F_{3}^{ * } }}{{\partial \varepsilon_{3} }}\), \(F_{23} = \frac{{\partial F_{3}^{ * } }}{{\partial \sigma_{3} }}\), \(F_{24} = \frac{{\partial F_{3}^{ * } }}{\partial n}\).

Assuming m, F0, En are functions of confining pressure and freeze-thaw cycles, then

$$\left\{ {\begin{array}{*{20}l} {dF_{0} = F_{3} d\sigma_{3} + F_{4} dn} \hfill \\ {dm = m_{3} d\sigma_{3} + m_{4} dn} \hfill \\ {dE_{n} = E{}_{3}d\sigma_{3} + E{}_{4}dn} \hfill \\ {d\mu_{n} = W{}_{3}d\sigma_{3} + W{}_{4}dn} \hfill \\ \end{array} } \right.$$
(27)

where \(F_{3} = \frac{{\partial F_{0} }}{{\partial \sigma_{3} }}\), \(m_{3} = \frac{\partial m}{{\partial \sigma_{3} }}\), \(F_{4} = \frac{{\partial F_{0} }}{\partial n}\), \(m_{4} = \frac{\partial m}{{\partial n}}\), \(E_{3} = \frac{{\partial E_{n} }}{{\partial \sigma_{3} }}\), \(E_{4} = \frac{{\partial E_{n} }}{\partial n}\), \(W_{3} = \frac{{\partial W_{n} }}{{\partial \sigma_{3} }}\), \(W_{4} = \frac{{\partial W_{n} }}{\partial n}\).

By substituting Eqs. (26) and (27) into Eq. (25) and simplifying, it can be obtained that

$$\left\{ {\begin{array}{*{20}l} {T_{1} d\sigma_{1} + T_{2} d\sigma_{3} + T_{3} d\varepsilon_{1} + T_{4} dn = 0} \hfill \\ {S_{1} d\sigma_{1} + S_{2} d\sigma_{3} + S_{3} d\varepsilon_{3} + S_{4} dn = 0} \hfill \\ \end{array} } \right.$$
(28)

where.

\(T_{1} = \alpha_{2} F_{12} - 1\), \(T_{2} = \alpha_{2} F_{13} + \alpha_{3} m_{3} + \alpha_{4} F^{\prime}_{3} + \alpha_{5} E_{3} + 2\mu_{n} + 2\sigma_{3} W_{3}\), \(T_{3} = \alpha_{1} + \alpha_{2} F_{11}\), \(T_{4} = \alpha_{3} m_{4} + \alpha_{4} F_{4} + \alpha_{5} E_{4} + 2\sigma_{3} W_{4}\), \(S_{1} = \beta_{2} F_{22} + \mu_{n}\), \(S_{2} = \beta_{2} F_{23} + \beta_{3} m_{3} + \beta_{4} F_{3} + \beta_{5} E_{3} + W_{3} \left( {\sigma_{1} + \sigma_{3} - \sigma_{r} } \right) + \mu_{n} - 1\), \(S_{3} = \beta_{1} + \beta_{1} F_{21}\), \(S_{4} = \beta_{3} m_{4} + \beta_{4} F_{4} + \beta_{5} E_{4} + W_{4} \left( {\sigma_{1} + \sigma_{3} - \sigma_{r} } \right)\).

Solving Eq. (28) can obtain

$$d\sigma_{1} = \frac{{ - S_{2} T_{3} d\varepsilon_{1} + S_{3} T_{2} d\varepsilon_{3} + \left( {T_{2} S_{4} - T_{4} S_{2} } \right)dn}}{{S_{2} T_{1} - S_{2} T_{2} }}$$
(29)

By combining Eq. (29) with condition (19), it can be obtained that

$$\frac{{d\sigma_{1} }}{{d\varepsilon_{1} }}\left| {_{{\varepsilon_{1} = \varepsilon_{{{\text{sc}}}} }} } \right. = \frac{{ - S_{2} T_{3} }}{{S_{2} T_{1} - S_{2} T_{2} }}\left| {_{{\varepsilon_{1} = \varepsilon_{{{\text{sc}}}} }} } \right. = 0$$
(30)

Eq. (30) can be simplified as

$$T_{3} = \left( {\alpha_{1} + \alpha_{2} F_{11} } \right)\left| {_{{\varepsilon_{1} = \varepsilon_{{{\text{sc}}}} }} } \right. = 0$$
(31)

where.

\(\alpha_{1} = \frac{{E_{{\text{n}}} }}{{E_{0} }}E_{{\text{n}}} \exp \left[ { - \left( {\frac{{F_{11}^{ * } }}{{F_{0} }}} \right)^{m} } \right]\), \(\alpha_{2} = - \frac{{mE_{{\text{n}}} }}{{F_{11}^{ * } E_{0} }}\left( {E_{{\text{n}}} \varepsilon_{1} + 2\mu \sigma_{3} - \sigma_{{\text{r}}} } \right)\left( {\frac{{F_{11}^{ * } }}{{F_{0} }}} \right)^{m} \exp \left[ { - \left( {\frac{{F_{11}^{ * } }}{{F_{0} }}} \right)^{m} } \right]\), \(F_{11} = \frac{{E_{{\text{n}}} F_{11}^{ * } }}{{E_{{\text{n}}} \varepsilon_{1} + 2\mu \sigma_{3} - \sigma_{{\text{r}}} }}\).

Solving Eq. (31) and substitute the result into Eq. (23) can obtain the relationship between the corresponding stress components in the limit state

$$\sigma_{1} = F_{0} \left( {1 - D_{{\text{n}}} } \right)\left( \frac{1}{m} \right)^{{\frac{{1}}{m}}} {\text{exp}}\left( { - \frac{1}{m}} \right) + \alpha \sigma_{3} + \left( {1 - \alpha } \right)\sigma_{{\text{r}}}$$
(32)

Equation (32) is the revised M-C strength criterion established in this article. According to Eq. 32, it can be inferred that: (1) This criterion considers the influence of freeze-thaw cycles and confining pressure on rock strength, and can characterize the relationship between internal stress parameters at rock limit states under different freeze-thaw cycles and confining pressure. (2) This criterion includes model parameters, which are determined by mechanical parameters, thus establishing the relationship between the criterion and mechanical parameters, especially considering the influence of residual effects. Therefore, the strength criterion can be determined while describing the entire process of rock failure, providing reliable assurance for the safety and economy of engineering design in cold regions.

Verification of revised Mohr-Coulomb strength criterion

To verify the rationality of the revised M-C strength criterion for rock considering freeze-thaw and residual stress, the data from Figs. 2 and 3 is substituted into Eq. (32) to obtain the theoretical data of the strength criterion and compared with the test strength data, as shown in Fig. 5.

Fig. 5
Fig. 5The alternative text for this image may have been generated using AI.
Full size image

Verification of revised M-C strength criterion.

From Fig. 5, it can be seen that the revised M-C strength criterion data in this paper does not exceed the test data and is relatively close to the test data. The difference between the two is less than 25% during freeze-thaw process, indicating that the established strength criteria can be safely used as a basis for rock elemental failure, verifying the rationality of the strength standard considering freeze-thaw and residual effects. From the graph, The freeze-thaw cycles increased from 0 to 40 times, and the maximum principal stress decreased by 62.22%, 58.57%, 51.84%, and 50.19% respectively at confining pressures of 0 MPa, 2 MPa, 4 MPa, and 6 MPa. This indicates that freeze-thaw exacerbates the damage to red sandstone. With the increase of freeze-thaw cycles, the maximum principal stress continuously decreases, but the rate of decrease gradually slows down. Moreover, under high confining pressure conditions, the rate of decrease in maximum principal stress also decreases, indicating that confining pressure suppresses damage to red sandstone.

Cite the test data from reference 33 and substitute it into Eq. (32) to obtain the theoretical strength values under different freeze-thaw cycles and confining pressure, and compare them with the M-C and D-P strength criteria. The results are shown in Fig. 6.

Fig. 6
Fig. 6The alternative text for this image may have been generated using AI.
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Comparison of data with different strength criteria.

From Fig. 6, it can be seen that when the confining pressure is small, both the criteria in this paper and the M-C criterion are in good agreement with the test results. However, with the increase of confining pressure, the strength obtained by M-C criterion is significantly greater than the test results. When the confining pressure is 6MPa, it exceeds 10% of the test data. This is mainly because M-C criterion is a linear relationship, which can not reflect the curve characteristics of the Mohr envelope of rock strength. The data obtained by the D-P criterion is smaller than the test data and has a significant difference. As the freeze-thaw cycles and confining pressure increase, the gap between the curves also increases, indicating that the D-P criterion is relatively conservative and cannot reflect the development of rock strength under freeze-thaw cycles and confining pressure. With the increase of confining pressure, the maximum principal stress obtained by the revised M-C strength criterion in this paper continuously increases, but its growth rate gradually slows down and remains lower than test data, with a difference of no more than 20%. The agreement with test curve is high, indicating that the revised M-C strength criterion in this paper can reflect the curve of the Mohr envelope and is closer to the actual situation, verifying the rationality of the strength criterion established in this paper. And as shown in Fig. 4, as the number of freeze-thaw cycles increases, the curve properties of the Mohr envelope become more and more obvious, indicating that the strength criterion in this paper can reflect the influence of freeze-thaw effects and residual strength on rock strength characteristics, and can be used as a standard to measure whether rock micro elements are damaged under freeze-thaw cycles and confining pressure. Compared with the M-C strength criterion, starting from the entire process of rock deformation and failure, residual strength is introduced into the strength criterion, so that under various freeze-thaw cycles and confining pressures, the theoretical strength data is below the test data, ensuring engineering safety. It shows that the strength criterion can reflect the curve of Mohr envelope of rock, which is closer to the actual situation. Compared with the D-P criterion, the theoretical data in this paper is more consistent with the test data, solving the drawback of the D-P criterion being more conservative.

Conclusion

The total area of rock micro elements under freeze-thaw and load is abstracted into two parts: undamaged and damage, the damage part bears residual stress, while the undamaged part bears effective stress, a damage constitutive model that simulates the entire process of deformation and failure of rock under freeze-thaw and load is established. Based on this, a method for establishing strength criteria of rock using multivariate function extremum is explored according to the concept of rock yield or failure. Therefore, the M-C strength criterion is reasonably revised, which has important theoretical significance and engineering practical value. The following conclusions can be drawn:

(1) The established damage constitutive model fully reflects the entire process of rock deformation and failure, especially the post peak softening and residual deformation characteristics. The model fully reflects the influence of freeze-thaw and stress states on rock strength. With the increase of freeze-thaw cycles or confining pressure, the compressibility of rock gradually increases, the peak strain continues to increase, and the stress drop after the peak of the stress-strain curve slows down. The rock gradually transitions from brittle failure to ductile failure mode. However, the elastic modulus, elastic limit, peak stress, and residual strength decrease with increasing freeze-thaw cycles and increase with increasing confining pressure.

(2) Starting from the concept of rock damage and the study of the whole process simulation simulation method of rock deformation and failure, the revised M-C strength criterion of rock is established by considering the freeze-thaw and residual effects. This criterion introduces the influence of freeze-thaw cycles factor, which can characterize the relationship between internal stress parameters in rock limit states under different freeze-thaw cycles. The introduction of residual stress into the strength criterion makes the theoretical data closer to the test data and more in line with practical engineering.