The constitutive model of rock is closely connected with the mechanical properties of rock. To achieve a more accurate quantitative analysis of the mechanical properties of rock after the action of freeze-thaw cycles, it is necessary to establish the constitutive models of rock subjected to freeze-thaw cycles from the view of rock damage. Based on the assumption of rock couple damage, this study established a statistical damage constitutive model of rock subjected to freeze-thaw cycles by combining the lognormal distribution, which is commonly used in engineering reliability analysis, and the strain strength theory. Then, the coordinates and derivative at the peak of the stress-strain curve of the rock after the action of freeze-thaw cycles were obtained through experiments to solve the statistical distribution parameters _{ε} and _{ε} reflects the strength of the rock, which shows a positive relation, and

Freeze and thaw effect is widely encountered in cold areas, which shows significant influences on the rock mass when constructing geotechnical engineering. In order to ensure the stability of the projects, in addition to considering the inherent mechanical properties of rock mass, it is also necessary to consider the deterioration of the rock mechanical properties under the low-temperature environment and the repeated freezing and thawing action, which has been a research focus in the field of rock mechanics. Extensive research studies have been carried out on the degradation of rock mechanical properties caused by freeze-thaw cycles. Typically, the changes in the compressive mechanical properties [

Since the concept of rock statistical damage constitutive was put forward by combining continuous damage theory and statistical strength theory, it has become an efficient method to studying the rock stress-strain relationships [

There exist multitudinous microdefects (cracks, pores, etc.) inside the rock. The water in these microdefects freezes into ice when the temperature around the rock decreases below 0°C, causing volume expansion, while it melts into water when the temperature rises above 0°C, in which progress part of the water will migrate [_{n} is the elastic modulus of the rock after different freeze-thaw cycles and _{0} is the initial elastic modulus of the rock.

For a quasi-brittle material such as rock, the strain caused by loading is often used to analyze the rock damage. Due to the rock discontinuity and the random distribution of load-bearing particles, it can be assumed that the strain limit of the rock microunits after the action of freeze-thaw cycles obeys a certain statistical distribution, and the failure occurs when the strain of the microunit exceeds the limit. In order to calculate the damage variable _{p} of the rock at a certain strain level after the action of freeze-thaw cycle, the initial number of rock microunits after the action of freeze-thaw cycles was denoted as _{0}, and the failure number of rock microunits under a certain strain level is _{p}. Definition:

For the statistical distribution that the strain limit of rock microunits after the action of freeze-thaw cycles obeys, the previous research studies mainly employed the Weibull distribution. However, rock is a kind of quasi-brittle material; due to the obvious characteristic length, the problem of size effect in Weibull statistical theory does not apply to quasi-brittle material [_{ε} and

In the process of the rock microunits, strain increases from 0 to _{1}, and the number of the damaged rock microunits is

From formulas (

The damage variables of the rock subjected to the freeze-thaw cycles and the loading separately were deduced above. According to the generalized equivalent strain principle proposed in [

From formulas (

When the rock is not subject to freeze-thaw cycles

When the rock is unloaded (

According to the Lemaitre equivalent strain principle [

From formulas (

As shown in Figure

The general compression stress-strain curve of rock after the action of freeze-thaw cycles.

It can be seen from Figure

Then, the following formulas can be obtained:

Assuming

From formulas (

Taking the logarithm of both sides of formula (

From formula (

The test results of red sandstone triaxial compression after different freeze-thaw cycles conducted in [

According to the test results of [

Parameter values of the rock damage constitutive model under different freeze-thaw cycles.

F-T cycles | ||||||||
---|---|---|---|---|---|---|---|---|

0 | 2 | 38.4 | 1.387 | 0.258 | 14.572 | 11.0 | 0.2144 | 2.6588 |

4 | 1.628 | 0.255 | 19.652 | 13.0 | 0.3018 | 2.8553 | ||

6 | 1.649 | 0.254 | 24.866 | 16.0 | 0.3078 | 3.0644 | ||

5 | 2 | 35.8 | 1.295 | 0.259 | 13.101 | 10.6 | 0.2295 | 2.6289 |

4 | 1.452 | 0.257 | 19.132 | 13.2 | 0.2099 | 2.8386 | ||

6 | 1.565 | 0.255 | 24.347 | 17.0 | 0.3499 | 3.1278 | ||

10 | 2 | 33.6 | 1.156 | 0.262 | 12.701 | 11.3 | 0.2078 | 2.6821 |

4 | 1.289 | 0.260 | 18.910 | 14.5 | 0.1950 | 2.9239 | ||

6 | 1.325 | 0.259 | 23.519 | 19.2 | 0.3465 | 3.2494 | ||

20 | 2 | 30.9 | 0.890 | 0.268 | 11.356 | 12.1 | 0.0985 | 2.6606 |

4 | 1.066 | 0.264 | 18.100 | 18.0 | 0.2996 | 3.1804 | ||

6 | 1.240 | 0.260 | 22.903 | 21.1 | 0.4157 | 3.3369 | ||

40 | 2 | 29.2 | 0.710 | 0.277 | 10.570 | 15.2 | 0.2325 | 2.9907 |

4 | 0.917 | 0.273 | 17.121 | 19.0 | 0.2627 | 3.2253 | ||

6 | 0.932 | 0.269 | 21.274 | 24.9 | 0.3832 | 3.5076 |

For example, when the number of freeze-thaw cycles

Then,

Substituting it into formula (

Finally, substituting

Then, according to formula (

The stress-strain curves of rock subjected to different freeze-thaw cycles. (a)

It can be seen from Figure

Through carrying out sensitivity analysis of parameters _{ε} and

By fixing _{n} = 2 GPa, _{3} = 0 MPa and supposing _{ε} = 0.8, 0.9, 1.0, 1.1, and 1.2, respectively, the different stress-strain curves were shown in Figure _{ε} increases, the elastic segments of the stress-strain curve basically overlap, the peak strength of the rock increases significantly, and the postpeak curves are roughly parallel. So, it can be considered that _{ε} reflects the strength of the rock, which is positively related with the strength of the rock.

The sensitivity analysis of parameters

Similarly, keeping _{n} = 2 GPa, _{ε} = 1, and _{3} = 0 MPa and supposing

The sensitivity analysis of parameters

In the lognormal distribution,

Through theoretical derivation, this paper established a new statistical damage constitutive model of rock subjected to freeze-thaw cycles based on the lognormal distribution. The model is simple in expression, the parameters are easy to solve, and the stress-strain curve of the rock after the action of different freeze-thaw cycles can be obtained, which has strong applicability.

The calculated theoretical curves by established model were compared with the experimental curves, which have similar trends and show a great coincidence, indicating that the statistical damage constitutive model is reasonable and valid. However, it cannot describe the compaction stage and the postpeak stage of the rock very well.

The sensitivity analysis of two lognormal distribution statistical parameters _{ε} and _{ε} reflects the strength of the rock, which shows a positive relation.

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was funded by the project (51774322) supported by National Natural Science Foundation of China and the project (2018JJ2500) supported by Hunan Provincial Natural Science Foundation of China.