A new damage constitutive model for frozen-thawed sandstone under triaxial conditions: Considering the characteristics of pre-peak compaction and post-peak residual strength

Abstract

The study of the constitutive model under stress loading for rock after undergoing freeze-thaw weathering cycles (FTWCs) treatment has important theoretical significance for the site operation and numerical calculation of rock mass engineering in cold regions. In this work, a series of FTWCs treatment tests of sandstone were carried out, and the evolution characteristics of the T₂ spectrum distribution curve for sandstone before and after FTWCs treatment were analyzed utilizing nuclear magnetic resonance (NMR) technology. The schematic diagram of freeze-thaw damage evolution for sandstone was drawn, and the damage evolution mechanism was analyzed. Evolution laws of the stress-strain curves, peak and residual strength, peak and residual strain, and Young's modulus of sandstone under different FTWCs and confining pressures were analyzed by conventional triaxial compression tests. A piecewise constitutive model under triaxial stress loading for sandstone after undergoing FTWCs treatment was established, and the model considered the significant influence of FTWCs on the nonlinear deformation in the compaction stage and residual strength. By introducing error analysis indexes, the proposed model and published models were compared with the experimental data, it was found that the proposed model's performance is better than the published models, which indicated that the proposed model has good performance and strong universality.

Keywords

Freeze-thaw weathering cycles deterioration of mechanical parameters constitutive model of rock freeze-thaw damage nuclear magnetic resonance technology

Introduction

The constitutive model of rock is a comprehensive description of the strength and deformation parameters of rock, which plays an important role in practical engineering and numerical simulation. It has always been a hot topic in the field of rock mechanics, scholars have done a lot of work on rock damage constitutive models and achieved great progress and fruitful results (Bruning et al., 2019; Li et al., 2012; Shen et al., 2019; Wang et al., 2021). However, due to the different environments in which the rocks are located, the mechanical properties and deformation behaviors under the effect of the external environment for rocks are quite different, which needs further research and exploration. In recent years, large-scale projects such as transportation construction (Li et al., 2022; Yahaghi et al., 2021) and mining (Feng et al., 2022; Wang et al., 2021) are gradually developing into high-altitude cold regions. Due to the heavy rain and snow and large temperature difference in high-altitude cold regions, the FTWCs induced by the periodic changes in temperature induces freeze-thaw damage for rock, which accelerates the propagation and coalescence of internal crack (Cao et al., 2022; Liu et al., 2022; Huang et al., 2022; Qiao et al., 2022). Under loading conditions, the rock strength deteriorates significantly and the deformation exhibits strong nonlinear characteristics (Gao et al., 2021; Hou et al., 2022). The damage evolution law of rock in the whole loading process after undergoing FTWCs was difficult to describe, which used the constitutive model of rock under conventional conditions. Therefore, it is necessary to establish the damage constitutive model under stress loading for rock subjected to FTWCs treatment, which can better analyze the damage evolution laws under stress loading for rock subjected to FTWCs treatment and has important theoretical and practical engineering significance.

Scholars have carried out relatively little research work on the damage constitutive model under stress loading for rock subjected to FTWCs treatment and achieved a few research results. Wen Fang et al. (2019) established a damage statistical constitutive model of rock under freeze-thaw and uniaxial loading based on the generalized strain equivalent theory. Shengtao Zhou et al. (2020) established a fractal damage constitutive model considering rock residual strength based on the fractal characteristics of the sectional macroscopic image. Based on internal state variables theory under the framework of thermodynamics, A stress-temperature-water coupled plastic incremental constitutive model of rock after undergoing FTWCs was proposed by Zhen Wang et al. (2019). Dexing Qu et al. (2018) considered the effect of FTWCs in an acidic environment on the compaction stage of stress-strain curves, by introducing the correction coefficient of the total damage, and the damage constitutive model of acidic freeze-thaw sandstone under loading was established. Feng Gao et al. (2021) proposed a new piecewise constitutive model under uniaxial stress loading for rock subjected to FTWCs treatment, according to the characteristics of compaction and post-compaction stages of the stress-strain curve under uniaxial stress loading for rock subjected to FTWCs treatment, and it describes the compaction, elastic deformation, yielding and failure stage of the sandstone very well.

All the above damage constitutive models of rocks subjected to FTWCs treatment are the uniaxial compression conditions, and the effect of confining pressure under triaxial compression conditions has not been considered. There is no doubt that actual rock mass engineering is always in a certain stress field. To be closer to the actual engineering, it is necessary to establish a damage constitutive model under triaxial stress loading for rock subjected to FTWCs treatment. Shibing Huang et al. (2018) derived a statistical damage constitutive model under triaxial stress loading for rock subjected to FTWCs treatment, and the model was applied to analyze the stability of a tunnel under the coupled thermo-hydro-mechanical condition in cold regions. Hang Lin et al. (2021) and Yani Lu et al. (2019) established a damage constitutive model under triaxial stress loading for rock subjected to FTWCs treatment based on Lognormal distribution and Weibull distribution, respectively. Those two models cannot describe the compaction stage and residual stage, but the elastic deformation, yielding and failure stage of the rock very well. Hongyan Liu et al. (2021) established a constitutive model under triaxial stress loading for rock subjected to FTWCs treatment based on deformation and propagation of the microcrack. Huimei Zhang et al. (2020) and Lei Shi et al. (2021) considered the residual strength characteristics under triaxial stress loading for sandstone that underwent FTWCs treatment, and a damage constitutive model under triaxial stress loading for sandstone subjected to FTWCs treatment was established. Huimei Zhang et al. (2021) and Xiangzhen Meng et al. (2021) proposed a constitutive model under triaxial stress loading for rock subjected to FTWCs treatment, based on the growth of crack law and the modified logistic equation, respectively. Neither of these models can describe the compaction stage of rock well, while the post-compaction stage can be described well.

From the above analysis, it can be seen that the constitutive model currently established either only considers the nonlinear deformation characteristics in the compaction stage (Gao et al., 2021; Qu et al., 2018), or only considers the post-peak residual strength characteristics (Zhang et al., 2020; Shi et al., 2021; Zhang et al., 2021; Meng et al., 2021). Therefore, it is necessary to establish a new constitutive model that can reflect both the nonlinear deformation characteristics in the compaction stage and post-peak residual strength characteristics. Firstly, a series of indoor FTWCs tests and conventional triaxial compression (CTC) tests were carried out on sandstones. Secondly, combined with the evolution characteristics of the T₂ spectrum distribution curve for sandstone before and after FTWCs treatment, the freeze-thaw damage mechanism of sandstone was deeply analyzed; Thirdly, the influence mechanism of the number of FTWCs and confining pressure on the evolution laws of sandstone mechanical parameters were analyzed. Finally, a new constitutive model considering both nonlinear deformation in the compaction stage and post-peak residual strength characteristics for rock subjected to FTWCs treatment was established, and the performance evaluation of the model was carried out based on the experimental data of this paper and the model of published papers.

Experimental materials and methods

Rock specimen preparation

The rock specimens used in the tests are sandstone with a fine grain texture, good homogeneity and no macroscopic visible joints and cracks. According to the International Society for Rock Mechanics (IRSM) suggested method for rock characterization, testing and monitoring (Ulusay, 2015), the natural rock blocks were processed into cylindrical specimens with a height-to-diameter ratio of 2:1 (diameter of 50 mm), and the flatness of the specimens both end surfaces was less than 0.05 mm.

Experimental procedures

All specimens were put into the drying oven for drying treatment, and then all the specimens were grouped and numbered. According to the number of FTWCs set in the tests, the specimens were classified into 5 groups: group A, B, C, D and E. Each group has 12 specimens that were carried out CTC tests with confining pressures of 3MPa, 6MPa, 9MPa and 12MPa, respectively.

The specimens of Group A (0 FTWCs) were used directly to the CTC tests. All the other specimens were placed in a vacuum pump and forced to be saturated with water for 4 h under a vacuum pressure of 0.1 MPa, and then immersed in water for 24 h. Meantime, the Ani-MR150 rock NMR imaging analysis system was used to NMR tests for specimens before FTWCs.

All the saturated specimens were put into the TDS-300 automatic freeze-thaw test machine for FTWCs tests. According to the temperature variation law of the sampling site, referring to the freeze-thaw cycles tests operating rules in the Code for rock tests in water and hydropower projects (SL/T 264-2020) (Changjiang River Scientific Research Institute, 2020), the FTWCs test parameters were set as follows: freezing temperature was -20 °C, thawing temperature was +20 °C, freezing and thaw time was 4 h, and the maximum number of FTWCs was 40.

When the number of FTWCs set in the test was reached, the corresponding specimens were taken out to measure height and diameter, and the Ani-MR150 rock NMR imaging analysis system was used to NMR tests. Then the CTC tests were carried out, and the displacement-control loading mode with a loading rate of 0.1 mm/min was adopted. The FTWCs tests and CTC tests were repeated until the end of 40 FTWCs tests. The testing process is shown in Figure 1.

Experimental results and analysis

T₂ spectrum distribution curves and freeze-thaw damage mechanism

Figure 2 shows the evolution characteristics of the T₂ spectrum distribution curves of sandstone before and after FTWCs treatment. According to the principle of NMR (Kenyon, 1992; Kenyon, 1997; Li et al., 2018), the transverse relaxation time T₂ is proportional to the pore size inside the rock, and the peak value of the T₂ curves is proportional to the number of pores of the corresponding size.

It can be seen from Figure 2 that the T₂ spectrum distribution curves of sandstone specimens before and after FTWCs treatment are dominated by three spectral peaks, which can respectively characterize the micropores, mesoporous and macropores pore. As shown in Figure 2(a), in the early stage of the FTWCs (before 10 cycles), the peak value of the first spectral peak mainly increased, and the peak value of the second and third spectral peaks did not change significantly, that is, in the early stage of the FTWCs, the micropores pores increase mainly. As shown in Figure 2(b) and Figure 2(c), with the increase in the number of FTWCs, the peak values of the first and second spectral peaks increased in the middle stage of the FTWCs (between 10 and 30 cycles), while the peak values of the third spectral peak did not change significantly, that is, in the middle stage of the FTWCs, the micropores and mesoporous pores increase mainly. As shown in Figure 2(d), the peak values of the first, second and third spectral peaks all increased in the late-stage FTWCs (between 30 and 40 cycles), that is, the micropores, mesoporous and macropores pores all increased in the late-stage FTWCs. In addition, the expansion of the T₂ spectrum distribution curves to the left, which indicates that some new micropore pores are generated. The above analysis results show that as the number of FTWCs increases, the internal pore structure shows a process of gradual evolution from micropores pores to mesopores pores, and then to macropores pores, meantime, some new micropore pores are also generated. That is, with the increase in the number of FTWCs, the internal pores of sandstone are gradually developed, the porosity is gradually increased, and the internal damage is gradually enhanced.

To deeply analyze the freeze-thaw damage evolution mechanism of sandstone subjected to FTWCs, a schematic diagram of the freeze-thaw damage evolution mechanism of sandstone was drawn in Figure 3. As shown in Figure 3(a), before the FTWCs treatment, the mineral particles of sandstone are relatively close to each other. As shown in Figure 3(b), the cross-section along the height direction was selected for analysis. Rock is a natural porous geological material that is composed of different kinds of micro-defects (Zhu et al., 2018). According to whether the micro-defects are connected, micro-defects can be divided into the closed type and branch type. When these micro-defects are filled with water, the water inside the rock freezes as the ambient temperature drops below 0 °C. As shown in Figure 3(c), for closed microdefects, due to the water-ice phase transition, the volume increase will generate frost pressure which will lead to the development of micro-defects (Hori and Morihiro, 1998; Park et al., 2015). As shown in Figure 3(d), for branch micro-defects, in addition to the expansion of micro-defects caused by water-ice phase transition, water migration will also take away some mineral particles due to the hydraulic gradient difference (Li et al., 2022; McGreevy and Whalley, 1985). As shown in Figure 3(b), when the ambient temperature increases, the pore ice inside the micro-defect melts into water, which will take away some mineral particles, meanwhile also creating a water-saturated condition for the next FTWCs. Such repeated circulation will lead to the micro-defects inside the sandstone to continue propagation, leading to the gradual connection between pores, and shows a process of gradual evolution from micropore pores to mesopores pores, and then to macropores pores, meantime, some new micropore pores are also generated, which is similar to the evolution mechanism shown by the T₂ spectrum distribution curve. Finally, as shown in Figure 3(e), the mineral particles of sandstone become loose, the porosity increases, and the damage is enhanced. The strength decreases and the deformation increases of the sandstone under loading.

Stress-strain curve

Figures 4 and 5 show the complete stress-strain curves of sandstone under different number of FTWCs and different confining pressure. All the stress-strain curves have similar characteristics. According to the progressive failure theory of rock (Martin and Chandler, 1994; Cai et al., 2004), the complete stress-strain curve can be classified into five stages: compaction stage, elastic deformation, yielding, failure and strain softening stage. However, with the increase of FTWCs and confining pressure, the complete stress-strain curves of sandstone show the following characteristics:

As shown in Figure 4, when the confining pressure is the same, the compaction stage of the whole stress-strain curve gradually became longer with the increase in the number of FTWCs, which is due to the micro-defects inside sandstone gradually developed caused by freeze-thaw weathering cycle treatment. In addition, the FTWCs treatment also affects the strength and post-peak deformation characteristics of sandstone, which is shown as follows: with the increase in the number of FTWCs, the peak strength and residual strength gradually decreased, the peak strain and residual strain gradually increased, and the post-peak brittleness gradually decreased.

As shown in Figure 5, under the same FTWCs, as the confining pressure increased, the compaction stage of the complete stress-strain curve gradually became shorter, which is due to some micro-defects inside the sandstone gradually closing under confining pressure. Meanwhile, the confining pressure has a significant influence on the strength and post-peak deformation characteristics of sandstone subjected to FTWCs treatment, which is shown as follows: as the confining pressure increased, the peak strength and residual strength of sandstone gradually increased, the peak strain and residual strain gradually increased, and the post-peak brittleness gradually decreased.

The above complete stress-strain curves characteristics show that, for the strength evolution laws of sandstone, the number of FTWCs and confining pressure show the opposite influence mechanism, that is, the increase of FTWCs leads to the sandstone strength decreased, while the increase of confining pressure leads to the sandstone strength increased. For the post-peak deformation behavior of sandstone, the number of FTWCs and confining pressure show the same influence mechanism. But for the pre-peak deformation behavior of sandstone, the number of FTWCs and confining pressure show the opposite influence mechanism, namely the increase of FTWCs leads to the compaction stage of complete stress-strain curve gradually longer, the nonlinear deformation characteristics is more and more obvious; while the increase of confining pressure leads to the compaction stage of stress-strain curve gradually shorter, nonlinear the deformation characteristics is less and less obvious. The measured mechanical parameters of sandstone are listed in Table 1.

Table 1.

Statistical table of mechanical parameters of sandstone under different number of FTWCs and different confining pressure.

Number of FTWCs (time)	Confining pressure (MPa)	Peak strength (MPa)	Peak strain (/)	Residual strength (MPa)	Residual strain (/)	Young’s modulus (GPa)	Poisson’s ratio (/)
0	3	47.99	0.00519	12.38	0.00648	13.17	0.246
	6	64.75	0.00646	23.90	0.00820	14.54	0.242
	9	75.46	0.00722	32.03	0.01035	15.06	0.239
	12	85.24	0.00763	35.79	0.01163	16.35	0.236
10	3	45.58	0.00591	13.24	0.00731	9.56	0.253
	6	57.22	0.00779	22.29	0.00934	10.54	0.248
	9	67.40	0.00825	29.90	0.01146	11.70	0.244
	12	79.55	0.00832	36.55	0.01251	12.79	0.241
20	3	42.89	0.00692	12.80	0.01078	8.31	0.260
	6	54.48	0.00836	20.33	0.01206	8.86	0.255
	9	64.06	0.00958	28.91	0.01450	9.54	0.249
	12	72.66	0.00988	39.95	0.01521	10.43	0.245
30	3	38.97	0.00862	11.27	0.01090	6.37	0.266
	6	50.81	0.00943	18.82	0.01358	7.17	0.262
	9	60.16	0.01093	28.74	0.01601	7.68	0.254
	12	65.37	0.01115	32.96	0.01691	8.57	0.249
40	3	31.78	0.00940	10.49	0.01185	4.96	0.274
	6	42.75	0.01123	20.65	0.01605	5.27	0.270
	9	49.90	0.01215	26.69	0.01768	6.75	0.263
	12	61.77	0.01301	34.18	0.01696	7.17	0.257

Deterioration characteristics of mechanical parameters

Figure 6 shows the evolution laws of the peak strength of sandstone specimens versus the number of FTWCs and confining pressure. The results show that the number of FTWCs and confining pressure have a significant influence on the peak strength of sandstone. According to Figure 6(a), under the same confining pressure, the greater the number of FTWCs is, the lower the peak strength. It can be explained as follows: the micro-defects inside the sandstone are gradually developed caused by FTWCs treatment, and these micro-defects are easier to propagate under axial loading, resulting in decreases in bearing capacity. On the other hand, under the same number of FTWCs, the peak strength of sandstone increased with confining pressure, as shown in Figure 6(b). This is because the confining pressure can promote the closure of micro-defects caused by FTWCs. The greater confining pressure is, these micro-defects are harder to propagate under axial loading, resulting in increases in bearing capacity.

Evolution laws of the peak strain of sandstone specimens versus the number of FTWCs and confining pressure are shown in Figure 7. In general, as the increase in the number of FTWCs and confining pressure, the peak strain increased gradually. The influence mechanism of the number of FTWCs and confining pressure on the peak strain of sandstone is the same.

Figure 8 shows the evolution laws of the residual strength of sandstone specimens versus the number of FTWCs and confining pressure. Similar to the evolution laws of peak strength, under the same confining pressure conditions, except for individual abnormal data, as the increase in the number of FTWCs, the residual strength gradually decreased. Under the same number of FTWCs, the residual strength increased with confining pressure.

Evolution laws of the residual strain of sandstone specimens versus the number of FTWCs and confining pressure are shown in Figure 9. Similar to the evolution laws of peak strain, under the same confining pressure conditions, except for individual abnormal data, as the increase in the number of FTWCs, the residual strain gradually increased. Under the same number of FTWCs, the residual strength increased with confining pressure.

Figure 10 shows the evolution laws of the Young’s modulus of sandstone specimens versus the number of FTWCs and confining pressure. Figure 10(a) shows that under the same confining pressure conditions, as the increase in the number of FTWCs, Young's modulus gradually decreased. This is because the micro-defects inside sandstone gradually developed under the action of FTWCs, resulting in the damage enhanced, and the decreases of deformation resistance of sandstone. On the other hand, under the same number of FTWCs, Young's modulus of sandstone increased with confining pressure, as shown in Figure 10b. This is because the confining pressure can promote the closure of micro-defects caused by FTWCs, enlarge the contact area between mineral grains, inhibit the shear slip of micro-defects, improve the frictional strength of closed micro-defects, and enhance the deformation resistance of sandstone.

Damage constitutive model for frozen-thawed sandstone under triaxial conditions

The results in Section ‘Stress-strain curve’ show that there is an obvious compaction stage in the complete stress-strain curves under triaxial loading for sandstone subjected to different number of FTWCs, and the post-peak strength of sandstone does not directly decrease to 0, maintaining the residual strength. Existing constitutive models either only consider the effect of FTWCs on the compaction stage, or only consider the effect of FTWCs on post-peak deformation behavior. Therefore, it is significant to establish a new constitutive model that considers both the compaction stage and the post-peak deformation behavior. Based on the different influence mechanisms of FTWCs and confining pressure on rock deformation behavior, the complete stress-strain curve was divided into two stages when establishing the constitutive model: the compaction stage (from the origin of coordinates to the full compaction point) and the post-compaction stage (after the full compaction point).

Constitutive model of compaction stage

There are more or less micro-defects inside the rock. After the FTWCs treatment, the micro-defects will further develop. The elastic modulus is small at the initial loading, as the loading progresses, the micro-defects inside the rock gradually close, and the elastic modulus increases continuously large until it is equal to the tangent elastic modulus of the rock. Based on the evolution law of elastic modulus in the compaction stage, assuming that there is no damage occurs, the compaction coefficient is introduced to characterize the deformation behaviors in the compaction stage. The elastic modulus in the compaction stage can be expressed as:

E_{C} = E_{i} + (E_{T} - E_{i}) \times C

(1)

where

E_{C}

is the deformation modulus in the compaction stage,

E_{i}

is the initial deformation modulus,

E_{T}

is the tangent elastic modulus, C is the compaction coefficient.

Due to the randomness and heterogeneity of micro-defects inside the rock, different types and sizes of micro-defects exhibit different behaviors during the compaction process. Therefore, the statistical model is the best model to describe the compaction behavior of micro-defects inside the rock. As a simple method, it is assumed that the external load (expressed as the specimen strain $ε$ ) is regarded as a random variable during the compaction process of micro-defects inside the rock. When the load reaches a certain strain level $ε$ , the compaction coefficient C is defined as follows:

C = \frac{N_{ε}}{N} = \frac{\int_{0}^{ε} N p (x) d x}{N} = \int_{0}^{ε} p (x) d x

(2)

where

N_{ε}

is the number of fully compacted micro-defects within a rock specimen at compressive strain

ε

;

N

is the total number of micro-defects within a rock specimen;

p (x)

is the probability density function of the number of closed micro-defects within a rock specimen. Referring to the statistical damage mechanics, the strength of micro-elements obeying Weibull distribution was commonly used to study the damage evolution characteristics of the rock failure process (Cao et al., 2010; Bian et al., 2019; Liu and Dai, 2018). In this paper, the specimen strain

ε

that obeys Weibull distribution was also used to study the compaction process of rock. Therefore, the probability density function of the specimen strain

ε

can be expressed as:

p (ε) = \frac{m}{f_{0}} {(\frac{ε}{f_{0}})}^{m - 1} \exp [- {(\frac{ε}{f_{0}})}^{m}]

(3)

where

m

and

f_{0}

are the Weibull distribution parameters,

ε

is the strain of the rock specimen. Substituting equation (3) into equation (2), the compaction coefficient C is:

C = 1 - \exp [- {(\frac{ε}{f_{0}})}^{m}]

(4)

Combining equations (1) and (4) with the stress-strain relationship ( $σ_{1} = E_{C} ε_{1}$ ), the constitutive equation for the compaction stage can be expressed as:

σ_{1} = E_{i} ε_{1} + (E_{T} - E_{i}) {1 - \exp [- {(\frac{ε_{1}}{f_{0}})}^{m}]} ε_{1}

(5)

To ensure the continuity of the complete stress-strain curve, equation (5) should satisfy the following conditions at the full compaction point:

{\begin{cases} σ = σ_{1 c} \begin{matrix}  \end{matrix} ε = ε_{1 c} \\ \frac{d σ}{d ε_{1}} | (σ = σ_{1 c} \begin{matrix}  \end{matrix} ε = ε_{1 c}) = E_{T} \end{cases}

(6)

where

σ_{1 c}

is the full compaction stress,

ε_{1 c}

is full compaction strain.

Substituting equation (6) into equation (5) leads to:

{\begin{cases} σ_{1 c} = E_{i} ε_{1 c} + (E_{T} - E_{i}) {1 - \exp [- {(\frac{ε_{1 c}}{f_{0}})}^{m}]} ε_{1 c} \\ E_{T} + (E_{T} - E_{i}) \exp [- {(\frac{ε_{1 c}}{f_{0}})}^{m}] [m {(\frac{ε_{1 c}}{f_{0}})}^{m} - 1] = E_{T} \end{cases}

(7)

By the solving system of equation (7), the Weibull distribution parameters $m$ and $f_{0}$ can be written as follows:

f_{0} = ε_{1c} m^{\frac{1}{m}}

(8)

m = 1 / \ln [\frac{(E_{T} - E_{i}) ε_{1c}}{E_{T} ε_{1c} - σ_{1c}}]

(9)

Substituting equation (6) into equation (5), the constitutive model of the compaction stage can be written as follows:

σ_{1} = E_{i} ε_{1} + (E_{T} - E_{i}) {1 - \exp [- \frac{1}{m} {(\frac{ε}{ε_{1 c}})}^{m}]} ε_{1}

(10)

Damage constitutive model of the post-compaction stage

For the post-compaction stage, with the continuous loading, the rock will undergo elastic deformation, yielding, failure and strain softening stage. During this process, the micro-cracks inside the rock gradually develop, penetrate, and form a macroscopic fracture plane, resulting in decreases in bearing capacity, but does not directly decrease to 0, maintaining the residual strength characteristics. According to the damage theory (Lemaitre, 1985) and deformation compatibility principle (Cao et al., 2006), the statistical damage constitutive model of rock in the post-compaction stage can be written as follows:

σ_{1} = E_{T} ε_{1} (1 - D) + D σ_{r}

(11)

where D is the damage variable,

σ_{r}

is the residual strength.

According to the statistical damage mechanics, the damage variable D is defined as the ratio of the number of damaged micro-units $L_{F}$ to the total number of micro-units L within a rock speciemen. The expression of the damage variable D is as follows:

D = \frac{L_{F}}{L} = \frac{\int_{0}^{F} L P (y) d y}{L} = \int_{0}^{F} P (y) d y

(12)

where

p (y)

is the probability density function of the number of damaged micro-units within a rock speciemen. It is assumed that the rock is macroscopically homogeneous isotropic, and the strength of micro-units F inside the rock obeys Weibull distribution. Therefore, the probability density function of the strength of micro-units F can be expressed as follows:

P (F) = \frac{M}{F_{0}} {(\frac{F}{F_{0}})}^{M - 1} \exp [- {(\frac{F}{F_{0}})}^{M}]

(13)

where M and

F_{0}

are the Weibull distribution parameters, F is the strength of micro-units in the post-compaction stage. Substituting equation (13) into equation (12), the damage variable D is:

D = 1 - \exp [- {(\frac{F}{F_{0}})}^{M}]

(14)

The micro-units strength of rock is often expressed by the rock failure criterion, but the currently published papers usually choose a certain criterion directly, and there is no detailed report on the reason why this criterion is chosen and the method of choosing the criterion. Therefore, this paper proposed to use the rock failure criterion to fit test data, and the optimal selection by comparing the fitting correlation of various rock failure criteria. Mohr-coulomb criterion (Zhao, 2000) and Hoek-Brown criterion (Hoek and Brown, 1997) are widely used in rock engineering, in this paper, those two criteria were selected to carry out strength fitting analysis on sandstone subjected to different number of FTWCs, and the results are shown in Figure 11. As can be seen from Figure 11, the correlation coefficients fitted by the Hoek-Brown criterion are higher than those of the Mohr-Coulomb criterion under any number of FTWCs, indicating that the Hoek-Brown criterion is more suitable to describe the strength characteristics of sandstone subjected to FTWCs treatment. Therefore, the Hoek-Brown criterion was selected to characterize the micro-units strength of rock F.

According to the effective stress invariant, the Hoek-Brown criterion can be expressed as follows:

\begin{array}{l} F (σ^{*}) = \frac{1}{3} m_{i} σ_{c i} I_{1}^{*} + 4 J_{2}^{*} \cos^{2} θ + m_{i} σ_{c i} \sqrt{J_{2}^{*}} (\cos θ + \frac{\sqrt{3}}{3} \sin θ) \\ I_{1}^{*} = σ_{1}^{*} + σ_{2}^{*} + σ_{3}^{*} \\ J_{2}^{*} = \frac{1}{6} [{(σ_{1}^{*} - σ_{2}^{*})}^{2} + {(σ_{2}^{*} - σ_{3}^{*})}^{2} + {(σ_{1}^{*} - σ_{3}^{*})}^{2}] \end{array}}

(15)

where

I_{1}^{*}

is the first effective stress invariant;

J_{2}^{*}

is the second effective-biased stress invariant;

θ

is the Lode angle. In the conventional triaxial test,

σ_{1} > σ_{2} = σ_{3}

θ

=30°. Equation (15) can be simplified as follows:

{\begin{cases} F (σ^{*}) = \frac{1}{3} m_{i} σ_{c i} I_{1}^{*} + 3 J_{2}^{*} + \frac{2 m_{i} σ_{c i} \sqrt{J_{2}^{*}}}{\sqrt{3}} \\ I_{1}^{*} = \frac{E_{T} ε_{1} (σ_{1} + 2 σ_{3})}{σ_{1} - 2 μ σ_{3}} \begin{matrix}  \end{matrix} J_{2}^{*} = \frac{{[E_{T} ε_{1} (σ_{1} - σ_{3})]}^{2}}{3 {[σ_{1} - 2 μ σ_{3}]}^{2}} \end{cases}

(16)

By simplifying equation (16), the strength of micro-elements F can be expressed as follows:

F (σ^{*}) = \frac{E_{T} ε_{1} σ_{1} m_{i} σ_{c i}}{σ_{1} - 2 μ σ_{3}} + {[\frac{E_{T} ε_{1} (σ_{1} - σ_{3})}{σ_{1} - 2 μ σ_{3}}]}^{2}

(17)

Substituting equation (17) into equation (14), the expression of the damage variable D is as follows:

D = 1 - \exp [- {(\frac{\frac{E_{T} ε_{1} σ_{1} m_{i} σ_{c i}}{(σ_{1} - 2 μ σ_{3})} + \frac{E_{T}^{2} ε_{1}^{2} {(σ_{1} - σ_{3})}^{2}}{{(σ_{1} - 2 μ σ_{3})}^{2}}}{F_{0}})}^{M}]

(18)

Substituting equation (18) into equation (11), the constitutive model for the post-compaction stage can be written as follows：

σ_{1} = (E_{T} ε_{1} - σ_{r}) \exp [- {(\frac{\frac{E_{T} ε_{1} σ_{1} m_{i} σ_{c i}}{(σ_{1} - 2 μ σ_{3})} + \frac{E_{T}^{2} ε_{1}^{2} {(σ_{1} - σ_{3})}^{2}}{{(σ_{1} - 2 μ σ_{3})}^{2}}}{F_{0}})}^{M}] + σ_{r}

(19)

Since the constitutive model is established piecewise, equation (19) should satisfy the continuous condition at the full compaction point, so equation (19) is transformed into as follows:

σ_{1} = [E_{T} (ε_{1} - ε_{1 c}) - σ_{r}] \exp [- {(\frac{\frac{E_{T} (ε_{1} - ε_{1 c}) (σ_{1} - σ_{1 c}) m_{i} σ_{c i}}{(σ_{1} - σ_{1 c} - 2 μ σ_{3})} + \frac{E_{T}^{2} {(ε_{1} - ε_{1 c})}^{2} {(σ_{1} - σ_{1 c} - σ_{3})}^{2}}{{(σ_{1} - σ_{1 c} - 2 μ σ_{3})}^{2}}}{F_{0}})}^{M}] + σ_{r} + σ_{1 c}

(20)

Substituting the full compaction strain into equation (20): $σ_{1} = [E_{T} (ε_{1 c} - ε_{1 c}) - σ_{r}] \times 1+ σ_{r} + σ_{1 c} = σ_{1 c}$ , satisfying continuity; By substituting the residual strain into equation (20): $σ_{1} = [E_{T} (ε_{1 r} - ε_{1 c}) - σ_{r}] \times 0+ σ_{r} + σ_{1 c} = σ_{r} + σ_{1 c}$ , does not satisfy the residual strength characteristics. That is, equation (20) satisfies the continuity of the full compaction point, but the value of residual strength obtained by equation (20) is equal to the full compaction stress plus the actual residual strength, which is inconsistent with the actual situation.

Therefore, the method of coordinate transformation was proposed in this paper to overcome the above problems. The method of coordinate transformation is shown in Figure 12. The elastic deformation segment of the stress-strain curve was reversely extended to intersect the X-axis at the point $O'$ , and a new coordinate system $X^{'} O Y^{'}$ was established. In the new coordinate system $X^{'} O Y^{'}$ , the compaction stage of the stress-strain curve is not considered, therefore, equation (20) can eliminate the use of $ε_{1} - ε_{1 c}$ and $σ_{1} - σ_{1 c}$ instead of $ε_{1}$ and $σ_{1}$ , the expression of the constitutive model is transformed into equation (19).

In the new coordinate system $X^{'} O Y^{'}$ , By substituting the full compaction strain into equation (19): $σ_{1} = [E_{T} ε_{1 c} - σ_{r}] \times 1+ σ_{r} = σ_{1 c}$ , satisfying continuity; By substituting the residual strain into equation (19): $σ_{1} = [E_{T} ε_{1 r} - σ_{r}] \times 0+ σ_{r} = σ_{r}$ , satisfying the residual strength characteristics. Therefore, the stress-strain curve obtained by coordinate transformation can well describe the characteristics of the complete stress-strain curve for sandstone subjected to FTWCs treatment.

The parameters M and F₀ in equation (19) can be obtained through the peak point of the complete stress-strain curves, and the following conditions should be satisfied at the peak point:

{\begin{cases} ε_{1} = ε_{1 p} \begin{matrix}  \end{matrix} σ_{1} = σ_{1 p} \\ \frac{d σ_{1}}{d ε_{1}} | (σ = σ_{1 p} \begin{matrix}  \end{matrix} ε = ε_{1 p}) = 0 \end{cases}

(21)

where

σ_{1 p}

is the peak stress,

ε_{1 p}

is peak strain.

Substituting equation (21) into equation (19) leads to:

{\begin{cases} \ln (\frac{E_{T} ε_{1 p} - σ_{r}}{σ_{1 p} - σ_{r}}) = {(\frac{\frac{E_{T} ε_{1 p} σ_{1 p} m_{i} σ_{c i}}{(σ_{1 p} - 2 μ σ_{3})} + \frac{E_{T}^{2} ε_{1 p}^{2} {(σ_{1 p} - σ_{3})}^{2}}{{(σ_{1 p} - 2 μ σ_{3})}^{2}}}{F_{0}})}^{M} \\ E_{T} - \frac{(E_{T} ε_{1 p} - σ_{r})}{ε_{1 p}} \times M {[\frac{\frac{E_{T} ε_{1 p} σ_{1 p} m_{i} σ_{c i}}{(σ_{1 p} - 2 μ σ_{3})} + \frac{E_{T}^{2} ε_{1 p}^{2} {(σ_{1 p} - σ_{3})}^{2}}{{(σ_{1 p} - 2 μ σ_{3})}^{2}}}{F_{0}}]}^{M} = 0 \end{cases}

(22)

By the solving system of equation (10), the Weibull distribution parameters M and F₀ can be written as follows:

M = \frac{\frac{E_{T} ε_{1 p}}{E_{T} ε_{1 p} - σ_{r}}}{\ln (\frac{E_{T} ε_{1 p} - σ_{r}}{σ_{1 p} - σ_{r}})}

(23)

F_{0} = {[\frac{M \times (E_{T} ε_{1 p} - σ_{r})}{E_{T} ε_{1 p}}]}^{\frac{1}{M}} \times [\frac{E_{T} ε_{1 p} σ_{1 p} m_{i} σ_{c i}}{(σ_{1 p} - 2 μ σ_{3})} + \frac{E_{T}^{2} ε_{1 p}^{2} {(σ_{1 p} - σ_{3})}^{2}}{{(σ_{1 p} - 2 μ σ_{3})}^{2}}]

(24)

By combining equation (10) and equation (19), considering the influence of the number of FTWCs on rock mechanical parameters, the constitutive model under triaxial stress loading for sandstone subjected to FTWCs treatment can be written as follows:

{\begin{array}{l} σ_{1} = E_{i} (n) ε_{1} + [E_{T} (n) - E_{i} (n)] {1 - \exp [- \frac{1}{m (n)} {(\frac{ε}{ε_{1c} (n)})}^{m (n)}]} ε_{1} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} & ε_{1} \leq ε_{1c} (n) \end{matrix} \end{matrix} \end{matrix} \\ σ_{1} = [E_{T} (n) ε_{1} - σ_{r} (n)] \exp [- {(\frac{\frac{E_{T} (n) ε_{1} σ_{1} m_{i} (n) σ_{c i} (n)}{[σ_{1} - 2 μ (n) σ_{3}]} + \frac{{[E_{T} (n)]}^{2} ε_{1}^{2} {(σ_{1} - σ_{3})}^{2}}{{[σ_{1} - 2 μ (n) σ_{3}]}^{2}}}{F_{0} (n)})}^{M}] + σ_{r} (n) \begin{matrix} ε_{1} > ε_{1c} (n) \end{matrix} \end{array}

(25)

Model validation and discussions

Validation of the constitutive model

By substituting the relevant parameters in Table 1 into equations (8), (9), (23) and (24), the model parameters under different working conditions can be obtained. Finally, the model parameters and relevant mechanical parameters were substituted into equation (25), and the theoretical stress-strain curves of sandstone under different number of FTWCs and different confining pressures were plotted in Figure 13. Figure 13 shows that the theoretical curves of the proposed model agree with the experimental data, especially the nonlinear deformation characteristics in the compaction stage, peak and residual strength characteristics. As shown in Figure 14, the theoretical curves of the existing model (Zhang et al., 2020; Shi et al., 2021) can only reflect the peak and residual strength characteristics, but cannot reflect the nonlinear deformation characteristics in the compaction stage. As shown in Figure 15, the theoretical curves of the existing model (Gao et al., 2021) can only reflect the nonlinear deformation characteristics in the compaction stage and peak strength characteristics, but cannot reflect the residual strength characteristics. Therefore, the proposed model overcomes the shortcomings that the existing models. The detailed drawing process of Figs. 13-15 see Appendix I.

Evaluation of constitutive model performance

Figure 13 only reflects the superiority of the proposed model in this paper from an intuitive level. To evaluate the advantages of the proposed model more objectively, the error analysis index of the model data and experimental data were introduced for comprehensive evaluation. In this paper, the commonly used error analysis index correlation coefficient (R), root mean square error (RMSE), variance accounted for (VAF) and mean absolute error (MAE) were selected for evaluation, and the calculation formula of four indexes are as follows:

R = \sum_{i = 1}^{N} (y_{e i} - {\bar{y}}_{e i}) (y_{m i} - {\bar{y}}_{m i}) / \sqrt{\sum_{i = 1}^{N} {(y_{e i} - {\bar{y}}_{e i})}^{2} \times \sum_{i = 1}^{N} {(y_{m i} - {\bar{y}}_{m i})}^{2}}

(26)

RMSE = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(y_{e i} - y_{m i})}^{2}}

(27)

VAF = [1 - \frac{Var (y_{e i} - y_{m i})}{Var (y_{e i})}] \times 100

(28)

MAE = \frac{1}{N} \sum_{i = 1}^{N} | y_{e i} - y_{m i} |

(29)

where

y_{e i}

represents the experimental value,

y_{m i}

is the model value,

{\bar{y}}_{e i}

represents the average of the experimental value,

{\bar{y}}_{m i}

represents the average of the model value, and N denotes the number of values.

By substituting the experimental data and model data into equations (26), (27), (28) and (29) to calculate the corresponding error analysis index values, as shown in Table.2.

Table 2.

Statistical table of error analysis index.

Number of FTWCs (time)	Confining pressure (MPa)	Error analysis index
		Proposed model				Published paper model (Shi et al., 2021; Zhang et al., 2020)				Published paper model (Gao et al., 2021)
		R	RMSE	VAF	MAE	R	RMSE	VAF	MAE	R	RMSE	VAF	MAE
0	3	0.916	6.307	83.343	2.044	0.843	12.140	64.133	9.224	0.794	10.861	55.173	5.929
	6	0.957	5.804	91.425	2.132	0.905	13.832	75.900	11.330	0.883	10.670	76.322	5.886
	9	0.966	5.712	93.171	2.363	0.929	12.567	83.275	9.916	0.899	11.450	79.901	6.656
	12	0.962	6.956	92.604	3.230	0.933	13.049	85.562	10.209	0.898	13.274	79.868	8.378
10	3	0.965	3.336	92.797	0.994	0.868	8.624	74.493	5.513	0.770	10.247	47.028	6.916
	6	0.896	6.425	80.247	2.723	0.850	9.047	72.035	5.605	0.751	14.438	52.074	11.564
	9	0.959	5.785	91.803	3.035	0.931	8.884	86.689	6.328	0.875	12.617	75.953	9.410
	12	0.974	4.589	94.664	1.852	0.959	6.546	91.683	3.939	0.871	13.587	73.583	9.563
20	3	0.976	2.586	95.109	0.886	0.899	7.253	80.521	4.838	0.803	9.609	54.919	6.630
	6	0.969	3.689	93.855	1.417	0.931	6.982	86.460	4.543	0.851	10.432	68.482	7.220
	9	0.959	5.229	91.877	2.412	0.934	7.930	87.092	5.458	0.867	12.035	74.414	8.822
	12	0.989	3.093	97.736	1.664	0.980	4.930	95.961	3.524	0.911	11.806	83.006	8.882
30	3	0.937	3.940	87.738	1.052	0.836	8.468	69.233	5.657	0.744	10.257	42.382	7.352
	6	0.974	3.058	94.131	1.138	0.929	6.421	85.516	4.298	0.838	9.644	62.798	6.948
	9	0.948	5.407	89.615	2.255	0.900	8.993	80.874	6.330	0.872	11.436	75.881	8.715
	12	0.970	4.223	94.075	1.922	0.936	7.470	87.198	5.200	0.885	11.645	78.010	8.964
40	3	0.886	4.078	78.514	1.264	0.781	7.366	60.619	4.908	0.687	9.495	38.862	7.442
	6	0.987	1.677	97.396	0.784	0.934	4.942	87.117	3.419	0.833	9.042	68.466	6.892
	9	0.949	4.064	89.842	1.920	0.906	6.139	81.776	4.151	0.836	11.673	69.585	9.915
	12	0.932	6.007	86.332	2.827	0.904	7.644	81.691	4.956	0.879	12.584	77.136	10.536

According to the judicial principle of error analysis index value on model performance, R and VAF are regarded as the positive index, the larger the index value, the better the model performance; RMSE and MAE are regarded as the inverse index, the smaller the index value, the better the model performance. By taking the reciprocal of the inverse index, the same type of index under the same number of FTWCs was normalized, and the stacked histogram of error analysis indexes as shown in Figure 16 was drawn. It can be seen from Figure 16 that for any number of FTWCs, the normalized value of the error analysis index of the proposed model is greater than that of the published paper model (Gao et al., 2021; Zhang et al., 2020; Shi et al., 2021), indicating that the performance of the proposed model in this paper has better performance.

Discussions

Compared with the published models, the proposed model can reflect both the nonlinear deformation characteristics in the compaction stage, peak and residual strength characteristics of the complete stress-strain curve for rock under different FTWCs and different confining pressures, but, in the post-peak stress drop stage, the proposed model has some difference with the experimental data. Since the characteristics of the curves are similar, some curves were selected to draw as shown in Figure 17. Figure 17 shows that in the pre-peak stage and residual stage, the absolute error between the theoretical value and the experimental value is very small, fluctuates in a small range up and down near the 0 line. However, in the post-peak strain softening stage, the absolute error between the theoretical value and the experimental value is large, which is a unimodal distribution, and the maximum absolute error reaches 30 MPa. The main reason is: the evolution characteristic of the Weibull distribution curve is a gradually decreasing process. Affected by the number of FTWCs and confining pressure, the stress-strain curve obtained from the test has a slow decline process after the peak point, but then quickly drops to the residual strength, so the proposed model is hard to better reflect the process. At present, the commonly used method is to modify the damage variable by introducing a coefficient (Zhu et al., 2019; Hou et al., 2022; Lin et al., 2019), but the effect is not good. It may be that the analysis of the post-peak deformation and failure mechanism is not deep enough, and it is difficult to establish a better damage variable to reflect the failure process. Therefore, we can start from the post-peak deformation and failure mechanism of rock subjected to freeze-thawing weathering cycles treatment, finding a new damage variable, and establishing a constitutive model that can reflect the whole process of the complete stress-strain curve.

Conclusion

In this work, the Freeze-thaw damage evolution mechanism of sandstone was investigated based on NMR tests of sandstones before and after different FTWCs treatment. CTC tests of sandstone subjected to different number of FTWCs treatment were conducted, and the evolution laws of stress-strain curves and mechanical parameters were investigated. A new piecewise constitutive model was proposed to describe the stress-strain relationships under triaxial stress loading for rock subjected to FTWCs treatment. The main conclusions can be summarized as follows:

As the increase in the number of FTWCs, the evolution mechanism of the freeze-thaw damage of sandstone was: the internal pore structure shows a process of gradual evolution from micropore pores to mesopore pores, and then to macropores pores, meantime, some new micropore pores are also generated.

For the strength and Young’s modulus evolution laws of sandstone, the number of FTWCs and confining pressure show the opposite influence mechanism. When the confining pressure is a constant, with the increase in the number of FTWCs, the peak and residual strength and Young’s modulus decrease. When the FTWCs is a constant, as the confining pressure increase, the peak and residual strength and Young’s modulus increase.

For the post-peak deformation behavior of sandstone, the number of FTWCs and confining pressure show the same influence mechanism. But for the pre-peak deformation behavior of sandstone, the number of FTWCs and confining pressure show the opposite influence mechanism, When the confining pressure is a constant, with the increase in the number of FTWCs, the compaction stage of the complete stress-strain curve gradually longer. When the FTWCs is a constant, the compaction stage of the stress-strain curve is gradually shortened as the confining pressure increases.

Compared with the reference model, the proposed model can reflect both the nonlinear deformation characteristics in the compaction stage, peak and residual strength characteristics of the complete stress-strain curve for rock under different FTWCs and different confining pressures. Meantime, for any number of FTWCs, the normalized value of the error analysis index of the proposed model is greater than that of the reference model, indicating that the performance of the proposed model in this paper has better performance.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No. 51774323), Hunan Provincial Natural Science Foundation of China (Grant No. 2020JJ4704 and 2020JJ4712) and Fundamental Research Funds for the Central Universities of Central South University (Grant No. 2021zzts0279).

ORCID iDs

Xin Xiong

Chun Xu

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A new damage constitutive model for frozen-thawed sandstone under triaxial conditions: Considering the characteristics of pre-peak compaction and post-peak residual strength

Abstract

Keywords

Introduction

Experimental materials and methods

Rock specimen preparation

Experimental procedures

Experimental results and analysis

T2 spectrum distribution curves and freeze-thaw damage mechanism

Stress-strain curve

Deterioration characteristics of mechanical parameters

Damage constitutive model for frozen-thawed sandstone under triaxial conditions

Constitutive model of compaction stage

Damage constitutive model of the post-compaction stage

Model validation and discussions

Validation of the constitutive model

Evaluation of constitutive model performance

Discussions

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

T₂ spectrum distribution curves and freeze-thaw damage mechanism