Experimental study on the mechanical properties of red sandstone with fractures under different loading rates

Abstract

In order to study the effects of crack inclination angle and loading rate on rock mechanical properties, creep characteristics, and failure characteristics. Taking homogeneous red sandstone with different fracture angles as the research object, uniaxial compression tests and uniaxial compression creep tests were conducted at different loading rates. The results showed that under the same fracture angle, the loading rate was positively correlated with the peak strength, elastic modulus, instantaneous strain, creep strain, and steady-state creep rate of the sample, while negatively correlated with the peak strain. At the same loading rate, the mechanical properties and creep properties of the sample were controlled by the crack inclination angle α. With the increase of α, the peak strength, peak strain, instantaneous strain, creep strain and steady-state creep rate decreased first and then increased, and the elastic modulus increased. On the basis of rock creep testing, it is also important to establish a creep model that conforms to the actual test situation for studying rock creep characteristics. However, many models currently used cannot accurately describe the three stages of rock creep, especially the accelerated creep stage. Therefore, based on Burgers elements, this paper introduces plastic damage bodies based on damage rates and software components based on fractional calculus, A new creep model was obtained and its rationality was verified through experimental results. The results showed that the fit between the model and experimental data was above 0.97, indicating that the model can better describe the three stages of rock creep, especially reflecting the non-linear characteristics of the accelerated creep stage.

Keywords

Fissure-bearing red sandstone loading rate mechanical properties creep properties damage characteristics non-linear creep model

Introduction

Mechanical and creep properties of rocks are important for rock engineering design and stability analysis (Huang et al., 2012; Jenabidehkordi, 2019; Zhang et al., 2019). However, as a non-homogeneous material, the discontinuities of rocks exist in the rock mass in the form of fine-scale cracks, and the cracks (Fei et al., 2023; Wen et al., 2021; Zhao et al., 2016) as well as the uncertainty-induced variations in the rock load (Huang and Huang, 2010; Fei et al., 2022; Zhang and Zhao, 2014) seriously affect the mechanical properties of rocks, so the study of the mechanical and creep properties of rocks at different loading rates as well as inclination angles is of practical significance for safety and long-term stability in engineering. Some scholars have summarized the effects of different loading rates and fracture inclination angles on the mechanical properties of different rocks by uniaxial compression tests (Kang et al., 2023; Li et al., 2023; Qin et al., 2020). Yang et al. (2022) set different crack inclinations on sandstone and performed creep tests to reveal the relationship between its creep properties and intact rock. Wei et al. (2020) investigated the distribution characteristics and evolution law of internal fractures in coal rock by triaxial compression creep test combined with CT scanning technology and numerical software. Chen et al. (2021) conducted creep tests on prefabricated fissured marble under different water pressures, analyzed the effect of water pressure on the creep properties of fissured rocks, and summarized the damage mechanism of the rocks. Shan et al. (2020) conducted a series of conventional triaxial compression and triaxial unloading creep tests on fissured red sandstone, analyzed the transient strength and deformation characteristics of the rock specimens, and explored the creep deformation and damage evolution laws of the rock specimens.

In order to further reveal the mechanical properties of rocks, scholars have described the mechanical behavior of rocks by constructing different mechanical models, Zhang et al. (2021) provided an overview of creep properties and ontological modeling using saltpeter as an entry point, and made recommendations for direction. Wang et al. (2020) established the rock creep damage evolution equation from the salt rock creep process and proposed a new rock creep damage constitutive model by combining with the damage theory. Ping et al. (2016) definition of a new nonlinear damage creep intrinsic model based on nonlinear damage creep characteristics and damage variables of rocks. Li et al. (2020) proposed a new method to calculate the initial damage by using the Burgers model to cascade the elastic-plastic body with the damage variables. Zhang et al. (2015) introduced two sets of internal variables to describe the internal structural adjustments of the material and derived a viscoelastic-viscoplastic intrinsic model with internal variables, which can well simulate the creep properties of the material. Wang et al. (2021) investigated the uniaxial creep of salt rock under long cycle cyclic loading and obtained the effects of different maximum cyclic stresses and cycles on the creep properties of salt rock.

Through the previous description, it is found that scholars are limited to single-variable studies on the mechanical properties of rocks under different loading rates or different crack inclinations, while relatively few studies have been conducted under the coupled effect of the two. Therefore, in this paper, the red sandstone with good homogeneity was selected as the test rock sample, and cracks ranging from 0° to 90° were prefabricated in the middle of the rock sample. Uniaxial compression and uniaxial compression creep tests were carried out on rock samples with different loading rates. The effects of different fracture inclinations and loading rates on the mechanical and creep properties of red sandstone are analyzed and discussed, and a new nonlinear creep damage constitutive model is developed. The results show that the model can accurately describe the whole process of rock creep, including the accelerated creep stage, and can better reflect the three development trends of creep in fissured red sandstone, which verifies the rationality of the model.

Overview of the trial

Specimen preparation

Because the rock has a relatively obvious discrete type, the red sandstone with good homogeneity is selected as the test rock sample in this test study. The water absorption rate of the selected red sandstone is 4.5% and the permeability is 27.48 md. According to the method suggested by ISRM (ISRM, 2007), standard cylindrical red sandstone samples with a diameter of 50 mm and a height of 100 mm were processed and polished. In order to reduce errors and ensure the accuracy of the test, the specific accuracy of the processed rock is required to be determined in accordance with the Hydraulic and Hydropower Rock Test Regulations (SL264-2001) (MWRPRC, 2007). The sample is processed into a single crack through sample with 5 crack angles (0º, 30º, 45º, 60º, 90º) by the method of wire cutting. In this paper, the influence of fracture inclination on the strength, deformation and failure characteristics of fractured red sandstone is studied. The crack length = 20 mm, width = 1 mm, as shown in Figure 1.

Test equipment

This mechanical test of red sandstone with cracks under different loading rates adopts the RST-2000 high temperature and high pressure multi-field coupling mechanical test system produced by Jilin Province Ruist Test Instrument Manufacturing Co., LTD. The test system (Figure 2) can complete uniaxial compression test of different kinds of rocks, and the maximum test force of the axial system is 2000 kN. Loading speed 0.01–10 kN/s; Maximum confining pressure 70 MPa; The sample size is: the axial measurement range is 0∼8 mm, the radial measurement range is 0∼4 mm; The measuring range is 0∼100 mm and the control rate is 0.1∼60 mm/min.

Uniaxial compression tests on fractured red sandstone at different loading rates

Test programme

In order to study the mechanical properties and failure characteristics of fractured red sandstone at different loading rates, a total of 18 sets of conventional uniaxial compression tests have been designed. The loading method is load control. Uniaxial compression tests were carried out at different loading rates (20 N/s, 50 N/s, 100 N/s) for complete samples and samples with different prefabricated through crack angles (0°, 30°, 45°, 60°, 90°). The data were recorded through the rock mechanics test system, analyzed and processed, and the test data were image processed. The effects of fracture dip Angle and loading rate on the mechanical properties and failure characteristics of fractured red sandstone under uniaxial compression are studied.

Stress-strain relationship curve analysis

By analyzing and collating the test data obtained from uniaxial compression of red sandstone with cracks under different loading rates, stress-strain curves under different crack inclination angles and loading rates are obtained, as shown in Figures 3 and 4.

As shown in Figures 3 and 4, crack inclination Angle and loading rate have no great influence on the trend of stress-strain curve of the sample, and both of them mainly go through four stages, namely: (1) crack compaction; (2) elastic deformation; (3) crack generation and propagation; (4) Post-peak failure.

At the same time, in the stage of crack generation and expansion, the stress-strain curve of some samples showed obvious stress drop before reaching the stress peak, and then the stress would gradually increase to reach the peak value. This is because the prefabricated crack would affect the deterioration process of the rock. When the stress concentration occurs at the prefabricated crack tip, the crack expansion leads to the instantaneous local damage of the rock, resulting in the release of energy. The phenomenon of “stress drop” appears, but at the same time, this process causes the fracture pressure to close and the stress will be redistributed, resulting in the increase of bearing capacity and stress growth.

It can be seen from Figures 3 and 4 that the stress-strain curve of fractured red sandstone with different loading rates increases approximately linearly before the peak value, but after the peak value, the stress drops sharply, and the sample loses its bearing capacity in a short period of time. At this time, a large amount of energy is released and accompanied by a large fracturing sound, showing obvious brittle failure characteristics.

Characteristics of peak strength, elastic modulus, and peak strain of fractured red sandstone under different loading rates

The relationship between peak strength and crack inclination of red sandstone under different loading rates is shown in Figure 5. It can be seen from Figure 5 that when there are fractures, the uniaxial compressive strength of rocks with fractures decreases significantly compared with intact samples. The peak strength of red sandstone with fractures generally decreases first and then increases with the increase of fracture inclination, and the lowest value occurs at 0°or 30°; when the fracture inclination is 90°, the peak strength of samples is the closest to that of intact samples. It shows that when the crack Angle is 90°, it has the least influence on the peak strength of the sample.

The relationship between peak strength and loading rate of red sandstones with different fracture inclination angles is shown in Figure 6. As can be seen from Figure 6, most red sandstones with fractures show a similar situation under the influence of loading rate, that is, when the loading rate increases from low to high, the peak strength of the sample increases, resulting in a strengthening effect.

The relationship between elastic modulus and crack inclination of red sandstone under different loading rates is shown in Figure 7. As can be seen from Figure 7, the presence of cracks has a greater impact on the elastic modulus, and the elastic modulus of the sample decreases compared with that of the complete sample. In the range of 0°∼90°, the elastic modulus of the sample increases with the increase of the crack Angle, and the elastic modulus is the smallest at 0° and the largest at 90°.

The relationship between elastic modulus and loading rate of red sandstone with different fracture inclination angles is shown in Figure 8. It can be seen from Figure 8 that the elastic modulus of fractured red sandstone increases with the increase of loading rate, showing a positive correlation, and loading rate has a strengthening effect on the elastic modulus.

The relationship between peak strain and crack inclination of red sandstone under different loading rates is shown in Figure 9. It can be seen from Figure 9 that when there are prefabricated cracks, the peak strain of red sandstone with prefabricated cracks is generally smaller than that of intact samples. At the same time, it can be seen that the peak strain of the sample decreases first and then increases with the increase of the prefabricated crack angle, The minimum peak strain is obtained at 30°. The maximum peak strain is obtained at 90°, which is closest to the complete sample.

The relationship between peak strain and loading rate of red sandstone under different loading rates is shown in Figure 10. As can be seen from Figure 10, when the crack inclination is the same, the peak strain decreases with the increase of loading rate, showing a negative correlation.

As the loading rate increases, the peak strength and elastic modulus are positively correlated with the loading rate, while the peak strain is negatively correlated with the loading rate. The reason is that at low loading rate, the initial cracks in the rock can be fully developed, and the initial cracks can be well developed. At higher loading rate, the initial cracks in the rock cannot be fully and effectively developed, and the development and expansion of cracks are inhibited. Therefore, the compressive strength and elastic modulus of rock samples are effectively improved with the increase of loading rate, the deformation resistance is also improved, and the peak strain is reduced.

Analysis of damage characteristics of red sandstone containing fractures at different loading rates

According to the experimental observation, the prefabricated crack angle has obvious influence on the failure characteristics of the sample. In this section, the sample with a loading rate of 50 N/s is taken as an example (as shown in Figure 11) to analyze its failure characteristics. Under the loading rate of 50 N/s, the failure of fractured red sandstone with the change of fracture inclination is roughly as follows:(1) Macro-cracks appear on the surface of the complete sample through the upper and lower surfaces, and shear failure occurs; (2) When the prefabricated crack angle is 0°, tensile wing cracks in the middle of the prefabricated crack are generated up and down, tensile cracks in the middle and anti-tensile cracks in the left end of the crack penetrate the sample, while secondary shear cracks in the right end of the prefabricated crack extend to the side of the sample through the upper and lower surfaces, resulting in tensile shear mixed failure. (3) When the prefabricated crack angle is 30°, a tensile wing crack is generated at the left end of the prefabricated crack and spreads diagonally after extending to a certain length. At the same time, secondary shear cracks are generated at both ends of the prefabricated crack and expand to the side of the sample. Finally, shear failure occurs due to the rapid penetration of the shear crack through the upper and lower surfaces of the sample. (4) When the prefabricated crack angle is 45°, the initial tensile crack occurs at both ends of the prefabricated crack and expands diagonally after expanding to a certain extent. Finally, shear failure occurs because the shear crack rapidly passes through the upper and lower surfaces of the sample. (5) When the prefabricated crack angle is 60°, the crack starts at both ends of the prefabricated crack and produces a tensile wing crack along the loading direction. After expanding to a certain length, the crack expands diagonally and passes through the upper and lower surfaces, resulting in shear failure. (6) When the prefabricated crack angle is 90°, macroscopic cracks appear on the surface through the upper and lower surfaces and develop along the prefabricated cracks.

It can be seen from Figure 11 that different prefabricated crack angle angles have an impact on the failure characteristics and mode of the sample. Under the same loading rate, when the prefabricated crack angle is small, more cracks are generated in the rock sample before complete failure, and the number of cracks gradually decreases with the increase of the angle.

At the same time, the loading rate also has an obvious effect on the failure characteristics of rock samples. Taking the red sandstone sample with a crack inclination of 45° as an example, its failure characteristics under different loading rates are shown in Figure 12. It can be seen from Figure 12 that for red sandstone samples with the same prefabricated crack angle, the increase of loading rate can reduce the number of cracks on the surface of the sample, and only the main cracks that cause the failure of the sample are shown in the end. This is mainly because when the loading rate is low, the prefabricated cracks and the initial defects existing in the sample themselves can have sufficient time to develop fully, so more cracks will be generated. When the loading rate is high, the initial cracks inside the rock cannot be fully developed, and the development of cracks is inhibited, so the number of cracks on the specimen surface is relatively small.

Uniaxial creep tests on fractured red sandstone at different loading rates

Test programme

Hierarchical loading has the advantages of short cycle and fast speed. Therefore, this paper adopts the method of hierarchical loading to study the influence of prefabricated crack angle and loading rate on the creep characteristics of red sandstone. According to uniaxial compressive strength, the applied loads are divided into four stages (0.5, 0.6, 0.7, 0.8 for step by step loading; Is the compressive strength value of the sample, which is determined according to conventional uniaxial test) as shown in Table 1. In this paper, the sample with different prefabricated crack angles (0°, 30°, 60°) is loaded to various levels of stress through different loading rates (20 N/s, 50 N/s, 100 N/s) by load loading method. In the loading process of the first several stress levels of red sandstone before accelerated creep failure, the time of decay creep stage is about 1 h, while in the loading process of the last stress level of failure, the time required for decay creep stage is reduced, and the red sandstone enters the accelerated creep stage and suffers accelerated creep failure after 2 to 3 hours (Gong and Chen, 2014). Therefore, in this paper, the constant load time of each stress stage is set at 2 h. According to the experimental data, the creep curve is drawn to study and analyze the creep characteristics of fractured red sandstone under different loading rates.

Table 1.

Design scheme for stress level of uniaxial compression creep test under staged loading.

Loading rate	$α (°)$	0.5 Compressive strength/MPa	0.6 Compressive strength/MPa	0.7 Compressive strength/MPa	0.8 Compressive strength/MPa
20 N/s	0	6.13	7.36	8.58	9.81
	30	4.12	4.95	5.77	6.60
	60	8.22	9.86	11.51	13.15
50 N/s	0	5.21	6.25	7.29	8.33
	30	5.03	6.03	7.04	8.05
	60	8.23	9.88	11.52	13.17
100 N/s	0	6.26	7.51	8.76	10.02
	30	5.88	7.05	8.23	9.40
	60	9.28	11.14	12.99	14.85

Test results

Under the conditions of fractional loading, the specimens showed obvious creep characteristics. The creep curve of the specimen during the whole creep process under uniaxial fractional loading is shown in Figure 13. It can be seen from the Figure that (1) the creep curve of the whole process under the creep test is a step type. During the process of stress loading to the preset stress level, the sample will deform to a certain extent, resulting in varying degrees of strain increase. Most of the axial strain is generated in each stress loading stage, among which the axial strain generated by stress loading in the first stage is the most obvious. When the stress level is kept constant for 2 h, the sample will creep slowly with the increase of time, and the deformation is composed of instantaneous deformation and creep deformation caused by loading. (2) When the stress level is low, the strain-time curve of the sample does not change significantly. With the increase of the stress level, the slope of the creep curve gradually increases, and decay creep and stable creep mainly occur. When the loading stress level exceeds the yield strength of the rock, accelerated creep stage will be entered and accelerated creep failure will occur. As shown in Figure 13, under the loading rate of 20 N/s, the sample with crack inclination of 0° appears accelerated creep stage, that is, accelerated creep failure will occur after 1.31 h at the stress level of 9.81 MPa of the fourth stage. The other samples did not have accelerated creep stage, indicating that the stress set in the test was low and could not meet the requirements of accelerating creep in other samples. (3) Through comparison, it can be seen that loading rate and crack angle have an impact on the creep curve, which is manifested in the impact on instantaneous strain, creep strain, creep rate, etc.

The creep curve obtained by the stepwise loading test method is stepped, so the creep curve must be transformed before it can be used. In this paper, Chen’s loading method (Liu, 1944) is used to process the creep curve, and the obtained creep curve is shown in Figures 14 to 16. The creep characteristic data such as instantaneous strain and creep strain at all levels are shown in Tables 2 to 4.

Table 2.

Creep characteristic indexes of fractured red sandstone at a loading rate of 20 N/s.

Loading Rate (N/s)	Crack Inclination Angle(°)	Stress/MPa	Instantaneous strain/%	Pre-creep strain/%	Post creep strain/%	Creep strain/%
20	0	6.13	0.1787	0.1787	0.1888	0.0101
		7.36	0.0206	0.2099	0.2212	0.0113
		8.58	0.0208	0.2420	0.2543	0.0123
		9.81	0.0181	0.2724	0.2978	0.0254
	30	4.12	0.1489	0.1489	0.1580	0.0091
		4.95	0.0174	0.1754	0.1844	0.0090
		5.77	0.0158	0.1997	0.2086	0.0089
		6.60	0.0133	0.2219	0.2321	0.0102
	60	8.22	0.2416	0.2416	0.2507	0.0091
		9.86	0.0205	0.2702	0.2793	0.0091
		11.51	0.0177	0.2970	0.3076	0.0106
		13.15	0.0169	0.3245	0.3370	0.0126

Table 3.

Creep characteristic indexes of fractured red sandstone at a loading rate of 50 N/s.

Loading Rate (N/s)	Crack Inclination Angle(°)	Stress/MPa	Instantaneous strain/%	Pre-creep strain/%	Post creep strain/%	Creep strain/%
50	0	5.21	0.1555	0.1555	0.1674	0.0119
		6.25	0.0189	0.1863	0.1982	0.0119
		7.29	0.0187	0.2164	0.2288	0.0124
		8.33	0.0155	0.2443	0.2613	0.0170
	30	5.03	0.1722	0.1722	0.1821	0.0099
		6.03	0.0158	0.1979	0.2077	0.0098
		7.04	0.0137	0.2214	0.2317	0.0103
		8.05	0.0117	0.2434	0.2545	0.0111
	60	8.23	0.2089	0.2089	0.2195	0.0106
		9.88	0.0193	0.2388	0.2495	0.0107
		11.52	0.0174	0.2669	0.2789	0.0120
		13.17	0.0164	0.2953	0.3095	0.0142

Table 4.

Creep characteristic indexes of fractured red sandstone at a loading rate of 100 N/s.

Loading Rate (N/s)	Crack Inclination Angle (°)	Stress/MPa	Instantaneous strain/%	Pre-creep strain/%	Post creep strain/%	Creep strain/%
100	0	6.26	0.2712	0.2712	0.2832	0.0120
		7.51	0.0272	0.3104	0.3232	0.0128
		8.76	0.0246	0.3478	0.3625	0.0147
		10.02	0.0231	0.3856	0.4048	0.0197
	30	5.88	0.1973	0.1973	0.2073	0.0100
		7.05	0.0216	0.2289	0.2389	0.0100
		8.23	0.0205	0.2593	0.2713	0.0120
		9.40	0.0203	0.2916	0.3052	0.0136
	60	9.28	0.2372	0.2372	0.2487	0.0115
		11.14	0.0233	0.2720	0.2834	0.0114
		12.99	0.0223	0.3057	0.3188	0.0132
		14.82	0.0218	0.3406	0.3551	0.0145

Analysis of results and discussion

Effect of stress level, fracture inclination and loading rate on instantaneous strain

Figures 17 to 19 show the variation of instantaneous strain of fractured red sandstone with stress level and crack inclination at the same loading rate. Figures 17(a), 18(a) and 19(a) show the relationship between the instantaneous strain and the loading stress level of red sandstone with fracture at the same loading rate. It can be seen from the Figure that with the increase of stress level, the instantaneous strain of red sandstone with fracture shows an overall trend of decreasing. This is because with the increase of axial load applied and the constant axial stress maintained during creep, the initial pores inside the sample are gradually densified, so the instantaneous strain is reduced. Figures 17(b), 18(b) and 19(b) show the relationship between instantaneous strain and crack inclination of fractured red sandstone at the same loading rate. It can be seen that different crack inclination angles have different influences on the instantaneous strain of samples. Under the same loading rate and the same stress level, the instantaneous strain first decreases and then increases with the increase of crack inclination Angle.

Figure 20 shows the instantaneous strain variation of fractured red sandstone with loading rate at the second to fourth stress levels. It can be seen from the Figure that, at the same first-order stress level and the same fracture inclination Angle, the instantaneous strain of red sandstone with fracture decreases slightly when the loading rate increases from 20 N/s to 50 N/s, while the instantaneous strain increases significantly when the loading rate increases to 100 N/s.

Effect of stress level, fissure inclination and loading rate on creep strain

Figures 21 to 23 show the creep strain variation with stress level and crack inclination at the same loading rate. It can be seen from the relationship between creep strain and loading stress level in Figures 21(a), 22(a) and 23(a) that under the same loading rate, the relationship curves between stress level and creep strain are similar, and the creep strain of specimens with different crack inclination angles increases with the increase of stress level. Meanwhile, it can be seen that under a small stress level, with the increase of stress level, The creep strain increases slowly or stays flat. When the compressive strength exceeds 70%∼80%, the creep strain increases rapidly. As can be seen from the relationship between crack creep strain and crack inclination Angle in Figures 21(b), 22(b) and 23(b), the relationship curves between crack inclination Angle and creep strain are similar under the same loading rate and the same stress level. With the increase of crack inclination Angle in the range of 0° to 60°, the creep strain at all stress levels shows a trend of decreasing and then increasing. That is, the creep strain decreases when the crack inclination is from 0° to 30°, and increases when the crack inclination is from 30° to 60°, and the range of change from 0° to 30° is greater than the range of change from 30° to 60°. Take the third-order stress level under the loading rate of 20 N/s as an example. The creep strain from 0° to 60° is 0.0123%, 0.0089% and 0.0106%, respectively; the creep strain from 0° to 30° is decreased by 0.0034% (27.6%); the creep strain from 30° to 60° is increased by 0.0017% (19.1%); At the same time, at the loading rate of 20 N/s, the sample with crack inclination of 0° has accelerated creep failure under the fourth stress level, so the creep strain is larger.

Figure 24 shows the creep strain variation of fractured red sandstone with loading rate under various stress levels. It can be seen from the Figure that under the same loading rate, the creep strain of fractured red sandstone increases with the increase of stress level. At the same first stress level, except for the sample of 0° with a loading rate of 20 N/s at the fourth stress level, the creep strain of other red sandstones with cracks increases with the increase of loading rate. The reason why the creep strain increases with the increase of the loading rate is that the faster the loading rate is, the uncompacted initial cracks of the internal structure during graded loading need to be completed in the creep stage. Therefore, the greater the loading rate is, the more initial cracks and so on need to be compacted in the creep stage, and the greater the creep strain will be generated.

Effect of crack angle and loading rate on steady state creep rate

In this creep test, all creep curves show decay creep and stable creep stages, so the stable creep stage is a very important transition stage. It is of great significance to study the variation law of stable creep rate for the stability of the project. Therefore, in order to better analyze the stable creep stage, this paper chooses the third stage of stress level with relatively obvious steady-state creep rate, and studies the steady-state creep rate within 2 h of the same creep time by calculating the steady-state creep rate. The steady creep rate of fractured red sandstone under the third order stress level at different loading rates is shown in Table 5.

Table 5.

Steady state creep rates of fractured red sandstone under different loading rates at the third level stress level.

Loading rate (N/s)	Crack inclination angle (°)	Steady state creep rate (10⁻⁵h⁻¹)
20	0	3.3
	30	1.9
	60	2.3
50	0	3.4
	30	2.0
	60	2.6
100	0	3.5
	30	2.5
	60	2.8

Figure 25 shows the relationship between steady state creep rate and crack inclination and loading rate. As can be seen from the relationship between steady-state creep rate and crack inclination angle in Figure 25(a), under the same loading rate, the steady-state creep rate firstly decreases and then increases with the increase of crack inclination angle between 0° and 60°, while the steady-state creep rate decreases between 0° and 30° and increases between 30° and 60°. As can be seen from the relationship between steady-state creep rate and loading rate in Figure 25(b), under the same crack inclination angle, steady-state creep rate increases with the increase of loading rate.

Long-term strength determination

In order to accurately obtain the long-term strength of the red sandstone with a prefabricated crack angle of 0° and at a loading rate of 20 N/s, the steady-state creep rate method is adopted to determine the long-term strength. The steady-state creep rate method considers that the maximum load corresponding to the steady state creep rate is the long-term strength. Therefore, the relationship curve between steady-state creep rate and stress level was established, as shown in Figure 26. Before the first inflection point, the stress level is decay creep, and then enters the steady-state creep stage. When it passes the second inflection point, the rate rises rapidly, and at this time, it is in the accelerated creep stage and failure occurs. The stress corresponding to the tangent intersection point derived from the two inflection points is the corresponding long-term strength of the sample. In this section, an empirical formula of power function type is used to fit Figure 26, and the initial creep rate is set to zero:

Fitting function:

y = A * \exp (- x / t_{1}) + y_{0}

(1)

where

A, t_{1}, y_{0}

is a function parameter.

As can be seen from Figure 26, the creep rate tends to increase exponentially with increasing stress levels. The expression fitted to the function is

y = 0.21751 + 1.74029 e^{- 7} * \exp (- x / (- 0.55363))

The long-term strength of the specimen was 9.1 MPa, which is 73.8% of the uniaxial compressive strength.

Study of a nonlinear creep damage intrinsic structure model

On the basis of rock creep test, it is also a very important aspect to study the creep characteristics of rock by establishing a creep model that conforms to the actual test situation and determining the corresponding parameters through parameter inversion. Due to the complexity of rock creep process, many existing models cannot accurately describe the three stages of rock creep, especially the accelerated creep stage. According to the uniaxial compression creep test of red sandstone containing cracks under different loading rates in Chapter 4, it can be concluded that the accelerated creep stage occurs in samples with a crack inclination angle of 0° at a loading rate of 20 N/s, In order to better describe the nonlinear changes in the accelerated creep stage of rocks, this chapter introduces a plastic damage element based on damage rate changes and a software element based on fractional calculus on the basis of the Burgess model, and establishes a nonlinear creep damage constitutive model that can describe the characteristics of the loading and creep stages of red sandstone with cracks.

Plastic damage bodies

In order to better describe the non-linear characteristics in the accelerated creep process, Cao et al. (2013) proposed an elastoplastic damage element based on the damage rate variation as shown in Figure 27, which abstracts the elastoplastic damage body as a micrometric body, as shown in Figure 28. The micro-element consists of two parts of material, damaged and undamaged, with a cross-sectional area of A for the micro-element as a whole and A1 and A2 for the damaged and undamaged sections respectively. $σ$ is the nominal stress (average stress) to which the elastoplastic damaged micro-element is subjected, while under the action of $σ$ , the micro-element is damaged, and since the damaged part cannot continue to bear the load, the stress should all be borne by the undamaged part, defining the stress to which it is subjected $σ^{'}$ as the effective stress, which is obtained from the force equilibrium and geometric relations as follows:

A σ = A_{2} σ^{'}

(2)

Defines the damage variable $D$ :

D = A_{1} / A = 1 - A_{2} / A

(3)

Based on the principles of damage mechanics, the relationship between the nominal stress $σ$ and the effective stress $σ^{'}$ can be obtained as follows:

σ = σ' (1 - D)

(4)

Assuming that the stress-strain in the undamaged part of the material obeys Hooke’s law of linear elasticity, the stress on the undamaged part of the material is:

σ' = E ε'

(5)

where

E

is the modulus of elasticity of the elastoplastic damaged body and

ε'

is the undamaged part of the strain.

Since the damaged part and the undamaged part belong to the same elastic-plastic element, based on the deformation coordination principle, the following can be obtained:

ε = ε^{'}

(6)

From equations (5) and (6), it follows that

σ^{'} = E ε

(7)

Substituting equation (7) into equation (4) yields

σ = E ε (1 - D)

(8)

Equation (8) is the intrinsic model of the elastoplastic damage body based on the Lematrie strain equivalence hypothesis (Lemaitre, 1984). From equation (8), it can be seen that the key to building the intrinsic model of the rock elastic-plastic damage body is to determine the damage variable $D$ .

Kachanov (Kachanov, 1992) proposed a formula for calculating creep damage in rocks:

\frac{d D (t)}{d t} = M {(\frac{σ}{1 - D (t)})}^{n}

(9)

where

M

and n are rock material constants.

When the initial condition is considered, i.e., $t = 0$ , $D (t) = 0$ can be obtained by integrating equation (9):

D (t) = 1 - {[1 - M (1 + n) σ^{n} t]}^{\frac{1}{n + 1}}

(10)

Let the moment of accelerated creep in the rock be $t_{c}$ , then the value of rock damage at this point is $D (t) = 1$ . Then, from equation (10), we have

t_{c} = \frac{1}{M (1 + n) σ^{n}}

(11)

From equations (10) and (11), we have

D (t) = 1 - {(1 - \frac{t}{t_{c}})}^{\frac{1}{n + 1}}

(12)

Where n is the rock material parameter and t is the creep time.

There are certain conditions for creep damage to occur in rocks, when the external stress $σ$ is less than the yield stress $σ_{s}$ of the rock, the damage factor is 0. Only when the external stress $σ$ is greater than the yield stress $σ_{s}$ of the rock will damage occur, so the rock creep damage evolution model is obtained by considering the influence of the rock creep threshold:

D (t) = {\begin{array}{l} 0 & σ < σ_{s} \\ 1 - {(1 - \frac{t}{t_{c}})}^{α} & σ \geq σ_{s} \end{array}

(13)

where

α = \frac{1}{n + 1}

Substituting equation (13) into equation (8), the intrinsic model of the elastoplastic damage body is shown in equation (14). As can be seen from equation (14), when the loading stress is less than the yield stress and the time tends to 0, the elastoplastic damage body degenerates into an elastic element, which can be used to describe the instantaneous elastic deformation of the rock, when only the elastic strain is generated, the rock does not occur damage; when the loading stress is greater than the yield stress, the time tends to creep accelerated damage time, the strain tends to infinity, which can well describe the characteristics of the accelerated creep of the rock. As the study is when the loading stress is greater than the yield stress when the elastoplastic body damage produced by the part of the description of the process of accelerated creep for the rock, so do not consider the elastic part of the elastoplastic damage body, only the study of the part of the plastic strain (Wu et al., 2016), which can be obtained from the formula for plastic damage body as equation (15):

ε = {\begin{array}{l} \frac{σ}{E} & σ < σ_{s} \\ \frac{σ}{E} {(1 - \frac{t}{t_{c}})}^{- α} & σ \geq σ_{s} \end{array}

(14)

ε = {\begin{array}{l} 0 & σ < σ_{s} \\ \frac{σ}{E} {(1 - \frac{t}{t_{c}})}^{- α} & σ \geq σ_{s} \end{array}

(15)

Software components based on fractional calculus

Fractional-order calculus modelling is widely used in physical mechanics modelling because of its ability to obtain physically meaningful and concise parametric equations by fitting experimental data alone. Yin et al. (2007) used the Riemann-Liouville type fractional order calculus operator theory (Nonnenmacher and Metzler, 1995) to construct a new element that represents a state object between an ideal solid and an ideal fluid, which is called a software element.

The Riemann-Liouville type fractional order calculus operator theory defines the $β$ -order integral of a function $f_{t}$

\frac{d^{- β} f (t)}{d t^{- β}} = D {t_{0}}_{t}^{- β} f (t) = \int_{t_{0}}^{t} \frac{(t - τ)}{Γ (β)} f (τ) d τ

(16)

Fractional order differentiation is defined as

\frac{d^{β} f (t)}{d t^{β}} = D {t_{0}}_{t}^{β} f (t) = \frac{d^{n}}{d t^{n}} [D {t_{0}}_{t}^{- (n - β)} f (t)]

(17)

in equations (16) and (17),

β > 0

, 且

n - 1 < β \leq n

and (n is a positive integer);

$Γ (β)$ is a Gamma function defined by

Γ (β) = \int_{0}^{\infty} e^{- t} t^{β - 1} d t Re (β) > 0

(18)

At the same time it introduces a new view and approach (Siegmar et al., 2002) that the stress-strain relationship for an ideal solid satisfies Hook’s law: $σ (t) - ε (t)$ , The stress-strain relationship for an ideal fluid satisfies Newton’s law: $σ (t) - d^{1} ε (t) / d t^{1}$ , rewrite $σ (t) - ε (t)$ as $σ (t) - d^{β} ε (t) / d t^{β}$ . This nicely contains the material between the ideal solid and the ideal fluid.

The intrinsic equation of the software component is

σ (t) = ξ d^{β} ε (t) / (d t^{β}) (0 \leq β \leq 1)

(19)

where

ξ, β

are material constants,

ξ

is the viscosity-like coefficient in the software element’s instanton equation, and

β

is the order of the software element. When

β = 0

, the software element is a hueco body representing an ideal solid; when

β = 1

, the software element is a damper element representing an ideal fluid.

When $σ (t) = const$ , i.e. when the stress is constant, the software element will be able to describe the creep behavior of the material by integrating equation (19) and obtaining the creep equation for the software element according to the theory of Riemann-Liouville type fractional order calculus operators as

ε (t) = \frac{σ}{ξ} \frac{t^{β}}{Γ (1 + β)} 0 \leq β \leq 1

(20)

A diagram of the software components is shown in Figure 29.

Creep modelling and validation

The introduced fractional order calculus based software components are tandem with the Hooke body to construct a modified Maxwell body, which is then tandem with the Kelvin body to obtain a modified Burgers model, and also tandem with the introduced plastic damage body to obtain the new model shown in Figure 30:

The creep equation for the nonlinear creep damage intrinsic model can be derived from the relationship of the creep model in series as

Degenerates to the modified Burgers model when $σ < σ_{s}$

ε = \frac{σ}{E_{1}} + \frac{σ}{η_{1}} \frac{t^{β}}{Γ (1 + β)} + \frac{σ}{E_{2}} (1 - \exp (- \frac{E_{2}}{η_{2}} t))

(21)

When $σ \geq σ_{s}$ , the plastic damage body involved in creep, the rock into the accelerated creep phase, the creep equation is obtained as

ε = \frac{σ}{E_{1}} + \frac{σ}{η_{1}} \frac{t^{β}}{Γ (1 + β)} + \frac{σ}{E_{2}} (1 - \exp (- \frac{E_{2}}{η_{2}} t)) + \frac{σ}{E_{3}} {(1 - \frac{t}{t_{c}})}^{- α}

(22)

So the creep equation for the non-linear damage creep model is

ε = {\begin{array}{l} \frac{σ}{E_{1}} + \frac{σ}{η_{1}} \frac{t^{β}}{Γ (1 + β)} + \frac{σ}{E_{2}} (1 - \exp (- \frac{E_{2}}{η_{2}} t)), & σ < σ_{s} \\ \frac{σ}{E_{1}} + \frac{σ}{η_{1}} \frac{t^{β}}{Γ (1 + β)} + \frac{σ}{E_{2}} (1 - \exp (- \frac{E_{2}}{η_{2}} t)) + \frac{σ}{E_{3}} {(1 - \frac{t}{t_{c}})}^{- α}, & σ \geq σ_{s} \end{array}

(23)

Based on the established non-linear creep damage intrinsic model, the uniaxial creep test data for a specimen with a fracture dip of 0° at a loading rate of 20 N/s were fitted using the L-M least squares method using the mathematical software Origin to obtain the model parameters at various stress levels.

Validation of the test data against the fitted curve is shown in Figure 31:

It can be seen that the square of the correlation coefficients for the fitting of creep parameters has reached above 0.97 from Table 6. At the same time, the new nonlinear creep damage constitutive model constructed can accurately reflect the complete creep process of rocks at all levels of stress compared to the Burgess model, especially the nonlinear characteristics of the accelerated creep stage. The fitting effect is relatively ideal, verifying the applicability of the model.

Table 6.

Creep model parameter identification results.

$σ$ /MPa	$E_{1}$ /MPa	$η_{1}$ /MPa·h	$β$	$E_{2}$ /MPa	$η_{2}$ /MPa·h	$E_{3}$ /MPa	$t_{c}$ /h	$α$	R²
6.13	34.5	825.6	0.423	5377.1	7490.5	—	—	—	0.99
7.36	37.1	804.3	0.219	8662.4	3745.1	—	—	—	0.98
8.58	39.9	614.2	0.136	1.6 × 10²⁵	8.5 × 10²⁴	—	—	—	0.97
9.81	45.7	733.8	0.271	3.24 × 10²⁵	9.4 × 10²⁴	449.0	1.31	0.08	0.99

Conclusion

The presence of prefabricated cracks decreases the peak strength, elastic modulus and peak strain of rock for intact samples. With the increase of crack inclination Angle, the peak strength and peak strain first decrease and then increase, the elastic modulus increases, and the number of cracks gradually decreases. When the fracture is 90°, the mechanical properties of red sandstone with fracture are the closest to those of intact samples. With the increase of loading rate, the peak strength and elastic modulus of fractured red sandstone increase, the peak strain decreases, and the number of cracks on the specimen surface decreases.

With the increase of stress level, the instantaneous strain decreases and the creep strain increases. The instantaneous strain, creep strain and steady creep speed decrease first and then increase with the increase of crack inclination. With the increase of loading rate, the instantaneous strain of fractured red sandstone decreases first and then increases, and the creep strain and steady creep rate increase.

According to the variation characteristics of the creep curve, a creep constitutive model which can describe the whole process of three-stage creep is obtained by introducing software elements based on fractional calculus and plastic damage elements based on damage rate changes. The creep test data of uniaxial compression are compared and analyzed by parameter inversion and fitting. It is determined that the model can well reflect the creep development characteristics of fractured red sandstone, which proves the rationality and effectiveness of the model.

Supplemental Material

sj-pdf-1-ijd-10.1177_10567895241277657 - Supplemental material for Experimental study on the mechanical properties of red sandstone with fractures under different loading rates

Supplemental material, sj-pdf-1-ijd-10.1177_10567895241277657 for Experimental study on the mechanical properties of red sandstone with fractures under different loading rates by Hui Wang, Zhichao Xu, Hongyuan Huai, Yunteng Yin, Jiacong Zeng, Zhihao Du and Hang Zhou in International Journal of Damage Mechanics

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 (No. 51804182), the Excellent Youth Innovation Team Program of Shandong Province Universities (No. 2019KJG007), the Natural Science Foundation of Shandong Province (No. ZR2020ME097), the Quality Improvement Program for Postgraduate Education in Shandong Province (No. SDYAL21062), and the Construction Project of Case Library of Shandong University of Science and Technology (Yzlts2021032). The financial supports are gratefully appreciated.

ORCID iD

Hui Wang

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