Ynlin Zho, Jinhi Liu, Chunshun Zhng, Houqun Zhng, Jin Lio, Sito Zhu, Linyng Zhng

a School of Resources, Environment and Safety Engineering, Hunan University of Science and Technology, Xiangtan 411201, China

b School of Emergency Technology and Management, North China Institute of Science and Technology, Beijing 101601, China

c School of Civil Engineering, Wuhan University, Wuhan 430072, China

d State Key Laboratory for Geomechanics and Deep Underground Engineering,School of Mechanics and Civil Engineering,China University of Mining and Technology,Xuzhou 221116,China

e School of Civil and Resource Engineering, University of Science and Technology Beijing, Beijing 100083, China

f Department of Civil and Architectural Engineering and Mechanics, University of Arizona, Tucson, AZ 85721, USA

Keywords:Post-peak stage Cyclic loading and unloading Hydromechanical coupling Sandstone Water pressure

A B S T R A C T This paper investigates mechanical behaviours of sandstone during post-peak cyclic loading and unloading subjected to hydromechanical coupling effect, confirming the peak and residual strengths reduction laws of sandstone with water pressure, and revealing the influence of water pressure on the upper limit stress and deformation characteristics of sandstone during post-peak cyclic loading and unloading.Regarding the rock strength, the experimental study confirms that the peak strength σp and residual strength σr decrease as water pressure P increases.Especially, the normalized strength parameters σp/σpk and σr/σre was negatively and linearly correlated with the P/σ3.Moreover, the Hoek-Brown strength criterion can be applied to describe the relationship between effective peak strength and effective confining stress.During post-peak cyclic loading and unloading, both the upper limit stress σp(i) and crack damage threshold stress σcd(i)of each cycle tend to decrease with the increasing cycle number.A hysteresis loop exists among the loading and unloading stress–strain curves,indicating the unloading deformation modulus Eunload is larger than the loading deformation modulus Eload.Based on experimental results,a post-peak strength prediction model related to water pressure and plastic shear strain is established.Ⓒ2023 Published by Elsevier B.V.on behalf of China University of Mining & Technology.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

The stress loading and unloading sequences are often encountered in rock engineering, such as earthquakes, drilling and blasting, mechanical excavation and mining seismicity, resulting in a complex mechanical response in the load history of the surrounding rock, particularly in water-bearing environments [1–4].For example, the stress redistribution of the surrounding rock caused by excavation, may cause the surrounding rock within a certain range around the tunnel in the post-peak state, and then the surrounding rock in the post-peak state will be reloaded due to the action of subsequent excavation or mining stress.The complex mining effect will cause the post-peak surrounding rock to continuously experience cyclic loads such as unloading, loading, reunloading, and re-loading, which may bring great harm to the safety and stability of the construction.Therefore,the investigation of the strength and deformation characteristics of rock during cyclic loading,especially regarding the mechanical properties of postpeak rock under the hydromechanical coupling effect,is significant not only in theoretical understanding but also in engineering practice for safe construction [5–14].

Previous studies have demonstrated that the mechanical behaviours of rocks under cyclic loading are different from those under monotonic loading [11,15–22].Remarkably, the accumulation of irreversible deformation and damage evolution can only be captured by the cyclic loading and unloading tests.Such tests also enable us to obtain a series of rock mechanical properties, such as the evolution of the rock stiffness and strength under complex loading and unloading paths.Therefore, many experimental and theoretical studies have employed cyclic loading and unloading tests to study rock deformation and damage characteristics [23–29].Through cyclic loading and unloading tests on various rocks,the stress-strain curves of rocks under various loading and unloading rates and confining pressure conditions were obtained[25–29].Moreover, the evolution patterns of the elastic modulus and strength parameters versus the number of cycles and loadingunloading rates have been revealed [11,14,21,25,29].The effects of different stress paths and loading rates on the deformation,peak stress and energy characteristics also have been analyzed[8,11,14,19,20,27–34].The results indicated the stress-strain curves of rock in the cyclic loading and unloading testing has a stress-strain closed loop which presents a hysteresis effect[14,34,35].Moreover, the nonlinear mechanical behaviours are caused by damage evolution and irreversible deformation accumulation[8,11].The studies on softening mechanical behaviour in the post-peak stage mainly focused on crack propagation,damage and fracture mechanics behaviours under cyclic loading and unloading conditions [14,32–39].The post-peak damage-controlled tests on sandstone, limestone and granite from Munoz and Taheri [38]found that the global loss of strength of rocks is mainly from irreversible deformation and stiffness degradation in the localized damage zone.Feng et al.[36] carried out post-peak cyclic loading tests on saturated and dry sandstones,and observed that the sandstone’s post-peak dilatancy and plastic shear strain tend to increase,but the post-peak strength decreases when the sandstone core was saturated.

Water pressure is known to influence the mechanical behaviours of rock.In particular, the increased fluid pressure in pores or fractures induces mechanical behaviours change and pore opening or fractures sliding in the rock.Such hydromechanical coupling effects have primarily been studied using laboratory experiments and numerical simulations[2,40–55].Those studies have indicated that pore pressure has an indisputable impact on mechanical behaviours and deterioration of strength and stiffness properties of rock material [42,44,47,48,50].For sandstone, Wang et al.[2] experimentally investigated the hydromechanical properties of the sandstone under various water pressures, indicating that compared with axial strain,the lateral strain is more sensitive to water pressure.The experimental results from Yang et al.[51] indicated that the elastic modulus and peak axial strain of saturated red sandstone drop linearly with increasing water pressure.Asahina et al.[56]found that water pressure in rock reduces the effective stress,causing instability between pre-generated fracture surfaces.Fortin et al.[57] found that compaction localization in saturated sandstone is closely related to acoustic signatures during loading.The above research results enriched the understanding of hydromechanical behaviours of sandstone.

Currently, the studies mainly focus on the strength and deformation behaviours of rock in the pre-peak stage under cyclic loading and unloading conditions.However, the study of post-peak strength and deformation characteristics was still limited.Moreover, previous rock cyclic loading and unloading tests have not considered the effect of water pressure on mechanical behaviours of rock,especially in the post-peak stage.This paper attempts to fill this gap through post-peak cyclic loading and unloading tests on sandstone under hydromechanical coupling effect and reveals sandstone’s post-peak strength and deformation characteristics in detail.This study will further improve the understanding of the mechanical behaviours of rock in the post-peak stage, especially under hydromechanical coupling condition.

2.Specimen preparation and test procedure

The core sandstone specimens with 50 mm in diameter were from the -240 working face of the main skip shaft at Hunan Baoshan Nonferrous Metals Mining Co.Ltd.,located in south China.The specimens were cut into 102 mm long and then ground to 100 mm to obtain two parallel surfaces on the specimen,following the specifications from the International Society for Rock Mechanics(ISRM)[58].Specimens were dried for 24 h at T=50°C to obtain the water content of natural sandstone, and then saturated by using deionized water for 48 h under a vacuum before experiments.Before testing, each specimen was subjected to a nuclear magnetic resonance (NMR) test at room temperature to obtain the pore distribution characteristics of sandstone.Thirty sandstone specimens were prepared to obtain the mechanical characteristics of porous sandstone during post-peak cyclic loading and unloading under various confining and water pressures.Some specimens are shown in Fig.1.

The post-peak cyclic loading and unloading on sandstone under hydromechanical coupling were conducted on the MTS 815 rock mechanics test system (Fig.2).Five main units of the MTS 815 are as follows: triaxial cell, loading and water supply units, deformation and pressure monitoring units, and data-acquisition unit.The maximum loading force is 4600 kN.The maximum confining and water pressures applied are 160 and 140 MPa, respectively.The axial deformation can be monitored with a pair of linear variable displacement transducers (LVDTs), and the circumferential deformation also can be measured with a circumferential extensometer connected to a roller chain assembly wrapped around the jacketed specimen.A special thermo-shrinking plastic membrane was used to seal up the saturated specimen circumferentially, and then the specimen was placed on the base pedestal in the triaxial cell,and then the upper and low ends of specimen were connected to the upstream and downstream water reservoirs,respectively.

Fig.2.MTS 815 rock test machine.

The post-peak cyclic loading and unloading test procedure is as follows: (1) Sandstone specimens are placed at the base central position of the triaxial pressure chamber,followed by a small axial load of 1.0 kN applied to fix the rock core.Then the confining pressure is applied to a predetermined value at a rate of 0.05 MPa/s.Subsequently,the axial stress is applied to the same value at a rate of 0.05 MPa/s to achieve an initial hydrostatic loading condition.(2) The water pressures applied on two ends of specimen by upstream and downstream reservoirs at the same time are increased to a predetermined value below the confining pressure.(3)During testing,confining pressure is maintained constant while a 0.005 mm/s axial loading rate is monotonically applied until the peak strength is reached.After that, the deviatoric stress level is unloaded to zero at an axial unloading rate of 0.005 mm/s through controlling axial displacement,while the confining and water pressures remain unchanged.(4) During the post-peak stage, the axial stress is applied in each subsequent cycle until the second peak stress is reached, and then the deviatoric stress level is again unloaded to zero without changing the confining and water pressures.Finally,the stress cycles of loading and unloading are continued until the residual strength of the specimen is reached.

During the above test,the loading and unloading rates are both 0.005 mm/s through controlling axial displacement.Also, the confining pressures are set to 10,20,and 30 MPa,and the water pressures are loaded to 20–80% of the confining pressures, i.e., from 2 to 24 MPa at an interval of 2 MPa, at a loading rate of 0.05 MPa/s,and then the constant water pressure is kept for 30 min in order that pressurized water can fully flow into the pores in sandstone.All these can be well coped with by the computer-implemented script in MTS 815.Noted that conventional triaxial compression tests under the four confining pressures of 5, 10, 20, and 30 MPa and uniaxial compression tests without water pressure are also performed on the saturated sandstone specimens, respectively, to determine the strength characteristics of sandstone specimens without water pressure before the cyclic loading and unloading tests.All the above tests follow ASTM D7012-14[58].To intuitively study the micro-fracture mechanism of sandstone under hydromechanical coupling, scanning electron microscopy (SEM) tests and X-ray diffraction(XRD) analysis of specimens before and after test were also comparatively conducted in this study.

3.Results and discussion

3.1.Analysis of nuclear magnetic data

The T2spectrum distribution of three representative sandstone specimens can be obtained by the AniMR-150 NMR test system.According to the mechanism of nuclear magnetic relaxation, the fluids within different pore types of rock have different relaxation times,located at different positions on the T2spectrum curves.The T2spectrum curve can be converted into the aperture distribution curve according to the corresponding relationship between T2value and pore size (r) [59].The aperture distribution curves of sandstone specimens are shown in Fig.3a.As shown in Fig.3a,the curves of aperture distribution of sandstone present a threepeak shape.Different peaks correspond to different pore types:the first, second and third peaks from left to right approximately denote the micropores,mesopores and macropores,with the corresponding T2<10 ms, 10 ms≤T2≤100 ms, and T2>100 ms, respectively.The macropores and mesopores have very good and relatively good connectivity, respectively.The distributions of micropores are relatively isolated, with poor connectivity [59].The porosity components contributed by micropores, mesopores and macropores are shown in Fig.3a.So it shows that the porosity contributed by macropores occupies 95.8%,and the average porosity of sandstone is approximately 12.6%.

Fig.3.Porosity distribution and cumulative porosity versus pore diameter and NMR image of the transverse section.

Fig.4.Curves of deviatoric stress versus axial and lateral strains of sandstone during post-peak cyclic loading and unloading under confining pressure of 10 MPa and different water pressures.

Fig.5.Curves of deviatoric stress versus axial and lateral strains of sandstone during post-peak cyclic loading and unloading under confining pressure of 20 MPa and different water pressures.

Fig.6.Curves of deviatoric stress versus axial and lateral strains of sandstone during post-peak cyclic loading and unloading under confining pressure of 30 MPa and different water pressures.

The NMR images can differentiate between rock and water because the water contains many more protons than the rock[55].As a result, the microstructure properties of sandstone can be discovered.Two-dimensional cross-section images of sandstone specimens were captured.The fluid locations within sandstone can be shown as light spots in the NMR images.The higher fluid contents are implied by brighter and larger light spots, indicating the presence of more and larger pores.Fig.3b and c show the diagram of transverse section imaging and the corresponding NMR image of the sandstone specimen.In the NMR image, some bright light spots of various sizes are separately distributed over the image, indicating that pores are distributed unevenly in the specimen.

3.2.Peak strength and peak strain characteristics during monotonically loading stage

According to conventional triaxial and uniaxial compression tests on saturated sandstone specimens without water pressure,the triaxial compression strength (TCS) σpk, and the residual strength, σre, of sandstone are measured to 57.1, 70.4, 88.1, 107,and 27.5,37.6,59.1,and 81.6 MPa subjected to the confining pressures of 5, 10, 20 and 30 MPa, with the uniaxial compression strength (UCS) of 34.6 MPa.

The curves of deviatoric stress versus axial and lateral strains of sandstone specimens during post-peak cyclic loading and unloading subjected to various confining and water pressures are illustrated in Figs.4–6.It can be observed that the outer profiles of the stress-strain curves of the cyclic loading and unloading tests are consistent with those of the conventional triaxial compression tests.According to the effective stress principle, under a conventional triaxial stress state with water pressure, the effective axial stress(i.e.,the effective maximum principal stress)and effective confining pressure(i.e.,the effective minimum principal stress)can be expressed:

where σ1and σ3are axial stress and confining pressure, i.e., the maximum and minimum principal stresses, respectively.

Based on the experimental results and Eq.(1),the variation laws of the effective peak strength with the effective confining pressureunder hydromechanical coupling are depicted in Fig.7, where,as a comparison,the experimental results of peak strength σpkversus confining pressure σ3subjected to conventional triaxial compression without water pressure are also plotted.

Fig.7. or σpk versus theor σ3.

For saturated sandstone specimens without water pressure,according to the Mohr-Coulomb yield criterion, the relationship between the peak strength σpkand the minimum principal stress(i.e., confining pressure)σ3is expressed as

where φ and c are internal friction angle and cohesion of sandstone.The internal friction angle and cohesion of sandstones without water pressure are 23.5° and 19.8 MPa, respectively.

For sandstone specimens with water pressure, the approximately linear relationship betweenandcan be observed based on most experimental results (see the red dash line in Fig.7).However, when σ3=10 MPa with P=8 MPa (i.e., P/σ3=0.8),σ3=20 MPa with P=16 MPa (i.e., P/σ3=0.8), and σ3=30 MPa with P=16–24 MPa (i.e., P/σ3=0.53–0.8), experimental data which are lower than the fitting straight line deviate from the above linear relationship (see the points in the red cycle).This indicates that when the P/σ3ratio reaches a critical value, the peak strength drops rapidly with a decrease in.The critical P/σ3ratio is 0.8,0.8 and 0.53 at σ3=10, 20, and 30 MPa, respectively.When P approaches the confining stress,the water pressure effect becomes significant, and the rock is probably close to hydraulic fracturing.

The nonlinear relationship between andfor sandstone specimens with water pressure can be approximately generalized by Hoek-Brown strength criterion as

where σcis the UCS of saturated sandstone specimen,measured to 34.6 MPa (Fig.7); msa material constant, and s=1 for intact rock.The Hoek-Brown strength envelope curve of sandstone specimens with water pressure is plotted in Fig.7, where m is determined to 7.23.

The peak strength σpis related to confining and water pressures.The relationship between peak strength σpand water pressure P is illustrated in Fig.8a.As shown in Fig.8a,σptends to be greater under higher confining pressure or lower water pressure.To study the combining effect of P and σ3on σp, two normalized parameters of P/σ3and σp/σpkare used.The relationship between the σp/σpkand the P/σ3is plotted in Fig.8b.There is a negative linear correlation between P/σ3and σp/σpk:

Fig.8.Relationship between peak strength σp and water pressure P, and σp/σpk versus P/σ3.

where σpkis the peak strength of sandstone subjected to conventional triaxial loading without water pressure; λ a material parameter determined to 0.780 by fitting.

To further verify the negative linear correlation between P/σ3and σp/σpk, the experimental data from Zhou et al.[60], Yang et al.[61], Li et al.[62], Xing et al.[63], Xu et al.[64], Liu et al.[65], and Li et al.[66] are also analyzed (Fig.9).The comparison results between this study and those of other researchers highlight the generality of the negative linear correlations of σp/σpkand σr/σreversus P/σ3.Moreover, the proposed negative linear correlations from sandstone are also applicable to other types of rocks,where the model parameter λ falls in between 0.32 and 0.78.

Fig.9.Experimental data and fitting line of σp/σpk versus P/σ3 from previous studies and the present study [60–66].

The curves of axial and lateral peak strains () versus water pressure at various confining pressures are presented in Fig.10.Generally, the axial peak straintends to decrease with increased water pressure or a decrease in confining pressure.The water pressure promotes rock damage and weakens its energy accumulation capacity, and the energy storage limit of rock is reduced, which leads to the decrease of axial peak strain with increased water pressure[62].The lateral peak strainis smaller than the axial peak strain.The lateral peak strains fluctuate with water pressure around the horizontal lines of=4.04×10-3, 3.82×10-3, and 2.47×10-3under confining pressures of 10, 20, and 30 MPa, respectively (Fig.10).For one thing,an increase in water pressure induces a peak strength drop,which causes a decrease in lateral peak strain;for another,an increase in water pressure itself can enhance the lateral strain, due to the decrease in effective confining pressure reduced by water pressure in standstone.Under the influence of the two above competing mechanisms,the change law ofwith the increase of water pressure is not apparent.

Fig.10.Peak strains and versus water pressure P at various confining pressures.

3.3.Post-peak strength of sandstone subjected to post-peak cyclic loading and unloading

During post-peak cyclic loading and unloading, rock strength characteristics can be characterized by the upper limit stress, σp(i), and crack damage threshold stress, σcd(i), in the ith cycle, and residual stress, σr.In detail, σcd(i), corresponding to the reversal point of volumetric strain at the onset of dilation [1,13], describes a deformation transition from the compaction-dominated state to the dilatancy-dominated state (Fig.11b).It is noted that in Fig.11, σpand σcdare the highest stress and crack damage threshold stress during monotonic loading to peak strength,respectively.The evolution of σp(i)with cycle number, i, subjected to different confining and water pressures are plotted in Fig.12a–c, where the corresponding highest peak stress σpis also marked.The peak stress σp(i)gradually decreases with the increasing cycle number in the post-peak region until the residual strength is reached.A normalized indicator of σp(i)/σpis used to study the strength degradation laws.The curves of σp(i)/σpversus cycle number i subjected to different confining and water pressures are plotted in Fig.12d–f.The relationship between σp(i)/σpand cycle number i can be described by a negative exponential function:

Fig.12.σp(i) and σp(i) /σp versus cycle number i at different σ3.

where α is a strength degradation parameter.A smaller α corresponds to a lower gradation speed.The α tends to decrease with increased confining pressure, signifying that strength decays in a slower ratio with an increasing number of cyclic loading and unloading when the confining pressure increases from 10 to 20 and 30 MPa.

As expected,crack damage threshold stress σcd(i)also decreases with the increasing cycle number.The σcd(i)/σp(i)ratio of the ith loading cycle is calculated and shown in Fig.13.The σcd(i)/σp(i)ratio ranging from 0.4 to 1.0 has an increasing tendency with the increase of cycle number,which indicates crack damage threshold stress σcd(i)is closer to peak stress σp(i)with the more cycle number.

Fig.13.σcd(i)/σp(i) versus cycle number i at different σ3.

At each loading cycle, the σp(i), σcd(i), and σrof the sandstone tend to increase with an increase in confining pressure.As an example, the σp(i)at σ3=30 MPa are 1.78–5.23 times those at σ3=10 MPa, and the residual strength σrat σ3=30 MPa is 1.79–3.0 times those at σ3=10 MPa, when the water pressures range from 2 to 8 MPa, respectively (Fig.15a).However, differing from the effect of confining pressure,water pressure significantly weakens the σp(i), σcd(i)and σrof sandstone.Taking the 3rd cycle at σ3=30 MPa, for example (Fig.12c), when water pressure increases from 2 to 22 MPa,the σp(3)decreases from 78.8 to 33.5 MPa(57.5%drop), and the corresponding σcd(6)decreases from 75.2 to 33.2 MPa (55.9% drop).Especially, when water pressure increases to 24 MPa, sandstone specimen experiencing a full stress-strain process, loses bearing capacity and is unable to finish reloading due to the strong hydraulic fracturing effect of high water pressure.

The degradation behaviours of residual strength σrwith an increasing water pressure are plotted in Fig.14a.Similar to the σp,the two normalized parameters of P/σ3and σr/σreare also used.The relationship of the σr/σreversus the P/σ3is plotted in Fig.14b.An approximately negative linear correlation between P/σ3and σr/σrecan be observed:

Fig.14.Relationship between residual strength σr and water pressure P, and σr/σre versus P/σ3.

where σreis the residual strength of sandstone subjected to conventional triaxial loading without water pressure.

According to the principle of effective stress, the effective confining pressure applied on post-peak broken sandstone decreases with an increasing water pressure, which leads to the degradation of residual strength due to the decrease of effective lateral confinement.It is observed that the line slope of the P/σ3versus σr/σreis larger than of the P/σ3versus σp/σpk, which indicates the effect of water pressure on residual strength is more obvious than on peak strength.The reason lies in that compared with intact sandstone,the water pressure can more fully be applied in pores and fractures for broken sandstone experiencing post-peak cyclic loading and unloading, due to the increase of pores and fractures in broken sandstone.

3.4.Deformation behaviours of sandstone subjected to post-peak cyclic loading and unloading

During loading,the axial(ε1)and lateral(ε3)strains of the sandstone specimens are composed of elastic strain component (and)and plastic strain component(and)(Fig.15).Elastic strain can be recovered after unloading.However, the strains from the closure of the micro-pores or micro-fractures inside sandstone specimens and the propagation of secondary cracks are irrecoverable after unloading, called plastic strain.Irreversible strains under cyclic unloading and loading refer to the difference in strains from the end to the beginning of a cycle.So plastic strain of a single cycle can be calculated as follows [38]:

Fig.15.Separation of elastic strain and plastic strain.

Based on the above analysis, the accumulative axial and lateral plastic strains,and, are axial and lateral strains at the end of a cycle, and the deviatoric stress is unloaded to zero.The accumulative plastic strain is the summation of plastic strain increment of each cycle:

The successive loading and unloading history induces the accumulation of irreversible deformation in the sandstone specimens.As shown in Fig.16a–c, with the increase of the cyclic loading and unloading,the axial and lateral residual strains are constantly accumulated, indicating that the pores and cracks in sandstone increase gradually with increasing loading-unloading cycles.

Fig.16.Accumulative plastic strain versus cycle number at different σ3.

During the rock specimens’ cyclic loading and unloading tests,the loading-unloading path in each cycle could not be repeated with a hysteresis loop existing among the loading and unloading stress-strain curves.This indicates that during loading and unloading,the sandstone has two deformation moduli:the loading deformation modulus, Eload, and the unloading deformation modulus,Eunload.To reflect the overall deformation behaviours of the sandstone during the post-peak cyclic loading and unloading,the secant slopes of the loading and unloading sections of the stress–strain curve of each cycle are taken as the loading deformation modulus and unloading deformation modulus of rock, respectively.Eload(i)and Eunload(i)in the ith cyclic loading and unloading are calculated based on the experimental results of different specimens under various confining and water pressures.Unified change rules of Eloadand Eunloadare observed under different confining and water pressures.For example, the variation of Eloadand Eunloadversus cycle number under confining pressure of 20 MPa with water pressures ranging from 2 to 16 MPa are plotted in Fig.17a and b.The Eloadtends to increase at a low rate with an increase in loading cycles during the initial 2–3 cycles.The reason lies in that the pores and fractures are reclosed during post-peak loading.However,during the subsequent unloading process,some of the pores and fractures were unable to be fully reopened due to the friction between the micro-crack interfaces, resulting in a denser internal structure and a slight increase in the overall stiffness of sandstone during the initial 2–3 cycles.As the number of cycles increases, Eloadtends to decrease, due to the gradual expansion of fractures and microcracks in sandstone during the post-peak stage.Then Eloaddecreases rapidly at the last loading cycle.The Eunloadtends to fluctuate gently with an increase in cycle number, without noticeable change law with cycle number.Moreover,Eunloadis larger than Eloaddue to a hysteresis loop among the loading and unloading stressstrain curves.

Fig.17.Eload and Eunload versus cycle number under confining pressure of 20 MPa with water pressures ranging from 2 MPa to 16 MPa.

3.5.Post-peak strength prediction model related to water pressure and plastic shear strain

According to the experimental results,plastic deformation continuously develops during post-peak cyclic loading and unloading.To characterize the plastic state of sandstone specimens, it is necessary to define the plastic internal variables.Generally,the plastic shear strainis taken as the internal variable [32]:

Fig.18.Eload versus the γp under confining pressure of 20 MPa with water pressures ranging from 2 MPa to 16 MPa.

Furthermore, the relationship between the σp(i)and the plastic shear strain γpcan be calculated, as presented in Fig.19.It can be found that the σp(i)gradually decreases with an increase in the γp.Through the interpolation processing to Fig.19,the relation curve of any γpand σp(i)under various confining pressures σ3and water pressures P can be captured.Two normalized indicators of σp(i)/σpkand P/σ3are used to study the strength degradation with the plastic shear strainγp.The curves of the σp(i)/σpkversus the P/σ3at different plastic shear strains γpare plotted in Fig.20a–g,through statistical analysis of all experimental data.It is observed that the relationship between the σp(i)/σpkand the P/σ3at a certain γpcan be fitted by a negative exponential function (Fig.20a–g):

Fig.19.σp(i) versus the plastic shear strain γp from experimental results at different σ3, and from prediction at different σ3.

Fig.20.Experimental results and fitting curves of the σp(i)/σpk versus P/σ3 at different γp.

where k is a strength decay parameter with an increase in P/σ3,the smaller the value of k, the lower the rate of strength decay; and m the σp(i)/σpkwithout water pressure in sandstone (The m and k as two model parameters related to plastic shear strains γp).

The m and k at different plastic shear strains γpare listed in Table 1.Interestingly,the strength decay parameter k tends to linearly increase with an increase in the γp, whereas m linearly decreases with an increase in the γpas shown in Fig.21a and b.The fitting equations are as follows:

Fig.21.m and k versus plastic shear strains γp.

Substituting Eq.(11) into Eq.(10), the post-peak strength prediction model related to water pressure and plastic shear strain can be expressed as:

The post-peak strength prediction model containing P/σ3,and γpreflects the effect of water pressure and plastic shear strain on the post-peak strength of sandstone, and an essential parameter of triaxial compression strength σpkwithout water pressure which can expediently be obtained by conventional triaxial compression tests.The measured strengths and predicted results based on the proposed model (Eq.(12)) under different confining and water pressures are comparatively plotted in Fig.19a–f.Their relative errors are also shown in Fig.19d–f, where the relative errors δ are obtained with the following equation:

where σpe(i), and σpp(i)are the experimental results and the predicted results from the proposed fitting equation(Eq.(12)),respectively.From Fig.19d–f, it is found that the predictions are in good agreement with the experimental results, with the relative errors ranging from 3.29% to 30.8% and an average error of 12.9%.

3.6.Failure mode of sandstone specimens

The ultimate failure mode of the loaded sandstone specimen is a critical feature reflecting its failure mechanism.The observation of failure specimens found that the sandstone specimens all experienced single shear fracture with various fracture angles (i.e., the angles between the shear fracture surfaces and the horizontal directions).Moreover, the fracture angle increases with increased water pressure at a constant confining pressure.Fig.22a–h shows sandstone specimens’ failure mode and fracture angle at a confining pressure of 20 MPa and various water pressures.When the water pressure increases from 2 to 16 MPa, the fracture angle increases from 51° to 72°.When water pressure is relatively low,the axial stress required for sandstone failure is higher.The internal friction effect becomes evident, which inhibits the fracture of the rock specimen and makes the fracture angle of the sandstone specimen smaller (Fig.22a–f).However, the water pressure is relatively high (P/σ3≥0.6), and the effective radial pressure and restraint capacity are reduced due to the higher water pressure applied in sandstone.This leads to the weaker internal friction effect, so the fracture plane with a larger fracture angle of above 70° occurs (Fig.22g and h).

Fig.22.Failure mode and fracture angle of sandstone specimens under post-peak cyclic loading and unloading conditions at confining pressure of 20 MPa and different water pressures.

3.7.Result analysis of XRD and SEM scanning

The XRD data of dry sandstone specimens and after hydromechanical coupling test are comparatively shown in Fig.23a and b,respectively.The main mineral components of dry sandstone are feldspar, quartz, andesite cuttings, amphibole, pyroxene and cement, and the cement is mainly calcite and iron oxide.The percentage content of various minerals in sandstone is decreased in some degree, and new mineral components are produced after hydromechanical coupling test, because some mineral can react in the water environment,e.g.,feldspar,cements(including calcite and iron oxide) and quartz.Under hydromechanical coupling, the calcareous types of cement are prone to dissolve, leading to the loss of mineral composition and the gradual exposure of mineral particles, resulting in some new minerals.

Fig.23.The XRD results of dry sandstone specimen and after hydromechanical coupling test.

The hydromechanical coupling action alters the micro-pore and fracture structure of sandstone.Due to the limit of space, as an example, the SEM results of dry sandstone specimen before test and failed sandstone specimens after post-peak cyclic loading and unloading under confining pressure of 20 MPa are shown in Fig.24a–i.For dry sandstone specimen, the mineral particles are arranged in a regular way, fully wrapped in the cement, and the overall structure is dense with some relatively small micro-pores(Fig.24a).During post-peak cyclic loading and unloading subjected to hydromechanical coupling, some calcareous and argillaceous cement are dissolved in sandstone, leading to cavity and microcrack formation (Fig.24b–g).As a result, the friction coefficient and cohesion of sandstone is reduced,and the link between cement and diagenetic mineral particles is weakened.With the water pressure increasing,the rupture and pulling out of the grains and shear cracks propagation become prominent due to hydraulic fracturing effect of higher water pressure (Fig.24h and i).

Fig.24.The SEM results of dry sandstone specimen before test and failed sandstone specimens after post-peak cyclic loading and unloading under confining pressure of 20 MPa and different water pressures.(Note: A: Mirco-pore; B: Mirco-crack; C: Cavity; D: Deep cavity; E: Shear cracks).

Table 1 m and k at different plastic shear strains γp.

4.Discussion

From experiments, we have confirmed the reduction laws of rock peak and residual strengths in sandstone under water pressure.Subsequently, we investigated the influence of water pressure on the strength and deformation characteristics of sandstone in the post-peak stage.Finally, we proposed a postpeak strength prediction model that accounts for water pressure and plastic shear strain.The study has important engineering value for the safe construction and disaster prediction, because the continuous mining effect will cause surrounding rock in post-peak state to successively experience cyclic loads such as unloading,loading, re-unloading, and re-loading.For example, the changes of stress state during the excavation of roadway can be described as follows: (a) The stress state before excavation of the original rock(equivalent to the initial loading state).(b)The stress redistribution of the surrounding rock caused by excavation and unloading, causing the surrounding rock within a certain range around the cavern to be in the post peak state (post-peak loading state).(c) The stress state under the action of subsequent excavation or mining stress (unloading state post peak).(d) The stress state released by surrounding rock deformation (unloading state), etc.For water-rich surrounding rock, the cyclic loading and unloading will cause the plastic deformation accumulation and seepage passage development in surrounding rock, the water inrush accident will come out,due to hydromechanical coupling.For example,Carboniferous Permian coal seams in coal mining areas are often threatened by Ordovician strong karst aquifer in the floor.In the process of coal seam mining, the floor rock will successively bear the cyclic loads of loading from advanced abutment pressure,mining unloading and reloading from the backfill in goaf [67].During mining advance in the working face, the floor strata in front of the working face coal wall is compressed by the advanced abutment pressure, which will cause floor strata failure with plastic deformation in certain range once the advanced abutment pressure exceeds the ultimate strength of the floor rock mass.As the mining further advances,the above floor strata in plastic state becomes the floor under the goaf,keeping unloading state due to the roof pressure relief.Once the goaf is filled by the falling gangue or the backfill, conducting the roof pressure to the damaged floor, and then the floor strata is reloaded.Furtherly,the periodic pressures resulting from roof collapses can also lead to cyclic loading and unloading of the floor strata.The floor strata are continuously destroyed under cyclic loading and unloading, resulting in many fractures in floor, and the water conductivity of floor is obviously changed.Eventually, the passage of water inrush may be formed under hydromechanical coupling.

5.Conclusions

To investigate the influence of water pressure on mechanical behaviours of sandstone in the post-peak stage, a series of postpeak cyclic loading and unloading tests under hydromechanical coupling was designed and carried out based on the MTS 815 rock mechanics test system.The strength,deformation and failure characteristics of rock specimens in the post-peak cyclic loading and unloading tests were analyzed in-depth, and some conclusions are drawn as follows:

(1) The upper limit stress σp(i)and crack damage threshold stress σcd(i)gradually decrease with the increasing cycle number in the post-peak region until the residual strength is reached.The relationship between σp(i)/σpand cycle number i can be described by a negative exponential function.

(2) Water pressure significantly weakens the peak strength σp,residual strength σr, and upper limit stress σp(i)in the ith loading cycle.The negative linear correlations of σp/σpkand σr/σreversus P/σ3are experimentally obtained.The nonlinear relationship between effective peak strength and effective minimum principal stress can be generalized by Hoek-Brown strength criterion.

(3) The axial and lateral strains of each loaded sandstone specimen increase as the cycle number increases,while the plasticity also accumulates gradually.The Eloadincreases a little in the initial 2–3 cycles, then decreases with an increase in cycle number.Moreover, the Eunloadis larger than the Eload.

(4) With plastic shear strain γpintroduced, a new post-peak strength model of sandstone related to water pressure and plastic shear strain is proposed.The predictions are in good agreement with the experimental results.

(5) Based on the XRD and SEM scanning tests, the change of mineral compositions, cavity and micro-crack formation and shear cracks propagation in sandstone are observed during post-peak cyclic loading and unloading under hydromechanical coupling.

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Nos.52274118 and 52274145) and the Construction Project of Chenzhou National Sustainable Development Agenda Innovation Demonstration Zone (No.2021sfQ18).