Mechanical behaviour of medium-grained sandstones exposed to differential cyclic loading with distinct loading and unloading rates

Zhengyng Song,Heinz Konietzky,Yunfeng Wu,Kun Du,Xin Ci

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

b Geotechnical Institute,TU Bergakademie Freiberg,Freiberg,09599,Germany

c School of Resources and Safety Engineering,Central South University,Changsha,410083,China

Keywords:Differential cyclic loading (DCL)Rock fatigue Loading/unloading rate Energy dissipation Damage accumulation

ABSTRACT This work aims to investigate the mechanical behaviour of medium-grained sandstones under cyclic loading with different loading/unloading rates.This type of cyclic loading is called differential cyclic loading (DCL) and is considered for testing rock behaviour.In the experiments,constant amplitude and multi-level cyclic loadings were performed.Three loading modes were designed to consider different relationships between loading and unloading rates.Axial strain evolution and energy dissipation were analysed for different loading/unloading rates and maximum cyclic load level.The correlations between P-wave velocities and strengths of rocks deduced from this research are compared with existing published data.The relationships between final strength and axial strain at failure under different loading patterns were also discussed and a rough assessment of the remaining fatigue life is introduced using the predicted value by fitting the axial peak strain.

1.Introduction

Disasters related to rock dynamics occur frequently during the underground excavation process.Events such as rock and coal bursts are often triggered under periodic unloading during tunnelling,mining drift or other underground constructions.Earthquakes (Brown and Hudson,1974; Osinov,2003),rockbursts(Bagde and Petroˇs,2005a;Cai et al.,2021),coal bursts(Linghu et al.,2021; Liu et al.,2021; Zhou et al.,2021),blasting (Yim and Krauthammer,2012),borehole drilling (Wood,2015),hydraulic fracturing (Zang et al.,2013; Ji et al.,2021),and shearer cutting in coal mines(Wang et al.,2015,2021)can produce seismic waves and periodic cyclic stresses.Thus,the rock masses in underground engineering normally experience permanently dynamic and repetitive disturbances (Cai et al.,2020; Song et al.,2022).As a typical discontinuous and highly heterogeneous material,rocks are particularly vulnerable to such type of repetitive stress and fail prematurely under chronic cyclic loading and vibrations(Geranmayeh Vaneghi et al.,2018; Zhao et al.,2019; Liu and Dai,2021; Song et al.,2021a,b; Wang et al.,2021b).This typical failure under cyclic loading is termed ‘rock fatigue’ (Cerfontaine and Collin,2018; Liu et al.,2018).This type of repetitive stress commonly results in loosening and decohesion of the rock grains(Martin and Chandler,1994; Guo et al.,2012).The damage mechanism of fatigue is similar to that of creep to some extent.Both fatigue and creep can be regarded as stress-and time-dependent responses of rock (Atkinson,1984; Cerfontaine and Collin,2018).In the past several decades,investigations on the fatigue behaviour of rocks have been carried out and reported widely.A recent up-todate literature review on rock fatigue is recommended and delivers the fundamental knowledge for understanding rock fatigue (Liu and Dai,2021).Based on these abundant investigations,it is widely accepted that stress level,waveform,stress path,loading frequency and loading pattern have pronounced impacts on the fatigue behaviour of rock(Bagde and Petroˇs,2005a;Erarslan et al.,2014; Ghamgosar and Erarslan,2016; Oneschkow,2016; Zhang et al.,2020).Therefore,the cyclic stress applied in the tests is important and can determine the damage evolution and failure behaviour of rock.Table 1 briefly summarises the publications of laboratory tests on cyclic loading of rock over the last two decades.As shown in Table 1,loading and unloading rates of these tests were always identical.In general,using the same loading and unloading rates can significantly simplify the configuration of stress in laboratory testing,and there is no need to assign different loading and unloading rates when the stress amplitude is varied.However,this simplification does not consider the real and fundamental characteristics of in situ cyclic stress encountered in engineering practices,such as actual seismic waves induced by earthquakes,excavation processes or blasting.

Table 1 Brief summary of typical laboratory cyclic tests on brittle geomaterials over the last two decades.

Fig.1 shows the measured in situ stresses of the Linglong gold mine tunnelling at 350 m below the ground in Shandong Province,China (Hu et al.,2021).The in situ stress fluctuates due to the impact of blasting and excavation advance.As plotted in Fig.1,three patterns of stress variations can be identified: (1) duration of loading phase is longer than that of unloading phase,(2)durations of loading and unloading phases are nearly equal,and(3)duration of loading phase is shorter than that of unloading phase.These pattern leads to different loading/unloading rates.Fig.1 demonstrates that the in situ loading and unloading rates can be either diverse or nearly identical.The simplified cyclic stress in laboratory testing without considering the difference between loading and unloading rates is not able to characterise the actual patterns of in situ stresses in the field.In this study,we have defined that the cyclic stress with distinct loading and unloading rates is termed‘differential cyclic loading’ (DCL).This type of cyclic stress is much closer to the pattern of in situ stress which should be essentially considered in laboratory testing.In practice,geomaterials in nature and engineering are exposed to a complex and random sequence of loading during their life cycles.For example,a simplified seismic stress signal recorded for the Wenchuan earthquake (i.e.Msof 8,occurred in Sichuan Province,China in 2008) is shown in Fig.1c(Chen et al.,2020),which clearly shows that the seismic waves in an earthquake have a cyclic pattern with random features and distinct loading and unloading rates characterised by the diagram of acceleration and time,similar to that of DCL.Investigations on the mechanical responses of rocks subjected to DCL can improve the understanding of fatigue failure mechanisms of rock structures exposed to earthquakes or other dynamic or cyclic loads.

The discontinuous and heterogeneous nature of rock leads to a wide scatter of physico-mechanical properties (Song et al.,2019a,b).This induces difficulties to design testing schemes when multilevel cyclic stress is used.Pre-failure often occurs in cyclic testing due to the inherent defects of the samples.Furthermore,compared with quasi-static monotonic loading,many damage indicators,such as axial residual strain (Xiao et al.,2010) and energy dissipation(Song et al.,2018a;Wang et al.,2021a),evolve in an imperceptible manner,especially when the cyclic stress is far below the shortterm strength.The growth rates of residual strain and cumulative energy dissipation can well characterise the extent of damage accumulation and depend on stress path.The previous investigations have elaborated the relation between damage rate and load level of rocks under cyclic loading with identical loading/unloading rates (Korsunsky et al.,2007; Meng et al.,2016; Jiang et al.,2017; Song et al.,2020; Zhang et al.,2020; Wang et al.,2022).However,due to the lack of related experimental data,features of rock damage evolution are unknown so far for situations with different loading and unloading rates.

This paper documents DCL tests performed on grey mediumgrained sandstones and reveals the fatigue damage of rocks under different loading and unloading rates during cyclic loading.This is of great significance for improving the understanding of fundamental fatigue mechanisms of rock under the regimes of DCL.Multiple cyclic load levels and various DCL schemes were designed and performed.The primary objective of this work is to reveal the relationships between damage accumulation rate and stress path as well as loading/unloading rates.The analyses focus on rock deformation and energy dissipation.

2.Experimental set-up

2.1.Rock sampling and properties

Twenty samples of grey medium-grained sandstones were drilled from a 570 m deep tunnel(Shanxi formation)of Qinshui coal field located in the south of Shanxi Province,China(Fig.2).The rock is quartz sandstone and the surfaces of the samples are intact and smooth.Macroscopic bedding,weak intercalation and pores cannot be observed on the surface.Mineral grains are closely packed with size of 0.05-0.2 mm.According to Fig.3c and d,X-ray diffraction(XRD)results show that the samples have a high content of quartz(87%),kaolinite (6.8%) and calcium carbonate (6.2%),indicating high stiffness and brittleness.The cylindrical samples have dimensions of 100 mm×50 mm(height×diameter).Both ends were polished according to recommendations of the International Society for Rock Mechanics and Rock Engineering (ISRM).Before laboratory testing,the geometry and physical properties of all samples were measured (see Table 2).Fig.3a and b shows the correlations between P-wave velocity and dry density.The P-wave velocity varies between 3700 km/s and 5200 km/s.This large scattering is attributed to the highly inhomogeneous properties of the rock.In the XRD analysis,crushed rock powder was used.

Fig.1.(a) In situ stress measurement devices; (b) Measured in situ stress in Linglong gold mine(Hu et al.,2021):three modes of stress variations are observed,i.e.DCL;and(c) Seismic spectrum of the Wenchuan Earthquake (Chen et al.,2020).

2.2.Testing apparatus and measuring system

Fig.2.(a)Prepared sandstone samples and(b)ultrasonic wave velocity measurement ahead of mechanical tests.

The laboratory tests were performed at the Advanced Testing Center of the Central South University,Changsha,China.Fig.4 shows the complete testing device,the measuring system as well as the thermostatic system to provide a constant temperature.The MTS 322 loading device is used,and it can perform monotonic and high-frequency(up to 50 Hz)cyclic testing.The loading capacity of the device can reach 500 kN and the stiffness is larger than 1370 kN/mm (Fig.4a).The linear variable differential transformer(LVDT) is set at the top of the testing device (Fig.4b).An infrared screening system was used to capture the thermal evolution of the samples during the testing (Fig.4c).The acoustic emission (AE)system is shown in Fig.4d.In this study,two channels were used and two small-scale piezoelectric sensors with full metal housing were attached directly to the surface of rock samples using a thermoplastic glue.Fig.4e illustrates the system to keep the temperature constant at room temperature of 26°C±0.5°C during the testing.Considering the high brittleness of the sandstone samples,the radial strain was not measured because the violent rupture could lead to serious dynamic fragmentation causing damage to the radial gauge.The axial strain was measured through the LVDT fixed at the top of the MTS 322 system (Fig.4b).The data acquisition frequency is 100 Hz(i.e.the time interval between two consecutive data points is 0.01 s).This guarantees robustness and accuracy of the datasets.The reasons for using LVDT located at the loading platen for strain measurement are as follows:

(1) The small-scale LVDT located in the middle part of sample is not used due to the potential serious damage to the LVDT induced by the violent rupture.

(2) The results from strain gauges cannot be synchronized precisely with the stress data,which would result in unrealistic stress-strain relations,including wrong determination of hysteresis loops and energy dissipation.

(3) Our experience(Song et al.,2018a)for working with concrete samples using both techniques suggests that deformation measurements at the loading platen are also reliable,although not perfect.

Due to the end effect and stiffness difference between loading platen and rock sample(stiffness of rigid loading platen is obviously larger than that of rock sample),the axial strain measured by platen may be larger than the values measured by the LVDT or strain gauges located (or glued) at the surfaces of middle part of the samples.This may slightly reduce the values of rock modulus due to the large strain.

2.3.Test schemes

2.3.1.Measurement of uniaxial compressive strength (UCS)

Four samples (G2,G9,G10 and G14) were used to measure the UCS of the rock samples prior to cyclic testing.The monotonic loading rate was set to 60 kN/min and the UCS values for the four samples are shown below(both force and stress at failure are given):

(1) G2: 239.53 kN/122.21 MPa;

(2) G9: 263.71 kN/134.55 MPa;

(3) G10: 166.88 kN/85.14 MPa;

(4) G14: 141.8 kN/72.34 MPa.

It is noticed that the UCS values of the four samples vary widely.In order to make the cyclic loading data robust,we used a reference UCS value of 110 kN(which is smaller than the four measured values)to design the testing procedure and to ensure that all samples can survive during all load cycles without unexpected failure.

2.3.2.Cyclic testing

Three cyclic loading modes have been used in this work.The frequency was set as constant 0.2 Hz regardless of constant amplitude or multi-level cyclic loading.This means that a single cycle has a duration of 5 s.The lowest loading rate in this research was 2.8 MPa/s and the highest can reach 33.62 MPa/s.These are larger than the threshold value (0.05 MPa/s) for static loading proposed by Zhao(2000),but much lower than the dynamic range(Zhao et al.,1999).In addition,the duration for stress-wave propagation along the sample length was around 3 × 10-5s which is much shorter than the loading duration (i.e.1-4 s per cycle),indicating a quasi-static situation.Therefore,the presented cyclic tests can be regarded as in quasi-static states.Fig.5 shows the applied stress waveforms of the three modes and the actual stress data from the MTS 322 loading system.

(1) Constant amplitude cyclic loading

Three samples (G5,G15 and G16) were tested under the constant amplitude cyclic loading (Table 3).Three hundred of cycles were initially performed.If the sample remained stable,the monotonic loading continued until the sample failed.The monotonic loading rate was 60 kN/min.

(2) Multi-level cyclic loading(slight increase of maximum load)

Fig.3.Mechanical properties of rock samples from the insights of ultrasonic measurement and XRD:(a)P-wave velocity along axial direction versus dry density,(b)P-wave velocity along radial direction versus dry density,(c) XRD patterns of rock samples,and (d) components of the rock samples.

Six samples were selected for the multi-level cyclic loading tests with slight increase of maximum load.Table 4 provides the details of testing configuration:G8 and G9 were tested under Mode 1 with the same loading path,while G6 and G13 followed Mode 2,and G1 and G17 followed Mode 3.At each cyclic loading stage (CLS),the minimum load was fixed to 44 kN and the maximum load was enhanced by 5.5 kN(5%of reference UCS)between two successive CLS.The frequency was constant as 0.2 Hz.In each CLS,30 cycles were performed.If the sample could remain stable after all cycles(30 cycles/CLS × 9 CLS=270 cycles),monotonic loading was applied until failure was reached.The monotonic loading rate was 60 kN/min.

(3) Multi-level cyclic loading (rapid increase of maximum load)

Six samples were selected for the multi-level cyclic loading tests with rapid increase of maximum load.Table 5 provides the detailsof testing configuration:G3 and G7 followed the Mode 1 with the same loading path,while G12 and G18 followed the Mode 2,and G11 and G14 followed the Mode 3.In each CLS,the minimum load was fixed to 44 kN and the maximum load was enhanced by 11 kN(10%of reference UCS)between two successive CLS.The frequency is always 0.2 Hz.In each CLS,30 cycles were performed.If the sample could remain stable after all cycles (30 cycles/CLS × 5 CLS=150 cycles),monotonic loading was applied until the sample failed.The monotonic loading rate was 60 kN/min.Fig.6 illustrates the corresponding loading paths of the aforementioned loading schemes.

Table 2 Geometrical and physical properties of the sandstone samples.

Table 3 Cyclic loading tests with constant amplitude.

Table 4 Multi-level cyclic loading tests with slight increase of maximum load.

Table 5 Multi-level cyclic loading tests with rapid increase of maximum load.

3.Results

3.1.Sample deformation

Numerous studies(Ray et al.,1999;Xiao et al.,2010;Song et al.,2020,2021c) have documented that the evolution of axial strain and its increasing rate are suitable indicators to characterise the damage accumulation of rocks and other brittle materials subjected to cyclic loading.The evolution of axial strain corresponding to the maximum and minimum loads can be used to characterise the damage intensity.Here we explain the procedure to calculate the fitted growth slope of axial strain.For each CLS,30 cycles were performed under the same load scheme.Then,in each CLS,we used linear relations to fit the axial strains at the maximum(termed the‘peak strain’in Fig.7a)and minimum stresses(termed the‘residual strain’ in Fig.7b) versus the number of cycles,e.g.axial strains for 1-30 cycles are linearly fitted and the linear slope is designated as‘fitted slope’.Then,this procedure was repeated for 31-60 cycles in the 2nd CLS.The fitted slope is a parameter to characterise the increasing rate of strain evolution.The calculations of fitted slopes for residual peak and axial strains are illustrated in Fig.7a and b,respectively.Normally,the slope for the axial strain in the first CLS exhibits a much larger value due to the compaction effect.Therefore,the first CLS was excluded from the evaluation.

3.1.1.Constant amplitude cyclic loading

G16,G5 and G15 were cyclically loaded with a constant stress amplitude (44-88 kN).The deformation behaviour is analysed for the first 150 cycles.We divided the 150 cycles into five groups(i.e.1-30 cycles,31-60 cycles,61-90 cycles,91-120 cycles,and 121-150 cycles).

Fig.4.(a) Overview of testing system,(b) central part (loading frame) of MTS 322 test system,(c) LVDT of MTS 322 test system,(d) infrared temperature screening systems,(e)system to keep temperature constant,(f) sample with AE sensors,(g) piezoelectric AE sensors with full metal housing,and (h) AE amplifier.

Fig.5.Illustration of three mode of cyclic loadings:(a)Mode 1:loading rate is four times the unloading rate; (b) Mode 2: loading rate is equal to unloading rate; and(c) Mode 3:unloading rate is four times the loading rate.

Fig.7c and d presents the curves for the fitted slope of axial peak and residual strains,respectively.Both subfigures clearly show that the increasing rates of axial strain decrease with the on-going cyclic loadings.All three modes (Modes 1,2 and 3) show a nonlinear decrease versus number of cycles.It is observed that the fitted slope becomes nearly constant after 120 cycles for all three loading modes (elastic behaviour).The curves of the three loading modes are different for the first 30 cycles,while they reach nearly identical values after 120 cycles.The results in Fig.7 illustrate that the difference between loading and unloading rates are limited and the effect on the evolution of damage is not pronounced in terms of the axial growth rate.

3.1.2.Multi-level cyclic loading

As described in Section 2.3.2,two groups of multi-level cyclic tests were performed.The detailed cyclic stress paths are summarised in Tables 4 and 5.The samples G3,G7,G12,G18,G11 and G14 are cyclically loaded and the incremental amplitude between two consecutive CLS is 11 kN.This amplitude is two times the value applied to the other group(5.5 kN),which includes the samples G8,G19,G6,G13,G1 and G17.

(1) Rapid increase of maximum cyclic load

For the group with rapid increase of maximum cyclic load,each sample was tested for 30 cycles in each CLS and five CLSs were performed in total.We analysed 150 cycles and the results are presented in Fig.8.The left,middle and right panels of Fig.8 show the results of Modes 1,2 and 3,respectively.The evolution of axial peak strain shows an obvious jump with the increase of maximum cyclic stress.For the residual strain,however,the increase of maximum cyclic stress does not lead to a clear jump.Tables 6 and 7 summarise the fitted slopes for axial strain evolution for all CLSs.

Fig.6.Three loading schemes: (a) Constant amplitude cyclic loading,(b) multi-level cyclic loading with slightly increased maximum load with an amplitude of 5.5 kN,and (c)multi-level cyclic loading with rapid increase of maximum load with an amplitude of 11 kN (10% of reference UCS).

According to the note of Table 5,sample G7 has experienced two cyclic loading with the same loading path due to the unexpected power outage.This led to a different evolution pattern of G7 in Fig.9.For other samples,the first CLS shows a larger fitted slope for axial strain which is attributed to the compaction effect as commonly reported in the literature (Song et al.,2018a,2020).However,for G7,this behaviour is missing.This clearly demonstrates that the initial cyclic loading can eliminate the compaction effect.This also signifies that the compaction effect during the first CLS is reversible and the sample deformation should be treated as plastic(damage).Due to this special loading path,it is reasonable to ignore the results of G7 and compare only that of other five samples.It is difficult to extract the differences or specific characteristics of the fitted slopes between the three loading modes.All the fitted slopes exhibit a drop at the transition from the first to the second CLS.Only G3 shows an increasing fitted slope versus maximum load,while the strain growth rates for other samples slightly fluctuate.This is also not consistent with our former test results of concrete,where an exponential relation was reported between strain parameter and maximum stress(Song et al.,2018a).This discrepancy may be caused by the applied load level and the material strength.In our former testing (Song et al.,2018a),the applied load is larger than 70% of the UCS.In this work,the cyclic load is much lower.Due to the obvious scattering of mechanical properties of the sandstone samples,the applied cyclic load is maintained below 50% of the UCS to obtain a delicate process of mechanical responses under gentle cyclic load level (the load is in elastic state).Therefore,the low cyclic loading stress could make the evolution of the fitted slope of axial strain rate not sensitive to the increase of the maximum cyclic stress.In summary,the fitted slope during axial strain evolution is independent of loading/unloading rate when the applied cyclic stress is moderate (in this work the load level is below 50% of the UCS).

(2) Slight increase of maximum cyclic load

The evolution of the axial strain versus number of cycles for the group of samples with slight increase of the maximum cyclic load is illustrated in Fig.10.Inthe tests,nine CLSs wereadopted(270cycles),instead of five CLSs(150 cycles)shown in Fig.8.The maximum cyclic loads for the first and final CLSs are identical for the two groups.The main difference is the increasing load amplitude between two consecutive CLS(11 kN or 5.5 kN).Similar to Fig.8,the evolution of axial peak strain also exhibits a staggered pattern with increasing maximum load.The residual strain evolves approximately linearly,which is slightly different from the pattern shown in Fig.8.Specifically,the evolution of the residual strain for different CLSs (see Fig.10) are continuous and connected with each other without visible jumps.This demonstrates that the slight increase of maximum cyclic load(i.e.5.5 kN)leads to alinear increase of residual strainwith increasing maximum cyclic stress.However,quantitative relationships cannot be derived from Fig.10.The fitted slopes of axial strain evolution for each CLS are listed in Tables 8 and 9.

Table 6 Fitted slopes of axial peak strain in different CLSs for groups with rapid increase of the maximum cyclic stress.

Table 7 Fitted slopes of axial residual strain in different CLSs for groups with rapid increase of the maximum cyclic stress.

Table 8 Fitted slopes of axial peak strain in different CLSs for groups with slight increase of the maximum cyclic stress.

Table 9 Fitted slopes of axial residual strain in different CLSs for groups with slight increase of the maximum cyclic stress.

Fig.7.Illustration of calculating the fitted growth slope of axial strain:Fitted slope of axial(a)peak and(b)residual strains,3rd CLS for G15 is shown as an example;Fitted slopes of(c) peak and (d) residual strains versus number of cycles for G16,G5 and G15.

Fig.11 shows the fitted slope for the evolution of axial strain versus the CLS together with the load level.The effect of loading/unloading rates on the evolution of axial strain is not obvious when exposed to slightly increased cyclic loads.It should be noted that,although the loading and unloading rates are different for the three modes as shown in Fig.11,load level and path are identical for each CLS.The similar tendency of the fitted slopes for all six samples indicates that the loading path (variation of the load levels) dominates the evolution patterns of axial strain rather than the variations of loading/unloading rates.This is similar to the conclusions drawn from Fig.9.In summary,regardless of the different amplitudes of stress increase,the evolution of the fitted slopes for axial strain can be considered as independent of the loading/unloading rates.In case of slight increase of maximum stress,the fitted slope for strain after the first CLS remains constant due to the elastic behaviour of rock.

Fig.8.Axial strain evolution versus number of cycles for the groups with rapid increase of maximum cyclic stress:Axial(a)peak and(d)residual strains for G3,axial(b)peak and(e)residual strains for G12,axial(c)peak and(f)residual strains for G4,axial(g)peak and(j)residual strains for G7,axial(h)peak and(k)residual strains for G18,and axial(i)peak and (l) residual strains for G11.

(3) Comprehensive comparison of the two groups of tests

Fig.12 shows the fitted slopes for axial strain versus maximum cyclic load for 12 samples exposed to multi-level cyclic loading.The black symbols represent the samples subjected to the rapid increase of maximum load,while the red symbols represent samples with slight increase of maximum load.As shown in Fig.12a and b,the data of the first CLS (compaction process) show larger magnitude and scattering than those in the other group.However,after the first CLS,the datasets for both groups exhibit different trends.The fitted slopes of axial strain forthe groups with rapid loading increase(greyareawith black symbols) are apparently larger than that of the other group(red area with red symbols) for both the axial peak and residual strains.An exception is for G7 due to the special loading path(two times of cyclic loading,see purple curve in Fig.12).Fig.12 also shows that a slight and gradual increase of maximum stress will reduce the damage accumulation rate which is characterised by the values of fitted slopes of axial strain.Furthermore,the first loading stage shows much larger values of fitted slope.

Except for the fitted slope of axial strain,many other strainrelated parameters can reflect the mechanical behaviour of rocks exposed to different cyclic loading patterns.Fig.13 a-d shows the axial peak and residual strains at the first and final cycles based on 15 cyclically loaded samples.G5,G15 and G16 are subjected to the constant amplitude cyclic loading and experience a higher load level (88 kN) than the other 12 samples under multi-level cyclic loading(66 kN)in the first CLS.Therefore,the data in Fig.13a and c shows that the axial peak and residual strains for G5,G15 and G16 are larger than those of others.For the final cycle,as shown in Fig.13b and d,scattering of data is obvious which was induced by the difference in loading paths as well as the scatter of rock physical properties.Therefore,the strains are plotted as the strains at the final cycle minus that at the first cycle (Fig.13e and f).It becomesclear that the loading paths have a prominent effect on the compression extent (characterised by axial strain) of the rock samples during testing(Fig.13).The samples G5,G15 and G16 show smaller axial peak strains (Fig.13e).This demonstrates that a stepwise increase of maximum stress could significantly increase the axial strain when compared to a constant cyclic load level.Same behaviour is observed in Fig.13f concerning the axial residual strain.It is difficult to reveal the detailed laws behind the evolution of axial strain based on the information provided in Fig.13.Because of the scattering of properties and complex loading paths,it is hard to draw convincing conclusions on the evolution of the axial strain.Whether the evolution of axial strain is stress path-dependent or not will be further analysed and discussed by considering the relation of axial strain for first and final cycles.

Fig.9.Fitted slopes of axial strain and maximum cyclic load versus CLS for the samples with rapid increase of the maximum cyclic stress: Fitted slope for axial (a) peak and (b)residual strains.

Fig.10.Evolution of axial strain versus number of cycles for the groups with slight increase of maximum cyclic stress:Axial(a)peak and(d)residual strains for G8,axial(b)peak and(e)residual strains for G6,axial(c)peak and(f)residual strains for G1,axial(g)peak and(j)residual strains for G19,axial(h)peak and(k)residual strains for G13,and axial(i)peak and (l) residual strains for G17.

In this study,a new parameter termed‘strain ratio’was defined as the ratio of the axial strain of the final cycle to that of the first cycle.Fig.14a and b shows the strain ratio for the axial peak and residual strains.The stepwise increase of maximum cyclic stress by multi-level cyclic loading results in a larger axial peak strain ratio than that by constant amplitude cyclic loading(Fig.14a).However,for the axial residual strain ratio,the disparities between the two loading regimes are reduced.The axial residual strain ratios of the samples subjected to the multi-level cyclic loading are larger than those in constant amplitude cyclic loading (Fig.14b).Fig.14c and d shows the correlations between the axial strains of the first and final cycles.Fig.14c shows that the datasets of the constant amplitude cyclic loading and multi-level cyclic loading can be linearly fitted,whereas a gap is observed between the two fitting lines.This gap is induced by different loading patterns (constant amplitude and multi-level cyclic loading).For the multi-level cyclic loading,although loading/unloading rates and increase of stress amplitude between two consecutive CLS are not identical,the datasets for all samples can be well fitted.A similar behaviour is observed for the axial residual strain as shown in Fig.14d.The gap of fitting lines in Fig.14d is smaller than that in Fig.14c,indicating that the stepwise increase of maximum cyclic stress also has some effects on the residual strain,but the effect is not that significant as for the peak strain.The correlations between axial strains for the first and final cycles show a linear relationship (Fig.14).Furthermore,the datasets belonging to the constant and multi-level cyclic loadings exhibit a gap which is induced by the difference in loading paths.For the 12 samples subjected to multi-level cyclic loading,although the detailed loading paths and number of cycles are not identical,the load levels for the first and final CLSs are the same.The maximum cyclic load of the first and final CLSs have more pronounced effect on the correlation of the first and final axial strains than the loading/unloading rates or number of cycles.This is quite significant for understanding and predicting the constitutive relation of rocks under cyclic loading regardless of the loading/unloading rates.The correlation between axial strains of the first and final cycles is dependent on the cyclic load level,but independent of the loading/unloading rate.

Fig.11.Fitted slopes for axial strain versus CLS for the groups with slight increase of the maximum cyclic stress:Axial(a)peak and(b)residual strains(G8 and G9 belong to Mode 1;G6 and G13 belong to Mode 2; G1 and G17 belong to Mode 3).

3.2.Energy dissipation

Energy dissipation or energy dissipation density in cyclic loading is normally defined as the value of area of hysteresis loop(the area enclosed by the loading and unloading stress-strain curves).The hysteresis loop is related to the energy exchange during the whole cyclic loading process,including heat transfer,fracture propagation,and inherent damage accumulation.Many investigations reported that the dissipated energy density in the cyclic loading can be used as an indicator to characterise the rate of damage accumulation (Song et al.,2018a; Zheng et al.,2020;Liu and Dai,2021).When the cyclic stress increases,the dissipated energy density rises correspondingly.The reader is referred to Song et al.(2018b) and Zhang et al.(2020) for detailed description of the method to calculate the dissipated energy density.Similarly,in cyclic shear tests,the dissipated energy is generally used as an indicator of frictional energy losses (Kasyap et al.,2021).The energy dissipation is found to be proportional to the slope of hysteresis loops.A higher energy dissipation always occurs under the maximum tangential load and corresponds to serious damage as characterised by a larger shear displacement.Kasyap et al.(2021) stated that the periodic opening and closure of macroscopic cracks in rock allow a large amount of energy to be dissipated,which leads to much larger energy dissipation and lower stiffness in crack-rich rock than the case with fewer cracks and lower stress amplitude.For the tests with smaller shear amplitudes,the microcracks may be initiated and propagate in the weak cementation of the samples,such as the fractured mineral grains.

3.2.1.Constant amplitude cyclic loading

Fig.15a and b shows the dissipated energy density for each counted cycle and the accumulated dissipated energy density.For all three loading modes,the dissipated energy density for the first several cycles is obviously larger than that in the rest of the loading cycles,and then it remains almost constant after approximately 10 cycles.In order to quantify the relationship between the dissipated energy density and number of cycles,the values of dissipated energy for 30 cycles in each CLS(1-30,31-60,…,121-150 cycles) were averaged,as shown in Fig.15c.The results in Fig.15c are different from those presented in Fig.7.The values of average dissipated energy after 30 cycles are approximately the same(Fig.15c).Whereas for the axial strain(Fig.7),the fitted slopes for peak and residual strains experience a consistent drop from the 1st to the 150th cycle.Based on these results,it is concluded that the three modes enter the constant dissipated energy stage at almost the same time (number of cycles).This indicates that the loading/unloading rates have a negligible influence on the evolution of energy dissipation when the stage of constant energy is reached.

3.2.2.Multi-level cyclic loading

Fig.16a and b presents the dissipated energy density of each counted cycle and the accumulated dissipated energy for the six samples tested by rapid increase of maximum stress.All five CLSs and 150 cycles are analysed.The corresponding results of the group with slight increase of maximum stress are presented in Fig.16c and d.A clear difference is observed by comparing Fig.16a and c that large values of dissipated energy appear in the first cycle of a new CLS with the stepwise increase of the maximum load in Fig.16a,while no such sudden increase is observed in Fig.16c.This specific difference is caused by the increasing amplitude between two consecutive CLS.The rapid increase(11 kN)of maximum load(Fig.16a) leads to more obviously enhanced dissipated energy in the first cycle of a new CLS when compared to samples under 5.5 kN incremental of stress(Fig.16c).This also explains the different rise of the curves (Fig.16b and d).The high energy dissipation for the first cycle of each CLS(refer to the area enclosed by the blue dashed lines) indicates violent energy exchange due to rapid stress increase.The stress levels for the first and final CLSs are identical in Fig.16a and c.With the rapid stress increase,large values of energy dissipation exist in all loading cycles (Fig.16a).This demonstrates that the slow increase of maximum cyclic load can be beneficial to reduce the energy dissipation (damage).All three loading modes confirm that the effect of stress amplitude on energy dissipation is loading/unloading rate-independent.In engineering practices,a slight stepwise increase of load is recommended to reduce the damage accumulation rate.

Fig.12.Fitted slopes for axial strain versus maximum cyclic load levels for 12 samples exposed to multi-level cyclic loading: (a) Peak and (b) residual strains.

Fig.13.Axial peak strains of (a) the first and (b) final cycles,axial residual strains of (c) the first and (d) final cycle,axial peak strain difference with growth of (e) peak and (f)residual strains.

The relationships between energy density dissipation and cyclic load level for 12 samples under multi-level cyclic loading are illustrated in Figs.17 and 18.The dissipated energy of 30 cycles in each CLS is averaged to represent the energy dissipation during one CLS.The averaged dissipated energy for the 2nd CLS is larger or approximately equal to that in the 1st CLS for all three loading modes (Fig.17).This indicates that the relationship between the averaged dissipated energy for each CLS and the maximum cyclic load is possibly positive correlated.However,Fig.18 shows a different relationship for the 1st-3rd CLS.The averaged dissipated energy in the 1st CLS is larger than that in the 2nd and 3rd CLS.This behaviour is due to the smaller amplitude of stress increase between the consecutive CLS.This signifies that a slight increase(amplitude of 5.5 kN)of maximum stress during cyclic loading can significantly reduce the level of energy dissipation for the 2nd CLS and slow down the accumulating rate of energy dissipation for all the remaining CLSs.When the data of the first CLS are excluded,it can be observed that regardless of the variations of loading/unloading rates and amplitude of stress increase,all samples show an exponential relationship with the maximum cyclic load.

Fig.19 shows the datasets of 12 samples under two multi-level cyclic loadings.Similar to Fig.12,the black and red symbols represent the tests using rapid and slow increase of maximum stresses,respectively.Smaller value of energy dissipation indicates a lower damage accumulation rate.A decreasing amplitude of stress rise can reduce the damage accumulation when the paths of load level(initial and final levels) are fixed.

Fig.14.Axial (a) peak and (b) residual strain ratios; Correlations between axial (c) peak and (d) residual strains of the first and final cycles.

It can be seen that the effect of loading/unloading rate on the energy dissipation during multi-level cyclic loading is negligible when the loading path is fixed.The amount of energy dissipation depends on the load level and path during the multi-level compressive cyclic loading.The evolution of energy dissipation under different loading/unloading rates and loading paths is highly consistent.The scattering of rock properties can lead to a slightly different energy dissipation due to the variations of stiffness,strength,micro-structure,and other strain-related parameters.As illustrated by the correlations in Fig.14,the axial strains of the first and final cycles are linearly related.However,whether the accumulated dissipated energy is related to the amount of energy consumption for the first cycle is still unclear.Fig.20 shows the correlations between dissipated energy for the first cycle and the total dissipated energy from the first to the final cycle.It is clear that the fittings of the three loading modes lead to three separate data clusters.This is distinct from the results of axial strains shown in Fig.14.In Fig.14,the correlations between axial strains for the 1st and final cycles are consistent for both groups of multi-level cyclic loadings.The three datasets are plotted in terms of dissipated energy (see Fig.20).This demonstrates that the impacts of loading path on the axial strain evolution and energy dissipation are different.The relationship between the axial strain for the first and final cycles depends on the load level.Once the load levels for the first and final CLSs are fixed,the relationships between the axial strains for the first and final cycles can be determined.For energy dissipation,the loading patterns(constant amplitude or multi-level cyclic loading) and the amplitude of stress increase influence the energy dissipation of the first cycle and total energy dissipation of all cycles.According to Fig.20,the loading/unloading rates do not have obvious impacts on the results.Once the load level and path are finalized,the total energy dissipation can be predicted based on the relation presented in Fig.19.The prediction method does not need to consider the scattering of physical properties for different rock samples.

4.Discussion and implications

The unconfined strength of rock (also known as UCS) is one of the essential parameters of rock mechanics in civil engineering,mining,and tunnelling (Jahed Armaghani et al.,2016).Generally,this parameter is measured by considering the physical and mineralogical properties of the rock.Therefore,a proper and reasonable method of testing the unconfined strength of rocks is essential and necessary.Table 10 reviews works in the literature and lists relevant researches concerning the correlations between rock strength and P-wave velocity.It is known that the P-wave velocity is widely used to predict the unconfined strength of rock.The researches listed in Table 10 are of diverse types,ranging from soft to brittle hard rock.Table 10 refers to the unconfined strengths of rock under monotonic loading.Here we present the results showing relations between final strengths (monotonic loading is performed after cyclic loading) of the sandstone and the P-wave velocities measured prior to the tests (Fig.21).P-wave velocities were measured in radial and axial directions.According to the coefficient of determination (R2),it is found that the P-wave velocity along the radial direction is closely related to the final strength of the rock.According to Table 10,the relation between UCS and P-wave velocity can be either linear or nonlinear.We use linear and exponential functions to fit our testing results.Exponential fitting provides a better result.The fitting results are also listed in Table 11.This demonstrates that using the P-wave velocity to predict the unconfined strength of rocks under cyclic loading is reliable.

The relationships between final strength of sandstone samples and axial strain up to the failure state are shown in Fig.22.The axial strains at failure shown in Fig.22 are measured at the peak load for all samples regardless of the loading regimes.In detail,for samples under monotonic loading only,the failure strain is referred to as the axial strain at the peak load during the monotonic loading.For the samples under both cyclic and monotonic loadings,the axial strain at failure is referred to as the axial strain measured at the peak load in the final monotonic loading phase.It indicates that samples exposed to monotonic loading show a different pattern compared to that subjected to multi-level cyclic loading with monotonic loading afterwards.Similar to Fig.14,a gap is also observed due to the different loading types.The results in Fig.22 indicate that samples under monotonic loading have larger axial strain at failure than the one under multi-level cyclic loading plus subsequent monotonic loading.This demonstrates that the effect of multi-level cyclic loading can reduce the evolution of axial strain prior to the application of monotonic stress up to the failure.The results in Fig.22 show that the final strength after cyclic loading is almost the same as those under monotonic loading only.This is possibly due to the cyclic load levels in this work are too low to induce obvious damage.The load level was chosen to guarantee that all samples remain in the elastic to quasi-elastic state,so as to have reliable comparisons between deformation and energy dissipation among different samples.

For further evaluation,Fig.22 summarises the axial strain at failure,as well as the fitted slopes of axial peak strain for different CLSs (Tables 6 and 8) under multi-level cyclic loading.After experiencing all CLSs,the samples are assumed to experience a cyclic loading up to the failure by applying the stress level of the final CLS(44-110 kN).The evolution of axial peak strain is assumed to followa linear pattern until reaching the failure state and the value of axial strain at failure induced bycyclic loading is the same as the final axial strain obtained in Fig.22.Therefore,the remaining cycles(remaining fatigue life)of the samples exposed to the stress level of final CLS can be calculated as the ratio of the axial strain difference (axial strain at failure minus the axial peak strain for the last cycle of final CLS) to the fitted slope of peak strain for the last CLS.The results are shown in Table 11.Fig.23 shows the predicted remaining fatigue life versus the final strength at failure for the two groups of multi-level cyclic loading.A good linear relation is observed which is highly consistent with the classic S-N (stress level and number of cycles up to failure)curves in fatigue life prediction.This verifies that using the method described above can well reproduce linear S-N curves.

Table 10 Correlations between UCS and P-wave velocity in the literature.

Table 11 Prediction of remaining fatigue life based on the fitted slope of axial peak strain.

Fig.15.Energy dissipation for G16,G5 and G15: (a) Evolution of dissipated energy density,(b) evolution of accumulated dissipated energy density,and (c) averaged dissipated energy at different cycles.

Fig.16.Energy dissipation for multi-level cyclic loading: (a) Dissipated energy density versus number of cycles for the group using rapid increasing stress,(b) accumulated dissipated energy density versus number of cycles for the group using rapid increasing stress,(c) dissipated energy density versus number of cycles for the group using slowly increasing stress,and (d) accumulated dissipated energy density versus number of cycles for the group using slowly increasing stress.

Fig.17.Averaged energy dissipation of each CLS versus maximum cyclic load for the group with rapid increasing maximum stress:(a)G3 and(d)G7 in Mode 1;(b)G12 and(e)G18 in Mode 2; and (c) G4 and (f) G11 in Mode 3.

Fig.18.Averaged energy dissipation of each CLS versus maximum cyclic load for the group of slightly increased maximum stress:(a)G8 and(d)G19 in Mode 1;(b)G6 and(e)G13 in Mode 2; and (c) G1 and (f) G17 in Mode 3.

Investigations on DCL can be further extended to shear behaviour of rocks and soils concerning tectonic movements,earthquakes,and geological features (Dang et al.,2018; Kasyap and Senetakis,2020,2021).Many articles found that cyclic shear behaviour and energy dissipation for rocks and gouge materials are loading rate-dependent (Jafari et al.,2003; Dang et al.,2016,2021,2022; Kasyap and Senetakis,2020),and the geomaterials with lower stiffness usually show a larger energy loss during cyclic shear loading.Different water contents will also affect the cyclic shear behaviour.The seismic waves induced by the earthquakes could also have distinct loading/unloading rates,i.e.differential shear cyclic loading.Further investigations in differential cyclic shear testing could be beneficial to the understanding of the failure mechanism exposed to earthquake loading.

Fig.19.Averaged dissipated energy versus maximum cyclic load levels for all 12 samples under multi-level cyclic loading.

Fig.20.Correlations between total dissipated energy density and dissipated energy density of the first cycle.

Fig.21.The final strengths of sandstone samples with P-wave velocities along(a)axial and(b)radial directions by linear fitting and with that along(c)axial and(d)radial directions by exponential fitting.

5.Conclusions

Fig.22.Final strength of sandstone samples versus axial strain up to failure under different loading modes.

Fig.23.Predicted remaining fatigue life versus the final strength at failure.

This work reported cyclic testing on sandstone samples by considering distinct loading and unloading rates (DCL).Different loading modes and paths are designed and performed.The conclusions of the study are drawn as follows:

(1) For the specific sandstone and applied load levels,it can be concluded that the evolution of axial strain and energy dissipation for the sandstones in this work is loading/unloading rate-independent.

(2) The dissipated energy is a more sensitive and delicate indicator than the axial strain during multi-level cyclic loading.Between the average dissipated energy for different loading stages and the maximum load level,an exponential relationship was found.Samples without former loading history show an obvious compaction effect which corresponds to a higher growth rate of axial strain at the first loading stage.The compaction effect will disappear for samples with a loading history.The compaction is of plastic nature(inelastic).

(3) Slight increase of the maximum cyclic load stepwise will obviously reduce the accumulation rate in terms of axial strain evolution and energy dissipation when load level and loading path are fixed.Axial strain in first and final cycles exhibits a good linear evolution trend.This relation is load level-dependent and loading/unloading rate-independent.The relationship between the dissipated energy in the first cycle and the total dissipated energy of all cycles also follows a linear law.This relation depends on the load level,but is independent of the loading/unloading rate.

(4) Using P-wave velocity to predict the strength of rock samples under monotonic loading after initial cyclic loading is still reliable.The results of P-wave velocity along the radial direction show a better consistency with rock strength than that along the axial direction.Samples experiencing only the monotonic loading have a larger axial strain at failure than those under multi-level cyclic loading and subsequent monotonic loading.The remaining fatigue life can be approximately predicted using the fitted slope of axial strain and the axial strain at failure.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This article is funded by China Postdoctoral Science Foundation(Grant No.2021M700012)and the Fundamental Research Funds for the Central Universities (Grant No.06500182),Funds from Joint National-Local Engineering Research Center for Safe and Precise Coal Mining (Grant No.EC2021004),the authors sincerely appreciate the assistance from Drs.Yichao Rui and Qingxin Chen in Central South University during the laboratory testing.