Intermittent disturbance mechanical behavior and fractionaldeterioration mechanical model of rock under complex true triaxial stress paths

Zhi Zheng *,Hongyu Xu ,Ki Zhng ,Gungling Feng ,Qing Zhng ,Yufei Zho

a State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures,Key Laboratory of Disaster Prevention and Structural Safety of Ministry of Education,College of Civil Engineering and Architecture,Guangxi University,Nanning 530004,China

b State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences,Wuhan 430071,China

c State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin,China Institute of Water Resources and Hydropower Research,Beijing 100080,China

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

Keywords:True triaxial static and disturbance test Mechanical properties Failure mechanism and precursor Intermittent disturbance effect Fractional mechanical model

ABSTRACT Mechanical excavation,blasting,adjacent rockburst and fracture slip that occur during mining excavation impose dynamic loads on the rock mass,leading to further fracture of damaged surrounding rock in three-dimensional high-stress and even causing disasters.Therefore,a novel complex true triaxial static-dynamic combined loading method reflecting underground excavation damage and then frequent intermittent disturbance failure is proposed.True triaxial static compression and intermittent disturbance tests are carried out on monzogabbro.The effects of intermediate principal stress and amplitude on the strength characteristics,deformation characteristics,failure characteristics,and precursors of monzogabbro are analyzed,intermediate principal stress and amplitude increase monzogabbro strength and tensile fracture mechanism.Rapid increases in microseismic parameters during rock loading can be precursors for intermittent rock disturbance.Based on the experimental result,the new damage fractional elements and method with considering crack initiation stress and crack unstable stress as initiation and acceleration condition of intermittent disturbance irreversible deformation are proposed.A novel three-dimensional disturbance fractional deterioration model considering the intermediate principal stress effect and intermittent disturbance damage effect is established,and the model predicted results align well with the experimental results.The sensitivity of stress states and model parameters is further explored,and the intermittent disturbance behaviors at different f are predicted.This study provides valuable theoretical bases for the stability analysis of deep mining engineering under dynamic loads.

1.Introduction

With the large-scale exploitation and exhaustion of shallow mineral resources,the utilization of deep mineral resources has gradually become the prevailing trend.The construction of large deep mines and other deep underground engineering has been steadily increasing on an annual basis [1,2].Deep rock mass engineering is excavated under the three-dimensional stress state(σ1＞σ2＞σ3,where σ1,σ2and σ3are the maximum,intermediate and minimum principal stress respectively),resulting in local geometry changes of the surrounding rock,continuous 3D stress redistribution,unloading,concentration,rotation,and other load changes,leading to significant static damage [3–5].Mechanical excavation,blasting,adjacent rockburst and fracture slip occur during coal mine excavation,resulting in dynamic loads on coal seams and rock masses (Fig.1a).For instance,shock waves generated by blasting or earthquakes propagate through the surrounding rocks,attenuate,and create low-amplitude,low-frequency micro-dynamic disturbances that impact the rocks(Fig.1b),resulting in partial failure of coal pillars and surrounding rocks and even serious engineering disasters[6].However,most of the mechanical models used as references for rock engineering excavation design and stability analysis are established based on the results of uniaxial or triaxial static tests.To conduct more comprehensive research on rocks within their corresponding environments and to better understand their mechanical properties,it is necessary to study the intermittent disturbance behavior and establish a suitable mechanical model specifically for rocks under true triaxial stress.Such research would provide valuable reference for the stability analysis of deep mineral resources development.

The study of rock mechanical properties based on uniaxial and conventional triaxial static tests has accumulated a large number of research results and achieved a more comprehensive and systematic understanding [7,8] and is not repeated here.To reasonably reflect the 3D stress state of deep rock engineering,researchers have carried out a series of true triaxial tests to study various mechanical properties of rocks and have made more indepth progress in rock strength characteristics,deformation and failure characteristics,post-peak characteristics,and rockburst[9–12].In the study of dynamic testing of rock,Attewell and Farmer [13] carried out many fatigue tests on dolomite to verify that the accumulated strain energy inside rock can destroy beyond the critical value.Bagde and Petroš [14] showed that microcracks are the main cause of fatigue damage and the stress’s loading rate affects sandstone’s dynamic strength under uniaxial dynamic cycling tests.To analyze the failure characteristics of rock under intermittent disturbance,Shi et al.[15] conducted uniaxial intermittent cyclic disturbance tests and performed comparative studies by the digital image correlation (DIC) system and acoustic emission(AE)system.Most fatigue tests have high amplitude characteristics,which differs from the loading state of deep engineering rock masses [16,17].In addition,most of the dynamic rockburst tests [18,19],high strain rate impact tests [20],and disturbance tests [21] are conducted without considering excavation-induced stress states of rock mass in practical engineering.For example,researchers studied the dynamic energy partition and dissipation of hydrostatically confined rocks subjected to high loading rate impacting through an improved split Hopkinson pressure bar(SHPB).The static-dynamic coupling test system studied the comprehensive effects of water saturation and static prestress on the dynamic and mechanical behavior of sandstone [22].Currently,most experiments are not suitable for characterizing the true three-dimensional stress state,stress paths,and dynamic-static combinations of excavation damage and intermittent disturbance damage of engineering surrounding rock [23].Therefore,further research is needed on the mechanical properties and fracture evolution mechanisms of deep engineering rock subjected to true triaxial intermittent disturbance.However,recent research findings have demonstrated the accuracy of the microseismic monitoring system for rock failure early warning in practical engineering applications.For instance,researchers have utilized the microseismic monitoring system to identify different stages of rock failure[24],enabling the implementation of preventive measures in advance.These studies highlight the significance of rock failure early warning and offer new research perspectives on the mechanical properties and fracture evolution mechanisms of deep engineering rock.

Mechanical properties and failure mechanism are the core of rock mechanics research,and the constitutive model that can correctly reflect the properties of rock has been of significant concern in the field of rock mechanics and engineering research.Numerous rock constitutive models have been established based on uniaxial and conventional triaxial tests [25–27],which play a crucial role in surface and shallow rock engineering in the design,construction and post-monitoring stages.The previous researchers have found that the deformation characteristics of rock under low strain rate cyclic loading are similar to creep characteristics[28],and all have three stages of attenuation,steady-state and accelerating during failure process,so the micro-dynamic disturbance model of rocks can be constructed based on the theory of creep mechanic model.The creep constitutive models of rocksare generally empirical models,component combination models and mechanism-based creep constitutive models[29,30].The element combination model consists of components such as Newtonian,Kelvin and Maxwell bodies,which can better express the physical meaning with precise parameters and simple application,but it cannot reflect the deformation characteristics of the nonlinear accelerating stage better.To further simulate the accelerating stage,research scholars combined the damage with the accelerating stage and achieved good results [31,32].In addition,the use of the fractional creep model provides a better reflection of the accelerating stage.Rogers [33]and Bagley and Torvik [34] extended fractional calculus theory further,laying a foundation for subsequent model establishment.Adolfsson et al.[35] proposed a viscoelastic fractional model,which can better characterize the stress relaxation and creep phenomena and satisfy the Clausius-Durham inequality(CDI).Muliana[36]developed a viscoelastic fractional model based on the separation of functions of nonlinear elastic strain measurements and applied to the behavior of homogeneous material.By introducing fractional derivatives,the rock creep model established by Zhang et al.[37] and Qu et al.[38] has good simulation of the full creep stage of rock under uniaxial or conventional triaxial compression,which indicates that the fractional model may have good applicability for studying the dynamic disturbance behavior of rock.Wu et al.[39]used the short memory method to divide the entire creep process into three memory stages and established a new rock fraction variable-order creep model.Based on statistical microscopic damage mechanics,Yao et al.[40] proposed a coupled damage mechanical model to simulate the stress–strain characteristics of rock disturbance under conventional triaxial tests.Zhu et al.[41]developed a damage constitutive model for rock stress relaxation and dynamic disturbance based on the uniaxial dynamic disturbance test and achieved good simulation results.Most of the above-mentioned models are based on the understanding of rock under uniaxial and conventional triaxial static or disturbance tests,without considering and suitable for characterizing the intermittent disturbance behaviors of rock under true triaxial stress.

To address the limitations of the previous research,a novel complex true triaxial static-dynamic combined loading method reflecting excavation damage of surrounding rock and then frequent intermittent disturbance failure is proposed.True triaxial static compression and intermittent disturbance tests were carried out on monzogabbro.The analysis focuses on the influences of intermediate principal stress σ2and amplitudeAon the strength,deformation,failure characteristics and precursors of monzogabbro.On the basis of the test results,a novel three-dimensional intermittent disturbance fractional deterioration model considering intermediate principal stress effect and intermittent disturbance damage effect is established.The sensitivity analysis of static-dynamic stress states and model parameters is further explored,and the intermittent disturbance behaviors at different frequenciesfare predicted.

2.Experimental methodology

2.1. Specimen preparation and test facilities

In the study,monzogabbro was carefully selected and processed into prismatic specimens measuring 100 mm × 100 mm ×200 mm.All specimens were derived from the same large rock sample to ensure homogeneity,integrity,and minimize testing dispersion.The typical specimen is shown in Fig.2a,which is mainly composed of plagioclase,potassium feldspar and pyroxene,with smaller amounts of amphibole and biotite (Fig.2b and c).

Fig.2.Typical monzogabbro specimen and high-pressure servo dynamic true triaxial testing machine [19].

This test system is a new high-pressure servo dynamic true triaxial testing machine [19],as shown in Fig.2d.The test machine has static and dynamic loading systems,which can perform independent servo-controlled static loading or dynamic loading on rock specimens in three directions.The maximum vertical loading pressure range is 0–5000 kN,the maximum horizontal loading pressure range is 0–3000 kN,disturbance frequency range is 0–20 Hz and disturbance amplitude is from 0 to 500 kN.The high-precision intelligent microseismic monitoring system used in the experiment was jointly developed by the Institute of Rock and Soil Mechanics,Chinese Academy of Sciences,and Hubei Haizhen Seismic Technology Co.,Ltd.It employs a 30 V/g constant voltage supply and high-precision acceleration sensors to acquire the microseismic signals.

2.2. Novel true triaxial multistage intermittent disturbance test method

A new three-stage true triaxial static and dynamic combination test method is proposed to reasonably reflect the stress conditions of excavation unloading and stress concentration in the surrounding rock and subsequently trigger intermittent disturbance failure.The test stress path is shown in Fig.3.The test method consists of three stages:Stage I is the in-situ stress recovery stage,which simulates the real 3D in-situ stress conditions;Stage II is the static stress damage stage(stress redistribution stage).During this stage,Z-axis stress σzincreases,Y-axis stress σyandX-axis stress σxincrease or decrease to reach the desired value,reflecting the true conditions of unloading and stress concentration that occur during the excavation of surrounding rock;Stage III is the intermittent disturbance failure stage,where a loading scheme combining multistage cyclic loading and unloading with intermittent disturbance(with the same frequencyfand different amplitudesA) is employed.Before conducting the disturbance test scheme,the strength data are determined through true triaxial static tests.At least one-stage cyclic loading and unloading is set before the crack initiation stress σciand crack unstable stress σcd.According to the strength taken from the static test,multistage loading and unloading levels including 0.4σp,0.5σp,0.6σp,0.7σp,0.8σp,0.85σp,0.90σp,0.95σp,0.975σpand σpare set(σpis failure strength).The stress is loaded and unloaded at a rate of 2 MPa/s.The specific experimental scheme is shown in Table 1.In order to facilitate the analysis of the mechanical and deformation characteristics of monzogabbro,the principal stress directions corresponding to σ1,σ2,and σ3in stageⅢare used as the reference for the following analysis(as shown in Fig.3).

Table 1 Experimental scheme of monzogabbro under true triaxial static compression and intermittent disturbance.

Fig.3.The experimental stress path.

3.Results and analysis of monzogabbro under TTSCT

3.1. Deformation and strength characteristics of monzogabbro under TTSCT

Fig.4 shows stress–strain curves of monzogabbro under TTSCT(true triaxial static compression test),where ε1,ε2and ε3represents maximum,intermediate,and minimum principal strains,respectively.As shown in Fig.4a,when σ2is small and close to σ3,the deformations of the ε2and ε3directions are close.As σ2increases,the difference in deformation in the ε2and ε3directions gradually increases.This indicates that σ2enhances the deformation anisotropy of monzogabbro.

Fig.4.Stress–strain curves of monzogabbro under TTSCT at different σ2 and the same σ3 (5 MPa).

Table 2 and Fig.5 depict the characteristic strengths of monzogabbro under TTSCT,where εciis crack initiation strain,εcdis crack unstable strain,εpis failure strain andEis elastic modulus.As shown in Fig.5,the σ2exhibits a significant effect on each characteristic strength of monzogabbro at the same σ3.Furthermore,the characteristic strengths σci,σcd,σpof monzogabbro increases with increasing σ2.When σ2is small,the σpincreases greatly with the increase of σ2,while the increase in σpbecomes marginal at higher σ2levels.In comparison to σpand σcd,the increase in σciwith respect to σ2is relatively smaller.

Table 2 Experimental mechanical parameters of monzogabbro under TTSCT.

Fig.5.The relationship between characteristic strength and σ2 of monzogabbro under TTSCT.

3.2. Failure characteristics of monzogabbro under TTSCT

Fig.6 shows the failure characteristics of monzogabbro under TTSCT at the different σ2and the same σ3.At the lower σ2levels,monzogabbro is primarily a mixed tensile-shear failure mode,with predominant shear failure.The resulting cracks exhibit rougher surfaces and contain more debris,indicating distinct frictional characteristics.Moreover,there are fewer non-through vertical cracks,and they have shorter lengths.As σ2increases,the number of vertical and subvertical tensile cracks within the rock rises,aligning parallel to the σ1direction.These vertical micro-cracks interconnect,giving rise to inclined shear zones that macroscopically resemble inclined fractures.The length of non-through cracks inside the rock increases,showing a tendency to penetrate further.σ2enhances the mechanism of meso tensile cracking.

Fig.6.The macro-meso failure characteristics of monzogabbro under TTSCT at different σ2 and the same σ3 (5 MPa).

3.3.Failure precursors of monzogabbro based on b-value under TTSCT

The changes in microseismic parameters (b-value and lgN/b)can reflect the internal damage and fracture process of the rock.When theb-value is larger,it indicates the occurrence of microfracture damage inside the rock.The frequency and magnitude of the microseismic signal generated by rock mass fracture follow a G-R relationship [42]:

whereMis the microseismic event magnitude;Nthe total number of microseismic events that have changed within ΔM;andaandbthe formula parameters.Microseismic parametersb-value and lgN/bevolution of monzogabbro under TTSCT at different σ2and the same σ3are shown in Fig.7.In the initial stage of stress loading,microcracks within monzogabbro undergo compression and crack closure,leading to a rapid increase in theb-value.Subsequently,as the stress increases,theb-value continues to fluctuate within a certain range.As the failure approaches,theb-value decreases rapidly with a high amplitude,and drops to the lowest value near zero.Compared withb-value,lgN/bdoes not change much before rock failure and only fluctuates within a small range.As the failure approaches,lgN/bincreases rapidly with high amplitude,and reaches the maximum value.Therefore,the microseismic signs of rock failure under true triaxial stress conditions are significant and can provide early warning of rock failure.Compared withbvalue,the fluctuation of lgN/bis smaller before rock failure,making it more suitable as a precursor to characterize rock failure.

Fig.7.Microseismic parameters b-value and lgN/b evolution of monzogabbro under TTSCT at different σ2 and the same σ3 (5 MPa).

4.Results and analysis of monzogabbro under TTIDT

4.1. Deformation characteristics of monzogabbro under TTIDT

4.1.1.Stress–straincharacteristicsofmonzogabbrounderTTIDT

Fig.8 illustrates the stress–strain curve characteristics of monzogabbro under TTIDT (true triaxial intermittent disturbance test)at different amplitudeAand the same σ2(40 MPa),σ3(5 MPa),frequencyf(0.5 Hz).In stage I and stage II,the stress–strain curve of monzogabbro exhibits similar characteristics to the static test conducted at the same σ2and σ3.In stage III,the stress–strain curve of monzogabbro under loading and unloading shows restorability due to the rock’s memory effect,particularly when the maximum value of σ1is low.When σ1approaches its peak,the stress–strain curves in the three directions form hysteresis loop and the hysteresis loop in the σ3direction is more significant.During the intermittent disturbance stage,irreversible strain occurs in all three directions,with the most significant strain response in the ε3direction.Under the condition of different amplitudeA,maximum principal strain ε1and minimum principal strain ε3exhibit substantial changes during loading and unloading,following similar patterns.

Fig.8.Stress–strain curves of monzogabbro under TTIDT at different amplitude A and the same σ2 (40 MPa),σ3 (5 MPa),frequency f (0.5 Hz).

4.1.2.Strain–timecharacteristicsofmonzogabbrounderTTIDT

Fig.9 shows the strain–time curves of monzogabbro under TTIDT at different amplitudeAand the same σ2(40 MPa),σ3(5 MPa),frequencyf(0.5 Hz).Compared with the σ3,the larger σ2leads to a smaller change in ε2of the monzogabbro,so the strain changes in the ε1direction and the ε3direction are much larger than that in the ε2direction.Due to the disturbance in the σ1direction,the strain in the ε1direction also responds accordingly,and the response degree of ε2and ε3is small due to the Poisson effect.When the maximum value of σ1during loading and unloading is low,the ε1curve in the intermittent disturbance process only appears in the attenuation and steady-state stages.In the attenuation stage,strain increases slowly while the strain rate decreases.In the steady-state stage,the strain growth rate approaches zero,and the strain remains stable.As the maximum value of σ1during loading and unloading increases,the internal damage degree of the rock deepens under the influence of the disturbance load,resulting in a larger strain growth rate in the steadystate stage and linear strain accumulation over time.When σ1approaches its peak,the rock undergoes the three stages of attenuation,steady-state and accelerating.In the accelerating stage,the strain of monzogabbro increases rapidly,ultimately leading to failure.Before rock failure,the intermittent disturbance induces significant irreversible deformation,which can serve as a warning sign of impending failure in engineering applications,reflecting the deterioration effect of intermittent disturbance on the rock.

Fig.9.Strain–time curves of monzogabbro under TTIDT at different amplitude A and the same σ2 (40 MPa),σ3 (5 MPa),frequency f (0.5 Hz).

4.1.3.DeformationmodulusdegradationofmonzogabbrounderTTIDT

The decay of the deformation modulus can reflect the damage caused by stress loading.The loading damage of the rock can be characterized by the change in the deformation modulus in the unloading stage at each level.The loading damageD1can be calculated:

whereKiis the deformation modulus of monzogabbro;K0the initial deformation modulus of monzogabbro,i.e.,the deformation modulus of the first stage of cyclic loading and unloading;andKidetermined by the absolute value of the ratio of the maximum principal stress increment dσ1ito the maximum principal strain increments dεiein each cyclic loading and unloading of the rock,as shown in Fig.10.From the relationship betweenD1and the stress-strength ratio λ under different test conditions in Fig.11,the growth rate ofD1increases continuously with increasing λ as an exponential function.When the stress is low,only a small number of cracks are generated inside the monzogabbro,and the damage increase is small.As the stress increases,more cracks are generated inside the monzogabbro,causing the damage increase.When failure approaches,a large number of cracks initiation,propagation and coalescence inside the monzogabbro,resulting in a rapid increase in damage.The relation between the loading damageD1and the stress-strength ratio λ as follows:

Fig.11.Experimental result and theoretical curve of the D1-λ curves (loading damage variable D1 versus stress-strength ratio λ) at different amplitude A and the same σ2(40 MPa),σ3 (5 MPa),frequency f (0.5 Hz).

wherep1,p2andp3are model parameters;and λ the stress-strength ratio.

The stress-strength ratio λ can be defined by:

where σ1iis maximum σ1atith cyclic loading and unloading.

4.2. Strength characteristics of monzogabbro under TTIDT

Fig.12 and Table 3 depict the relationship between the failure strength σpand the amplitudeAof monzogabbro under TTIDT at different amplitudeAand the same σ2(40 MPa),σ3(5 MPa),frequencyf(0.5 Hz).As shown in Fig.12,the failure strength of monzogabbro exhibits a decreasing trend with increasingA,indicating a clear amplitude effect.The damage generated inside the monzogabbro is earlier and more serious compared with that at low amplitude,which accelerates the process of rock damage.Under the same stress conditions,the larger the amplitudeAis,the lower the bearing capacity of the rock,the faster the rate of deterioration,and the earlier the time of damage(as shown in Fig.13).According to Table 3,even when the applied disturbance amplitudeAis relatively small,the bearing capacity is reduced,leading to premature failure of monzogabbro.Furthermore,at the same σ2and σ3,comparing the failure strength of the static test in Fig.5,the disturbance failure strength of monzogabbro is significantly decreased.

Table 3 Experimental mechanical parameters of monzogabbro under TTIDT.

Fig.12.The relationship between the failure strength σp and the amplitude A of monzogabbro under TTIDT at different amplitude A and the same σ2 (40 MPa),σ3(5 MPa),frequency f (0.5 Hz).

Fig.13.Stress–time curves of monzogabbro under TTIDT at different amplitude A and the same σ2 (40 MPa),σ3 (5 MPa),frequency f (0.5 Hz).

4.3. Failure characteristics of monzogabbro under TTIDT

Fig.14 shows the macro-meso failure characteristics of monzogabbro under TTIDT at different amplitudeAand the same σ2(40 MPa),σ3(5 MPa),frequencyf(0.5 Hz).The failure characteristics of monzogabbro under TTIDT are mainly tensile-shear fracture,and asAincreases,the tensile fracture mechanism strengthens.AtA=0.1 MPa,the monzogabbro with lowamplitude intermittent disturbance experiences mainly shear failure,accompanied by some small vertical cracks surrounding the main fracture.The fracture surface exhibits severe damage caused by shear friction,resulting in the presence of a significant amount of rock powder,as shown in Fig.14.With increasingA,the occurrence of tensile cracks within the monzogabbro becomes more prominent,accompanied by the development of numerous vertical cracks with a tendency to penetrate around the primary through-cracks.AtA=3 MPa,the monzogabbro mainly suffers from tensile failure,a large number of vertical cracks are distributed inside the rock,and the friction damage is lower.Furthermore,compared with the true triaxial static result at the same σ2and σ3in Fig.6,the disturbance amplitude enhances the tensile fracture mechanism of the rock.The direction of crack propagation in monzogabbro remains consistent,advancing towards σ1and expanding towards σ3.

Fig.14.The macro-meso failure characteristics of monzogabbro under TTIDT at different amplitude A and the same σ2 (40 MPa),σ3 (5 MPa),frequency f (0.5 Hz).

4.4.Failure precursors of monzogabbro based on b-value under TTIDT

Microseismic parametersb-value and lgN/bevolution of monzogabbro under TTIDT at different amplitudeAand the same σ2are shown in Fig.15.Compared to the static test,theb-value and lgN/bvalue during the initial two stages of the intermittent disturbance test exhibit similar trends.Upon entering intermittent disturbance stage,theb-value shows an increase during stress loading,followed by a decrease as the stress is unloaded after intermittent disturbance.It is noteworthy that the monzogabbro experiences greater internal damage following intermittent disturbance loading,when compared to the static test.This suggests that the microcracks generated within the monzogabbro during loading undergo earlier mutual expansion and form larger fractures after intermittent disturbance and stress unloading,reducing rock strength.Furthermore,compared to the static test,theb-value and lgN/bvalue curves in the intermittent disturbance test exhibit more complex fluctuations before rock failure.The lgN/bvalue exceeds 1 multiple times before rock failure,deviating from the evolution pattern observed in the static test.This is because the influence of stress redistribution and cyclic loading–unloading disturbances leads to numerous fracture events inside the rock before failure,resulting in significant fluctuations in theb-value and lgN/bvalue.Regarding precursor characteristics,the static test exhibits more prominent precursor patterns with smoother fluctuation change compared to the intermittent disturbance test.Although there are certain differences in the specific evolution of microseismic parameters between the two types of tests,the overall trend is consistent when approaching failure,withb-value rapidly and significantly decreasing to very low values (approaching 0),and lgN/brapidly and significantly increasing to very high values(greater than 1).

Fig.15.Microseismic parameters b-value and lgN/b evolution of monzogabbro under TTIDT at different amplitude A and the same σ2 (40 MPa),σ3 (5 MPa),frequency f(0.5 Hz).

5.A novel three-dimensional fractional deterioration mechanical model for rock true triaxial intermittent disturbance failure

5.1. Model establishment

(1) From the results of the true triaxial intermittent disturbance test in section 4 above,it can be seen that the instantaneous elastic deformation of monzogabbro (i.e.,attenuation stage)is simulated by introducing spring elements,and the fractional elements are introduced to simulate the strain characteristics of the steady-state and accelerating stages of monzogabbro.The spring element and the fractional element are more accurate in simulating the deformation characteristics of the monzogabbro under static loading.Still,the disturbed deformation of monzogabbro under disturbance loading cannot be adequately expressed,so further introduction of the disturbance element is needed.

(2) From section 4 above,it is known that during stress loading,the static load causes damageD1to monzogabbro,which can be transformed into a damage spring element by combining it with a spring element.Applying multiple intermittent disturbance loads to monzogabbro produces intermittent disturbance damageD2,which is transformed into a damage fractional element by combining it with a fractional element that simulates the deformation characteristics of the fully disturbed process.Fig.9 shows that in the first few stages of monzogabbro damage,the monzogabbro will produce large irreversible deformation under the intermittent disturbance load,so the disturbance element for disturbance damageD3related to σ1andNr(number of disturbances) is introduced to establish the damage disturbance element.

(3) Research scholars have shown that damage occurs only when the stress of monzogabbro exceeds a certain level,i.e.,there is a threshold stress as the limit between damage and non-damage.From the above 4 sections,it can be seen that after the loading stress reaches the monzogabbro’s σci,monzogabbro gradually produces disturbance deformation and enters the attenuation stage.The influence of the threshold stress (i.e.,σci) must be considered in the simulation of fractional components in the attenuation and steady-state stages.Similarly,when the loading stress reaches the disturbance threshold stress,monzogabbro enters the accelerating stage,and the fractional element that simulates the accelerating stage also needs to consider the effect of the threshold stress associated with the disturbance threshold stress.

Summarizing (1) to (3),a damage fractional mechanical model consisting of a damage spring,two damage fractional elements and a damage disturbance element can be established in this study(as shown in the Fig.16).FciandFcdis the yield functions corresponding to σciand σcd,β and γ are fractional derivative orders,η is the viscosity coefficient,εe(t) is the elastic body strain,εveis the disturbance viscoelastic-plastic body strain,εvpis the disturbance viscoplastic body strain,εdis the disturbance strain,E1andE2are elastic modulus.

Fig.16.Three-dimensional fractional deterioration mechanical model for intermittent disturbance failure under true triaxial stress.

5.2. Three-dimensional yield function

The classical 2D strength criterion Mohr-Coulomb strength criterion without considering the σ2effect cannot correctly model the rock strength characteristics of rock under true triaxial stress,so a true triaxial strength criterion that considers the σ2effect is needed.In this study,the 3D strength criterion Modified Weibols-Cook (MWC) criterion is used to establish the yield function,and the MWC criterion is expressed:

wherec1,c2andc3are test parameters;the second invariant of the stress tensor;and σmthe average stress.

Combining Eqs.(5),(6)and(7),the yield functionFis obtained:

Fig.17 shows that the characteristic stresses such as σci,σcdand σpof monzogabbro under TTSCT have a good fitting relationship with the MWC criterion.Substituting σ2,σ3and the characteristic strength under different amplitude states into Eqs.(6) and (7),σmandare obtained,respectively.The yield function corresponding to each characteristic stress is obtained by fitting calculation,as shown in Table 4 and Fig.17.

Table 4 Fitting parameters of the characteristic stress of monzogabbro under the TTSCT.

Fig.17.The characteristic stress experiment results and Modified Weibols-Cook Criterion (MWC) fitting curves of monzogabbro under TTSCT.

5.3. The new damage fractional element

Fractional calculus provides a more precise characterization of the time-dependent behavior of materials,allowing for the representation of values at the present and historical moments.When using fractional derivatives to construct rock instanton relations,fewer parameters and monomials are used,and the parameters can be obtained more simply by experiment.Constitutive equations constructed with few parameters can perfectly reflect theproperties of describing the instantaneous elasticity,viscosity,and plasticity of the object and have a higher agreement with the test results than the traditional element theory.The Riemann-Liouville type represents a primary form of fractional calculus and is widely employed in practice.The β-order integral of functionf(t) is defined as:

wheretis the time;the fractional calculus operator;L[] the Laplace transform of functionf(t);sthe function parameter;0＜n-1＜β≤n.Letf(0)=0,Eq.(10) can be rewritten as:

The fractional element constitutive equation can be written as:

where 0 ≤β ≤1.When σ(t)=σ0,Eq.(12)can be transformed into:

where Γ() denotes the gamma function.When β equal to different values,the element can be converted accordingly (Fig.18).

Fig.18.Basic mechanical elements.

When intermittent disturbance is applied to rock,as disturbance number increases,the internal damage of the rock accumulates and the disturbance deformation rate gradually increases.The attenuation of ηβof the fractional element in the model can reflect this phenomenon.

The rock will produce damage in each disturbance period,so the disturbance damageD2can be expressed by cumulative plastic deformation:

where εpiis the cumulative plastic strain of each disturbance period;and εfthe final cumulative plastic strain.According to the disturbance damage variable proposed by previous research scholars[43],the disturbance damage variableD2is a function of the cumulative plastic deformation εpas the independent variable:

where α and χ are material parameters.

According to the Manson-Coffin empirical equation,the total strain ε can be divided into εeand εp:

wherec1,c2,k1,k2,β1,and β2represent material parameters;andNrthe number of disturbances.

A relationship between disturbances and εpcan be established as follows:

LetA1=– (αc2)χ,B=–β2χ,C=(1–k2)β2χ,D2be reformulated as:

whereA1,BandCrepresent material parameters.The disturbance damage fractional element ε(t) can be expressed as:

5.4. Three-dimensional damage fractional mechanical model

(1) One-dimensional constitutive equation

Elastic constitutive equation:

Viscoelastic-plastic constitutive equation:

where σcicorresponding to σs1in the proposed model.

Viscoplastic constitutive equation:

where σcdcorresponding to σs2in the proposed model,γ is the fractional derivative order.

Intermittent disturbance stress σdcan be written as:

whereTis the intermittent disturbance period;t1the static stress loading and unloading time;andna positive integer.Disturbance element constitutive equation εdcan be written as:

Disturbance element damageD3can be written as:

where κ and ψ are model parameters;σpthe failure strength;andNfthe disturbance number at rock failure.

Combining Eqs.(21),(24),(27)and the one-dimensional constitutive equation can be obtained:

(2) Three-dimensional constitutive equation

In a 3D stress space,the stress tensor σijis:

whereSijis the deviatoric stress tensor;δijthe kronecker tensor.

The strain tensor εijat any position inside rock is:

whereeijis the deviatoric strain tensor;and εmthe volumetric strain.

σmandSijconform to the generalized Hooke’s law:

Substituting Eq.(32) into Eq.(35),the 3D elastic constitutive equation is obtained:

whereKis the bulk modulus andGthe shear modulus.When the loading stress of the rock does not reach the yield surface corresponding to crack initiation stress σcistate,it is in elastic stage,and there is no disturbance plastic strain.When the load exceeds the σci,the rock produces partially recoverable viscoelastic-plastic disturbance deformation.According to the Perzyna equation,disturbance viscoplastic strain rate was expressed:

whereFis a 3D yield function;φ(F) is a nonlinear function of the independent variableF,Qa 3D plastic potential function,according to the associated flow rule,Q=F.In 3D strain space,represents disturbed viscoplastic strain development direction.

Therefore,the viscoelastic-plastic strain rate is:

whereF0=1.

Viscoelastic-plastic three-dimensional constitutive equation:

When the load on the rock does not reach the crack unstable stress yield surface,the rock will produce elastic strain and viscoelastic-plastic disturbance deformation,but it does not reach the failure state.When the load exceeds the crack unstable stress yield surface,the rock produces irrecoverable viscoplastic deformation and eventually fails.The viscoplastic strain rate:

Viscoplastic three-dimensional constitutive equation:

The deviator stressS1can be expressed as:

Substituting Eq.(44)into Eqs.(36),(40)and(43),the ε1of rock under the true triaxial disturbance state can be obtained as follows:

6.Verification of the proposed model based on true triaxial intermittent disturbance test result

The proposed mechanical model,based on the MWC criterion,was identified and validated using 1stOpt software,utilizing test results of monzogabbro under TTIDT,and comparison curves of monzogabbro between proposed model and experimental result were plotted in Fig.19.Table 5 shows the model parameters.According to Fig.19,the model theoretical curve is in good agreement with the experimental result,and the proposed mechanical model can accurately reflect the intermittent disturbance deformation characteristics of monzogabbro under true triaxial stress.

Table 5 The proposed model fitting parameters under TTIDT.

Fig.19.Comparison of the strain–time curves of monzogabbro between proposed model and experimental result under TTIDT.

7.Discussion

According to the model parameter identification results in Table 5,a test condition is used to bring the fractional mechanical model into the constitutive equation,and parameter sensitivity analysis was performed.The conditions are σ1–σ3=193 MPa,σ2=40 MPa,σ3=5 MPa,A=3 MPa,f=0.5 Hz,K=17.63 GPa,G=63.77 GPa,ηβ=207.53 GPa,and β=0.18.

7.1. Sensitivity analysis of stress parameters and disturbance parameters

(1) Effect of deviatoric stress σ1–σ3

Keep other parameters unchanged,only the σ1–σ3is changed,and σ1–σ3in the test conditions are set to 185,189,193,and 197 MPa.Fig.20a is the disturbance plastic strain curves of different deviatoric stresses under test conditions.According to Fig.20a,with increasing value of σ1–σ3,there is a corresponding increase in the instantaneous elastic strain,total strain,and strain rate.

Fig.20.Sensitivity analysis of three-dimensional stress.

(2) Effect of intermediate principal stress σ2

Keeping other parameters unchanged,only the σ2is changed,and the values of σ2in the test conditions are 20,30,40 and 50 MPa.Fig.20b is the disturbance plastic strain curves at different σ2under test conditions.According to Fig.20b,with the increase in σ2,the strain (instantaneous elastic strain and total strain) and strain rate are smaller.

(3) Effect of amplitudeA

Keeping other parameters unchanged,only theAis changed,amplitudeAis 2,3,4,and 5 MPa,and the disturbance plastic strain curves of differentAunder the test conditions can be obtained,as shown in Fig.20c.With increasingA,the strain curve changes more.

(4) Effect of frequencyf

Keeping other parameters unchanged,only thefis changed,andfin the test conditions is set to 0.2,0.4,0.5,and 0.8 Hz,respectively.The disturbance plastic strain curves of different amplitudes under the test conditions can be obtained,as shown in Fig.20d.According to Fig.18d,with increasingf,the time of the same number of disturbances of the strain curve is longer.

7.2. Sensitivity analysis of model parameters

(1) Effect of volume modulusK

Keeping other parameters unchanged,only theKis changed,and the values ofKin the test conditions are 16.63,17.63,18.63 and 19.63 GPa,respectively.Fig.21a is the disturbance plastic strain curves of different volume modulus under the test conditions.According to Fig.21a,with increasingK,the instantaneous elastic strain and the total strain are smaller.

Fig.21.Sensitivity analysis of model parameters.

(2) Effect of shear modulusG

Keeping other parameters unchanged,only theGis changed,and the values ofGin the test conditions are 59.77,63.77,67.77 and 71.77 GPa,respectively.Fig.21b is the disturbance plastic strain curves of different shear modulus under test conditions.According to Fig.21b,with increasingG,the instantaneous elastic strain and the total strain are smaller.

(3) Effect of viscosity coefficient ηβ

Keeping other parameters unchanged,the values of ηβin the test conditions are 197.53,207.53,217.53 and 227.53 GPa·s,respectively.Fig.21c is the disturbance plastic strain curves of different ηβunder the test conditions.According to the analysis of Fig.21c,with the increase in ηβ,the duration of the decay stage is shorter,the strain rate in the steady-state stage is smaller,and the total strain is smaller.

(4) Effect of β

Keeping other parameters unchanged,only the β is changed,and the values of β in the test conditions are 0.08,0.18,0.28 and 0.38.Fig.21d is the disturbance plastic strain curves of different β under the test conditions.According to Fig.21d,with increasing β,the duration of the attenuation stage is shorter,the greater the strain rate is at steady-state and the accelerating stages,and the larger total strain.

7.3.Prediction of intermittent disturbance behaviors of rock under true triaxial stress with the proposed model

The limited availability of experimental results has led to a poor understanding of the intermittent disturbance behaviors of rocks under true triaxial stress.To better investigate intermittent disturbance behaviors of rock,theoretical prediction research is explored based on the proposed model.Mechanical parameters of monzogabbro under the conditions of σ2=40 MPa,σ3=5 MPa,A=3 MPa,andf=0.5 Hz,fis changed.By changingf(0.2,0.4,0.6 and 0.8 Hz),the strain curves of monzogabbro at λ=0.439,λ=0.658,λ=0.944 and λ=1 are predicted.Fig.22 shows the predicted strain curves of monzogabbro at different frequencies,which further expanded the understanding of intermittent disturbance characteristics.

Fig.22.The predicted strain curves of monzogabbro under TTIDT at different frequency f and the same σ2 (40 MPa),σ3 (5 MPa),amplitude A (3 MPa).

8.Conclusions

In this study,a novel true triaxial multistage intermittent disturbance test method was provided and conducted to investigate monzogabbro’s strength,deformation,failure characteristics and precursors.Based on experimental results and fractional theory,a three-dimensional fractional deterioration model for intermittent disturbance damage under true triaxial stress was established.The main conclusions are as follows:

(1) With increasing σ2,the crack initiation stress,crack unstable stress,and failure strength of monzogabbro gradually increase.The failure strain also gradually increases,and the tensile fracture mechanism is enhanced.

(2) With increasing amplitudeA,the bearing capacity of monzogabbro decreases,and the tensile fracture mechanism is enhanced.With disturbance times and loading levels increasing,internal damage accumulation increases,deformation modulus decreases rapidly,and irreversible strain mutation occurs before failure.

(3) The microseismic precursory signs before rock intermittent disturbance failure are significant withb-value rapidly and significantly decreasing to very low values (approaching 0),and lgN/brapidly and significantly increasing to very high values (greater than 1).

(4) A novel fractional deterioration mechanical model of rock under true triaxial intermittent disturbance considering intermediate principal stress and disturbance damage effects is established,which predicted results align well with the test results.Static-dynamic stress state (σ1–σ3,σ2,A,f)and model parameters (K,G,ηβ,β) have a significant influence on rock intermittent disturbance behavior.

Acknowledgements

The authors greatly acknowledge the financial support from the National Natural Science Foundation of China (No.52109119),the Guangxi Natural Science Foundation (No.2021GXNSFBA075030),the Guangxi Science and Technology Project (No.Guike AD20325002),the Chinese Postdoctoral Science Fund Project (No.2022 M723408),and the Open Research Fund of State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin(China Institute of Water Resources and Hydropower Research)(No.IWHR-SKL-202202).