Yijing Zong,Lijun Hn,Qinbin Meng,Yingcho Wng

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

b School of Transportation Engineering,Jiangsu Vocational Institute of Architectural Technology,Xuzhou 221116,China

Keywords:Rock mechanics Damage Re-loading Strength characteristics Strain hardening

ABSTRACT To study the strength properties and evolution laws of cracked sandstone samples in re-loading tests,strength and damage evolution properties of intact sandstone samples were first analyzed through the triaxial compression tests carried on TAW-2000 microcomputer control electro-hydraulic servo rock triaxial test system.Damage evolution models were established based on dilatancy properties realizing the real-time and quantitative evaluation of samples damage state in loading process.On this basis,samples with different damage were obtained by pre-peak,peak point,post-peak and residual strength stage unloading tests in the loading process of intact samples.The characteristics of the stress-strain curves and strength evolution laws were studied through the re-loading tests of samples with different damage under different stress states.The experimental results showed that the slope of stress-strain curves,peak strength and residual strength of cracked samples increased linearly with confining pressure and decreased linearly with damage.The equivalent cohesion decreased with damage in the exponential decay curves.The mechanics properties of samples transformed from strain softening to strain hardening with damage.

1.Introduction

The post-peak behaviors of rock have been the focus of rock mechanics.The stability of surrounding rock in deep roadways is strongly associated with the post-peak state of rock;therefore,it is of great significance to study the post-peak deformation and failure characteristics of rock.The research methods of the post-peak mechanical properties of rock are mainly experimental research and theoretical research.In experimental study,the post-peak strength and deformation characteristics of rock are studied by means of triaxial compression of rock or similar material,postpeak loading and unloading tests and cyclic loading and unloading tests [1-5].For instance,the post-peak strength and deformation characteristics of rock were studied by uniaxial and triaxial compression tests of limestone,siltstone and fine sandstone rock,which showed that post-peak characteristic curves of rock were mainly the sliding of the fracture surfaces[6,7].The initial softened state reflected the change from continuum to blocks and the loading section.The failure process of rock was from intact rock to large blocks and small blocks,contributing to the evolution of the mechanical rock properties.In theoretical research,the constitutive model reflecting the post-peak mechanical properties of rock was established based on test data,elastic and plastic mechanics theory and the characteristics of the test curves [8-12].For instance,post-peak strain softening model of rock under confining pressure was established based on the strength degradation angle and the decrease index of post-peak strength[13,14].On this basis,a simulation program was developed with Fish language in Fast Lagrangian Analysis of Continua (FLAC).

The above research results have presented a general understanding of post-peak characteristics and failure processes of rock.However,there are many obvious insufficiencies in the mechanical properties,aging evolution processes and coupling mechanisms of deep cracked surrounding rock and support,making it impossible to provide a reliable theoretical and quantitative basis for the design of the rational supporting structure,parameters and time.

Therefore,the strength and damage evolution properties of intact sandstone samples were first analyzed through triaxial compression tests carried on a TAW-2000 microcomputer control electro-hydraulic servo rock triaxial test system.The damage evolution model was established based on dilatancy properties to realize the real-time quantitative evaluation of the damage state of samples in the loading process.On this basis,samples with different damage were obtained by pre-peak,peak point,post-peak and residual strength stage unloading tests in the loading process of intact samples.Then,the characteristics of stress-strain curves and strength evolution laws were studied through the re-loading tests of cracked samples with different damage under different stress states.The influences of confining pressure and damage on the peak strength,residual strength,equivalent cohesion strength and friction angle of cracked samples were also analyzed to reveal the transformation laws from strain softening to strain hardening of the mechanical properties of samples with damage.

2.Triaxial compression tests of intact samples

2.1.Test program and process

Red sandstone,taken from Linyi in Shandong province,is of fine,blocky structure with good homogeneity.The cementing type of the sandstone is pore cementation with grain point-line contact.Red sandstone was processed into standard samples with a diameter of 50 mm and length of 100 mm according to ISRM and Determination Methods for Physical and Mechanical Properties of Coal and Rock.

Triaxial compression tests of the sandstone were carried out on a TAW-2000 rock mechanics electro-hydraulic servo testing system,with a maximum axial force of 2000 kN,maximum confining pressure of 60 MPa,axial deformation measurement ranges from 0 to 10 mm,and radial deformation measurement ranges from 0 to 5.0 mm.

The confining pressures of triaxial compression tests were 5,10,15,20 and 30 MPa.The experimental process was as follows:the confining pressure was loaded to the set value at a rate of 0.05 MPa/s after the installation of samples.With the constant confining pressure,the axial pressure was increased until the samples failed by the displacement control method with the loading rate of 0.002 mm/s.The axial pressure-loading rate of the T-6 sample after the peak strength was 0.001 mm/s.Loading control methods of samples are shown in Table 1.

Fig.1 shows the complete stress-strain curves of samples under triaxial compression,and the test results are as shown in Table 2.

2.2.Strength properties

Table 2 and Fig.1 show that the peak strength of the three samples under uniaxial compression are 70.14,69.19 and 68.34 MPa.The dispersion degree of which is only 2.60%,showing the good homogeneity of the samples.The rock can well meet the requirement for tests.

The peak strength and residual strength of samples under triaxial compression increase linearly with the increasing confining pressure.Based on the Mohr-Coulomb strength criterion,the peak strength and residual strength of samples under confining pressures of 5,10,15 and 20 MPa are analyzed by linear regression statistics,as shown in Fig.2.

Fig.1.Complete stress-strain curves of sandstone samples under triaxial compression.

Multiple regression result shows that the influence coefficient of confining pressure on peak strength is 3.064 but 1.302 on residual strength,illustrating the higher sensitivity of peak strength to confining pressure,which shows good agreement with those of related experiments[15,16].Based on the Mohr-Coulomb strength criterion and regression parameters,the cohesion and internal friction angle of rock samples can be determined to be 31.31 MPa and 30.51°,respectively.In the residual deformation stage,the internal macro-fractures in the samples will cause the decrease of the equivalent cohesion and internal friction angle.

Calculated from residual strength,the equivalent cohesion is 21.88 MPa,30.12%lower than the peak strength calculation result,while the internal friction angle is 7.54°,with a drop of 75.29%,which demonstrates that the cracked rock samples still have significant carrying capacity.Rock sample failure shows obvious ductility characteristics.

2.3.Damage evolution properties

Fig.3 shows the relationship between radial strain,volumetric strain and axial strain of sandstone samples under different confining pressure.

As shown as in Fig.3,the axial strain,radial strain and volumetric strain of samples increase linearly with increasing axial stress before the yield point,and samples are under compression.With the expansion and extension of the internal cracks in rock samples,the volume strain decreases gradually to the negative,and the states of rock sample turn from compression to dilatancy,macroscopically showing obvious dilatancy characteristics.The rock dilatancy is the result of the initiation and propagation of micro fractures in rock,reflecting rock damage under external load.Therefore,the damage evolution laws of rocks can be studied in the entire loading process based on the dilatation properties.

Table 1 Samples specification and loading control methods.

Table 2 Uniaxial and triaxial compression test results of sandstone samples.

Fig.2.Relationship between peak strength,residual strength and confining pressure.

We consider the maximum compressive volumetric strains εvdas the volumetric dilatation starting point and assume that there is no damage before that point.The damage variable D is defined as the ratio of volumetric dilatation strain at some point in the strain-stress curves to the volumetric dilatation strain of the initial residual strength stage [17-19].The damage variable D can be expressed as Eq.(1).where D,εv,εvd,εvr,ε1and ε1dare the damage variable,volumetric strain,the maximum compressive volumetric strain,volumetric strain of initial residual strength stage,axial strain and axial strain corresponding to the maximum compressive volumetric strains,respectively.

The samples at the residual strength stage still have great residual strength due to the incomplete failure.Therefore,Eq.(1)can be corrected as:

where K is the correction factor.

where σrand σpare the residual strength and the peak strength,respectively.

The damage variable of samples under different confining pressure can be obtained by substituting Eqs.(3)into (2).

Fig.4 shows the comparison of AE damage with volumetric dilatation damage under uniaxial compression.As shown in Fig.4,volumetric dilatation damage has preferably coherence with AE damage.Therefore,it is feasible to determine the damage variable of samples in the whole loading process.

3.Mechanical properties of cracked samples in re-loading tests

3.1.Preparation of cracked samples

The preparation of cracked rock samples is as follows:

The axial pressure and confining pressure were applied to samples under hydrostatic pressure at a loading rate of 0.05 MPa/s to the design value 30 MPa.

With a constant confining pressure,the axial pressure was applied to the unloading points at the loading rate of 0.002 mm/s in the pre-peak stage and 0.001 mm/s in the post-peak stage.Cracked samples with different damage were obtained after the alternate unloading of the axial pressure and confining pressure at the unloading points,the damage of which could be determined by Eq.(2).

The partial unloading curves are shown in Fig.5,and the mechanical parameters of samples and damage values at unloading points are shown in Table 3.

Fig.3.Relationship between radial strain,volumetric strain and axial strain of intact samples.

Fig.4.AE damage and volumetric dilatation damage of sandstone samples under uniaxial compression.

Fig.5.Unloading stress-strain curves of intact samples under different stress states.

As shown in Fig.5 and Table 3,the curves of rock samples at different unloading points are very similar,and the peak strength is 175.02-181.71 MPa,with the peak strength dispersion of 3.76%.The stress ratio deviation of actual stress and design stress at unloading points is only -1.10% to 1.41%.The deviation of axial strain,radial strain and volumetric strain at the unloading point and the corresponding positions on the curves of intact samples is-8.95% to 7.86%,-7.99% to 6.21% and -15.13% to 7.21%,respectively.Consequently,unloading tests can be carried out at the design unloading points to obtain the cracked samples with certain damage.

3.2.Mechanical properties of cracked samples in re-loading tests

The re-loading tests of cracked samples with different damage were carried in a TAW-2000 testing system as described in Section 2.1.Stress-strain curves of cracked samples in re-loading tests are shown in Fig.6,and test results are shown in Table 4.

3.3.Characteristics of stress-strain curves in re-loading tests

It can be seen from Fig.6 that confining pressure and damage have very significant influence on the mechanical properties of rock samples in re-loading tests.The stress-strain curve shape of cracked samples T-7 to T-34 with low damage (less than 0.345)are similar to that of intact samples.The deviatoric stress has good linear with axial strain,radial strain and volumetric strain before the yield points.

After the yield points,the stress-strain curves show convex morphology due to the initiation and expansion of internal cracks in samples,with the decreasing slope and increasing yield deformation.With the increase of confining pressure,the slope of the stress-strain curves,the peak strength and the residual strength of cracked samples with different damage are greatly improved.Cracked rock samples with low damage show some brittle failure characteristics.With increasing damage,the stress-strain curves post-peak gradually level off,and the mechanics properties of cracked rock mass transformed from elastic-brittleness to elasticplasticity.The slope of stress-strain curves,peak strength and residual strength of cracked samples are greatly reduced.

The post-peak stress-strain curves of cracked samples with damage greater than 0.378 first manifest strain softening property.With the increase of deformation,the irregular fracture surfaces gradually transmit the stress and due to fissures filling of small rocks and grains,resulting in strain hardening property of cracked rock samples.The relationship between the peak radial strain,volumetric strain and confining pressure,showed that the damage degree took on nonlinear characteristics.The growth rates of cracked samples bearing capacity increase with the confining pressure and damage,showing significant strain hardening characteristics.

Table 3 Mechanical parameters of samples at unloading points.

Fig.6.Re-loading complete stress-strain curves of cracked samples.

3.4.Strength properties of cracked samples in re-loading

(1)Peak strength

As shown in Fig.7,the peak strength of cracked samples with different damage increases linearly with confining pressure.

Fitting results of the relationship between peak strength and confining pressure of cracked samples are listed in Table 5.Confining pressure has different actions on the peak strength of cracked samples with different damage.The peak strength of 95% postpeak unloading samples(D=0.234)is most sensitive to the confining pressure with the influence coefficient A of 3.669,while that of samples unloading at initial points in residual strength stage is 1.830.The parameters of B can be regarded as the uniaxial compressive strength of cracked samples,decreasing linearly with damage,which has the similar variation tendency to peak strength of samples under triaxial tests.

The peak strength degradation (Dσ)of cracked samples,the ratio of the difference between the peak strength of intact sample and cracked sample and the peak strength of intact samples decrease with the increase of confining pressure at an exponential attenuation function with confining pressure,as shown in Fig.8.

(2)Residual strength

Table 4 indicates that the residual strength of cracked samples with different damage increases linearly with confining pressure,as shown in Fig.9.

3.5.Strength evolution laws of cracked samples in re-loading tests

(1)Evolution of peak strength

As shown in Fig.10,damage has a significant influence on the peak strength of cracked samples,resulting in the linear decrease of peak strength of cracked samples.The peak strength degradation rate of samples with damage (the slope of the fitting curves shown in Fig.10)under the confining pressure of 5,10,15 and 20 MPa is -210.4,-183.4,-213.6 and -228.6,respectively,which indicates that confining pressure has less effect on the peak strength degradation of cracked samples with different damage.

(2)Evolution of residual strength

As listed in Table 4,the relationship between the residual strength and the damage of cracked samples under different confining pressure has no obvious regularity.The ratio of residual strength and peak strength(σr/σp)may serve as an index to assess the bearing capability of cracked samples after the re-failure.The relationship between σr/σpand the cracked sample damage is shown in Fig.11.

It can be seen from Fig.11 that the ratio of σr/σpincreases with damage in quadratic parabola.Therefore,the degradation of residual strength compared to peak strength decreases with damage,showing significant strain hardening characteristics.

Table 4 Results of cracked samples in re-loading tests.

Fig.7.Relationship between peak strength and confining pressure of cracked samples.

Table 5 Fitting results of the relationship between peak strength and confining pressure of cracked samples.

Fig.8.Relationship between peak strength degradation and confining pressure of cracked samples with different damage.

Fig.9.Relationship between residual strength and confining pressure of cracked samples.

(3)Evolution of equivalent cohesion and internal friction angle

As shown in Fig.12,equivalent cohesion of cracked samples decreases with damage in the exponential decay curve,which is consistent with the research results in study by Niu et al.[20].There is a quadratic parabola relationship between equivalent internal friction angle and damage.When the damage is less than 0.345,the equivalent internal friction of cracked samples changes little with damage,having a maximum of 34.865°and a minimum of 30.260°,while it decreases rapidly when the damage is greater than 0.345.

Fig.10.Relationship between peak strength and damage.

Fig.11.Relationship between σr/σpand damage.

Fig.12.Relationship between equivalent cohesion,internal friction angle and damage.

Fig.13.Corresponding relation between surrounding rock units in broken zone and complete stress-strain curve.

4.Application of test results

In this paper,the evolution laws of cracked surrounding rock in the broken zone of a deep roadway were revealed by re-loading tests of cracked samples with different damage.The corresponding relationship between surrounding rock units in broken zone and complete stress-strain curves was established as shown in Fig.13.Test results can be applied in the following two aspects.

(1)The bearing capacity of the surrounding rock at different positions in the broken zone can be obtained according to the test results and corresponding relation,which will provide the basic data for failure mechanisms analysis and roadway support design.

(2)The strength parameters attenuation model of cracked rock samples was established according to the test results.On this basis,the post-peak strength parameters attenuation model of cracked samples represented by plastic shear strain was obtained by substituting the plastic shear strain into the attenuation model,which can be applied to theoretical and numerical simulation analysis for failure mechanisms of surrounding rock in deep roadway.

5.Conclusions

(1)The peak strength and residual strength of intact samples under triaxial compression increase linearly with the increasing confining pressure.Based on the dilatancy characteristics showed in the loading processes,the damage constitutive model was established to conduct the real-time and quantitative evaluation of the samples damage state.

(2)Cracked samples with different damage were obtained by pre-peak,peak point,post-peak and residual strength stage unloading tests in the loading process of intact samples,which could be used in the re-loading tests to study the strength properties and evolution laws of cracked samples.

(3)The slope of the stress-strain curves,peak strength and residual strength of cracked samples increased linearly with confining pressure and decreased linearly with damage value.The cracked samples T-7 to T-34 with low damage(less than 0.345)show strain softening characteristics,while the mechanics properties of samples transformed from strain softening to strain hardening with the damage when the damage is greater than 0.345.

(4)Equivalent cohesion of cracked samples decreases with damage in the exponential decay curve,while the relationship between the equivalent internal friction and damage is consistent with a quadratic parabola.

Acknowledgments

Financial support for this work is gratefully acknowledged,provided by the National Natural Science Foundation of China (Nos.51323004,51574223),the China Postdoctoral Science Foundation(No.2015M571843),the Natural Science Foundation of Jiangsu Province of China(No.BK20160208),the University Nature Science Research Projects of Jiangsu Province of China (No.17KJB440003)and the Open Research Fund of Research Center of Jiangsu Collaborative Innovation Center for Building Energy Saving and Construction Technology of China (No.SJXTY1502).