Introduction

In underground engineering, tunnel excavation leads to stress redistribution and plastic deformation of rock mass. The deformation of soft rock can reach thousands of millimeters, while hard rock may have rock burst or dilatancy. Above phenomena cause serious damage to the supporting structure and bring serial problems to the tunnel support and construction1,2. In order to ensure the safety and stability of tunnel, it is of great significance to further study the mechanical property and failure mechanism of surrounding rock3.

According to the failure process, the existing researches mainly focus on three aspects of the complete failure process of the rock, i.e., the elastic stage before failure stage, the state reaching the peak strength, and the post-failure stage after the peak strength. In the elastic stage, the deformation of the rock is mainly manifested as reversible elastic deformation, and the state of peak strength reaches the maximum value of the rock strength, when the internal structure of the rock begins to change significantly. These two aspects have been discussed by numerous researches, and comparatively, the researches on the post-failure stage are not that sufficient.

Plenty laboratory uniaxial, triaxial and large-scale field tests show that after the rock or rock mass reaches the peak stress stage, the subsequent deformation leads to the rapid deterioration of the strength of rock or rock mass, which is called ‘strain-softening’ behaviour4. Some researchers analysed the variation law of strength parameters of rock during the strain-softening stage. Based on the Mohr-Coulomb criterion, Martin and Chandler5 established the relationship between the strength parameter of rock and the damage variable. It is believed that the increase of plastic strain leads to a gradual decrease of cohesion, while the internal friction angle first increases and then decreases.

Hajiabdolmajid et al.6 proposed the CFWS (cohesion weakening and friction strengthening) model, which considered that during the the plastic stage, cohesion decreases linearly while the friction angle increases linearly with the increase of plastic shear strain. According to the experimental data of sandstone, mudstone and shaly sandstone, Pourhosseini and Shabanimashcool7 presented a CWFC (cohesion weakening and friction constant) model. Jiang et al.8 and Min et al.9 proposed a nonlinear strain-softening model, which includes a nonlinear variation of elastic modulus, cohesion and internal friction angle under different confining pressure and plastic strain. Based on Hoek-Brown failure criterion, Peng et al.10,11 and He et al.12 emphasised rock damage mechanical model and focused on the residual strength of rock. Walton et al.13 and Zhao et al.14 analysed the dilatancy characteristics of limestone, quartzite, sandstone and coal rock in the strain-softening stage from brittleness to plasticity.

Some researches applied the strain-softening behaviour into the tunnel. For example, Xiao et al.15 considered a unified strain-hardening and strain-softening elastoplastic constitutive model and investigated nonlinear characteristics of the stress–strain curve, peak strength and residual strength. Based on the Hoek-Brown failure criterion, Cui et al.16,17 presented a new calculation procedure for strain-softening behaviour in rock mass for deep tunnels. Wu et al.18 proposed an improved strain-softening model, which effectively described the nonlinear evolution of residual strength of rock mass surrounding the tunnels under the effect of confining stress.

The above researches investigated the strain-softening behaviour of rock or rock mass from a macroscopic standpoint, without revealing the failure process and mechanism of rock at the microscopic level. Essentially, the mesoscopic numerical simulation method plays an important role in illustrating the deformation failure and crack propagation processes in rock.

Based on the discrete element method UDEC, Gao et al.19 used the grain model to numerically analyse the mesoscopic cracks propagation behaviour of rock crushable grains. By the discrete element method PFC, Sinha and Walton20 simulated the triaxial compression test of brittle rock on the basis of the bonded block model (BBM) and analysed the progressive failure process and crack propagation evolution of brittle rock. Potyondy21 and Baud et al.22 studied the influence of pore defects on rock strength and crack propagation. Saadat and Taheri23 simulated the progressive strain-softening response of grain contact in granite from the mesoscopic level, effectively demonstrating the macroscopic behavior of granite sample.

Overall, in the failure process of the rock, the evolutions of the mechanical parameters of the hard and soft rocks in the strain-softening stage under various confining stresses are totally different24,25,26,27,28. However, current researches have not comprehensively investigated the strain-softening behaviours of hard and soft rocks from both the macroscopic and mesoscopic perspectives.

In order to overcome the deficiency, in this analysis, from the macroscopic perspective, on the basis of stress–strain curves of granite and sandstone under different confining pressures through laboratory triaxial compression test, the peak strength, residual strength, and the elastic modulus of granite and sandstone are derived. The critical plastic shear strain and post-peak reduction modulus of granite and sandstone are discussed. The evolution of strength parameters versus the plastic shear strain in the strain-softening stage is proposed. Based on these relations, the method to develop the strain-softening model of hard and soft rocks is presented. From the mesoscopic perspective, according to the results of laboratory triaxial compression test, the parameter calibration of discrete element method PFC for the samples of the granite and sandstone is completed. The evolutions of mesoscopic crack propagation and mesoscopic particle displacement field in the complete failure process are solved out. The evolution of mesoscopic crack propagation reaching the residual stage with different confining pressure is discussed. Combining the mesoscopic crack propagation, typical stresses in the failure stage such as the points of initial crack. dilatancy, peak stress, residual stress are investigated. Through the combination of macroscopic experimental method and mesoscopic numerical method, the strain-softening behaviour of hard and soft rocks is thoroughly analysed.

Experiments of granite and sandstone

In this analysis, granite and sandstone are regarded as the typical rocks of hard rock and soft rock, respectively. Granite and sandstone are made into cylindrical standard samples with a diameter of 50 mm and a height of 100 mm respectively, as shown in Fig. 1. The sample minerals of granite consist mainly of quartz, feldspar (potassium feldspar and plagioclase), and minor amounts of dark minerals such as biotite and hornblende. The sample minerals of sandstone consist mainly of quartz grains, feldspar fragments, and small amounts of mica and cementation such as calcareous or siliceous. The two groups of samples have no obvious cracks on the surface. The sampling process is standardized and meticulous, showing good integrity and conforming to the test standards.

Fig. 1
figure 1

Rock sample.

Full size image

Triaxial compression tests are conducted by rock test system that self-developed by Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan. As illustrated in Fig. 2, the system is mainly composed of self-balancing triaxial pressure chamber, high pressure servo pump, deformation measurement system, motor and loading control system. Granite sample and sandstone sample are each divided into groups to carry on triaxial compression tests under different confining pressures, with the triaxial compression loading rate of 1 × 10− 5 m / s. The physical and mechanical parameters of granite and sandstone are plotted in Table 1.

Fig. 2
figure 2

Rock test system.

Full size image
Table 1 Physical and mechanical parameters of rock samples.
Full size table

Figures 3 and 4 show the stress–strain curves for granite sample and sandstone sample by triaxial compression tests, respectively. Under low confining pressure, there is a significant brittle drop after the peak of granite sample. As the confining pressure increases, the post-peak ductility characteristic increases. In comparison to the granite sample, the post-peak ductility of sandstone sample is more remarkable under the same confining pressure, with the smaller variation between peak strength and residual strength.

Fig. 3
figure 3

Stress–strain curves for granite by triaxial compression tests.

Full size image
Fig. 4
figure 4

Stress–strain curves for sandstone by triaxial compression tests.

Full size image

Figures 5 and 6 show the failure conditions of granite sample and sandstone sample respectively. When confining stress is 0, granite sample exhibits multiple tensile cracks parallel to the axial direction, which are connected to multiple shear cracks. With the increase of confining pressure, shear cracks prevail in the sample. Different from granite sample, shear cracks are predominant in sandstone sample under all the confining pressures, and there is merely an oblique shear band. As the confining pressure increases, the longitudinal associated cracks appear at a certain angle to the oblique shear band. The angle between the shear crack of the sample and the horizontal surface is defined as the shear damage angle, which is in the range of 0° to 90°.

According to Figs. 5 and 6, as the confining pressure increases, the shear damage angle of granite sample and sandstone sample gradually decreases. The shear damage angle of granite sample is distributed between 68 ° and 83 °, while that of sandstone sample ranges from 46 ° to 72 °, indicating that the shear damage angle of granite sample is greater than that of sandstone sample under different confining pressures.

Fig. 5
figure 5

Failure condition of granite sample.

Full size image
Fig. 6
figure 6

Failure condition of sandstone sample.

Full size image

Strain-softening model of granite and sandstone

Post-peak reduction modulus

The post-peak reduction modulus M is defined as the average negative slope of the curve from peak stage to residual stage, which is a key parameter to study the strain-softening behaviour, as shown in Fig. 7. The greater M means more significant drop brittleness in the strain-softening stage and greater difference between peak strength and residual strength. When M is infinitely and 0, the strain-softening behaviour corresponds to two extreme conditions, one is elastic-brittle-plastic and the other is elastic-perfectly-plastic. Figure 8 declares the fitting results of post-peak reduction modulus for granite sample and sandstone sample by triaxial compression tests.

Fig. 7
figure 7

Schematic diagram of rock post-peak reduction modulus.

Full size image
Fig. 8
figure 8

Fitting results of rock post-peak reduction modulus versus confining pressure for granite sample and sandstone sample.

Full size image

It can be found from Fig. 8 that the post-peak reduction modulus of granite sample significantly exceeds that of sandstone sample, which indicates granite sample strength decreases rapidly without an obvious softening stage after reaching peak failure. Instead, due to higher sensitivity to confining pressure, already small post-peak reduction modulus of sandstone sample keeps decreasing under increasing confining pressure. In this process, the ductile plastic characteristic of sandstone sample gradually appears.

Critical plastic shear strain

On the basis of the whole stress–strain curves obtained from triaxial compression tests, the critical plastic shear strain of granite sample and sandstone sample under different confining pressures is analysed. As shown in Fig. 9, similar to other researches29,30,31, the definition of plastic shear strain here is the difference between the maximum and minimum plastic principal strain:

$${\eta _{\mathrm{i}}}=\varepsilon _{{{\mathrm{1,i}}}}^{{\text{p}}} - \varepsilon _{{{\text{3,i}}}}^{{\text{p}}}$$
(1)

where ηi represents the plastic shear strain at a point in the post-peak stage. \(\varepsilon _{{1,\text{i}}}^{\text{p}}\) is the maximum plastic principal strain in the post-peak stage, which means, the difference between the maximum total strain and the maximum elastic strain at this point. \(\varepsilon _{{3,\text{i}}}^{\text{p}}\) represents the minimum plastic principal strain in the post-peak stage, in other words, the difference between the minimum total strain and the minimum elastic strain at this point29,30,31.

Fig. 9
figure 9

Schematic diagram for definition of plastic shear strain.

Full size image

As shown in Fig. 9, the critical plastic shear strain η* is defined as the plastic shear strain when the rock reaches the residual stage:

$${\eta ^ * }=\varepsilon _{{1,\operatorname{res} }}^{{\text{p}}} - \varepsilon _{{{\text{3,res}}}}^{{\text{p}}}$$
(2)

where \(\varepsilon _{{1,\text{res}}}^{\text{p}}\) is the maximum plastic principal strain when reaching the residual stage, \(\varepsilon _{{3,\text{res}}}^{\text{p}}\) is the minimum plastic principal strain when reaching the residual stage. E1 is the axial modulus of elasticity and E3 is the circumflex modulus of elasticity.

From the Fig. 9 and Eq. (2), it is found that the smaller η* is, the faster rock reaches residual stage, while, the larger η* is, the later the rock reaches residual stage. Figure 10 describes the fitting results of the critical plastic shear strain of granite sample and sandstone sample. As can be seen from Fig. 10, consistent with most conclusions32,33, with the increase of confining stress, the critical plastic shear strain gradually reduces, and the stress–strain curve reaches the residual stage more quickly.

In general, granite, as brittle rock, is more prone to brittle failure than sandstone, which means the critical plastic shear strain of granite should be smaller than that of sandstone under the same confining pressure. However, as shown in Fig. 10, the critical plastic shear strain of granite sample exceeds that of sandstone sample slightly. This may be due to the large energy released by granite sample during the peak failure in triaxial compression test, which results in a larger circumferential strain measurement value, and ultimately a larger critical plastic shear strain value of granite sample.

Fig. 10
figure 10

Fitting results of critical plastic shear strain versus confining stress for for granite sample and sandstone sample.

Full size image

Strength parameters at peak and residual stage

On the basis of M-C strength criterion (Eq. (3)) and H-B strength criterion (Eq. (4)), the strength parameters at peak and residual stages of granite and sandstone samples are analysed.

$${\sigma _1}=\frac{{1+\sin \varphi }}{{1 - \sin \varphi }}{\sigma _3}+\frac{{2c\cos \varphi }}{{1 - \sin \varphi }}$$
(3)

where σ1 and σ3 represent the rock maximum and minimum principal stresses at failure stage respectively, and c is cohesion of rock, φ is internal friction angle of rock.

$${\sigma _{\text{1}}}={\sigma _{\text{3}}}+{\sigma _{{\text{ci}}}}{\left( {{m_{\text{b}}}{\sigma _3}/{\sigma _{{\text{ci}}}}+s} \right)^a}$$
(4)

where σ1 and σ3 represent the rock maximum and minimum principal stresses at failure stage respectively, and σci is the uniaxial compressive strength of intact rock, mb, s, α are rock mass quality coefficients. The expressions are as follows:

$${m_{\text{b}}}={m_{\text{i}}}\exp \left( {\frac{{{\text{GSI}} - 100}}{{28 - 14D}}} \right)$$
(5)
$$s=\exp \left( {\frac{{{\text{GSI}} - 100}}{{9 - 3D}}} \right)$$
(6)
$$a=\frac{1}{2}+\frac{1}{6}\left[ {\exp \left( {\frac{{ - {\text{GSI}}}}{{15}}} \right) - \exp \left( {\frac{{ - 20}}{3}} \right)} \right]$$
(7)

Intact rocks are used in the laboratory tests. For the intact rock have no subjected to blast damage and stress relaxation, D is taken as 0, geological strength index (GSI) is regarded as 100, mb equals mi, s and α are taken as 1 and 0.5 respectively. Then, the corresponding generalised H-B strength criterion can be simplified as:

$${\sigma _1}={\sigma _3}+{\sigma _{{\text{ci}}}}\sqrt {{m_{\text{i}}}{\sigma _{\text{3}}}{\text{/}}{\sigma _{{\text{ci}}}}+1}$$
(8)

By substituting the peak compressive strength of the rock sample under different confining pressures into Eq. (8), the strength parameters of the intact rock mi and σci can be obtained. When the rock exceeds the peak point and reaches the residual stage, GSI decreases from 100 to the smaller value GSIr (i.e., GSI in the residual stage). Since the parameters mi and σci are given, Eq. (4) can be rewritten as:

$${\sigma _{\text{1}}}={\sigma _{\text{3}}}+{\sigma _{{\text{ci}}}}{\left( {{\sigma _3}{m_{\text{i}}}\exp \left( {\frac{{{\text{GS}}{{\text{I}}^{\text{r}}} - 100}}{{28}}} \right)/{\sigma _{{\text{ci}}}}+\exp \left( {\frac{{{\text{GS}}{{\text{I}}^{\text{r}}} - 100}}{9}} \right)} \right)^{\frac{1}{2}+\frac{1}{6}\left[ {\exp \left( {\frac{{ - {\text{GS}}{{\text{I}}^{\text{r}}}}}{{15}}} \right) - \exp \left( {\frac{{ - 20}}{3}} \right)} \right]}}$$
(9)

By substituting the residual compressive strength of the sample under different confining pressures into Eq. (9), the GSIr, while reaching the residual stage, can be solved.

According to the stress–strain curves of granite sample and sandstone sample in Figs. 3 and 4, the peak compressive strength σ1peak and residual compressive strength σ1res under different confining pressures are collected. Based on M-C and H-B strength criteria, the results are linearly fitted respectively. The results are shown in Fig. 11, and obtained peak strength parameters and residual strength parameters of M-C and H-B strength criteria are listed in Tables 2 and 3.

Fig. 11
figure 11

Peak, residual compressive strengths versus confining stress for (a) granite sample and (b) sandstone sample.

Full size image
Table 2 Strength parameters of M-C and H-B strength criteria for granite sample.
Full size table
Table 3 Strength parameters of M-C and H-B strength criteria for sandstone sample.
Full size table

It is observed from Table 2 that strength drop of granite sample from peak stage to residual stage is significantly greater than that of sandstone sample. When granite sample reaches the residual stage, the corresponding GSIr is 61.65, which is less than 82.2 of sandstone sample at same stage. This means that the fragmentation degree of granite sample in the test is greater than that of sandstone sample, which is consistent with Figs. 5 and 6: under uniaxial compression, there are more cracks of granite sample than that of sandstone sample.

As shown in Tables 2 and 3, after the peak failure of granite sample and sandstone sample, the cohesion decreases rapidly while internal friction angle changes slightly. In other words, loss of cohesion due to particle crushing is the main reason for descended strength in the strain-softening stage while the internal friction force varies little in this stage.

Evolution of strength parameters in softening stage

Based on the M-C strength criterion, the evolution of strength parameters, such as internal friction angle and cohesion, are derived by following steps:

  1. (1)

    Taking granite sample as an example, the relationship between the maximum principal stress σ1 and plastic shear strain η under different confining pressures is established (Fig. 12a).

  2. (2)

    The relationship between the maximum principal stress σ1 and confining pressure σ3 under different plastic shear strains η is proposed (Fig. 12b).

  3. (3)

    Based on the above relationship, the evolutions of cohesion c and internal friction angle φ of granite sample and sandstone sample with different plastic shear strains ηare obtained by using M-C strength criterion, which are shown in Fig. 13. The method can be referred to section “Strength parameters at peak and residual stage”.

Fig. 12
figure 12

Evolution law of rock strength parameters. (a) Maximum principal stress σ1 versus plastic shear strain η under different confining pressures σ3 for rock, (b) maximum principal stress σ1 versus confining pressure σ3 under different plastic shear strains η for rock

Full size image
Fig. 13
figure 13

Evolution of cohesion c and internal friction angle φ with plastic shear strain η for (a) granite sample and (b) sandstone sample.

Full size image

As shown in Fig. 13a, for the granite sample, with the plastic shear strain increasing, the cohesion gradually decreases, while the internal friction angle first increases to the peak and then decreases. The variation is similar to the research results of Martin and Chandler5 except that he believes that the internal friction angle is 0 when the plastic shear strain is 0. However, according to the relationship between granite strength and confining pressure, the internal friction angle value calculated in this analysis is similar to that of the residual internal friction angle.

According to Fig. 13b, similar to granite sample, for the sandstone sample, the cohesion also decreases with the increase of plastic shear strain. However, when plastic shear strain enhances, internal friction angle grows with the increasing rate continuously decreasing. This phenomenon is similar to the cohesion weakening and friction strengthening model (CWFS) that proposed by Hajiabdolmajid et al.6, showing a trend of gradually decreasing cohesive and increasing internal friction angle with the increase of plastic shear strain.

The above conclusions show that the evolution of rock strength parameters in the failure process is strongly related to rock types, and hard rock and soft rock exhibit completely different variations.

Method to develop strain-softening model

The method to obtain the strain-softening model can be summarised as,

  1. (1)

    Based on the stress–strain curves, obtain the post-peak reduction modulus and critical plastic shear strain with different confining stresses. Gain the variation of the maximum principal stress versus the minimum principal stress (i.e., the confining pressure) at the peak stage and residual stage. Obtain the variation of the maximum principal stress versus the minimum principal stress with different plastic shear strains in the strain-softening stage.

  2. (2)

    Based on the variation of the maximum principal stress versus the minimum principal stress (i.e., the confining stress) at the peak stage and residual stage, obtain the strength parameters by fitting M-C and H-B strength criteria.

  3. (3)

    According to the variation of the maximum principal stress versus the minimum principal stress with different plastic shear strains, obtain the evolution of the strength parameters in the strain-softening stage.

The following chart is shown in Fig. 14.

Fig. 14
figure 14

Flow chart for method to develop strain-softening model.

Full size image

Failure processes of granite and sandstone by discrete element method

Numerical modelling

In this section, the discrete element method PFC2D34 is utilised to simulate the uniaxial compression test, biaxial compression test and direct tensile test of granite model and sandstone model. The size of the numerical model sample is 50 mm × 100 mm. Too few particles in the model will cause the numerical results to not accurately reflect the actual situation, too many particles will lead to too long time for numerical test calculation, and the particle radius in the model is set between 0.4 ~ 0.6 mm. The schematic diagram of the particle model is shown in Fig. 15. In the numerical simulation of the uniaxial compression test, the loading rate of the upper and lower loading plates is 1 × 10− 3 m/s.

Fig. 15
figure 15

Schematic diagram of numerical model.

Full size image

Calibration of parameters

Based on the laboratory test results of granite sample and sandstone sample under different confining pressures, the mesoscopic parameters of flat-joint model are used to calibrate the macroscopic mechanical parameters of granite model and sandstone model. The calibration process of the mesoscopic parameters is shown in Fig. 16.

Firstly, the macroscopic parameters such as elastic modulus and Poisson’s ratio are adopted to calibrate the mesoscopic parameters such as normal contact stiffness of particles kn, the ratio of normal contact stiffness to shear contact stiffness kn/ks, normal contact stiffness of the contact model \(\overline{k_{\mathrm{n}}}\), and the ratio of normal contact stiffness to shear contact stiffness of the contact model \(\overline{k_{\mathrm{n}}} /\overline{k_{\mathrm{s}}}\).

Secondly, the tensile strength \(\overline{\sigma _t}\) and cohesion \(\overline{\sigma _c}\) of contact model are calibrated according to the tensile strength and uniaxial compressive strength.

Finally, according to the peak compressive strength under different confining pressures, the friction coefficient of the particles u and the internal friction angle of the model \(\overline {\varphi }\) are calibrated. Then, the calibration of mesoscopic parameters is accomplished, the process is displayed in Table 4.

Fig. 16
figure 16

Mesoscopic parameters calibration process.

Full size image
Table 4 Calibration of mesoscopic parameters of the flat-joint model.
Full size table

Based on the above calibration results, numerical simulations of triaxial compression tests of granite model and sandstone model are conducted. Figure 17 shows the comparison of the stress–strain curves with different confining stresses for granite model and sandstone model by experimental method and discrete element method.

Fig. 17
figure 17

Comparison of the stress–strain curves by experimental method and discrete element method. (a) granite model, (b) sandstone model

Full size image

For both granite sample and sandstone sample under different confining pressures, the mechanical parameters of numerical simulation are similar to the laboratory test results. The curves of the two methods fit well, which means the mesoscopic mechanical parameters in the flat-joint model can reasonably simulate the macroscopic characteristic of granite and sandstone.

Analysis of mesoscopic crack propagation of granite and sandstone

Evolution of mesoscopic crack propagation in the complete failure process

By the discrete element method, Figs. 18 and 19 respectively show the axial stress, volumetric strain and number of mesoscopic cracks of granite model and sandstone model versus the vertical strain when the confining stress is 5 MPa. The evolution of mesoscopic cracks propagation for granite model and sandstone model is added in Figs. 18 and 19 as well. It is noted that the mesoscopic cracks contain two types, i.e., the tension cracks and shear cracks. The number of the tension and shear cracks can be gained by programming in MATLAB35. The blue line represents the tensile crack while the green line represents the shear crack.

Fig. 18
figure 18

Axial stress, volumetric strain and number of mesoscopic cracks of granite model versus the vertical strain when the confining stress is 5 MPa.

Full size image
Fig. 19
figure 19

Axial stress, volumetric strain and number of mesoscopic cracks of sandstone model versus the vertical strain when the confining stress is 5 MPa.

Full size image

To clearly reflect the mesoscopic behaviour of the entire process of rock failure, five feature points are defined, namely the point of initial crack (A), point of dilatancy (B), point of peak stress (C), point of zero volumetric strain (D), and point of residual stress (E). Correspondingly, OA refers to the stage of rock compaction stage, AB represents the stage of crack initiation stage, BC refers to the stage of stable crack propagation, and CE represents the rapid crack propagation.

As B is the point of dilatancy, the volume of the model begins to dilate after B. It is shown from Figs. 18 and 19 that, for all the confining stresses and for both the granite model and sandstone model, the number of the shear cracks starts to become larger than zero from the point of B. This implies that the dilatancy behaviour of the rock is dominated by the shear slip between the particles of rock, and the point of dilatancy can also be regarded as the initiation of shear cracks. As shown in the Figs. 18 and 19, compared to granite model, the sandstone model exhibits more shear cracks throughout the complete failure process. For the sandstone model, before the peak point, the cracks are mainly composed of the tensile cracks, but after the peak point, the shear crack fast increases.

Figure 20 plots the typical failure patterns by experimental method and discrete element method arriving at the residual stage when confining pressure is 5 MPa. As shown in Fig. 20, for confining pressure is 5 MPa, when reaching the residual stage, the granite model and sandstone model represent completely different failure patterns. The granite model exhibits a single shear band running through the model top to the bottom, which is a typical shear failure pattern. On the other hand, the sandstone model shows multiple slightly shorter shear bands, which is a wedge failure pattern. These two failure patterns are consistent with the macroscopic experimental results (Fig. 20).

Fig. 20
figure 20

Typical failure patterns by experimental method and discrete element method element method arriving at the residual stage when confining pressure is 5 MPa (a) for granite and (b) for sandstone.

Full size image

Figure 21 shows the displacement vectors of the granite model and sandstone model at five feature points under the confining pressure of 5 MPa. When the granite model reaches the point of zero volumetric strain, two clear parallel shear bands appear on the left and right sides. Along the two bands, the particles in the model are divided into three regions. Movement directions of the particles of the left and right regions roughly parallel to the shear band. The deformation in the model is mainly caused by the relative sliding of particles in the three regions along the shear band. When granite model reaches the residual stage, the deformation and failure are greater, and the two shear bands decreases to a single evident shear band. Thus, shear failure pattern occurs in the granite model. On the other hand, when the sandstone model reaches the point of zero volumetric strain, four shear bands appears, dividing the model into five regions, one region is the internal rhomboid shaped region, and the other four are the outer regions at the four corners. Four shear bands divide the five regions. The deformations in the outer regions, which are caused by the sliding along the shear bands, are more remarkable than the internal rhomboid shaped region. As a result, when reaching the residual stage, wedge failure pattern occurs in the sandstone model. During the process from the point of peak stress to the point of zero volumetric strain, the maximum displacement of granite model particles is smaller than that of sandstone model particles. The former is 5.2 × 10− 5 mm and the latter is 14.5 × 10− 5 mm.

The above phenomenon indicates that after entering the softening stage, small strain occurs and shear band appears in granite model, followed by immediately splitting failure. However, in the sandstone model, the larger plastic strain is required to form the shear band.

Fig. 21
figure 21

Displacement vectors of the granite model and sandstone model at five feature points under the confining pressure of 5 MPa (a) for granite model, (b) sandstone model.

Full size image

Evolution of mesoscopic crack propagation reaching the residual stage versus confining stresses

Figure 22 shows the evolution of mesoscopic crack propagation of the granite model and sandstone model reaching the residual stage under different confining pressures. As shown in Fig. 22, the blue line represents tensile crack while the red line represents shear crack. It can be observed from Fig. 22a that the tensile cracks prevail in the granite model under low confining pressure. As the confining pressure gradually increases, the oblique shear band becomes more prominent, and the number of small shear bands also increases. As the confining pressure increases to 20 MPa, it is found that the shear failure pattern is predominant in the granite model. By the experimental method, under the uniaxial compression or low confining pressure, multiple tensile cracks which are parallel to the axial direction appear in the granite sample. However, as the confining pressure increases, the sample still experiences shear failure pattern. This indicates that the results of the mesoscopic discrete element method are consistent with those of macroscopic experimental method.

Fig. 22
figure 22

Evolution of mesoscopic crack propagation with different confining stresses for (a) granite model and (b) sandstone model.

Full size image

The failure pattern of sandstone model is completely different from that of the granite model. With different confining stress, the granite model primarily undergoes shear failure pattern. With the increase of the confining pressure, the sandstone model initially exhibits the wedge failure pattern but latter the shear failure pattern. Specifically, under low confining pressure, the sandstone model undergoes wedge failure pattern due to the formation of four shear bands. However, as the confining pressure becomes larger, the wedge failure pattern is gradually replaced with the shear failure pattern by appearing merely one oblique shear band, and as the confining stress exceeds 15 MPa, another shear band becomes large and evident. And the shear bands may intersect with each other, resulting in the X-shaped shear failure, which coincidences with the experimental results.

Both the mesoscopic discrete element method and macroscopic experimental method indicate that when the residual stage is reached, the angle between the shear band and the horizontal surface of sandstone model is smaller than that of granite model. The mutual long and short shear bands occur among multiple blocks in the sandstone model, whereas for the granite model, a predominant shear band penetrates from the top of the granite model to the bottom.

The reason behind the observation is that, in the experiment when reaching the residual stage, the difference between the confining pressure and axial stress of sandstone model is not significant, whereas the axial stress of the granite model is several times more than its confining pressure. Therefore, the failure pattern of sandstone is influenced by both confining pressure and axial stress, while granite is mainly affected by axial stress.

Typical stress of failure process

Four feature points are analysed in the entire failure process, i.e., stress at the point of initial crack σci, stress at the point of dilatancy σcd, stress at the point of peak stress σpeak and stress at the point of residual stress σres. The dimensionless forms of the four typical stresses are denoted as σci/σpeak, σcd/σpeak, σres /σpeak. The schematic graph of the typical stresses is plotted in Fig. 23. The relations between the three feature stresses and confining pressure are plotted in Figs. 24, 25 and 26. The related results of the typical stresses are displayed in Table 5.

Fig. 23
figure 23

Stress parameters calibration at feature points for (a) granite model and (b) sandstone model.

Full size image
Table 5 Stress parameters calibration at feature points for flat-joint model.
Full size table
Fig. 24
figure 24

Evolution of stress at the point of initial crack σci with different confining stresses for (a) granite model and (b) sandstone model.

Full size image

Figure 24 shows the evolution of stress at the point of initial crack with different confining stresses for the granite model and sandstone model. According to Table 5, for the granite model, the stress at the point of initial crack σci under different confining pressures ranges from 24.3 MPa to 68.2 MPa. Meanwhile, the ratio of stress at the point of initial crack to stress at the point of peak stress σci/σpeak is distributed between 13.56% and 19.56%. As for the sandstone model, the stress at the point of initial crack under different confining pressures σci ranges from 6.6 MPa to 24.8 MPa, while the ratio of stress at the point of initial crack to stress at the point of peak stress σci/σpeak is between 26.13% and 33.23%.

The above phenomena indicate that, under various confining pressures, the granite model exhibits the greater stress at the point of initial crack than the sandstone model does. As for the ratio of stress at the point of initial crack to stress at the point of peak stress σci/σpeak, value of the former is smaller than that of the latter. This shows that the granite model exhibits microcracks earlier than the sandstone model under confining pressure.

Figure 25 shows the evolution of stress at the point of dilatancy with different confining stresses for the granite model and sandstone model. For the granite model, the stress at the point of dilatancy σcd ranges from 153.1 MPa to 328.6 MPa. The ratio of the stress at the point of dilatancy to stress at the point of peak stress σcd /σpeak ranges from 91.7 to 99.27%. Above high ratio value indicates that the granite model experiences peak failure immediately after reaching the point of dilatancy. Meanwhile, for sandstone model, the stress at the point of dilatancy σcd ranges from 9.2 MPa to 68.5 MPa, and the ratio of the stress at the point of dilatancy to stress at the point of peak stress σcd /σpeak is distributed between 36.65 and 91.63%. The above parameters both significantly increase with the increase of confining pressure, indicating that confining pressure has a remarkable impact on the dilatancy behaviour of the sandstone model. Under the low confining pressure, the sandstone model reaches the point of dilatancy earlier. As confining pressure increases, the point of dilatancy becomes closer to the peak failure point.

Fig. 25
figure 25

Evolution of stress at the point of dilatancy σcd with different confining stresses for (a) granite model and (b) sandstone model.

Full size image

Figure 26 describes evolution of stress at the point of residual stress σres with different confining stresses for the granite model and sandstone model. Under different confining pressures, for the granite model, the stress at the point of residual stress σres is between 0.1 MPa to 89.1 MPa. The ratio of the stress at the point of residual stress to stress at the point of peak stress σres /σpeak ranges from 0.03 to 25.56%. As shown in Fig. 26a, the granite model exhibits a highly linear fit between the stress at the point of residual stress and confining stress, with a correlation coefficient as high as 97%. The stress at the point of residual stress keeps increasing with the increase of confining pressure. On the other hand, for the sandstone model, the stress at the point of residual stress σres ranges from 0.1 MPa to 20.7 MPa, and the ratio of the stress at the point of residual stress at the point of peak stress σres /σpeak ranges from 0.26 to 37.33%. Figure 26b describes that the relationship between the stress at the point of residual stress and confining pressure in the sandstone model is nonlinear. When the confining pressure increases from 0 to 15 MPa, the stress at the point of residual stress increases with the increase of confining pressure. Once the confining pressure exceeds 15 MPa, the residual stress of the sandstone model begins to drop with the increase of confining pressure. Under the uniaxial compression condition, both the granite model and sandstone model exhibit obvious brittle rock characteristic. After reaching the point of peak stress, the strength of granite and sandstone rapidly drops to nearly zero.

Fig. 26
figure 26

Evolution of stress at the point of residual stress σres with different confining stresses for (a) granite model and (b) sandstone model.

Full size image

Conclusion

From the macroscopic perspective, on the basis of stress–strain curves, the strength at different stages and the elastic modulus of granite and sandstone are derived. The evolutions of strength parameters versus the plastic shear strain in the strain-softening stage are proposed. Furthermore, the method to develop the strain-softening model of hard and soft rocks is presented. From the mesoscopic perspective, the parameter calibration for the samples of the granite and sandstone is completed by discrete element method PFC. The evolutions of mesoscopic crack propagation and particle displacement field in the complete failure process are solved out. Typical stresses in the failure stage are investigated. Through the combination of macroscopic experimental method and mesoscopic numerical method, the strain-softening behaviour of hard and soft rocks is thoroughly analysed. The conclusions are as follows:

  1. (1)

    In the strain-softening stage (from the peak state to the residual state), the strengths of hard rock and soft rock exhibit different drop degrees. Both the drop degree and the fragmentation degree of hard rock are significantly greater than those of the soft rock. Loss of cohesion due to the particle crushing is the main reason for the descended strength in the strain-softening stage while the internal friction angle varies slightly in this stage.

  2. (2)

    The evolution of rock strength parameters in the failure process is strongly related to rock types, and hard rock and soft rock exhibit completely different variations. For the hard rock, with the plastic shear strain increasing, the cohesion gradually decreases, while the internal friction angle first increases to the peak and then decreases. For the soft rock, the cohesion also decreases with the increase of plastic shear strain. However, when plastic shear strain enhances, internal friction angle grows with the increasing rate continuously decreasing.

  3. (3)

    From the mesoscopic numerical method, it is found that the failure pattern of sandstone model is completely different from that of the granite model. The pattern of the granite model is basically not affected by the confining pressure, whereas that of the sandstone model is greatly influenced by the confining pressure. With different confining stress, the granite model primarily undergoes shear failure pattern. With the increase of the confining pressure, the sandstone model initially exhibits the wedge failure pattern but latter the shear failure pattern. These two failure patterns from the mesoscopic observations are consistent with the macroscopic experimental results.

  4. (4)

    Both the mesoscopic numerical method and macroscopic experimental method indicate that when the residual stage is reached, the angle between the shear band and the horizontal surface of sandstone model is smaller than that of granite model. The reason behind the observation is that, in the experiment when reaching the residual stage, the difference between the confining pressure and axial stress of sandstone model is not significant, whereas the axial stress of the granite model is several times more than its confining pressure. Therefore, the failure pattern of sandstone is influenced by both confining pressure and axial stress, while granite is mainly affected by axial stress.

  5. (5)

    Confining pressure has a remarkable impact on the dilatancy behaviour of the sandstone model. Under the low confining pressure, the sandstone model reaches the point of dilatancy earlier. As confining pressure increases, the point of dilatancy becomes closer to the peak failure point. In contrast, the effect of the confining pressure on the occurrence of dilatancy behaviour of the sandstone model is slight. For all the confining pressure, the granite model experiences peak failure immediately after reaching the point of dilatancy.

  6. (6)

    For the granite model, the stress at the point of residual stress is linearly increased with the increase of the confining stress. For the sandstone model, the stress at the point of residual stress initially increases but latter drops with the increase of the confining stress.