Study of Post-Peak Strain Softening Mechanical Behaviour of Rock Material Based on Hoek – Brown Criterion

In order to build the post-peak strain softeningmodel of rock, the evolution laws of rock parameters (m, s) were obtained by using the evolutionary mode of piecewise linear function regarding the maximum principle stress. Based on the nonlinear Hoek–Brown criterion, the analytical relationship of the rock strength parameters (m, s), cohesion (c), and friction angle (φ) has been developed by theoretical derivation. According to the analysis on the four different types of rock, it is found that, within the range from 0 to σ3min, the peak hardness of the rock becomes smaller as the confining pressure increases and the degree of rock fragmentation decreases as well. (e post-peak stress-strain curves obtained from the developed softening model are in good agreement with the laboratory test results under different confining pressures. In conclusion, the analytical method is reasonable, and it can predict the post-peak mechanical behaviour of rock well, which provides a new thought for the rocksoftening simulation.


Introduction
It is well known that the rock material is one type of heterogeneous brittle material with complex mechanical properties, which is significantly different from the metal material.In terms of post-peak strain softening characteristics, it is impossible to describe by the classical strength theory [1].Many researchers have studied from different aspects regarding the post-peak strain softening behaviour.Rummel and Fairhurst [2] were one of the earliest researchers to study the general capabilities of post-peak compression tests of the rock.Tutluoglu et al. [3] gathered a lot of data related to the postfailure part of the stressstrain curve to study the relationship between prefailure and postfailure mechanical properties of the rock material.But the confining pressure on the post-peak mechanical behaviour of rock was not considered.Furthermore, the laboratory tests [4][5][6][7][8] show that the confining pressure greatly affected the post-peak mechanical behaviour of rock.Meng et al. [9] proposed a new evaluation method based on the magnitude and velocity of the post-peak stress drop, which accurately accounts for the influence of the confining pressure on strain softening.Alejano and Alonso [10] proposed a conveniently simple formulation of the dilatancy angle, which is capable of representing the rock sample's strain softening behaviour in compressive tests.Walton et al. [11] performed a series of uniaxial and triaxial tests and studied post-peak behaviour about the strength, deformability, and dilatancy at low confinement.Fang and Harrison [12] utilized the concept of a degradation index to describe the variation in degradation that rock exhibits when it is subjected to various confining pressures.Zhao et al. [13] established evolution laws of the strength parameters of soft mudstone at the post-peak stage by considering stiffness degradation.By introducing the concept of generalized friction angle and generalized cohesion, Lu et al. [14] constructed the post-peak computing model combined with FLAC code.Leontiev and Huacasi [15] and Filho and Leontiev [16] have conducted research from different angles on the post-peak mechanics of rock.
In order to study the post-peak strain softening mechanical properties of rock, the mechanical models described above are based on the Mohr-Coulomb criterion.However, the extent of rock damage was not studied in their studies.It is seldom to establish the post-peak strain softening model based on the Hoek-Brown criterion.Tan et al. [17] considered degradation and expansion behaviour of the rock constructing the post-peak strain softening model based on the Hoek-Brown criterion.e rock parameters (m, s) in the Hoek-Brown criterion re ect the failure condition of rock reasonably.In this paper, piecewise linear function was used to obtain the evolution of rock strength parameters (m, s).Accordingly, the analytical relationships among rock strength parameters (m, s), friction angle (φ), and cohesion (c) were developed, and a new post-peak strain softening model was proposed.
e veri cation with laboratory test results shows that the analytical model is reasonable to predict the post-peak rock behaviour.

Simplification of Strain Softening Process
Figure 1 shows a series of stress-strain curves of Hawkesbury sandstone specimens under di erent con ning pressures [18].It can be observed that the rock failure is a staged development process.With the increase of con ning pressure, the rock post-peak strain softening behaviour is gradually transformed into ideal elastic-plastic behaviour.In order to process the main deformation and failure characteristics more informative in mathematics [19,20], Figure 1 has been simpli ed to Figure 2. In Figure 2, the dash line indicates the stress-strain curve obtained by the experiment and the solid line represents the ideal stress-strain curve.e strain softening model is typically divided into three stages: pre-peak elastic deformation stage, post-peak strain softening stage, and residual stage.

Analysis of the Rock Parameters of the Hoek-Brown Criterion and the Post-Peak Constitutive Model
In the post-peak deformation stage, the development of strain softening is induced by the change of the rock strength parameters [12][13][14].

Strength Criterion.
According to the Hoek-Brown criterion [21], where σ 1 is the e ective major principal stress, σ 3 is the e ective minor principal stress, η is the strain softening parameter, and σ ci is the uniaxial compressive strength of intact rock.m and s are rock strength parameters; s represents the degree of fragmentation of rock ranging from 0.0 to 1.0.For unfractured rock mass (i.e., rock), s 1.0.m represents the hardness of rock ranging from 0.0000001 to 25.For intact hard rock, m 2.5.

Strain Softening Parameters.
ere are usually two methods for selecting strain softening parameters.e rst one is based on the increment method: wherein _ ε p i (i 1, 2, 3) represents the changing rate of the ith principal plastic strain.
In another method, the strain softening parameter is considered as a function of the internal variable.e major principal strain ε 1 , the major principal plastic strain ε 1p , the plastic shear strain c 1 , and the equivalent plastic shear strain c 2 are frequently used in this method: ( As the rock parameters (m, s) vary with the increase of the major principal strain, the major principal strain ε 1 from the above method is used as the softening parameter in this paper, which leads to ease of analytical calculation.2 Advances in Civil Engineering

Analysis of Rock Strength Parameters (m, s) and the Post-Peak Constitutive Model.
e relationship between postpeak strength parameters and strain softening parameter, namely, the evolution of strength parameters, can be obtained by the laboratory experiment and numerical simulation.For simplicity, the mentioned relationship is often assumed as the form of a piecewise linear function [22][23][24][25], which can be expressed as where η, η p , and η r represent the strength parameter, strength parameter at the peak, and strength parameter at the starting point of the residual stage, respectively; c p and c * are the critical softening parameter at the peak and strength parameter at the starting point of the residual stage, as shown in Figure 3.
According to (4), the evolution of rock strength parameters (m, s) can be obtained: where m p and s p are the peak values of m and s; m r and s r are the peak residual values of m and s; and ε p and ε r are the major principal strain values at the peak and at the starting point of the residual stage, respectively.By substituting ( 5) and ( 6) into (1), the integration in (1) may be carried out analytically, and the post-peak strain softening constitutive model can be obtained accordingly.

Analytical Study of Hoek-Brown Parameters
Before calculation by ( 5) and ( 6), the strength parameters (m, s) at the peak and the starting point of the residual stage should be determined.In rock engineering, strength parameters (m, s) are generally determined by Geological Strength Index (GSI).However, it is di cult to use such a method to indicate strength parameters for rock specimens.erefore, the paper proposes to use the Mohr-Coulomb strength criterion to obtain strength parameters (c, φ) at the peak and at the starting point of the residual stage under di erent con ning pressures.e strength parameters (m, s) can be obtained via the relationship between themselves and the strength parameters (c, φ) afterwards.

Strength Parameters (c, φ) at the Peak and the Starting
Point of the Residual Stage.Cohesion c and friction angle φ at the peak and the starting point of the residual stage under di erent con ning pressures are various [26].e paper assumes that any point in the post-peak stage is satis ed with the Mohr-Coulomb ultimate failure condition during the elastic unloading process.In this case, the unloading path L is parallel with the secant M at the peak, as shown in Figure 4. e same plastic shear strain (ε ps ), under di erent con ning pressures and the same unloading path, corresponds to a series of di erent ultimate stress states: (σ 1 ′ , σ 3 ′ ), (σ 1 ″ , σ 3 ″ ), and (σ 1 ‴ , σ 3 ‴ ).By using the above method, the groups of di erent ultimate stress at the peak and the starting point of the residual stage can be obtained and displayed as Mohr stress circles in Figure 5.According to several Mohr stress circles, the envelope can be drawn.us, the corresponding cohesion c and friction angle φ at the peak and the starting point of the residual stage under di erent con ning pressures can be obtained by analysis of the Mohr strength straight lines through the tangent points of envelopes.

Analytical Derivation.
Uniaxial compressive strength of rock mass may be expressed as [27] Uniaxial tensile strength of rock mass may be written as [12] By combining with the Mohr-Coulomb criterion, cohesion c and friction angle φ can be obtained [28]: By combining ( 7)- (10), the relationship among m, s, σ ci , c, and φ can be obtained: Equations ( 11) and ( 12) can be modi ed to the following equations: where By combining ( 13)-( 16), the following results can be expressed: Substituting ( 17) and ( 18) into ( 15) and ( 16), the following equations are obtained: After substituting the strength parameters (c, φ) at the peak and the starting point of the residual stage into (19) and (20), the rock strength parameters (s, m) at the peak and the starting point of the residual stage can be calculated accordingly.
Equations ( 19) and (20) show that the rock strength parameters (s, m) are related to the uniaxial compressive strength of intact rock σ ci , friction angle φ, and cohesion c, which can be expressed as In terms of several groups of (σ 1 , σ 3 ) under di erent con ning pressures, σ ci is calculated [10]: where x σ 3 , y (σ 1 − σ 3 ) 2 , and n is the number of groups of data.

Application of the Developed Analytical Method.
In order to verify the developed post-peak strain softening model, the triaxial compression test results of the Hawkesbury sandstone [18] were used.e strength parameters (c, φ) at the peak and the starting point of the residual stage for the Hawkesbury sandstone were calculated via the above analytical method, and the results are shown in Table 1.According to (22), σ ci of the Hawkesbury sandstone is 21.21 MPa.Substituting the data in Table 1 into (19) and (20), the corresponding rock strength parameters (m, s) at the peak and the starting point of the residual stage are obtained and listed in Table 2.
It can be observed from Table 2 that the rock strength parameter s is greater than 1.0, which is contradicted with the de ned range of s (from 0 to 1.0).e reason is that the cohesion c and the friction angle φ are obtained based on the Mohr-Coulomb strength criterion.
ere is a maximum limit of con ning pressure (σ 3max ) for linear tting between the Mohr-Coulomb criterion and Hoek-Brown criterion [29], as shown in Figure 6.
Figure 5: Mohr diagram for the tested specimens at the same plastic shear strain ε ps under di erent con ning pressures [9].

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In terms of the upper limit of the maximum con ning pressure (σ 3max ) obtained in rock engineering, it is mainly in uenced by the engineering experience and classi cation.It is suggested in this paper that the value of the maximum con ning pressure (σ 3max ) can be reached when the strength parameter (s) value is equal to 1.0.In this elastic state, the rock specimen is in peak strength under the con ning pressure.Furthermore, the internal structure of the specimen is unfractured; therefore, Hoek-Brown and Mohr-Coulomb criteria could be used to t its stress-strain curve.
On the contrary, once this upper limit of the maximum con ning pressure is exceeded, it is unmeaningful to use the Hoek-Brown criterion or Mohr-Coulomb criterion since the calculated results display deviation from the real ones.In other words, it is no necessary to de ne a value to s.
As indicated in Tables 1 and 2, the cohesion and friction angle at the peak and the starting point of the residual stage change with the con ning pressure.It may be concluded that cohesion (c) and friction angle (φ) are a function of the con ning pressure σ 3 .By conducting secondary tting of c, ϕ, and σ 3 in Table 1, the tting result is shown in Figures 7  and 8, and the tting functions are Substituting ( 23) and ( 24) into ( 19), the maximum con ning pressure σ 3max is calculated to be 12.1 MPa when s 1 in (19).
Within a certain range of con ning pressure and increasing con ning pressure at the peak, m p becomes smaller while s p becomes greater.It can be concluded that the increase of con ning pressure results in smaller hardness at the peak and better rock quality.With the increase of con ning pressure, rock develops gradually from brittle to ductile, which results in increased plasticity, reduced brittleness, and reduced hardness.It is interesting to note that as a result of the increase of con ning pressure, the internal of the damage becomes smaller and the fragmentation in uences less when the rock reaches the peak.
By substituting the corresponding values of m and p for the con ning pressures of 2, 5, 8, and 12 MPa into ( 5) and ( 6), the analytical results of such two equations can be obtained and substituted into (1).us, the post-peak strain softening model of the Hawkesbury sandstone can be established.Figure 9 displays the comparison between the whole stress-strain curves of numerical simulation and triaxial compression tests for the Hawkesbury sandstone.

Discussion of Results.
In order to verify the analytical results, the same analytical method was carried out with laboratory results of the representative rock types: Indiana limestone [26], Tennessee marble [7], and Hegang granite [7].Table 3 presents the axial strains and the rock parameters (m, s) of di erent rock types corresponding to the peak strength and residual strength.e values of S p /S r are related to di erent con ning pressures.As can be seen from Figures 9-12, numerical simulation is consistent with experimental data trends within a certain range of con ning pressure, and the numerical simulation value agrees well with actual experimental values, especially for peak strength and residual strength.It indicates that the post-peak strain softening model established in the paper is reasonable, and it can accurately describe the mechanical post-peak behaviour of rock.
For better understanding of the post-peak softening characteristics and the degree of fragmentation and the degree of brittleness of rock, the brittleness index is proposed for S p /S r .Figure 13 illustrates the relationship among S p /S r , con ning pressure, and rock type.
As the con ning pressure decreases, S p /S r gradually decreases and tends to become stable nally.In other words, as the con ning pressure increases, the rock gradually develops brittleness to ductility.In uenced by the con ning pressure, the performance of brittleness of di erent types of rock is varying.e results also show that the reaction of the Hawkesbury sandstone is the most violent, followed by the Hegang granite and Tennessee marble.

Conclusion
A reasonable post-peak strain softening model plays an important role in the analysis of rock behaviour.Based on the nonlinear Hoek-Brown criterion, a post-peak strain softening model was proposed in this paper, and the conclusions are listed below:

Figure 3 :
Figure 3: Variation of the strength parameter η in the whole stressstrain curve.

Figure 4 :
Figure 4: Post-peak strain softening simpli ed model stress states under the same unloading path.

Figure 7 :
Figure 7: Relationship of c or φ changing with con ning pressure at the peak.
illustrate the comparison chart between the numerical simulation and experimental data for the Hawkesbury sandstone, Tennessee marble, and Hegang granite, respectively.

R 2 = 0. 95 Figure 8 :Figure 9 :
Figure 8: Relationship of c or φ changing with con ning pressure at the starting point of the residual stage.

Table 1 :
Data of the Hawkesbury sandstone in triaxial compression tests.

Table 2 :
Rock strength parameters (m, s) at the peak position and at the starting residual stage.

Table 3 :
Rock strength parameters (m, s) of different rock types at the peak position and at the starting point of the residual stage.