Study on the damage-softening constitutive model of rock and experimental verification

Sheng-Qi Yang·Bo Hu·Peng Xu

Abstract A damage-softening model is presented to describe the stress–strain curve of rock.By comparing the Hoek–Brown(H–B) and Mohr–Coulomb(M–C)yield criterion,the equivalent M–C yield criterion is selected as the strength criterion in this model.To better characterize the rock damage and failure processes with considering the relationship between damage and deformation,the concept of yield stress ratio is introduced to describe the yield-strengthening deformation before rock peak stress.Damage events are described by two cumulative damage evolution laws.The evolution equations of tensile and shear damage are presented based on the equivalent plastic strains, and the maximum value between tensile and shear damage represents the total damage for rock.Considering that rock cannot bear tensile load after tensile failure but still has a certain shear strength,its tensile and shear strengths are small after shear failure.The elastic modulus is affected by tensile damage,whereas the angle of internal friction,the cohesion, and dilation angles are influenced by shear damage.The proposed damage-softening model describes the strain softening,brittle stress drop, and residual strength of rock after peak stress, and finally the model is implemented in FLAC3D.Comparing the test and the numerical calculation results,the damage-softening model better describes the total stress–strain curve of rock.

Keywords Rock deformation·Damage evolution·Softening·Numerical calculation

List of symbols

D Damage variable

DtTensile damage

DsShear damage

E?Elastic modulus at yield strengthening stage

E,EsYoung’s modulus

E0Initial value of elastic modulus

K,G Bulk and shear modulus

k1Ratio of yield stress to peak strength

k2Ratio of elastic modulus at yield strengthening stage to Young’s modulus

H–B Hoek–Brown yield criterion

M–C Mohr–Coulomb yield criterion

miDimensionless empirical constant

fsShear yield plane

gtTensile yield plane

h Boundary plane between shear and tensile yield plane

T Tensile

T–S Tensile–shear

C–S Compressive–shear

c Cohesion

c?Equivalent cohesion

cpCohesion at peak strength point

crCohesion at residual strength stage

? Angle of internal friction

??Equivalent internal friction angle

?pAngle of internal friction at peak strength point

?rAngle of internal friction at residual strength stage

ψ Dilation angle

σ1,σ3Major and minor principal stresses

σcUniaxial compressive strength of rock

σ3maxThe maximum confining pressure

σtTensile strength

σyieldYield stress

σpeakPeak stress

σtTensile strength

σt0Initial value of tensile strength

ingpsEquivalent shear plastic strain

ingptEquivalent tensile plastic strain

ingpsLCritical equivalent plastic strain of rock entering the residual deformation stage

ω Shear strength parameter[i.e.,fraction angle(?),cohesion(c), and dilation angle(ψ)]

ωpInitial values of shear strength parameters

ωrResidual values of shear strength parameters

?ingIStrain matrix of the unit

D Stiffness matrix of the unitIncrement of shear plastic strain

?KsIncrement of equivalent shear plastic strain

?KtIncrement of equivalent tensile plastic strain

1 Introduction

The mechanical behaviour of rock and rock mass is the basis of stability analysis for many rock engineering projects,such as tunnel excavations,coal mining,coal seam gas extraction,shale gas development, and underground storage construction[1–5].Rock materials have dilation and strain softening phenomena other than linear elasticity,especially at the post-peak region[6–8].To better understand the rock failure process,it is necessary to investigate the complete stress–strain curve of rock.However,it is a very difficult task to provide a reasonable constitutive model that characterizes the complete stress–strain curve of rock,especially describing the brittle stress drop and strain softening after peak stress.

As is well known,the compressive deformation process of rock can be divided into three parts.(1)The elastic deformation stage before peak stress,in which rock deformation includes pre-existing crack closure,linearly elastic deformation, and yield-strengthening deformation.During this stage,variations in the mechanical parameters of rock are unapparent.Therefore,these three deformation measures can simply be classified as an elastic deformation stage before peak stress;(2)The softening stage after peak stress,in which the compressive stress decreases with the increasing of strain, and the bearing capacity and mechanical property of rock decreases significantly;(3)The residual deformation stage,in which the compressive stress does not change significantly with increasing compressive strain and shows ideal plastic deformation characteristics.As a result,the mechanical properties of rock before and after the peak stress point are significantly different.It is very important to research the post-peak mechanical properties of rocks because the deep surrounding rock of caverns tends to be in the post-peak deformation region after excavations.It has been pointed out by Li et al.[9]that the constitutive model of rock is mainly identified as macro-mechanical- and micromechanical-based models.The macro-mechanical models are established based on physical phenomena,such as the relationships between stress and strain and the internal variables calculated by the thermodynamics of irreversible processes[10–12].The latter phenomena are based on micro mechanics,such as the propagation of micro cracks or flaws in material[13–17].Most physical phenomena are based on continuum and fracture mechanics.The micro mechanical model based on the deformation and growth of assumed and simplified elliptic micro cracks(i.e.,single-line micro crack or penny-shaped micro cracks)reveals the mechanism of the damage and failure process of rocks[18–25].However,the shape of the micro cracks in rock materials are truly non-simple single cracks or penny-shaped cracks.The simplification may induce deviations of the theoretical model from the actual results and evaluations of the damage state[26].

The strength criterion of rock is a factor which should be considered in establishing a constitutive model.Many strength criteria,such as the three-dimensional nonlinear strength criterion[27–30],can characterize the rock damage and failure processes of rocks and evaluate the effects of intermediate principal stress.The Mohr–Coulomb(M–C)yield criterion is widely used in numerical simulations of geotechnical materials.The mechanical properties of rock after peak stress can be determined by the M–C yield criterion[31,32].The residual strength of rock is mainly related to the angle of internal friction(?) and cohesion(c).Therefore,the variations of c and ? at the post-peak deformation region reflect the characteristics of residual strength.However,the variations differ with rock types.The peak and residual strength characteristics of different rocks were analysed by M–C criterion,as shown in Fig.1[33–38].The results show that the strength between soft rock(i.e.,coal and mudstone) and hard rock(i.e.,marble and granite)varied greatly.In contrast,the cohesion after peak stress decreased significantly,whereas the variations in the angle of internal friction were different.The residual friction angles(?r) of marble and s and stone increase especially significantly,while those of coal and granite did not change significantly.The friction angles of siltstone and mudstone exhibit distinct decreases.

Figure 1 presents the relative relationships between the cohesion/friction angle at the peak stress point and that at the residual deformation stage of the different rocks.It is clear that the cohesion of rock decreases gradually and the cohesion(cp)at the peak strength point is larger than that(cr)at the residual strength stage,whereas the friction angle at the peak point(?p) and that at the residual strength stage(?r)are especially different.When ?r?p,the residual strength increases with the increasing of confining pressure and even exceeds the peak strength,which is the characteristic of the transition from brittleness to ductility.

Fig.1 Peak and residual strength parameters of different rocks

To investigate the post-peak mechanical properties of rock,the mechanical parameters of rock after peak stress should be researched.Moreover,the dilation angle(ψ),which is used to describe the volume expansion,has great influence on the post-peak mechanical behaviour.The dilationangle is always regarded as0 in non-associative flow law,whereas it equals the friction angle in associative flow law.However,it is unreasonable to assume that the dilation angle is a constant in a geotechnical numerical calculation[39].A large number of engineering cases have been summarized by Hoek and Brown[40] and they proposed that the value of the rock shear expansion angle can be determined based on the quality of rock mass.The dilation angle was determined by Vemeer and Borst[41]to be20°smaller than the internal friction angle and the y proposed a value for the rock shear expansion angle.

Asa solution to the creep problem for metals,Kachanov[42]introduced a damage concept and Lemaitre[43]elaborated on it.The damage concept was applied by Kyoya et al.[44]to rock mass who the n called it damage tensor.The phenomenon of strain softening of rock or concrete can be explained using the damage mechanics theory.The deterioration process of mechanical parameters of rock can be explained by the damage viewpoint[45,46].Damage in rock gradually increases and the dilation angle should decrease during deformation.Therefore,this research presents a damage-softening model considering the degradation of dilation angle in addition to the deterioration of elastic modulus,tensile strength,friction angle, and cohesion.

2 Construction of the dam age-softening model

The variations of the mechanical parameters of the rock obtained from experimentation are complicated and cannot directly be used in an engineering application.Therefore,the test results need to be simplified to obtain the variations of friction angle,cohesion,dilatation angle, and damage variable(D).In this section,according to the strength yield criterion,we present a damage evolution equation and damage weaken equation for post-peak deformation as a damage-softening model;the model considers the variations of the mechanical parameters of the rock during the postpeak deformation process.The model is composed of three parts:(1)the strength yield criterion;(2)the cumulated damage evolution rate calculated by the plastic parameter;(3)the damage evolution equations of the elastic parameters.

2.1 Strength yield criterion

The Hoek–Brown(H–B)yield criterion reflects the nonlinear relationship between the major and minor principal stresses,which is used to predict the failure of rock and rock mass, and it also reflects the influences of structure planes and stress states on the strength.The parameters of the H–B yield criterion are calculated by a lab experiment,which is written as[47]:

where σ1and σ3are major and minor principal stresses,respectively;σcis the uniaxial compressive strength of rock;miis a dimensionless empirical constant.Figure 2 presents the experimental results of granite under traditional triaxial compression.It is clear that the H–B yield criterion characterizes the strength property of rock under low and high confining pressure.

Fig.2 Hoek–Brown peak strength envelope for granites[48]

Fig.3 Relationships between major and minor principal stresses for H–B and equivalent M–C criteria[48]

However,it is difficult to determine the parameters of the H–B criterion, and the implementation of numerical calculations is infeasible.Although the M–C yield criterion cannot accurately reflect the strength characteristics of rocks,it can characterize the strength of the semi-brittle materials and its accuracy is adequate for engineering.Hence,the M–C yield criterion is still widely used in most numerical calculations for geotechnical engineering.Its equation can be written as

where c and ? are the cohesion and internal friction angleat the peak strength point,respectively.

The M–C yield criterion which does not reflect the tensile failure properties of rocks can predict shear failure of rocks,but the calculated tensile strengths are larger than that of the real rock materials.Therefore,it is necessary to establish a tensile yield criterion which can characterizes the tensile failure strengths of rocks based on the M–C yield criterion.It is termed the modified M–C yield criterion,as shown in Fig.3.

Subsequently,Hoek and Brown[48]presented equivalent strength parameters of M–C,which were equivalent to the H–B criterion within the ranges of different stresses(i.e.,c?and ??are equivalent cohesion and equivalent internal friction angle).These equivalent strength parameters are written as[48–52]:

where σ3n? σ3max/σci;σ3maxis the maximum confining pressure.When the range of the confining pressure is small(i.e.,σ3n?0),the equivalent M–C yield criterion characterizes the tensile–shear strength characteristic of rock under low confining pressure.However,at a high range of confining pressure,the compressive–shear strength calculated by the equivalent M–C yield criterion is relatively large,whereas the strength curve of the equivalent M–C yield criterion describes the compressive–shear strength of rock under high confining pressure(i.e.,σ3n?1).The surrounding rocks near a cavern are always under low confining pressure,while the distant surrounding rocks are under high confining pressure.Therefore,to accurately research the deformation and failure behaviours of a cavern and to characterize the tensile–shear strength properties of rocks,the confining pressure range should be small.

The ratio of the tensile strength to the compression strength of rock is calculated by the H–B criterion,written as Eq.(5),

where σcand σtare compressive and tensile strengths,respectively.The tensile strength of rock calculated by the H–B criterion is always large,especially when miis relatively small.The ratio of the tensile strength to the compression strength of rock is larger than that of the experimental results.Hence,the calculated safety coefficient of geotechnical engineering is high, and the nonlinear expression of the H–B criterion is difficult to implement in numerical calculations.

Fig.4 Equivalent M–C model domains used in the failure criterion.Note:T tensile;T–S tensile–shear;C–S compressive–shear

The modified equivalent M–C yield criterion best approaches the H–B criterion within a given confining pressure range.The established tensile–compressive–shear strength criterion approximately characterizes the tensile,tensile–shear, and compressive–shear strength characteristics of rock,but the tensile strength should be modified as follows[48,51,53,54].

The modified equivalent M–C yield criterion can theoretically explain the tensile,tensile–shear, and compressive–shear failures of rock and can distinguish the above three failure modes in the (σ1,σ3)plane stress space,as shown in Fig.4.Finally,the strength criterion is selected as the strength yield criterion for the proposed damage-softening model.The modified equivalent M–C criterion is composed of a shear yield plane(fs),tensile yield plane(gt), and the boundary plane between the m(h).

The M–C shear yield criterion is

The tensile yield criterion is

The boundary plane between shear and tensile plane is

2.2 The cumulative damage evolution law

Rock material does not show ideal elasto-plastic property.However,it shows yield strengthening characteristics before reaching the peak stress point during conventional triaxial compression.Specifically,when the compressive stress exceeds the yield stress,its elastic modulus decreases,but axial stress still increases with increased strain.Therefore,the yield stress ratio is defined as the ratio of yield stress to the peak stress of rock based on the M–C strength criterion.

where σyieldand σpeakrepresent yield stress and peak strength,respectively;k1and k2are material constants that reflect the ratio of yield stress to peak strength and the reduction ratio of elastic modulus at the yield-strengthening stage are generally taken as 0.75–0.80;E?and Esare the elastic modulus at the yield-strengthening stage and Young’s modulus,respectively.

The damage variable represents the damage degree in rock.It is important to research the damage evolution during rock deformation to evaluate the loading capacity of rock.D is relatively small within the elastic deformation region and it is approximately equal to 0 before peak stress[9].Damage evolution can be simplified as in Fig.5,in which D remains 0 before peak stress and the n increases linearly within the softening region and finally increases to the maximum at the residual deformation stage.

To reasonably characterize the damage and failure processes of rock and to build a relationship between damage and deformation of rock,a cumulative damage evolution law is proposed.The damage variable was used in the cumulative damage evolution law and represented the degree of damage in the rock.When D?0,it means that rock deforms elastically;D?1 means that rock yield and failure occur at the residual deformation stage.To distinguish the tensile damage and shear damage,Dtand Dsare used to represent the two damage events,respectively, and the larger of the two variables is used to describe the total damage variable D.There are variations between Dtand Ds,especially under shear yield conditions.Dsincreases linearly with increasing shear equivalent plastic strain,whereas Dtequals1.0 under tensile yield conditions(because rock is a brittle material,differing from metal)as written in the following equations.

Fig.5 Damage evolution curve of rock without the crack-closure process

where ingpsand ingptare the shear and tensile equivalent plastic strains,which are calculated by the associative flow rule and the plastic flow surface.The ingpsLis the critical shear equivalent plastic strain of rock entering the residual deformation stage.

2.3 The damage-softening law

The physical properties of material will change when yield damage occurs and its strength and deformation parameters will weaken correspondingly.The cohesion of rock will decrease and its friction angle will change at the post-peak deformation region.Here,we just consider the damage evolution laws of elastic modulus,tensile strength,friction angle,cohesion, and dilatation angle.We note that the influence of tensile damage on the physical properties of rock is different than that of shear damage.The rock,after tensile failure occurs,is not subject to tensile loading but is subject to shear loading.However,the tensile and the shear strengths of rock are small after shear failure occurs.This assumes that the elastic modulus of rock is just affected by the tensile damage,where as the friction angle,cohesion, and dilatation angle are just affected by the shear damage.

where ω is the shear strength parameter(i.e.,?,c, and ψ;ωpand ωrare the initial and residual values of the shear strength parameters,respectively.)

where E is elastic modulus;E0is the initial value of elastic modulus;σtis tensile strength;σt0is the initial value of tensile strength.

The mechanical parameters of rock are modified using the above damage-softening equations.The established damagesoftening model of rock exhibits different mechanical characteristics under tensile and compressive conditions,especially under the compression shear condition.Rock entering the residual deformation stage still exhibits a certain residual shear strength when shear damage has occurred,whereas the residual strength and elastic modulus of rock decrease to 0 when the rock enters the residual deformation stage and tensile damage occurs.The model shows similar mechanical behaviour to that of real rock materials at the post-peak stage.

3 Numerical implementation of the rock dam age-softening model

The proposed damage-softening model is implemented in FLAC3D that calculates using the finite difference method.Therefore,the 、three-dimensional difference format of the proposed damage-softening model(i.e.,the relationship between the stress increment and the strain increment)should be provided.When the stress levels are low,rock deforms elastically, and the difference format of the elastic constitutive model can be written as[55]:

in which

where K and G are the bulk and shear modulus,respectively; and ? and ψ are the friction angle and dilation angle,respectively.The shear plastic increment()is calculated as:

The increment of equivalent plastic shear strain can be defined as:

If gt<0 and h?0,it means that the integration point tensile plastic yield occurs in the current loading step.The n,the trial calculated stress should be modified according to the tensile yield law to let the trial calculated stress return to the tensile yield surface.The modified stresses can be expressed as the following equation:

Correspondingly,the tensile plastic increment()can be written as:

Fig.6 flow diagram of rock damage-softening constitutive model in FLAC3D

The increment of equivalent plastic tensile strain(?Kt)can be defined as:

The tensile and shear damage variables(Dtand Ds,respectively)can be calculated according to the calculated plastic shear strain.The n,the mechanical parameters will be modified and saved according to the tensile and shear yield laws;subsequently,the next trial calculation will start.The programming of the proposed damage-softening model is presented in Fig.6.

Table 1 presents the model parameters of the damage softening model,including input and output parameters.Compared with the traditional M–C model,the residual strength parameters and strain-softening parameters are introduced in the damage-softening model based on the M–C model.Therefore,the damage-softening model can sufficiently describe the strain softening and residual strength properties of rock.Moreover,the proposed model can quantitatively provide the tensile and shear damage values of units that describe the damage degree of rock during the deformation process under loading.The parameters in Table1 can be obtained according to the indoor triaxial compressive tests and their calculation methods are presented in Sect.2.It is clear that the form of the damage-softening model is simple, and its physical parameters are clear and easily obtained.

Table1 Parameters in the rock damage-softening constitutive model

4 Experimental verification o f the dam age-softening model

4.1 Experimental verification of brittle rock

S and stone,showing obvious brittle failure characteristics under compression,is a typical kind of sedimentary rock,which is mainly composed of s and grains.Figure 7 plots the peak and residual strengths of siltstone under traditional triaxial compression.The experimental results are from the reference[56].It is clear that the strength curves agree with the equivalent M–C strength curves.Its cohesion obviously decreases,but the variation in friction angle is not significant under compression conditions.The peak strength is always larger than the residual strength with increasing confinement;therefore,siltstone does not show brittle-ductile transition under traditional triaxial compression.The experimental and numerically calculated stress–strain curves of siltstone are shown in Fig.8[56].The parameters of the damage-softening model are presented in Table 2.From Fig.8,the compressive stress–strain curves calculated by the damage-softening model show three linear stages,including the brittle stress drop during the post-peak softening process.It is clear that the proposed model sufficiently describes the peak and residual strengths,as well as the brittle deformation at the post-peak stage.

Fig.7 Peak and residual strengths of siltstone

Figure 8 presents the comparison results of the experiment and numerical calculations for siltstone under confining pressure of 5 MPa.The model and experimental curves differ greatly during the pre-peak deformation process because the real rock has high porosity and large elastic compression occurs,whereas their peak and residual strengths,as well as the post-peak deformations,are similar.The damage evolution curve is also plotted in Fig.8,where the stress–strain curve is almost linear before peak stress and damage in rock is kept at 0.When the sample failure occurred at the peak stress point,a sudden stress drop occurred, and the damage variable had a sudden rise showing the same brittle characteristic during this stage.Subsequently,it entered the residual deformation stage and the damage variable increased to 1.The deviation of the peak strain between simulation and experiment was obvious,but the deviation of the peak strength at different confining pressures was small.

Fig.8 Axial stress–strain curves of siltstone under triaxial compression with different confining stresses

Table2 Parameters used in the damage-softening constitutive model of rock(siltstone)

4.2 Experimental verification of brittle-ductile rock

Marble is a typical kind of metamorphic rock,with high density and homogeneous properties.Under traditional triaxial compression,it shows obvious characteristics of brittlenessductility transition.The study shows that the brittlenessductility transformation property of marble is related to its residual strength[57].Figure 9 presents the peak and residual strength properties of marble,in which the strength characteristics are in good agreement with the equivalent M–C strength criterion.In the process from peak point to residual deformation,the cohesion decreases,but the friction angle increases gradually.The difference values between the peak strength and the residual strength become small,showing a marked trend of the transition from brittle failure to ductility under triaxial compression.

Fig.9 Peak and residual strength of marble

The damage-softening model is also used to simulate the experimental results of marble.Table 3 presents the parameters of the model for numerical calculation.The result comparisons for the experiments and numerical calculation are plotted in Fig.10.It is clear that the calculation results are in good agreement with the tests,meaning that the damage softening model sufficiently describes the characteristics of peak strength,post-peak softening, and residual strength of marble.The calculated and experimental stress–strain curves of the marble under 5-MPa confinement are plotted in Fig.10, and the damage evolution curve is also plotted during the compression process.The calculated stress–strain curve has four linear stages.During compression,the marble was compacted gradually, and its axial strain increased linearly with deviatoric stress.Its yield strengthening occurred before peak stress.During these two stages,the re was no damage in the marble as assumed in the proposed model and the damage variable remained 0.The stress–strain curve strain softening occurred, and the damage increased linearly as strain increased at the post-peak stage.Subsequently,it entered the residual deformation stage and stress had no obvious variation and damage increased to 1.The damage variable corresponds to the softening deformation of the rock after peak strength,which accurately reflects the stress state of the rock.

5 Discussion

To better understand the effects of the variety of relevant parameters in the proposed model(Table 4),the influence of critical parameters on the stress–strain curve is discussed.Figure 11 presents the influence of the parameters in the damage-softening model on the stress–strain curve and damage evolution during compression.It is clear that the yield strengthening stage is influenced by the yield_ratio.The higher(close to 1.0)the yield_ratio,the shorter the yield strengthening stage in Fig.11a.In Fig.11b,the ratio affects the deterioration of the elastic modulus at the post-peak stage,showing the higher the ratio,the slower the deterioration of the elastic modulus.The higher the cohesion during the residual deformation stage(Cr),the larger the residual strength in Fig.11c.The friction angle during the residual deformation stage(?r)enhances the residual strength and slows the deterioration process of the elastic modulus at the post-peak stage.

Table3 Parameters used in the damage-softening constitutive model of marble

Fig.10 Axial stress–strain curves of marble under different confining pressures obtained by experiment and model

Fig.11 Sensitivity analysis of the parameters in the presented model

The phenomenon of strain softening for rock can be explained using the damage mechanics the , and the deterioration process of the mechanical parameters of rock can be explained by the damage viewpoint.Dilation angle,which is used to describe the volume expansion,has great influence on the post-peak mechanical behaviour.It is unreasonable to assume that the dilation angle is a constant in geotechnical numerical calculations.Asa result,damage in rock increases gradually and the dilation angle should decrease during deformation.This research presents a damage-softening model considering degradation of dilation angle in addition to the deterioration of elastic modulus,tensile strength,friction angle, and cohesion.Moreover,the concept of yield stress ratio based on the M–C strength criterion is introduced to explain the yield-strengthening deformation before peak stress of the rock.The physical meaning of the model is clear, and it can be conveniently implemented in numerical calculations.The proposed model can describe the stress–strain curve of the hard rock showing brittle stress drop after peak stress and characterizes the strain softening and plastic flow behaviour of rock under high confinement.However,the proposed model cannot explain the failure mechanism of rock from a micro-mechanical view.The damage evolution during compression is linear and does not show the nonlinear deformation behaviour of rock.

Table4 Sensitivity analysis of the parameters in a damage-softening constitutive model

(2)Tensile and shear damage evolution laws are built to make links with rock deformation.The mechanical parameters are deteriorated according to the damage laws based on continuum damage mechanics.The elastic modulus is affected by tensile damage,whereas the cohesion,friction angle, and dilation angle are controlled by shear damage law.

(3)The damage-softening model is implemented in FLAC3D and the comparisons of results from experiments and numerical simulations show that the proposed model can effectively characterize different rock deformations.

AcknowledgementsThis research was supported by the National Natural Science Foundation of China(Grants51734009&51179189),the Fifth “333”Project of Jiangsu Province(2016) and the China Postdoctoral Science Foundation(Grant 2018M 642360).The authors would like to express their sincere gratitude to the editor and two anonymous reviewers for their valuable comments which have greatly improved this paper.

Compliance with ethical standards

6 Conclusions

The aim of this study is to present a simple model to describe rock deformation,especially the post-peak deformation.As a result,a damage-softening model was developed for characterizing the yield-strengthening deformation before peak stress,strain softening, and the brittle stress-drop characteristics of rock.Finally,the proposed model is implemented in FLAC3D and is verified by various experimental results.

(1)The concept of yield stress ratio based on the M–C strength criterion is introduced to explain the yield strengthening deformation before peak stress of rock.The modified equivalent M–C yield criterion is used in the model to explain theoretically tensile,tensile–shear, and compressive shear failure of rock.

Conflict of interestOn behalf of all authors,the corresponding author states that the re is no conflict of interest.