New true-triaxial rock strength criteria considering intrinsic material characteristics

Qiang Zhang·Cheng Li·Xiaowei Quan·Yanning Wang·Liyuan Yu·Binsong Jiang

1 Introduction

Rock strength is the foundation for rock engineering,including tunnel support design,stability evaluation of rock slopes,and estimation of the bearing capacity of the rocks.The classic Mohr–Coulomb(MC)[1],Griffith[2],and Hoek–Brown(HB)[3]criteria are mainly used for theoretical and numerical simulations,due to their simple expression forms.Most of the triaxial test data are located on the generalized compression meridian curve(σ1≥σ2=σ3).Considering the strength character under high confining pressure,You[4]proposed the exponent strength criterion(EP)for conventional stress state with an exponent function form.A significant disadvantage of these criteria,which are classified as 2D criteria in this paper,is that the intermediate and minimum principal stresses are equal,while the actual in-situ rock is normally subjected to an orthotropic stress state where the three principal stresses are different.For a givenσ3,a large number of tri-axial tests show that,asσ2increases,the rock strength firstly increases to a certain value,then gradually decreases[5–9].

In the present study,various types of 3D strength criteria including three orthotropic principal stresses are proposed.Those criteria mainly aim at neither the meridian function nor the deviatoric function.Hosein[10]proposed a new polyaxial criterion by introducingσ2,which had the similar expression form as the EP criterion.Lu and Du[11]proposed a nonlinear 3D criterion by employing a power meridian curve,and the intermediate principal stress was considered in the same way as SMP.You[4]improved the EP criterion to a true triaxial criterion(TT)by introducing intermediate principal stress effect with another exponent function.Unfortunately,TT criterion shows unconvex at the extension corner,and the additional two strength parameters has no physical meaning.The second type modifies the commonly used 2D criteria consideringσ2,while the most commonly used MC and HB criteria are usually employed.For MC criterion:the modified MC criteria of MCWW and MCJP were proposed by considering the curve transition form on the deviatoric plane by Zhang et al.[12]and Lee et al.[13],respectively,and Singh et al.[14]derived the piecewise expression of the modified MC criterion considering the ductile character under high confining pressure.Meanwhile,Yu[15]proposed the linear unified strength criterion.For HB criterion:the HBWW[13],Z–Z[16],3D HB[17,18],and HBMN[19]criteria are generally used for comparison with new proposed criteria.These proposed criteria were validated using true triaxial test data collected from previous studies,but some of the test data were incomplete or incorrect as they came from secondary sources,which may influence the accuracy[20–23].

The former strength criteria mainly focus on neither the deviatoric nor the meridian plane.However,a suitable strength criterion should depend on both the Lode angle dependence function and the meridian function.However,none of those proposed criteria show significant advantages over others.Therefore,with the consideration of intrinsic rock strength characterization,this paper proposed the new criteria with more suitable expression form of Lode angle dependence function and deviatoric function.Finally,14 groups of original triaxial test data of different types of rock collected from previous studies are employed to validate the proposed criteria.

2 Description of strength criterion

2.1 General class of strength criteria

The strength criterion can be expressed in the principle stress space.Considering the symmetry about the hydrostatic axis,Roscoe invariantspandqare employed to describe the space relation between the strength surface and stress state.The principal stressσ1≥σ2≥σ3(compression stress is denoted as positive and tensile stress negative)can be expressed as follows

In this way,the strength surface can be expressed as functions of the former three variables,and can be expressed in a unified form[24],as shown

wheref(p)andL(θ)are functions ofpandθ,respectively.

2.2 Meridian function

2.2.1 Classic 2D strength criteria

Although various expression functions were employed to describe rock strength characters,the meridian curves can be classified as linear,parabolic and exponent functions.In this paper,the 2D criteria of MC,HB,and EP are employed to represent the commonly used meridian forms.

(1)MC criterion

Due to its simple expression form,the linear MC criterion is widely used for geotechnical materials,and Fig.1 shows the failure surface on the meridian and deviatoric plane.In principal stress space,it can be formulated as follows

Fig.1 Failure surface on the meridian and deviatoric plane of MC criterion.a Meridian plane.b Deviatoric plane

Fig.2 Failure surface on the meridian and deviatoric plane of HB criterion.a Meridian plane.b Deviatoric plane

Fig.3 Failure surface on the meridian and deviatoric plane of EP criterion.a Meridian plane.b Deviatoric plane

(2)HB criterion

Based on a large number of test data,Hoek and Brown proposed the empirical HB criterion,whose strength parameters are based on the geological strength index(GSI).The trace of compression and extension on the meridian plane clearly show curve characteristics,and the deviatoric trace is approximately linear,as shown in Fig.2.In the principal stress space,the generalized form can be formulated as follows

wherefHB(σ3)=σc(miσ3/σc+s)α,σcis the uniaxial compression strength,m i,s,andαare strength parameters.

(3)EP criterion

The exponent EP criterion for conventional stress state is proposed by You considering the difference between the generalized compression and extension strength with the same hydrostatic pressure.The failure surface on the meridian and deviatoric plane is shown in Fig.3.In a principal stress space,it can be expressed by

Clearly,Figs.1 and 2 denote that the trace of former three criteria remains(approximately)linear on the deviatoric plane.Meanwhile,the shape of HB and MC criteria keep self-similar with the increasing hydrostatic pressure.The ratio between compression and extension trends to be 1 when hydrostatic pressure is big enough for HB and EP criteria.But that of MC criterion remains a constant of(3+sinϕ)/(3− sinϕ).Unfortunately,the difference between generalized compression and extension strength monotonously increase as hydrostatic pressure for HB and MC criteria.

However,both the compression and extension meridian trace remain curved lines for EP criterion,the difference between the generalized compression trace and extension trace undergoing a process of first increasing then decreasing.The hydrostatic pressureIc,where the difference between compression and extension has the biggest value,can be easily adjusted by parameterK.Moreover,the compression and extension strength tends to be equal when the hydrostatic pressure is sufficiently high.So,EP criterion can well describe the rock strength characters on meridian plane.So,the meridian trace function of EP is employed in this paper.

2.2.2 Meridian function of EP[19]

In thep−qspace,the Roscoe invariantspandqare functions of the first and second invariants of the stress tensors,respectively.For triaxial stress state,they can be written as follows

On the generalized compression and extension meridian planes,σ2=σ3andσ2=σ1,respectively.Therefore,the Roscoe invariantspandqcan be rewritten as follows

The compression and extension traces on the meridian plane can be expressed by a tangent modulus.substituting Eq.(7)into Eq.(5),the following slopes can be obtained

where the superscriptsdenotes secant values,and the subscriptscandedenote triaxial compression and extension,respectively.The tangent modulus of meridian curves can be obtained by differentiation

Fig.4 Trace of meridian curves and its representation by tangent

which leads to:

Figure 4 shows the tangent line of compression and extension meridian plane forEP criterion in thep−qspace.Based on Eqs.(5),(11),and(12),the intersection between the tangent line andpaxial can be obtained as

In this way,the triaxial compression and extension meridian curves can be formulated by tangent moduli as

2.3 Deviatoric function

According to the stability postulate of Drucker,the strength criterion should satisfy requirements for aspect ratio,continuous,differentiable and convex.At least,the continuity and convexity must be satisfied for stable materials.However,most of the generally used 2D criteria are singular on the compression and extension meridian planes.This may induce convergence problems for numerical simulation.Moreover,only the maximum and minimum principal stresses are employed without considering the intermediate principal stress effect for the former 2D criteria.Lots of experiments show thatσ1experiences firstly increase then decrease process asσ2varying fromσ3toσ1.It also means that radiusqgenerally decrease fromqctoqtalong a curved path,but not(approximate)linear forms,on the deviatoric plane.In order to represent the intermediate principal stress effect,various Lode angle dependence functions were proposed[15,25–30].Among those Lode angle dependence functions,three widely used functions are unconditionally convex and smooth for 0.5≤qt/qc≤1.0 without introducing additional parameters as shown in Eqs.(15)–(17).Clearly,LMNis formed using the pricewise trigonometric function,which is smooth and continuous atθ= ± π/6.Meanwhile,LWWandLYMHare derived by mapping a quarter of the elliptic function to−π/6≤θ≤ π/6,andLWWandLYMHare complementary functionsto each other,which will be illustrated in the following section.All of the above three Lode angle dependence functions have only two basic parameters,i.e.Lodeangleand extension–compression ratio,and the extension–compression ratio can bedetermined by its intersection points with the meridian plane.Moreover,none of other strength parameters were introduced.The deviatoric cross sections of the three criteria are shown in Fig.5,and they are employed to modify the EP criterion in the following section.

Fig.5 Deviatoric cross sections of Lode angle dependence functions L MN,L WW,and L YMH

Fig.6 Modifying method of classic criteria with Lode angle dependence function

3 Proposed new true triaxial strength criterion

3.1 Modifying method

Figure 6 illustrates the modifying method with the common 2D criterion and Lode angle dependence function.For the commonly used strength criteria,the radiusqof the strength surface on the deviatoric plane approximately decreases linearly from the compression to extension state,without considering the effect ofσ2.As formerly mentioned,the radiusqon the deviatoric plane indirectly reflects the effect ofσ2.For a triaxial stresses state,the invariant valueqshould be located betweenqc(σ1≥σ2=σ3)andqt(σ1=σ2≥σ3)with the samepandθ,and it also should be changed fromqctoqtas the lode angle changing from−π/6 to π/6.Moreover,Eq.(13)shows that the traces of the generalized compression and extension state have the same intersection with thepaxis as the sameσ3.In this way,the actualvalueqcan be simply calculated by tangent modulus and hydrostatic pressure.And the actual tangent modulus should lies betweenqtandqc,which is determined by Lode angle dependence functions of Eqs.(15)–(17).Then,by connecting the generalized compression and extension states on the deviatoric plane along Lode angle dependence function,the intermediate principal stress effect would be included.When the transition from compression to extension occurs,the modified criteria can be formulated by submitting the Lode angle dependence function and meridian plane into Eq.(2).

Fig.7 Failure surface on the principal space and deviatoric plane of EPWW criterion.a Principal stress space.b Deviatoric plane

Figure 7 shows the EPWW criterion in both principal stresses space and deviatoric plane,in which the effects of minimum principal stress,hydrostatic pressure,and intermediate effect are clearly shown.Similarly,the new proposed strength criterion can also be obtained by rotating the tangent along the Lode angle dependence function from the extension meridian curve to the compression meridian curve.In this way,it can be formulated as follows

where¯Lis the complementary function ofL,which should satisfy

3.2 Relation between Fc and Fe

In view of the Lode angle dependence functions of Eqs.(16)and(17),they have similar expression forms.The complementary functionLWWcan be expressed as follows

By submitting Eq.(21)into Eq.(19),the modified criterion based on the Lode angle dependence function¯LWWand extension meridian curve can be formulated as follows

Using Eq.(21)and¯β,Eq.(22)can be rewritten as follows

4 Identification of modified criteria

In this section,14 different types of rock triaxial test data are employed to validate the proposed true triaxial strength criteria.All of these test data were collected from the original sources,as listed in Table1.It should be noted that the average strength ofσ1 was employed for groups of data with the sameσ2 andσ3.

4.1 Evaluation method for best- fitting strength parameters

The proposed true triaxial criteria have the same strength parameters as the 2D ones.In order to illustrate the reasonable of proposed true triaxial strength criteria(EPWW,EPMN,and EPYMH),the modification of MC and HB by selected three Lode angle dependence functions are carried out in the same way,and also analyzed using 14 types of rock.Equations(18)and(19)are non-linear explicit functions ofp,q,andθ.Therefore,a grid search method is employed to determine the best-fitting strength parameters for each criterion.The accuracy of fitting to the tested triaxial data may be evaluated by the residual standard deviation(RSD),defined by the following

4.2 Discussion on the best-fitting results

The RSD of each group criteria for the 14 types of rock are shown in Fig.8.It should be noted that the comparison of the RSD values between the different rock types bears little meaning for their different strength degrees.The proposed true triaxial criteria significantly decrease the RSD foral most all of the rock types in comparison to the 2D criteria for MC,HB,and EP.

Tables 2–4 show the calculated results for the best-fitting strength parameters of the 14 types of rock.Among the three Lode angle dependence functions,eight types of rock have the minimum RSD values based on theLWWLode angle dependence function for the MC group criteria,six for both the HB group criteria,and 12 for the EP group criteria.This demonstrates that the WW Lode angle dependence function has more reasonable forms for rock strength criteria than the MNand YMHLode angle dependence func-tions.At the same time,EPWW criterion has the lowest RSD for Solnhofen limestone,Shirahama sandstone,KTB amphibolite,Mizuho trachyte,Yamaguchimarble,Manazuru andesite,Orikabe monzonite,Apache Leap tuff,and Maha Sarakhamsalt.Additionally,MCWWhasthelowest RSDfor Yuubari shale,and HBWW has the lowest RSD for Westerly granite and In ad a granite.Thus the lowest RSD group of rock denotes that EP has a more suitable meridian expression for rock as they have the same Lode angle dependence functionLWW.Furthermore,among all of the three group criteria,the EPWW criterion results are the lowest among the RSD values for all rock types,except Yuubari shale for MCWW,Dense marble for HBYMH,Westerly granite and Inadagranite for HBWW,and Dunham dolomite for EPYMH.In this way,we can conclude that the EP criterion and WW Lode angle dependence function show more suitable states among the three commonly used 2D criteria and three Lode angle dependence functions.

Table 1 Rock types and data source

Fig.8 Prediction errors for MC,HB,EP,and their generalized 3D forms.a MC,MCWW,MCMN,and MCYMH.b HB,HBWW,HBMN,and HBYMH.c EP,EPWW,EPMN,and EPYMH

In Fig.9,the strength predictions for the 14 types of rock,based on the EPWW criterion,are compared with the experimentaldata in theσ1−σ2plane.In the figure,the symbols represent the actual experimental data,while the curves are the prediction values.The experimental data with the sameσ3are listed with the same symbols.The effect of the intermediate principal stress on the rock strength is clearly shown.It is evident that the proposed new criteria fit the experimental data quite well.In particular,the predictions for Shirahama sandstone,Yuubari shale,Mizuho trachyte,Yamaguchi marble,and Maha Sarakham salt are all very accurate.Additionally,the EPWW criterion shows accurate prediction values forthe lowervalues ofσ3,such as Dunham dolomite,Solnhofen limestone,Mizuho trachyte,Yamaguchi marble,Manazuru andesite,and Inada granite.The prediction uniaxial compression strength by EPWW shows high accuracy with the experiment values,as the errors of the uniaxial compression between the actual experiments and prediction values are＜1.5%for Dunham dolomite,Solnhofen limestone,Mizuho trachyte,Manazuru andesite,and Apache Leap tuff.Obviously,among the twelve compared criteria,EPWW criterion results in much lower RSD,and the bestfitting strength parameters are more reasonable.

Fig.8 continued

Table 2 Best fitting parameters for MC,MCWW,MCMN,and MCYMH criteria

Table 3 Best fitting parameters for HB,HBWW,HBMN and HBYMH criteria

Table 4 Best fitting parameters for EP,EPWW,EPMN and EPYMH criteria

Fig.9 Best-fitting the polyaxial test data with the EPWW criterion.a Dunham dolomite.b Solnhofen limestone.c Shirahama sandstone.d Yuubari shale.e KTB amphibolite.f Mizuho trachyte.g Dense marble.h Westerly granite.i Yamaguchi marble.j Manazuru andesite.k Inada granite.l Orikabe monzonite.m Apache Leap tuff.n Maha Sarakham salt

Fig.9 continued

Fig.9 continued

5 Conclusion

This paper evaluated the proposed new true triaxial criteria,and solved their non-smoothness and non-convexity problems in a simple way.Based on the study,the following conclusions can be drawn.

The suitability of a criterion mainly dependson the meridian and deviatoric functions.The transition functions of the MC,HB,and EP criteria on deviatoric plane are(approximately)linear,which does not reflect the effect of the intermediate principal stress.This is the common disadvantage of classic 2D strength criteria.Among the former three 2D strength criteria,the EP criterion has good expression forms on the meridian plane,which can accurately denote the hydrostatic pressure effect.

Among various Lode angle dependence functions,theLWW,LMN,andLYMHsatisfy the requirements of convexity and differential for all ranges ofβ.They are employed to describe the intermediate principal stress effect.And new true triaxial strength criteria of EPWW,EPMN,and EPYMH are proposed.Moreover,the relationship of proposed criteria that based on compression(Fc)and extension(Fe)meridian trace is discussed.It shows thatFcbased onLWW(LYMH)equalsFebased onLYMH(LWW).Meanwhile,FcandFebased onLMNequal each other due to the self-similar characteristics ofLMN.

In addition,14 types of original triaxial experiment data obtained from original research sources were employed to validate the proposed criteria.The fitting accuracy is comprised with 2D criteria.The results show that the EP criterion andLWWwere more suitable for describing the meridian curves and deviatoric envelope of the rock strength characteristics than the other criteria and Lode angle dependence functions,respectively.Meanwhile,the strength prediction characters were analyzed,and EPWW criterion shows the highest accuracy for strength prediction among the compared twelve criteria.

AcknowledgementsThis project was supported by the National Natural Science Foundation of China(Grants 51204168,51579239),the China Postdoctoral Science Foundation funded project(Grants 2013M531424,2015M580493),the National Basic Research 973 Program of China(Grants 2013CB036003,2014CB046306),and the Fundamental Research Funds for the Central Universities(Grant 2012QNB23).

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