A simpli fied approach to directly consider intact rock anisotropy in Hoek-Brown failure criterion

Mohamed A.Ismael,Hassan F.Imam,Yasser El-Shayeb

Mining Engineering Department,Faculty of Engineering,Cairo University,Cairo,Egypt

A R T I C L E I N F O

Anisotropy Hoek-Brown failure criterion Rock mechanics Anisotropic parameter Degree of anisotropy

Many rock types have naturally occurring inherent anisotropic planes,such as bedding planes,foliation,or flow structures.Such characteristic induces directional features and anisotropy in rocks’strength and deformational properties.The Hoek-Brown(H-B)failure criterion is an empirical strength criterion widely applied to rock mechanics and engineering.A direct modification to H-B failure criterion to account for rock anisotropy is considered as the base of the research.Such modification introduced a new definition of the anisotropy as direct parameter named the anisotropic parameter(Kβ).However,the computation of this parameter takes much experimental work and cannot be calculated in a simple way.The aim of this paper is to study the trend of the relation between the degree of anisotropy(Rc)and the minimum value of anisotropic parameter(Kmin),and to predict the Kmindirectly from the uniaxial compression tests instead of triaxial tests,and also to decrease the amount of experimental work.

?2014 Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.Production and hosting by Elsevier B.V.All rights reserved.

*Corresponding author.

E-mail addresses:mohamed.ismael.mining@gmail.com,me_egypt2030@yahoo.com(M.A.Ismael).

Peer review under responsibility of Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.

1674-7755?2014 Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.Production and hosting by Elsevier B.V.All rights reserved.

http://dx.doi.org/10.1016/j.jrmge.2014.06.003

1.Introduction

The study of both intact rock and rock mass behaviors is a hot issue in the design of engineering applications on/in rocks.Rock mass behavior is usually assumed to be isotropic.According to Hudson and Harrison(2000),one of the engineering myths is that rocks are often supposed to be always safe under any engineering activities because of the assumptions of high strength and stiffness.The consequential difficulties which are faced at studying rocks are their inherent characteristics of discontinuity,anisotropy,inhomogeneity,and inelasticity.

Hence,the concept of design on/in rocks is that most of rock masses are under stresses and forces-regardless of the source of stresses-which in fluence the stability of the rock materials.These stresses and forces affect the rocks by inducing changes in the rock structures and forming discontinuities in rock masses.Discontinuity is a collective term used to include joints,fractures,bedding planes,rock cleavage,foliation,shear zones,faults,etc.The planes of anisotropy are also classified as discontinuities and fractures.These planes of anisotropy are known as inherent planes in rock material,e.g.pre-existing planes of weakness,bedding planes,or planes induced by the loading associated with construction or operation of the project,e.g.initiation and propagation of cracks(Bobet et al.,2009).

Inherent anisotropy is considered as the nature of rocks which affects the behavior and properties of rocks in particular for the metamorphic and sedimentary rocks.Fromrock mechanics point of view,anisotropic nature of rocks causes the difference of the rock strength with respect to the orientation of loading and inherent planes(Saroglou and Tsiambaos,2008).

Hoek-Brown(H-B)failure criterion is one of the main criteria of rock failure mechanism and is used widely for design purposes in/on rock masses.This criterion is used to predict the rock strength with respect to the principal stresses(σ1,σ3).Most of modifications to H-B failure criterion for anisotropy are dealing with the anisotropic nature of rocks by modifying the rock constants(m,s)according to the angle between load orientation and inherent planes(β).These approaches are named indirect modifications to H-B failure criterion for anisotropy.

On the other hand,a reported modification to H-B failure criterion by Saroglou and Tsiambaos(2008)had considered the anisotropic nature of rocks directly in the original H-B failure criterion.Therefore,this approach is considered as direct modification to H-B failure criterion for anisotropy which de fines a new parameter,the anisotropic parameter(Kβ),and inserts it in the original equation of H-B failure criterion.

The main problem is how to define the trend and find out the relationship between the direct anisotropic parameter and rock strength classification proposed by Ramamurthy(1993).The strength anisotropy classification depends on the degree of anisotropy(Rc),which is the ratio of the maximum to the minimumvalues of uniaxial compressive strength(UCS)of anisotropic intact rock samples.The lowerthe value of Rc,the closerthe rock nature to be isotropic.

In order to compute the value of anisotropic parameter(Kβ)at any given angleβ,we need to make a triaxial compression test for the intact rock samples.On the contrary,evaluating the degree of anisotropy(Rc)for rock samples requires uniaxial compressive test for anisotropic intact rock samples.So,it is important to estimate the value of anisotropic parameter using uniaxial compression test rather than triaxial compression test.Consequently,a proposed relation between the direct anisotropic parameter and rock strength classification is introduced.The relationship between the anisotropic parameter and the UCS is analyzed.

Fig.1.Variation of the axial strength relative to the discontinuity inclinationβ(after Hoek and Brown(1980)).

2.Anisotropic nature of rocks

An anisotropic rock has different properties in different directions.This nature is governed by the special structure characteristics of rocks which are the uncontrollable nature of rocks:the minerals forming the rocks,the orientations of the minerals’crystals,and the interaction between different grains.

This anisotropic nature of rocks can be stated,according to Bagheripour et al.(2011),as follows:

(1)Most foliated metamorphic rocks,such as schist,slates,gneisses and phyllites,contain a natural orientation in their flat/long minerals or a banding phenomenon which results in anisotropy in their mechanical properties.

(2)Stratified sedimentary rocks like sandstone,shale or sandstone-shale alteration often display anisotropic behaviors due to presence of bedding planes.

(3)Anisotropy can also be exhibited by igneous rocks having flow structures as may be observed in rhyolites(Matsukura et al.,2002).

These anisotropies are often referred to as inherent anisotropy,and the corresponding rocks are sometimes categorized as intact anisotropic rock.

The strength anisotropy definition means that the directional characteristics of rocks,such as deformability modulus,strength,brittleness,permeability and discontinuity frequency,are the function of angle between load orientation and inherent planes of anisotropy,β(Fig.1).

Previous studies tried to solve the problem of representation of anisotropy in rock strength criterion.Attempts of classification of the rocks were made based on the anisotropic properties(Tsidzi,1990;Ramamurthy,1993);these classifications are deterministic for the degree of anisotropy in the examined rocks.These studies began with Jaeger definition(Jaeger,1960),followed by Donath experimental work(Donath,1964)which dealt with the shear failure and behavior in anisotropic rocks,and the Walsh-Brace theory(Walsh and Brace,1964)of fracture criterion for brittle anisotropic rocks.Hoek(1964)proposed a definition of the failure and fracture in the anisotropic rocks based on the“single weakness plane of failure”theory of Jaeger.In addition to these efforts,McLamore and Gray(1967)studied the mechanical behavior of the anisotropic sedimentary rocks.Similarly,Attewell and Sandford(1974)contributed to this field by preparing an experimental and mechanical study for the shear strength of the anisotropic brittle rocks.Hoek and Brown(1980)proposed a modification to the failure criterion considering rock anisotropy.Colak and Unlu(2004)presented the variations of rock parameter mifor the intact transverse anisotropic rock in H-B failure criterion.Last but not least,there was a direct modification which was discussed by Saroglou and Tsiambaos(2008),where a direct modification was added to the original H-B failure criterion to take the anisotropy into account.

3.Hoek-Brown failure criterion

The H-B failure criterion(Hoek et al.,2002)was developed using empirical data and considered the relation between the principal stresses.This relation is governed by three parameters(m,s,σci)which are the material constants and the compressive strength of rock material,respectively.Constants s and m define structural pattern,or quality,of rock mass and rock type,respectively.The constant s predominantly affects the ‘tensile’strength of the rock and its influence is the most important at very low confinement(σ3).

The original criterion depends on the compressive strengthσciand the material constants m and s.For inhomogeneous and anisotropic rocks,it is not easy to determine the rock strength which is very important for the applications of engineering in/on rocks(civil and mining).The empirical failure criterion initially proposed by Hoek and Brown for intact rock is described as

where σ1and σ3are the major and minor principal stresses,respectively;andαis the material constant.In case that the rock material is intact,the equation of the H-B failure criterion for intact rock can be rewritten as

where m is equal to the constant mifor intact rock,α=0.5 and s=1.

3.1.Original Hoek-Brown failure criterion

The determined values of the H-B criterion parameters for intact rock(σci,mi,s)are defined based on the results of uniaxial and triaxial tests,when loading is applied perpendicularly to the planes of anisotropy(foliation or bedding).The values ofσciand miwill be significantly different in case that failure will occur in the direction of such a plane(Colak and Unlu,2004).

Hoek and Brown(1980)considered that the ‘single plane of weakness’theory(Jaeger,1960)is sufficient for the prediction of strength,when the rock behaves in anisotropy way due to the presence of a single plane of weakness(e.g.discontinuity plane),but did not describe adequately the strength behavior of intact rock possessing inherent anisotropy,due to the presence of bedding or foliation,as Ramamurthy(1993)stated,as in the case of siltstones,schists,gneisses,etc.

Consequently,in order to predict the strength of intact anisotropic rock,Hoek and Brown(1980)suggested that the value of the constants m and s of their empirical criterion should be altered accordingly based on the orientation of the foliation plane relative to the principal loading axis,β.Rather than following the approach proposed by McLamore and Gray(1967),Hoek and Brown adopted a series of empirical equations to modify the material constants m and s which have been used in the original formulation.The resulting equations for m and s are

where miis the value of m for the intact rock,A and P are constants,andθandζare expressed as

where ζmis the value ofβ at minimum m,ζsis the value ofβ at minimum s,and A2,A3,P2and P3are constants.The values of m and s for different values of the discontinuity angleβare calculated by means of the linear regression analysis(Hoek and Brown,1980)in the underground excavation in rocks.The variations of the constants m and s relative to the orientation of the joints are given by

where constants A and P can be computed by

The variation of the values m/miand s relative to the orientation angle,β,for Martinsburg slate(Donath,1964)are presented in

Fig.2,in which the values of s atβ=30°and 45°are also given.The negative value of s obtained in such cases has no physical significance and is set to zero in order to avoid mathematical complications in the subsequent analysis(Hoek and Brown,1980).

3.2.Direct modification to Hoek-Brown failure criterion

Saroglou and Tsiambaos(2008)found that it is obvious that the original failure criterion modifications for the anisotropic rock material depend on the trial and error(the back analysis),besides the large number of experiments which may be not provided for each case.Also there are no modifications about natural(inherent)anisotropy such as schistosity,foliation and bedding planes.

The modification to H-B failure criterion has been carried out by incorporation of a new parameter,called“anisotropic parameter”Kβ,representing the effect of strength anisotropy:

Fig.2.(a)Variation of m/miand s vs.angleβand samples of anisotropic rock:(b)with intact pieces(s=1);(c)and(d)with crushed pieces(0＜s＜1)(after Bagheripour et al.(2011)).

Fig.3.The sequence of the tests and data processing(after Saroglou and Tsiambaos(2008)).

where σcβis the UCS at an angle of loading β.The direct modification to the H-B failure criterion,which is considered as an experimental model(Fig.3)in order to account for the effect of anisotropy on strength,is based on the inclusion of directional properties(Saroglou and Tsiambaos,2008).The modification is based on(1)the variation of the UCS of intact rock due to presence of foliation(σcβ);(2)the anisotropic strength parameter Kβ that denotes the range between the minimum and the maximum strengths of intact anisotropic rock;and(3)the constant s is equal to the unity(=1)due to the intact state of rock(Hoek et al.,2002).

The minimum value of Kβ,Kmin=K30,45,occurs at angle βmin.This angle βminbetween the axis of major principal stress(σ1)and the foliation planes ranges from 30°to 45°(Donath,1966).

Although this modification is introduced in the failure criterion,there still need a large number of experiments for calculation of Kβ.Also,the modification of Saroglou et al.is dedicated to the metamorphic rocks neglecting sedimentary rocks.There is no linkage between the descriptive analyses of anisotropy and the anisotropic parameter Kβ.

3.3.The variation of the anisotropic parameter vs.the anisotropic strength

Ramamurthy(1993)had given a classification of strength anisotropy based on the ratio Rc,which describes the degree of anisotropy using a scalar quantity.Using degree of anisotropy,it is possible to calculate the strength for each anisotropic rock type directly based on the anisotropic parameter Kβand the constant midepending on the type of rock and the texture(Saroglou and Tsiambaos,2008).

It is evident that the ratio K90/Kminis greater for the rocks with a high degree of anisotropy,e.g.Penrhyn slate,gneiss,and reduces significantly for the rocks with a low degree of anisotropy,e.g.schist and marble,where K90is the value of the parameter Kβwhen loading is perpendicular to the foliation,equal to unity(=1),and Kminis its value at the orientation of minimum strength atβ =30°-45°.

Both the anisotropic parameter Kβand the strength anisotropy depend on the angle of loading relative tothe planes of the inherent anisotropy(Table 2).As Rcis the ratio of the compressive strength at β=90°to the minimum value of compressive strength,the variation between the parameter Kβand anisotropic strength should be compared.

Table 1Anisotropic parameter and mivalues for various rock types(Saroglou and Tsiambaos,2008).

4.Simplified approach to consider rock anisotropy in Hoek-Brown criterion

The methodology would be arranged the same as that referred by Saroglou and Tsiambaos(2008).The procedureof Kβcalculations is going to be applied to data obtained from confining compressive test on other 10 anisotropic intact rock samples.These 10 rock samples are Martinsburg slate(Donath,1964),South African slate(Hoek,1964),Green River shale II(McLamore and Gray,1967),fractured sandstone(Horino and Ellickson,1970),model rock(weak sandstone)(Bagheripour and Mostyn,1996),sandstone and siltstone(Yas?ar,2001),model rock(Bagheripour et al.,2011),slates S and Z(Saeidi et al.,2013).

According to Fig.4,the paper’s main focus is to plot the relation between the calculated anisotropic strength of tested anisotropic intact rocks and the minimum value of the anisotropic parameter Kmin.This methodology aims at linking the anisotropic strength classification introduced by Ramamurthy(1993)and the associated anisotropic parameter.The proposed relation would be used directly once the anisotropic strength is known or computed;also the authors show the compatibility between the anisotropic strength Rcand the ratio K90/Kmin.Mathematically,the proposed relation is introduced to generally predict the minimum value of anisotropic parameter Kmingiven that the value of anisotropic strength Rcis obtained.It can be represented by

Table 2Parameter K90/Kminvalues and Rcvalues for Penrhyn slate,gneiss A,gneiss B,schist and marble(Saroglou and Tsiambaos,2008).

The results would be the values of the anisotropic parameter for each orientation;these results are extracted from the tested rock samples.There would be a linear relation between the anisotropic strength Rcand the ratio of K90/Kminfor the tested rock samples.It can be noted that the relation would satisfy the solution for finding the value of Kminfor any given anisotropic rock,once the anisotropic strength is given or calculated.Finally,it is logic that the relationship between Kminand 1/Rcis in direct proportion.

4.1.Limitations of proposed relation

The limitations of the proposed relation are described as follows:

(1)The relationship shows continuity within the ranges of the anisotropic strength classification proposed by Ramamurthy(1993).

不僅僅是海吉星，在整個長沙縣經(jīng)濟(jì)社會發(fā)展進(jìn)程中，高速公路帶來的交通優(yōu)勢功不可沒。境內(nèi)有京港澳高速、平汝高速、長株高速、長永高速、機(jī)場高速、長沙繞城高速東北、東南段和正在建設(shè)之中的江背至干杉高速公路等多條高速公路，另有G107、S207、S103等國省干線公路穿境而過。可以說，湖南省再沒有，全國也少見像長沙縣這樣具有如此便利的交通條件的地方。

(2)For the isotropic rock(Rc=1),the value of Kminmay equal the unity(=1),according to the anisotropic strength classification proposed by Ramamurthy(1993).

(3)The value of Kmincannot be negative or zero under any cir

cumstances,according to Saroglou and Tsiambaos(2008).

(4)The anisotropic strength(Rc)is the base of ranging the values of

Kmin.

(5)Theminimum valueofKminisunde fined,becausethe maximum value of the anisotropic strength(Rc)is not defined.

Fig.4.Schematic diagram for the proposed relation which links Kmin,Rcand degree of foliation.

4.2.Processing of the results

The gathered data can be listed in Table 3 for deduction of the relationship between the anisotropic strength and the minimum value of the anisotropic parameter.

4.2.1.Notes of the gathered data

The notes of the gathered data are described as follows:

(1)The minimum value of the UCS is usually obtained atβ =30°,except in case of schist(atβ =45°).

(2)The minimum value of the anisotropic parameter(Kmin)is usually obtained at β =30°,except in cases of gneiss A(at β =45°),Green River shale II(at β =45°),and slate S(at β =60°).

(3)The maximum value of the UCS is mostly obtained atβ =90°,except in cases of Green River shale II(atβ =0°)and South African slate(atβ =0°).

(4)The studied samples of anisotropic rocks include 5 low anisotropic rocks,4 medium anisotropic rocks,4 high anisotropic rocks,and 2 very high anisotropic rocks,according to the anisotropic strength classification(Ramamurthy,1993).

4.2.2.Regression and curve fitting for the outputs

The linear formulation resulting from the output data in Table 4 is represented in Fig.5,and can be expressed by

Table 4The expected ranges of Kminusing Rcclassification.

The coefficient of determination R2for Eq.(13)is 0.9826.Also as is indicated by Eq.(13),the variables Rcand K90/Kminpresent a direct proportion.In case of perfect isotropic samples(Rc=1),the value K90/Kminis 0.694×1+0.53=1.224.

Accordingly,the practical maximum value of ratio K90/Kminshows that the value for perfectly isotropic intact rock sample is not equal to 1 as it is assumed previously.

Also a linear regression is carried out in order to evaluate the best fit of the proposed relation between Rcand Kminfor the studied rock samples.The linear formulation is represented as a power function as shown in Fig.6,and can be written as

Table 3Gathered data of 15 tested rock samples to deduce the proposed relation between Rcand Kmin.

The coefficient of determination of R2for Eq.(14)is 0.9538.Also as indicated byEq.(14),there is an inverse proportion between the two variables Rcand Kmin.Also the maximum value of Kmincan be obtained when the value of Rcis maximum,in case of perfectly isotropic samples(Rc=1),and the value Kminis 0.9×1-0.79=0.9.

Finally,it shows that the maximum value of Kminfor perfect isotropic rocks is not the unity(=1)as it is assumed previously.

Thus,a modification to the anisotropic strength classification can be conducted using this relation to obtain the range of Kminvalues according to the obtained ranges of Rc.This relation is subjected to continuous modification in terms of extra experimental data,as shown in Table 4 and Fig.7.

On the other hand,it is depicted from Table 5 and Fig.8 that the values of Kminobtained by the proposed relation agree well with the computed values obtained from the tested anisotropic rock samples.Nevertheless,there is a slight deviation in the expected values of Kminthan the computed values especially for the range of Kminlarger than 0.4 as it is noticed in Fig.8.

Furthermore,there is a close relation between the computed and the expected values of Kminfor high to very high anisotropy rock samples(Kmin＜0.4).Thus,the proposed relation showsa good ability in prediction of the values of Kminusing the UCS(Rci)rather than the excessive triaxial testing to extract the value of Kminfrom tested rock samples.

Fig.5.Linear regression for the Rcand ratio K90/Kminof the studied rock samples.

Fig.6.Linear regression for Rcand Kminof the studied rock samples.

5.Conclusions

It is obvious that the anisotropy characteristics of rocks have effects on their mechanical properties,which depend on the load orientation relative to the inherent anisotropic planes.While the main role of the anisotropy,from rock mechanics point of view,is represented in the differentiation of the rock strength associated with the orientation of loading and inherent planes.Also,the strength anisotropy definition means that the directional characteristics of the rocks,such as deformability modulus,strength,brittleness,permeability and discontinuity frequency,are functions of the angle between load orientation and inherent planes of anisotropy,β.

Fig.7.Extent of Kminvalues depending on Rcclassification.Legend can be referred to Table 4.

Table 5Comparison of measured and computed values of Kminfor tested rock samples.

The indirect modification to H-B failure criterion aims at altering the values of the rock parameters,depending on a large number of experiments which may be not provided for each case.On the other hand,the direct modification to H-B failure criterion to account for the anisotropy is the insertion of the anisotropic parameter into the criterion.The authors use the terminology,the anisotropic strength parameter Kβwhich presents the range between the minimum and the maximum strengths of the intact anisotropic rock.The minimum value of the anisotropic strength parameter,Kmin,can be obtained when loading is performed at the angle βmin.This angle “βmin”is defined as the angle between the major principal stress(σ1)and the foliation planes,ranging between 30°and 45°.

The proposed relation between the minimum value of the anisotropic parameter Kminand the anisotropic strength classification Rcis obtained through conducting the experimental work on 15 anisotropic rock samples of low to very high anisotropy.It is found that there is a good match between the expected values of Kminusing the proposed relation and the computed values of Kminfor the same tested rock samples.Also,for low and medium anisotropy rock samples,further investigations could be helpful to revel the significant differences in values of Kmin.

Fig.8.Histogram of the measured and computed values of Kminfor tested rock samples.

Conflict of interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

We are grateful for the helpful discussions with Prof.Hany M.Helal(Former Higher Education Minister,Egypt)and also for his useful comments.

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