Engineering Geology 222 (2017) 72–83

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Engineering Geology

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Geomechanical types and mechanical analyses of rockbursts

Tianbin Li ⁎, Chunchi Ma, Minglei Zhu, Lubo Meng, Guoqing ChenState Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu, Sichuan 610059, PR China

⁎ Corresponding author.E-mail addresses: ltb@cdut.edu.cn (T. Li), nanjiguang1

http://dx.doi.org/10.1016/j.enggeo.2017.03.0110013-7952/© 2017 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 16 June 2016Received in revised form 16 February 2017Accepted 11 March 2017Available online 16 March 2017

Rockbursts generally occur in highly stressed rockmasses because of underground openings. To ensure the safetyof construction works, factors that play a role in predicting and preventing rockbursts should be analyzed. Basedon numerous geological and mechanical analyses of rockbursts in China, six basic geomechanical types ofrockbursts are classified with unique developing characteristics (rock-mass structure, crack type, failure plane,energy release, etc.), which are further divided into stress rockburst and stress-structure rockburst. The stressrockbursts are characterized by the development of micro-cracks using fracture mechanical theories; the forma-tion of a macroscopic failure plane indicates that a state of maximum energy release is achieved (along the pre-ferred crack path). The stress-structural rockbursts are considered a structural failure when propagating cracksintersect the existing discontinuities, which are analyzed using the catastrophe models. Finally, occurrencecriteria of the six geomechanical types of rockbursts are proposed.

© 2017 Elsevier B.V. All rights reserved.

Keywords:RockburstGeomechanical typesOccurrence criterionFracture mechanicsCatastrophe models

1. Introduction

A rockburst is a dynamic instability phenomenon occurring at un-derground openings with hard-brittle lithology and high ground stress-es. The unloading mechanism leads to the adjustment of groundstresses (increase in tangential stress and decrease in radial stress)and sudden release of stored energy in rock masses, thus resulting inphenomena such as loosening and bursting, slabbing, ejecting, andeven throwing of rock blocks (Li et al., 2010). Over the past three de-cades, engineers have been troubled by rockbursts occurring in severalsignificant underground projects in China. The projects include diver-sion tunnels in the Tianshengqiao and Taipingyi hydropower stations,underground works in the Ertan hydropower station, Qinling extra-long tunnel of the Xikang railway, Erlangshan tunnel of the Sichuan–Tibet highway, and diversion tunnels in the Jinping II hydropower sta-tion along the Yalong River (Gong et al., 2012; Li et al., 2012). To ensurethe safety of constructionworks in deep underground projects in the fu-ture it is necessary to understand the factors that play a role inpredicting and preventing rockbursts. The occurrences of rockbursts,which are strongly related to the geological and mechanical propertiesof rockmasses, need to be interpreted usingmathematical andmechan-ical analyses.

Most studies on the mechanism and prediction of rockbursts arebased on the applications of the strength, rigidity, energy, fractal, frac-ture damage, and catastrophe theories etc., and numerical analyses

017@sina.com (C. Ma).

(Ma et al., 2016, 2015). Hoek and Brown (1980) and Russense (1974)proposed the stress–strength criteria of rockbursts for practical rock en-gineering. Zhao et al. (2017) modified these criteria based on observa-tions in a division tunnel. In the rigidity theories, Petukov (1979)observed severely ruptured rock specimens during sudden unloadingin an experiment conducted on a flexible testing machine. Qian(2014) calculated the stiffness of rock pillars and surrounding rocksand established a forecast model for the strain rockburst of pillars. Inthe energy theories, Cook et al. (1966) explained the mechanism ofrockbursts as external forces breaking the mechanical equilibrium ofsurrounding rocks; thus, more energy is released than dissipated.Kidybiński (1981) proposed a proneness index of rockbursts (wetindex) using the ratio of the elastic strain energy induced by sudden re-lease to the dissipated energy. Song et al. (2012) predicted rockburstbased on the dissipative structure theory. By applying the fracture me-chanical theories, studies confirmed that the maximum compressivestress could help determine the initiation of cracks;moreover, the strainenergy and dissipation energy along the crack path should be evaluatedto determine the direction of crack propagation (Bobet, 2000; Sharon etal., 1996). The elementary catastrophe theories are widely used in anal-yses of the rockburst system, e.g., in the cusp catastrophemodel (Pan etal., 2006). Previous studies show that the basic types of rockbursts aregenerally classified into strain burst, slip (fault-slip) burst, and theircombination (Chen et al., 2015; Fan et al., 2016; Feng et al., 2012). How-ever, these are not sufficiently related to the geomechanical nature ofrockbursts and do not clearly address the process mechanism ofrockbursts.

To solve this problem, studies on rockbursts were conducted inErlangshan tunnel of the Sichuan–Tibet highway, diversion tunnels in

List of symbols

σij∞ Far field stresses

σij Additional stresses around flawKII Stress intensity factor for mode II loadingβ Inclined angle of flawG Energy release ratef Internal frictional coefficientE Elastic modulusI Moment of inertiaGs Shear modulusμ Poisson's ratiop Tangential compressive stressσN Cohesion/Radial rock pressureω Deflection of rock beamD Bending rigidity of rock beamk Bending curvature of rock beamU Structural strain energyW External workV Total potential energyΔV Energy release during rockburst

73T. Li et al. / Engineering Geology 222 (2017) 72–83

the Jinping II hydropower station, and Futangba tunnel of the Duwenexpressway in Southwest China. The authors concluded that studieson rockburst mechanism must be based on detailed analyses of therockburst types and geological characteristics of rockbursts, followed

Table 1Typical underground works in China and characteristics of rockbursts.

Underground works Situations Lithology Grou

Cangling tunnel of the Taijinexpressway (Wang et al.,2006)

Total length is 7.5 km; themaximum depth is 756 m.

Tuff and granite. The mhorizstress

Diversion tunnels in theJinping II hydropowerstation (Li et al., 2010)

Total length is 17.2 km; generaldepth is 1500–2000 m; themaximum depth is 2525 m.

Triassic marble,sandy slate andchlorite schist.

The mprincMPa.

Diversion tunnels in theTianshengqiao IIhydropower station (Lee etal., 1996)

Total length is 9.7 km; averagedepth is 400 m; the maximumdepth is 800 m.

Tertiarylimestone anddolomite.

The mprinc21–2

Diversion tunnels in theTaipingyi hydropowerstation (Zhou and Hong,1995)

Total length is 10.5 km; generaldepth is 200–600 m.

Granite andgranodiorite.

The mprinc31.3 M

Diversion tunnels in theFutang hydropowerstation (Wu, 2003)

Total length is 19.3 km; generaldepth is 450–700 m.

Granite. The mprinc18.4 M

Erlangshan tunnel of theSichuan–Tibet highway(Wang et al., 1999)

Total length is 4.16 km; themaximum depth is 770 m.

Limestone,sandstone,siltstone andmudstone.

The mprinc17.5–

Futangba tunnel of theDuwen expressway (Li,2006)

Total length is 5.3 km. Granite. The mprinc20.8 M

Nibashan tunnel of the Yaxiexpressway (Deng, 2009)

Total length is 10 km; themaximum depth is 1650 m.

Rhyolite,andesite anddolomite.

The mprinc30–4

Qinling tunnel of theXikang railway (Gu et al.,2002)

Total length is 18 km; themaximum depth is 1600 m.

Migmatiticgneiss andmigmatiticgranite.

The mhorizstress

Underground works in theErtan hydropower station(Peng, 1998)

General depth is 220–480 m. Syenite andgabbro.

The mprincimainMPa.

by a thorough mathematical and mechanical analysis. By employingthis approach, geomechanical types of rockbusts with occurrencecriteria were proposed, thereby providing a suitable method to predictand prevent rockbursts.

2. Developing characteristics of rockbursts

Based on the rockburst records of underground works in our re-search areas, Table 1 gives the general field characteristics of rockbursts.

We proposed a grading scheme for the rockbursts—slight, moderate,strong, and extreme strong—in Table 2 with correspondingcharacteristics.

The following are the rockburst phenomena that can be easilydistinguished:

(1) Sheeting or bed spalling: Rock masses surrounding an under-ground opening peel off layer-by-layer, producing sheets orplates. The thickness of an individual sheet can lie in the range0.5–10 cm. The failure plane is usually flat, exhibiting conchoidalradial patterns of fracture.

(2) Buckling break: Surrounding rock masses buckle towards thefree surface affected by the high tangential stress.With the prog-ress of fracture and energy release, rock masses fail via buckling,leading to ejection. In general, the failure plane is relatively flat incentral parts and jagged along edges.

(3) Dome-like or wedge burst: Surrounding rock masses fail viashear fracture because the local stresses concentrate, follow-ed by ejection phenomena. The failure plane is dome-like orform wedges by combining the shear and tensile-shearfractures.

nd stresses Rockburst characteristics

aximumontal principalis 12.27 MPa.

Rockbursts occurred as shear slip or spalling off in flakes andlenticles with popping sounds.

aximumipal stress is 72

The minimum depth of rockbursts is 400 m. Rockbursts mostlyoccurred within 40 m from the tunnel face and in the range 2–6 hafter excavation; in the sidewall with 170 cases, tunnel top 130cases, and tunnel face 100 cases.

aximumipal stress is6 MPa.

Rockbursts occurred in the dry sections with moderate-jointedrock masses, and generally in the upper-left and bottom-rightparts of tunnel.

aximumipal stress isPa.

Rock masses behaved as spalling off with ringing sounds;otherwise, behaved as shear slip with thundering sounds.Small-scale rockburst occurred in the complete granites;otherwise, occurred in the jointed rock masses.

aximumipal stress isPa.

Large-scale rockbursts occurred with thundering sounds;otherwise, occurred with ringing sounds. The shapes of failureplanes are right-angle, step-like or nest-like.

aximumipal stress is35.3 MPa.

N200 rockburst cases occurred close to the tunnel face, covering anaccumulative length of 1095 m. They are controlled by thestructure of rock masses.

aximumipal stress isPa.

Rockbursts mostly occurred at the exit section of tunnel. Thecovered length is 2692 m, accounting for 25% of total length of thetwin tunnels.

aximumipal stress is5 MPa.

The strongest rockburst occurred in the side wall with length over40 m.

aximumontal principalis 27.3 MPa.

Slight rockbursts produced sheets. Moderate rockbursts producedflakes and lenticles. Bursting pieces of strong rockbursts can be 3.4m and in various shapes.

aximumpal stress inchamber is 64.4

Rockbursts can be characterized as three types: with poppingsounds, popping and ejection of small pieces, and throwing of largerock blocks.

Table 2Grading scheme and characteristics of rockbursts.

Rockburst gradesAcousticfeatures Movement Timing

Affecteddepth (m)

Shapes of burstingpieces

Stress/strengthpmax/Rc

Wetindex

Impact onworks

Slight (grade I) Crackling Spalling or bulging Scattered in timehistory

b0.5 Sheets or lenticles 0.3–0.5 b2.0 Little

Moderate (gradeII)

Crackling andringing

Spalling, buckling or slightejection

Progressivelydeveloping

0.5–1.0 Lenticles or plates 0.5–0.7 2.0–5.0 Large

Strong (grade III) Rumbling Strong ejection or bursting Progressively or quicklydeveloping

1.0–2.0 Plates and massive rocks 0.7–0.9 2.0–5.0 Great

Extreme Strong(grade IV)

Loudlyrumbling

Extreme strong ejection ormulti-bursting

Quickly developing tothe deep

N2.0 Fragments, plates andmassive rocks

N0.9 N5.0 Very great

Note: pmax denotes a maximum tangential stress of rock masses; Rc denotes an uniaxial compressive strength.

74 T. Li et al. / Engineering Geology 222 (2017) 72–83

The following are the geometrical shapes of the failure planes ofrockbursts:

(1) Flat failure plane: The plane is flat with a rough surface, showingradial and parallel patterns.

(2) Stepped failure plane (Fig. 1a): The average height of an individ-ual step is in the range 0.5–10 cm, reaching up to a maximum of20 cm.

(3) Conchoidal failure plane (Fig. 1b): The plane takes the shape of ashell, showing tiny radial stripes.

(4) Dome-like failure plane (Fig. 1c): The shape is largely spherical orellipsoidal.

(5) Arch-like failure plane (Fig. 1d): The shape is curved and con-trolled by two sets of intersecting joint planes. Rock “powders”remaining in the failure planes are caused by the sudden releaseof high stresses.

3. Geomechanical types of rockbursts

Based on the geomechanical characteristics of rockbursts de-scribed previously, six geomechanical types of rockbursts are pro-posed: (1) Tensile cracking and spalling, (2) Tensile cracking andtoppling, (3) Tensile cracking and sliding, (4) Tensile shearing andbursting, (5) Buckling and breaking, and (6) Arc shearing andbursting.

Fig. 1. Geometrical shapes of failure planes of rockbursts.

3.1. Tensile cracking and spalling

Because of the increase in the tangential compressive stress and de-crease in the radial stress (unloading effect), tensile cracks develop nearthe free surface, resulting in flaky spalling of rock masses (Fig. 2a). Thistype of rockburst mostly occurs in the rock masses wherein the micro-fractures are approximately parallel to the free surface. This type ofrockburst may occur in a single layer or progressively in multiple layersand has an insignificant impact on the undergroundworkswith negligi-ble amount of energy released in a small range (corresponding to thelower grade of rockburst). The failure plane is flat or conchoidal withstep-like boundaries (Fig. 2b).

3.2. Tensile cracking and toppling

Because of the unloading effect induced by underground opening,tensile cracks initiate and develop approximately parallel to the freesurface. Consequently, the upper ends of cracks ultimately cut throughto the free surface or intersect the existing discontinuities; thus, rockmasses expand and topple under radial pressure (unloading rebound).This type of rockburstmainly occurs in layered rockmasses with specif-ic ductility. The release of energy is little so that the affected range is lim-ited. Rockbursts generally locate at side walls with stepped or irregularfailure planes (Fig. 3a and b).

3.3. Tensile cracking and sliding

The ground stress releases along the existing fractured zones afterexcavation, inducing slip in shear. Moreover, tensile cracks developand connect the shear fractures. Finally, rockburst occurs in the formof layered spalling or wedge burst. This type of rockburst largely occursin layered or massive rock masses, and side walls with existing frac-tures. The energy release is moderate, leading to rockbursts with slightormoderate grade. The failure plane is largely stepped or curved (Fig. 4aand b).

Fig. 2. Tensile cracking and spalling type of rockburst.

Fig. 3. Tensile cracking and toppling type of rockburst. Fig. 5. Tensile shearing and bursting type of rockburst.

75T. Li et al. / Engineering Geology 222 (2017) 72–83

3.4. Tensile shearing and bursting

After excavation in highly stressed rock masses, tensile cracks andtensile-shear fractures develop, and progressively expand towardsboth sides of the free surface, resulting in wedge burst. This type ofrockburst mostly occurs in massive (relatively intact) rock masses ormasses with a thick-layered structure. The affected range of rockburstis relatively extensive with a considerable amount of energy released.Moreover, substantial ejection phenomena are observed. A moderateor strong grade of rockburst occurs with largely stepped or arc-like fail-ure plane (Fig. 5a and b).

3.5. Buckling and breaking

As the surrounding rock masses expand and bend towards the freesurface because of the unloading effect, stresses increase and concen-trate in the bending areas. Consequently, rock masses break and burstvia buckling when the tensile stress exceeds the strength. This type ofrockburst mostly occurs in layered rock masses, wherein both ends ofthe rock layers are not outcropping in the free surface. Ringing soundsand ejection phenomena can be observed during the rockburst withgenerally moderate grade. The failure plane is stepped or conchoidal(Fig. 6a and b).

3.6. Arc shearing and bursting

After excavation in rock masses with initial high energy, the strainenergy constantly accumulates because of stress redistribution and dif-ferentiation. Moreover, the rock masses expand and rebound towardsthe free surface, where existing micro fractures gradually expand.Hence, strain energy suddenly releases, resulting in rockburst via arc-form shearing. This type of rockburst occurs in intact or massive rockmasses at a specific depth, releasing a large amount of energy with con-siderable ejection of rock pieces. Several rockbursts with strong or ex-tremely strong grade can occur at the same position in time history,

Fig. 4. Tensile cracking and sliding type of rockburst.

progressively extending into the deeper parts of the rock masses. Thefailure plane is mainly dome-like or elliptical in shape (Fig. 7). Theshear slip at the lower part can induce tensile opening at the upperpart (Fig. 7b). The thickness of the bursting pieces is largely in therange 1.0–2.0 m.

Classifying the geomechanical types of rockbursts is beneficial to un-derstand themechanism of rockbursts in practice. Table 3 gives the im-portant developing characteristics of rockbursts, which are comparedwith the strain burst and slip burst proposed in previous studies.

Strain burst generally occurs in relatively intact rockmasses, resem-bling the tensile cracking and spalling type. As the cracking and spallingcontinue to propagate, the “shallow-hole” failure plane is formed withirregular and stepped boundaries (referring to Fig. 2b). The existingmacroscopic discontinuities generally control the slip burst, resemblingthe tensile cracking and sliding type. The four other geomechanicaltypes of rockbursts are clearly not included in strain burst or slip burstby the following reasons: tensile shearing and bursting and arc shearingand bursting generally develop in intact rockmasses with the evolutionof micro cracks; tensile cracking and toppling and buckling and break-ing are structural failures affected by the combined effects of cracksand existing macroscopic discontinuities.

Hence, the six geomechanical types of rockbursts are classified intotwo major types: the stress rockbursts (including tensile cracking andspalling, tensile shearing and bursting, and arc shearing and bursting)and stress-structure rockbursts (including tensile cracking and sliding,tensile cracking and toppling, and buckling and breaking). As the twotypes of rockbursts possess different mechanisms, fracture mechanics(analyzing the evolution of stress rockbursts) and catastrophe theory(analyzing the mutation of stress-structure rockbursts) are applied inChapters 4 and 5.

4. Fracture mechanical analyses of stress rockbursts

Underground openings cause unloading of the surrounding rockmasses and the stress adjustment/redistribution. Moreover, stressesaround the existing micro-fractures (flaws) accumulate, thereby

Fig. 6. Buckling and breaking type of rockburst.

Fig. 7. Arc shearing and bursting type of rockburst.

76 T. Li et al. / Engineering Geology 222 (2017) 72–83

initiating newer cracks. Rockbursts in hard brittle rocks in the compres-sive stress states, can be characterized by crack initiation, propagation,and coalescence, which can be analyzed as a case of compressive frac-turing in fracture mechanical theories.

4.1. Crack types

With the release of the radial stress, the tangential stress positivelyaffects the compressive fracturing of surrounding rock masses; thetwo-dimensional stress state of flaws helps explain the growth of crackseffectively. Fig. 8 shows an inclined flaw in the infinite linear-elasticplate. The flaw is closed because of the compressive pressure p (viewedas the tangential compressive stress), with a length 2a, negligiblewidth,and angleβ inclined to the load direction. The far field stressesσij

∞ (inde-pendent from flaws) can be expressed in the Cartesian and polar formsas follows:

σ∞x ¼ pcos2β

σ∞y ¼ psin2β

τ∞xy ¼ p sinβ cosβ

8><>:

σ∞rr ¼ pcos2 β−θð Þ

σ∞θθ ¼ psin2 β−θð Þ

τ∞rθ ¼ p sin β−θð Þ cos β−θð Þ

8<: ð1Þ

The additional stresses due to the flaw can be expressed in the polarform, using the Westergaard function (Atkinson, 1987).

σ rr ¼ 12ffiffiffiffiffiffiffiffi2πr

p K II sinθ2

3 cosθ−1ð Þ ¼ K IIffiffiffiffiffiffiffiffi2πr

p f rr

σθθ ¼ −32ffiffiffiffiffiffiffiffi2πr

p K II sinθ cosθ2¼ K IIffiffiffiffiffiffiffiffi

2πrp f θθ

τrθ ¼ 12ffiffiffiffiffiffiffiffi2πr

p K II cosθ2

3 cosθ−1ð Þ ¼ K IIffiffiffiffiffiffiffiffi2πr

p f rθ

8>>>>>><>>>>>>:

ð2Þ

K II ¼ τ∞xy− fσ∞y

� � ffiffiffiffiffiffiπa

p ð3Þ

In the above equations, σrr, σθθ and τrθ are the radial, tangential (re-ferring to the flaw) and shear stresses, respectively, shown in Fig. 8. Thepolar coordinates (θ and r) define the location and distance from the tip

Table 3Geomechanical types of rockbursts and corresponding characteristics.

Characteristics Tensile cracking and spallingTensile cracking andtoppling Tensile cra

Structure of rockmasses

Existing micro fractures parallel tothe free surface

Layered structure Layered orfractured z

Crackingproperty

Tensile Tensile Tensile an

Failure plane Flat or conchoidal with step-likeboundaries

Irregular or stepped Stepped or

Energy release Negligible Little Moderate

of theflaw, respectively. KII denotes the stress intensity factor under thisloading condition. f is the internal friction coefficient of the rockmasses.

Newly initiated cracks under compressive load can generally be clas-sified as tensile and shear cracks (Lajtai, 1974; Mahanta et al., 2016;Wong and Einstein, 2009). The total stress field, obtained by superposi-tion of the far-field stresses (Eq. (1)) and flaw-induced additionalstresses (Eq. (2)), can explain the crack types effectively; otherwise,considering only the additional stresses is insufficient. Fig. 9 shows thecommon tensile cracks originating from the flaw; contour lines of theminimum principal stress are observed (Fig. 9a). Both areas of the ten-sile and compressive stresses are illustrated (symbols “+” and “−” de-note the tensile and compressive stress areas, respectively). Thethickness of the lines helps differentiate the stress levels (thicker linesdenote higher values). Tensile cracks follow the criterion of maximumtensile stress, characterized by two trend lines corresponding to thepaths of type I crack (cracking angle θ0= arccos(1/3)) and type IIcrack (Fig. 9b).

The shear crack paths originating from the flaws under compressiveload can be interpreted using the concept of the maximum-energy re-lease rate. The driving force “g” of a crack path can be defined as follows(Lawn, 1993).

g ¼ G−R ð4Þ

In this equation,G is the energy release rate along the crack paths, andR is the energy dissipation rate. When G exceeds R, the driving force g in-creases; thus, the crack is in theunstable state. Rock failures canbe seen asa series of crack phenomena, where the effect of strain softening deterio-rates the storage capacity of strain energy (energy release rate G in-creases), resulting in rockbursts when the driving force of cracks isconsiderable. The maximum driving force is equivalent to the maximumenergy release rate along a crack path (assuming an isotropic systemwith uniform energy dissipation rates along diverse paths). The energyrelease rate G can be expressed:

G θ;βð Þ ¼ Gθθ 1−μ2� �=E þ Grθ 1−μ2� �

=EGθθ ¼ K II f θθ þ σ∞

θθGrθ ¼ K II f rθ þ τ∞rθ

8<: ð5Þ

cking and slidingBuckling andbreaking

Tensile shearing andbursting

Arc shearing andbursting

massive with existingone

layered orlayer-like

Intact or massive Intact or massive

d shear Tensile Tensile, shear andtensile-shear

Shear and tensile

curved Stepped orconchoidal

Stepped or arc-like Elliptical ordome-like

Moderate Large Large or verylarge

Fig. 8. Stresses of flaw under compressive load.Fig. 10. Types of shear cracks originating from flaw.

77T. Li et al. / Engineering Geology 222 (2017) 72–83

In this equation, Gθθ and Grθ are the tangential and shear compo-nents of the energy release rate. fθθ and frθ are the tangential and shearcomponents of the additional stresses (Eq. (2)). μ is the Poisson'sratio, and E is the elastic modulus. Thus, the energy release rate G(θ,β) is controlled by the inclined angle (β) of the flaw and crack path(angle θ).

Fig. 10 shows the contour lines of the maximum shear stress of theflaw. Two stress concentration zones exist at each flaw tip: one in thetensile stress area (characterized by the trend line with angle θ0), andthe other in the compressive stress area. The dashed area (Fig. 10a) rep-resents the maximum energy release rate G, comprising the higher Grθ

and Gθθ (Gθθ in the tensile stress area is considered in G). The shearcracks originate from the flaw (Fig. 10b) with a cracking angle θ0 ¼ 1

2

arccosðffiffiffi2

p=3Þ.

4.2. Development of rockburst failure planes

The failure planes of rockbursts exhibit the following processes inorder: crack initiation, propagation, and coalescence. The threegeomechanical types of rockburst—tensile cracking and spalling, arcshearing and bursting, and tensile shearing and bursting—are analyzedas follows.

4.2.1. Tensile cracking and spalling typeFig. 11 shows the development of this type of rockburst. (a) Initia-

tion: Flaws are induced because of the unloading effect of the rockmasses. (b) Propagation: Type I or II tensile cracks grow from theflaws under the increasing tangential compressive stress. (c) Coales-cence: The cracks continuously extend along the existing micro frac-tures (or the bedding planes parallel to the excavation surface), thusresulting in slabbing and spalling.

Fig. 9. Types of tensile cracks originating from flaw.

The occurrence criterion for this type of rockburst can be derivedbased on the maximum tensile stress around the flaws. The tangentialcompressive stress of the rock masses should follow the condition:

p≥2K ΙC

−3ffiffiffiffiffiffiπa

psinβ cosβ cos θ0=2ð Þ sinθ0

ð6Þ

In this equation, 2a andβ denote the length and inclined angle of theflaw, which can be qualitatively assessed in application. The crackingangle θ0= arccos(1/3) and KIC is the anti-tensile property of the rock.

4.2.2. Arc shearing and bursting typeAs described in Section 4.1, the energy release rateG(β) is influenced

by the inclined angle β of the flaw. Fig. 12 shows the relationship be-tween β and G (Atkinson, 1987). As β increases and approaches βm,the energy release rate continuously increases to the maximum Gm. Asβ increases to beyond βm, the energy release rate decreases. Hence, βm

is the preferred angle of the flaw, which is expressed:

βm ¼ 12

arctan1f

ð7Þ

The potential failure plane of the rockburst (by connection of in-clined flaws) is presented and terminated at the preferred angle βm

(the dotted line denotes a less possible contribution to the failureplanewith β2 N βm andG2 b Gm). In this state, the effect of themaximumenergy release rate is obtained, which is a crucial condition forrockburst.

This type of rockburst develops as seen in Fig. 13. (a) Initiation: Thepotential failure plane emerges because of the unloading effect. (b)Propagation: The shear cracks grow from the inclined flaws, extendalong the potential failure plane, and terminate at the preferred angle.(c) Coalescence: The shearing and slipping in the lower region of therock masses induces tensile opening in the upper region.

The occurrence criterion for this type of rockburst can be interpretedas the crack growth in pursuit of the maximum energy release rate G(θ,

Fig. 11. Development of failure plane in tensile cracking and spalling type.

Fig. 12. Relationship between β and G and the potential failure plane.

Fig. 14. Development of failure plane in tensile shearing and bursting type.

78 T. Li et al. / Engineering Geology 222 (2017) 72–83

β). The tangential compressive stress of the rock masses should followthe condition:

p≥2K IICffiffiffiffiffiffi

πap

sinβm cosβm cos θ0=2ð Þ 3 cosθ0−1ð Þ ð8Þ

In this equation, βm denotes the preferred angle with respect to Eq.(7), KIIC is the anti-shear property of the rock, and the cracking angle θ0¼ 1

2 arccosðffiffiffi2

p=3Þ.

4.2.3. Tensile shearing and bursting typeThis type of rockburst develops as seen in Fig. 14. (a) Initiation: The

potential failure plane is formed because of the unloading effect. (b)Propagation: The shear and tensile cracks grow from the inclinedflaws. (c) Coalescence: The shear and tensile cracks extend to connectwith each other along the potential failure plane and terminate at thepreferred angle. An arc-like or stepped failure plane of the rockburst isformed.

As this type of rockburst develops along the potential failure plane,the occurrence criterion can refer to the arc shearing and burstingtype (Eq. (8)).

The formation of themacroscopic failure plane indicates that a stateof maximum energy release rate has been reached, particularly in thetwo geomechanical types: arc shearing and bursting, and tensile shear-ing and bursting. Under these circumstances, considerable amount ofenergy release contributes to the development of the fracture planesand ejection of rocks.

5. Catastrophe theoretical analyses of stress-structure rockbursts

In the catastrophe theories, rockbursts are considered structurallyunstable (Dyskin and Germanovich, 1993; Pan et al., 2006, 2001). Gen-eral laws of state changes occurring when the dynamic system shiftsdiscretely from one stable state to another, can be studied using amath-ematical model. Catastrophe models provide many practical uses, in-cluding the deduction of the catastrophe criterion based on thetopological mathematical methods. Among them, the cusp catastrophemodel is the most commonly used method because it is concise,

Fig. 13. Development of failure plane in arc shearing and bursting type.

including only two control variables (Pan et al., 1994; Qin et al., 2001).The standard function of the cusp catastrophe model is as follows:

V xð Þ ¼ x4

4þ ax2

2þ bx ð9Þ

where x denotes the state variable, and a and b denote the control var-iables. The critical points are defined by deriving Eq. (9), leading to theequation of folded plane S.

V 0 xð Þ ¼ x3 þ axþ b ¼ 0 ð10Þ

The bifurcation set (corresponding to the sharp corner area in thecontrol plane a–b, Fig. 15) is a projection of the folded plane S and sat-isfies the equation:

V ″ xð Þ ¼ 3x2 þ a ¼ 0 ð11Þ

Solving Eqs. (8) and (9) by eliminating the variable x, the bifurcationset of points is obtained:

Δ ¼ 4a3 þ 27b2 ð12Þ

Fig. 15 illustrates the cusp catastrophemodel. The control variables aand b and state variable x are represented along the three coordinateaxes. The folded surface S and projection area EFG satisfy Eq. (12).Area inside EFG is mutational (unstable), on the contrary the stablearea. When the potential function of the rock structure changes alongthe path AB, the system passes through the mutational area, and

Fig. 15. Diagram of cusp catastrophe model.

Fig. 17.Mechanical model for buckling and breaking type.

79T. Li et al. / Engineering Geology 222 (2017) 72–83

consequently, rockburst occurs. When the function changes along thepath CD, the rockburst does not occur because the system changes con-tinuously along the path. The proposed geomechanical types of stress-structure rockbursts based on catastrophe models are analyzed asfollows.

5.1. Buckling and breaking type

Three stages of development can be distinguished (Fig. 16). (a)Under the increasing tangential compressive stress, the tensile cracksoriginate from the layered or layer-like rock masses. (b) Formation ofthe rock slabs: Because of the propagation of the tensile cracks, longand thin rock slabs are formed with gradually increasing flexibility,leading to an unstable statewherein the slabs slightly bend. (c) Bucklingand breaking: The bending of rock slabs continuously develops,resulting in a buckling and breaking type of rockburst.

The rock beam and spring mechanical model is proposed for analy-ses. Fig. 17 shows the rock beam with unit width, length L, thicknessh, deflection ω, tangential compressive stress p, and cohesion σN. Thedashed line indicates the curved shape of the rock beam.

The deflection of the rock beam can be expressed as follows:

ω ¼ u sinπsL

ð13Þ

where s is the length of the arc, and u is themaximumaxial deflection atthe center of the beam. The cohesion σN is proportional to the degree ofbending based on the spring model.

σN ¼ σ t F ωð Þ ð14Þ

where 0 ≤ F(ω) = ω/u ≤ 1, and σt is the maximum cohesion (approxi-mating the tensile strength). The cohesion σN is a function of location;the maximum cohesion deals at the center of the rock beam.

The external work W and structural strain energy U determine thetotal potential energy V of the rock beam,which can be expressed as fol-lows:

V ¼ U−Wp þWN ð15Þ

where, Wp is the work done by the tangential stress; WN is the workdone by the cohesion. U, WN and Wp can be expressed as follows:

U ¼ D2

Z L

0k2ds WN ¼

Z L

0σNωds Wp ¼ phδ ¼ 1

2

Z L

0ph

dωds

� �2

ds ð16Þ

Here, the bending rigidity of the rock beam D=Eh3/12(1−μ2), thebending curvature k=(d2ω/ds2)[1+(dω/ds)2](1/2), μ is the Poisson'sratio, and δ is the displacement along the rock beam. By applying theTaylor expansion to the above integration at u=0, the potential energy

Fig. 16. Development of buckling and breaking type.

function can be obtained as follows:

V uð Þ ¼ DL32

πL

� �6u4 þ DL

4πL

� �4−

π2ph4L

u2 þ Lhσ t

2πuþ ο u4� � ð17Þ

Based on the standard form of the cusp catastrophe model, the po-tential function can be simplified:

V ¼ 14x4 þ a

2x2 þ bx ð18Þ

where

x ¼ DL8

πL

� �6 14

u

a ¼ DL2

πL

� �4−

π2ph2L

DL8

πL

� �6 −12

b ¼ Lhσ t

2πDL8

πL

� �6 −14

8>>>>>>>><>>>>>>>>:

ð19Þ

In terms of the cusp catastrophe model, the mutational condition ofthe rock beam occurs at 4a3 + 27b2 ≤ 0. Hence:

4DL2

πL

� �4−

π2ph2L

3DL8

πL

� �6 −32

þ 27Lhσ t

2π

� �2 DL8

πL

� �6 −12

≤0 ð20Þ

and the necessary condition satisfying the above equation is:

DL2

πL

� �4−

π2ph2L

≤0 ð21Þ

Hence, the conditions (occurrence criterion) leading to the bucklingand breaking type are as follows:

p≥π2D

L2h

4DL2

πL

� �4−

π2ph2L

3DL8

πL

� �6 −32

þ 27Lhσ t

2π

� �2 DL8

πL

� �6 −12

≤0

8>>><>>>:

ð22Þ

Because rockbursts are accompanied by energy release, the energygap before and after themutation of the system can be used to estimate

Fig. 18. Energy change through the mutation.

Fig. 20. Development of tensile cracking and toppling type.

80 T. Li et al. / Engineering Geology 222 (2017) 72–83

the energy release (Fig. 18).

ΔV ¼ 14Δx4 þ 1

2aΔx2 þ bΔx ð23Þ

By substitutingΔx ¼ 3ð− a3Þ

12 into Eq. (23), the following is obtained:

ΔV ¼ 34a2 þ b

ffiffiffiffiffiffiffiffiffiffi−3a

pð24Þ

Eqs. (24) and (19) show that the energy release during a rockburst iscontrolled by the variables h, L, E, μ, σt and p.

5.2. Tensile cracking and toppling type

Three developing stages are distinguished: (Fig. 19). (a) The tensilecracks develop in the layered rockmasses. (b) The tensile cracks contin-uously extend and intersect the existing discontinuities, loosening therock slabs from the surrounding rock masses. Consequently, the stressis redistributed and the surrounding rock pressures act on the rockslabs, thereby slightly bending. (c) The bending develops and causestensile fracture, thereby toppling the rock slabs. The effect of the sur-rounding rock pressures can be considered as energy being suppliedto the rock slabs, which facilitates this type of rockburst. A previous ex-periment successfully reproduced the phenomenon of ejectiverockburst by considering the mechanism of energy supply (Gu, 2014).

The cantilever-beam model is selected for analyses (Fig. 20) withunit width, thickness h, length L, elastic modulus E and weightmg. Thestress state of the model is simplified by considering the radial pressure(stress) of the surrounding rock σN. The total energy comprises threeparts: the structural strain energy U, work done by the radial pressureWN and work done by gravity Wg.

V ¼ U−WN−Wg ð25Þ

U ¼ D2

Z L

0f ″

21þ f 0

2� �−1

dx WN ¼Z L

0σN f xð Þdx Wg

¼ mg2

Z L

01−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− f 0

2q� �

dx ð26Þ

where D is the bending rigidity, f(x) is the displacement perpendicularto the cantilever beam, and the displacement along the cantilever

beam δ ¼Z L

0ð1−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− f 0

2q

Þdx.The following equation is obtained by applying the Taylor expansion

to Eq. (25) at x = 0:

V ¼ k4x4 þ k3x3 þ k2x2 þ k1x ð27Þ

Fig. 19.Mechanical model of tensile cracking and toppling type.

where

k1 ¼ σN

4L2; k2 ¼ σN

4L

k3 ¼ −mgσ2

NL4

24D2 ; k4 ¼ mgσ2NL

3

16D2

8><>: ð28Þ

By converting Eq. (27) to the standard form by eliminating x3 withthe help of s=x−b and b=k3/4k4, the following equation is obtained.

V ¼ g4x4 þ g2x

2 þ g1x ð29Þ

where

g1 ¼ −4b3k4 þ 3b2k3−2bk2 þ k1g2 ¼ 6b2k4−3bk3 þ k2g4 ¼ k4

8<: ð30Þ

After the conversion,V ¼ 14 s

4 þ us2 þ vs is obtained with the param-eters u= g2/2 g4 and v= g1/4 g4. In catastrophe theory, the mutationalcriterion is given as follows:

4u3 þ 27v2 ≤0 ð31Þ

The necessary condition to satisfy the above equation is u ≤ 0. Thus,the necessary and sufficient conditions (occurrence criterion) for the

Fig. 21. Development of the tensile cracking and sliding type.

Fig. 22.Mechanical model of tensile cracking and sliding type.

81T. Li et al. / Engineering Geology 222 (2017) 72–83

tensile cracking and toppling type are as follows:

σN ≥48D2

mgL4

44D2

mgL2−

L2σN

12

!3

þ 27EI

54σNL3 þ

σNL2

108mg

!2

≤0

8>>>><>>>>:

ð32Þ

where I = h3/12. Finally the energy gap before and after the rockburstis:

ΔV ¼ 34u2 þ v

ffiffiffiffiffiffiffiffiffiffiffi−3u

pð33Þ

Eqs. (33), (30) and (28) show that the energy release duringrockburst is controlled by the variables h, L, E, μ, mg and σN.

5.3. Tensile cracking and sliding type

The development can be subdivided into three stages (Fig. 21). (a)The tensile cracks develop under the increasing tangential compressivestress; (b) The cracks grow and intersect the existing fractured zone,resulting in a shear slip along the fractured zone; (c) The shear failureoccurs, and the rock block moves away from the surrounding rockmasses.

Themechanicalmodel for the shearing slip iswell defined by consid-ering the fracture zone as a strain-softening medium. As shown in Fig.22, the thickness of the strain-softening medium is h, the normal forceis N, the length andweight of the rock block are L andmg, the creep dis-placement along the failure plane is u, and the inclined angle is β.

Applying theWeibull constitutive model (Lawn, 1993) to define thestrain-softening medium leads to:

τ ¼ Gsuh

exp − u=u0ð Þmb� � ð34Þ

where, u0 is the averagemeasure of u,mb is the brittleness index, and Gs

is the initial shear modulus.Deriving Eq. (34) gives the displacement corresponding to the peak

intensity:

up ¼ u0 1=mbð Þ 1=mbð Þ ð35Þ

The peak intensity is given:

τp ¼ Gsu0 1=mbð Þ 1=mbð Þe− 1=mbð Þh i

=h ð36Þ

The displacement of the inflection point at the stress–strain curve isdetermined from the second derivative of the constitutive equation:

u1 ¼ mb þ 1mb

� � 1=mbð Þu0 ð37Þ

The strain energy and sliding potential energy constitute the totalpotential energy of the shear system. The potential function can be

obtained:

V ¼Z u

0u=hð Þ � LGsdu−mgu sinβ ð38Þ

V′ can be expressed as follows:

V0 ¼ u=hð Þ � LGse− u=u0ð Þmb−mg sinβ ð39Þ

By applying the Taylor expansion at u1=((m+1)/m)(1/mb)u0, thestandard form can be converted with the following parameters:

x ¼ u−u1

u1; ξ ¼ mg sinβ

GsLu1e mbþ1ð Þ=mb½ �

a ¼ −6

mb þ 1ð Þ2; b ¼ 6

mb mb þ 1ð Þ21−ξð Þ

8>><>>: ð40Þ

According to 4a3 + 27b2 ≤ 0, the condition for the minimum force(occurrence criterion) corresponding to the limited equilibrium of therockburst is given:

mg sinβ≥ 1þ 2ffiffiffi2

pmb

3 mb þ 1ð Þ

" #mb þ 1mb

� � 1mb GsLu0

he� mbþ1ð Þ=mb½ � ð41Þ

The following is the expression for the limited shear stress at thestate of the limited equilibrium:

τ f ¼mg sinβ

L¼ 1þ 2

ffiffiffi2

pmb

3 mb þ 1ð Þ

" #mb þ 1mb

� � 1mb Gsu0

he� mbþ1ð Þ=mb½ � ð42Þ

Finally the energy gap before and after the rockburst is:

ΔV ¼ 34a2 þ b

ffiffiffiffiffiffiffiffiffiffi−3a

pð43Þ

Eqs. (43) and (40) show that the energy release during the rockburstis controlled by variables h, L, Gs, u0, β , mb and mg.

In this section, catastrophe models are established based on under-standing the development of rockburst types. Compared with previouscatastrophe studies, the variables related to a specific geomechanicaltype of rockburst are identified and contribute to the establishment ofoccurrence criteria.

6. Application and discussion

We believe that the study on geomechanical types of rockburstsfrom qualitative analyses and combining it with the mechanical de-scription is one of the effective ways to predict and prevent rockburst.We applied the rockburst criteria in field conditions and selected thebuckling and breaking type for the case study. The rockburst was ob-served in the exploratory tunnel of Jinping II hydropower station, locat-ed at pile No. 3732.4m. The field lithology is largelymarble with layeredstructure (Table 4 lists the relevant mechanical and geometrical param-eters). The failure plane of the rockburst exhibited typical characteris-tics of the buckling and breaking type (referring to Fig. 6). Byparameterizing the occurrence criterion (Eq. (22)), the field conditionsare found to satisfy the criterion (Eq. (44)): in the necessary conditionthe maximum tangential compressive stress of 20.9 MPa exceeds thethreshold of 19.2 MPa; the sufficient condition equals −0.6 which isless than zero. Hence, identifying the mechanical and geometrical pa-rameters correctly can lead to the appropriate assessment of rockburst.In addition, the rockburst criteria of the other five geomechanical types

Table 4Parameterizing criterion of buckling and breaking type.

Lithology Mechanical properties Geometry

Marble Density ρ(kg/m3)

Uniaxial compressive strength Rc(MPa)

Uniaxial tensile strength σt

(MPa)Young's modulus E(MPa)

Poisson's ratioμ

Length L(m)

Thickness h(m)

2650 110–120 2–5 12–15 0.2–0.25 2–3 0.08–0.1

82 T. Li et al. / Engineering Geology 222 (2017) 72–83

are given in Eqs. (6), (8), (32) and (42).

p ¼ 20:9≥π2D

L2h¼ 19:2

4DL2

πL

� �4−

π2ph2L

3DL8

πL

� �6 −32

þ 27Lhσ t

2π

� �2 DL8

πL

� �6 −12

¼ −0:6≤0

8>>><>>>:

ð44Þ

In this paper, the term “geomechanical” refers to the combination ofthe two methods of engineering geological and mechanical analyses. Inthe former, the grading scheme of the rockburst (Table 2) is establishedto recognize and assess the rockburst phenomena; the proposals of therockburst geomechanical types help better understand rockburstmech-anisms, and provide a chance to predict rockbursts corresponding tospecific rock-mass structures and developing characteristics (Table 3).In the latter, the mechanical analyses and catastrophe models helpbetter understand rockbursts by illustrating the process mechanismand establishing a series of occurrence criteria to guide and predictrockbursts quantitatively.

However, because of the complexity of the rockburst in practicalconditions, only the occurrence criteria are insufficient to analyze andassess the rockbursts, and hence, requiring engineering geological in-vestigations and comprehensive judgement. In addition, the applicationof the occurrence criteria still needs more consideration in future stud-ies, such as the parameters that correspond to the geometry in stress-structural rockbursts and strength in stress rockbursts, which cannotbe easily identified and correlated in field conditions. By simplifyingthe form of criteria and strengthening its practicality is required inorder to be operated more efficiently.

7. Conclusions

The major conclusions of this study are as follows:

(1) Six geomechanical types of rockbursts are proposed after abun-dant geological investigations and mechanical analyses: tensilecracking and spalling, tensile cracking and toppling, tensilecracking and sliding, buckling and breaking, tensile shearingand bursting, and arc shearing and bursting. These six types canbe classified into two major types: stress rockburst and stress–structure rockburst.

The stress rockburst occurs in relatively intact rock masses, whichcan be characterized by the development of crack initiation, crackgrowth, and crack coalescence, such as those occurring in the tensilecracking and spalling, arc shearing and bursting, and tensile shearingand bursting. The stress-structure rockburst occurs in the form of astructural failure, because of the combined effects of crack developmentand presence of macroscopic discontinuities, such as those occurring inthe tensile cracking and toppling, tensile cracking and sliding, and buck-ling and breaking.

(2) The fracturemechanical theories are applied to analyze the stressrockbursts by defining the fracturing process. Based on the totalstress field around the existing flaw, the tensile cracking andspalling type can be interpreted as the development of tensilecracks. The arc shearing and bursting, and tensile shearing andbursting types, develop along the potential fracture plane andterminate at the preferred angle, indicating that a state of

maximum energy release rate is achieved. The occurrencecriteria for the stress rockbursts are proposed.

(3) The catastrophe theories are applied to analyze the stress-struc-ture rockbursts. Based on the cusp catastrophemodel and energypotential function, the buckling and breaking type is defined asbending failure under the tangential compressive stress. Thebending failure occurs in the tensile cracking and toppling typebecause of the radial pressures of the surrounding rock. In thetensile cracking and sliding type, the shear failure is definedalong the existing fractured zones induced by the growth and in-terconnected tensile cracks. The occurrence criteria for thestress-structure rockbursts are proposed.

Acknowledgments

The authors appreciate all of peoplewho contribute to this paper, par-ticularly the people providing supports from the poor construction envi-ronments. The authors also express sincere gratitude to Professor NiekRengers and Professor Theo van Asch for their improvement to this paperin English writing. This work was finacially supported by the NationalNatural Science Foundation of China (grant numbers 41230635 and41172279); the open fund from the State Key Laboratory of Geo-hazardPrevention and Geo-environment Protection, China (grant numbersSKLGP2009Z002, SKLGP2013Z004 and SKLGP2015Z004).

References

Atkinson, B.K., 1987. Fracture Mechanics of Rock. Academic Press, New York.Bobet, A., 2000. The initiation of secondary cracks in compression. Eng. Fract. Mech. 66

(2), 187–219.Chen, B.R., Feng, X.T., Li, Q.P., Luo, R.Z., Li, S., 2015. Rock burst intensity classification based

on the radiated energy with damage intensity at Jingping II hydropower station.China. Rock Mech. Rock Eng. 48 (1), 289–303.

Cook, N.G.W., Hoek, E., Pretorius, J.P.G., Ortlepp, W.D., Salamon, M.D.G., 1966. Rock me-chanics applied to the study of rockbursts. J. South. Afr. Inst. Min. Metall. 66 (10),435–528.

Deng, L., 2009. Study on the Key Rock Mass Engineering Problems of Nibashan HighwayTunnel. Doctor Degree Thesis, Southwest Jiaotong University, Chengdu (in Chinese).http://cdmd.cnki.com.cn/Article/CDMD-10613-2010115652.htm

Dyskin, A.V., Germanovich, L.N., 1993. Model of rockburst caused by cracks growing nearfree surface. Rockbursts and Seismicity in Mines 93. Proceedings of the 3rd Interna-tional Symposium on Rockbursts and Seismicity in Mines, Ontario, Canada,pp. 169–175.

Fan, Y., Lu, W.B., Zhou, Y.H., Yan, P., Leng, Z.D., Chen, M., 2016. Influence of tunnelingmethods on the strainburst characteristics during the excavation of deep rockmasses. Eng. Geol. 201, 85–95.

Feng, X.T., Chen, B.R., Ming, H.J., et al., 2012. Evolution law and mechanism of rockburstsin deep tunnels: immediate rockburst. Chin. J. Rock Mech. Eng. 31 (3):434–444 (inChinese). http://www.rockmech.org/EN/abstract/abstract27414.shtml.

Gong, Q.M., Yin, L.J., Wu, S.Y., Zhao, J., Ting, Y., 2012. Rock burst and slabbing failure and itsinfluence on TBM excavation at headrace tunnels in Jinping II hydropower station.Eng. Geol. 124, 98–108.

Gu, J.C., 2014. Mechanism of ejective rockburst and model testing technology. Chin.J. Rock Mech. Eng. 33 (6):1081–1089 (in Chinese). http://www.rockmech.org/EN/abstract/abstract28663.shtml.

Gu, M.C., He, F.L., Cheng, C.Z., 2002. Study on rockburst in Qingling tunnel. Chin. J. RockMech. Eng. 34:1–11 (in Chinese). http://www.ixueshu.com/document/6a8f83fbcfe479dc318947a18e7f9386.html.

Hoek, E., Brown, E.T., 1980. Underground Excavation in Rock. Inst. Min. Metall., London.Kidybiński, A., 1981. Bursting liability indices of coal. Int. J. Rock Mech. Min. Sci. 18 (6),

295–304.Lajtai, E.Z., 1974. Brittle fracture in compression. Int. J. Fract. 10 (4), 525–536.Lawn, B., 1993. Fracture of Brittle Solids. Cambridge University Press, Cambridge.Lee, C.F., Wang, S.J., Yang, Z.F., 1996. Geotechnical aspects of rock tunnelling in China.

Tunn. Undergr. Sp. Tech. 11, 445–454.

83T. Li et al. / Engineering Geology 222 (2017) 72–83

Li, K., 2006. Rockburst and its prevention at Futangba tunnel of the Duwen expressway.Subgrade Eng. 1:123–125 (in Chinese). http://www.cnki.com.cn/Article/CJFDTOTAL-LJGC200601052.htm.

Li, T.B., Meng, L.B., Zheng, J.G., Xiao, X.P., 2010. The features of rockburst and itsgeomechanical patterns in a hydroelectric power station in west China. GeologicallyActive: Proceedings of the 11th IAEG Congress, Auckland, New Zealand,pp. 3203–3211.

Li, S.J., Feng, X.T., Li, Z.H., Chen, B.R., Zhang, C.Q., Zhou, H., 2012. In situ monitoring ofrockburst nucleation and evolution in the deeply buried tunnels of Jinping II hydro-power station. Eng. Geol. 137, 85–96.

Ma, C.C., Li, T.B., Cheng, G.Q., Cheng, Z.Q., 2015. Amicro particlemodel for hard brittle rockand the effect of unloading rock burst. Chin. J. Rock Mech. Eng. 34, 1–11 (in Chinese).http://www.rockmech.org/EN/Y2015/V34/I02/217

Ma, C.C., Li, T.B., Xing, H.L., Zhang, H., Wang, M.J., Liu, T.Y., Cheng, G.Q., Cheng, Z.Q., 2016.Brittle rock modeling approach and its validation using excavation-induced micro-seismicity. Rock Mech. Rock. Eng. 49 (8), 3175–3188.

Mahanta, B., Singh, T.N., Ranjith, P.G., 2016. Influence of thermal treatment on mode Ifracture toughness of certain Indian rocks. Eng. Geol. 210, 103–114.

Pan, Y.S., Zhang, M.T., Li, G.Z., 1994. The study of chamer rockburst by the cusp model ofcatastrophe theory. Appl. Math. Mech. 15, 893–900.

Pan, Y., Liu, Y., Gu, S.F., 2001. Fold catastrophe model of mining fault rockburst. Int. J. CoalSci. Tech. 20 (1), 43–48.

Pan, Y., Zhang, Y., Yu, G.M., 2006. Mechanism and catastrophe theory analysis of circulartunnel rockburst. Appl. Math. Mech. 27 (6), 841–852.

Peng, J.X., 1998. Rockburst and its prevention at underground works of the Ertan hydro-power station. (7), 39–40 (in Chinese). http://www.cnki.com.cn/Article/CJFDTotal-SLFD807.012.htm

Petukov, I.M., 1979. Rockbursts in Kizel Coalfield Mines. Perm Publ, Perm.

Qian, Q.H., 2014. Definition, mechanism, classification and quantitative forecast model forrockburst and pressure bump. Rock Soil Mech. 35 (1):1–6 (in Chinese). http://www.cnki.com.cn/Article/CJFDTotal-YTLX201401001.htm.

Qin, S., Jiao, J.J., Wang, S., 2001. A cusp catastrophe model of instability of slip-bucklingslope. Rock Mech. Rock. Eng. 34 (2), 119–134.

Russense, B.F., 1974. Analyses of Rockburst in Tunnels in Valley Sides. M. Sc. Thesis. Nor-wegian Inst. of Technology, Trondheim (in Norwegian).

Sharon, E., Gross, S.P., Fineberg, J., 1996. Energy dissipation in dynamic fracture. Phys. Rev.Lett. 76, 2117.

Song, D.Z., Wang, E.Y., Li, N., Jin, M.Y., Xue, S.P., 2012. Rock burst prevention based on dis-sipative structure theory. Int. J. Rock Mech. Min. Sci. 22 (2), 159–163.

Wang, L.S., Li, T.B., Xu, J., 1999. Rockburst and its intensity classification of Erlangshanhighway tunnel. Highway 28 (2):41–45 (in Chinese). http://202.115.138.30/2004/gongcheng/Upload/file/dxal/5/8.pdf.

Wang, Q., Tang, Y.B., Li, Z., 2006. Engineering geological characteristics and preventionmeasures for rock bursts in cangling tunnel, Zhejiang province. J. Eng. Geol. 14 (2):276–280 (in Chinese). http://www.gcdz.org/CN/abstract/abstract9088.shtml#.

Wong, L.N.Y., Einstein, H.H., 2009. Systematic evaluation of cracking behavior in speci-mens containing single flaws under uniaxial compression. Int. J. Rock Mech. Min.Sci. 46, 239–249.

Wu, Y., 2003. Construction technology of rock burst prevention at diversion tunnels of theFutang hydropower station. Sichuan Water Power, 22(2), 91–94 (in Chinese). http://www.ixueshu.com/document/12186935795fec06.html

Zhao, G.B., Wang, D.Y., Gao, B., Wang, S.J., 2017. Modifying rock burst criteria based on ob-servations in a division tunnel. Eng. Geol. 216, 153–160.

Zhou, P.D., Hong, K.R., 1995. The rockburst features of Taipingyi tunnel and preventionmethods. Chin. J. Rock Mech. Eng. 42 (2):171–178 (in Chinese). http://www.rockmech.org/CN/abstract/abstract26004.shtml.