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Lecture 14:Lecture 14:Brittle Fracture & Brittle Fracture & Brittle Fracture & Brittle Fracture &

StressStress--Controlled FailureControlled Failure

1 of 46 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Author’s Note:Author’s Note:The lecture slides provided here are taken from the course “Geotechnical Engineering Practice”, which is part of the 4th year Geological Engineering program at the University of British Columbia (V C d ) Th k i i d (Vancouver, Canada). The course covers rock engineering and geotechnical design methodologies, building on those already taken by the students covering Introductory Rock Mechanics and Advanced Rock Mechanics Rock Mechanics.

Although the slides have been modified in part to add context, they of course are missing the detailed narrative that accompanies any l l d h h l lecture. It is also recognized that these lectures summarize, reproduce and build on the work of others for which gratitude is extended. Where possible, efforts have been made to acknowledge th v ri us s urc s ith list f r f r nc s b in pr vid d t th the various sources, with a list of references being provided at the end of each lecture.

Errors, omissions, comments, etc., can be forwarded to the


Errors, omissions, comments, etc., can be forwarded to the author at: [email protected]

Brittle Brittle ––vsvs-- Plastic Failure MechanismsPlastic Failure Mechanisms


SpallingSpalling and and RockburstingRockbursting


Kaiser et al. (2000)

Stress Driven Spalling in Tunnelling Stress Driven Spalling in Tunnelling -- IssuesIssues

Falling slabs of rock a hazard to workers.

Problem for TBM as gripper pads cannot be seated on the side wall.


RockburstingRockburstingR kb t A dd d i l t f il f k h k f t Rockburst: A sudden and violent failure of rock where rock fragments are ejected into the excavation. Energy is released as seismic energy radiated in the form of strain waves.

Strainburst: A self-initiated rockburst that develops due to an disequilibrium between high stresses and the rock strength (i.e. dynamic unstable fracturing).g ( y f g)

Usually occurs after blasting, as blasted face is unable to adjust to the immediately stress increase (i.e. ground “relief”) Immediate unloading of confinement from a triaxial to uniaxial stress condition, stored energy released as seismic energyIf burst partly occurs along geological

d b ll d Strainburst in the Faido multi-function tunnel (Gotthard Base Tunnel, Switzerland).

structures, damage may be controlled or limitedAlso commonly occurs when drifting through contact between a brittle and relative soft rock (i e hi hly dependent on local mine rock


rock (i.e. highly dependent on local mine rock stiffness)

RockburstingRockburstingPillar burst: Unstable pillar failure caused by some combination of fracturing, sliding, and buckling that extends deeper into the rock mass than a strain burst and often involves the core or foundation of a pillar. As such, there is often a loss of pillar load-carrying capacity that can liberate a considerable amount of energy.

Complete collapse of pillars – b tsevere burst

Domino effect – collapse of one pillar overstress adjacent pillarsNormally occurs when sill or

h i k t d fill crown shrinkage or cut and fill stopes reach critical sizePillar bursts are cause by the sudden change in potential energy as the hangingwall and energy as the hangingwall and footwall rapidly converge during failure process


RockburstingRockburstingSlip burst: Slip bursts are defined both by mechanism (stick-slip shear movement on a discontinuity) and the regional nature of driving forces. These bursts are less likely to be triggered by a particular blast, and more likely to occur afterwards. Slip occurs when the ratio of shear to normal (effective) stress along the fault plane reaches a critical value (its shear strength).

Similar to mechanics of an earthquake 99

7)qMost mining operations, the fault slippage intersects the mine openingsIn most cases, mining activity causes ya

ttet

al.

(19

g yslip by removing normal stress, although some local intensification of shear stress may also occurChanges in stress along a fault are ft li k d t i ti iti b

Why

Sip bursts at the L k F id Mi

often linked to mine activities by time-dependent deformation processes. These time-dependent processes can act over long periods of time regardless of continued


Lucky Friday Mine (Idaho).

of time, regardless of continued mining

Failure Criterion in Solid MechanicsFailure Criterion in Solid MechanicsTo understand the mechanisms at work leading to stress-induced ground control problems (spalling & bursting), we need to understand the basic principals of rock strength and brittle f fracture processes.

Traditionally there have been two approaches to analyzing rock Traditionally, there have been two approaches to analyzing rock strength:

experimental approach(i.e. phenomenological)

stress based

energy basedstrain based

mechanistic approach


Analysis of Rock StrengthAnalysis of Rock Strength

Phenomenological Approach

Relies on eneralization of

Mechanistic Approach

Derives its theories from Relies on generalization of large scale observations.

Derives its theories from elements of fracture at the microscopic scale.

Theories include:

• Maximum Stress theory• Tresca theory

Theories include: Theories include:

• Griffith Crack theoryTresca theory

• Coulomb theory• Mohr-Coulomb failure criterion

• Linear Elastic Fracture Mechanics (LEFM)

• Hoek-Brown failure criterion


Maximum (Minimum) Stress TheoryMaximum (Minimum) Stress TheoryTh i h i b bl h id l The compressive strength is probably the most widely used and quoted rock engineering parameter. Under uniaxial loading conditions, the maximum stress that the rock sample can sustain is referred to as the uniaxial rock sample can sustain is referred to as the uniaxial compressive strength, UCS.

It i i t t t li It is important to realize that the compressive strength is not an intrinsic property Intrinsic property. Intrinsic material properties do not depend on the specimen geometry or the loading geometry or the loading conditions used in the test: the uniaxial compressive strength does.


p gHarrison & Hudson (2000)

Maximum (Minimum) Stress TheoryMaximum (Minimum) Stress Theory

compression

limittension

failure occurs if 1 > UCS

limit

or if 3 < t

1 n3

UCStt


Maximum Maximum Stress Theory Stress Theory –– Limiting FactorLimiting Factor

compressionlimit

tensionlimit

1

Works okay in hydrostatic tension!

n

1 2 3

2 3

y

UCSt

Predicts failure in hydrostatic compression, which is not correct, therefore does not

l k!always work!


Hydrostatic CompressionHydrostatic Compressionl d d l d h h Applying non-deviatoric stresses produces a volume decrease which

eventually changes the rock fabric permanently as pores are crushed. Although such collapse produces an inflection in the stress vs strain response the rock will always accept additional -vs- strain response the rock will always accept additional

hydrostatic load.

I existing cracks close and minerals are compressed;

II elastic rock compression, consisting of pore deformation and grain compression at an and grain compression at an approximately linear rate;

III pore collapse;

IV intergrain locking and infinite IV intergrain locking and infinite compression as the only compressible elements remaining are the grains themselves


themselves.

Goodman (1989)

Deviatoric CompressionDeviatoric Compression

Deviatoric stresses are much di ti th th more disruptive than the

corresponding levels of hydrostatic stress. This is because they allow for the u y w fmaterial to deform in one direction more than the others (i.e. in the direction of the smaller load) In effect this smaller load). In effect, this allows fracturing, rupture and shearing of the rock to occur.

deformation


Goodman (1989)

TrescaTresca Theory (Pure Shear)Theory (Pure Shear)

1 due to symmetry, theory states that material will fail at 45°angles

failure occurs if

2

3

45°

max

max > So3

critical circle

123 123 n


TrescaTresca Theory Theory –– Limiting FactorLimiting Factor

1

2

3

45°

3

for uniaxial cases:c = t

max Which is not true for rock!!

shear limit

nc


Coulomb Coulomb Theory (Shear)Theory (Shear)

1

failure occurs if : > c + tan

2

45° + /2

max > c + tan

3

c

n

90° +

123t


Coulomb Coulomb Theory Theory –– Limiting FactorLimiting Factor

in uniaxial tension,coulomb theory

predictsfailure at an angle…f il i

c

gfailure in tension

13

… but uniaxial tensilefailure occurs along a plane perpendicular

to loading nn


MohrMohr--Coulomb Failure CriterionCoulomb Failure Criterion

1

failure occurs if :> c t n

2

45° + /2

max > c + tan

345 + /2

c

tensiontensioncutoffcutoff

n

90° +

123t


MohrMohr--Coulomb Failure CriterionCoulomb Failure Criterion

Theory tells us that failurec

yoccurs in shear ….

…. this agrees well with geological evidence where faulting is

1

ggenerally said to occur at angles of 30° (thrust) or 60° (normal)

3

45° + /2 = 60° 30°

3

60° So geometry seems to works!


Shear Failure EvolutionShear Failure Evolution

However:

• shear fractures are not easily found prior to failure

all of our observations • all of our observations where we say intact failure occurred in shear have been made after the fact

peak

(i.e. post-failure)

h h h • this may mean that shear does not occur at peak but post-peak


Mechanistic ControlsMechanistic ControlsThe Mohr-Coulomb criterion is most suitable for cohesionless materials, shear along discontinuity surfaces (e.g. along a pre-existing fault plane), and when rocks fail in a more ductile manner. Mechanistically though:

- Friction develops only on differential movement. Such movement can take place freely in a cohesionless material, but hardly in a cohesive one like rock prior to the development of a failure plane. In other words, mobilization of friction only becomes a factor once a failure plane is in the latter stages of friction only becomes a factor once a failure plane is in the latter stages of development;

- Many brittle failures observed in the lab and underground appear to be largely controlled by the development of microfractures. Since these fractures largely controlled by the development of microfractures. Since these fractures initiate on a microscopic scale at stresses below the peak strength, the dismissal of all processes undetectable to the naked eye and prior to peak strength leaves the phenomenological approach lacking.

This is not to say that phenomenological approaches like Mohr-Coulomb are not useful. Remember: Mohr-Coulomb is probably the most widely used failure criterion in industry, but its limitations need to be recognized.


Analysis of Brittle Rock StrengthAnalysis of Brittle Rock Strength

Phenomenological Approach

Relies on eneralization of

Mechanistic Approach

Derives its theories from Relies on generalization of large scale observations.

Derives its theories from elements of fracture at the microscopic scale.

Theories include:

• Maximum Stress theory• Tresca theory

Theories include: Theories include:

• Griffith Crack theoryTresca theory

• Coulomb theory• Mohr-Coulomb failure criterion

• Linear Elastic Fracture Mechanics (LEFM)

• Hoek-Brown failure criterion


Mechanistic Brittle Fracture TheoriesMechanistic Brittle Fracture Theories

F

F

Fmax At the atomic level, the development of interatomic r

rormax

pforces is controlled by the atomic spacing which can be altered by means of external loading

ro

FFmax … on extension, the

structure fractures where

external loading …

on

ror

structure fractures where the interatomic force is exhausted (i.e. the theoretical tensile strength)

tens

i

bonds become unstable


Mechanistic Brittle Fracture TheoriesMechanistic Brittle Fracture Theories

F Fr

F

roC ≈ ∞

ro

In compression …

F

on

… displacement is countered by an inexhaustible repulsive force

Fmax

tion

rotens

i

r

ion

attr

act

n

Thus, interatomic bonds will only break when pulled apart

com

pres

si

repu

lsio (i.e. in tension).


Theoretical StrengthTheoretical StrengthStrength is therefore a function of the

F

F

Fmax

gcohesive forces between atoms, where if F > Fmax, then the interatomic bonds will break. As such, we can derive the following: r

rmaxro

following: ro

F

Now for most rocks the Young’s on

Fmaxion

Now for most rocks, the Young s modulus, E, is of the order 10-100 GPa. If so, then the theoretical tensile strength of these rocks should be 1-10

rotens

io

r

on

attr

acti

n gGPa.

com

pres

sio

repu

lsio

n

However, this is at least 1000 timesgreater than the true tensile strength


g gof rock!!!

Griffith TheoryGriffith TheoryTo explain this discrepancy, Griffith (1920) postulated that in the case of a linear elastic material, brittle fracture is initiated through tensile stress concentrations at the tips of small, thin cracks randomly distributed within an otherwise isotropic material an otherwise isotropic material.


Griffith TheoryGriffith Theory

When the stresses around a Griffith crack increase due to an additional load, the corresponding increase in the potential energy may be balanced by either an increase in the strain energy and/or by an increase in the

k f (i th h k xt i )

In a compressive stress field Griffith (1924)

crack surface energy (i.e. through crack extension).

In a compressive stress field, Griffith (1924) suggested that although the applied stress may be compressive, the local stresses at the crack tipswould be tensile. Solving for a 2-D plane stress condition, it was found that the applied compressive stress required for crack growth was 8 times greater than in tension:

E = Young’s Modulus = crack surface energyc = crack half-length


Linear Elastic Fracture MechanicsLinear Elastic Fracture MechanicsGriffith’s energy instability concept forms the basis for the study of fracture mechanics in which the mechanics, in which the loading applied to a crack tip is analyzed to determine whether or not the crack tensile in-plane out of plane

1) Associated with a crack tip in a loaded material is a stress intensity factor, KI, corresponding to the induced stress state surrounding the crack (and

will propagate.in planeshear

out of planetearing

I p g glikewise KII and KIII depending on the crack displacement mode).

2) For a given crack, the boundary material will have a critical stress intensity factor, KIc, corresponding to the material strength at the crack tip.

4) The crack will continue to propagate as long as the above expression is met, and won’t stop until: K < K

3) The criterion for crack propagation can then be written as KI ≥ KIc. Laboratory testing for the KIc parameter is referred to as fracture toughness testing.


and won t stop until: KI < KIc.Ingraffea (1987)

Crack Propagation in TensionCrack Propagation in Tension

For a crack aligned perpendicular to a uniaxial tensile load, the maximum tensile stress concentration on the crack boundary is stress concentration on the crack boundary is at the tip of the long axis. This results in crack growth occurring perpendicular to the direction of the applied tension, enlarging the pp , g gcrack continuously until a free surface is reached.

Assuming that the solid is isotropic, the orientation of the growing crack remains constant and the magnitude of the local stress at the most highly stressed point on the crack surface increases as the crack lengthens.


Brace & Bombolakis (1963)

Crack Propagation in CompressionCrack Propagation in CompressionUnder uniaxial compressive loading conditions, the highest tangential stress concentration on an elliptical crack boundary was inclined 30°to the major principal stress. As these cracks develop, they will

li h l i h h j i i l rotate to align themselves with the major principal stress, 1.


Lajtai (1971)

Crack Propagation in CompressionCrack Propagation in CompressionExperimentally, it has been shown that brittle fractures propagate in the direction of 1. Cracks develop in this way to allow the newly forming crack faces to open/dilate in the direction of least

1

faces to open/dilate in the direction of least resistance (i.e. normal to 1 in the direction of 3).

This is most easily accommodated in uniaxial i i 0 F l l f compression since 3 = 0. For example, along a free

surface!!3


Crack Propagation in CompressionCrack Propagation in Compression

Opening OpeningPW

PH

Kaiser et al. (2000)


Laboratory Testing of Damage InitiationLaboratory Testing of Damage InitiationCorrelating the measured stress-strain behavior of a rock sample during uniaxial compression to the opening and closing of “Griffith” cracks, several important stages in the progressive failure of the sample can be detected.

Stage I - Existing cracks preferentially aligned to the applied load will close.

cd

Stage II - Near linear elastic stress-strain behaviour occurs.

ci

II

Stage III – New cracks initiate and propagate in a stable fashion ( ).ci

IV

IIIStage IV - Cracks begin to coalesce and propagate in an unstable fashion ( ).cd

IIII


Martin (1997)

Crack Crack Initiation & PropagationInitiation & PropagationEberhardt et al. (1998b)

A high degree of correlation can be established between stress-strain data and acoustic emission (AE) response in terms of identifying the onset of crack

( k h) l b l


initiation (i.e. crack growth) in laboratory test samples.

Crack Interaction Crack Interaction & & CoalescenceCoalescence

crack i ldi fcrackinteraction

yielding ofbridgingmaterial

crack tipstressesincrease

cracks propagate& interact

cracks coalesce;energy released

localization& developmentof rupture

surface

Eberhardt et al. (1998a)


Unstable Crack PropagationUnstable Crack PropagationAs crack induced damage accumulates the stress level associated with crack

Stable propagation: controlling the applied load can stop crack growth

As crack-induced damage accumulates, the stress level associated with crack initiation remains essentially unchanged; however, the stress level required for rupture reduces dramatically.

Stable propagation: controlling the applied load can stop crack growth.

Unstable propagation: relationship between the applied stress and the crack length ceases to exist and other parameters, such as the crack growth velocity take control of the propagation processgrowth velocity, take control of the propagation process.

Under such conditions, crack propagation would p p gbe expected to continue even if loading was stopped and held constant.

Bieniawski (1967) correlated the threshold 7) Bieniawski (1967) correlated the threshold for unstable crack growth, also referred to as the point of critical energy release and the crack damage threshold, with the point of reversal in the volumetric stressia

wski

(196

7


point of reversal in the volumetric stress-strain curve.Bi

en

Stiffness, Energy & FailureStiffness, Energy & FailureTh i l d l t f f il t bl k ti i

Capacity of the pillar

The violence and completeness of failure once unstable crack propagation is reached will depend on the relationship between the stiffness of the loaded component and that of the surrounding rock.

p y pto sustain load = Pmax

A

BPillar still has capacity to support load post-

k l ABpeak along AB

However, the post peak behaviour is also influenced by the surrounding rock stiffness (through which


( gthe pillar is being loaded).

Unstable Crack Propagation Unstable Crack Propagation –– Pillar LoadingPillar Loading

Thus, if we have an increase of convergence s beyond Pmax, to accommodate this displacement, the load on the pillar must reduce from PA to PB.

s

The amount of energy, Wpillar, absorbed in the process is given by A 1absorbed in the process is given by the area ABED.

However, in displacing by s from point A, the mine rock only unloads

FPF

A

2p , yto F and releases stored strain energy, Wmine, given by the area AFED.

C I thi W W d PB

B

D E

C In this case, Wmine > Wpillar, and catastophic failure occurs at, or shortly after, peak strength because the energy released by the mine rock during unloading is greater than that

s 3


during unloading is greater than that which can be absorbed by the pillar. Hoek & Brown (1980)

Failure Failure Around Around Underground ExcavationsUnderground ExcavationsMartin et al. (1999)

Observations from underground mining in massive brittle rocks suggest that failure initiates ggwhen the maximum tangential boundary stress reaches approximately 40% of the unconfined compressive strengthunconfined compressive strength.

maxmax = 0.4 = 0.4 UCSUCScici == 0.40.4 UCSUCS

(199

9)rh

ardt

et

al. This correlates with experimental

studies of brittle rock failure that show that stress-induced damage initiates at approximately 40%.


Eber

Failure Around Underground ExcavationsFailure Around Underground Excavations

cici = 0.4 = 0.4 UCSUCS


Martin et al. (1999)

Example: Tunnel Example: Tunnel SpallingSpalling & Depth of Failure& Depth of FailureP bl A 14 di t 100 d t l i t Problem: A 14-m diameter, 100-m deep tunnel is to

be excavated in a weak but massive sedimentary rock unit with an average compressive strength of 25 MPa. The compr ss str ngth of 5 M a. h tunnel will be excavated by a tunnel boring machine. In-situ stress tests revealed that the major principal stress is horizontal and three times higher than the vertical stress three times higher than the vertical stress. This has raised concerns of potential ground control problems related to stress-induced fracturing and slabbing of the rock. g g

As such, the designers need t tim t th t ti l to estimate the potential depth of stress-induced slabbing in order to select the proper rock support


p p ppmeasures.

Example: Tunnel Example: Tunnel SpallingSpalling & Depth of Failure& Depth of FailureAssuming a vertical stress of 2.5 MPa(calculated from the overburden), and r ur n), an adopting a horizontal to vertical stress ratio of 3, a maximum tangential stress of 20 g fMPa in the tunnel roof is calculated.

51D f

8.02520max MPa

c

5.1a

f

mmD f 1285.1

maD f 4Using Martin et al. (1999)’s empirical relationship This means that,

potentially, the l bbi d


slabbing may extend 4 m into the roof.

Lecture ReferencesLecture ReferencesBieniawski ZT (1967) Mechanism of brittle rock fracture: Part I Theory of the fracture processBieniawski, ZT (1967). Mechanism of brittle rock fracture: Part I - Theory of the fracture process.International Journal of Rock Mechanics and Mining Sciences & Geomechanical Abstracts 4(4): 395-406.

Brace, WF & Bombolakis, EG (1963). A note on brittle crack growth in compression. Journal ofGeophysical Research 68(12): 3709 3713Geophysical Research, 68(12): 3709-3713.

Eberhardt, E, Stead, D, Stimpson, B & Lajtai, EZ (1998a). The effect of neighbouring cracks onelliptical crack initiation and propagation in uniaxial and triaxial stress fields. Engineering FractureMechanics 59(2): 103-115.

Eberhardt, E, Stead, D, Stimpson, B & Read, RS (1998b). Identifying crack initiation andpropagation thresholds in brittle rock. Canadian Geotechnical Journal 35(2): 222-233.

Eberhardt, E, Stead, D & Stimpson, B (1999). Quantifying pre-peak progressive fracture damagein rock during uniaxial loading. International Journal of Rock Mechanics and Mining Sciences 36(3):n rock dur ng un ax al load ng. Internat onal Journal of Rock Mechan cs and M n ng Sc ences 36(3)361-380.

Goodman, RE (1989). Introduction to Rock Mechanics. John Wiley & Sons: New York.

Griffith, AA (1920). The phenomena of rupture and flow in solids. Philosophical Transactions of theR l S i f L d S i A M h i l d Ph i l S i 221(587) 163 198Royal Society of London, Series A, Mathematical and Physical Sciences, 221(587): 163-198.

Griffith, AA (1924). The theory of rupture. In Proceedings of the First International Congress forApplied Mechanics, Delft, pp. 55-63.

Harrison JP & Hudson JA (2000) Engineering Rock Mechanics – Part 2: Illustrative Worked


Harrison, JP & Hudson, JA (2000). Engineering Rock Mechanics Part 2: Illustrative WorkedExamples. Elsevier Science: Oxford.

Lecture ReferencesLecture ReferencesHoek, E & Brown, ET (1980). Underground Excavations in Rock. Institution of Mining andMetallurgy: London.

Ingraffea, AR (1987). Theory of crack initiation and propagation in rock. In Fracture Mechanics ofRock. Academic Press Inc. Ltd.: London, pp. 71-110.

Lajtai, EZ (1971). A theoretical and experimental evaluation of the Griffith theory of brittlefracture. Tectonophysics, 11: 129-156.

Kaiser, PK, Diederichs, MS, Martin, D, Sharpe, J & Steiner, W (2000). Underground works inhard rock tunnelling and mining In GeoEng2000 Melbourne Technomic Publishing Company:hard rock tunnelling and mining. In GeoEng2000, Melbourne. Technomic Publishing Company:Lancaster, pp. 841-926.

Martin, CD, Kaiser, PK & McCreath, DR (1999). Hoek-Brown parameters for predicting the depthof brittle failure around tunnels. Canadian Geotechnical Journal 36(1): 136-151.

Whyatt, JK, Blake, W & Williams, TJ (1997). Classification of large seismic events at the LuckyFriday Mine. Transactions of the Institution of Mining and Metallurgy, Section A: Mining Industry,106: A148–A162.


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