Lecture 5: Brittle Fracture & Stress-Controlled FailureLecture 5: Brittle Fracture &...

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EOSC433EOSC433: :

Geotechnical Engineering Geotechnical Engineering Practice & DesignPractice & Design

Lecture 5: Lecture 5: Brittle Fracture & Brittle Fracture &

StressStress--Controlled FailureControlled Failure

1 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)

Stress and FailureStress and Failure


The excavation of an underground opening in stressed rock results in the deformation and weakening of the host rock. The analysis of this response is essential in rock mechanics design, since the resulting imbalance in the energy of the system results in the progressive degradation of the rock mass strength

In general, there are two approaches to stress and failure :

experimental approach(i.e. phenomenological)

stress based

energy basedstrain based

mechanistic approach

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Analysis of Rock StrengthAnalysis of Rock Strength


Phenomenological Approach

Relies on generalization of large scale observations.

Mechanistic Approach

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

• Maximum Stress theory• Tresca theory• Coulomb theory• Mohr-Coulomb failure criterion• Hoek-Brown failure criterion

Theories include: Theories include:

• Griffith Crack theory• Linear Elastic Fracture

Mechanics (LEFM)

Compressive StrengthCompressive Strength


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 uniaxialcompressive strength, σUCS.

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

Harrison & Hudson (2000)

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Mechanics (LEFM)

Phenomenological Failure CriteriaPhenomenological Failure Criteria


Maximum (Minimum) Stress TheoryMaximum (Minimum) Stress Theory

τ

σUCS

compressionlimit

σ1

failure occurs if σ1 > σUCS

σ1 σn

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τ

σUCS

compressionlimit

failure occurs if σ1 > σUCS

σ1 σn

tensionlimit

σt

or if σ3 < σt

σ3 σ3

Hydrostatic CompressionHydrostatic Compression


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 hydrostatic load.

I existing cracks close and minerals are compressed;

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

III pore collapse;

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

Goodman (1989)

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τ

σn

σUCS

compressionlimit

tensionlimit

σt

Predicts failure where none can occur, therefore

does not work in hydrostatic compression!

σ1 = σ2 = σ3

−σ1 = −σ2 = −σ3

Works okay in hydrostatic tension!

DeviatoricDeviatoric CompressionCompression


Deviatoric stresses are much more disruptive than the corresponding levels of hydrostatic stress. This is because they allow for the material to deform in one direction more than the others (i.e. in the direction of the smaller load). In effect, this allows fracturing, rupture and shearing of the rock to occur.

deformation

Goodman (1989)

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Mechanics (LEFM)



TrescaTresca TheoryTheory

τ

critical circle

σ1

τmax

σ2

failure occurs if τmax > So

σ3

σ1

σ2

σ3

45°

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

σn

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TrescaTresca TheoryTheory

τ

σ1

σ2

σ3

45°

σnσc

τmax

for uniaxial cases:σc = σt

Which is not true for rock!!

shear limit










Mechanics (LEFM)

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Coulomb TheoryCoulomb Theory

σ1

σ2

σ345° + φ/2

σn

τφ

90° + φc

σ1σ2σ3σt

failure occurs if :τmax > c + σtan φ



Coulomb TheoryCoulomb Theory

τφ

c

σ1σ3

90° + φ

… but uniaxial tensilefailure occurs along a plane perpendicular

to loading σn

in uniaxial tension,coulomb theory

predictsfailure at an angle…failure in

tension

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Mechanics (LEFM)



MohrMohr--Coulomb Failure CriterionCoulomb Failure Criterion

σ1

σ2

σ345° + φ/2

σn

τ failure occurs if :τmax > c + σtan φ

90° + φc

σ1σ2σ3σt

φ

mohr-coulo

mb criterio

n

tensiontensioncutoffcutoff

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σn

theory tells us that as we gradually load a sample, the stresses increase until failure occurs in shear ….

c

φ

σc

mohr-coulo

mb criterio

n




ε

it is widely believed thatfailure occurs in shear ….

σc

σ1

σ3

60°

…. this agrees well with geological evidence where faulting is

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

φ

45° + φ/2 = 60°≈ 30°

So geometry seems to works!

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Shear Failure EvolutionShear Failure Evolution


However:

• shear fractures are not easily found prior to failure

• all of our observations where we say intact failure occurred in shear have been made after the fact (i.e. post-failure)

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

σ

ε

σpeak

Mechanistic ControlsMechanistic Controls


The 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 development;

- Many brittle failures observed in the lab and underground appear to be 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.

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Mechanics (LEFM)

Rock Failure CriterionRock Failure Criterion

Generalized Hoek-Brown

Mohr-Coulomb


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Strength of MaterialsStrength of Materials


Historical data:Galileo (early 1600) – oldest tensile/bending tests;

Coulomb (1773) – generally accepted mode of fracturing in compression by shear;

Voight (1894) – testing of brittle failure in tension under squeezing;

Mohr (1900-1914) – failure envelope with two image points denoting the conjugate directions of shear planes;

Von Karman (1910-1911) –triaxial compression tests demonstrating plastic behaviour of marble with visible slip lines.

Strength of MaterialsStrength of Materials


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Analysis of Brittle Rock StrengthAnalysis of Brittle Rock Strength









Mechanics (LEFM)

Mechanistic Brittle Fracture TheoriesMechanistic Brittle Fracture Theories


F

ror

Fmax … on extension, the structure fractures where the interatomic force is exhausted (i.e. the theoretical tensile strength)

F

F

rormax

Fmax At the atomic level, the development of interatomicforces is controlled by the atomic spacing which can be altered by means of external loading …

bonds become unstable

tens

ion

ro

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Mechanistic Brittle Fracture TheoriesMechanistic Brittle Fracture Theories


F

rotens

ion

r

… displacement is countered by an inexhaustible repulsive force

F

roC ≈ ∞

F

com

pres

sion

Fmax

attr

actio

nre

puls

ion

ro

F

In compression …

Thus, interatomic bonds will only break when pulled apart (i.e. in tension).

Theoretical StrengthTheoretical Strength


F

F

rmax

F

ro

Fmax

Strength is therefore a function of the cohesive forces between atoms, where if F > Fmax, then the interatomic bonds will break. As such, we can derive the following:

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 GPa.

rotens

ion

r

com

pres

sion

Fmax

attr

actio

nre

puls

ion

ro

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

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Griffith TheoryGriffith Theory


To 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.



Using the “Theorem of Minimum Potential Energy”, Griffith (1920) established that 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 crack surface energy (i.e. through crack extension).

Solving for a 2-D plane stress condition, crack extension will occur when:

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

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Griffith-based relationships derived for tensile stress fields have proven practical for fracture studies involving such solid materials as metals, glass and ceramics. However, these relationships are less relevant in rock engineering problems which predominantly involve compressive stress fields.

σ

σ

Griffith (1924) therefore expanded his original formulation to include compressive stress fields. Griffith suggested that although the applied stress may be compressive, the local stresses at the crack tips would be tensile. Reformulating Griffith’s original equation, it was found that the applied compressive stress required for crack growth was 8 times greater:

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

Linear Elastic Fracture MechanicsLinear Elastic Fracture Mechanics


Griffith’s theory assumes that crack growth occurs when the maximum tensile stress concentration, occurring on a critical flaw boundary, reaches the tensile strength of the material surrounding the flaw. Over time, this stress-strength relationship has evolved into linear elastic fracture mechanics (LEFM).

Fracture mechanics concepts assume that cracks in a solid material can be stressed in three different modes:

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CanadaCanada’’s Nuclear Waste Disposal Concepts Nuclear Waste Disposal Concept


Nuclear Waste DisposalNuclear Waste Disposal


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Nuclear Waste Nuclear Waste –– Geologic DisposalGeologic Disposal


USA – Yucca Mountain Swiss – Opalinus Clay

Germans – Salt

CanadaCanada’’s Nuclear Waste Disposal Concepts Nuclear Waste Disposal Concept


Canada – Granite

AECL’s URL

AECL’s URL = Atomic Energy of Canada Limited’s Underground Research Laboratory

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AECLAECL’’ss Underground Research LaboratoryUnderground Research Laboratory


Martin (1997)



Martin (1997)

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Martin (1997)



240 m Level240 m Level σσ33

σσ11

420 m Level420 m LevelMartin (1997)

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AECLAECL’’ss URL URL –– Brittle FailureBrittle Failure


300mm diameter

1.2m diameterMartin (1997)



In thin sectionIn thin section::

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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 at the tip of the long axis. This results in crack growth occurring perpendicular to the direction of the applied tension, enlarging the crack continuously until a free surface is reached (Brace & Bombolakis, 1963).

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.

Crack Propagation in CompressionCrack Propagation in Compression


Experimentally, it has been shown that brittle fractures propagate in the direction of σ1. Cracks develop in this way to allow the newly forming crack faces to open/dilate in the direction of least resistance (i.e. normal to σ1 in the direction of σ3).

This is most easily accommodated in uniaxialcompression since σ3 = 0. For example, along a free surface!!

σ1

σ3

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In thin sectionIn thin section::

Damage Around an Underground ExcavationDamage Around an Underground Excavation


σ1 = 55 MPa

σ3 = 14 MPa

1.75 m

final shape

stages in notchdevelopment

microseismicevents

σσ33σσ11

420 m Level420 m Level

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Crack Interaction and CoalescenceCrack Interaction and Coalescence


crackinteraction

localizedstresses increase

cracks propagateand interact

Eberhardt et al. (1998)

cracks coalesceand energy is released

coalescence ofbridging material

yielding and



Opening OpeningPW

PH

σ

Kaiser et al. (2000)

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Mar

tin

(199

7)

Laboratory Testing of Damage InitiationLaboratory Testing of Damage Initiation


Correlating 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. Amongst these, crack initiation represents the stress where microfracturing begins and is marked as the point where the lateral or volumetric strain curves depart from linearity.

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Crack/Damage InitiationCrack/Damage Initiation


A high degree of correlation was established between stress-strain data and acoustic emission (AE) response in terms of identifying the onset of damage initiation (i.e. crack growth) in laboratory tested samples.


Brittle Fracture DamageBrittle Fracture Damage


Friction (Friction (°°))

Cohesion (%)Cohesion (%)

Normalized DamageNormalized Damage

Laboratory Compression TestsLaboratory Compression TestsM

arti

n (1

997)

increasing damageincreasing damage

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σ1 = 55 MPa

σ3 = 14 MPa

1.75 m

final shapestages in notchdevelopment

microseismicevents

final shape

notch position

Damage Initiation ThresholdDamage Initiation Threshold


100 (0.44)85 (0.40)CrackInitiation

227 (1)210 (1)Peak Strength

Granodiorite(MPa)

Granite(MPa)

σσcici = 0.4 = 0.4 σσUCSUCS


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57 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)M

arti

n (1

997)



σσcici = 0.4 = 0.4 σσUCSUCS Kais

er e

t al

.(20

00)

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Kaiser et al. (2000)

σσcici = 0.4 = 0.4 σσUCSUCS



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