1
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
2 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
2
Analysis of Rock StrengthAnalysis of Rock Strength
3 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
4 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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)
3
Analysis of Rock StrengthAnalysis of Rock Strength
5 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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)
Phenomenological Failure CriteriaPhenomenological Failure Criteria
6 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Maximum (Minimum) Stress TheoryMaximum (Minimum) Stress Theory
τ
σUCS
compressionlimit
σ1
failure occurs if σ1 > σUCS
σ1 σn
4
Phenomenological Failure CriteriaPhenomenological Failure Criteria
7 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Maximum (Minimum) Stress TheoryMaximum (Minimum) Stress Theory
τ
σUCS
compressionlimit
failure occurs if σ1 > σUCS
σ1 σn
tensionlimit
σt
or if σ3 < σt
σ3 σ3
Hydrostatic CompressionHydrostatic Compression
8 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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)
5
Phenomenological Failure CriteriaPhenomenological Failure Criteria
9 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Maximum (Minimum) Stress TheoryMaximum (Minimum) Stress Theory
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
10 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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)
6
Analysis of Rock StrengthAnalysis of Rock Strength
11 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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)
Phenomenological Failure CriteriaPhenomenological Failure Criteria
12 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
7
Phenomenological Failure CriteriaPhenomenological Failure Criteria
13 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
TrescaTresca TheoryTheory
τ
σ1
σ2
σ3
45°
σnσc
τmax
for uniaxial cases:σc = σt
Which is not true for rock!!
shear limit
Analysis of Rock StrengthAnalysis of Rock Strength
14 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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)
8
Phenomenological Failure CriteriaPhenomenological Failure Criteria
15 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Coulomb TheoryCoulomb Theory
σ1
σ2
σ345° + φ/2
σn
τφ
90° + φc
σ1σ2σ3σt
failure occurs if :τmax > c + σtan φ
Phenomenological Failure CriteriaPhenomenological Failure Criteria
16 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
9
Analysis of Rock StrengthAnalysis of Rock Strength
17 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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)
Phenomenological Failure CriteriaPhenomenological Failure Criteria
18 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
10
Phenomenological Failure CriteriaPhenomenological Failure Criteria
19 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
MohrMohr--Coulomb Failure CriterionCoulomb Failure Criterion
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
Phenomenological Failure CriteriaPhenomenological Failure Criteria
20 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
MohrMohr--Coulomb Failure CriterionCoulomb Failure Criterion
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!
11
Shear Failure EvolutionShear Failure Evolution
21 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
22 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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.
12
Analysis of Rock StrengthAnalysis of Rock Strength
23 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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)
Rock Failure CriterionRock Failure Criterion
Generalized Hoek-Brown
Mohr-Coulomb
24 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
13
Strength of MaterialsStrength of Materials
25 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
26 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
14
Analysis of Brittle Rock StrengthAnalysis of Brittle Rock Strength
27 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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)
Mechanistic Brittle Fracture TheoriesMechanistic Brittle Fracture Theories
28 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
15
Mechanistic Brittle Fracture TheoriesMechanistic Brittle Fracture Theories
29 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
30 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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!!!
16
Griffith TheoryGriffith Theory
31 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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.
Griffith TheoryGriffith Theory
32 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
17
Griffith TheoryGriffith Theory
33 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
34 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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:
18
CanadaCanada’’s Nuclear Waste Disposal Concepts Nuclear Waste Disposal Concept
35 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Nuclear Waste DisposalNuclear Waste Disposal
36 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
19
Nuclear Waste Nuclear Waste –– Geologic DisposalGeologic Disposal
37 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
USA – Yucca Mountain Swiss – Opalinus Clay
Germans – Salt
CanadaCanada’’s Nuclear Waste Disposal Concepts Nuclear Waste Disposal Concept
38 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Canada – Granite
AECL’s URL
AECL’s URL = Atomic Energy of Canada Limited’s Underground Research Laboratory
20
AECLAECL’’ss Underground Research LaboratoryUnderground Research Laboratory
39 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Martin (1997)
AECLAECL’’ss Underground Research LaboratoryUnderground Research Laboratory
40 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Martin (1997)
21
AECLAECL’’ss Underground Research LaboratoryUnderground Research Laboratory
41 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Martin (1997)
AECLAECL’’ss Underground Research LaboratoryUnderground Research Laboratory
42 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
240 m Level240 m Level σσ33
σσ11
420 m Level420 m LevelMartin (1997)
22
AECLAECL’’ss URL URL –– Brittle FailureBrittle Failure
43 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
300mm diameter
1.2m diameterMartin (1997)
AECLAECL’’ss URL URL –– Brittle FailureBrittle Failure
44 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
In thin sectionIn thin section::
23
Crack Propagation in TensionCrack Propagation in Tension
45 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
46 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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
24
AECLAECL’’ss URL URL –– Brittle FailureBrittle Failure
47 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
In thin sectionIn thin section::
Damage Around an Underground ExcavationDamage Around an Underground Excavation
48 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
σ1 = 55 MPa
σ3 = 14 MPa
1.75 m
final shape
stages in notchdevelopment
microseismicevents
σσ33σσ11
420 m Level420 m Level
25
Crack Interaction and CoalescenceCrack Interaction and Coalescence
49 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
crackinteraction
localizedstresses increase
cracks propagateand interact
Eberhardt et al. (1998)
cracks coalesceand energy is released
coalescence ofbridging material
yielding and
Damage Around an Underground ExcavationDamage Around an Underground Excavation
50 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Opening OpeningPW
PH
σ
Kaiser et al. (2000)
26
Damage Around an Underground ExcavationDamage Around an Underground Excavation
51 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Mar
tin
(199
7)
Laboratory Testing of Damage InitiationLaboratory Testing of Damage Initiation
52 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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.
27
Crack/Damage InitiationCrack/Damage Initiation
53 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
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.
Eberhardt et al. (1998)
Brittle Fracture DamageBrittle Fracture Damage
54 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Friction (Friction (°°))
Cohesion (%)Cohesion (%)
Normalized DamageNormalized Damage
Laboratory Compression TestsLaboratory Compression TestsM
arti
n (1
997)
increasing damageincreasing damage
28
Damage Around an Underground ExcavationDamage Around an Underground Excavation
55 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
σ1 = 55 MPa
σ3 = 14 MPa
1.75 m
final shapestages in notchdevelopment
microseismicevents
final shape
notch position
Damage Initiation ThresholdDamage Initiation Threshold
56 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
100 (0.44)85 (0.40)CrackInitiation
227 (1)210 (1)Peak Strength
Granodiorite(MPa)
Granite(MPa)
σσcici = 0.4 = 0.4 σσUCSUCS
Eberhardt et al. (1998)
29
Damage Around an Underground ExcavationDamage Around an Underground Excavation
57 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)M
arti
n (1
997)
Damage Around an Underground ExcavationDamage Around an Underground Excavation
58 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
σσcici = 0.4 = 0.4 σσUCSUCS Kais
er e
t al
.(20
00)
30
Damage Around an Underground ExcavationDamage Around an Underground Excavation
59 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Kaiser et al. (2000)
σσcici = 0.4 = 0.4 σσUCSUCS
Damage Around an Underground ExcavationDamage Around an Underground Excavation
60 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06)
Top Related