1 EOSC433 EOSC433 : : Geotechnical Engineering Geotechnical Engineering Practice & Design Practice & Design Lecture 5: Lecture 5: Brittle Fracture & Brittle Fracture & Stress Stress-Controlled Failure Controlled Failure 1 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06) Stress and Failure Stress 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 based strain based mechanistic approach
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Lecture 5: Brittle Fracture & Stress-Controlled FailureLecture 5: Brittle Fracture & Stress-Controlled Failure 1 of 58 Dr. Erik Eberhardt EOSC 433 (Term 2, 2005/06) Stress and Failure
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EOSC433EOSC433: :
Geotechnical Engineering Geotechnical Engineering Practice & DesignPractice & Design
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
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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.
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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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Analysis of Rock StrengthAnalysis of Rock Strength
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Phenomenological Approach
Relies on generalization of large scale observations.
Mechanistic Approach
Derives its theories from elements of fracture at the microscopic scale.
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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
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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.
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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
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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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Analysis of Rock StrengthAnalysis of Rock Strength
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Phenomenological Approach
Relies on generalization of large scale observations.
Mechanistic Approach
Derives its theories from elements of fracture at the microscopic scale.
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
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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
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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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Analysis of Rock StrengthAnalysis of Rock Strength
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Phenomenological Approach
Relies on generalization of large scale observations.
Mechanistic Approach
Derives its theories from elements of fracture at the microscopic scale.
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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
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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
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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
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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:
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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:
Linear Elastic Fracture MechanicsLinear Elastic Fracture Mechanics
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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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In thin sectionIn thin section::
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Crack Propagation in TensionCrack Propagation in Tension
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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
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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!!
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In thin sectionIn thin section::
Damage Around an Underground ExcavationDamage Around an Underground Excavation
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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
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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
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Opening OpeningPW
PH
σ
Kaiser et al. (2000)
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Damage Around an Underground ExcavationDamage Around an Underground Excavation
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Mar
tin
(199
7)
Laboratory Testing of Damage InitiationLaboratory Testing of Damage Initiation
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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
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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
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