Aula-L11-BrittleFracture.pdf

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1 of 11 Erik Eberhardt – UBC Geological Engineering EOSC 433 (2008)

EOSC433EOSC433::

Geotechnical EngineeringGeotechnical EngineeringPractice & DesignPractice & Design

Lecture 11:Lecture 11:

Rock Strength &Rock Strength &StressStress--Controlled FailureControlled Failure


Stress and FailureStress and Failure

The excavation of an underground opening in stressed rock resultsin the deformation and weakening of the host rock. The analysisof this response is essential in rock mechanics design, since theresulting imbalance in the energy of the system results in theprogressive 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




Analysis of Rock StrengthAnalysis of Rock Strength

Phenomenological ApproachRelies on generalization oflarge scale observations.

Mechanistic ApproachDerives its theories fromelements of fracture at themicroscopic 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 FractureMechanics (LEFM)


Compressive StrengthCompressive Strength

The compressive strength is probably the most widelyused and quoted rock engineering parameter. Underuniaxial loading conditions, the maximum stress that therock sample can sustain is referred to as the uniaxialcompressive strength, UCS.

It is important to realizethat the compressivestrength is not an intrinsicproperty. Intrinsicmaterial properties do notdepend on the specimengeometry or the loadingconditions used in thetest: the uniaxialcompressive strength does.

Harrison & Hudson (2000)




Hydrostatic CompressionHydrostatic Compression

Applying non-deviatoric stresses produces a volume decrease which

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

I existing cracks close andminerals are compressed;

II elastic rock compression,consisting of pore deformationand grain compression at anapproximately linear rate;

III pore collapse;

IV intergrain locking and infinitecompression as the only

compressible elementsremaining are the grainsthemselves.

Goodman (1989)


DeviatoricDeviatoric CompressionCompression

Deviatoric stresses are muchmore disruptive than thecorresponding levels ofhydrostatic stress. This isbecause they allow for thematerial to deform in onedirection more than the others(i.e. in the direction of thesmaller load). In effect, thisallows fracturing, rupture andshearing of the rock to occur.

deformation

Goodman (1989)




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 takeplace freely in a cohesionless material, but hardly in a cohesive one like rockprior to the development of a failure plane. In other words, mobilization offriction only becomes a factor once a failure plane is in the latter stages ofdevelopment;

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

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


Linear Elastic Fracture MechanicsLinear Elastic Fracture Mechanics

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

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




Crack Propagation in TensionCrack Propagation in Tension

For a crack aligned perpendicular to auniaxial tensile load, the maximum tensilestress concentration on the crack boundary isat the tip of the long axis. This results incrack growth occurring perpendicular to thedirection of the applied tension, enlarging thecrack continuously until a free surface isreached (Brace & Bombolakis, 1963).

Assuming that the solid is isotropic, theorientation of the growing crack remainsconstant and the magnitude of the localstress at the most highly stressed point on

the crack surface increases as the cracklengthens.


Crack Propagation in CompressionCrack Propagation in Compression

Experimentally, it has been shown that brittlefractures propagate in the direction of 1. Cracksdevelop in this way to allow the newly forming crackfaces to open/dilate in the direction of leastresistance (i.e. normal to 1 in the direction of 3).

This is most easily accommodated in uniaxialcompression since 3 = 0. For example, along a freesurface!!

1

3




Laboratory Testing of Damage InitiationLaboratory Testing of Damage InitiationCorrelating the measured stress-strain behavior of a rock sample duringuniaxial compression, to the opening and closing of “Griffith” cracksseveral important stages in the progressive failure of the sample can bedetected. Amongst these, crack initiation represents the stress wheremicrofracturing begins and is marked as the point where the lateral orvolumetric strain curves depart from linearity.

Aula-L11-BrittleFracture.pdf

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