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Lecture 11 Brittle deformation fracture

mechanics

we have looked at fractures joints, veins,

faults

these are brittle phenomena caused by brittle

deformation

brittle deformation permanent change

that occurs in a solid material due to growth of

fractures and/or sliding on them once they have

formed

why and how does brittle deformation take place?

solid composed of atoms or ions bonded to one

another through chemical bonds which can be

visualized as tiny springs

each chemical bond has an equilibrium length

any two chemical bonds connected to same atom

have an equilibrium angle between them

during elastic strain..bonds holding atoms

together in solid, stretch, shorten, and/or

bend, but they do not break once stress is

removed, the bonds return to equilibrium

elastic strain is recoverable

rock cannot develop large elastic strains (only a

few percent) must deform in a ductile way (does

not break) must deform in a brittle way (does

break)

during brittle deformation.stresses become large

enough to bend, then break atomic bonds. new

fracture forms or old surface slips

fractures can be between grains or across grains

what exactly happens when something breaks?

discussing solids. (liquids and gases dont

break)

breaks bonds at atomic scale

how do structural geologists examine how rocks

break?

experimental apparatus that uses cylindrical

samples

from Structural Analysis, CD-ROM, DePaor

Donath apparatus

cylinder of rock is placed in jacket (0.5 inch

diameter 1.0 inch length)

fluid is pumped into area surrounding specimen

to generate confining pressure similar to burial

at depth

specimen is loaded by piston

from Davis and Reynolds, 1996

can do different types of experiments

axial compression vertical axial compressive

stress gt confining pressure

axial extension confining pressure gt vertical

axial compressive stress

tensile strength rocks pulled apart

from Davis and Reynolds, 1996

called triaxial deformation experimentsthis is

misleading most do not permit three

principal stresses to vary independently

four categories of brittle deformation processes

1) tensile cracking opening and propagation

of cracks into unfractured material

2) shear rupture initiation of macroscopic

shear fracture

3) frictional sliding sliding on preexisting

fracture

4) cataclastic flow macroscopic ductile flow

from grain-scale fracturing

and frictional sliding

1) tensile cracking

cracks on atomic scale

crystal lattice (atoms and bonds)

crack

one model is for crack surface to break at once

tensile stress necessary is equal to

strength of each chemical bond multiplied by

number of bonds

this theoretical strength is 500 to 5000

MPa .very large number! measurement of rock

strength in Earths crust suggests tensile

cracking occurs at about 10 MPa or less

this is known as the strength paradox

engineers realized far-field stress (stress

applied at a distance from area of interest) is

concentrated at sides of flaws (holes) in

an elastic (recoverable) medium

concentration along an ellipse-shaped flaw

is (2a 1)/b with a as long axis and b as

short axis of ellipse

stress concentration at ends of elliptical hole

depend on axial ratio axial ratio of

81--concentration factor of 17 axial ratio of

321--concentration factor of 65

Griffith crack theory

in 1920s A.W. Griffith applied this idea to

fracture formation

all materials contain preexisting microcracks

where stress is concentrated microcracks

propagate and grow even under low far-field stress

crack with largest axial ratio will propagate

first

rocks in Earths crust are weak because they

contain Griffith cracks

in 1930s a new approach linear elastic fracture

mechanics

all cracks have nearly infinite axial ratio

(cracks are sharp) do not propagate under very

small stresses because tips are blunted by a

crack-tip process zone

crack

process zone plastic deformation

predicts that a longer crack will propagate

before a shorter one

Griffith crack theory and linear elastic fracture

mechanics imply cracks do not form

instantaneously. begin at small flaw and grow

outward

not all bonds break at once

theoretical strength is not reality

what happens during tensile cracking? look at

laboratory experiments rock cylinders

stretched along axis

largest crack forms throughgoing crack (when

crack reaches edges of sample, the sample

separates into two pieces)

opening of microcracks

hydraulic fracturing can create tensile

fracturing in a rock cylinder even if remote

principal stresses are compressive

by increasing the fluid pressure in pores and

cracks

fluid pressure in cracks creates tensile stress

at crack tip

crack propagates and increases volume of crack

fluid pressure decreases as fluid has greater

volume

crack stops propagating when fluid pressure

drops below that necessary to crack rock

crack continues to crack when additional fluid

builds up fluid pressure

tensile cracking driven by hydraulic fracturing

occurs in pulses in response to influx of fluid

2) shear rupture (shear fracture)

surface across which rock loses continuity when

shear stresses parallel to surface are

sufficiently large

in rock cylinder experiments, shear fractures

form at acute angle to far-field ?1 (?1 gt ?2

?3 )

normal stress component across surface generates

frictional resistance if shear stress

component exceeds resistance

evolves into fault

laboratory triaxial-loading

what happened in the rock cylinder during

experiment?

failure strength for shear fracture not a

definition of stress state when single crack

propagates, but stress state when many cracks

coalesce to form throughgoing rupture

two shear ruptures can form (conjugates) each

at 30 to axial stress angle between two is 60

acute bisectrix of fractures parallels

far-field ?1

in reality, only one orientation will continue as

it offsets other

3) frictional sliding

friction resistance to sliding on

surface frictional sliding movement on surface

occurs when shear stress parallel to

surface gt frictional resistance to sliding

frictional resistance to sliding proportional

to normal stress component across surface

why? fault surfaces have bumps on them

(asperities) that act to hold rock

surfaces in place increase in normal

stress pushes asperities into opposing

wall more deeply

4) cataclasis and cataclastic flow (discuss later)

cataclasis microfracturing, frictional sliding

of grains, and rotation and transport of

grains -- similar to grinding corn between

two mill stones grains roll, rotate, break, and

grind into cornmeal

for what stress states will brittle deformation

occur?

talking about three different phenomena

tensile cracking shear fracture development

frictional sliding

1) tensile-cracking

criteria for tensile-cracking from linear elastic

fracture mechanics

K1 is stress intensity factor st is far-field

tensile stress Y is geometry of crack

(dimensionless) c is half the length of the crack

K1 stY(pc)1/2

crack grows when K1 reaches value of K1c which

is critical stress intensity factor or

fracture toughness (tensile strength)

leads to stc, critical tensile stress instant

when crack grows

stc K1c /(Y(pc)1/2)

tensile stress depends on fracture toughness,

crack shape, and length

2) shear fracture

let us return to rock cylinder laboratory

experiments

piece of rock cut into cylinder with length 2-4

times diameter sample placed between two steel

pistons which are forced together applied

stress changes length, diameter, volume of

sample, which are measured by strain gauges

attached to sample

at first, when stress is removed, sample

returns to original shape recoverable

characteristic of elastic deformation

(rubber band)

but, if enough stress is applied, sample

fractures (breaks)

conduct triaxial loading experiments to

determine applied stress at which sample breaks

?a axial stress, ?1

?c confining stress, ?3

first experiment set confining pressure low and

increase axial load (stress) until sample

breaks second experiment set confining pressure

higher and increase axial load (stress) until

new sample breaks keep repeating experiments

you will generate a series of pairs of confining

stresses and associated axial stresses at which

samples break

- 40 540 500 MPa
- 800 650 MPa
- 400 1400 1000 MPa

we use these as s1 and s3 and plot Mohr

circles get a sequence of circles offset from

one another

diameters are stress difference centers are

stress sum/2

failure envelope

?s

Mohr circles that define stress states

where samples fracture (critical stress states)

together define the failure envelope for a

particular rock

?n

failure envelope is tangent to circles of all

critical stress states and is a straight

line can also draw failure envelope in negative

quadrant for ?s (mirror image about ?n

axis)

what does this straight line mean?

corresponds to Coulomb fracture criterion

Charles Coulomb in 18th century proposed that

formation of shear stress parallel to failure

relates to normal stress by

?s C tan f (?n) (empirical)

?s shear stress parallel to fracture at

failure C cohesion of rock (constant) ?n

normal stress across shear zone at instant of

failure tan f µ coefficient of internal

friction (constant of proportionality)

this has form of y mx b (equation

of a line)

y ?s x ?n b intercept on ?s

axis m slope tan f µ

so Coulomb criterion plots as straight line on ?n

, ?s plot

return to our Mohr circle with Coulomb criterion

plotted

orientation of planes that break is specified by

2? of point that is both on circle and line

from Rowland and Duebendorfer, 1994

further work by Otto Mohr on shear-fracture

criteria showed that straight line for Coulomb

criterion is valid only for limited range

of confining pressures at lower confining

pressures curves to steeper slope at higher

confining pressures curves to shallower slope

defines parabola

angle of fracture plane relative to stress

components changes as function of stress state

plot of either Coulomb or Mohr-Coulomb criterion

defines failure envelope on Mohr diagram

failure envelope separates fields of stable and

unstable stress

can we do this for very high and very low

confining pressures?

or when one of the principal stresses is tensile?

high confining pressures begin to have plastic

deformation cannot have failure

envelopeimplies brittle can approximate

yield envelopesample yields plastically

two parallel lines that parallel ?n axis known

as Von Mises criterion (independent of

differential stress)

tensile stress necessary to cause tensile

failure represented by a point, the tensile

strength, on ?n axis to left of ?s

from Griffith crack theory, depends on flaws..

highly variable

tensile strength (experiments show less than

compressive strength)

can create composite failure envelope from

empirical criteria

types of fracture (right) for composite curve

(above)

both from van der Pluijm and Marshak, 1997

3) frictional sliding

friction requires certain critical shear stress

to be reached before sliding initiates on

preexisting fracture

failure criterion for frictional sliding

experimental data show that this

plots as sloping straight line on Mohr

diagram failure criterion for

frictional sliding is largely

independent of rock type

?s / ?n constant

Byerlees law

for ?n lt 200 MPa ?s 0.85 ?n for 200 MPa lt ?n

lt 2000 MPa ?s 50 MPa 0.6 ?n

the important question will new fractures form

or will existing fractures?

examine failure envelopes to decide

figure below shows both Byerlees law for

frictional sliding and Coulomb shear fracture

envelope for Blair Dolomite

slope and intercept of two envelopes are

different for specific orientations of

preexisting fractures, Mohr circle touches

frictional envelope first

preexisting fractures willbefore new

fracture forms

what is effect of fluids on failure?

all rocks contain pores and cracks below water

table these are filled with fluid (usually

water, but sometimes oil or gas)

for permeable rock (interconnected pore

space), fluid can flow easily pressure is

hydrostatic Pf ?fluidgh

pore pressure exceeds hydrostatic if permeability

is low fluid trapped in sandstone lens

surrounded by shale pore pressure in sandstone

can approach lithostatic (pressure approaches

that of overlying rock)

Pf ?rockgh

when Pf gt hydrostatic, fluid is overpressured

pore pressure is outward push that counteracts

inward compression

if Pf gt ?3

hydraulic fracturing (discussed earlier)

how does pore pressure affect Mohr circle?

pore pressure affects rock equally in all

directions back to our rock cylinder

experiments axially load rock pump fluid into

sample for Pf

Pf reduces both ?1 and ?3

?1- Pf and ?3- Pf

diameter Mohrs circle unchanged, but center

moves left

yields effective stress, ?n-Pf

?s c µ(?n - Pf)

for Coulomb failure

pore pressure weakens rock

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