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Lecture 11: Brittle deformation; fracture mechanics

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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

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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"

why and how does brittle deformation take place?

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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 don#t break"

…breaks bonds at atomic scale…

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how do structural geologists e$amine how rocks break%

…e$perimental apparatus that uses cylindrical samples

from &tructural 'nalysis, ()*+-, )e.aor 

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)onath apparatuscylinder of rock is placed in jacket

!0 inch diameter1 2 inch length"

 fluid is pumped into area

 surrounding specimen to

 generate confining pressure

 similar to burial at depth

specimen is loaded by piston

from: )avis and +eynolds, 2334

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can do different types of e$periments

a$ial compression:

vertical a$ial compressive stress 5 confining pressure

called tria$ial deformation e$periments…this is misleading…

  most do not permit three principal stresses to vary independently

from: )avis and +eynolds, 2334

a$ial e$tension:confining pressure 5 vertical a$ial compressive stress

tensile strength:

rocks pulled apart

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four categories of brittle deformation processes

2" tensile cracking:

opening and propagation of cracks intounfractured material

6" shear rupture:

initiation of macroscopic shear fracture

7" frictional sliding:

sliding on pree$isting fracture

8" cataclastic flow:macroscopic ductile flow from

grain*scale fracturing and

frictional sliding

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2" 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 bondsthis theoretical strength is 90 to 0 -.a

…very large number…

measurement of rock strength in ;arth#s crust suggests

tensile cracking occurs at about 2 -.a or less

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this is known as the strength parado$

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

a

b

stress concentration at ends of elliptical hole depend on a$ial ratio:

a$ial ratio of 2"/b  with a as long a$is and b as short a$is of ellipse

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in 236s '? @riffith applied this idea to fracture formation

• all materials contain pree$isting microcracks where

stress is concentrated• microcracks propagate and grow even under low far*field stress

crack with largest a$ial ratio will propagate first

@riffith crack theory

rocks in Earth’s crust are weak because they contain Griffith cracksin 237s a new approach: linear elastic fracture mechanics

…all cracks have nearly infinite a$ial ratio !cracks are sharp"

…do not propagate under very small stresses because

tips are blunted by a crack-tip process zone

 process zone

   plastic deformation

crack 

 predicts that a longer crack will propagate before a shorter one

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@riffith crack theory and linear elastic fracture mechanics

imply cracks do not form instantaneously…

 begin at small flaw and grow outward

what happens during tensile cracking%

…look at laboratory e$periments

  rock cylinders stretched along a$is

opening of 

microcracks

largest crack forms

throughgoing crack 

!when crack reaches

edges of sample, the

sample separates into

two pieces"

not all bonds break at once theoretical strength is not reality

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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

tensile cracking driven by hydraulic fracturing occurs

in pulses in response to influ$ of fluid

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 belowthat necessary to crack rock 

crack continues to crack when additional fluid builds up

 fluid pressure

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6" shear rupture !shear fracture"

surface across which rock loses continuity when shear stresses

 parallel to surface are sufficiently large

in rock cylinder e$periments, shear fractures form atacute angle to far*field σ2  !σ2 5 σ6 A σ7 "

normal stress component across surface generates frictional resistance1

  if shear stress component e$ceeds resistance

σ2

σ6 A σ7

volume

decreases

volume

increases

!dilatancy*

cracking"

cracks coalesce

to form fault

!failure"

laboratory tria!ial-loading 

evolves into fault 

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what happened in the rock cylinder during e!periment?

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 7B to a$ial stress1 angle between two is 4B

  acute bisectri$ of fractures parallels far*field σ2

σ2

in reality, only one orientation will continue as it offsets other 

cracks form cracks coalescefrom: van der .luijm and -arshak, 233=

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7" frictional sliding

friction: resistance to sliding on surface

frictional sliding: movement on surface occurs when

   shear stress parallel to surface " 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

8" cataclasis and cataclastic flow !discuss later"

cataclasis: microfracturing, frictional sliding of grains, androtation and transport of grains

  ** similar to grinding corn between two mill stones:

grains roll, rotate, break, and grind into cornmeal

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for what stress states will brittle deformation occur%

 talking about three different phenomena:

  tensile cracking$ shear fracture development$ frictional sliding 

2" tensile*cracking

criteria for tensile*cracking from linear elastic fracture mechanics:

C 2 is stress intensity factor 

σt is far*field tensile stressD is geometry of crack !dimensionless"

c is half the length of the crack 

C 2 A σtD!Ec"2/6

crack grows when C 2 reaches value of C 2c  which is

  critical stress intensity factor or fracture toughness !tensile strength"

σtcA C 2c /!D!Ec"2/6"

…leads to σtc, critical tensile stress: instant when crack grows

tensile stress depends on fracture toughness, crack shape, and length

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6" shear fracture

let us return to rock cylinder laboratory e!periments#

• piece of rock cut into cylinder with length 6*8 times diameter1• sample placed between two steel pistons which are forced together1

• 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"

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but, if enough stress is applied, sample fractures %breaks&

…conduct tria$ial loading e$periments to determine

applied stress at which sample breaks

σc

σa   σaA a$ial stress, σ2 

σcA confining stress, σ7

first e$periment… set confining pressure low and increase

a$ial load !stress" until sample breaks

second e$periment… set confining pressure higher and increase

a$ial load !stress" until new sample breaks

keep repeating e$periments…

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you will generate a series of pairs of confining stresses and

associated a$ial stresses at which samples break…

  8 08 0 -.a

20

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σs

σn

0

0

2 20

-ohr circles that define

stress states where

samples fracture

!critical stress states"

together define the

failure envelope

for a particular rock 

failure envelope

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 σs  !mirror image about σn a$is"

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what does this straight line mean?

#corresponds to 'oulomb fracture criterion

(harles (oulomb in 2 tan φ (σn"  %empirical&

σs 

A shear stress parallel to fracture at failure

( A cohesion of rock !constant"

σn A normal stress across shear zone at instant of failure

tan φ A F A coefficient of internal friction !constant of proportionality"

this has form of y ( m! ) b %e*uation of a line&

y A σs $ A σn  b A intercept on σs a$is m A slope A tan φ A F

so (oulomb criterion plots as straight line on σn , σs plot

t t - h i l ith ( l b it i l tt d

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return to our -ohr circle with (oulomb criterion plotted

orientation of planes

that break is specified

 by 6θ of point that is both on circle and line

from: +owland and )uebendorfer, 2338

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further work by tto -ohr on shear*fracture criteria showed

  that straight line for (oulomb 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

σs

σn

0

0

220

this is ohr-'oulomb criterion

angle of fracture plane

relative to stress components

changes as function of stress state

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 plot of either (oulomb or -ohr*(oulomb criterion defines

failure envelope on -ohr diagram

no failure

  %stable&

-ohr circle

inside envelope

σs

σn

 failure at two points

  %brittle failure&

-ohr circle

tangent to envelope

σs

σn

impossible

  %unstable&

-ohr circle

outside envelope

σs

σn

failure envelope separates fields of GstableH and GunstableH stress

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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 deformation1

• cannot have GfailureH envelope…implies brittle• can appro$imate GyieldH envelope… sample yields plastically

σs

σn

tensile stress: necessary to cause tensile failure represented by a

 point, the tensile strength, on σn

 a$is to left of σs

 from Griffith crack theory, depends on flaws

  highly variable 

σn

σs

tensile strength %e!periments show less than compressive strength&

two parallel lines that parallel σn a$is

known as Ion -ises criterion  !independent of differential stress"

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can create composite failure envelope from empirical criteria

types of fracture !right"

  for composite curve

  !above"

 both from: van der .luijm and -arshak, 233=

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7" frictional sliding

friction requires certain critical shear stress to be reached

 before sliding initiates on pree$isting fracture

 failure criterion for frictional sliding 

  e$perimental data show that this plots as

  sloping straight line on -ohr diagram

  failure criterion for frictional sliding

  is largely independent of rock type

  σs / σn A constant

  Jyerlee#s law

for σn K 6 -.a: σs A 4 σn 

from: van der .luijm and -arshak, 233=

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the important question…

will new fractures form or will e$isting fractures slide%

e!amine failure envelopes to decide

figure below shows both Jyerlee#s law for frictional sliding and

  (oulomb shear fracture envelope for Jlair )olomite

slope and intercept of two envelopes are different…

for specific orientations of pree$isting fractures,

-ohr circle touches frictional envelope first

from: van der .luijm and -arshak, 233=

 pree!isting fractures will slide before new fracture forms

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what is effect of fluids on failure%

all rocks contain pores and cracks1

 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

.f  A ρfluidgh 

 pore pressure e$ceeds 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"

.f  A ρrock gh

when .f  5 hydrostatic, fluid is overpressured

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 pore pressure is outward push that counteracts inward compression

.f 

σif .f  5 σ7

how does pore pressure affect -ohr circle% pore pressure affects rock equally in all directions…

  back to our rock cylinder e$periments…

a$ially load rock1 pump fluid into sample for .f 

fluid

.f σ7

σ2

.f  reduces both σ2 and σ7 

yields effective stress, σn*.f 

σ2* .f  and σ7* .f  

diameter ohr’s circle unchanged, but center moves left 

σ2* .f σ2

σ7

* .f 

σ7

σd A constant

pore pressure weakens rock

σs A c > F!σn * .f "

for (oulomb failure

hydraulic fracturing !discussed earlier"