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COULOMB STRESSCOULOMB STRESS

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cracks form cracks coalesce

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3

P

1

(1 + 3 )

(1 - 3 )

(1 - 3 )2 2

2

n

s

point P represents a plane that is degrees from 3plots 2 counterclockwise from 1 (note original diagram in upper right)

s

n

1

3

D

Mohr circle

A

BC

sP

nP

normal and shear stresses on point P are nP and sP, respectivelynP = 1/2(1 - 3 ) + 1/2(1 - 3 )cos 2sP = 1/2(1 - 3 ) sin 2

13

P: plane to left

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conduct triaxial loading experiments to determine applied stress at which sample breaks

c

a a= axial stress, 1c= confining stress, 3

first experiment set confining pressure low and increaseaxial load (stress) until sample breaks

second experiment set confining pressure higher and increaseaxial load (stress) until new sample breaks

keep repeating experiments

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you will generate a series of pairs of confining stresses andassociated axial stresses at which samples break

40 540 500 MPa150 800 650 MPa400 1400 1000 MPa

ac

we use these as 1 and 3 and plot Mohr circles

get a sequence of circlesoffset from one another

a- c s

500

500

1000n

1500

diameters are stress difference;centers are stress sum/2

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s

n500

500

1000 1500

Mohr circles that definestress states wheresamples fracture

(critical stress states) together define the

failure envelopefor a particular rock

failure envelope

failure envelope is tangent to circles of all critical stress statesand is a straight line

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what does this straight line mean?

corresponds to Coulomb fracture criterion

Charles Coulomb in 18th century proposed that formation ofshear stress parallel to failure relates to normal stress by

s = C + tan (n) (empirical)

s = shear stress parallel to fracture at failureC = cohesion of rock (constant)n = normal stress across shear zone at instant of failuretan = = 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 =

so Coulomb criterion plots as straight line on n , s plot

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suppose 1 is oriented east-west, horizontal, and equal to 40 MPa

3 is vertical and equal to 20 MPa

determine normal and shear stresses on a fault plane thatstrikes north-south and dips 55 west

5540 MPa

20 MPaangle, , between 3 and fault plane is 35;

2 = 70

1

3

but which direction on Mohr circle?

55

35

3

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convention for shear stresses for Mohr circle diagram: sinistral is considered positive (+) dextral considered negative (-)

positive (sinistral) negative (dextral)

1

3

our example is dextral andthus, negative

10/25

to construct Mohr circle, we know

1 is equal to 40 MPa3 is equal to 20 MPa

2 = 70 and is negative

10

20 4070

10 n= 33.4 s = -9.4

P

n

s

1 + 32

distance from origin of center of circle(40+20)/2=30

1 - 3 diameter of circle: 40-20=20

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stress acts on every surface that passes through the pointwe can use three mutually perpendicular planes to describe

the stress state at the point, which we approximate as a cube

each of the three planeshas one normal component

& two shear components

therefore, 9 components necessaryto define stress at a point

3 normal6 shear

convention for describing: iji= plane on which component acts (defined by perpendicular)j = axis to which component is parallel

i = 1, 2, 3 j=1, 2, 3

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the nine components (six of which are independent) can be writtenin matrix form:

11 12 13

21 22 23

31 32 33

ij =

this is the stress tensor

describing stress as tensor allows relative ease of manipulation;can change reference frame (rotation; translation)

components on diagonal are normal stresses; off are shear stresses

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Earthquake ruptures before the Izmit earthquakeEarthquake ruptures before the Izmit earthquake

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Coulomb stress field prior to the Izmit earthquakeCoulomb stress field prior to the Izmit earthquake

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Resolved stresses on the Izmit and Dzce faults before August 99Resolved stresses on the Izmit and Dzce faults before August 99

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Coulomb stress with varying coefficient of friction valuesCoulomb stress with varying coefficient of friction values

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Coulomb stress changes due to the Izmit earthquakeCoulomb stress changes due to the Izmit earthquake

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Stress changes on the Dzce fault due to the previous earthquakeStress changes on the Dzce fault due to the previous earthquakess

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Present day Coulomb stress changes due to the past earthquakesPresent day Coulomb stress changes due to the past earthquakes

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