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εij =εxx εxy

εyx εyy

⎡

⎣ ⎢

⎤

⎦ ⎥=

−0.1 0

0 0.2

⎡

⎣ ⎢

⎤

⎦ ⎥

(10,10)

(9,12)

(10,0)(9,0)

y

X

(0,10)

(0,11)

This is the trace of the strain tensor. In general the trace of the strain tensor gives area change in 2-D and volume change in 3-D

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εxx +ε yy = −0.1+ 0.2 = 0.1

The principal axes are directions along which the starting vector and ending vector are parallel

Pure shear = principal axes do not rotate with time

Principal Axes

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ε1 = maximum stretch direction

Intermediate stretch direction

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ε2

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ε3 Minimum stretch direction (or most contractional)

The principal axes are all mutually orthogonal to one another

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εij =εxx εxy

εyx εyy

⎡

⎣ ⎢

⎤

⎦ ⎥=

0.1 0.1

0.1 0.1

⎡

⎣ ⎢

⎤

⎦ ⎥

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Δx = 0.1x + 0.1y

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Δy = 0.1x + 0.1y

(10 ,0)becomes (11,1)

(10,-10) remains fixed, as does (-10, 10)

(0, 10) becomes (1,11)

(10,10) becomes (12,12) etc...

(12,12)

(10,0)

(11,1)

y

X

(10,10)

(0,10)

(1,11)

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εij =εxx εxy

εyx εyy

⎡

⎣ ⎢

⎤

⎦ ⎥=

0.2 0.0

0.0 0.0

⎡

⎣ ⎢

⎤

⎦ ⎥

In principal axis coordinate system this tensor can be written:

Simple Shear

• In Simple shear the principal axes rotate with increasing shear

• Simple shear applies only to finite strain

MarkerThis part of marker

not disformed

Rotational strain

Stress = Force/Area

Force is measured in units of mass*acceleration

1 N (Newton) = 1 kg * m * s-2

another common unit for force is the pound

Pressure is a number. It corresponds to a special kind of stress.

Stress is a tensor, but it has the same units as pressure (Pa)

1000 Pa = 1 kPa1,000,000 Pa = 1 MPa (about 10 bars)

Traction is a Vector

• Tractions are vectors = force/area

• Traction can be resolved into two components

Normal component to plane = normal stress

Tangential component = shear stress

The stress tensor

• The stress tensor is symmetric

• The stress tensor has 3 principal axes

• The principal axes are mutually orthogonal

• principal axis = direction in which the traction vector is parallel to normal to plane => no shear stress resolved on that plane

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σ 2

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σ1

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σ 3

= maximum compressive principal stress

= intermediate compressive principal stress

= minimum compressive principal stress

Normal Stress and Shear Stress

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σn = Normal Stress resolved on plane

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τ = shear stress resolved on plane

Anderson Faulting Theory

• If σ1 is vertical then a new fault will be a normal fault (extensional)

• If σ1 is horizontal and σ3 is vertical then reverse (thrust) fault (contractional faulting)

• If σ1 and σ3 are both horizontal then strike-slip (transcurrent) fault

Fault Angles and Principal Stresses

σ2 in the plane of the fault

σ1 20°-40° from the plane of the fault

σ3 50°-70° from the plane of the fault

σ1

σ1

σ3

σ3

θ

θ

σn = (σ1+σ3)/2 - [(σ1-σ3)/2] cos 2

τ = [(σ1-σ3)/2] sin 2

THESE ARE ALSO THE EQUATIONS FOR A CIRCLE WITH A RADIUS OF (σ1-σ3)/2AND A CENTER (σ1+σ3)/2 TO THE RIGHT OF WHERE THE AXES CROSS!!!!

τ

σn

φ = atan μfrictional yield envelope

frictional yield envelope

τ

σnσ

1

σ3

σ1

σ3

2θ

Let’s Look at internal friction angles, coefficients of friction,

and theta

• If =10° (so =tan=0.18), then 2=80°, so =40° and σ1 axis is 40° from the fault plane.

• If =20° (so =tan=0.36), then 2=70°, so =35° and σ1 axis is 35° from the fault plane.

• If =30° (so =tan=0.58), then 2=60°, so =30° and σ1 axis is 30° from the fault plane.

• If =40° (so =tan=0.84), then 2=50°, so =25° and σ1 axis is 25° from the fault plane.

Cohesion

• Cohesion = shear strength that remains even when normal tractions are zero

• Byerlee’s law with cohesion

• The cohesion represents the intercept value

Pre-existing faults

• If there are pre-existing faults, then figure in previous slide predicts a range of orientations of faults, with respect to maximum principal stress direction that can slip

• If there are no pre-existing faults, then only one orientation is possible

Role of Fluid Pressure or Pore Pressure

• Hydrostatic Pressure: Phydrostatic = water g z

• Lithostatic pressure is when entire weight of the overlying rock (density rock) is being supported

• Plithostatic = rock g z

Fluid Pressures and Tractions

• Fluid Pressures can support normal tractions but not shear tractions!

• Elevated fluid pressures make the Mohr circle move to the left

Effective Stress

• Effective Stress = total stress minus the fluid Pressure

σ1' = σ1 - Pf

σ2' = σ2 - Pf

σ3' = σ3 - Pf

• Shear Tractions are not affected!

σ

τ

σnσ

1

σ3

1

σ3

Pf

τ

σn

tensile crack or joint (with

a single orientation normal

to minimum stress axis)

conjugate tensile

fractures (joints)

conjugate faulting

Joints

• The