this is the trace of the strain tensor. in general the trace of the strain tensor gives area change...
TRANSCRIPT
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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
