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 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 = maximum stretch direction Intermediate stretch direction Minimum stretch direction (or most contractional) The principal axes are all mutually orthogonal to one another

(10,0)becomes (11,1) (10,-10) remains fixed, as does (-10, 10) (0, 10) becomes (1,11) (10,10) becomes (12,12) etc...

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

Marker This part of marker not disformed Rotational strain

Stress = Force/Area Force is measured in units of mass*acceleration 1 N (Newton) = 1kg * 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 kPa 1,000,000 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

= maximum compressive principal stress = intermediate compressive principal stress = minimum compressive principal stress

Normal Stress and Shear Stress = Normal Stress resolved on plane = shear stress resolved on plane

Anderson Faulting Theory If  1 is vertical then a new fault will be a normal fault (extensional) If  1 is horizontal and  3 is vertical then reverse (thrust) fault (contractional faulting) If  1 and  3 are both horizontal then strike- slip (transcurrent) fault

Fault Angles and Principal Stresses  2 in the plane of the fault  1 20°-40° from the plane of the fault  3 50°-70° from the plane of the fault

 n = (  1 +  3 )/2 - [(  1 -  3 )/2] cos 2   = [(  1 -  3 )/2] sin 2  THESE ARE ALSO THE EQUATIONS FOR A CIRCLE WITH A RADIUS OF (  1 -  3 )/2 AND A CENTER (  1 +  3 )/2 TO THE RIGHT OF WHERE THE AXES CROSS!!!!

Let’s Look at internal friction angles, coefficients of friction, and theta If  =10° (so  =tan  =0.18), then 2  =80°, so  =40° and  1 axis is 40° from the fault plane. If  =20° (so  =tan  =0.36), then 2  =70°, so  =35° and  1 axis is 35° from the fault plane.

If  =30° (so  =tan  =0.58), then 2  =60°, so  =30° and  1 axis is 30° from the fault plane. If  =40° (so  =tan  =0.84), then 2  =50°, so  =25° and  1 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: P hydrostatic =  water g z Lithostatic pressure is when entire weight of the overlying rock (density  rock ) is being supported P lithostatic =  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  1 ' =  1 - P f  2 ' =  2 - P f  3 ' =  3 - P f Shear Tractions are not affected!

Joints The