3. 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

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

6. (10,0)becomes (11,1) (10,-10) remains fixed, as does (-10, 10) (0, 10) becomes (1,11) (10,10) becomes (12,12) etc...

7. In principal axis coordinate system this tensor can be written:

8. Simple Shear In Simple shear the principal axes rotate with increasing shear Simple shear applies only to finite strain

9. Marker This part of marker not disformed Rotational strain

21. 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

24. 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)

25. 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

27. 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

28. = maximum compressive principal stress = intermediate compressive principal stress = minimum compressive principal stress

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

32. 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

33. 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

34.  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!!!!

37. 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.

38. 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.

39. Cohesion Cohesion = shear strength that remains even when normal tractions are zero Byerlee’s law with cohesion The cohesion represents the intercept value

41. 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

45. 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

46. 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

47. 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!

55. Joints The