Fault Mechanics Laboratory Pore pressure scale Lowers normal stress, moves stress circle to left Doesn Doesn t change shear Deviatoric stress not affected This example: failure will be by tensile cracks Mohr Mohr s circles in mohr detail Failure at large scale Anderson theory Earthquakes without the shaking Brittle deformation and crack theory Dislocations Next week - rheology Normal and shear stresses Vertical σ1 Horizontal Ordinary space σ3 Arbitrary angle is θ 2D picture, but σ2 = σ3 Laboratory scale Much of this is done by rock mechanics specialists Engineering geology needs their input We use bits of it Low and high pressure apparatus Back to Mohr diagram for stress A whole different space - normal versus shear stresses This should be very familiar Now, mapping back onto σ1 - σ3 at some angle to σ1 identify that plane by its pole (perpendicular) Maximum shear stress across that plane is at 45º Normal stress still more than half-way at 45º Plane We space, we get Mohr stress Three dimensional Mohr Ordinary 2 points same shear, different normal points equal normal, opposite shear Complementary angles 2 For triaxial state of stress, 3 nested circles This makes the angles of maximum normal/shearme ssy Best done with tensors Mohr Coulomb failure envelope Shear stress!!c =!0 + " tan$!0 $ 2# "2 P "1 Normal stress " Optimum combination Coefficient of of normal/shear internal friction friction 1
Coulomb failure envelope Angle of failure Tangent to failure envelope Angle relative to pole (perpendicular) to σ1 Shear stress!!c =!0 + " tan$!0 $ 2# "2 90! " = 180! 2# critical 90 + " # critical = 2 "1 P Normal stress " For! = 30, a typical value, " critical = 60 Mohr-Coulomb failure envelope The envelope not actually linear Failure mechanism changes as function of pressure Still, more or less what you should by used to A series of mechanisms pressure relative to rock strength determines which Confining Tensile fracture by σ1 plane parallel to σ1 Dominated Fracture Transitional It It s What do we mean by failure? Failure What do we mean by failure? cracking up Cracks What do we mean by failure? II Coulomb shear failure Note the angle on the Coulomb cylinder: be sure you understand why it behaves that way Brittle-ductile transition Almost always gradual Plastic yielding Von Mises criterion What do we mean by failure? Macroscopic fault mechanics: At the outcrop to regional scale Structural geologists Seismologists Seismologists The world is not simple simple Anderson theory Shear zones Actual geology more complicated than laboratory, even though that is bad enough Vertical stress is given by ρgh in each case More We or less lithostatic drop the lithostatic, only deal with deviatoric stress What kind of fault you get depends on the relative values of the three deviatoric stresses Thrust, normal, strike slip 2
Map and cross section views Faults are seldom straight And the structures differ with depth Anderson theory: simplified version Anderson theory In effect, style of fault slip is controlled by which stress direction is least confined Fault angle not specified by Anderson theory Room problems, failure law, fluids, heterogeneities can all enter Faults shouldn t t cross? What does Anderson predict? Assume rocks are riddled with faults Slip at angle with minimum tectonic stress Not the same as Mohr-Coulomb Coefficient of friction controls fault angle Friction on faults Only a problem for brittle fracture You can also crush intersections Coefficient of friction! yx = f s! yy Friction on faults Does Anderson theory work? If you conflate Mohr-Coulomb theory at tan φ 30º º with Anderson theory, you predict normal faults dipping at 60º, thrust faults at 30º, and strike-slip faults at 30º to the regional stress field. But Large thrust faults dip as little as 2º San Andreas perpendicular to maximum horizontal stress direction Worst of all, large normal faults dip as little as 6º Brittle deformation and cracks Dig now into microscopic scale Engineering - don t t let the wings fall off planes Modes of cracks Tensile 2 2 shear modes Stress intensity factors Wing cracks 3
Tensile and shear cracks Wing cracks Stress concentration factors Mostly look at brittle deformation Tensile - pull apart (Mode I) Shear II - sliding parallel to crack (Mode II) Shear III - tearing perpendicular to crack (Mode III) Mode II and III cracks are unstable Tensile stress concentration bends them out of plane You can hear this happening Down to even more microscopic level: Dislocations Explained (maybe) Point and line crystal flaws Single defects can move Aids diffusion And deformation To some extent Also can nucleate line defects, etc. Single atoms out of place Extras, deficits Can move through lattice Point defects Line defects (dislocations) A whole series of defects arranged in a more or less linear array Two types Edge dislocations Screw dislocations Can get serious deformation when these move through lattice Edge dislocations Lattice plane stops abruptly along line Line vector is tip of dislocation Burgers vector: points in direction of offset of lattice plane Slip at right angles to line vector, parallel to Burgers vector Edge dislocations move Have to break, reform bonds at dislocation tip as planes rearrange Always an odd plane out 4
Screw locations on the move Incremental slippage as dislocation moves Creates ramp in lattice planes Dislocation glide Both can coexist Edge on front face, screw on left Dislocations in quartz The world seen in a corn cob The real world - not a pretty picture Rheology - a Continuum Mechanics Viewpoint A mechanistic view Combine springs and dashpots Represent elastic and viscous aspects of rock Recoverable and permanent deformation Time varying behavior Temperature dependent Can make quite complicated mixes More of this next week 5