Rock Deformation and Strength

• Structural geology involves studying geologic structures produced by deformation.
• Structural geologists describe structures geometrically and reconstruct the sequence of deformational events
• Apply principles of physics, engineering, math, and materials science and use concepts of stress and strain.
• They solve problems using geometry, algebra, and trigonometry

Fault Activity

• To determine how long a fault has been active, determine total slip/slip rate.
• A fault with a slip rate of 2.0 mm/yr and a total slip of 1.0 km has been active for 500,000 years.
• Paleo seismic studies show faults slip episodically, producing an average of 3.3 m displacement in a M 7.2 earthquake.
• Recurrence interval = slip per earthquake/slip rate
• Earthquakes occur every 1650 years given a 2.0 mm/yr sliprate and 3.3m displacement per event.
• Faults form due to tectonic forces, accumulating stress until a critical threshold is reached.
• The recurrence pattern suggests a stick-slip behavior rather than continuous creep.

Deformation

• Deformation is changes in shape, size, displacement, and rotation in rocks moving from an undeformed state to a deformed state when sufficient stress is applied.
• Homogenous deformation involves distortion (strain), rigid body translation, and rigid body rotation
• deformation=distortion(strain)±rotation±translation
• Deformation occurs due to stresses acting on rocks
• Geologic structures include fractures, faults, folds, and fabrics (foliations, lineation)

Brittle vs Ductile Deformation

• Brittle deformation includes fracturing and faulting which implies a loss of cohesion and abrupt displacements
• Related forces are present during brittle deformation and invokes elastic deformation, yield stress, strength, and brittle failure concepts from experimental fracture mechanics.
• Ductile deformation implies no loss of cohesion and occurs gradually at higher temperatures (metamorphic conditions resulting in tectonite fabrics (foliations/lineations) with shape changes due to strain.

Factors Affecting Deformation

• Mode of deformation and rock strength depend on temperature, rock composition, grain size, confining pressure, differential pressure/stress, deformation rate, and pore fluid pressure.
• Rock strength is the maximum differential stress a rock can support before failure.

Analyzing Geological Structures

• Geometric description is a statement of what is objectively seen with minimal interpretations.
• Kinematic description and analysis interpret how a structure developed through time.
• Mechanical or dynamical modes use models.

Plate Margins

• Three principal plate margin types: convergent (subduction and collision zones), divergent (mid-ocean ridges, continental rifts), and transform (strike-slip boundaries)

Lithosphere

• The lithosphere is comprised of the crust and uppermost mantle, and thickness ranges from 0-150 km
• Oceanic lithosphere includes oceanic crust of 6km + mantle lithosphere (0-100 km).
• Continental lithosphere thickness ranges from 15-80 km

Scalars/Vectors/Tensors

Scalars: only have magnitude (T, mass, length, density).

• Vectors have both magnitude and direction (velocity, displacement, acceleration, force)
• Tensors: infinite array of vectors (stress, permeability, magnetic susceptibility, conductivity, speed of light in a crystal).

Forces

• Forces within rocks include internal interatomic forces in the crystal lattice which determine material properties (strength, rigidity).
• External forces include body forces acting on each particle of mass in a volume (gravity) and surface forces where one body acts on another (plate collision).
• Traction is force applied to an area (Σ = F/A)

Stress Units

• The unit of stress is pascals (1 Pa= 1N/m^2)
• 1 MPa = 10^6 Pa = 10 bars
• 100 MPa= 1 kbar.

Describing Stress

• Force is vector (a push or pull).
• Traction (force intensity acting on a specific surface).
• Surface stress (balanced equal and opposite tractions)
• State of stress (surface stresses at all possible orientations).

2-D Stress

• Stress ellipse defined by principal stresses.
• Rocks fail when applied stress exceeds their strength, when the maximum differential stress is σ1-σ3

Stress Terms

• ε = a traction
• σn = normal stress
• σε (sometimes (t)) = shear stress
• σ1 = the maximum principal stress
• σ2 = the intermediate principal stress
• σ3 = the least principal stress
• σxx, σyy, σzz = the normal stresses acting on 3 perpendicular planes in an x-y-z coordinate system
• σxz, σxy, σyz,..etc = the shear stresses acting on 3 perpendicular planes in an x-y-z coordinate system, the first subscript defines the plane, the second indicates the direction of shear
• δ = differential stress (σ1 - σ3)
• Mean normal stress = (σ1 + σ2 + σ3)/3
• θ = the angle between the normal (pole) to a plane and the σ1 axis
• α = the angle between the plane itself and the σ1 axis

Fundamental Stress Equations (2-D)

• In terms of the principal stresses:
• ση = (σ1 + σ3)/2 + ((σ1- σ3)/2) * Cos (2θ)
• σε = ((σ1- σ3)/2) * Sin (2θ)
• ((σ1 + σ3)/2) = center of the Mohr circle
• ((σ1 - σ3)/2) = radius of the Mohr circle
• Lithostatic stress is stress produced by the overlying rock column

Mohr Diagram

• When we know the state of stress, the Mohr circle can calculate sigma n and sigma s for any plane passing through that point, just as long as we know the orientation of that plane relative to the principle stresses.
• States of Stress Terminology:
  • Hydrostatic pressure: all principal stresses are equal
  • Uniaxial compression
  • Uniaxial tension
• axial compression
• axial tension
• triaxial stress
• pure shear stress
• deviatoric stress
• differential stress
• effective stress
• Vertical stress ¿pgh

Brittle Deformation Modes

• Brittle deformation can be classified by joints or faults
• Mode I- extension fractures: relative motion perpendicular
• Mode II- shear fractures: relative motion perpendicular

Fault Terminology

• Fault plane, hanging wall, footwall
• Displacement
• High angle, low angle, listric

Fracture Criterion

• Fracture criterion: a mathematical description of fracture conditions depending on critical stresses required to produce failure and orientation of the fracture plane, which represents an idealized fracture model.

Fracture Mechanics

• Fracture mechanics is based on laboratory experiments performed on rocks at various P,T, & strain rates
  • Dependent on confining pressure and fluid pressure.
  • It is also affected by preexisting weaknesses in rocks and is controlled by magnitude of sigma 1 and 3 and records; sigma 1 and 3 and theta

Tensile Failure Criteria

• Tensile strenght of a rock: -10 to -40 Mpa
• Represented as a vertical line on Mohr space.
• Mohr circle that touches this line is a critical Mohr circle (depends on sigma 3)
• Fracture angle where circle is tangent to failure line
• • α₁ = 0°, θ₁ = 90° - always perpendicular to 03 and parallel to 01

Shear Failure Criteria

• Linear approximation to tangent line for a set of critical Mohr circles
• Diameter increases linearly with mean normal stress
• Slope of line is related to predicted angle of failure
• The Mohr coulomb fracture criterion: predicts that fault normals will always be at about 60 to the maximum compressive stress sigma1 and a fault surface will be about 30 to sigma 1. Conjugate shear fractures: two fault surfaces about 30 on either side of sigma 1

Shear failure is based on function of mean normal stress and rock cohesion (20-80 Mpa). −C= cohesion (intercept) (20-100 Mpa), µ= coefficient of internal friction (0.58 +/- 0.12) φ= angle of internal friction (30 +/- 5). all Mohr circle within the space bound by these lines are stable critical stress state, one of two possible shear fractures will develop αf ~ 30° (45- φ/2), θf ~ 60°(45+ φ /2)

Fracture Orientation

• sigma s/ sigma n ratio is maximized at 30, or theta f at 60
• Mode I: joints fissures, veins, dikes
• Mode II: faults, deformation bands, shear zones
• Mode IV: stylolite, anti cracks, closing fractures

Andersonian Mechanics

• One of the principal stresses is always vertical (the other two must be horizontal), and rocks fail by either tensile failure or coulomb failure criterion.
• Predicts the general orientations of all faults and extension fractures and the problems: principal stresses do not need to be vertical, does not account for non coulomb behavior of some faults, and does not consider pre existing weaknesses.

Griffeth Crack Theory

• Rocks are a lot weaker than theoretical predictions, are riddled with microscopic elliptical cracks that act to strongly concentrate stresses
• Provides a general theory for both tensile and shear failure
• Stresses at tips of cracks-more elliptical=higher stress concentration

Shear Fractures

• Linking up of tiny extension fractures
• Minor volume increase
• Wing cracks at terminations and edges

Pore Fluids and Effective Stress

• Pf has enormous weakening effect on rock strength
• Effective stress= applied normal stress – Pf (equivalent to shifting Mohr circle to the left by and equal amount to Pf)
• Shear stresses not affected
• A = Hubbert Rubey Pore Fluid Pressure Ratio = Pf /σn, λ can vary from ~0.35 to 1.0

Lithosphere/Crust

• Lithosphere includes the crust and the upper most mantle. It lies over the asthenosphere, is 0-150 km thick, mechanical, thermal, rigid
• Earth's crust- compositional layers. Include the mantle, core and crust
• oceanic lithosphere is oceanic crust (6km) + mantle lithosphere (0-100 km). it is mafic
• The continental crust is (15-80 km). it is more variable and compositionally diverse.

Stress

• Stress causes deformations in the earth.
• Units are N= 1kg/(m sec^2), Mpa, bar/kbar
• Tensor: an infinite array of vectors
• Scalars have magnitude only (T, mass, length, density) Vectors have magnitude and direction. (velocity, displacement, acceleration, force)

2-D stress

Portray the 2-D state of a stress at a point:

• stress ellipse
• two principal stresses
• normal and shear stresses on any 2 perpendicular planes
• tensor notation (4 components)
• Mohr circle is a complete representation of the state of stress at a point.
• Axes are σε ση
• Angles in a physical space are doubled in Mohr space.

Possible States of Stress

Possible states of stress in the earth:

• hydrostatic pressure
• unixaial stress
• uniaxial tension
• axial compression
• axial extension
• triaxial stress
• pure shear stress
• deviatoric stress
• differential stress
• effective stress
• Ways to determine the orientations and magnitudes of stresses in the earth:
• borehole breakout
• overcoring
• hydraulic fracturing
• volcanic vent and fracture alignments
• Lithostatic stress is the stress produced by the overlying rock column.

This lesson covers factors influencing rock deformation, stress types, and material properties. It also defines traction and the Mohr-Coulomb fracture criterion. Key concepts include ductile deformation, lithostatic rock stress, and geologic boundary