4  Module III: Theoretical Underpinnings

4.1 Key Concept Summary

Understanding stress, strain, and rheology is fundamental to interpreting geological structures. Stress (force per unit area) causes strain (deformation), and the relationship between them is controlled by rheology (material properties). The state of stress determines what type of structures form, while material properties control whether deformation is brittle or ductile.

4.2 Self-Test Questions

4.2.1 Stress Fundamentals

Can you explain what stress is and how it differs from force?

Think about:

  • Definition: force per unit area
  • Units: Pascals (Pa)
  • Why is stress a tensor quantity?
  • What is a traction vector?

Can you describe the stress tensor and its properties?

Consider:

  • Why does it have 9 components but only 6 are independent?
  • What does symmetry mean (\(\sigma_{ij} = \sigma_{ji}\))?
  • What are normal stresses vs. shear stresses?
  • How do you visualize stress on a cubic element?

Can you explain principal stresses and their significance?

Think about:

  • Definition: orientations where shear stresses vanish
  • The three principal stresses: \(\sigma_1 \geq \sigma_2 \geq \sigma_3\)
  • Why is one principal stress perpendicular to Earth’s surface?
  • How do principal stress orientations control structure type?

Can you describe the three tectonic stress regimes?

Consider:

  • Normal faulting: \(\sigma_1\) vertical, \(\sigma_3\) horizontal (extension)
  • Strike-slip: \(\sigma_2\) vertical (wrench)
  • Thrust faulting: \(\sigma_3\) vertical (compression)
  • How do you identify the regime from focal mechanisms?

Can you explain and use the Mohr circle for stress?

Think about:

  • What does the circle represent?
  • How do you plot stress on any plane?
  • What is the failure envelope?
  • How does it relate to the Mohr-Coulomb criterion?

Can you describe Anderson’s theory of faulting?

Consider:

  • How does fault orientation relate to principal stresses?
  • Why do faults form at ~30° to \(\sigma_1\)?
  • Role of friction coefficient
  • Conjugate fault sets

Can you explain how stress is measured in the Earth?

Think about:

  • Hydraulic fracturing
  • Borehole breakouts
  • Earthquake focal mechanisms
  • Over-coring

4.2.2 Strain and Strain Rate

Can you distinguish between the four forms of deformation?

Consider:

  • Translation (change in position)
  • Rotation (change in orientation, no deformation)
  • Distortion (change in shape)
  • Dilation (change in volume)
  • Which involve actual strain?

Can you explain the strain tensor and its components?

Think about:

  • How is strain defined mathematically?
  • Normal strains vs. shear strains
  • What is volumetric strain (dilatation)?
  • What does incompressibility mean (\(\delta = 0\))?

Can you describe principal strains and the strain ellipsoid?

Consider:

  • Three principal strains: \(\varepsilon_1 \geq \varepsilon_2 \geq \varepsilon_3\)
  • Relationship to principal stress directions
  • Strain ellipsoid visualization
  • Shortening vs. extension axes

Can you broadly explain strain invariants and their significance?

Think about:

  • First invariant (volumetric strain)
  • Second invariant (magnitude of deviatoric strain)
  • Why are they “invariant”?
  • Their use in global strain rate maps

Do you know that people use the Flinn diagram to classify strain types?

(We didn’t spend time on this, so a broad overview is OK)

  • Axes: strain ratios
  • K-value: \((\varepsilon_1-\varepsilon_2)/(\varepsilon_2-\varepsilon_3)\)
  • Uniaxial extension (constriction): \(K = \infty\)
  • Plane strain: \(K = 1\)
  • Uniaxial shortening (flattening): \(K = 0\)
  • Intermediate values

Can you distinguish between strain and strain rate?

Think about:

  • Strain: cumulative deformation
  • Strain rate: rate of deformation (units: \(s^{-1}\))
  • Relationship: \(D_{ij} = \frac{1}{2}[\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}]\)
  • Why is strain rate important for understanding active tectonics?

Can you describe methods for measuring strain in rocks?

Consider:

  • Deformed fossils
  • Stretched pebbles
  • Reduction spots
  • Deformed oolites
  • Boudin structures
  • What assumptions are required?

4.2.3 Rheology

Can you explain what rheology is and why it matters?

Think about:

  • Definition: study of how materials deform under stress
  • Controls whether deformation is brittle or ductile
  • Determines the stress-strain or stress-strain rate relationship

Can you describe elastic behavior and the key parameters?

Consider:

  • Hooke’s Law: \(\sigma = E \cdot \varepsilon\) (1D)
  • Young’s modulus (\(E\)): stiffness
  • Poisson’s ratio (\(\nu\)): lateral vs. axial strain
  • Shear modulus (\(\mu\) or \(G\))
  • Bulk modulus (\(K\))
  • Deformation is reversible

Can you describe viscous behavior and contrast it with elastic?

Think about:

  • Stress proportional to strain rate, not strain
  • Newtonian: \(\tau = \eta \cdot \dot{\gamma}\)
  • Viscosity (\(\eta\)): resistance to flow (Pa·s)
  • Deformation is irreversible
  • Time-dependent behavior

Can you explain non-Newtonian viscosity and power-law creep?

Consider:

  • Power-law: \(\eta = K(II_D)^{n-1}\)
  • \(n > 1\): shear thinning (most rocks)
  • \(n < 1\): shear thickening
  • Effective viscosity depends on strain rate

Can you describe plastic behavior?

Think about:

  • Yield stress (\(\tau_Y\)): threshold for deformation
  • Stress limited after yield
  • Frictional: \(\tau_Y \approx \mu P\) (granular materials)
  • Effective viscosity: \(\eta_{\text{eff}} = \tau_Y/II_D\)

Can you identify and explain the major deformation mechanisms in rocks?

Brittle mechanisms:

  • Fracturing
  • Cataclastic flow
  • Dominant in upper crust

Ductile (crystal plastic) mechanisms:

  • Diffusion creep (Coble, Nabarro-Herring)
  • Dislocation creep (glide and climb)
  • Grain boundary sliding
  • Dynamic recrystallization
  • Dominant in lower crust and mantle

Can you explain how physical factors affect rock strength and deformation style?

Consider:

  • Temperature: increases ductility, decreases strength
  • Confining pressure: increases strength, promotes ductility
  • Pore fluid pressure: decreases effective stress
  • Strain rate: higher rates favor brittle behavior
  • Grain size: smaller grains weaken rock
  • Water content: weakens minerals

Can you describe the brittle-ductile transition?

Think about:

  • Depth-dependent transition
  • Upper crust: brittle, strength increases with depth (pressure effect)
  • Lower crust: ductile, strength decreases with depth (temperature effect)
  • Byerlee’s Law describes the transition
  • Why is this important for earthquake distribution?

Can you interpret a rock deformation map?

Consider:

  • Axes: temperature vs. stress
  • Fields: elastic, brittle fracture, various crystal plastic mechanisms
  • How does the dominant mechanism change with conditions?