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Tectonics I. Tectonics I Vocabulary of stress and strain Elastic, ductile and viscous deformation Mohr ’ s circle and yield stresses Failure, friction and faults Brittle to ductile transition Anderson theory and f ault types around the solar system Tectonics II

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Tectonics I

    • Vocabulary of stress and strain
    • Elastic, ductile and viscous deformation
    • Mohr’s circle and yield stresses
    • Failure, friction and faults
    • Brittle to ductile transition
    • Anderson theory and fault types around the solar system
  • Tectonics II
    • Generating tectonic stresses on planets
    • Slope failure and landslides
    • Viscoelastic behavior and the Maxwell time
    • Non-brittle deformation, folds and boudinage etc…

Compositional vs. mechanical terms

    • Crust, mantle, core are compositionally different
      • Earth has two types of crust
    • Lithosphere, Asthenosphere, Mesosphere, Outer Core and Inner Core are mechanically different
      • Earth’s lithosphere is divided into plates…

How is the lithosphere defined?

    • Behaves elastically over geologic time
    • Warm rocks flow viscously
      • Most of the mantle flows over geologic time
    • Cold rocks behave elastically
      • Crust and upper mantle
  • Rocks start to flow at half their melting temperature
    • Thermal conductivity of rock is ~3.3 W/m/K
    • At what depth is T=Tm/2

Melosh, 2011

Relative movement of blocks of crustal material

Mars –

Extension and compression

Earth –

Pretty much everything

Moon & Mercury –

Wrinkle Ridges

Europa – Extension and strike-slip

Enceladus - Extension


The same thing that supports topography allows tectonics to occur

    • Materials have strength
    • Consider a cylindrical mountain, width w and height h
    • How long would strength-less topography last?


Weight of the mountain


F=ma for material in the hemisphere


Conserve volume

Solution for h:

i.e. mountains 10km across would collapse in ~13s


Response of materials to stress (σ) – elastic deformation





Linear (normal) strain (ε) = ΔL/L

Shear Strain (ε) = ΔL/L

G is shear modulus (rigidity)

E is Young’s modulus

Volumetric strain = ΔV/V

K is the bulk modulus


Stress is a 2nd order tensor

    • Combining this quantity with a vector describing the orientation of a plane gives the traction (a vector) acting on that plane

i describes the orientation of a plane of interest

j describes the component of the traction on that plane

These components are arranged in a 3x3 matrix

Are normal stresses, causing normal strain

(Pressure is )

Are shear stresses, causing shear strain

We’re only interested in deformation, not rigid body rotation so:


The components of the tensor depend on the coordinate system used…

  • There is at least one special coordinate system where the components of the stress tensor are only non-zero on the diagonal i.e. there are NO shear stresses on planes perpendicular to these coordinate axes

Shear stresses in one coordinate system can appear as normal stresses in another


These are principle stresses that act parallel to the principle axes

The tractions on these planes have only one component – the normal component

Pressure again:



Principle stresses produce strains in those directions

    • Principle strains – all longitudinal
  • Stretching a material in one direction usually means it wants to contract in orthogonal directions
    • Quantified with Poisson’s ratio
    • This property of real materials means shear stain is always present
  • Extensional strain of σ1/E in one direction implies orthogonal compression of –ν σ1/E
    • Where ν is Poisson’s ratio
    • Range 0.0-0.5

Linear strain (ε) = ΔL/L

E is Young’s modulus



Where λ is the Lamé parameter

G is the shear modulus



Groups of two of the previous parameters describe the elastic response of a homogenous isotropic solid

    • Conversions between parameters are straightforward

Materials fail under too much stress

    • Elastic response up to the yield stress
    • Brittle or ductile failure after that
  • Material usually fails because of shear stresses first
    • Wait! I thought there were no shear stresses when using principle axis…
    • How big is the shear stress?

Strain hardening

Special case of plastic flow

Strain Softening

Ductile (distributed) failure

Brittle failure


How much shear stress is there?

    • Depends on orientation relative to the principle stresses
    • In two dimensions…
    • Normal and shear stresses form

a Mohr circle

Maximum shear stress:

On a plane orientated at 45° to the principle axis

Depends on difference in max/min principle stresses

Unaffected (mostly) by the intermediate principle stress


(Tresca criterion)

  • Consider differential stress
    • Failure when:
    • Failure when:
  • Increase confining pressure
    • Increases yield stress
    • Promotes ductile failure

(Von Mises criterion)

  • Increase temperature
    • Decrease yield stress
    • Promotes ductile failure

Low confining pressure

    • Weaker rock with brittle faulting
  • High confining pressure (+ high temperatures)
    • Stronger rock with ductile deformation

What sets this yield strength?

  • Mineral crystals are strong, but rocks are packed with microfractures
  • Crack are long and thin
    • Approximated as ellipses
    • a >> b
    • Effective stress concentrators
    • Larger cracks are easier to grow






Failure envelopes

    • When shear stress exceeds a critical value then failure occurs
    • Critical shear stress increases with increasing pressure
    • Rocks have finite strength even with no confining pressure
    • Coulomb failure envelope
      • Yo is rock cohesion (20-50 MPa)
      • fF is the coefficient of internal friction (~0.6)

What about fractured rock?

Cohesion = 0

Tensile strength =0

Byerlee’s Law:

Melosh, 2011


Why do faults stick and slip?

  • Basically because the coefficients of static and dynamic friction are different
  • Stick-slip faults store energy to release as Earthquakes
    • Shear-strain increases with time as:
    • Stress on the fault is:
      • G is the shear modulus
      • σfd (dynamic friction) left over from previous break
      • Fault can handle stresses up to σfs before it breaks (Static friction)
    • Breaks after time:
    • Fault locks when stress falls to σfd (dynamic friction)
    • If σfd < σfs then you get stick-slip behavior

Brittle to ductile transition

    • Confining pressure increases with Depth (rocks get stronger)
    • Temperature increases with depth and promotes rock flow
    • Upper 100m – Griffith cracks
    • P~0.1-1 Kbars, z < 8-15km, shear fractures
    • P~10 kbar, z < 30-40km distributed deformation (ductile)
    • This transition sets the depth of faults

Melosh, 2011



Back to Mohr circles…

  • Coulomb failure criterion is a straight line
    • Intercept is cohesive strength
    • Slope = angle of internal friction
    • Tan(slope) = fs
  • In geologic settings
    • Coefficient of internal friction ~0.6
    • Angle of internal friction ~30°
  • Angle of intersection gives fault orientation
  • So θ is ~60°
  • θ is the angle between the fault plane and the minimum principle stress,

Anderson theory of faulting

    • All faults explained with shear stresses
    • No shear stresses on a free surface means that one principle stress axis is perpendicular to it.
    • Three principle stresses
      • σ1 > σ2 > σ3
      • σ1 bisects the acute angle (2 x 30°)
      • σ2 parallel to both shear plains
      • σ3 bisects the obtuse angle (2 x 60°)
    • So there are only three possibilities
    • One of these principle stresses is the one that is perpendicular to the free surface.
    • Note all the forces here are compressive…. Only their strengths differ


Before we talk about faults….
  • Fault geometry
    • Dip measures the steepness of the fault plane
    • Strike measures its orientation

Extensional Tectonics

  • Crust gets pulled apart
  • Final landscape occupies more area than initial
  • Can occur in settings of
    • Uplift (e.g. volcanic dome)
    • Edge of subsidence basins (e.g. collapsing ice sheet)

Steeply dipping

Shallowly dipping

Horst and Graben
    • Graben are down-dropped blocks of crust
    • Parallel sides
    • Fault planes typically dip at 60 degrees
    • Horst are the parallel blocks remaining between grabens
    • Width of graben gives depth of fracturing
    • On Mars fault planes intersect at depths of 0.5-5km
In reality graben fields are complex…
    • Different episodes can produce different orientations
    • Old graben can be reactivated

Lakshmi -Venus

Ceraunius Fossae - Mars


Compressional Tectonics

  • Crust gets pushed together
  • Final landscape occupies less area than initial
  • Can occur in settings of
    • Center of subsidence basins (e.g. lunar maria)
  • Overthrust – dip < 20 & large displacements
  • Blindthrust – fault has not yet broken the surface

Steeply dipping

Shallowly dipping


Right-lateral (Dextral)

Left-lateral (Sinistral)

Shear Tectonics

  • Strike Slip faults
    • Shear forces cause build up of strain
    • Displacement resisted by friction
    • Fault eventually breaks
  • Vertical Strike-slip faults = wrench faults
  • Oblique normal and thrust faults have a strike-slip component



Tectonics I

    • Vocabulary of stress and strain
    • Elastic, ductile and viscous deformation
    • Mohr’s circle and yield stresses
    • Failure, friction and faults
    • Brittle to ductile transition
    • Anderson theory and fault types around the solar system
  • Tectonics II
    • Generating tectonic stresses on planets
    • Slope failure and landslides
    • Viscoelastic behavior and the Maxwell time
    • Non-brittle deformation, folds and boudinage etc…
How to faults break?
  • Shear zone starts with formation of Riedel shears (R and R’)
    • Orientation controlled by angle of internal friction
  • Formation of P-shears
    • Mirror image of R shears
    • Links of R-shears to complete the shear zone

Revere St., San Francisco

(Hayward Fault)

Wrinkle ridges
    • Surface expression of blind thrust faults (or eroded thrust faults)
    • Associated with topographic steps
    • Upper sediments can be folded without breaking
    • Fault spacing used to constrain the brittle to ductile transition on Mars

Montesi and Zuber, 2003.


Where η is the dynamic viscosity

  • Rocks flow as well as flex
    • Stress is related to strain rate
    • Viscous deformation is irreversible
    • Motion of lattice defects, requires activation energies
    • Viscous flow is highly temperature dependant



Back to our mountain example


Solution for h:

Works in reverse too…

In the case of post-glacial rebound

τ ~ 5000years

w ~ 300km

Implies η ~ 1021 Pa s – pretty good

How to quantify τfs
    • Sliding block experiments
    • Increase slope until slide occurs
    • Normal stress is:
    • Shear stress is:
    • Sliding starts when:
  • Experiments show:
    • Amonton’s law – the harder you press the fault together the stronger it is
    • So fs=tan(Φ)
    • fs is about 0.85 for many geologic materials
  • In general:
    • Coulomb behavior – linear increase in strength with confining pressure
    • Co is the cohesion
    • Φ is the angle of internal friction
    • In loose granular stuff Φ is the angle of repose (~35 degrees) and Co is 0.
Effect of pore pressure
    • Reduces normal stress…
    • And cohesion term…
    • Material fails under lower stresses
    • Pore pressure – interconnected full pores
    • Density of water < rock
    • Max pore pressure is ~40% of overburden
  • Landslides on the Earth are commonly triggered by changes in pore pressure