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## Tectonics I

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- 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?

w

Weight of the mountain

h

F=ma for material in the hemisphere

v

Conserve volume

Solution for h:

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

Response of materials to stress (σ) – elastic deformation

L

ΔL

ΔL

L

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

- 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:

Where:

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

ΔL

L

Where λ is the Lamé parameter

G is the shear modulus

or

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

- 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

- 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

σ

b

a

σ

- 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

- 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

- 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

Golembek

- 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,

- 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

σ2

Before we talk about faults….

- Fault geometry
- Dip measures the steepness of the fault plane
- Strike measures its orientation

Largest principle (σ1) stress perpendicular to surface

- Typical dips at ~60°

- 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

Smallest (σ3) principle stress perpendicular to surface

- Typical dips of 30°

- 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

Intermediate (σ2) principle stress perpendicular to surface

- Fault planes typically vertical

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

Europa

- 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

w

h

Back to our mountain example

v

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

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