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Tectonics I

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

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  1. Tectonics I

  2. 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…

  3. 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…

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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:

  9. 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:

  10. 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

  11. Groups of two of the previous parameters describe the elastic response of a homogenous isotropic solid • Conversions between parameters are straightforward

  12. Typical numbers (Turcotte & Schubert)

  13. 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

  14. 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

  15. (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

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

  17. 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

  18. 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

  19. 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

  20. 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 Golembek

  21. 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,

  22. 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 σ2

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

  24. Largest principle (σ1) stress perpendicular to surface • Typical dips at ~60°

  25. 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

  26. 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

  27. In reality graben fields are complex… • Different episodes can produce different orientations • Old graben can be reactivated Lakshmi -Venus Ceraunius Fossae - Mars

  28. Smallest (σ3) principle stress perpendicular to surface • Typical dips of 30°

  29. 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

  30. Montesi and Zuber, 2003.

  31. Intermediate (σ2) principle stress perpendicular to surface • Fault planes typically vertical

  32. 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 Europa

  33. 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…

  34. Random extras

  35. 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)

  36. 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.

  37. 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

  38. 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.

  39. 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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