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## The Mechanics of the crust

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**The Mechanics of the crust**How do rocks deform in the crust ? • Mechanisms • Bearing strength Must consider: Brittle crust Ductile crust**The Brittle and the Ductile regime**Brittle Ductile**cataclastic intermediate betweenbrittle and ductileFault**Gouge =>**Triaxial experiment allow to impose various confining (blue)**and deviatoric (red – blue) normal stresses**The regime stress-strain curves evolveswhen the confining**pressure increases**The effect of increasing confining pressure**Confining stress**Two distinct types of plasticity**Strain Hardening Plasticity At depth : Strain Hardening Plasticity : can accommodate permanent strain without losing the ability to resist load At The surface : Strain Softening Plasticity: its ability to resist load decreases with permanent strain Strain Softening Plasticity**The mode of failure evolves when increasing the confining**pressure Cataclastic Brittle**IMPORTANT**The loss of resistance of the upper crust While the lower crust is still resistant Is responsible for the the earthquakes instability : For a given applied force the displacement should be infinite Infinite displacement in the upper crust upper crust force**The Brittle regime**Responsible for the faults close to the surface**Common Observation :Conjugate Shear Fractures**Conjugate fractures Are pair of fault which Slip at the same time They have opposite shear senses**On rock experimentsconjugates fault are also observed inthe**lab.**Byerlee’s Law rule for rock friction deduced from triaxial**experiment(t-s : mohr space) t NOT STABLE STABLE s**Impossible Stress State**• Any stress state whose circle lies outside the envelope is an unstable stress state, and is not physically possible • Before stress reaches this state, the sample would have failed**Failure Envelope**The Shaded stable Area is bounded by the failure envelope in black**Stable Stress State**Any stress state lying within the envelope is stable**Defining Stress State**Stress state tangent to the envelope defines the failure state A fault forms…**Coulomb Criterion The failure enveloppe is linear**The further away from the origin the circle center is, the larger is the radius of the circle The bigger is the maximum compression**Coulomb’s Criterion**• t = σs= C + μ σn • C is a constant that specifies the shear stress necessary to cause failure if the normal stress is zero (order 10 Mpa) • The two fractures occur at an angle fº, and correspond to the tangency points of the circle representing the stress state at failure with the Coulomb failure envelope**Byerlee’s Law rule for rock friction deduced from triaxial**experiment(t-s : mohr space) • For σn < 200 MPa, • For 200MPa < σn < 5000MPa where: t = shear stress (MPa) sn = normal stress (MPa)**Possibles Applications : Thrust are usualy diping under 30°**Continental deformation**The dip angle can serve to define the friction associated**with the earthquake Exercice what is the friction angle Here subduction zone**Exercice : Conjugates faults**Plot on a stereonet the conjugates faults Fa) Strike : 25°E, dip : 35°E Fb) Strike : 30°W, dip : 15°W Measure the angle between the fault planes Deduce the internal friction angle and the principal directions of compression and extension**Normal Faults close to the surface**Here the two normal Faults are conjugate Faults they typically form an angle of 60°**Conjugates Fault and Stress**• Conjugate shear fractures develop at about = 30 degrees from the maximum compressional stress : 1 • 1 bisects the acute angle of about 60o between the two fractures • The minimum compressional stress 3 bisects the obtuse angle between the two fractures**Conjugates Faults and the Principal Stresses**• Reverse faults are more likely to form if 3 is vertical and constant (at a standard state), while horizontal, compressive 1 and2 increase in value compared to the standard state • Normal faults form if 1 is vertical and constant, while horizontal 3 and2 decrease in value, or if horizontal 3 is tensile • Strike-slip faults form if 2 is vertical and constant, while horizontal 1 and2 increase and decreases in value, respectively NEAR the SURFACE**Brace-Goetze strength profiles**Brittle Ductile After Kohlstedt et al., 1995**Frictional Sliding**• Frictional force does not depend on the shape of the object • Both objects, of the same mass, have the same sliding force, despite having different areas of contact**Amonton’s Law**• Frictional resistance to sliding normal stress component across the surface • First “published” account this empirical law of friction was made by the French physicist Guillaume Amonton in 1699, although Leonardo da Vinci’s notes indicate he knew of the result about 200 years earlier • If normal stress increases, the asperities are pushed more deeply into the opposing surface, and increasing resistance to sliding**Fracture Surface**• Fracture surface, showing voids and asperities (Figure 6.23a, text) • As another, also bumpy, surface tries to slide over the first surface, their asperities interact, causing friction**Real Area of Contact**• The bumps mean that only a small part of the surfaces are actually in contact • Dark areas are real area of contact (RAC) (Figure 6.23c, text)**Surface Anchors**• The forces normal to these surfaces will be concentrated on the small areas in contact • Asperities cumulatively act as small anchors, retarding any slippage along the surface (Figure 6.23b, text)**Criteria for Frictional Sliding**• Before the initiation of frictional sliding, enough shear must be present to overcome friction • We can define a criterion for frictional sliding to represent the necessary shear • Experimental work has shown that, independent of rock type, the following criterion holds • σs/σn = constant**Movement of Stress Along σ3 Axis**• When represented on a Mohr diagram, the Mohr circle moves to the left along the normal stress axis • Figure 6.27 in text**Pore Pressure and Shear Fracturing**• Pore pressure also plays a role on shear fracturing • Since pore pressure counteracts the confining pressure, we can rewrite the equation for shear stress to take pore pressure into account: • σs = c + μ(σn - Pf)