The Frictional Regime Processes in Structural Geology & Tectonics Ben van der Pluijm WW Norton+Authors, unless noted otherwise 1/25/2016 10:08 AM
We Discuss The Frictional Regime Processes of Brittle Deformation Tensile Cracking Axial Experiment Formation of Shear Fractures Failure Criteria Faults and Stress Andersonian Theory Frictional Sliding Byerlee s Law Stress and Sliding Role of Fluids Frictional Regime PSG&T 2
Generalized Fault Model and Deformation Regimes P l /s d T( o C) Rock Regime San Andreas Fault, Carrizo Plain Frictional Regime PSG&T 3
Extra: Frictional and Plastic Deformation Regimes Frictional regime where deformation is localized and involves fractures. normal stress and pressure dependent temperature and strain insensitive shear stress is function of normal stress Plastic regime where deformation is distributed and involves crystal flow. normal stress and pressure insensitive temperature and strain dependent shear stress is function of temperature and strain rate Frictional Regime PSG&T 4
Brittle Deformation and Structures Brittle deformation is permanent change in a solid material due to formation of cracks and fractures and/or due to sliding on fractures once they formed. Categories of brittle deformation: Tensile cracking Shear rupture Frictional sliding Cataclastic flow Frictional Regime PSG&T 5
Atomic View: Elasticity and Failure Atomic bonds are elastic, like springs Frictional Regime PSG&T 7
Rock Strength Paradox Strength Paradox: Rocks allow only few % elastic strain (e) before permanent deformation (failure), which occurs at low stresses. s = E. e (next) = 5x10 10. 0.1 = 5x10 9 Pa = 5x10 3 MPa Thus, theoretical strength of rock is 1000 s MPa, but the observed tensile strength is only 10 s MPa. Frictional Regime PSG&T 8
Elasticity Elastic Behavior: s = E. e E = Young s Modulus e = strain = (l-l o )/l o (rubber band) s s = G. g G=shear modulus g = shear strain Other elastic parameters: Bulk Modulus (K): K = s/dv; Shear Modulus (Rigidity, G): G = s s /g Poisson s ratio: n = e perpendicular to s /e parallel to s Frictional Regime PSG&T 9
Tensile Cracking and Stress Concentration Remote v. local stress: stress concentration, C, is: (2b/a) +1 Crack 1 x.02 mm: C = 100! Greater as cracks grow, so larger is weaker. Frictional Regime PSG&T 10
Axial Experiments: Griffith Cracks Extension Compression Effect of preexisting (or Griffith) cracks: preferred activation, then longest cracks grow because of stress concentration. Frictional Regime PSG&T 11
Crack Modes Tensile cracks (Mode I) Shear cracks (Mode II and III) Shear cracks are not faults. When they propagate, they rotate into Mode I orientation ( wing cracks ) Frictional Regime PSG&T 12
Formation of Shear Fractures Shear fracture is surface across which rock loses continuity when shear stress parallel to surface is sufficiently large. (I, II) (III) dilatancy Shear fractures are not same as large tensile cracks, as their eventual orientation shows. Initial state I, II III IV Differential stress at failure is called failure stress (σ f ). Frictional Regime PSG&T 13
Failure Criteria in Mohr Space Tensile crack: Griffith criterion (A) Shear fracture: Mohr-Coulomb criterion (B-C) Frictional-Plastic transition (D) Plastic shear zone (E) Frictional Regime PSG&T 14
Shear Failure Criteria 1 Coulomb Criterion Coulomb failure criterion: s s = C + ms n σ s is shear stress parallel to fracture surface at failure C is cohesion, a constant that specifies shear stress necessary to cause failure if normal stress across potential fracture plane equals zero σ n is normal stress across shear fracture at instant of failure μ is a constant: coefficient of internal friction Fracture surfaces (two!) at ~30 o to s 1 and ~60 o to s 3. Yet, s s is maximum at 45 o. Why? Frictional Regime PSG&T 15
Why 30 o instead of 45 o Fracture Angle with s 1? Fracture surfaces at ~30 o to s1 and ~60 o to s 3. Yet, s s is maximum at 45 o. s1 Frictional Regime PSG&T 16
Shear Failure Criteria 2 Tensile Stress Parabolic failure envelope: steeper near tensile field and shallower at high s n (Mohr criterion) Fractures toward 90 o in tension, fractures around 30 o in compression, relative to σ 1 Combined: Mohr-Coulomb failure criterion failure envelope Frictional Regime PSG&T 17
Shear Failure Criteria 3 High Stress At high confining stresses: Von Mises criterion Plastic deformation occurs at high (normal) stress, which is shear stress independent Frictional Regime PSG&T 18
Composite failure envelope Note the changing angle of rupture with respect to σ 1 Frictional Regime PSG&T 19
Fault Types: Normal, Reverse, Lateral-slip Faults Frictional Regime PSG&T 20
Predictions from Anderson s Theory of Faulting Earth solid-air contact is free surface, meaning no shear stresses, so orthogonal principal stresses. Andersonian predictions: High-angle normal faults Low-angle reverse faults Lateral-slip faults Unexplained: Low-angle normal faults Non-conjugate fault systems Frictional Regime PSG&T 21
Sliding on a Surface (Friction Criterion) Frictional sliding refers to movement on a pre-existing surface when shear parallel to surface exceeds sliding resistance. Amonton s Laws of Friction: Frictional force is function of normal force. Frictional force is independent of (apparent) area of contact. Frictional force is (mostly) independent of material used. (17th C) (15th C da Vinci experiments) Frictional Regime PSG&T 22
Surface Roughness (Asperities) The bumps and irregularities (roughness) that protrude on natural surfaces are called asperities, which anchor surfaces. Real v. Apparent Area of Contact Larger mass (larger F), deeper penetration Frictional Regime PSG&T 23
Surface Roughness Sliding has to exceed resistance (strength) of asperity (+dilation). Frictional Regime PSG&T 24
Frictional Sliding Criteria (Byerlee s Law) s s / s n = constant = m = coefficient of friction Byerlee s Law depends on σ n. For σ n < 200 MPa, the best-fitting criterion is σ s = 0.85.σ n. For 200 MPa <σ n < 2000 MPa, the best-fitting criterion is σ s = 50 MPa + 0.6.σ n. m = 0.6-0.85, or ~0.75 (static friction) Frictional Regime PSG&T 25
Sliding on Existing Fractures or Create New Fractures? s 3 s 1 Instead of new failure surface (A), sliding on existing surfaces (gray area). Preexisting surfaces from B to E will slide with decreasing friction coefficients. (Blair Dolomite experiment) Frictional Regime PSG&T 26
Principal Stresses and Fault Types = r.g.h = r.g.h = r.g.h Normal Fault Reverse Fault Lateral-Slip Fault Differential stress for faulting is increasingly greater from normal, to lateral slip to reverse faulting. s s Normal ρ g h Reverse s n Frictional Regime PSG&T 27
Stress Conditions for Sliding s s = m. s n Consider m = 0.75 (Friction Law) s 1 = s 3. 4 Substitute for principal stresses, fracture angle and rearrange gives: s 1 = s 3. [m 2 +1) 1/2 + m] 2 Normal fault: s 1 = (ρ g h), so (ρ g h) = s 3 x 4 (s 1 s 3 ) = (ρ g h) 0.25(ρ g h) = 0.75(ρ g h) Reverse fault: s 3 = (ρ g h) s 1 = (ρ g h) x 4 (s 1 s 3 ) = 4(ρ g h) - (ρ g h) = 3(ρ g h) σ d β (ρ g h) β is 3, 1.2, and 0.75 for reverse, strike-slip, and normal faulting Frictional Regime PSG&T 28
Stress Conditions for Sliding (s 1 s 3 ) m = 0.75 σ d β (ρ g h) β is 3, 1.2, and 0.75 for reverse, strike-slip, and normal faulting borehole measurement, sliding on strike-slip fault, but stable on reverse fault Frictional Regime PSG&T 29
Fluids and Fracturing Hydrostatic (fluid) pressure P f = ρ g h, where ρ is density of water (1000 kg/m 3 ), g is gravitational constant (9.8 m/s 2 ), and h is depth. Lithostatic pressure P l = ρ g h, weight of overlying column of rock (ρ = 2500 3000 kg/m 3 ). Frictional Regime PSG&T 30
Fluid Pressure and Effective Stress outward push σ s = C + μ.(σ n P f ) [fracturing] σ s = μ.(σ n P f ) [sliding] (σ n P f ) is commonly labeled σ n *, the effective stress. (hydraulic fracturing or fracking ) So, m effective = m (1 P f /s n ) m effective m Frictional Regime PSG&T 31
Limiting Stress Conditions for Sliding, with Pore-fluid (s 1 s 3 ) σ d β (ρ g h). (1 λ) l = 0.9 (90% of lithostatic pressure) β is 3, 1.2, and 0.75 for reverse, strike-slip, and normal faulting (with m = 0.75) λ = P f /P l, ratio of pore-fluid pressure and lithostatic pressure (λ ranges from 0.35 (1000/2750) for hydrostatic fluid pressure to 1 for lithostatic fluid pressure) Increasing λ increases fracturing potential: fracking, injection well EQs Frictional Regime PSG&T 32
Extra: Friction Coefficient from Stress Measurement KTB, Germany (9100m) Calculate μ eff from KTB deep borehole measurements at strike-slip conditions: s 1 = s 3. [m 2 +1) 1/2 + m] 2 m eff = 0.6 s 1 = s 3. 3.15 σ d β (ρ g h) So, crust critically stressed for strikeslip faulting Frictional Regime PSG&T 33