# Mechanics of Earthquakes and Faulting

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1 Mechanics of Earthquakes and Faulting Standard Solids and Fracture Fluids: Mechanical, Chemical Effects Effective Stress Dilatancy Hardening and Stability Mead, 1925 (Mead, W. J., The geologic role of dilatancy. Jour. Geol. 33, , 1925.) Volumetric work and stability. Frank, 1965 (Frank, F. C., On dilatancy in relation to seismic sources. Rev. Geophys. 3, , 1965) Basic Elasticity Lecture 5, 7 Sep. 2017

2 Mead, 1925 (Geologic Role of Dilatancy) Shear Localization Strain homogeneity depends on whether dilatancy is restricted Homogeneous strain if dilatancy is not opposed Strain localization if deformed under finite confining pressure W = τ p dγ + σ dθ τ = σ ( µ p + dθ / dγ ) Shear strength depends on friction and dilatancy rate Deformation mode (degree of strain localization) minimizes dilatancy rate Shear Bands Form if:

3 Frank, Volumetric work, shear localization and stability. Applies to Friction and Fracture Volume Strain, θ Strain hardening when: Shear Strain, γ Strain softening when:

4 Volume Strain, θ Shear Strain, γ Marone, Raleigh & Scholz, 1990

5 Volume Strain, θ Shear Strain, γ Marone, Raleigh & Scholz, 1990

6 Standard Solids There are many types of rheologies and failure laws based on idealized stress-strainfailure relations. In general they are simple, however, one has to keep in mind what is meant by failure, since different definitions are used for different rheologies. Ideal plastic (also called perfectly plastic or rigid plastic) has no deformation (zero strain) until a threshold stress and then infinite strain at that yield stress. Elasto-plastic is elastic (strain) up to a threshold stress and then infinite strain at that yield stress. Strain hardening refers to a variety of cases in which the yield strength (stress required to cause further deformation) increases with strain. Strain softening is the opposite of strain hardening. It occurs commonly in brittle deformation prior to catastrophic failure and during shear localization. Visco-plastic or elasto-viscous refer to cases in which yield strength depends strongly on strain-rate. Recall that essentially all real materials exhibit some form of strain-rate dependent strength, but in viscous materials this effect is substantial.

7 Shear Fracture Energy from Postfailure Behavior Differential Stress, (σ 1 - P c ) Fracture Dilatancy: cracks forming and opening Axial strain Hardening, Modulus Increase: cracks closing Brace, Paulding & Scholz, 1966; Scholz 1968.

8 Shear Fracture Energy from Postfailure Behavior Fracture (σ 1 - P c ) Strain Lockner et al., 1991

9 Shear Fracture Energy from Postfailure Behavior Lockner et al., 1991 Inferred shear stress vs. slip relation for slip-weakening model. (based on Wong, 1982) Wong, 1982, found that shear stress dropped ~ 0.2 GPa over a slip distance of ~50 microns. Exercise: Estimate G from these data and compare it to the values reported in Scholz (Table 1.1) and Wong, 1982.

10 σ effective = σ n - Pp σ 3 σ 1 Pp σ 1 σ 3 Exercise: Follow through the implications of Kronecker s delta to see that pore pressure only influences normal stresses and not shear stresses. Hint: see the equations for stress transformation that led to Mohr s circle. σ = ( σ 1 + σ 2 ) 2 ( + σ σ 1 2) 2 cos2α

11 Void space filled with a fluid at pressure Pp σ But what if Ar A? p o p σ Fluids play a role by opposing the normal stress

12 Effective Stress Law Coupled Effects σ Applied Stress p o Pore Pressure Strength, Stability p Dilatancy Shear Rate σ Pore Fluid, Pp Pore Fluid, Pp

13 Dilatancy: Volumetric Strain: Assume no change in solid volume Dilatancy Rate: Shear Rate Pore Fluid, Pp Pore Fluid, Pp

14 Dilatancy: Undrained loading Volumetric Strain: Assume no change in solid volume Dilatancy Rate: Dilatancy Hardening if : or Shear Rate Pore Fluid, Pp Pore Fluid, Pp

15 Elasticity: σ ij = 2µε ij + λδ ij θ where λ and µ are Lame s constants, δ is Kronecher s delta (δ ij = 1 for i = j and δ ij = 0 for i j) and θ is the volumetric strain. Lame s constants are (can be related to) Elastic moduli Shear Modulus, Bulk Modulus, Young s Modulus µ = E 2 1 +υ deformation (normalized by the initial length) for a given change in shear stress G = dτ/(dx/l) = dτ/dγ ( ) = G The shear modulus, G or µ, is the shear dτ dx L

16 K = λ µ K is the bulk modulus. The bulk modulus is the change in volume (normalized by the initial volume) for a given change in hydrostatic pressure: K = dp/(dv/v i ) = dp/ dθ After Compression V final V initial dv = V final - V initial

17 ε = u/l, linear strain u σ = E ε, where E is Young s Modulus. L Note that Modulus has units of stress (Pa) Young s Modulus is important in many problems. Think of it as a generalized (i.e., complex) spring constant z τ zx τ xz τ zz τ zy τ yz τyy As in Hooke s law, which relates force and displacement through a spring constant, the modulus relates stress and strain. τ xy τ yx y x τ xx

18 Elasticity: σ ij = 2µε ij + λδ ij θ where λ and µ are Lame s constants, δ is Kronecher s delta (δ ij = 1 for i = j and δ ij = 0 for i j) and θ is the volumetric strain. Lame s constants are (can be related to) Elastic moduli µ = E 2( 1 +υ ) = G The shear modulus, G or µ, is the shear deformation (normalized by the initial length) for a given change in shear stress G = dτ/(dx/l) = dτ/dγ K = λ µ K is the bulk modulus. The bulk modulus is the change in volume (normalized by the initial volume) for a given change in hydrostatic pressure: K = dp/(dv/v) =dp/dθ λ = υe 1 +υ ( )( 1 2υ) λ can be related to E and υ

19 u L There are nine components of the strain tensor ε xx, ε xy, ε xz ε yx, ε yy, ε yz ε zx, ε zy, ε zz z τ zz Poisson s ratio ν is an elastic parameter that describes the lateral expansion due to a vertical deformation: τ zx τ xz τ zy τ yz τ xy τ yx τyy ν = -ε zz /ε xx Poisson s ratio is 0.5 for water and 0.25 for a typical granite. x τ xx y

20 Body waves (P, S) P waves involve volume change, but not shear distortion Particle motion is parallel to wave propagation direction S waves involve shear distortion but not volume change Particle motion is perpendicular to wave propagation direction (SH and SV)

21 Seismic waves are elastic disturbances We can study them using our knowledge of elasticity. Equations of motion: Navier s equation: mass x acceleration = body forces + surface forces In 1-D:

22 Seismic waves are elastic disturbances We can study them using our knowledge of elasticity. Equations of motion: Navier s equation: mass x acceleration = body forces + surface forces In 1-D: S-Waves (no volume change) P-Waves (no shear distortion)

23 P-wave velocity near Earth s surface is ~ 6 km/s. Seismic frequency band is about 2 mhz to 20 Hz (0.002 to 20 Hz) V = λ f Assume strains are small and elastic (ε < 1 micro strain), what is particle motion for a 1 Hz P-wave? ΔL/L = ΔL ~ P-Waves (no shear distortion)

24 Seismic Velocity Seismic velocity is a material property (like density). P waves always travel faster than S waves. Seismic velocities depend on quantities like chemical composition, pressure, temperature, etc. Faster Velocities Lower temperatures Higher pressures Solid phases Slower Velocities Higher temperatures Lower pressures Liquid phases

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