At a depth in continental crust of density

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Earth, Climate & Energy Physics · Seismology, tectonics and geodynamics

The Solid Earth, Ch Stress, strain and elastic waves in the Earth

Problem

At a depth z=6 kmz = 6\,\mathrm{km} in continental crust of density ρ=2800 kg m−3\rho = 2800\,\mathrm{kg\,m^{-3}}, the vertical stress equals the weight of the overlying rock, σzz=−ρgz\sigma_{zz} = -\rho gz (tension positive), and is a principal stress because the free surface above carries no shear traction. In the horizontal plane the stress tensor (tension positive, in MPa) is σxx=−120,σyy=−90,σxy=20,\sigma_{xx} = -120, \qquad \sigma_{yy} = -90, \qquad \sigma_{xy} = 20, with σxz=σyz=0\sigma_{xz} = \sigma_{yz} = 0. Pore pressure is hydrostatic, pf=ρwgzp_f = \rho_wgz with ρw=1000 kg m−3\rho_w = 1000\,\mathrm{kg\,m^{-3}}. The fault has cohesion S0=20 MPaS_0 = 20\,\mathrm{MPa} and coefficient of friction μf=0.6\mu_f = 0.6. Take g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}}. ((a)) Find the two horizontal principal stresses, and hence the three principal stresses of the full three-dimensional state as compressive magnitudes S1≥S2≥S3S_1\ge S_2\ge S_3. Find the mean pressure pˉ=−σkk/3\bar p=-\sigma_{kk}/3, compare it with the lithostatic value ρgz\rho gz, and comment on its size relative to the deviatoric stresses. ((b)) State which of Anderson's three faulting regimes this stress state predicts, and why. ((c)) Using effective stresses (subtracting the hydrostatic pore pressure), decide whether the Coulomb criterion predicts failure on the optimally oriented plane. If it does not, find the extra pore-fluid pressure, above hydrostatic, needed to cause failure. ((d)) Give the dip of the fault predicted by this coefficient of friction.

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