Derive, from the Mohr circle construction, the general result that a fault forms at…

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

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

Problem

Derive, from the Mohr circle construction, the general result that a fault forms at an angle 45∘−φf/245^\circ - \varphi_f/2 from the direction of greatest compression S1S_1, where tan⁡φf=μf\tan\varphi_f = \mu_f is the coefficient of friction on the failure plane and S0S_0 is the cohesion. Start from σn(θ)=12(S1+S3)+12(S1−S3)cos⁡2θ\sigma_n(\theta) = \tfrac12(S_1+S_3) + \tfrac12(S_1-S_3)\cos2\theta, τ(θ)=12(S1−S3)sin⁡2θ\tau(\theta) = \tfrac12(S_1-S_3)\sin2\theta, where θ\theta is the angle between the plane's normal and S1S_1, and the Coulomb line τ=S0+μfσn\tau = S_0 + \mu_f\sigma_n. ((a)) Show that the tangency condition between the line and the circle gives 2θ=90∘+φf2\theta = 90^\circ + \varphi_f, and hence that the fault plane itself lies at 45∘−φf/245^\circ - \varphi_f/2 from S1S_1. ((b)) Evaluate φf\varphi_f and the fault angle for gouge-lined faults with μf=0.85\mu_f = 0.85.

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