Page 281 - Physical Principles of Sedimentary Basin Analysis

8.3 Stick–slip faulting 263

angle 2θ in the Mohr’s circle. An inspection of Figure 8.4a shows that 2θ = π/2 + φ, and
we therefore get that cos 2θ = cos(π/2+φ) =− sin φ and sin 2θ = sin(π/2+φ) = cos φ.
2
2
The trigonometric relations sin φ = tan φ/(1 + tan φ) 1/2 and cos φ = 1/(1 + tan φ) 1/2
together with tan φ = μ then gives
μ 1
and . (8.6)
cos 2θ =− 2 sin 2θ = 2
1 + μ 1 + μ
The condition for failure τ = μσ is

1 μ μ
(σ 1 − σ 3 ) sin 2θ = (σ 1 + σ 3 ) + (σ 1 − σ 3 ) cos 2θ (8.7)
2 2 2
and when sin 2θ and cos 2θ are replaced by expressions (8.6), we get the ratio (8.5)aftera
little algebra.

8.3 Stick–slip faulting

Fault movements are rarely by a steady sliding or creep. Instead faults accumulate stress
and elastic energy over long time, which is suddenly released in earthquakes. The faults are
therefore locked, except for the rare events of earthquakes. The accumulated elastic energy
is released as seismic waves that shake the ground and as heat from friction along the fault.
Figure 8.5 shows a simple model for the stick–slip behavior of a fault, where the fault is
initially in a relaxed situation, without any shear stress in the fault plane. The fault is then
subjected to shear strain at a constant rate, which accumulates until the corresponding shear
stress exceeds the static friction of the fault. The fault then slips, and the fault is opposed
by dynamic friction during motion. The static friction is the shear stress that opposes the
initialization of movement, and the dynamic friction is the shear stress that opposes the
motion. When the fault slips, and the shear stress along the fault equals the static friction
τ s ,wehave

τ s = τ d + G (8.8)

where τ d is the dynamic friction, G is the shear modulus and is the amount of the shear
strain necessary for the fault to slip. Relation (8.8) between static and dynamic friction

h

h
w
(a) (b) (c)
Figure 8.5. Three stages in stick–slip faulting: (a) The fault is relaxed (right after fault slip). (b) The
fault accumulates stress. (c) The fault has slipped and the elastic strain is converted to a permanent
displacement.