A block of unit area slides on a rate-and-state friction surface at effective normal…

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

The Solid Earth, Ch Faults, the earthquake cycle and seismic hazard

Problem

A block of unit area slides on a rate-and-state friction surface at effective normal stress

\sigma_n'

, with direct effect

a

, evolution effect

b

, and characteristic slip distance

D_c

. The friction law and the ageing evolution of the state variable

\theta

are

\mu_f = \mu_{f0} + a\ln(V/V_0) + b\ln(V_0\theta/D_c), \qquad \dot\theta = 1 - \frac{V\theta}{D_c}

The block is pulled by a spring of stiffness

k

(force per unit area per unit slip deficit) whose far end moves at the plate rate

V_{\rm pl}

, so that steady sliding at

V=V_{\rm pl}

\theta=\theta_{\rm ss}=D_c/V_{\rm pl}

is a solution. The demonstration left to you in the spring-slider section linearises the force balance

k(V_{\rm pl}-V) = \sigma_n'\dot\mu_f

and the evolution equation about this state and finds that perturbations proportional to

e^{st}

grow only for

k < k_c \equiv \sigma_n'(b-a)/D_c

. ((a)) Starting from that linearisation, measure

s

in units of

V_{\rm pl}/D_c

and show that at

k=k_c

a perturbation neither grows nor decays but oscillates. Find its period, in units of

D_c/V_{\rm pl}

, in terms of

a

and

b

. ((b)) For the granite values

a=0.010

and

b=0.014

, evaluate that period. Then find the period and the e-folding decay time of the oscillation at

k=2k_c

, in the same units. ((c)) By replacing the spring stiffness with the elastic stiffness

\mu/L

of a slipping patch of size

L

in a medium of shear modulus

\mu

, obtain the nucleation length

L_c \approx \mu D_c/[\sigma_n'(b-a)]

below which such a patch is held stable by its own elastic surroundings. ((d)) A fault surface has

\sigma_n' = 150\,\mathrm{MPa}

b - a = 0.006

and a laboratory value

D_c = 20\,\mathrm{\mu m}

, embedded in crust of shear modulus

\mu = 30\,\mathrm{GPa}

. Evaluate

k_c

and

L_c

. ((e)) Surface roughness and gouge on a natural fault make the effective

D_c

some three orders of magnitude larger than the laboratory value; take

D_c = 2\,\mathrm{cm}

for the same fault and re-evaluate

L_c

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