6.4 Fault Mechanics
Byerlee's Law
In 1978, James Byerlee demonstrated through extensive laboratory experiments that the frictional strength of most rock types follows a remarkably simple empirical relationship, largely independent of rock type, surface roughness, or slip rate. This universal behavior, known as Byerlee's law, holds for a wide range of crustal rocks including granite, basalt, sandstone, and limestone:
Byerlee's law (two regimes):
\[ \tau = 0.85\,\sigma_n \quad \text{for } \sigma_n < 200 \text{ MPa} \]
\[ \tau = 50 + 0.6\,\sigma_n \quad \text{for } \sigma_n > 200 \text{ MPa} \]
where $\\tau$ is the shear stress at failure (MPa) and $\\sigma_n$ is the normal stress acting on the fault surface (MPa). The transition at ~200 MPa corresponds to a depth of approximately 5–8 km, depending on the stress regime.
The low-pressure regime ($\sigma_n < 200$ MPa) has zero cohesion and a friction coefficient of 0.85, corresponding to a friction angle of approximately 40°. The high-pressure regime has a cohesion of 50 MPa and a lower friction coefficient of 0.6 (friction angle ~31°), reflecting the transition from brittle fracture to pressure-dependent frictional sliding.
Notable exceptions to Byerlee's law include clay-rich fault gouges (montmorillonite, talc, serpentinite), which have friction coefficients as low as 0.1–0.3. These weak minerals play a critical role in controlling fault behavior, particularly along subduction zone megathrusts and creeping fault segments.
Mohr-Coulomb Failure Criterion
The Mohr-Coulomb failure criterion is the fundamental constitutive law for brittle rock failure and frictional sliding. It states that failure occurs on a plane when the shear stress on that plane exceeds a critical value that depends linearly on the normal stress:
Mohr-Coulomb criterion:
\[ \tau = c + \mu\,\sigma_n = c + \sigma_n\tan\phi \]
where $\\tau$ is shear stress, $c$ is cohesion (shear strength at zero normal stress),$\\mu = \\tan\\phi$ is the coefficient of internal friction, $\\phi$ is the angle of internal friction, and $\\sigma_n$ is the normal stress on the failure plane.
On a Mohr diagram ($\tau$ vs. $\sigma_n$), the failure criterion plots as a straight line with slope $\mu$ and intercept $c$. A stress state is stable when the Mohr circle lies entirely below the failure envelope. Failure occurs when the Mohr circle becomes tangent to the envelope. The orientation of the failure plane with respect to the maximum principal stress $\sigma_1$ is:
Angle of failure plane:
\[ \theta = 45° + \frac{\phi}{2} \]
For typical rock with $\\phi \\approx 30°$, the failure plane makes an angle of $\\theta = 60°$ with the $\\sigma_1$ direction, or equivalently, the fault dips at 60° from the direction of maximum compressive stress.
In the presence of pore fluid pressure $P_f$, the effective normal stress is reduced:$\sigma_n' = \sigma_n - P_f$. High pore pressures dramatically reduce fault strength, which is invoked to explain the apparent weakness of some major faults (e.g., the San Andreas "heat flow paradox", where the expected frictional heating from Byerlee-level friction is not observed).
Anderson's Theory of Faulting
E.M. Anderson (1905, 1951) recognized that because the Earth's surface is a free surface (no shear stress), one of the three principal stresses must be vertical at shallow depths. The orientation of the three principal stresses ($\sigma_1 > \sigma_2 > \sigma_3$) relative to the vertical determines which type of fault will form:
| Fault Type | Vertical Stress | $\sigma_1$ Direction | Fault Dip | Tectonic Setting |
|---|---|---|---|---|
| Normal fault | $\sigma_1$ = vertical | Vertical | ~60° | Extensional (rifts, divergent boundaries) |
| Thrust (reverse) fault | $\sigma_3$ = vertical | Horizontal | ~30° | Compressional (collision, subduction) |
| Strike-slip fault | $\sigma_2$ = vertical | Horizontal | ~90° | Transform boundaries |
For strike-slip faults, both $\sigma_1$ and $\sigma_3$ are horizontal, with$\sigma_2$ vertical (equal to the lithostatic load). The fault plane is vertical, and the maximum compressive stress makes an angle of ~30° with the fault trace (for $\phi \approx 30°$). This prediction is well confirmed by focal mechanism studies and in-situ stress measurements near transform faults.
Anderson's framework, while elegant, is a simplification. It assumes homogeneous, isotropic crust with no pre-existing weaknesses. In reality, faults often reactivate pre-existing structures at non-Andersonian angles, and stress rotations near major faults can cause local departures from the idealized geometry.
Rate-and-State Friction Law
The rate-and-state friction constitutive law, developed by Dieterich (1979) and Ruina (1983), is the modern framework for understanding how fault friction depends on both slip velocity and the history of contact. It explains why some faults produce earthquakes while others creep stably, and provides the theoretical basis for earthquake nucleation, aftershock sequences, and slow slip events.
Rate-and-state friction law:
\[ \mu = \mu_0 + a\ln\!\left(\frac{V}{V_0}\right) + b\ln\!\left(\frac{V_0\,\theta}{D_c}\right) \]
where $\\mu$ is the friction coefficient, $\\mu_0$ is a reference friction at velocity $V_0$, $V$ is the current slip velocity, $\\theta$ is the state variable (related to contact age), $D_c$ is the critical slip distance, and $a$ and $b$ are dimensionless constitutive parameters.
The two key parameters $a$ and $b$ control the frictional response to velocity perturbations:
Parameter a (Direct Effect)
The direct effect: an instantaneous increase in friction when slip velocity increases. This reflects the thermally activated creep of asperity contacts under load. Typical values:$a$ = 0.005–0.015. This effect is always stabilizing.
Parameter b (Evolution Effect)
The evolution effect: a gradual change in friction as the state variable evolves toward a new steady state. Reflects the growth or destruction of real contact area with time. Typical values: $b$ = 0.005–0.020. Can be destabilizing if $b > a$.
Stability Criterion: (a − b)
The sign of $(a - b)$ determines fault behavior. When (a − b) < 0 (velocity weakening), friction decreases with increasing slip velocity, creating the potential for unstable, seismic slip — earthquakes. When (a − b) > 0 (velocity strengthening), friction increases with velocity, promoting stable, aseismic creep. The transition from velocity weakening to velocity strengthening with depth defines the base of the seismogenic zone.
Earthquake Nucleation & Instability
Earthquake nucleation occurs when a fault patch in the velocity-weakening regime transitions from quasi-static (slow) to dynamic (fast) slip. The spring-slider analog provides physical intuition: a block on a frictional surface, loaded by a spring of stiffness $k$, will slide stably if the spring is stiff enough to "keep up" with the fault weakening, and unstably (earthquake) if the spring is too compliant.
Instability condition (earthquake nucleation):
\[ k < k_c = \frac{(b - a)\,\sigma_n}{D_c} \]
where $k$ is the elastic stiffness of the loading system (inversely proportional to fault patch size),$k_c$ is the critical stiffness, $(b - a)$ is the velocity-weakening magnitude,$\\sigma_n$ is normal stress, and $D_c$ is the critical slip distance. The instability requires $(b - a) > 0$ (velocity weakening).
Since the effective stiffness $k$ scales inversely with the size of the slipping patch ($k \sim G/L$ where $G$ is shear modulus and $L$ is patch dimension), the instability criterion defines a minimum nucleation length:
Critical nucleation length:
\[ L_c = \frac{G\,D_c}{(b - a)\,\sigma_n} \]
For typical crustal values ($G \\sim 30$ GPa, $D_c \\sim 10$–100 µm,$(b-a) \\sim 0.005$, $\\sigma_n \\sim 100$ MPa), $L_c \\sim 0.6$–6 m. The nucleation patch must grow beyond this size for dynamic rupture to initiate.
Seismogenic Zone Depth
The seismogenic zone is the depth range within which earthquakes nucleate on faults. Its extent is controlled primarily by temperature, which governs the transition from velocity-weakening (seismic) to velocity-strengthening (aseismic) friction. For quartz-rich continental crust:
| Boundary | Temperature | Typical Depth | Controlling Process |
|---|---|---|---|
| Upper limit | ~100–150°C | ~2–4 km | Transition from velocity-strengthening clay-rich gouge to velocity-weakening quartzofeldspathic rock |
| Lower limit | ~300–350°C | ~10–15 km | Onset of quartz plasticity; transition to velocity-strengthening behavior and distributed ductile shear |
The seismogenic zone thickness directly controls the maximum earthquake magnitude on a fault, because the rupture area is bounded by the seismogenic depth multiplied by the fault length. On the San Andreas Fault, microseismicity defines a seismogenic zone extending from ~2 km to ~12–15 km depth, consistent with the ~300–350°C isotherm as measured in deep boreholes (SAFOD project).
The seismogenic zone is deeper in oceanic lithosphere, where the olivine-dominated rheology permits brittle failure to ~600°C under certain conditions. In subduction zones, the seismogenic megathrust extends from ~10 km to ~40–50 km depth, bounded by the 150°C and 350–450°C isotherms on the plate interface.
Fault Zone Architecture
Mature fault zones exhibit a characteristic hierarchical structure when examined in cross-section. From the center outward, the following zones are typically identified:
1. Fault Core (Principal Slip Zone)
The narrowest zone (millimeters to centimeters thick) where most of the displacement is concentrated. Contains ultra-fine-grained gouge and ultracataclasite, often with a distinct foliation from shearing. In exhumed faults, a thin layer of pseudotachylyte (frictional melt) may record past seismic events. The principal slip surface may be only a few millimeters wide.
2. Fault Gouge / Cataclasite Zone
A broader zone (centimeters to meters) of intensely comminuted rock surrounding the core. Gouge is non-cohesive, clay-rich granular material; cataclasite is a cohesive, lithified equivalent. Grain size decreases toward the fault core, reflecting progressive comminution with cumulative fault displacement. Mineralogy is critical: clay minerals such as smectite and chrysotile develop through alteration and control frictional properties.
3. Damage Zone
A zone of elevated fracture density, subsidiary faults, and deformation bands extending meters to hundreds of meters from the fault core. Fracture density decays as a power law with distance from the fault. The damage zone is mechanically significant because it controls permeability and fluid flow around the fault, and its width influences the effective compliance (stiffness) of the fault.
4. Host Rock (Protolith)
Undeformed or minimally deformed wall rock beyond the damage zone. The transition from damage zone to host rock is typically gradational. Background fracture densities in the host rock may be elevated above regional values due to far-field stress effects from the fault.
Earthquake Stress Drop
The stress drop $\Delta\sigma$ in an earthquake is the difference between the shear stress on the fault before and after rupture. It is a fundamental source parameter that controls the radiated seismic energy and the amplitude of strong ground motion. For a fault with characteristic dimension $L$ and average slip $\bar{D}$:
Static stress drop:
\[ \Delta\sigma = C\,\frac{\mu\,\bar{D}}{L} \]
where $C$ is a geometry-dependent constant (e.g., $C = 7\\pi/16$ for a circular crack),$\\mu$ is the shear modulus (~30 GPa for crustal rock), $\\bar{D}$ is the average coseismic slip, and $L$ is the fault dimension (radius for a circular rupture, length for a rectangular one).
A remarkable observation in seismology is that earthquake stress drops are approximately constant at ~1–10 MPa across a huge range of earthquake magnitudes (from Mw −2 laboratory events to Mw 9 great earthquakes). This "constant stress drop" scaling implies that slip is proportional to rupture dimension ($\bar{D} \propto L$), which underlies the self-similar scaling of earthquake source spectra.
~3 MPa
Median Stress Drop (intraplate)
~1 MPa
Median Stress Drop (interplate)
10−4–10−3
Typical Strain Drop ($\bar{D}/L$)
Key Parameters Summary
| Parameter | Symbol | Typical Value | Significance |
|---|---|---|---|
| Byerlee friction (low $\sigma_n$) | $\mu$ | 0.85 | Universal rock friction |
| Direct effect | a | 0.005–0.015 | Instantaneous velocity effect |
| Evolution effect | b | 0.005–0.020 | State-dependent friction evolution |
| Critical slip distance | Dc | 1–100 µm | Length scale for state evolution |
| Stress drop | $\Delta\sigma$ | 1–10 MPa | Approximately scale-invariant |
| Seismogenic temperature | Tmax | ~350°C | Base of seismogenic zone (quartz) |