Fault Wear by Damage Evolution During Steady-State Slip

Abstract

Slip along faults generates wear products such as gouge layers and cataclasite zones that range in thickness from sub-millimeter to tens of meters. The properties of these zones apparently control fault strength and slip stability. Here we present a new model of wear in a three-body configuration that utilizes the damage rheology approach and considers the process as a microfracturing or damage front propagating from the gouge zone into the solid rock. The derivations for steady-state conditions lead to a scaling relation for the damage front velocity considered as the wear-rate. The model predicts that the wear-rate is a function of the shear-stress and may vanish when the shear-stress drops below the microfracturing strength of the fault host rock. The simulated results successfully fit the measured friction and wear during shear experiments along faults made of carbonate and tonalite. The model is also valid for relatively large confining pressures, small damage-induced change of the bulk modulus and significant degradation of the shear modulus, which are assumed for seismogenic zones of earthquake faults. The presented formulation indicates that wear dynamics in brittle materials in general and in natural faults in particular can be understood by the concept of a “propagating damage front” and the evolution of a third-body layer.

Appendix

1.1 Damage Rheology Model

Key derivations of the damage rheology model leading to the estimate of the speed of the propagating damage front are presented in this “Appendix”. Following thermodynamic balance relations and the Onsager (1931) principle, Lyakhovsky et al. (1997) developed a kinetic equation for the damage state variable, α (weakening and healing) which is a function of the progressive deformation. Non-linear elasticity that connects the effective elastic moduli to a damage variable and loading conditions allows accounting for the transition from damage accumulation to healing. This transition is controlled by the strain invariants ratio $\xi = {{I_{1} } \mathord{\left/ {\vphantom {{I_{1} } {\sqrt {I_{2} } }}} \right. \kern-0pt} {\sqrt {I_{2} } }}$, where I ₁ = ɛ _kk and I ₂ = ɛ _ij ɛ _ij are the invariants of the elastic strain tensor ɛ _ij . The ξ value is a conjugate quantity to the ratio between shear and normal stress expressed in terms of strains, instead of stresses. The rate of damage/healing accumulation is given by Lyakhovsky et al. (1997):

$$ \frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = \left\{ {\begin{array}{*{20}c} {C_{\text{d}} I_{2} \left( {\xi - \xi_{0} } \right)} & {{\text{for }}\xi > \xi_{0} } & {} \\ {C_{1} \exp \left( {\frac{\alpha }{{C_{2} }}} \right)I_{2} \left( {\xi - \xi_{0} } \right)} & {{\text{for }}\xi < \xi_{0} } & {} \\ \end{array} } \right.\!\!\!\!. $$

(4)

where the coefficient C _d is the rate of positive damage evolution (material degradation) that is constrained by laboratory experiments (Lyakhovsky et al. 1997; Hamiel et al. 2004, 2006). The value ξ = ξ ₀ controls the onset of damage accumulation or transition from material healing to weakening associated with microcrack nucleation and growth. The value of the critical strain invariants ratio also called “modified internal friction” (see Fig. 3 in Lyakhovsky et al. 1997) is explicitly related to the internal friction angle of Byerlee’s law (Byerlee 1978) and Poisson ratio of the intact rock.

The rate of damage recovery (healing) is assumed to depend exponentially on α and produces logarithmic healing with time in agreement with the behavior observed in laboratory experiments with rocks and other materials (e.g., Dieterich and Kilgore 1996; Scholz 2002; Muhuri et al. 2003; Johnson and Jia 2005). Lyakhovsky et al. (2005) showed that the local damage model reproduces the main phenomenological features of the rate- and state-dependent friction, and constrained the healing parameters C ₁, C ₂ by comparing the model calculations with empiric parameters of the slow-rate frictional sliding (e.g., Dieterich 1972, 1979; Marone 1998).

The damage accumulation under constant stress in a simplified 1-D model with effective elastic moduli degrading proportionally to (1 − α) follows a power law solution (e.g., Ben-Zion and Lyakhovsky 2002; Turcotte et al. 2003):

$$ \alpha (t) = 1 - \left( {1 - 3\frac{{C_{\text{d}} \sigma^{2} }}{{G_{0}^{2} }}t} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} . $$

(5)

where G ₀ is the elastic moduli of the intact rock, σ, the applied stress, and C _d the damage rate coefficient. This solution allows us to introduce a time scale, t _f, which is the time-to-failure when α = 1 (total damage) (e.g., Paterson and Wong 2005) that becomes

$$ t_{f} = \frac{{G_{0}^{2} }}{{3C_{d} \sigma^{2} }}. $$

(6)

This parameter controls the time scale of all processes associated with accumulated damage, including the growth rate of narrow fracture zones.

Recently, Lyakhovsky et al. (2011) developed a gradient-type damage rheology formulation that incorporates non-local behavior by enriching the local constitutive relations with a gradient of the damage state variable. This addition modifies the kinetic equation for the damage evolution. In addition to the source term controlling the damage growth in Eq. (4), the damage accumulation for ξ > ξ ₀ in non-local formulation includes damage diffusion term with a coefficient D:

$$ \frac{\partial \alpha }{\partial t} = C_{\text{d}} I_{2} \left( {\xi - \xi_{0} } \right) + D\nabla^{2} \alpha . $$

(7)

The non-local gradient-type damage kinetic Eq. (7) has the form of a Fisher-KPP reaction–diffusion type equation for which the general one-dimensional form is

$$ \frac{\partial u}{\partial t} = \frac{{\partial^{2} u}}{{\partial x^{2} }} + f(u) $$

(8)

The solution of the Fisher-KPP Eq. (8) exhibits a traveling wave or fronts switching between equilibrium states f(u) = 0 (see text). Travelling fronts propagating with the speed c exist in homogeneous media when $c \ge c_{*} \equiv 2\sqrt {f^{\prime}(0)}$. We calculate the source term f(α) = C _d I ₂(ξ − ξ ₀) for α = 0 under loading condition that mimics the experimental set-up with the normal stress σ _n and shear stress τ, and then estimate the speed of the propagating damage front.

Adopting damage model with (1 − α) reduction of the elastic moduli, the relations between stress and strain components (axial, ε _a, and transversal, ε _τ ) are (E ₀, G ₀, Young and shear moduli of the intact rock):

$$ \varepsilon_{\text{a}} = \frac{{\sigma_{\text{n}} }}{{E_{0} \left( {1 - \alpha } \right)}};\,\varepsilon_{\tau } = \frac{\tau }{{2G_{0} \left( {1 - \alpha } \right)}} $$

(9)

Using relations (9), the second strain invariant, I ₂, and strain invariant ratio, ξ, are,

$$ I_{2} = \varepsilon_{\text{a}}^{2} + 2\varepsilon_{\tau }^{2} = \frac{1}{{\left( {1 - \alpha } \right)^{2} }}\left[ {\frac{1}{{E_{0}^{2} }}\sigma_{\text{n}}^{2} + \frac{1}{{2G_{0}^{2} }}\tau^{2} } \right] $$

(10)

$$ \xi = \frac{{\varepsilon_{\text{a}} }}{{\sqrt {\varepsilon_{\text{a}}^{2} + 2\varepsilon_{\tau }^{2} } }} = \frac{ - 1}{{\sqrt {1 + 2\left( {\frac{{E_{0} \tau }}{{2G_{0} \sigma_{n} }}} \right)^{2} } }} $$

(11)

The minus sign in (11) implies that the compaction strains are negative. Note that without any shear loading (τ = 0), the strain invariants ratio is ξ = −1, which is slightly below typical ξ ₀ range (Lyakhovsky et al. 1997). This implies that damage is not accumulated at loading by normal stress alone without shear loading. Substituting (10, 11) into equation for the damage accumulation (7), leads to the following relation for the source term:

$$ f(\alpha ) = C_{\text{d}} \frac{1}{{\left( {1 - \alpha } \right)^{2} }}\left[ {\frac{1}{{E_{0}^{2} }}\sigma_{\text{n}}^{2} + \frac{1}{{2G_{0}^{2} }}\tau^{2} } \right]\left[ {\frac{ - 1}{{\sqrt {1 + 2\left( {\frac{{E_{0} \tau }}{{2G_{0} \sigma_{n} }}} \right)^{2} } }} - \xi_{0} } \right] $$

(12)

The speed of the travelling damage front, which in the present work is defined as the wear-rate (text), is controlled by the value $c_{*} = 2\sqrt {D \cdot f^{\prime}(0)}$ equal to

$$ c_{*} = 2\sqrt {2C_{\text{d}} D} \frac{{\sigma_{\text{n}} }}{{E_{0} }}\sqrt { - \xi_{0} \left[ {1 + 2\left( {\frac{{E_{0} \tau }}{{2G_{0} \sigma_{n} }}} \right)^{2} } \right] - \sqrt {1 + 2\left( {\frac{{E_{0} \tau }}{{2G_{0} \sigma_{n} }}} \right)^{2} } } $$

(13)

Taking μ _S = τ/σ _n for the steady-state friction coefficient, Eq. (13) is rearranged to:

$$ c_{*} = 2\sqrt {2C_{\text{d}} D} \frac{{\sigma_{\text{n}} }}{{E_{0} }}\sqrt { - \xi_{0} \left( {1 + A\mu_{\text{S}}^{2} } \right) - \sqrt {1 + A\mu_{\text{S}}^{2} } } $$

(14)

where A = 2(E ₀/2G ₀)² = 2(1 + ν)², and ν is the Poisson ratio. Note that typical values of the strain invariant ratio, ξ ₀, controlling the onset of damage accumulation in (4) vary between ξ ₀ = −1.2 and ξ ₀ = −0.6. These values are taken from Fig. 3 of Lyakhovsky et al. (1997) for friction angle 30°, ν = 0.2 and for friction angle 40°, ν = 0.3. The steady-state friction coefficient should be above certain critical or threshold values (μ _S ≥ μ _cr) related to the material strength to give positive expression under the radical in (14). A negative value for μ _S < μ _cr implies that the applied shear stress is below the level needed for the onset of damage accumulation and the source term in damage growth Eq. (7) is negative; no any damage is accumulated, and no wear is anticipated.

The Poisson ratio for most rocks varies within the small range ν = 0.2–0.3. Variation of the steady-state friction, μ _S, is also limited. It decreases from static friction values about 0.6–0.8 to dynamic values about 0.2–0.4. Accounting for this limited range of the material properties, the speed of the travelling damage front or wear-rate calculated using (14) only slightly depends on the Poisson ratio (black lines in Fig. 7). Hence, the wear-rate may be approximated by much simpler equation (red line in Fig. 7):

$$ {\text{wear\_rate}} \propto \sigma_{\text{n}} \sqrt {\mu_{\text{S}}^{2} - \mu_{\text{cr}}^{2} } $$

(15)

The units of the speed of the travelling damage front (13, 14) are length per time, while in the laboratory experiments wear-rate measured the thinning of the sample per unit slip. As the area of the surface is constant (Boneh et al. 2013), this thinning is proportional to the wear volume generation. Such unit conversion could be done under steady-state conditions with constant slip rate.

We note here that Eq. (14) was derived for the loading conditions of experimental set-up. More appropriate conditions for deep seismogenic zones should account for relatively large confining pressure, small damage-induced change of the bulk modulus and significant degradation of the shear modulus. Similar derivations lead directly to the more simple form (15) instead of (14) for the natural conditions.