A Continuum Damage–Breakage Faulting Model and Solid-Granular Transitions

Abstract

We present a thermodynamically-based formulation for mechanical modeling of faulting processes in the seismogenic brittle crust using a continuum damage–breakage rheology. The model combines previous results of a continuum damage framework for brittle solids with continuum breakage mechanics for granular flow. The formulation accounts for the density of distributed cracking and other internal flaws in damaged rocks with a scalar damage parameter, and addresses the grain size distribution of a granular phase in a failure slip zone with a breakage parameter. The stress–strain relation and kinetics of the damage and breakage processes are governed by the total energy function of the system, which combines the energy of the damaged solid with the energy of the granular material. A dynamic brittle instability is associated with a critical level of damage in the solid, leading to loss of convexity of the solid energy function and transition to a granular phase associated with lower energy level. A non-local formulation provides an intrinsic length scale associated with the internal damage structure, which leads to a finite length scale for damage localization that eliminates the unrealistic singular localization of local models. Shear heating during deformation can lead to a secondary finite-width internal localization. The formulation provides a framework for studying multiple aspects of brittle deformation, including potential feedback between evolving elastic moduli and properties of the slip localization zone and subsequent rupture behavior. The model has a more general transition from slow deformation to dynamic rupture than that associated with frictional sliding on a single pre-existing failure zone, and gives time and length scales for the onset of the dynamic fracturing process. Several features including the existence of finite localization width and transition from slow to rapid dynamic slip are illustrated using numerical simulations. A configuration having an existing narrow slip zone with localized damage produces for appropriate loading conditions an overall cyclic stick–slip motion. The simulated frictional response includes transitions from friction coefficient of ~0.7 at low slip velocity to dynamic friction below 0.4 at slip rates above ~0.1 m/s, followed by rapidly increasing friction for slip rates above ~1 m/s, consistent with laboratory observations.

Appendix: Thermodynamic Formulation

The total energy of a solid with a unit mass includes internal and kinetic components:

$$ E = E_{\text{k}} + U. $$

(46)

The specific kinetic energy of the solid is $E_{\text{k}} = v_{i} v_{i} /2$ with v _i being velocity and the specific internal energy, U, is expressed through the specific free energy F, temperature T and entropy S as U = F + TS. The energy balance equation dictates that the change in the energy of a system consists of a sum of different terms representing different energy forms. These terms are the total stress power, $\sigma_{ij} e_{ij}$, divergence of the heat flux $J_{i}^{(q)}$, internal heat source per unit mass, r, and interstitial work flux $J_{i}^{(i)}$:

$$ \rho \frac{{{\text{d}}U}}{{{\text{d}}t}} = \rho \frac{\text{d}}{{{\text{d}}t}}\left( {F + TS} \right) = \sigma_{ij} e_{ij} - \nabla_{i} J_{i}^{(q)} + \rho r - \nabla_{i} J_{i}^{(i)} . $$

(47)

The entropy balance equation includes entropy flux $J_{i}^{(s)}$, internal heat source, and non-negative local entropy production Γ:

$$ \rho \frac{{{\text{d}}S}}{{{\text{d}}t}} = - \nabla_{i} J_{i}^{(s)} + \frac{\rho r}{T} + \Upgamma ,\quad \Upgamma \ge 0. $$

(48)

The non-negative local entropy production results from all the dissipative irreversible processes in the medium including internal friction and damage evolution. From Eq. (7) of the main text, the change in the free energy, Gibbs equation, can be expressed as:

$$ {\text{d}}F = - S{\text{d}}T + \frac{\partial F}{{\partial \varepsilon_{ij} }}{\text{d}}\varepsilon_{ij} + \frac{\partial F}{\partial \alpha }{\text{d}}\alpha + \frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}{\text{d}}\left( {\nabla_{i} \alpha } \right) + \frac{\partial F}{\partial B}{\text{d}}B. $$

(49)

In a system with internal motions, the material time derivative is given by ${\text{d(}} \cdot )/{\text{d}}t = \partial (\cdot )/\partial t + v_{i} \nabla_{i} (\cdot )$. Similarly,

$$ \nabla_{i} \left( {\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}} \right) = \nabla_{i} \left( {\frac{\partial \alpha }{\partial t} + v_{k} \nabla_{k} \alpha } \right) = \frac{{{\text{d}}\left( {\nabla_{i} \alpha } \right)}}{{{\text{d}}t}} + \left( {\nabla_{i} v_{k} } \right)\nabla_{k} \alpha . $$

(50)

From (50), the time derivative of the last term in (49) can be written as:

$$ \frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}\frac{{{\text{d}}\left( {\nabla_{i} \alpha } \right)}}{{{\text{d}}t}} = - \left[ {\nabla_{i} \left( {\frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}} \right)\frac{{{\text{d}}\alpha }}{{{\text{d}}t}} + e_{ik} \frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}\nabla_{k} \alpha } \right] + \nabla_{i} \left( {\frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}} \right). $$

(51)

Combining the above equations, the non-negative energy dissipation ($D_{\text{E}} \ge 0$) is represented as the sum of the local entropy production and interstitial work flux:

$$ \begin{aligned} D_{\text{E}} & = T\varGamma + \nabla_{i} J_{i}^{(i)} = \left( {\sigma_{ij} - \rho \frac{\partial F}{{\partial \varepsilon_{ij} }} + \rho \frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}\nabla_{j} \alpha } \right)e_{ij} + T\nabla_{i} J_{i}^{(s)} - \nabla_{i} J_{i}^{(q)} \\ & \quad - \rho \nabla_{i} \left( {\frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}} \right) + \rho \frac{\partial F}{{\partial \varepsilon_{ij} }}\frac{{\partial \varepsilon_{ij}^{(p)} }}{\partial t} - \rho \left( {\frac{\partial F}{\partial \alpha } - \nabla_{i} \frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}} \right)\frac{{{\text{d}}\alpha }}{{{\text{d}}t}} - \rho \frac{\partial F}{\partial B}\frac{{{\text{d}}B}}{{{\text{d}}t}}. \\ \end{aligned} $$

(52)

If all the dissipative processes in the system are frozen, the local entropy production and interstitial working are zero. This condition leads to the definition of the stress tensor:

$$ \sigma_{ij} = \rho \left[ {\frac{\partial F}{{\partial \varepsilon_{ij} }} - \frac{1}{2}\left( {\frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}\nabla_{j} \alpha + \frac{\partial F}{{\partial \left( {\nabla_{j} \alpha } \right)}}\nabla_{i} \alpha } \right)} \right]. $$

(53)

In addition to the usual term $\partial F/\partial \varepsilon_{ij}$ (e.g., Malvern, 1969), there are terms associated with $\nabla \alpha$, or “structural stresses”. These stress components are associated with heterogeneous damage distribution and can be important in zones with high $\nabla \alpha$ values.

The presence of the extra flux $J_{i}^{(i)}$ in (47) requires a posteriori definition of the energy partitioning between local entropy production converted into heat and heat-free interstitial working. We assume that the two components of the energy dissipation involving conductive heat transport and irreversible deformation are associated with heat, while the interstitial working associated with the damage–breakage evolution does not affect the heat balance of the system. This means that the local entropy production is:

$$ \Upgamma = {\Upgamma}_{\text{H}} + {\Upgamma}_{\text{V}}. $$

(54)

Using the definition $J_{i}^{(s)} = J_{i}^{(q)} /T$, the term related to the conductive heat transport is:

$$ {\Upgamma}_{\text{H}} = - \frac{{J_{i}^{(q)} \cdot \nabla_{i} T}}{{T^{2} }}, $$

(55)

and the term related to the irreversible deformation is:

$$ {\Upgamma}_{\text{V}} = \frac{\rho }{T}\frac{\partial F}{{\partial \varepsilon_{ij} }}\frac{{{\text{d}}\varepsilon_{ij}^{(p)} }}{{{\text{d}}t}}. $$

(56)

Non-negativity of the local entropy production gives rise to the Fourier law for the thermal conductivity

$$ J_{i}^{(q)} = - k \cdot \nabla_{i} T, $$

(57)

where k is thermal conductivity. The non-negativity of the second term related to the irreversible deformation also known as shear heating, requires positively defined scalar or tensor of the effective viscosity in the constitutive relations connecting stress with the rate of irreversible strain accumulation. Finally, the heat balance equation controlling the rate of the temperature change includes two usual terms, i.e., conductive heat transport and shear heating:

$$ \rho C_{\text{p}} \frac{\partial T}{\partial t} = k\nabla^{2} T + \rho \frac{\partial F}{{\partial \varepsilon_{ij} }}\frac{{{\text{d}}\varepsilon_{ij}^{(p)} }}{{{\text{d}}t}}. $$

(58)

The interstitial working flux associated with damage–breakage evolution is defined as:

$$ \nabla_{i} J_{i}^{(i)} = - \rho \left[ {\left( {\frac{\partial F}{\partial \alpha } - \nabla_{i} \frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}} \right)\frac{{{\text{d}}\alpha }}{{{\text{d}}t}} + \frac{\partial F}{\partial B}\frac{{{\text{d}}B}}{{{\text{d}}t}}} \right] - \rho \nabla_{i} \left( {\frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}} \right). $$

(59)

Since the heat conduction, shear heating and interstitial working are independent physical processes, each should provide non-negative energy dissipation. This implies that the interstitial working is also non-negative ($\nabla_{i} J_{i}^{(i)} \ge 0$). The divergence term on the right side of (59) is associated with transport of the heterogeneous damage production. Being integrated over the whole system volume, according to Gauss theorem, its impact is equal to the flux through the external boundary of the system. This quantity should be defined as an additional boundary condition in the non-local system. Therefore, the first term in (59) standing for the energy dissipation associated with the damage–breakage evolution should be non-negative. Adopting Onsager (1931) reciprocal relations between thermodynamic forces and fluxes, we write phenomenological equations for the kinetics of the state variables α and Β as a set of two coupled differential equations:

$$ \begin{aligned} & \frac{{{\text{d}}B}}{{{\text{d}}t}} = C_{\text{BB}} \frac{\partial F}{\partial B} + C_{{{\text{B}\upalpha }}} \left( {\frac{\partial F}{\partial \alpha } - \nabla_{i} \frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}} \right) \\ & \frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = C_{{{\upalpha \text{B}}}} \frac{\partial F}{\partial B} + C_{\upalpha \upalpha } \left( {\frac{\partial F}{\partial \alpha } - \nabla_{i} \frac{\partial F}{{\partial \left( {\nabla_{i} \alpha } \right)}}} \right). \\ \end{aligned} $$

(60)

The kinetic coefficients (or functions), C _BB, C _Bα, C _αB, and C _αα, connect the thermodynamic forces associated with certain state variable with thermodynamic fluxes. The diagonal terms with C _BB and C _αα are associated with a given state variable, while the off-diagonal terms with C _αB and C _Bα provide coupling between evolving damage with breakage and vice versa.