Abstract
Earthquake fault zones are more complex, both geometrically and rheologically, than an idealized infinitely thin plane embedded in linear elastic material. To incorporate nonlinear material behaviour, natural complexities and multi-physics coupling within and outside of fault zones, here we present a first-order hyperbolic and thermodynamically compatible mathematical model for a continuum in a gravitational field which provides a unified description of nonlinear elasto-plasticity, material damage and of viscous Newtonian flows with phase transition between solid and liquid phases. The fault geometry and secondary cracks are described via a scalar function ξ ∈ [0, 1] that indicates the local level of material damage. The model also permits the representation of arbitrarily complex geometries via a diffuse interface approach based on the solid volume fraction function α ∈ [0, 1]. Neither of the two scalar fields ξ and α needs to be mesh-aligned, allowing thus faults and cracks with complex topology and the use of adaptive Cartesian meshes (AMR). The model shares common features with phase-field approaches, but substantially extends them. We show a wide range of numerical applications that are relevant for dynamic earthquake rupture in fault zones, including the co-seismic generation of secondary off-fault shear cracks, tensile rock fracture in the Brazilian disc test, as well as a natural convection problem in molten rock-like material.This article is part of the theme issue ‘Fracture dynamics of solid materials: from particles to the globe’.
Highlights
Multiple scales, multi-physics interactions and nonlinearities govern earthquake source processes, rendering the understanding of how faults slip a grand challenge of seismology [1,2]
We emphasize that the presented diffuse interface approach is not merely a way to regularize otherwise singular problems as posed by earthquake shear crack nucleation and propagation along zero-thickness interfaces, but potentially allows us to fully model volumetric fault zone shearing during earthquake rupture, which includes spontaneous partition of fault slip into intensely localized shear deformation within weaker fault-core gouge and more distributed damage within fault rocks and foliated gouges
We show that the continuum model can reproduce and extend classical solutions, while introducing dynamic differences (i) on the scale of pre-damaged/low-rigidity fault zone, such as out-of-plane rupture rotation, limiting peak slip rates, non-frictional control of rupture speed; and (ii) on the scale of the intact host rock, such as conjugate shear cracking in tensile lobes and amplification of velocity pulses in the emitted wavefield
Summary
Multi-physics interactions and nonlinearities govern earthquake source processes, rendering the understanding of how faults slip a grand challenge of seismology [1,2]. As in phase-field approaches, a crack or failure front is represented not as a discontinuity of zero thickness but as a diffuse interface across which ξ changes continuously from 0 (intact material) to 1 (fully damaged material) resulting in gradual but rapid degradation of material stiffness. Despite this conceptual similarity, the model developed here is very different from the phasefield models. In the first-order hyperbolic diffuse interface framework presented here, this can be achieved by taking into account higher gradients of the state variables such as curvature and torsion in the form of independent state variables [38,39]
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