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’.

地震断层带在几何和流变学上都比嵌入线弹性材料中的理想化无限薄平面更复杂。为了结合断层带内外的非线性材料行为、自然复杂性和多物理场耦合，我们在此提出了引力场连续体的一阶双曲线和热力学兼容数学模型，该模型提供了对非线性弹塑性的统一描述，材料损坏和粘性牛顿流动的固相和液相之间的相变。通过标量函数ξ描述断层几何形状和次生裂纹 ∈ [0, 1]，表示当地的材料损坏程度。该模型还允许通过基于固体体积分数函数α  ∈ [0, 1] 的扩散界面方法来表示任意复杂的几何形状。两个标量场ξ和α都不是需要进行网格对齐，从而允许具有复杂拓扑的故障和裂缝以及自适应笛卡尔网格 (AMR) 的使用。该模型与相场方法具有共同特征，但大大扩展了它们。我们展示了与断层带动态地震破裂相关的广泛数值应用，包括同震产生的次生断层剪切裂缝、巴西圆盘试验中的拉伸岩石断裂，以及自然对流问题熔岩状物质。

本文是主题“固体材料的断裂动力学：从粒子到地球”的一部分。