Cracking of rocks and rock-like materials exhibits a rich variety of patterns where tensile (mode I) and shear (mode II) fractures are often interwoven. These mixed-mode fractures usually involve microcracking and frictional sliding. Nevertheless, computational models that can efficiently simulate a combination of cohesive and frictional fractures remain scarce. In this work, we develop a double-phase-field formulation that employs two different phase fields to describe cohesive tensile fracture and frictional shear fracture individually. The formulation rigorously combines the two phase-fields through three approaches: (i) crack-direction-based decomposition of the strain energy into the tensile, shear, and pure compression parts, (ii) contact-dependent calculation of the potential energy, and (iii) energy-based determination of the dominant fracturing mode in each contact condition. The proposed model is validated, both qualitatively and quantitatively, with experimental data on mixed-mode fracture in rocks. A standout feature of the double phase-field model is that it can simulate, and naturally distinguish between, tensile and shear fractures without complex algorithms.

中文翻译：

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岩石中混合模式断裂的双相场公式

岩石和类岩石材料的开裂表现出丰富多样的模式，其中拉伸（模式 I）和剪切（模式 II）断裂通常交织在一起。这些混合模式裂缝通常涉及微裂纹和摩擦滑动。然而，能够有效模拟内聚和摩擦断裂组合的计算模型仍然很少。在这项工作中，我们开发了一种双相场公式，它使用两个不同的相场来分别描述内聚拉伸断裂和摩擦剪切断裂。该公式通过三种方法将两个相场严格结合起来：(i) 基于裂纹方向的应变能分解为拉伸、剪切和纯压缩部分，(ii) 势能的接触相关计算，(iii) 基于能量确定每种接触条件下的主要压裂模式。利用岩石中混合模式断裂的实验数据，对所提出的模型进行了定性和定量验证。双相场模型的一个突出特点是它可以模拟并自然区分拉伸和剪切断裂，而无需复杂的算法。