Fracture is a very challenging and complicated problem with various applications in engineering and physics. Although it has been extensively studied within the context of mesh-based numerical techniques, such as the finite element method (FEM), the research activity within the Smoothed Particle Hydrodynamics (SPH) community remains scarce. SPH is a particle-based numerical method used to discretize equations of continuum media. Its meshfree nature makes it ideal to simulate fracture scenarios that involve extreme deformations. However, to model fracture, SPH researchers have mostly relied on ad-hoc empirical local damage models, cohesive zone approaches, or pseudo-spring models, which come with a set of drawbacks and limitations. On the other hand, phase field models of brittle fracture have recently gained popularity in academic circles and provide significant improvements compared to previous approaches. These improvements include the derivation from fundamental fracture theories, the introduction of non-locality, and the ability to model multiple crack initiation, propagation, branching, and coalescence, in situations where no prior knowledge of the crack paths is available. Nevertheless, phase field for fracture has not been studied within SPH. In this proof-of-concept paper we develop and implement a phase field model of brittle fracture within the context of SPH. Comprehensive mathematical and implementation details are provided, and several challenging numerical examples are computed and illustrate the proposed method’s ability to accurately and efficiently simulate complex fracture scenarios.

断裂是一个非常具有挑战性和复杂性的问题，在工程和物理学中具有各种应用。尽管它已在基于网格的数值技术（例如有限元法 (FEM)）的背景下进行了广泛研究，但平滑粒子流体动力学 (SPH) 社区内的研究活动仍然很少。SPH 是一种基于粒子的数值方法，用于离散连续介质方程。它的无网格特性使其非常适合模拟涉及极端变形的断裂场景。然而，为了模拟断裂，SPH 研究人员主要依赖临时经验局部损伤模型、内聚区方法或伪弹簧模型，这些模型具有一系列缺点和局限性。另一方面，脆性断裂的相场模型最近在学术界获得了普及，并且与以前的方法相比提供了显着的改进。这些改进包括从基本断裂理论推导，引入非局部性，以及在没有先验裂纹路径知识的情况下模拟多个裂纹萌生、扩展、分支和聚结的能力。然而，在 SPH 中尚未研究断裂的相场。在这篇概念验证论文中，我们在 SPH 的背景下开发并实施了脆性断裂的相场模型。提供了全面的数学和实施细节，并计算了几个具有挑战性的数值示例，并说明了所提出的方法准确有效地模拟复杂裂缝场景的能力。