Three-dimensional (3D) phase-field models based on the cell-based smooth finite element method (CS-FEM) are established in this paper. The present model consists of three key ingredients: 1) a phase-field model of fracture in elasto-plastic solids, 2) a staggered scheme for nonlinear interactions between the phase-field equations and those for the elasto-plastic solid mechanics, and 3) its underline solution platform of the cell-based smoothed finite element method, CS-FEM. The phase-field model allows effective capturing of the evolution process of complex fracture morphology in elasto-plastic solids. The use of the staggered algorithm gives a practical means to couple the phase field and displacement field in fracturing elasto-plastic solids. The CS-FEM offers important softer model behavior in solving the governing equations. For wide applicability of our Phase-field CS-FEM approach, it is implemented in the commercial software ABAQUS via User Defined Element (UEL) subroutine. Numerical examples of both 2D and 3D benchmark problems with complex crack topologies are presented to verify the accuracy of the solutions. Good agreements are obtained with the experimental observations and existing numerical results in the literature. The phase-field CS-FEM approach offers an effective alternative means to model 3D fracturing in elasto-plastic solids.

本文建立了基于单元的光滑有限元方法（CS-FEM）的三维（3D）相场模型。本模型由三个关键要素组成：1) 弹塑性固体中断裂的相场模型，2) 相场方程与弹塑性固体力学方程之间非线性相互作用的交错方案，以及 3 ) 其基于单元的平滑有限元方法 CS-FEM 的下划线求解平台。相场模型可以有效捕捉弹塑性固体中复杂断裂形态的演变过程。交错算法的使用提供了一种在弹塑性固体压裂中耦合相场和位移场的实用方法。CS-FEM 在求解控制方程时提供了重要的软模型行为。为了我们的相场 CS-FEM 方法的广泛适用性，它通过用户定义元素 (UEL) 子程序在商业软件 ABAQUS 中实现。给出了具有复杂裂纹拓扑结构的 2D 和 3D 基准问题的数值示例，以验证解决方案的准确性。与实验观察和文献中现有的数值结果取得了良好的一致性。相场 CS-FEM 方法提供了一种有效的替代方法来模拟弹塑性固体中的 3D 压裂。与实验观察和文献中现有的数值结果取得了良好的一致性。相场 CS-FEM 方法提供了一种有效的替代方法来模拟弹塑性固体中的 3D 压裂。与实验观察和文献中现有的数值结果取得了良好的一致性。相场 CS-FEM 方法提供了一种有效的替代方法来模拟弹塑性固体中的 3D 压裂。