# Three Dimensional CS-FEM Phase-Field Modeling Technique for Brittle Fracture in Elastic Solids
## 1 Introduction
This method is based on the variational theory of fracture [24] addressing the shortcomings of the original [PERSON]'s theory, which could not predict crack nucleation or complex crack paths. The elegance of this method lies in the fact that it can successfully predict complex crack behaviors such as crack branching, curved crack paths, and crack merging even in three dimensions without any ad-hoc criterion. This gives it an immediate advantage over the traditional methods where the prediction of such phenomenons is quite complex. Apart from the primary applications of brittle fracture [28], the phase-field approach has been developed and applied to many complex fracture mechanics problems including large deformations [29], plasticity [30], multiphysics [31; 32], and dynamics [33; 34]. It has also been previously implemented in the commercial code ABAQUS by using its user subroutine features [27; 28]. The phase-field model, however, is not without its disadvantages, including having a very refined mesh in the expected crack propagation region. Advanced fracture algorithms such as the screened Poisson's equations [35; 36; 37] for crack propagation are developed to overcome such shortcomings. However, here we limit ourselves to the application of the phase-field method and discuss its useful features as well as disadvantages. In this work, we integrate the cell based S-FEM with phase-field model to simulate 3-D crack propagation. This integration is done on the commercial platform of ABAQUS. ABAQUS is one of the most widely used commercial codes and its excellent inbuilt solvers and sophisticated visualization tools are particularly attractive for implementing user developed element formulations. A CS-FEM formulation for the eight-node hexahedral elements (3D-CS-FEM-H8) is used, similar to the one used in [PERSON] et al. [38]. We consider each node to have an additional degree of freedom (phase-field) in addition to the standard displacements. The phase-field and the displacement variables are solved in a staggered fashion to attenuate instability [27]. The 3D CS-FEM-H8 has already been proven to have faster convergence rate and better accuracy than the standard FEM, thus its assimilation with the phase-field method which computes complex crack topologies without any ad-hoc criterion, produces another efficient technique to model crack propagation. One of the significant disadvantages of implementing this method in ABAQUS is that the software does not let users access data for any surrounding elements, thus we deviate from the classical way of calculating strains/stresses in CS-FEM [1; 38], and develop another novel, simple yet quite effective approach. It is noteworthy that this is also the reason for our inability to implement S-FEM models of higher accuracy into the software. However, with all these shortcomings, as demonstrated by a number of examples, this method proves to be quite an effective tool to predict complex crack behaviors. This paper is outlined as follows: In Section 2, we outline the phase-field model of fracture and the 3D-CS-FEM-H8, and discuss the implementation details of the combined method into ABAQUS. In Section 3, we provide numerical examples to validate our method. In Section 4, we conclude the paper with a summary of our findings and proposed future works.