# Three Dimensional CS-FEM Phase-Field Modeling Technique for Brittle Fracture in Elastic Solids
## 1 Introduction  
In the past decade, [PERSON] et al. [1; 2] generalized the gradient smoothing approach in meshfree method [3] and proposed the smoothed finite element method(S-FEM) to overcome some of the inherent shortcomings of the classical finite element method(FEM) such as overly stiff behavior, sensitivity to mesh distortions, and stress inaccuracy. The SFEM combines the FEM with the traditional meshfree methods in producing more accurate results with higher efficiency [1; 2]. It uses the base mesh of the FEM and reconstructs the strain field using the gradient smoothing technique. Thus, in contrast to the weak formulations of the FEM [4], the weakened weak formulation used here [1; 2] further softens the model [5], giving it a closer to exact stiffness [6]. The gradient smoothing operation facilitates creation of various smoothing domains based on elements, nodes, edges (2D), and faces (3D) over which the stress/strain is evaluated [1]. This, in turn, produces a wide variety of results and gives the analyst much needed freedom to design models as per the requirement. For example, the node based smoothing domains (NS-FEM) produce upper bound solutions for force driven problems [7]. The edge/face based smoothing domains (ES/FS-FEM) produce very accurate results using even triangular and tetrahedral mesh [8]. A combination of the NS-FEM and the ES-FEM produces very accurate close to exact solutions in the error norm [6; 9]. Amongst other applications [10; 11; 12; 13; 14], the S-FEM has been effectively applied to fracture problems for quasi-static crack propagation [15], anisotropic materials [16], dynamic fracture [1] and for singular geometries of arbitrary order [17; 18]. Although these studies have proven to be quite accurate and efficient when compared to the FEM, they treat the crack as an discrete entity, a formulation which has its own inherent problems. Fracture is the primary cause of failure in the majority of engineering structures. An initially existing small crack often leads to catastrophic failures by propagation under external loads. Over the past few decades, several theories and computational techniques have been developed to accurately model fracture and predict crack propagation in engineering systems. Classically, cracks are dealt with discretely in theoretical/computational fracture mechanics. The discontinuities have been separately modeled and special methodologies have been developed to accurately model the singular stress field near the crack tip. The collapsed quadrilateral or triangular element in the FEM [4] is the most widely used, however several other methods such as the boundary element method (BEM) [19], mesh-free methods [3; 20], the extended finite element method [21], and SFEM [18; 22; 23] have proven to be equally efficient. BEM uses a mesh only on the boundaries, thereby reducing the computational complexity by one order, but is unable to treat material non-linearities. S-FEM uses a base mesh of linear elements with an enrichment only around the crack-tip, thereby mitigating the computational cost of using a quadratic mesh throughout the domain [18]. However, in S-FEM, as in the classical FEM, the crack path is mesh dependent; cracks can only propagate along edges of elements and the tracking of crack front for three-dimensional problems is computationally very expensive. The X-FEM overcomes this mesh dependency but uses additional degrees of freedom and the integration in the weak form for the elements containing the crack becomes particularly very complex in three-dimensional fracture problems. An alternate way of modeling fracture has gained popularity in the computational community in the past decade or two. In this method, the crack is modeled as a diffused entity and represented by a continuous scalar variable called the phase-field [24]. This variable differentiates between the broken and intact material phases and, contrary to the discrete approach, provides a smooth transition between them [25; 26; 27; 28].