# Numerical Modeling of Crack Growth under Mixed-Mode Loading
## 1 Introduction  
Fracture mechanics is an efficient tool to optimize mechanical component performance, as failure and fracture are frequent industrial equipment problems. In reality, with regards to the ultimate load capability and progression of pre-existing cracks, computational methods are commonly used to assess the durability of cracked structures. The arbitrariness of the crack trajectory, however, requires appropriate computational methodologies, since the computational instability mechanisms influence the structural dynamic by decreasing strength and speeding up the mechanisms of crack propagation. Many of the simulations frameworks available in the literature are utilized using extended finite element method (XFEM) because of its capability to simulate complicated components, providing accuracy in the estimation of interfacial parameters among various materials. Numerical analyses such as finite element analysis have become an incredibly useful method for modeling the propagation of a crack and for determining associated parameters such as the energy release rate and stress intensity factors (SIFs). Over the last few decades, improvements to the finite element method (FEM) have continuously been made to improve the handling of complex material models and experimental conditions. During the meantime, new approaches and methods in several areas of study have been suggested and developed rapidly, including the FEM, Discrete Element Method (DEM) [1, 2, 3], Element Free Galerkin (EFG) method [4], XFEM [5, 6, 7], cohesive element method [8, 9], and phase-field method [10]. Similarly to the XFEM, [11] proposes a novel implementation of the phase-field regularized cohesive zone model (PF-CZM), in which the additional degrees of freedom characterizing crack behavior are only associated with nodes within a small sub-domain. Similarities and differences between the gradient-enhanced damage models and phase-field models were discussed in details by [12]. In several applications, understanding interface mechanics is essential for determining and predicting overall material behavior. It is possible to capturethe complex behavior of any interface by introducing a quantity called traction density proposed by [12]. Cracks created in structural parts by cyclic loads can move directly or change their direction according to the loading mode. A variety of commercial tools for pre-defined cracks have been developed in either constant or variable amplitude load conditions to evaluate the planar load mode of cracks, for example, AFGROW software for aerospace applications [13] and NASGRO software for fracture and fatigue crack growth analysis [14], which employ handbook solution for the SIFs and suitable fatigue crack growth (FCG) rate equations in the literature. Industrial structures, however, are mainly affected by mixed-mode load conditions that generate non-planar cracking path pathways. The above-mentioned commercial tools do not apply to the specimen experiencing mixed-mode loading. This means that the software modeling the realistic crack growth path and predicting fatigue life in a mixed-mode are in need for the industrial sector. In the case of loading in a mixed-mode, a better approximation of cyclic life requires a cyclic summation method based on an equivalent SIF, which is able to cover both the formation of arbitrary crack types and the effects of load interactions. A Finite Element methodology called XFEM has been introduced by [PERSON] and [PERSON] [15] to address complications in FEM crack initiation and propagation simulations. In this process, during the crack growth stages, there is no need to re-mesh, reducing the required time for computation and associated error while updating the mesh and mapping the structure. In several works, XFEM was validated and can be accurately designed to model discontinuities such as cracks [16; 17]. The implementation of XFEM using ANSYS can accurately model stationary cracks as well as growing cracks.