The paper "Extended Finite Element Method for Cohesive Crack Growth" by Nicolas Moës and Ted Belytschko introduces an extension of the finite element method (X-FEM) to model cohesive crack growth without remeshing. The method is designed to handle displacement discontinuities that do not align with interelement surfaces, making it suitable for simulating the growth of arbitrary cohesive cracks. The authors focus on the energetic approach, where the stress intensity factors at the tip of the cohesive zone are set to zero, avoiding the need to evaluate stresses at the mathematical tip of the crack. The paper begins by discussing the limitations of linear elastic fracture mechanics (LEFM) and the need for alternative models to account for the fracture process zone (FPZ). It reviews various numerical approaches to modeling cohesive crack growth, including boundary integral and boundary element formulations, as well as meshless methods. The X-FEM is then presented as a flexible method that can model discontinuities in the displacement field along the crack path, allowing for crack growth without remeshing. The authors derive the variational formulation for the cohesive crack model, including the equilibrium, kinematic, and constitutive equations. They introduce the concept of a load factor that controls the growth of the cohesive zone, ensuring that the Mode I stress intensity factor (SIF) vanishes at the mathematical tip of the cohesive zone. The direction of crack propagation is determined using linear fracture theory, with criteria such as the principle of local symmetry and the maximum hoop stress criterion. The numerical implementation of the X-FEM is described, including the enrichment of the finite element approximation with jump and branch functions to model the crack discontinuity. The method is applied to two numerical studies: a three-point bending specimen and a four-point shear specimen. The results show that the X-FEM approach is more accurate than classical finite element methods with node release, requiring fewer elements per characteristic length of the material. The study also demonstrates the effectiveness of the method in simulating cohesive crack growth in concrete, including the size effect and snap-back phenomena. The paper concludes by highlighting the advantages of the X-FEM approach, particularly its ability to handle arbitrary crack locations without remeshing, and provides insights into the robustness and accuracy of the method.The paper "Extended Finite Element Method for Cohesive Crack Growth" by Nicolas Moës and Ted Belytschko introduces an extension of the finite element method (X-FEM) to model cohesive crack growth without remeshing. The method is designed to handle displacement discontinuities that do not align with interelement surfaces, making it suitable for simulating the growth of arbitrary cohesive cracks. The authors focus on the energetic approach, where the stress intensity factors at the tip of the cohesive zone are set to zero, avoiding the need to evaluate stresses at the mathematical tip of the crack. The paper begins by discussing the limitations of linear elastic fracture mechanics (LEFM) and the need for alternative models to account for the fracture process zone (FPZ). It reviews various numerical approaches to modeling cohesive crack growth, including boundary integral and boundary element formulations, as well as meshless methods. The X-FEM is then presented as a flexible method that can model discontinuities in the displacement field along the crack path, allowing for crack growth without remeshing. The authors derive the variational formulation for the cohesive crack model, including the equilibrium, kinematic, and constitutive equations. They introduce the concept of a load factor that controls the growth of the cohesive zone, ensuring that the Mode I stress intensity factor (SIF) vanishes at the mathematical tip of the cohesive zone. The direction of crack propagation is determined using linear fracture theory, with criteria such as the principle of local symmetry and the maximum hoop stress criterion. The numerical implementation of the X-FEM is described, including the enrichment of the finite element approximation with jump and branch functions to model the crack discontinuity. The method is applied to two numerical studies: a three-point bending specimen and a four-point shear specimen. The results show that the X-FEM approach is more accurate than classical finite element methods with node release, requiring fewer elements per characteristic length of the material. The study also demonstrates the effectiveness of the method in simulating cohesive crack growth in concrete, including the size effect and snap-back phenomena. The paper concludes by highlighting the advantages of the X-FEM approach, particularly its ability to handle arbitrary crack locations without remeshing, and provides insights into the robustness and accuracy of the method.