The extended finite element method (X-FEM) is introduced for modeling cohesive crack growth in materials. This method allows for the simulation of crack propagation without requiring remeshing, as it can handle arbitrary discontinuities in the mesh. The key idea is to incorporate discontinuous functions into the finite element approximation, enabling the modeling of cracks that do not conform to the mesh. The method is based on the partition of unity property of finite elements, which allows for the local enrichment of the finite element approximation. The cohesive crack model is governed by a traction-displacement relationship across the crack faces near the tip. The growth of the cohesive zone is determined by requiring the stress intensity factors at the tip of the cohesive zone to vanish. This approach avoids the need to evaluate stresses at the mathematical tip of the crack, which is a common issue in traditional finite element methods. The X-FEM formulation is applied to simulate cohesive crack growth in concrete. The method is validated through numerical studies, including simulations of a three-point bending beam and a four-point shear specimen. The results show that the X-FEM approach is more accurate and efficient than traditional finite element methods, requiring fewer elements per characteristic length of the material. The method also avoids remeshing as the crack propagates, making it suitable for modeling arbitrary crack paths. The X-FEM approach is particularly effective in capturing the size effect in concrete, where the load-deflection curves are influenced by the size of the specimen. The method is also capable of simulating snap-back behavior, which is a common phenomenon in cohesive crack growth. The results demonstrate that the X-FEM method provides accurate and reliable predictions of crack propagation and the associated mechanical behavior of materials. The method is also robust in handling different cohesive laws and is suitable for both linear and nonlinear material behavior. Overall, the X-FEM method offers a powerful tool for modeling cohesive crack growth in a wide range of materials and applications.The extended finite element method (X-FEM) is introduced for modeling cohesive crack growth in materials. This method allows for the simulation of crack propagation without requiring remeshing, as it can handle arbitrary discontinuities in the mesh. The key idea is to incorporate discontinuous functions into the finite element approximation, enabling the modeling of cracks that do not conform to the mesh. The method is based on the partition of unity property of finite elements, which allows for the local enrichment of the finite element approximation. The cohesive crack model is governed by a traction-displacement relationship across the crack faces near the tip. The growth of the cohesive zone is determined by requiring the stress intensity factors at the tip of the cohesive zone to vanish. This approach avoids the need to evaluate stresses at the mathematical tip of the crack, which is a common issue in traditional finite element methods. The X-FEM formulation is applied to simulate cohesive crack growth in concrete. The method is validated through numerical studies, including simulations of a three-point bending beam and a four-point shear specimen. The results show that the X-FEM approach is more accurate and efficient than traditional finite element methods, requiring fewer elements per characteristic length of the material. The method also avoids remeshing as the crack propagates, making it suitable for modeling arbitrary crack paths. The X-FEM approach is particularly effective in capturing the size effect in concrete, where the load-deflection curves are influenced by the size of the specimen. The method is also capable of simulating snap-back behavior, which is a common phenomenon in cohesive crack growth. The results demonstrate that the X-FEM method provides accurate and reliable predictions of crack propagation and the associated mechanical behavior of materials. The method is also robust in handling different cohesive laws and is suitable for both linear and nonlinear material behavior. Overall, the X-FEM method offers a powerful tool for modeling cohesive crack growth in a wide range of materials and applications.