The capability of the explicit numerical methods to simulate accurately the real cutting process is investigated in this research work. Smoothed particle hydrodynamics - SPH, classical Lagrangian finite element method - FEM and Multi-Material Arbitrary Lagrangian Eulerian - ALE methods are chosen for the modeling and simulation of the orthogonal metal cutting process of AISI H13 in LS-DYNA. The cutting tool is modeled as a rigid FEM body that incrementally penetrates into the flexible deformable workpiece. At each numerical model, the dynamic elastoplastic behavior of the workpiece material is investigated by taking into account the Johnson-Cook (J-C) constitutive strength material model. The influence of the J-C parameter values found in literature to the models is explored. The obtained numerical SPH, FEM and ALE results of the estimated cutting and thrust forces, stress, plastic strain and thermal distributions are compared with results found in the literature. This comparison, leads to valuable conclusions for the performance of the three methods, concerning the approximation accuracy, model development complexity and computational time demands. Based on these conclusions the SPH method is chosen to simulate the experimentally performed orthogonal cut of AISI 1045. The obtained SPH numerical results outline its advantages among the other explicit simulation methods.