Dynamic fracture deals with the degradation and failure of materials and structures under conditions where the inertia effect must be included in the analysis. In this contribution, a dynamic description of a simple local damage approach and its detailed finite element implementation is presented towards brittle and quasi-brittle localization failure under time-dependent loading conditions. To eliminate the issue of the mesh-dependence, the softening parameter of the introduced model is controlled through a numerical factor that is determined based on material constants and the element length. The developed local damage model integrated with the smooth Rankine equivalent strain form results in an effective approach as it offers a physically realistic description of the damage localization in quasi-brittle media. The governing equation of the nonlinear behavior, when damage is involved, is spatially discretized by a nonlinear finite element method, and is efficiently solved using the standard Newton–Raphson method, while the popular Newmark-beta time integration method is applied to its time discretization. We also present the estimation of the crack tip velocity and dissipated energy. To show the accuracy and performance of the developed local damage approach, we study the dynamic crack propagation, and fragmentation branching in materials with brittle and quasi-brittle fracture. The obtained results are compared with experimental data and other numerical solutions available in literature. In terms of computational efficiency, the developed approach requires much less computational time than other numerical models in the same class of the smeared damage approach, i.e., gradient-enhanced and phase-field models, since it just needs to solve a single equation of motion.