Within the framework of Continuum Damage Mechanics some isotropic scalar damage models for concrete are revisited, with emphasis on a recent one proposed by the authors. This scalar damage model is based on the assumption that a stress split is required to capture the unilateral behavior exhibited by concrete when passing from tension to compression. Similar assumptions are pursued on many scalar damage models, yet with many differences being encountered on the strategies adopted for the implementation of such split, which sometimes is performed over the strain tensor. In this paper a discussion on the implications of those splits is conducted, as well as on the norms that define the elastic domain in the stress space. For the proposed damage model a strain-driven formalism is adopted, but the stress split of the Cauchy stress tensor. This strategy improves the algorithmic efficiency as much as required for the seismic analysis of large-scale problems, and circumvents many of the drawbacks present in similar damage models. Besides, two scalar damage variables are introduced as internal variables, as well as an inelastic strain tensor. Efficiency of the proposed constitutive model is illustrated through numerical applications. Algorithmic implementation is also detailed.