(eng): Quasistatic rate-independent damage combined with linearized plasticity with hardening at small strains is investigated. Fractional-step time discretization is devised with the purpose of obtaining a numerically efficient scheme, possibly converging to a physically relevant stress-driven solution, which however is to be verified a posteriori using a suitable integrated variant of the maximum-dissipation principle. Gradient theories both for damage and for plasticity are considered to make the scheme numerically stable with guaranteed convergence within the class of weak solutions. After finite-element approximation, this scheme is computationally implemented and illustrative 2-dimensional simulations are performed.