The modeling of concrete has always been of great interest. In this work, a yield criterion with three surfaces is proposed to capture the plastic yielding behavior under different loading conditions, including uniaxial tension/compression, biaxial tension/compression and tension-compression. Material failure, characterized by the degradation and finally the complete loss of material integrity, can be modeled by the continuum damage approach. Within finite element methods, the nonlocal enhancement of integral-type or gradient-type is often required for the well-posedness of the partial differential equation system. In this work, a localizing gradient damage model has been adopted to obtain mesh-insensitive material responses, while ruling out the unphysical broadening of the damage zone often observed in constant length scale gradient damage models. Following a consistent derivation, the plasticity-damage coupled model has been implemented into an in-house finite element framework. Several representative and demonstrative examples serve to illustrate the capability of the proposed description in concrete modeling, followed by the conclusion, where some final insights are provided.