To study the dynamic behavior and impact damage and failure of concrete materials and structures, a dynamic damage model was reformulated under the framework of the non-local peridynamic theory in this paper. The stretch rate of material bonds, equivalent to the strain rate in classical continuum mechanics, was introduced, and a rate-dependent peridynamic model describing the dynamic damage and failure of concrete materials was then proposed, taking both the dynamic failure of bonds and the rate-sensitivity of damage evolution under different stretch rate into account. The connection between the stretch rate and strength and tensile and compressive failure of concrete materials was established in the peridynamic model. To verify the proposed dynamic damage model for concrete failure, several typical examples simulating the mixed-mode fracture of concrete specimen were investigated. The failure mode, crack propagation paths and the crack propagation speed in the specimen in different cases were captured, which agree well with the experimental results and available numerical results. After model validation, the effect of the number and length of pre-existing cracks on the dynamic failure of concrete components was investigated further.