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In this work, we developed a bond-based cohesive peridynamics model (CPDM) and apply it to simulate inelastic fracture by using the meso-scale Xu-Needleman cohesive potential . By doing so, we have successfully developed a bond-based cohesive continuum mechanics model with intrinsic stress/strain measures as well as consistent and built-in macro-scale constitutive relations. The main novelties of this work are: (1) We have shown that the cohesive stress of the proposed nonlocal cohesive continuum mechanics model is exactly the same as the nonlocal peridynamic stress; (2) For the first time, we have applied an irreversible built-in cohesive stress-strain relation in a bond-based cohesive peridynamics to model inelastic material behaviors without prescribing phenomenological plasticity stress-strain relations; (3) The cohesive bond force possesses both axial and tangential components, and they contribute a nonlinear constitutive relation with variable Poisson's ratios; (4) The bond-based cohesive constitutive model is consistent with the cohesive fracture criterion, and (5) We have shown that the proposed method is able to model inelastic fracture and simulate ductile fracture of small scale yielding in the nonlocal cohesive continua. Several numerical examples have been presented to be compared with the finite element based continuum cohesive zone model, which shows that the proposed approach is a simple, efficient and effective method to model inelastic fracture in the nonlocal cohesive media.