Fiber characterization is a major challenge in conventional peridynamic (PD) simulations since the regular discretized grid restricts the direction of bond-pairs between material points. In this work, a new computational model for composite materials with arbitrary fiber orientations (AFOM) is proposed to extend the application scope of conventional PD models in engineering structures. In order to match the practical carbon fiber structure, the fiber bond is analyzed as a special type of research object in AFOM. The most unique feature is that two types of material points are utilized to model a composite structure, and a mapping relation is proposed to achieve data exchange for these component materials. Thanks to this operation, the developed AFOM can remove the limitation of conventional bond-based or ordinary state-based PD in terms of reinforcement characteristics, and the deformation and progressive damage behaviors of composite materials with general fiber orientations can be easily captured. It has been demonstrated from the deformation examples that the proposed AFOM can describe the anisotropic properties of composite structures with general layups well. The damage examples of composite laminates further demonstrate that the proposed AFOM can adaptively replicate the failure characteristics of anisotropic materials without any theoretical limitations.