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In this work, the geometry-constraint-based non-ordinary state-based peridynamic (GC-NOSBPD) model is further generalized to study the mechanical responses and rate-dependent fracture behaviors of viscoelastic materials. The viscoelasticity is generated by directly incorporating the classical viscoelastic constitutive relation in the form of the Generalized Maxwell Model, which is expressed as Prony-series type expansion, into the GC-NOSBPD framework. A strain-based scalar damage variable only driven by elastic energy, whose evolution follows a rate-independent law, is proposed to describe the progressive growth of the mode-I crack. Combining the rate-independent damage variable and viscoelasticity results in a rate-dependent damage model in the hypothesis that the fracture of viscoelastic material is driven by both elastic and viscous components of the energy. The fidelity of this model is established in the absence of damage by comparison with the benchmark solution for the viscoelastic response of a 3D bar subjected to various load histories. Further, the proposed model is employed to simulate two quasi-static fracture tests under various loading rates: a three-point bend fracture test for concrete and a double-cantilever-beam delamination test for rubber interface. These numerical results agree well with experimental tests, showing that the proposed model can effectively capture rate-dependent crack propagation. [ABSTRACT FROM AUTHOR]