Failure prediction of polymeric material and structures is an important engineering task from experimental evaluation as well as from numerical simulation point of view. To investigate the fracture behavior and the fatigue failure of polymeric material, this work adopts two different numerical methodologies to study crack initiation and propagation when the material is subjected to monotonic and cyclic fatigue loading. As a smeared crack approximation, the phase-field model does not depend on any explicit criterion to evolve cracks but yields good agreement compared to experimental validations. Another phenomenological approach to characterize crack growth based on a discrete approximation is the material force or configurational force approach, which largely depends on post-processing techniques. Both of them are developed according to the classical GRIFFITH criterion for brittle fracture. Nevertheless, regarding fatigue fracture phenomena, a fatigue induced degrading fracture toughness is assumed to evolve cracks, which basically captures the fatigue failure characteristics. This work implements these two methodologies, the phase-field approach and the material force method, into the Finite Element framework and simulates several demonstrative numerical examples, yielding good agreement by comparing to each other as well as to experimental results. Consequently, potential perspectives are proposed to close this paper.