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This investigation delves into the plastic deformation mechanisms responsible for the distinctive chip formation of Invar 36 alloy, marked by features of cleavage and folding instabilities during machining. This study unveils the flow characteristics of Invar 36 alloy by conducting cutting experiments and material property tests, leading to the formulation of flow stress models. Employing polycrystalline finite element simulation, the research illuminates how intergranular compression can induce surface bulging, subsequently triggering the initiation of cracks under tensile stress. As cutting speeds escalate, an intensification in plastic flow begets increasingly pronounced and frequent chip serrations. When speeds surpass 200 m/min, elevated temperatures and strain rates foster material buckling and adhesion to the tool's rake face, fostering the emergence of large-scale and subsidiary folds. This “fan-like” chip architecture precipitates a self-blocking effect, which in turn generates significant cutting forces and severe oscillations. An analysis via a mathematical-physical model reveals that the oscillation in cutting force incite the material's inertial responses, catalyzing abrupt shifts in shear strain and shear strain rate, and eventually leading to cracks and segmentation within the cutting zone. Finally, the research briefly discussed on the ramifications of these alterations in the cutting flow mechanism on tool wear and surface quality in the practical machining of Invar 36 alloy, alongside an outline of prospective research.