This study investigates flange fracture formation in unconventional incremental radial extrusion. This manufacturing technique involves using rings with a gradually increasing inside diameter for constraining the free flow of material in the radial direction. As a result, the shaped flange has a constant thickness and a significantly larger diameter than that formed using the standard extrusion process conducted without the use of rings. EN AW 6060 aluminum alloy tube sections were used as the billet material, and the extrusion process was conducted under cold forming conditions at ambient temperature. For the determination of material fracture initiation, a new method was proposed involving the analysis of strain, strain rate and values of the normalized Cockcroft–Latham fracture criterion integral. The main advantage of the new method is that it allows for the prediction of fracture initiation via only FEM results analysis, i.e., it is not necessary to carry out additional experiments aimed at calibrating or determining limit parameters of a given material. It was shown that the occurrence of differences in the distribution of the above-mentioned parameters coincided with flange fracture initiation.