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The paper states main provisions of the method for assessing the accumulation of damage in the structure of polymer materials under tensile stresses. The proposed method involves fractal analyses of deformation curves recorded with a high frequency. The minimum coverage method was used to determine the coordinates of critical points identified by the minimum levels of the fractality index. To determine the change in the fractality index depending on the relative elongation, we used a sequence of nested partitions. The amplitude variation was calculated for each of them as a sum of the differences between the maximum and minimum increase in tensile stresses in each analyzed time interval. The fractality index values were determined in the previous time interval of 0.16 sec. with a step of 0.01 sec. The paper provides the results of implementing the proposed method, which makes it possible to determine critical levels of stresses and relative deformations of polymer samples under tensile loads. We used ten samples of epoxy polymer based on modified Etal-247 resin cured by Etal-1472 hardener, including one sample with initial structural defects, as objects of study. We have identified the "critical" levels of tensile stresses and relative elongations, at which the failure rate grows, assessed by the minimum levels of fractality indices of the deformation curves. [ABSTRACT FROM AUTHOR]