Examples of the estimation of the moment of loss of stability of the deformation process, which is defined as the time of the beginning of deformation with decreasing applied stress under monotonic loading, are considered. Two examples of processes are considered: the necking in sample during the uniaxial tensile testing and the riveted joint of the fuselage of the aircraft. In the second case, in addition to the above method for estimating the moment of loss of stability, we also considered the possibility of applying the criterion derived from the results of finite element modeling, related to the ratio of effective stresses, defined as the tensile force divided by the cross-sectional area of the cross-section of the model still retaining its integrity, to effective deformation, defined as the maximum displacement of the ends of the stretched sample divided by the initial length of the unloaded model. When these two methods are used, fairly close results are obtained, what is demonstrated in obtaining for both cases similar estimates of the calculated values of the number of loading cycles, which is necessary for the crack in the aluminum 2024-T3 used in manufacturing the fuselage skin, to reach the critical length. Due to the variety of load levels and loading methods, the need for obtaining from the corresponding experiments a significant number of dependencies, required for the calculation, is noted. The method of acoustic emission is recommended as the main method for determining the starting moment of the motion of the fatigue crack. The need to refer to experimental methods for determining the laws of crack growth and the material parameters responsible for this is due to the dependence of the growth of microdefects on a wide range of external conditions: type of environment, temperature, pressure, etc. When using the proposed criteria, a methodology can be developed for assessing the onset of a critical state of parts used in various engineering areas that contain defects and are exposed to external loads. This will allow time to predict the time of their possible failure.