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Fatigue strength conditions presented in terms of normalized equivalentstress functionals defined on loading processes are used to unite the stages of materialdamage with fatigue crack initiation and multiple crack propagation under arbitraryloading history. Examples of employing the local form of the functionals associated withthe Palmgren-Miner linear damage accumulation rule and the power-type S–N diagramto a periodic crack system are given and shortcomings of the local approach arepointed out. A non-local approach free from the shortcomings is described. Equationsfor curvilinear crack growth rate vectors taking into account the whole damage historyahead of the crack are presented for multiple cracks under mixed-mode loading.