PSI - Issue 48

Shanyavskiy A. et al. / Procedia Structural Integrity 48 (2023) 119–126 Shanyavskiy et al/ Structural Integrity Procedia 00 (2023) 000–000

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4. Fatigue damage model and numerical calculation of damage development in a gear 4.1. Multi-mode two-criteria model of fatigue fracture

For mathematical modeling of the process of fatigue failure of gears of the gearbox in engine reducer and evaluation of the in-service life, we will use the scheme for describing the complete fatigue curve based on the Basquin relation, which establishes a relationship between the stress level and the number of cycles to failure. An analysis of the experimental fatigue curves shows that the nature of the decrease in fatigue strength with increasing the number of cycles is similar for the left (LCF, HCF) and right (VHCF) branches, which allows us to formulate a hypothesis about the similarity of their description (Burago and Nikitin (2016)). Within the framework of this model, it is assumed that the left and right branches of the complete fatigue curve are described by the following relations: (1) where σ –1 is the classical fatigue limit, σ̃ –1 is the fatigue limit in the VHCF region, σ eq is the equivalent stress, which for the case of uniaxial loading coincides with the amplitude of the cyclic stress, β is the exponent in power relation. In order to distinguish between parameters and variables related to different branches, we introduce the following indices: L is for the left branch, V is for the right one. The mode of repeated static loading with an amplitude of the order of the ultimate strength of the material σ U , as a rule, lasts the first 10 2 -10 3 loading cycles. Choosing, for definiteness, the value 10 3 as the beginning of the descending left branch of the fatigue curve, we obtain the value of the coefficient   1 3 10        U L L . Taking for definiteness the value of 10 8 cycles as the beginning of the descending right branch of the dual fatigue curves, we obtain the value of the coefficient   1 1 8 ~ 10         V V . In the case of a complex stress-strain state, the equivalent stress can be determined in accordance with experimentally validated multiaxial fatigue failure criteria to describe various crack opening mechanisms: microcracks of normal separation or shear. In the proposed model, two criteria are chosen: SWT (normal opening microcrack mechanism) presented by Smith et al. (1970) in the form expressed by Gates and Fatemi (2016) and CSV (shear microcrack mechanism) suggested by Carpinteri et al. (2011). The SWT criterion presents a variant of taking into account the main tensile stress: 2 1 1 max   n eq , main cyclic stress. To describe the development of shear-type microcracks, a criterion proposed by Carpinteri et al. (2011) was chosen that takes into account the features of their formation:     2 2 2 3 2 n n eq           , where Δ τ n /2 is the amplitude of the maximum shear stress acting on a certain area with normal n , 2 n   is the amplitude of the cyclic stress in the tension phase     min min max max n n n n n H H         acting on the same area. To describe the cyclic degradation of a material, the concept of a distributed damage function ψ is introduced, which takes values from 0 to 1 (Lemaitre and Chaboche (1994) and Murakami (2012)) and is conditionally equal to the relative density of microdefects in a small volume of a deformable sample. The change in the damage function with increasing the number of loading cycles is described by the kinetic equation proposed in Nikitin et al. (2020):               1 1 , d dN B (2) where B ( σ , Δ σ ) is the coefficient depending on the stress-strain state in the cycle, Δ σ is the range of the cyclic stress, γ is an experimentally determined parameter describing the rate of damage accumulation. The expressions for the coefficients B ( σ , Δ σ ) were obtained in Nikitin et al. (2020) and Nikitin et al. (2022) and have the following form:       / 1 2 10 1 1 1 3               L U eq L B B for σ –1 +Δ σ –1 < σ eq < σ U L      and N L eq    1 V N      V eq    1 ~ where max 1  is the value of the maximum main tensile stress,   max 1 max max 1 1    H  , Δ σ 1 /2 is the amplitude of the

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