PSI - Issue 43

A.D. Nikitin et al. / Procedia Structural Integrity 43 (2023) 53–58 A.D. Nikitin et al. / Structural Integrity Procedia 00 (2022) 000 – 000

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interpretations of the full SN-curve shape. Another interpretation of the full SN-curve is proposed by Mughrabi (2006) as a staircase curve. In the case of the staircase model, following the Wohler ideas, the bifurcation area between HCF and VHCF is represented by horizontal line that is not physically correct. Based on the crack initiation mechanism criterion the full SN-curve can be assumed as bi-modal distribution. One of the branches is corresponding to the surface crack initiation while the second one is associated with subsurface crack nucleation. In works Nikitin et al. (2020) this multi-mode distribution is assumed and represented by bi-modal SN-curve, Fig.1. The left branch corresponds to surface crack initiation mechanism while the right branch is associated with subsurface crack nucleation. Both branches of such bi-modal curve show a monotonic, smooth behavior with permanent decreasing of fatigue strength versus number of cycles. The left branch is described by the Basquin type equation L a u L N     − = + linked the stress amplitude, material constants and number of cycles to failure. The model is developed for a physically small volume that is small enough to avoid the texture effect and large enough to avoid the effect of dislocation structures. In this case the macroscopic behavior of such volume can be described in terms of continuum mechanics. Let’s assume the isotropic, elastic behavior of this structural element. The fatig ue behavior of the material under high amplitude and short fatigue life will be described by Basquin type equation of the left branch. The fatigue properties in the range of large durability are described by the similar equation devoted for the right branch. The parameters of equation can be found from the uniaxial tensile SN curve shape. Under low cycle fatigue loading conditions, large volumes of specimen or structure are experiencing a plastic deformation. The total fatigue life is determined mainly by crack growth stage. The stress amplitude under such loading conditions is between yield stress and ultimate tensile strength. For most metallic materials the fatigue strength at short fatigue life 10 2 – 10 3 cycles are not significantly lower than ultimate tensile strength. When the number of cycles is increasing up to 10 6 – 10 7 cycles, the left branch approaching the horizontal asymptote at classical fatigue limit. The left branch ends by bifurcation area, Fig.1 where the crack initiation mechanism is changing from surface to subsurface. This transition is usually observed in the range of 10 7 – 10 8 cycles for most of structural materials. The new crack initiation mechanism causes a new decrease of a fatigue strength versus number of cycles. Taken these limit condition into account the following expression for Basquin equation can be withdrawn for left and right branches. Left branch L eq u L N     − = + , 3 10 ( ) L L B u     = − Right branch (1) V eq u V N     − = + , 8 10 ( ) V V u u     = − where eq  is equivalent stress, u  is classic fatigue limit, u  is VHCF fatigue limit, B  is ultimate tensile strength, L  , V  are constants of material. Here and further the subscripts L and V will be used to distinguished parameters of left and right (VHCF) branches respectively. The powers L  and V  for the left and right branches are determined as the slop of SN-curves obtained under uniaxial tensile loads in HCF and VHCF ranges. Therefore, the introduced physically small volumes demonstrates a monotonic decreasing of the fatigue strength. The number of cycles to failure depends on material properties and loading parameters (equivalent stress). Let’s assume that the fatigue damage accumulation in this physically small volume can be caused by normal and shear mechanisms. These mechanisms can act simultaneity that is requiring two damage functions describing these processes: one is associated with normal crack opening mechanism, the second one is associated with shear crack opening. Let’s introduce the damage function having its value from 0 to 1. When the material is virgin the damage function equals 0. In the case of destroyed material the function turns to 1. In the general case the damage function is depending on mean stress, alternative stress range and loading history N. The loading history is not linear and can be described by kinetic equation: (2) where B is a coefficient depending on the stress state only. The elastic moduli of the material are depending on the damage function value as following: material degradation at ψ < ψ * , λ ( ψ ) = λ 0 (1 – κψ ), μ ( ψ ) = μ 0 (1 – κψ ) full fracture at ψ * ≤ ψ ≤ 1, λ = 0, μ = 0 1 ( , ) / (1 ) N B        −   =  −