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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4

where ψ * < 1 - critical value of damage function corresponding to the full fracture state. At the beginning of cyclic loading when damage function equal to 0 the elastic moduli are corresponding to virgin material. When damage function approaches 1 the elastic moduli tend to zero. To realize the through calculations method the value of elastic moduli is never reach 0, but take a small value, for example, 0.001 of initial value. Narrow localized zones where elastic moduli turn to such small value assumed as quasi- cracks. The terms ‘quasi’ is used to outline that material is still present at these locations but it cannot carry the external load anymore and acts like a free surface. Comparing the equation (1) and equation (2) the boundary conditions can be defined. When the stress state is established, the number of cycles to failure N f can be determined based on equation (1). In the terms of damage function, it means that its value should reach ‘1’. Therefore, the integration limits for the damage function in the equation (2) is from 0 to 1 while the limits for number of cycles is from 0 to N f . Based on these assumptions, the analytical expressions for coefficient B can be found for the left and right branches. Left branch eq  According to introduced limits the fatigue failure cannot be observed withing the reasonable for service life number of cycles under equivalent stress lower than VHCF fatigue limit. In the case of extremely high loads when the equivalent stress is higher than ultimate tensile strength the material crashed in one half cycle that is also out of interest for the present study. These conditions can be mathematically introduced by using Heavyside function. Here and further, we will use the following designation: ( ) f fH f = , ( ) H f is Heaviside function. The parameter 0 1    determines the rate of damage accumulation in each material and can be adjust by comparing experimental and numerical simulation results for the shape of SN-curve under uniaxial tensile loads. The system of equations (1) – (3) forms the basis of the multi-regime model. The type and rate of fatigue damage accumulation is depending on the combination and intensively of the external loads. The value of equivalent stress is included in the expression for coefficients B that are determining the kinetic equation for damage function. Therefore, the choice of the multiaxial criteria for describing the basic mechanisms of material failure (normal crack, shear crack) will affects the kinetic equation parameters and rate of damage accumulation. If the numerical simulation algorithm allows the two sets of calculation (for normal crack and shear crack criterion) withing the single settlement step, the two mechanisms will be in competition as we observe in the experiments. In the present work the criterion for normal crack opening was used in the form of Smith – Watson – Topper (SWT) (4) where max 1  – the maximum mean tensile stress, 1   is the range of the main tensile stress. Therefore, the expression for the equivalent stress in the case of normal crack opening mechanism will be (5) The shear crack opening can be successfully described by Carpinteri – Spagnoli – Vantadori (CSV) criterion. This criterion was already used for standard specimen calculation during the calibration and validation of shear mechanism. (6) where n   - range of shear stress on the plane where it reaches the maximum value, n   - the range of the maximum normal stress on the same plane. The expression for equivalent stress in this case can be written in the following form max 1 1 / 2 L u L N      −  = + 2 2 ( / 2) 3( / 2) L n n u L N      −  +  = + 1/ 8 10 / (     − − ) / (1 ) / 2 V  − V eq u u u B B  −   = =  max 1 1 / 2 n eq     = =  , max max max 1 1 1 ( ) H    = u eq B     +    , u 5 10 ( L  − ) u   = B u   − , 1/ 3 10 / (     − − B u ) / (1 ) / 2 L  − L eq u B B  −    = =  Right branch (3) u u u       +  ,

max ( )

n n n H   

 = 

2 / 2) 3(

2

(

/ 2)

eq   

= = 

n 

+ 

n 

,

(7)