PSI - Issue 13

B.A. Stratula et al. / Procedia Structural Integrity 13 (2018) 1402–1407 Author name / StructuralIntegrity Procedia 00 (2018) 000 – 000

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According to this criterion, the development of fatigue damage in the cyclic loading process occurs along the plane with the normal n (with components n k , k = 1,2,3 ), at which the maximum of the combination ( / 2 ) n F n      reaches a certain critical value. Such a plane is called critical plane. In this combination n   - the range of the shear stress in the cycle, n  - the normal stress on this plane, σ - the stress tensor, which determines the stress state in the material particle of the deformed body, n     n σ n , ( ) n       σ n n σ n n . To determine the durability of uniaxial cyclic loading of the sample up to its fatigue fracture, there is a Baskin relation that analytically represents the fatigue curve for various coefficients of the cycle asymmetry (left branch of the bimodal fatigue curve, Shanyavsky (2007), Fig. 1): u c N       where u  - fatigue limit, c  - fatigue strength factor,  - fatigue strength exponent, N - number of cycles before fracture. Fig. 1 shows schematically the amplitude a  of the cyclic loading process on the ordinate axis for various coefficients of the cycle asymmetry as a function of N.

Fig. 1. Bimodal fatigue curve for LCF-HCF and VHCF modes

The generalization of the Baskin relation to the multiaxial case for the Findley criterion has the form: ( / 2 ) F n n F n MAX F F S A N         where F  <0, F  , F S , F A are the parameters determined from the experimental data. Calculation of fatigue durability by the Findley criterion requires the determination of the orientation of the plane passing through a given material point where the maximum expression of the Findley function is reached: 2 n F n F       . For a multiaxial stress state this is not an easy task, which, as a rule, is to be solved numerically. We construct its analytic solution for in-phase and antiphase cycling loading. 1.1. In-phase multiaxial cyclic loading Let us consider a triaxial cyclic loading in a coordinate system connected with the principal axes of the stress tensor. We assume that these axes do not change during the cycle, and the principal values of the stress tensor vary according to a harmonic law without a phase shift with respect to each other:  where the additional index m denotes the stresses averaged per the cycle, and index a denotes its amplitudes. The ranges of the principal stresses in the cycle are 1 1 1 sin m a t t       , ( ) 2 2 2 sin m a t t       , ( ) 3 t t       , 1,2,3  3 3 sin m a ( ) 0 a

1 1 2 0 a      , 3 3 2 0 a      . We choose the principal axes so that the maxima of the principal stresses satisfy the inequalities: 1 2 3      , where 1 1 1 m a      , 2 2 2 m a      , 3 3 3 m a      . Also introduce the following notation: 12 1 2     , 13 1 3     , 23 2 3      , 2 2 2 0 a      ,