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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2 3 1 x x   ,  From these three pairs ( 2 x , 3 x ) choose one, where function 2 3 ( , ) F x x takes maximum value. So, for the case of multiaxial stress state the critical plane orientation is found analytically. Knowing components of normal to the critical plane it is possible to calculate Findley’s function and corresponding numbers of cycles to fracture N, i.e. fatigue durability for structural element, undergoing multiaxial stress state, Bourago et al. (2011). 1.2. Antiphase multiaxial cyclic loading An important practical example of antiphase cyclic multiaxial loading is fatigue tests for pure torsion or bending with torsion. It is not difficult to see that in a coordinate system connected with the principal axes of the stress tensor, an anti-phase cyclic loading with a harmonic law of time variation can always be represented in the form: 2 23 1 / 4  23 / 2 x     , 3 23 1 / 4  23 / 2 x   

1 1 sin m a t t       , ( ) 1

2 2 sin m a t t       , ( ) 2

3 t t              , 1,2,3 0 a   In this case 1 1 2 0 a      , 2 2 2 0 a      , 3 3 2 0 a       . The expression for comparison with the in-phase cyclic mode does not change: 3 3 m a 3 3 m a ( ) t sin( ) sin

2 2 2 12 1 2 ) n n    The normal stress at the plane with the normal n in the anti-phase mode depends on the time as: 2 2 2 2 2 2 1 1 2 2 3 3 1 1 2 2 3 3 ( ) ( ) ( )sin n m m m a a a t n n n n n n t               Therefore, for the determination of the Findley function, we consider two cases. I. If 2 2 2 1 1 2 2 3 3 0 a a a n n n       , then 2 2 2 1 1 2 2 3 3 [0, ] max n t T n n n         for sin 1 t   , where 1 1 1 m a      , 2 2 2 m a      , 3 3 3 m a      . We renumber the values, k = 1,2,3 so that they satisfy the inequalities 1 2 3      . After that, the same analysis and calculation of the orientation of the critical plane for the renumbered values k  , k   , k =1,2,3, as for the in phase regime, is carried out. The received values of the normal components are checked for the condition 2 2 2 1 1 2 2 3 3 0 a a a n n n       . If this condition is not satisfied, then we consider the following alternative case. II. If 2 2 2 1 1 2 2 3 3 0 a a a n n n       , then 2 2 2 1 1 2 2 3 3 [0, ] max n t T n n n         for sin 1 t    , where 1 1 1 m a      , 2 2 2 m a      , 3 3 3 m a      . In this case we renumber the values k  , k =1,2,3 so that they satisfy the inequalities 1 2 3      . After that, the same analysis and calculation of the orientation of the critical plane for the renumbered values k  , k   , k =1,2,3 as for the in-phase regime, is carried out. The received values of the normal components are checked for the condition 2 2 2 1 1 2 2 3 3 0 a a a n n n       . If both conditions are satisfied, then the orientation of the normal should be chosen, at which the value of the Findley function has a greater value. 1.3. Determination of the parameters of a multiaxial fatigue fracture criterion Earlier, in Bourago et al. (2011), a procedure was proposed for determining the parameters of classical multiaxial criteria (Sines and Crossland) in the regime of LCF or HCF according to the results of uniaxial tests with two different coefficients of cycle asymmetry (left branch of the bimodal uniaxial fatigue curve, Shanyavsky (2007), Fig.1). We present the results of this procedure for determining the parameters of the multiaxial Findley criterion. 2 2 2 13 1 3 ) n n  2 2 2 23 2 3 ) n n  (    (   ( n 