PSI - Issue 23

I.S. Nikitin et al. / Procedia Structural Integrity 23 (2019) 131–136 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

132

2

cyclic loadings with arbitrary phase shifts is developed. The procedure is based on the well-known criterion proposed by Papadopoulos (2001) and is valid, as the criterion itself, for the high-cycle domain of fatigue (HCF). 2. Papadopoulos’ fatigue criterion

Fig. 1.A critical plane image (Nikitin (2016))

Let us consider a material particle in a uniform stress state described by a stress tensor ( ) t σ that varies cyclically in time. Let us further choose a frame that is associated with the principal stresses 1  , 2  , 3  and choose a plane with a unit normal vector n . The multiaxial fatigue criterion is:

max a T

S AN

 

H  

 



,max

0

n

Coefficients 0 , , S A   and  are defined by tension-compression fatigue tests with two different asymmetry ratios 0 R  and 1 R  as it was shown by Burago et al. (2011), N is the number of cycles to crack initiation. The quantity max a T n is a maximal value of a shear stress among all existing planes with normal vectors n that contain the specified material particle in a loading cycle:

2        n n 2 ( ) ( , ) / , d a

( , ) max ( , , ) min ( , , ) a t P t P t t               n n n

T

2

a

0

,max H  is an equivalent peak hydrostatic stress at the specific place within a loading cycle

The quantity

. Shear stress vector on the plane with normal vector n is

( ) ( )      τ σ n n σ n n . The

,max max ( ) / 3 H kk t P t    

following equation may be obtained for the value of shear stress range (Nikitin et al. (2017)):       2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 1 3 1 3 2 3 2 3 ( ) a T n n n n n n                   n

3. Particular loading cases

3.1. Tri-axial tension-compression case Let us assume that principal stresses are varied in time by harmonic law with frequency  and arbitrary phase shifts 2  , 3  :