PSI - Issue 23

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

135

5

t t

2 1    t t

2 

2 

  

sin      

  





33   cos( t t             ) cos( ) 2 sin a a 1 2

2 1

Where the times 1 t and 2 t correspond to minimum and maximum values of shear stress at the plane with a unit normal vector n ; 1 2 , [0, ] t t T  , 2 / T    . The value of shear stress range may be written as :

2     2 1 t t









  ,

X

2           sin sin X X

4sin

sin

sin

2   

  

2

2

2



a

a a

a







2

t t

   ,









3 3 1 X x x    .

2 3 3 2 1 2 X x x x    ,

2 3 2 3 4 X x x x x     ,

1 2 2

where

S a a B X 

2        sin a X

2 B X X      C a a a 

X

sin2 , 

cos

cos2

2    a







Equations to calculate the times are:   1 arctg / 1 4 S C B B t T k            ,









2 S C B B k  

arctg / 2

S C B B

arctg /

  

  

  

2 t T

k  

1

 

1,2,3 k  .

,

,

0

4



2 i i x n  at the times 1 t and 2 t are:

Corresponding values of the unit- vector’s components





 a              2 2 sin 2 2 sin 0 0 2 2 2 sin a

x

, 2 0 x  , 1

2 3 1 x x x    .

3

a

0

,max H    was found:

In a similar way a time within the loading cycle that determines the maximal value of

,max H m    .

4. Comparison of the criterion with experimental results In order to rate the criterion a comparison with other one in Carpinteri et al. (2002) and experimental results by Nishihara and Kawamoto (1945) was made for bending-torsion experiments shown in Table 1. First two columns are bending and torsion stress amplitudes. The third column is the phase shift between them. From fourth to sevenths are data for the angle  that is the angle between the longitudinal axis of the specimen and the critical plane orientation n (fig. 1). The experimental and Carpinteri’s results are taken from literature; the numerical results were calculated by the direct selection from the Papadopoulos criterion as it is, namely all possible combinations of discrete time and angle values were studied. The combination with the largest found value was chosen as the critical one ; the analytical values for the angle α were calculated by applying the analytical solution that is given above. Proposed semi-analytical procedure is 200 times faster than numerical direct selection.