PSI - Issue 23

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

133

3









2 m a t         2 2 2 cos

3 3 cos m a t         3 3

,

1 1 1 cos , m a t      

where 1 m  , 2 m  , 3 m  are mean principal stress components and 1 a  , 1 a  , 1 a  are amplitudes of these stress components. Ranges of the principal stress tensor components within a loading cycle are:   1 1 1 2 cos cos , a t t              2 2 1 2 2 2 cos cos , a t t                  3 3 1 3 2 3 cos cos a t t            Where the times 1 t and 2 t correspond to minimum and maximum values of shear stress at the plane with unit normal vector n ; 1 2 , [0, ] t t T  , 2 / T    . The shear stress range can be written as:       2 2 2 2 2 2 2 2 2 2 2 2 2 12 12 1 2 13 13 1 3 23 23 2 3 4 sin sin sin A n n A n n A n n                  

t t

  

A

A

A

2      , 2 12 2 a

2      , 2 13 3 a

2      , 2

2

2

2

1 2 2

,

12

13

23

23

23

2 a 

3 a 

23 23 arctg  

     

t t

1 t T

arctg



arctg







   



/ 2

,

,

,

,

13

2 1

12

23









13

12

where:

13 1 a a        , 3 3 cos

12 a a        , 1 2 2 cos

23 a a         , 2 2 3 3 cos cos

3 a a     , 3 sin

2 a a     , 3 2 2 sin

23 a a         , 2 2 3 3 sin sin

S A A 

2 2 n n A  12 1 2 13  2

2 2 n n A 

2 2 23 2 3 n n

sin2

sin2

sin2





2

2

,

12

13 1 3

23

C A A 

2 2 n n A  12 1 2 13  2

2 2 n n A 

2 2 23 2 3 n n

cos2

cos2

cos2





2

2

,

12

13 1 3

23









2 2 0 13 sin b A     , 1 13

2 23 sin b A     , 2 2 23 0













12 b A 

A

A

2 2 sin 13

sin

2 2 sin 12

0 13  

 

 

 

0 12   

2

2

.

23

0

23

Expressions for the times and frequency are:   1 arctg / 1 4 S C A A t T k           , 2 4 t T    









2 S C A A k  

arctg / 2

S C A A

arctg /

  

0  

k  

1

1,2,3 k  .

,

,



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

Corresponding values of the unit normal vector’s components

2 ) / (4 ) x bb b b bb b    , 1 2 2 12 1 2 12 (2 1

2 ) / (4 ) x bb bb bb b    . 1 2 1 12 1 2 12 (2 2