PSI - Issue 23

Nikitin I.S. et al. / Procedia Structural Integrity 23 (2019) 119–124 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

122

4

3. Analytical results for constitutive relations For the general case, from these conditions the integration intervals were determined according to  and  in the formulas for the increments of the components of the plastic strain tensor (1). The integration was performed for the case where the maximum shear stress 13 S slightly exceeds the limit value. It can be written as 13 1 S    , 1   . At the same time other main shear stress components are 12 0 1 S   and 23 0 1 S   . Let’s use the following definitions of stress-strain states of a material point, similar to those used in Mohel et al. (1983). It is called active loading if 2 1 0    and   2 0    at all planes. It is called unloading if 2 1 0    and   2 0    at all planes. If 2 1 0    at all planes and some planes has   2 0    and some has   2 0    it is called partial loading . On 12 S - 13 S diagram the partial loading zone is located between    13 13 12 12 1 1 / S k S S S    and 13 0 S  ,   12 12 / 1 k S S   . The active loading zone is located above it, while the unloading zone is located below it. Thus, the description of the processes of active and partial loading in the above definitions was obtained. Integrating of (1) leads to following formulas. In the active loading zone:   11 0 12 12 13 13 12 12 (1 ) / 4 с S S E S E S      

22 0 12 12 13 13 (1 ) / 4 с S S D S     

(2)

11 22 ( )  

 

    

33

In the partial loading zone:  11 0 12 12 13 13 13 12 12 12 (1 ) / 2 с S S d E S d E S      

(3)

22 0 12 12 13 13 13 (1 ) / 2 с S S d D S    















 S S      , 13  2 2 12 12 1

2

13 12 4 1 1 E S S S        , 13 12

E S S

2 1 1   

Coefficients 12 13 13 , , E E D are

12 D E   .

12

12

13

Coefficients 12 d and 13 d are within the limits

13 0 d  , 12

/ 2 d   , and depends on 12 S and k .

k    the coefficients are

1 k  and 1

While 12 0 S  ,

1 k

k  

( 1)( 

) 

( 1)( 

) 

 

( 1 2 ) k   

1

1









d

and 12 d

arccos

arccos









.

13

k

k

k

k

k

1

1

1

1











1 k    the coefficients are

1 k  and

While 12 0 S  ,

k

k

(1 )(

)

(1 )(

)

1 k     

   

(2 1) k   

1









d

and 12 d

arcsin 1 1

arcsin









.

13

k

k

k

k

1

1

1









k    the coefficients are

1 k  and 1

While 12 0 S  ,