PSI - Issue 13

N.S. Selyutina et al. / Procedia Structural Integrity 13 (2018) 700–704 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

702

3

          



     ds

    

( ) s

t

1,

1,

1





0 y



t







    

     ds

( ) s

1

t

( ) t





(2)

, 1

1,



1

0 y





t





   

   





    

   

( ) s

1

t

ds



0 y





t





Equality ( ) 1  t  in (2) relates with the elastic deformation accumulation before the starting time t * of the macroscopic plastic flow. Gradual decrease of the relaxation function in the range 0 ( ) 1   t  corresponds to the material transition into the plastic stage of deformation. During the plastic stage of deformation, * t t  , the relaxation function satisfies the condition



      t t y    0 

     ds

t s ( ) ( )

1

(3)

1.



Let us define the actual stress ( ) t  in a deformed specimen by the following form ( ) 2 ( ) ( ), t g t t   

(4)

where is an “effective” shear stress that takes into account the plastic relaxation;  is a scalar 0   corresponds to the plastic deformation without hardening. Considering the stages of elastic and plastic deformations separately, we can obtain from (4) the following stress-strain relation in the case of a linear growth of strain ( ) ( ) t H t t     (the constant strain rate) parameter ( 1 0    ) characterizing the level of hardening; the case ( ) g t G 1      ( ) t

2 ( ), G t 

 * t t t t t 

,

    2

( ) ( ), t 

( ( )) t  

(5)



  

   

1





G

,





*





t 

 ( ) / t





.

where we take into account that

3. Comparison with classical Johnson-Cook model

Stress-strain behaviour of metals subjected by dynamic loading is determined using Johnson-Cook model:     , 1 ln 0                C A B n p y Here, A, B, C, n are the constant parameters of Johnson-Cook model; ε p is the equivalent plastic strain;   is the plastic strain rate; 1 0 1s     (Johnson and Cook (1985)). Many authors have chosen parameter 0   in the range from 10 -3 s -1 to 1 s -1 . Other words, definition of yield stress by empirical Johnson-Cook model depend on strain rate. Presented in (Selyutina and Petrov, 2018) comparison of Johnson-Cook model parameters with parameters of the structural-temporal plasticity model, which is invariant to history loading, shows their dependence not only τ and α but parameter of 0   :         (6)

1



   

      

   

   

  

E









y A   ,

(7)

C

 1 ( ) 1 1        0

ln

1









y

0

3 1

1

 



Apply Johnson-Cook model with parameters of (line 2) for experimental data on MP800HY steel (Singh et al. 2011). Line 2 has a good correspondence with experimental data at strain rate 0 10 s    (line 3) and 0 0.1s   