PSI - Issue 28

Alexey N. Fedorenko et al. / Procedia Structural Integrity 28 (2020) 804–810 Fedorenko A., Fedulov B., Lomakin E. / Structural Integrity Procedia 00 (2019) 000–000

806

3

failure stress in transverse direction, Y T – tension failure stress in transverse direction, and S is shear failure stress. It should be noted, that limiting values X c , X T , Y c , Y T , S in (3) are considered as fixed parameters only in case of static load of undamaged material. Damage occurs once one of the stress components attains corresponding value, and then mentioned above values are considered as functions on strains and strain rates to model subsequent damage behavior and hardening (Fedorenko et al., 2019). Next section shows an approach for equivalent transformation of X c , X T , Y c , Y T , S as functions of damage parameters and damage rate. 3. Strain-rate hardening Let us assume that stress components at which damage accumulates is determined at each time point by a set of parameters { ψ 1 , ψ 2, d ψ 1 /dt, d ψ 2 /dt }, i.e. by the damage parameters and damage rate. As shown in (3), for stress components we can consider X c , X T , Y c , Y T and S as strength functions, and write following relations:

1 ( , , , ), ( , , , ),             2 1 2

T X X X   X

T

1

1

2

2

C

C

2 ( , , , ), ( , , , ),             2 1 1

T Y Y Y Y S S   

(4)

T

1

1

2

2

C C

2 1   ( , , , ).     2 1

A number of experimental results show that the tensile loading diagrams are not dependent on the strain rate (Koerber, 2010-2011), so the dependence on damage rate of T X and T Y is not considered. Using that assumption, next step is a representation of strength function as a product of static ( St ) and dynamic ( Dyn ) parts:

1 ( ), ( )

X X

X X

 

Dyn    X

T

T

St

( ), 

1

1

C

C

C

2        2 ( ), ( ) ( ). C Dy D n yn S 2 ( ),

Y Y 

(5)

T

T

( ) Y Y Y  St

C t

C

S

S

S



2

2

Exact form of relations (5) are chosen similar to Johnson and Cook (1985) ones, however, the stresses are not expressed in terms of equivalent plastic strains and its rates, but in terms of the damage parameters and damage rates. Final form of proposed strength functions is expressed below:

X

A

,

X



T

T

1 / X B 

1 

T

N

 

  

 

 

X

A

 

  

X

1  

1

,

X

X C   



C

0

1  

C

1 / X 

Y B Y A B   

1 

 

C

Y

n

(6)

2  (1 ) , T 

Y

T

T

T





N

   

      

  

  

  

  

Y

Y C

2  

n

Y Y A B

(1    Y

)

1

ln

,

C





2 

0 2  

C

C

C

Y

N

   

   





  

  

  

S

  

  

2  

n

S

,

S

S

(1

)

1 sinh ln C 

S A 

B







2 

0 2  

S

i T n ,

i C n ,

0 i   should be defined to fit experimental

S A ,

S B , S n , C

i T A ,

i T B ,

i C A ,

i C B ,

where parameters

i , N i and