PSI - Issue 14

Bimal Das et al. / Procedia Structural Integrity 14 (2019) 619–626 Das et al./ Structural Integrity Procedia 00 (2018) 000 – 000

624

6

constitutive equations assuming isotropic and homogeneous describing small deformation, gene ralized Hooke’s law, yield criterion, associated plastic flow rule, consistency condition are presented as:

e p ε ε ε  

(1)

ν

 

  

e

e

σ 2G ε 

tr( ε )1

(2)

 

1 2 ν 

3 2



  : S Y     

f

S





(3)

f

s

3 2







p

d d 

dp

 



d (4) Where, d  is the stress increment, E is the elastic modulus, d  is the plastic multiplier, dp is the magnitude of plastic strain increment. In the rate independent plasticity, the load point must lie on the yield surface during the plastic deformation. This requirement is taken into account by the consistency condition described as: . d d 0 σ σ f f p p       (5) Kinematic hardening rule dictates the magnitude, direction and orientation of the yield surface by incorporating a stress tensor namely back stress tensor to capture the bauschinger effect. To predict the asymmetric strain controlled behavior, a nonlinear hardening rule proposed by Ohno et al. [11] is utilized in this work. Khutia et al. [12] and Das et al. [13] showed that Ohno-Wang model has good capability in predicting the asymmetric stress-strain response. They modified the dynamic recovery term of Armstrong and Frederick model, which activates after the magnitude of the back stress reaches a critical value. The rule is composed of several kinematic hardening rules having M number of back stress components, which are expressed as: M (i)    4.1. Kinematic hardening rule

i 1 d     

d

(6)

(i) m

  i

  i ( ) 

  

   

2 3

f



  i

  i (i)   γ 

(i)

p

p d : ε

d

C

d ε





 

(7)

  i

  i

r

f(

)



(i)

(i) (i) r C γ 

(8)

4.2. Isotropic hardening rule The cyclic softening response of P91 steel has been incorporated through the following equation:

0 σ

( ) r p   

(9)

0

Where, 0  is the yield surface size at zero plastic strain and ( ) r p is the isotropic hardening function. (10) and Q  is the maximum change in the size of the yield surface and b defines the rate at which the size of the yield surface changes as the plastic straining changes. 0 ( ) (1 ) bp r p Q e      