PSI - Issue 14

A.N. Savkin et al. / Procedia Structural Integrity 14 (2019) 684–687 Author name / Structural Integrity Procedia 00 (2018) 000–000

686

3

ε ε ε , * p * e *     

(2)

* е ε  — elastic strain increment; * p ε  — plastic strain increment. In this paper we accept von Mises yield criterion in combination with isotropic-kinematic hardening rule:   σ α σ ε 0, * y * p     f (3) Translation of yield surface is based on Chaboche summing rule, every individual translation follows Frederic– Armstrong kinematic hardening model:   γ α ψ ε , α α α * p       i i i i i i C (4)

α — sum of translations;

i α — individual translation;

i i С , γ — material constants; ψ — loading parameter

(«-1» — compression, «+1» — tension).

Limit surface radius increasing/decreasing follows simple saturation model:         ε , σ ε σ σ σ 1 exp * p 0 y y 0 y * y p b      

(5)

0 y σ — initial yield stress;

 y σ — asymptotic yield stress; b — isotropic hardening constant.

All needed material constants can be determined during standard multistep experimental procedure. Proposed integration algorithm of incremental constitutive expressions (3) can be interpreted as return mapping (closest point projection) scheme in one-dimensional implementation which is widely used in FEA. The first step of algorithm is called elastic predictor. After determining total strain increment it is assumed that its purely elastic. The stress and hardening parameters are predicted elastically as:

α α , σ σ * n

ε ε *        ε , * 1 n tr E

tr *

(6)

  . * p

tr

p

n

n

  σ α σ y 1 tr tr *

If trial stress is within the elastic domain (on the stress curve),

0,

then stress and other

n f

  



1



n



internal variables stay unchanged:   * tr tr * * σ σ , α α, ε  

ε



(7)

tr

*

1

1

p

p

n

n





1

n



  σ α σ y 1 tr tr *

then material response becomes plastic

If trial stress is outside the elastic domain

0,

n f

  



1



n



and plastic correction and internal variables updating is needed (mapping):             . ε ε , ε γ α ψ ε α α , ε σ σ 1 * p * p 1 * p 1 * p tr 1 1 * p tr * * 1                    n n n i n i i i n n n C E

(8)

Actual stress and plastic variables can be determined from yield condition which is a nonlinear equation in terms of plastic strain increment: