PSI - Issue 42

D.I. Fedorenkov et al. / Procedia Structural Integrity 42 (2022) 537–544 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

539

3

Nomenclature continues 

β β

damage parameter rate; critical damage parameter;

back stress tensor rate; back stress tensor; effective triaxiality function; energy density release rate; Young’s modulus of elasticity ; hydrostatic stress;

c 

inf , R  material parameters of the isotropic hardening; material constants of the kinematic hardening; a, b

v R

p

material parameters of the Lemaitre damage model; Y

, r s

ductility; reduction;

δ

E

Poisson's ratio; asymmetry factor.

ψ



number of cycles before fracture;

R

R N

2. Models 2.1. Isotropic hardening

The choice of an isotropic hardening model is the choice of a function approximating the uniaxial tension diagram. In this case, the Voce exponential hardening law is used. Its main advantage is the possibility of analytical differentiation by equivalent plastic strain e pl  without using iterative methods: 0 inf 1 exp( ) e pl R       = +  − −    . (1) 2.2. Kinematic hardening The kinematic hardening law describes the character of the hysteresis loop change. In this work, the Armstrong Frederick model was used, which is most widely used in modeling cyclic loading. The model is able to represent the hysteresis loop shapes quite well and has a small number of constants. Aygün et al. (2021) showed the ability of the model to simulate experiments under uniaxial tension quite well, including the plastic strain accumulation during cyclic loading. In a study by Dafalias et al. (2008) the possibility of using the model to predict plasticity under cyclic loading of thick-walled cylindrical and spherical vessels under various combinations of thermal and mechanical loads was demonstrated. In the traditional formulation, the law is:

e

a  −   β = ε β b



.

(2)

pl

pl

2.3. Lemaitre damage model The reason for choosing the Lemaitre damage model is as follows: relates the damage parameter rate to the plastic strain rate based on thermodynamic theory, taking into account the multiaxial stress state due to the triaxiality function, contains few constants. In general, the Lemaitre model is described as follows:

1 s Y r    −     − 

,

(3)

  =

1

e

v R





,

(4)

Y − =

2 2 (1 ) E  −