PSI - Issue 13

A. Prokhorov et al. / Procedia Structural Integrity 13 (2018) 1521–1526 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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4

small strains, the following kinematic decomposition takes place: p   e ε ε ε ,

(4)

where p ε - plastic strain tensor. It is assumed that plastic strains obeys associated flow-rule:

F 

y

ε

p

 

,

(5)

σ



where  - plastic multiplier,

y F - plastic potential,

0   ,

0 y F  . Plastic multiplier satisfies Kuhn-Tacker

conditions:

0 y F   . Plastic potential is computed according to the von Mises plasticity criterion: 3 : ( ) 0 2 y ys pe F      d d σ σ ,

(6)

2 : 3

p p ε ε - effective plastic strain. In this

where d σ - deviatoric part of stress tensor,

pe  

( ) ys pe   - yield function,

work, isotropic hardening of the material is described by:

0 ys h pe        , ( ) ( ) ys pe

(7)

0 ys  - initial yield stress. Functions

( ) h pe   for both materials are calculated from stress- plastic strain curves

where

presented in figure1(b). Kinematic boundary conditions and zero initial conditions were used for the simulation of the tensile test:

1   u 0 ,   u u , 0

(8)

(9)

2

0 0 p t   ε , e t   ε , 0 0 t   σ . 0 0

(10) (11)

(12) We will use the following equation for the determination of the surface temperature as the first approximation:

p W

T T

 



,

(13)

0

int

c



where 0 T - initial temperature of the specimen, int  - Taylor-Quinney coefficient which represents the amount of the dissipated heat and remains constant during the loading process, : p d p W d   σ ε - plastic work,  - density, c - specific heat. 3.2. Large plastic strains In case of the large plastic strains, Cauchy stress tensor is replaced by the second Piola-Kirchhoff stress tensor P and equilibrium equation has the form:   P 0 . (14) Green-Lagrange full strain tensor is determined according to the relation: