Issue 24

E.I. Kraus et alii, Frattura ed Integrità Strutturale, 24 (2013) 138-150; DOI: 10.3221/IGF-ESIS.24.15

- conservation of energy: - the strain rate tensor:  Stress tensor is represented in the conventional ij ij    e    ( ) , i j u u  , j i 0, 5 ij  

ij 

   

P s

where

,

ij

ij

- ij s : deviator of the stress tensor - P: pressure -  : density - u: speed Prandtl-Reuss stress-strain relations 2 , ij ij ij s d s G         ij 

ij 

3        with the plastic strain increments given by Levy-Mises Equation 2 0 ij ij s s ij  ij

2 3

Y  where G -shear modulus, 0

Y - dynamic yield stress, for definition scalar d   known procedure used to bring to the yield

circle. The plastic flow will be described by conserving the deviator of stress tensor at the elastic limit. We will assume that after the deformation increase, the stresses change and exceed the yield circle. This contradicts the yield condition and cannot be realized. Instead, we will assume that a plastic flow takes place in the material and the stresses remain at the elastic limit, i.e. on the yield circle. The plastic part of the deformations is perpendicular to the yield surface that leads to limitation of exactly those stresses which are connected with this part of the deformations. Thus, summarizing the assumption about the yield, we can write down the yield condition in the terms of the stress tensor deviator. An algorithm realizing such a condition was proposed by M.L. Wilkins [1]. If a redundant change of the stresses in a particular element has led to a law violation, then the main values of the stress tensor deviator have to be corrected in such a way, that the condition is met again. e consider the three-term Mie-Grüneisen equation of state with the solid-phase free energy being determined as 2/3 2 , , 0 0 ( ) 1 ( , ) ( ) ln 2 x v l v e V V F V T E V c T c T T V                 (1) where V is the specific volume,   x E V is the “cold” energy, T is the temperature, , 3 / v l c R A  is the specific heat of the lattice at constant volume, A is the mean atomic weight, R is the gas constant,   V  is the Debye temperature, and c is the experimental value of the electron heat capacity under standard conditions. The elastic (cold) component of energy E x ( V ) is related exclusively to interaction forces between the body atoms and is equal (including the energy of zero vibrations) to the specific internal energy at the absolute zero temperature. The thermodynamic model of a few-parameter equation of state is based on the dependence of the Grüneisen parameter  on the volume W E QUATION OF STATE , 0 v e

P

2

2

2

2

t

,0



 

1  



( ) V

a

(2)





s 

aV V

K

3 (1

/ )

(

2 / 3)

s

0

s 

0 v K V c  / s



s K is the adiabatic modulus of volume compression, v

c is the specific heat at constant volume,

where

,

and ,0 t P is the thermal pressure in the initial state. The general expression for the volume dependence of the Grüneisen parameter has the form

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