Issue 27

A. Kostina et alii, Frattura ed Integrità Strutturale, 27 (2014) 28-37; DOI: 10.3221/IGF-ESIS.27.04

e              p p 

(6)

Elastic strains are defined by linear Hooke’s law:

0 0 e K     

(7) (8)

 

d   

e

d 



G

2

  - deviator

 - spherical part of the stress tensor, d

where K - isotropic elastic modulus, G - elastic shear modulus, 0

0 e  - spherical part of the elastic strain tensor, e d

  - deviator part of the elastic strain.

part of the stress,

Dissipation function for a medium with defects can be represented in the following form [13]: : ( ) : 0 p T p q p T                   

(9)

where  is a free energy,  - density, q - heat flux vector, T - temperature. Dividing the thermal and mechanical problems and basing on the Onsager principle, we can obtain from (9) constitutive equations for calculating kinetics of plastic and structural strains: ( ) p p p                  (10) ( ) p p p p                (11) The kinetic coefficients   , p  and p   have the following form:

1 

1

 





S   



c

Exp  

1

 

a



1

1 

1 ( , , , , )   

 

p

  

 

H p p

S

p

c

c

y

Exp  

1

  

a

2

1

 



p





p

 , p

p   - characteristic relaxation times,  - stress intensity tensor, c S 1 a , 2

 ,

a - material constants,

y S -

where 

yield stress, p - intensity of p  , , c c

(   

  - material

p  - scaling factors, ( , , , ) H p p  

  



p p p p 

f

2

1)



c

c

c

c

function ( it can be considered as “degree of system nonequilibrium”). It is supposed that thermodynamic force p        can be written as:

  

   

 

   

   

c p p p p

  

 

p 

p 

 

  

1

 

 



f

1

(12)

 

p   



c  G p

p

2

c   

c

c

where ( ) f p denotes a power function for modeling of nonlinear hardening process:

a

c p              c p p k p

f

(13)

k is a scaling factor, a is the exponent. Eqs. (6)-(8) and (10)-(13) represent a closed system for a plastically deformed solid with nonlinear hardening. From the first thermodynamic law, we can obtain the expression for calculation the  parameter in the following form:

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