Issue 49

N. Burago et alii, Frattura ed Integrità Strutturale, 49 (2019) 212-224; DOI: 10.3221/IGF-ESIS.49.22

2

  T

K ln 

  















(   



) (    ε ε ε ε

T T

))

(

)

 

p

p

0

0 

0 



2

2

k

d   

q

( ) D H k   p p



 

T T        dt T 

( ) H k





p e e





p

where H is the Heaviside's function. It is assumed that the elastic components of the strain deviator are small compared to unity. The effect of thermal expansion is taken into account by a member with a coefficient  . The component of the dissipation rate, which is responsible for the plastic flow, is assumed to be a homogeneous function of the first order of the rate of plastic deformation, which corresponds to the case of an elastoplastic medium. Plastic deformation increases when active loading condition 0 0 ( ) 0 p p T        ε ε e is satisfied. It is also assumed that the material is plastically incompressible i.e. the dissipation rate depends only on the plastic strain rate deviator, which is usually well performed for metals. The resistence of the medium is represented by the modules of elasticity and yield strength. In addition to temperature, deformation and plastic deformation it also depends on the additional structural parameter of the state  . This parameter is called damage. Following relations utilize the damage parameter: 0 ( ) g      , 0 ( ) K K K g   , 0 ( ) p p p k k g   , where 0  and 0 K are elastic modules and 0 p k is a yield strength for intact material. The functions g  , K g and p g are decreasing from 1 to 0 while   . They provide a decrease in the resistance of the medium with an increase in the damage that occurs when the condition of destruction 0 0 ( ) 0 p T         ε ε e is fulfilled. The kinetics of the destruction process is determined by the dependence of the dissipation rate on the growth rate of damage. Non-negative functions 0  , 0 K , 0 p k , k  and q k depend on 0 T  ε and 0 p ε . So the constitutive relations take the following form:

 











  

U T 

  

k T   

p    σ I σ

q

q

T p K ln     

T

  









2 (     ε ε )

(   

T T

)

σ

(2)

p

0

0 

0 



 



 

d







 



1

( )    H

p k  

  

( ) H k

e

e e





p

p

p

p

dt

Boundary conditions are



0                             u n u σ n q n 0 ( 0      0 ( ) 0 0 ( ) un S t n un S \ S t u S t  u T S t u u S \ S t  T T S \ S t    

x



)   

p

x

σ n n

n

x



(3)







p

x





x x



q

T

n

1, 2   . Initial conditions are

where

0

0 x V t         x x u u

 

0    

(4)

T T

0

ε

p

0

0

Thus it is required to solve the initial-boundary problem for the system of equations (1), (2) with the boundary (3) and initial (4) conditions.

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