PSI - Issue 18

Plekhov O. et al. / Procedia Structural Integrity 18 (2019) 711–718 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

713

3

Furthermore, by following common practice we can write a decomposition:

.

( , ) e    

( ) e 

( ) e 

( ) T F T Ts

E



2

2

If we suggest that the specific heat is a constant value at room temperature ( ) c T c const   , then we can obtain an expression for the function ( ) F T in the form   0 0 ( ) ln ( ) F T c T T T T T     . If we also suppose that in the reference configuration 0   and 0 e   at 0 T T  , we can write:





2

(3)

,

( ) ln e S s p c T T E      .

( ) ( / 2 ( e 

) ) e 

( F c T T T T T E p Ts p E       ) ln ( )

T T

 

1

0

0

0

1

1

0

The energy balance equation has the following form:

(4)

q r     ,

e

x

where x q - heat flux. Substitution of (3) into (4):

1 x x cT TE W E p q r W E p p q r            , 1 ( ) '( ) e p p

(5)

where p W p     - the plastic work rate. Here, we take into account the assumption that the full strain rate    can be represented as the sum of three components: elastic   e  , plastic   p  and structural   p strain rates. The equation defining the fraction of the plastic work rate converted into heating  ( (1 )   ) is function characterizes the energy storage rate) can be expressed in the form: ( ) p

(6)

,

p

1 (    

) p Tp F p TF p W

where

1 '( ) p F E p Ts p   and 1 '( ) Tp F s p   . If we can neglect entropy associated with a structural parameter equation (6) can be written as 1 '( )

1 1 '( )

. p E p p W

  

It allows us to determine '( )

(1 ) p E p p W    and, consequently, estimate equation (3), describing the free energy of

system under consideration. To derive a constitutive equations for multidimensional case, we suppose that considered process of inelastic deformation is described in terms of two internal variables - p and r . An additional scalar parameter r characterizes isotropic hardening of the material. The second law of thermodynamics under isothermal conditions has the form:

F

F r r





.

: σ ε σ p

(  

) :

0

p

 

p





We introduce the following notations: R F r    ,

F     P σ p . Vectors of thermodynamic forces X and

fluxes J can be defined as   , , p = r  J p . We will postulate the existence of a convex function Ψ , which defines thermodynamic fluxes J by the relation:   , , = R  X σ P ,

(7)

X ,

J

=

Ψ   