Issue 29

M. Marino, Frattura ed Integrità Strutturale, 29 (2014) 96-110; DOI: 10.3221/IGF-ESIS.29.10

Interior forces are split in non-dissipative (denoted by the superscript nd ) and dissipative terms (superscript d ). The constitutive laws are chosen by introducing the free-energy = ( ) S   (providing the non-dissipative terms) and the pseudo-potential of dissipation = ( ) S    (providing the dissipative ones),

 

 

 

nd

d

nd

d

= b b b nd 

d

e 

e 



e 



i 

i 



i 





(7)

=

=

,

=

=

,

=

β β 





e   

e 

i   

i 

 

The free-energy  of the alloy is chosen as: ( , ) := ( ) ( ) e el e ch       β β where: • el  is the elastic free-energy contribution: 4

(8)

1 2

 

( ) :=    



e 

j 

: E d d 

e 

d

:

el

e

el

j

e

j

=1

where  and

el  are the reference values at = t 

of Cauchy stress and elastic free-energy, respectively. Assuming an

at =0 

undeformed material at the initial configuration, then it results = = 0 el   ch  is the free-energy contribution, related to the phase change: ( ) := ( , , ) I ( ) ch C β β r β β        where the indicator function I C •

.

ensures condition (4) to be satisfied. Moreover, r represents the phase transformation

rate vector, defined as:

a g y

( , ) v

    

   

t

3





x, y,

) := ( , ) | |, g y x v

x y

1 2 3 , = ( , , ) v v v v

(

with ,





r

v



g y 

( , ) v

 Phase change activates on the basis of temperature and stress states, being here regulated by activation functions:









( , ) :=[1 ( )] ( g y H v H y  v



d 





 

r   r



( ) ( H v H y

)/

)/



D

R

3

2









2 v ( , ) := ( , ) g y  v v ( , ) :=[1 ( )] ( g y   a g y  

 

d 









H v H y

( ) ( H v H y

)/

)/

D

R

3

g y 

( , ) v

/ / D R    as well as transformation stresses

/ / d r    depend on temperature T and they can be straight

Transformation strains

obtained from experimental data, as reported in Fig. 1. Since present work addresses isothermal conditions, transformation stresses and strains are here fixed parameters. Nevertheless, when non-isothermal conditions are addressed, values of transformation stresses and strains at the reference temperature T (at = t  ) can be considered. The pseudo-potential of dissipation  is chosen as: ( , ) := ( ) ( ) i ch fr i       β β     (9) where: • ch  is the pseudo-potential of dissipation related to the phase change: 2 ( ) := 2 ch  β β   • fr  is the pseudo-potential of dissipation related to the flow rule: ( ) := I ( ) ori fr i o i        

101