Issue 29

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

:= ( , ) ori ori d    β    .

with I o

the indicator function of the zero value and

It is worth pointing out that dimensional multiplicative unitary coefficients have to be considered in previous relationships, when necessary, in order to respect the unit of measure of the free-energy (namely, work per unit volume) and of the pseudo-potential of dissipation (that is, power per unit volume). These coefficients have been omitted here for the sake of compactness. Accordingly, since derivation with respect to a finite quantity (for instance, e  ) is formally equivalent to the one with respect to a perturbation (for instance, e d  ), constitutive choices (7), (8), and (9) give the interior forces as equal to: 4



  

) ori

(10)

e 





j 

e 

i 

I (   o i



E d

=

j

j

=1

β r  

( , , ) I ( ) C β       β 0

(11)

Governing equations Assuming strain  as control variable, the governing equations of the SMA thermodynamical problem, obtained from equilibrium relationships (6) and constitutive choices (7), (8), and (9), give the evolution of stress  and alloy composition β . Stress  results from:

4



(12)

=  



j 

e 

with = 

d d d  

i 

i 

2  = ( (| |) H d



3  (| |))



E d

d

H d

d

,

j

e

j

=1

Alloy composition β is found by means of a single-step prediction-projection procedure. Tentative values of volume fractions = j j j d       (with =1, 2, 3 j ) are computed first, where:



 

= d d d    

3 

(13)

1

2



   



  

)/ }| | d   

2  3 

3  ={[1 ( )] ( H H  2  ={[1 ( )] ( H H 





2  ( ) (

 

d

H H

) /

(14)

d

D

r

R



  





) / }| | d   

 





3  ( ) (

  

d

H H

) /

(15)

d

D

r

R

and then β is obtained as

  

β 

β β  

   

1 2 3 =( , , ) if    t

 

C C

=

,

(16)

β

pr

if

β

the orthogonal projection of β 

pr

pr

pr

pr t

being on C . Finally, it is worth pointing out that constitutive choices in Eq. (7) and the convexity of the pseudo-potential of dissipation in Eq. (9) allow to a-priori satisfy the inequality constraint prescribed by the second law of thermodynamics, [15-17]. 1  2  3  =( , , ) β

Figure 2 : Applied strain  vs. time  .

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