Issue 29

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

• = r t t  . It results = r 

  and = < r   

r  

2 =1 

3 = 0 

, and Eq. (13-15) give (since

and

)





     

      





1 



1 



d

= / de

= / de

/ ( , , ) = / e e β        

    





R

R

R











(22)



2  d   



2 



 

= / de

=1 / de

r

  

R

R

R





0

3 

3 

d

= 0

= 0

C  β 

and thereby = β β 

resulting r   denote reference stress and strain values at the onset of the reverse transformation. Moreover, from Eq. (12) and since from Eq. (16). Accordingly, stress r   and the corresponding strain

= r r t de    

= r t 

= r r t 

i d 

= e d de 

= E de   

 

2 3 = = 0 d d  



2 E de

= 0

, it results

,

, and

. Accordingly,

and

are

2

the starting stress and the starting strain of the reverse transformation. • [ , ) r r t end t t t  . At each step, it results 2 < 0 d  and 3 = 0 d 

i 

de  ,

e d 

d

=

=0

, and thereby Eq. (12) gives

, and

= = r t    . Accordingly, a plateau in the stress-strain response is obtained during the martensitic de-orientation, occurring at the reverse transformation stress r t  . During the overall transformation, the evolution laws for the volume fractions result in 1 2 = = / R e         as long as 2 > 0  (and thereby 1 <1  ). Accordingly, a complete reverse martensitic transformation (corresponding to 2  going from 1 to 0) is obtained in the time interval r t  . Thereby, time = = r r end end t t t dt   corresponds to 1 =1  and 2 = 0  (but 2 < 0 d  and 3 = 0 d  ) or, in other words, to a complete reverse transformation. In this situation, variations of the tentative volume fractions and alloy compositions are obtained from Eq. (13-15) and (16), and therefore



     



1 



d

= / de

    

    



1 / de 



1           0

R

R



β 





= = 0 pr

(23)

= / de 

 

C  

2 





d

= / de

β β

R

R



0

3 

d

= 0

    , with the strain produced during the martensitic lattice de-orientation (during the

Moreover, actual strain is = r t 

R

=     . Accordingly, considering a complete loading

[ , ] r r t end

t t  

time interval

) being fully inelastic and equal to

i

R

), the total inelastic strain at the end of the test is D R      .

[0, ] r end t

 

unloading cycle (that is,

1 [ , 2 ] r end t t



t

•

. At each step, alloy composition is governed by the projection described in Eq. (23). It results

e 

de  ,

i d 

= E de    .

2 3 = = 0 d d  

d

=

=0

, and thereby Eq. (12) gives

, and

1

0 dt  , it results

It is worth pointing out that, in the limit

,                  , , , d  d  d  r  r  r  r       

d  t

d 

d 

r 

r 

d 

r 

t

t

t



= /  



= /  

      

0 dt  , the obtained constitutive

d 

r 

E

E

E

/

where

and

. In summary, in the limit

d

d

D

1

1

2

relationship during a tensile traction-release test is (see Fig. 3)

 





d  

d 



E

if

1





d    



Loading ε [0,ε ] : ( )   

d 







(24)

if

<

      

D

max







d 









d 





E

(

) if

>



D d

D

2

2      ( d  E     







) if > 

r 

D d







Unloading ε [ε ,0] : ( )   

r 

r  





r 







(25)

if if

<

  

R

max

r    







E

R

1

and the corresponding evolution of single-oriented martensitic microstructure 2  is

106