Issue 29

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

• [0, ) d t t   . At each step, it results

    .

i d 

= e d de 

2 3 = = 0 d d  

= E de  

=0

and thereby Eq. (12) gives

,

,

d

1

Hence, from Eq. (13-15) and (16), it results = = in β β β . • = d t t  . It results = d    and = > d d     

, and thereby Eq. (13-15) and (16) give (since 2 = 0  )





      

      





= / de 

= / = / de de 

1  



1 

1    2 3



d

/ ( , , ) = / e e β        

    





D

D

D









(19)



 



2 



2 



d

= / de

r

  

D

D

D





0

3 

d

= 0

= 0

Accordingly, it results =(1 / , / , 0) t D D de de C β         (see Fig. 4). Thereby, stress d   and the corresponding strain d   denote reference stress and strain values at the onset of the direct transformation. Moreover, from Eq. (12) and since 2 3 = = 0 d d   , it results =0 i d  , = e d de  , and 1 = = d t E de     . Accordingly, 1 = d d t E de     and = d d t de     are the starting stress and the starting strain of the direct transformation, respectively. • [ , ) d d t end t t t  . At each step, it results 2 > 0 d  and 3 = 0 d  , and thereby Eq. (12) gives = i d de  , =0 e d  , and = = d t    . Accordingly, a plateau in the stress-strain response is obtained during the martensitic orientation that occurs at the direct transformation stress d t  (see Fig. 3). During the overall transformation, the evolution laws for the volume fractions result in 1 2 = = / D e          as long as 2 <1  (and thereby 1 > 0  ). Accordingly, a complete direct martensitic transformation (corresponding to 2  going from 0 to 1) is obtained in the time interval d t  (see Fig. 4). Thereby, time = = d d end end t t dt t   corresponds to 2 =1  and 1 = 0  (but 2 > 0 d  and 3 = 0 d  ), or, in other words, to a complete direct transformation. In this situation and as shown in Fig. 4, variations of tentative volume fractions are the same as in Eq. (19), and Eq. (16) gives / 0 = 1 / = = 1 0 0 D pr D de de C β β β β                            (20)  and ( ) = t β β

 





    , with the strain produced during the martensitic lattice orientation being fully

Moreover, actual strain is = d t 

D

  (see Fig. 3).

i 

=

inelastic, and equal to

D

1 [ , ] d end

t t t  . At each step, alloy composition is governed by the projection described in Eq. (20). It results

•

     

2 = ( d t

d 

= e d de 

i d 

2 3 = = 0 d d  

E  

= 0

)

, and thereby Eq. (12) gives

,

, and

(see Fig. 3).

D t

= d de   ). Consider

1 1 [ , 2 ] t t   , the material element undergoes unloading conditions (in fact,

Traction unloading. For

the following relevant stress and strain values:

r  

















r  

d

r 

 

r 



r 

2 E de      1 d  d 

 



2 E de

E de E de

=

=

=

,

=

max

1

E

2





1 (    r  

   

2 = ( d t E  

d 





d 





) E E de

=

)

with

and

. Moreover, let introduce the time interval

max

max

D t

2

e   

= / r R



t

and the time values:



r 







= , r t  

dt t 

= , t t    r r

r

r

r

r

r

r

 

(21)

t

t

t

t

t

=

,

=

max

t

end

end

t

e



< 2 r end

t

t is assumed. Model predictions are as follows:

where

1

1 ( , ) r t t t   . At each step, it results

2     d     , from r

i d 

e 

de  ,

2 3 = = 0 d d  

= E de  

d

=0

=

•

, and thereby Eq. (12) gives

,

   , volume fractions follow Eq. (20). Otherwise, if

. Accordingly, it results = =(0,1, 0) t β β . In fact, if > d

1 3 = = = 0 d d d       . 2

Eq. (13-15) it results

105