Issue 23

C. Maletta et alii, Frattura ed Integrità Strutturale, 23 (2013) 13-24; DOI: 10.3221/IGF-ESIS.23.02

  

K

a



(5)

Ie

e

where the distance Δr is given by: * Δ A r r r  

(6)

with



  

K

1

*

r

 

(7)

I

AM

f  

2



tc

where AM  is the direct A → M transformation stress and it is assumed to be constant ( AM AM AM s f      ); tc f can be regarded as a transformation constraint factor, i.e. it is defined based on the plastic constraint factor in LEFM [29], and varies in the range between 1 and   1 1 2    , with the lower and upper bounds corresponding to the plane stress and plane strain conditions, respectively. The stress distribution in the martensitic region can be obtained by modified relations for bilinear materials and it is given by the following equation:         1 1 2 1 1 1 2 1 2 2 1 2 AM Ie M M L A tc K r r E f r                          (8) where  represents the Young’s modulus ratio ( / M A E E   ) while   1     and 1 / 2   for plane stress and    1 1 2       and 3 / 2   under plane strain conditions. The martensitic radius M r , can be calculated by using the condition   1 AM M M tc r f    :

2

1 2 2   

  

K

2



r

 

(9)

Ie

M

AM

f

E

2







tc

L A

A r can be calculated by imposing the equilibrium condition at the crack tip, as described in [23], and,

The austenitic radius

thus, the following relation can be obtained:

2

K

    

1

1

1

  

*   

2 r r

Ie

(10)









A

AM

AM

f

E

/

  



1    

f

2

1



tc

L A

tc

Several analysis have been carried out in [23] by considering a central crack with length 2 a in an infinite plate subjected to mode I loading conditions. In particular, the effects of the main thermo-mechanical parameters of the alloy, in terms of the transformation strain ( L  ) and stress ( AM  ), on the crack tip stress distribution and transformation region have been analyzed, as illustrated in Fig. 4. In addition, comparisons with FE results are also illustrated in the figure and good agreements are observed. In particular, Fig. 4.a illustrates the crack tip stress distribution for a SMA with Young’s moduli 50 A E GPa  and 25 M E GPa  , while Fig. 4.b illustrates the values of / A r a and / M r a as a function of the transformation strain ( L  ) for two values of the transformation stress ( AM  ). A marked decrease of / A r a is clearly observed together with a decrease of / M r a when increasing the transformation stress AM  , as a direct consequence of the increased values of local stresses near the crack tip in the austenitic region. Furthermore, a slight increase of / A r a together with a reduction of / M r a is observed when increasing the transformation strain and these effects become more evident when reducing the transformation stress. The estimates of the analytical model on crack tip stress-induced transformation have been also compared with experimental literature data in [24]; these latter have been obtained by synchrotron X-ray microdiffraction experiments of a miniature CT specimen, under opening mode conditions with a constant load P =2860 N and a crack length-to-width ratio / 0.55 a W  [5]. In particular, in Fig. 5 the contours of the austenitic radius, A r , under both plane stress and plane strain conditions, are compared with the microdiffraction patterns. The experimentally observed transformation region is between plane stress and plane strain contours and this is the expected result as X-ray observations represents volume-

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