Issue 38

D. Marhabi et alii, Frattura ed Integrità Strutturale, 38 (2016) 36-46; DOI: 10.3221/IGF-ESIS.38.05

2

2

   

   

    

 

2

2

2

b *     R    

a *     R    

b *     R    



 



1 0

a R R             * *    b



1 4

2

2

m Rb ,

2

3

Rb













u dud

cos

cos

sin

  



I e



E

E





(7c)

3

3 2

  

   

Rb a b R R E R R E )                                R a b 2 * * * * 8  2



Eq Rb ,



The Integral transformation in circular section evaluated by



1 2  0   

2

2

2

    



2

   

   

2





Rb 2 ,  



R m Rb 

1 4

1 ,

m Rb

3

Rb





du d

(1 cos 2 )  

(1 sin 2 )  





u

(7d)



I c

  

E E

8

E

E

2



2

The over-energy (7a) under dissymmetrical rotating bending when a R *

1  , gives:

3

3

     

 

    

 

a R R * *                 b

a b R R * *            

2

Rb 2 2 )   

Rb 2

2









(

 

m Rb ,

Eq Rb ,

2

2



2





 









, 1 

, 1 

Te

Rb

(8)

E

4

b a E R R * * 8 1   

Rb d ,

              

 



For us, this result is necessary to identify critical crack in the cylindrical specimen. An Asymptotic Method and over-Energy expressed by Critical Stress The basic concept in the influence area of no damage crack is expressed by * * W WRb,d Te  We allow writing two limits equations near the small half axis and the large half axis of the ellipse (Fig. 1). With this asymptotic method, we have:

2*

       

b        

2 * 2









Eq Rb R ) , 

For

0,

(

Eq

2*

(9)

a        

2 * 2













For

,

(

)

Eq

Rb R

2

 

The equality (8) and (9) are considered to define the critical stress Eq. (10). It can be deduced by the application of the postulate (Eq. 2b):     Eq Rb Eq Rb Rb d Te Rb B C A , 2 * 2 * 2 0          (10a)

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