Issue 50

P. Livieri et alii, Frattura ed Integrità Strutturale, 50 (2019) 613-622; DOI: 10.3221/IGF-ESIS.50.52

O ORE -B URNS ON A CIRCULAR CRACK

L

et Ω be a fixed set. We can reconstruct the OB integral along the front crack ( ) 

  by its values along  by the

equation

( , ', )  





(1, '/ , )  

K Q

K Q

(3)

I

n

I

where 0   If we “read” the boundary point Q’ in terms of an angle α that is, for example Ω is star shaped with respect to the origin, (3) takes the simplest form (   ) n Q   with

( , , )   





(1, , )  

K

K

(4)

I

n

I

In the particular case when Ω is a disk of radius a centred at the origin of the plane ( , ) x y , we denote by (x,y) the system in dimensionless coordinates x= x a and y= y a (see Fig. 2).

Figure 2 : reference circular crack

By definition, for the unit disk Ω’, by (1) and (2) it follows:

a

2

( , ) ( , ) x y h x y 



(5)

( ) 



K

dx dy

I

,0



  2



2









y  

x

cos

sin

2 2

 

x y

1

With

(6)

( , ) 1/ ( , ) h x y f x y 





d





f x y

( , )

(7)



  2



2







y  

x

cos

sin









    0, ,  





By introducing the change of variables



0, / 2 

(1 sin )cos cos sin cos cos (1 sin )sin r r            x r y r   

 

(8)



where

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