Issue 47

P. Foti et alii, Frattura ed Integrità Strutturale, 47 (2019) 104-125; DOI: 10.3221/IGF-ESIS.47.09

1





 1  2 1 

   1 K 

 

 

2

  , 

2

2

2

2

(1) 

(1)    (1)    

(1) (1)

(1) (1)

(1) (1) zz rr

(1)

 

rr      

    

   

r   

 





 

1 W r

r

(9)

2

2(1 )



rr

zz

zz

E

2

1







   2 K 

 

 

2

  , 

2

2

2

2

2  2 1 

(2) 

(2)    (2)    

(2) (2)

(2) (2)

(2) (2) zz rr

(2)

 

rr      

    

   

r   

 





 

2 W r

r

2

2(1 )

(10)



rr

zz

zz

E

2

1 ,

  

1 2  

 

2

(1) (2)

(1) (2)

(1) (2) zz zz

    

rr     rr

   

 



K K 







12 W r

r

[



1 2

E

(11)





(1) (2)

(1) (2)

(1) (2) zz rr

(2) (1)

(2) (1)

(2) (1) zz rr

(1) (2)

rr      

    

   

rr     

    

   

r       r













 

2(1 )



zz

zz

In order to evaluate the averaged value of the strain energy density, a sector-shaped cylinder of radius 0 R along the notch tip line, called ‘control volume’, is considered. For more consideration about the shape of the control volume, we remand to ref [25–27]. The control volume radius 0 R depends only on the material and its value decreases with increasing brittleness. In plane problems, both in mode I and mixed mode (I+II) loading, the control volume becomes a circle or a circular sector with radius 0 R respectively in the case of cracks and pointed V-notches, as shown in Fig. 3.

Figure 3 : Control volume (area) for: a) sharp V-notch; b) crack.

R can be estimated for the crack case under plane strain and plane stress conditions as follows [28–30]:

The radius 0

2

IC t        K

(1 )(5 8 ) 4     



R

plane strain

(12)

0

2

C t        K

(5 3 ) 4   



R

plane stress

(13)

0

While in the case of a pointed V-notch the critical radius can be assessed by [8]:

1 

1 

  

  

2 2(1 )

1 

  

  

2

1     C t    

I K 

K I

1  2(1 )

C

1 1

1





R

(14)

0

)    

 

)   

EW

1 4 (

1 2 (

C

The values of  and 1

I depending on  and on the stresses field are reported in Tab. 1. In the case of mode I loading

1 C K equals the fracture toughness IC K .

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