Issue 45

L.M. Viespoli et alii, Frattura ed Integrità Strutturale, 45 (2018) 121-134; DOI: 10.3221/IGF-ESIS.45.10





1



 

    r

2

2

2

2

2 2( 1) 2 (2)  

(2)    (2)    

(2) (2)

(2) (2)

(2) (2) rr zz

(2)

( , ) 

2

2 1  

2 W r

r

K

 

rr      

    

   







2

rr

zz

zz





2

E

1 ( , )  



12 W r

r

K K

2

(1) (2)

(1) (2)

(1) (2) zz zz

    

rr rr    

   





1 2  

 

1 2



E





 



(1) (2)

(1) (2)

(1) (2)

(1) (2) rr zz

(1) (2)

(1) (2) zz rr

(1) (2) r r  

2 1  













rr      

    

rr    

   

zz    

   

    

zz





The integration of the strain energy density on a volume of radius R around the notch tip yields the elastic deformation energy:

   R



  A E R WdA W r 1 0       

   ,

   ,

   ,







W r 2

W r 12

rdrd

 

being the integration field symmetric to the notch bisector, the term W 12

gives a null contribution, the resulting energy is:

  

  

I

I

1

1

  E R E R E R      

 1



1

2 2

2

2 2





K R 1

K R 2

2

1

2

 1

 2

E

E

4

4

where:





          r  1 2 1



2

2

  

  

2

2

   1

    1 1

    1 1

  1

  1

    1 1

  



 

      

    rr  

    

  rr  









I

d

2

rr

zz

zz

zz

1









          r  2 2 1



2

2

  

  

2

2

   2

    2 2

    2 2

  2

  2

    2 2

  



 

      

    rr  

    

  rr  









I

d

2

rr

zz

zz

zz

2





and the integration area is:

    R A R rdrd R 2 0       



The average of the strain energy density on the area A gives:                E R W e K R e K R A R E E 2 1 2 1 2 2 1 2 1 1 2 2 1 1

with:

     1

I

   2

1



e

1

4

     2

I

   2

2



e

2

4

125