Issue 47

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

1 I Pl. Strain

1 I Pl. Stress

  [rad]

1 

2  [°]

0.10   1.1550 1.1497 1.1335 1.1063 1.0678 0.9582 0.8137 0.7343 0.6536

0.15   1.0925 1.0880 1.0738 1.0499 1.0156 0.9173 0.7859 0.7129 0.6380

0.20   1.0200 1.0162 1.0044

0.25   0.9375 0.9346 0.9254

0.30   0.8450 0.8431 0.8366

0.35   0.7425 0.7416 0.7382

0.40   0.6300 0.6303 0.6301 0.6282 0.6235 0.6024 0.5624 0.5344 0.5013

0.30   1.0250 1.0216 1.0108 0.9918 0.9642 0.8826 0.7701 0.7058 0.6386

0

1

0.5000 0.5002 0.5014 0.5050 0.5122 0.5445 0.6157 0.6736 0.7520

15 30 45 60 90

23/24 11/12

7/8 5/6 3/4 2/3 5/8

0.9841 0.9090 0.8247 0.7311 0.9547 0.8850 0.8066 0.7194 0.8690 0.8134 0.7504 0.6801

120 135 150

0.7524 0.6867 0.6186

0.7134 0.6558 0.5952

0.6687 0.6201 0.5678

0.6184 0.5796 0.5366

7/12

I for pointed V-notches under plane stress and plane strain conditions.

Table 1 : Parameters 1

The elastic deformation energy in the control volume around the notch tip is given as follows:





R





( ) R A E W dA W r        1 0 

( , ) 



( , ) 



( , ) 





2 W r

12 W r

rdrd

(15)

The integration field is symmetric with respect to the notch bisector; this condition sets to zero the contribution of 12 W . Therefore:

I

I

1

1

     

     

1

2

2

2

1 

2 

2

2

   

K R 

 

K R 

E E E

(16)

  R

  R

  R

1

2

1

2

1 

2 

E

E

4

4

Where   1 I 

  2 I 

and

are:













2

2

2

2



(1) 

(1)    (1)    

(1) (1)

(1) (1)

(1) (1) zz rr

(1)

 

rr      

    

   

r   







 



I

d

(17)

2

2(1 )

  



rr

zz

zz

1

















2

2

2

2



(2) 

(2)    (2)    

(2) (2)

(2) (2)

(2) (2) zz rr

(2)

 

rr      

    

   

r   







 



I

d

(18)

2

2(1 )

  



rr

zz

zz

2





Their values, assessed for different geometries and stresses field, are reported in Tab. 1 as a function of Poisson’s ratio. The value of the area on which the integration is carried out is given by:



rdrd R    

0   0 R

2





A

(19)

  R

0



 being expressed in radians. The averaged elastic deformation energy on the area results to be:

E

1

1

    R R

  1 e K R  2 1

  2 e K R  2 2

1 



2 



2(

1)

2(

1)

A E    

  

W

(20)

0

0

E

Being:

4 I

  1 



e

(21)

  

1 2

1  

109