Issue 20

R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

      

   

2

 

) k k R

(1

  

  

  

  

2

2 3 18 

2

2

x k R

R y

y

2

y

x

0    y





F G   





3

1

x

2

2

2

2

r

r

r

r

2

   

2

 

) k k R

(1

 

     

  

  

2

2 3 10 

2

2

x k R

R y

y

2

y

x

0      1 y

1   

F G   





1

(2)

y

2

2

2

2

r

r

r

r

2

   

   

2 k k R xy )

 

(1

  

2 4 R xy r

2 R y  6 8 12 3  2

2 2

k

2

R y

y

x



0    y

 



x

xy

4

2

4

r

r

r

2

2

where r x y   . The x-y coordinate system has origin in the inclusion centre (Fig. 2). Further, the coefficients , x y k k depend on the elastic constants of the base material ( 1 1 , E  ) and of the inclusion ( 2 2 , E  ) [6]:   (3 1) (1 3 ) c E E E      

2

2

1

1 2



k

x

2 2 (8 2 6 1 ) c c  

2        2 2 (1 )  E

1 2  

2  2 6 2 ) c c

  

E c

E E

(2

2

1

1

1

2

1 2

(3a)





2  2 E c    

1  (8 3 )

2 1 c E E 

(3 )



k

y

2 2 (8 2 6 1 ) c c  

2        2 2 (1 )  E

1 2  

2  2 6 2 ) c c

  

E c

E E

(2

2

1

1

1

2

1 2

2 1 1 c   

with

, and

2 8 (3 2 ) , y R y  2 2

2 4

R y

24





F

G

(3b)

4

6

r

r

Figure 2 : Circular elastic inclusion in an infinite elastic plane under remote uniform tensile stress 0 y  .

The elastic stress field in such heterogeneous materials can be computed by exploiting both Eqs 1-3 for a single inclusion and the superposition principle, provided that the inclusions are assumed to be non-interacting (as reasonably occurs for widely spaced inclusions). By considering point P belonging to the base material (Fig. 3), the resulting stress field is determined approximately by summing up the effects of the inclusions (such as particles 1, 2, 3, 4, etc. in Fig. 3), that is:

8