Issue 29

S. Terravecchia et al., Frattura ed Integrità Strutturale, 29 (2014) 61-73; DOI: 10.3221/IGF-ESIS.29.07

     

     

*      







* S N G N 

* G N G   d

 



(29)

d

d

b

b

gt

b

b

gt

c

c

, g tc

b

'

'

'

'







'   ' b c

b

g

bC

g

g

bb

bc

bb g ,

g

bC g are provided distinguishing between the singular and the regular part

The following contributions

bc

            

3 4

1

 



(reg)   bb g

1 x    

[1 ]   

g

g

[ ] Log x Log

x g 

(30a)

bb

bb

bb

(sing)

(reg )

16(1 ) 

3 4

3

 



(reg )   bc g



g

g

g

(30b)

bc

bc

bc

(sing)

(reg)

x  

16(1 ) 1

     

3 4

2

 











g

g

g

g

(30c)

bC bC

bC

bC

(sing )

(reg )

(reg)

16(1 ) 

x   

1













S N g 





 

 

 



g

g

N g

N g

N g

d

d

d

d

b

b bb

bc

bC

b

b bb

b

b bc

b

b bC

b

(sing)

(sing)

(sing)

( reg)

(reg)

(reg)









b

b

b

b

(31)

  

3 4 

1 ( )  









 

 



N g

N g

N g

d

d

d

b bb

b

b bc

b

b bC

b

(reg)

(reg )

(reg)

16(1 ) 

2







b

b

b

The sum of the effects in terms of distributions bb g , bc g bC

g eliminates the strong singularity 1/ r , while the weak

( ) Log r present in the

bb g distribution is eliminated in the weighing operation by integration by parts.

singularity

Eqn. (31) gives the coefficient in closed form. Numerical results Let us analyze the two-dimensional plate of Fig.1 having the following physical and mechanical characteristics: 1, 0.3, 0.1, 1 E s       .

y

y

u

g

x

x

x

x

a) b) Figure 4: Distribution on the boundary a) x u and b) x g .

The plate is subjected to a linear displacement distribution  u , including the nodal displacement C

 u , and to the linear

/ n    g u , all imposed on the boundary. The displacement and displacement gradient

displacement normal derivatives

fields imposed on the boundary have the distribution shown in Fig.4a,b. The field response is derived by the SIs after the solution being obtained. For the example here considered, characterized by the presence of eight nodes on the boundary and four corner (Fig. 1b), the know nodal vector U has sixteen components, whereas C U defined by

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