Issue 24

S. Psakhie et alii, Frattura ed Integrità Strutturale, 24 (2013) 26-59; DOI: 10.3221/IGF-ESIS.24.04

 

  

i N 

i N 

1

i xx

2















, ij x cos sin 



S q

S q

cos

ij ij ij

, ij x

ij ij ij

, ij x

 





j

j

1

1

i

1 2 i N G S q  i ij ij  

  

i N 

2







, ij x cos sin 









S q

cos

(A2)

 

 

, ij x

ij ij

, ij x

i j

i j

 





j

j

1

1

i

 

2      1 i G

i N 

1

i 

2



S q

cos

mean

ij ij

, ij x



K





j

1

i

i

To understand the meaning of the first contribution in (A2)

    

      

i N 

2







S q

cos

 

ij ij

, ij x

i j

1 1 2 j  

G

i



N

i

   the notion of average strain tensor in the volume of discrete element i has to be introduced. Considering relative normal and shear displacements of interacting elements i and j as components of the vector of relative displacement the expression for average strains i   in the volume of element i can be formally written by analogy with i   . In considered two-dimensional problem statement it has the following form: , ij x , ij x 1 cos sin  i ij ij i j j S q    

1 j i   N i

1

 

 

i 



cos cos 

ij 



cos sin 

ij 





q S

(A3)

 

 











ij ij

ij

ij

,

,

,

,

i j

i j



According to (A3) the first contribution in the right part of (A2) has the meaning of corresponding average strain tensor component:

1 j i   N i

1

ij ij q S  

 

i 

2









, ij x cos sin 



cos

.

(A4)



 

 

, ij x

, ij x

xx

i j

i j



To understand the meaning of the sum

1 j i   N i

1

2



S q

cos

ij ij

, ij x



in the second contribution in (A2) the procedure of homogenization of unit second-rank tensor 1 0 0 ˆ 0 1 0 0 0 1 e  has to be considered. General expression for average value i e  of unit tensor component i e 

in the volume of discrete

element i ( i e

i



e

) has the form:







r

1

1





i



 



e

e d 

e

d

,

(A5)





  



r



i

i





i

i

where the identity



r r

e     

e

e







  was applied (the Kronecker delta  

and the Einstein summation convention are employed here); r 

is projection of a

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