Issue 24

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

radius-vector to corresponding axis of laboratory system of coordinates. The volume integral in (A5) can be rewritten as a surface integral by applying the Gauss divergence theorem: 1 1 1 r 







i



 

 

e

e

d

x e n d   

x n dS  

,

(A6)





  





r



i

i

i S





i

i

i

where S i

is the surface of element i ; n 

is the unit outward normal to infinitesimal surface area dS . The element i is loaded

by forces acting at discrete interaction surfaces S ij integral can be replaced by the following sum:

(10). So, its total surface S i

can be represented as a set of S ij

, and surface

i N 

1

1



i





cos cos 

ij 

e

x n dS  

S q

,

(A7)







ij ij

ij

,

,







j

1

i

i

S

i

is total number of interaction surfaces ( 1 i N ij j

i    ). In accordance with S S

where  ij, 

is defined as shown in the Fig.5; N i

(A7) the sum in the second contribution in the right part of (A2) is:

1 j i   N i

1

i    i e e

2



S q

.

(A8)

cos

1

ij ij

, ij x

xx

xx



So, the expression (A2) can be rewritten in the form: 2 2 1 i i i i i xx i xx mean xx i G G K              

(A9)

which is Hooke's law for average stress and strain tensor components i xx  and i xx  . Corresponding expressions for other components of average stress tensor can be derived in analogous fashion:

 

  

i N 

i N 

1

i 

2







, ij x cos sin 









S q

S q

sin

yy

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

sin

(A10)

 

 

, ij x

ij ij

, ij x

i j

i j

 





j

j

1

1

i

  

    

1    

  

i N 

G

G

2

1

2

i 

i 

i 

i   

2





1  

S q

G

sin

2

i

i

mean

ij ij

, ij x

i yy

mean

yy



K

K





j

1

i

i

i

 

i N 

i N 

1

 

 

 

 

i xy

2









ij 

, ij x cos sin 



ij 



S q

S q

cos

ij ij

, ij x

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

 

 

, ij x

ij ij

, ij x

i j

i j

 





j

j

1

1

i

(A11)

 

2      1 i G

i N 

1

i 

, ij x cos sin 

ij 



S q

x

mean

ij ij

,



K





j

1

i

i

  

  

G

2

i G e  i xy

1   i xy

i 

i 

i 







G

2

2

i

mean

i xy

xy

K

i

Relations (A9)-(A11) are derived on the basis of assumption that total surface S i

of the element i is “occupied” by

of automaton/element i is free (i.e. neighbouring elements j can be formally considered as surfaces of interaction

interacting neighbours j . However, if a part of surface S i

), corresponding unoccupied surfaces S ik

“occupy” only part of S i

with virtual neighbours having zero stiffness. In that case specific forces at unoccupied surfaces S ik  0). Therefore equalities (A9)-(A11) are correct for discrete elements with partially free surface as well. In accordance with above mentioned the number of real has to be assigned as zero:  ij =  ij =0 (this doesn’t hold true for corresponding pair strains:  i ( j )  0 and  i ( j )

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