Issue 24

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

neighbors of element i ( real i N ) can be used in expression (16)-(16  ) when calculating average stress tensor components. At the same time a total number of neighbours (including virtual ones) have to be used in expression (A3). Thus, average stress and strain tensor components are connected by Hooke’s law when expressions (A1) are used for assigning central (normal) and tangential (shear) response of particle to mechanical impact of neighbours. This result confirms correctness of the proposed approach to description of interaction of discrete elements (or movable cellular automata) simulating locally isotropic linearly elastic medium.

A PPENDIX B

P

roposed expressions   i ij ij mean M       

i 



  

i

mean

(B1)

 

ij 

ij 

M

i

for scaling specific central and tangential forces of response of the element/automaton i to the impact of the neighbor j provide a fulfillment of radial return algorithm of Wilkins (25)-(26) for average stresses in the volume of the element i . Actually, substitution of relations (B1) in the expression (16  ) for  =  = x leads to the following equation:   2 , , , 1 cos cos sin i i N N i xx ij ij ij ij x ij ij ij ij x ij x S q S q                





 







j

j

1

1

i

 

  

i N 

i N 

1

2













, ij x cos sin 



M S q

S q

cos

(B2)

i

ij ij ij

, ij x

ij ij ij

, ij x

 





j

j

1

1

i

1 j i   N i

1

 1  



i 

2



M

S q

cos

i

mean

ij ij

, ij x



According to (16  ) the sum

 

  

i N 

i N 

1

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

in the first contribution in the right part of (B2) has a meaning of corresponding average stress tensor component before stress returning procedure (“elastic” stress i xx  ). In accordance with (A7)-(A8) the second sum

1 j i   N i

1

2



S q

cos

ij ij

, ij x



in the right part of (B2) is a diagonal component i xx Therefore, the expression (B2) can be rewritten in the form:       1 i i i i xx i xx i mean xx i i i xx mean i mean M M e M              

e =1 of unit second-rank tensor.

(B3)

which coincides with expression (26) for the case  =  = x . Corresponding expressions for other components of average stress tensor can be derived in analogous fashion:       1 i i i i yy i yy i mean yy i i i yy mean i mean M M e M               (B4)

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