Issue 24

S. Psakhie et alii, Frattura ed Integrità Strutturale, 24 (2013) 26-59 ; DOI: 10.3221/IGF-ESIS.24.04 54     2 , , , 1 1 2 , , , 1 1 2 , 1 1 cos cos sin 1 2 cos cos sin 2 1 1 cos i i i i i N N i xx ij ij ij ij x ij ij ij ij x ij x j j i N N i ij ij ij x ij ij ij x ij x i j i j j j i N i i mean ij ij ij x j i i S q S q G S q S q G S q K                                                     (A2) To understand the meaning of the first contribution in (A2)     2 , 1 , , 1 cos 1 2 cos sin i i N ij ij ij x i j j i N i ij ij ij x ij x i j j S q G S q                          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:     , , , , 1 1 cos cos cos sin i N i ij ij ij ij ij ij i j i j j i q S                      (A3) According to (A3) the first contribution in the right part of (A2) has the meaning of corresponding average strain tensor component:     2 , , , 1 1 cos cos sin i N i ij ij ij x ij x ij x xx i j i j j i q S                . (A4) To understand the meaning of the sum 2 , 1 1 cos i N ij ij ij x j i S q     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 i e e    ) has the form: 1 1 i i i i i r e e d e d r                  , (A5) where the identity r e e e r            was applied (the Kronecker delta   and the Einstein summation convention are employed here); r  is projection of a