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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