Issue 29
N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11
2 J
J
I
J
cos3 sin 3
6
I
e
1
2
(A7)
,
,
3
2
f
f
f
1
2
3
sin 3
3 3
The latter equation yields by (A3) and (A4):
6
6 cos3
3
2
(A8)
f
f
f
f
3
2
2
1
sin 3
6
3
follows by (A6). On the other hand, Eq. (A7) can be recast as
whence
2
J
6
sin 3 cos3 f
(A9)
3
f
3
2
and , by differentiating with respect to ε , it leads to: sin 3 cos3 3 cos3 sin 3 f f f f Noting that by (A9), (A3) and (A7) it turns out that: 2 sin 3 cos3 J f f 3 2 3 2
J
J
6
2 6
(A10)
3
3
2
3
3
3
2
6
(A11)
2 3
J
f f f f
f f f
3
f
3
1
2
2
1
1
2
2
1
and by (A6), (A5) and (A3) it turns out that: 2 1 1 2 2 1 it is a simple matter to verify that (A10) yields: 2 1 cos3 3 2 sin 3 1
3 f f f f f f ,
(A12)
ij
f f f f
(A13)
f f
i
j
2
2
1
where
cos3 2 2 2 2 cos3
0
(A14)
3sin 3
3sin 3 3cos3
0
and
, more effective from a computational point of view, are:
Alternative expressions for
1 cos3
3 J
J
6
2
2
sin 3 1 sin 3
J
J
J
1
2 cos3 2cos 3 5
J
cos3
6
(A15)
3
2
2
2
2
2 3
2
sin 3
J
J
J
J
J
J
3 6
18cos3
2
3
3
2
3
3
3
4
and
derived in (A8), (A6) and (A13), or the alternative expressions in (A15), get greatly
The expressions for
simplified when computed in an orthonormal basis of eigenvectors of ε . The tensor 1
f is spherical, thus it is represented
diag 1;1;1 3 where
diag • denotes the diagonal matrix with the enclosed eigenvalues. Recalling (A7), 2 f is
by
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