Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11 124 3 1 2 2 1 2 3 2 cos3 6 , , sin 3 sin 3 3 3 J I J J                  I e f f f (A7) The latter equation yields by (A3) and (A4): 2 3 2 2 1 6 6 cos3 3 sin 3 6 3             f f f f (A8) whence   follows by (A6). On the other hand, Eq. (A7) can be recast as 3 3 2 2 6 sin 3 cos3 J        f f (A9) and , by differentiating with respect to ε , it leads to:   3 3 3 2 3 2 2 3 6 2 6 sin 3 cos3 3 cos3 sin 3 J J                     f f f f (A10) Noting that by (A9), (A3) and (A7) it turns out that:     2 3 3 2 3 1 2 2 1 1 2 2 1 sin 3 cos3 6 2 3 3 J J                      f f f f f f f f f f   (A11) and by (A6), (A5) and (A3) it turns out that:   2 1 1 2 2 3 1 ,                     f f f f f f  (A12) it is a simple matter to verify that (A10) yields:   1 2 2 1 2 1 cos3 3 2 sin 3 ij i j                f f f f f f    (A13) where cos3 2 2 0 2 2 cos3 3sin 3 0 3sin 3 3cos3                        (A14) Alternative expressions for   and   , more effective from a computational point of view, are:   3 2 2 2 3 2 2 2 2 2 3 2 2 3 3 2 3 3 3 4 1 cos3 6 sin 3 1 1 cos3 6 cos3 2cos 3 5 sin 3 sin 3 3 6 18cos3 J J J J J J J J J J J J                                                            (A15) The expressions for   and   derived in (A8), (A6) and (A13), or the alternative expressions in (A15), get greatly simplified when computed in an orthonormal basis of eigenvectors of ε . The tensor 1 f is spherical, thus it is represented by   diag 1;1;1 3 where   diag • denotes the diagonal matrix with the enclosed eigenvalues. Recalling (A7), 2 f is