Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11 125 represented by   1 2 3 diag ; , ; e e e  where i , 1,..., 3 i e  are the eigenvalues of e , which are the roots of the characteristic polynomial of e : 3 2 3 0 J J       (A16) Using (A4), it is a simple matter to verify that those roots are   2 3 cos 2 3 , k       0, -1,1 . k  As a consequence, 2 f is represented by:     2 cos 0 0 2 0 cos 2 3 0 3 0 0 cos 2 3                   f (A17) In passing, it is noted that  may be assumed to belong to [0, 3]  , implying that the eigenvalues of e are sorted in descending order. Substituting 1 f and 2 f into (A8), it turns out that 3 f is represented by:     3 sin 0 0 2 0 sin 2 3 0 3 0 0 sin 2 3                    f (A18) Analogously, substituting the above representations into (A13), the representation of   is obtained. The only nonzero terms turn out to be:                                   2 1111 2233 3322 2 2222 3311 1133 2 3333 1122 2211 2 2323 2332 3223 3232 2 3131 3113 1331 1313 2 sin 2 3 2 2 sin 2 3 3 2 2 sin 2 3 3 1 cot 2 1 cot 2                                                                                     2 1212 1221 2112 2121 2 3 1 2 cot 2 3                             (A19) Expressions (A18) was reported in [45]; to the best of Authors’ knowledge, (A19) is new. It is emphasized that the singularity of   at integer multiples of 3  , due to the sin3  term in the denominator of (A8) or (A15), turns out to be removable, because it disappears in (A18). On the other hand, the “shear components” of   are indeed singular at integer multiples of 3  , as apparent by (A19). However,   only appears in (24), multiplied by D   . Using (23) and recalling that ' g vanishes at integer multiple of / 3  , it turns out that the product D    has removable singularities there, thus allowing for a stable numerical computation of D  . A PPENDIX B : GALLERY OF DEVIATORIC YIELD FUNCTIONS Von Mises yield function eferring to the form reported in (14) and (25), the von Mises yield function is obtained by assuming (see [45], for instance):   ˆ 1 g   σ (B1) R