Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11





, 1,..., 3 i e  are the eigenvalues of e , which are the roots of the characteristic

,  where

; e e e

represented by polynomial of e : 3 2 J 

diag ;

1 2 3

i

J     

0

(A16)

3





  0, -1,1 . k  As a consequence,



2 3 cos 



2 3 , k 

Using (A4), it is a simple matter to verify that those roots are

2 f is represented by:





    

cos

0  0

0 0 

2 0 cos 









2 3 

(A17)

f

  

2

3







2 3 

0

cos

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:





    

sin

0

0 0

2 0 sin 2 3 





 







(A18)

f

  

3

3





sin 2 3   

0

0

Analogously, substituting the above representations into (A13), the representation of 

 is obtained. The only nonzero

terms turn out to be:    

2 sin 2







     







2 

1111

2233

3322

3

2

2 3 2

   

  













     









  

sin 2

2 

2222

3311

1133

3

2













     









  

sin 2

2 

3333

1122

2211

3

3 1 cot 

(A19)



























       

2 

2323

2332

3223

3232

2

1 cot

2 3

  

  



























       





2 

3131

3113

1331

1313

2

1

2 3

 

 



























  

       

cot

2 

1212

1221

2112

2121

2

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

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