Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11 123 [40] Caselli, F., Bisegna, P., Polar decomposition based corotational framework for triangular shell elements with distributed loads, Int. J. Numer. Methods Eng., 95 (2013) 499–528. [41] Caselli, F., Bisegna, P., A corotational flat triangular element for large strain analysis of thin shells with application to soft biological tissues, Comput. Mech., (2014) DOI: 10.1007/s00466-014-1038-9. [42] Batoz, J.L., Bathe K.J., Ho L.W., A study of three-node triangular plate bending elements, Int. J. Numer. Methods Eng., 15 (1980) 1771–1812. [43] Del Piero, G., Some properties of the set of fourth-order tensors, with application to elasticity, J. Elast., 9 (1979) 245–261. [44] Lode, W., Versuche über den Einfuss der mittleren Hauptspannung auf das Fliessen der Metalle Eisen Kupfer und Nickel, Zeitung Phys., 36 (1926) 913–939. [45] Asensio, G., Moreno, C., Linearization and return mapping algorithms for elastoplasticity models, Int. J. Numer. Methods Eng., 53 (2002) 331–374. A PPENDIX A : H AIGH -W ESTERGAARD COORDINATES AND THEIR DERIVATIVES he analysis of isotropic yield functions can be pursued by disregarding the orientation of the stress/strain principal axes and using three isotropic invariants. The most common invariants are: 2 3 1 2 3 1 1 tr , tr , tr 2 3 I J J    ε e e (A1) where ε is a symmetric second-order tensor and e is its deviatoric part , defined by: 1 1 3 3 I            e ε I I I ε  (A2) I is the second-order identity tensor,  is the usual dyadic product between second-order tensors, and  is the fourth- order identity on second-order symmetric tensors. The gradients and the Hessians of those invariants are given by:   1 1 2 2 2 2 3 3 , 1 , 3 2 2 , 3 3 I I J J J J J                     I e I I e I e I I e e I I e     (A3) Here (•)  denotes the derivative with respect to ε ,  is the fourth-order null tensor,  denote the square tensor product between second-order tensors [43]. The analysis presented herein is however greatly simplified by using another set of invariants, known as Haigh- Westergaard coordinates [29, 44], defined by: 3 1 2 3/2 2 1 3 3 , 2 , arccos 3 2 3 J I J J             (A4) The gradients and Hessians of  and  are straightforwardly derived: 2 2 2 2 3 , 3 , J J J J                     I  (A5) whereas the derivation of   and   is somewhat more involved. To this end, the argument in [45] is briefly sketched, and the trihedron 1 2 3 { , , } f f f of second-order symmetric tensors is introduced: 1 2 3 , ,           f f f (A6) Using (A4), (A5) and (A3), it turns out that: T