Issue 42

M. Peron et alii, Frattura ed Integrità Strutturale, 42 (2017) 196-204; DOI: 10.3221/IGF-ESIS.42.21 198 (4) (5) Figure 2 : Control volumes in the Lazzarin et al. approach (a) and in the Treifi et al. approach (b) . Treifi et al. approach The second method has been proposed by Treifi et al. [29] and it is based on the evaluation of the averaged SED on two different control volumes (semi-circular sectors with a central angle equal to γ ) centred at the notch tip and characterized by a radius R (Fig. 2b). Known the SED values ( a W and b W ) by means of a FE analysis, and defined the control radius ( R ), it is possible to obtain a system of two equations in two unknowns ( K 1 and K 2 ):         1 2 1 2 1 2 1 2 2 2 1, 2, 12, 2 2 1 2 1 2 , 1 2 1 2 2 2 1 2 1 1 2 1 2 2 2 1, 2, 12, 2 2 1 2 1 2 , 1 2 1 2 2 2 1 2 1 1 2 1 2 2 1 2 ( ) 2 1 2 ( ) s s s a FE s s s b FE I I I K K K K W MK NK QK K E R R R I I I K K K K W MK NK QK K E R R R                                                                (6) where I 1, s , I 2, s and I 12, s are the integrals of the angular stress functions [29], which depend on the notch opening angle, 2 α , the angle defined by the semi-circular sector, γ , and the Poisson's ratio ν . In this case also the contribution of the mixed term ( K 1 . K 2 ሻ is present because the integration for the strain energy evaluation is not performed on a control volume symmetrical with respect to the notch bisector line (Fig. 2b). Due to the presence of the mixed term it is possible to decouple the contributions of the loading modes, obtaining a solution of the system even in the crack case. Solving the system of equations, the values of the NSIFs can be determined:       2 2 , , , , , , 2 1 4 4 a FE b FE a FE b FE a FE b FE MN W W W W W W Q K M       (7) , , 2 1 2 a FE b FE W W K QK   (8) New approach based on the deviatoric SED In the present contribution a new method is proposed. It is based on the evaluation of the total and deviatoric SED averaged in a single control volume (circular sector) centred at the notch tip and characterized by a radius R (Fig. 3a). Two independent equations can be obtained: one linked to the total SED and the other to the deviatoric one. In this way it is possible to obtain a solution of the system even in the case of a crack. , , 1 a b FE b a FE a b b a DW D W K D C D C    2 , 1 2 a FE a a W C K K D  