Issue 42

M. Peron et alii, Frattura ed Integrità Strutturale, 42 (2017) 196-204; DOI: 10.3221/IGF-ESIS.42.21 199 As previously, knowing the SED values ( and ), by means of a FE analysis, and defining the control radius ( R ), it is possible to obtain a system of two equations in two unknowns ( K 1 and K 2 ): (9) where I 1, dev and I 2, dev are the integrals of the angular stress functions related to the deviatoric strain energy density [18], which depend on the notch opening angle, 2 α , and the Poisson's ratio ν . Solving the system of equations, the values of the NSIFs can be determined: (10) (11) Figure 3 : Control volumes in the new approach (a) and in the modified version of the new approach (b) . As discussed earlier, the total SED can be derived directly from nodal displacements, so that also coarse meshes are able to give sufficiently accurate values for it. On the other hand, the calculation of the deviatoric SED by means of a FE code is based on the von Mises equivalent stress averaged within the element [18]. This quantity is more sensitive to the refinement level of the adopted mesh, so the new proposed method could not be mesh-insensitive. With the aim to improve the results obtained from the application of the new method (based on the deviatoric SED) in the case of coarse meshes, a modified version is proposed. The approach is similar to the previous but it is applied to a control volume consisting of a circular ring (Fig. 3b). Being the calculation of the deviatoric SED by means of a FE code based on the von Mises equivalent stress averaged within the element, that is a parameter sensitive to the refinement level of the adopted mesh, it could be useful to exclude from the calculation the area characterized by the highest stress gradient (i.e. the region close to the notch tip) in the case of coarse meshes. The control volume results to be constituted by a circular ring characterized by an outer radius R a and by an inner radius R b (Fig. 3b). As before, knowing the SED values ( W and dev W ), by means of a FE analysis, and defining the control radii ( R a and R b ), it is possible to obtain a system of two equations in two unknowns ( K 1 and K 2 ):     1 1 2 2 1 1 2 2 2 2 1 1 2 2 2 2 2 2 2 2 1 2 2 2 1, 1 2, 2 , 2 2 2 2 2 2 1 2 1 1 1 1 1 2 2 2 1 1 1 1 1 2 2 3 FE a b a b a b dev dev dev FE a b a b a b I K I K W R R R R E R R I K I K W R R R R E R R                                                                              (12) Solving this system of equations, as already shown in the previous cases, the values of the NSIFs can be determined. W dev W         1 2 1 2 2 2 2 2 1 1 2 2 1 2 2 1 2 1 1 2 2 2 1, 2, 2 2 1 2 , 1 2 2 1 2 1 1 2 1 2 2 2 1 3 2 2 FE dev dev dev FE dev dev I K I K W SK TK E R R I I K K W S K T K E R R                                              , 1 dev FE dev FE dev dev TW T W K TS T S    2 , 1 2 dev FE dev dev W S K K T  