Issue 49

G. Meneghetti et alii, Frattura ed Integrità Strutturale, 49 (2019) 53-64; DOI: 10.3221/IGF-ESIS.49.06 62 node linear quadrilateral elements (PLANE 25 of Ansys ® element library). Exact values of the averaged SED, Wഥ ୊୉୑ (Eqn. (8)), have been evaluated by adopting the direct approach with very refined meshes. Fig. 7d reports the ratio between approximate ( Wഥ ୒ୗ , nodal stress approach Eqn. (17)) and exact ( FEM W , direct approach Eqn. (8)) averaged SED values for all mode mixity ratios MM. The ratio Wഥ ୒ୗ /Wഥ ୊୉୑ is seen to converge to unity inside a ±10% scatter-band. In particular, Fig. 7d shows that convergence occurs for a mesh density ratio a/d greater than 3 for all mode mixity ratios MM taken into account. The obtained results show that the minimum crack size to FE size ratio a/d to apply the nodal stress approach (Eqn. (17)) remains constant regardless of the mode mixity ratio MM. This differs from what was obtained previously dealing with cracks subjected to in-plane mixed mode I+II loading, for which the minimum mesh density ratio a/d to apply the nodal stress approach increased with increasing the mode mixity ratio MM, since mode II loading is more demanding in terms of mesh density ratio a/d than mode I loading.   Figure 7 : Ratio between approximate ( Wഥ ୒ୗ ) and exact ( Wഥ ୊୉୑ ) averaged SED versus the mesh density ratio for (a), (b) and (c) the in- plane mixed mode I+II crack problem of Fig. 1a and (d) the out-of-plane mixed mode I+III crack problem of Fig. 1b. Wഥ ୒ୗ according to the nodal stress approach, Eqn. (17); Wഥ ୊୉୑ according to the direct approach, Eqn. (8). C ONCLUSIONS he present contribution has reviewed the nodal stress approach recently proposed to estimate the averaged strain energy density (SED) of mixed-mode I+II and I+III crack tip fields including the T-stress contribution. The method takes five FE nodal stresses calculated with coarse FE meshes made of four-node, linear quadrilateral finite elements: three of them are the singular, linear elastic crack tip opening, sliding and tearing peak stresses, respectively, which take into account the stress intensity factor contribution; the remaining two ones are the nodal stresses evaluated along the crack free edges at a selected distance from the crack tip and take into account the T-stress contribution. The conclusions can be summarised as follows:  Taking advantage of the closed-form expression of the averaged SED (Eqn. 7), the nodal stress approach has been formalised according to Eqn. (17); 0,00 0,50 1,00 1,50 2,00 2,50 3,00 1 10 100 1000 W NS /W FEM a/d 2α = 0 ° MM = 0 R 0 = 0.28 mm a/R 0 >1 +10% -10% (a)  3 0,00 0,50 1,00 1,50 2,00 2,50 3,00 1 10 100 1000 W NS /W FEM a/d 2α = 0 ° MM = 0.50 R 0 = 0.28 mm a/R 0 >1 +10% -10% (b)  10 0,00 0,50 1,00 1,50 2,00 2,50 3,00 1 10 100 1000 W NS /W FEM a/d 2α = 0 ° MM = 1.00 R 0 = 0.28 mm a/R 0 >1 +10% -10% (c)  14 0,00 0,50 1,00 1,50 2,00 2,50 3,00 1 10 100 1000 W NS /W FEM a/d 2α = 0 ° 0 ≤ MM ≤ 1 R 0 = 0.28 mm a/R 0  10 +10% -10%  3 (d) T