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E. Maggiolini et alii, Frattura ed Integrità Strutturale, 25 (2013) 117-123; DOI: 10.3221/IGF-ESIS.25.17 123 Figure 9: Three-dimensional geometry. Dimension σ eff,IG σ eff,int σ eff,sum Error [MPa] [MPa] [MPa] [%] 2D 1.86 1.65 1.65 -0.24 3D 1.81 1.62 1.71 5.52 Table 5: Comparison between 2D-3D of the same geometry. C ONCLUSIONS he Implicit Gradient method has already been demonstrated to be effective particularly in fatigue strength assessment of notches or joints. What was not available was a sound procedure for the effective stress evaluation in most FEM software, without integration or a PDE solver. With this method, through the definition of proper weighted coefficients N, it is possible to determinate the integral approximation σ eff,sum . This quantity is related to the Implicit Gradient effective value and their relations depends on local geometry; such relationship has been investigated in this paper. R EFERENCES [1] Tovo, R. Livieri, P., An implicit gradient application to fatigue of sharp notches and weldments, Engineering Fracture Mechanics, 74 (2007) 515–526. [2] Tovo, R., Livieri, P., An implicit gradient application to fatigue of complex structures, Engineering Fracture Mechanics, 75(7) (2008) 1804–1814. [3] Tanaka, K., Engineering formulae for fatigue strength reduction due to crack-like notches, International Journal of Fracture, 22 (1983) R39–R46. [4] Livieri, P. Tovo, R., Fatigue limit evaluation of notches, small cracks and defects: an engineering approach, Fatigue and Fracture of Engineering Materials and Structures, 27 (2004) 1037–1049. [5] Cristofori, A., Livieri, P., Tovo, R., An Application of the Implicit Gradient Method to welded structures under multiaxial fatigue loadings, International Journal of Fatigue, 31(1) (2009) 12–19. [6] Peerlings, R.H.J., de Borst, R., Brekelmans, W.A.M., de Vree, J.H.P., Gradient enhanced damage for quasi brittle material, International Journal of Numerical Methods in Engineering, 39 (1996) 3391–3403. [7] Dalquist, G., Björck, Å., Numerical Methods in Scientific Computing, Dover Publications, Inc., Mineola, New York, 1 (1974). T