Issue 45

L.M. Viespoli et alii, Frattura ed Integrità Strutturale, 45 (2018) 121-134; DOI: 10.3221/IGF-ESIS.45.10 124 The constant value  2 is added so that, in case of opening angle   2 0 ,  I K K 1 and  II K K 2 . As discussed, the use of the NSIFs for the fatigue life assessment of a notched component present two main difficulties: the critical value dependent on the notch opening angle and the necessity of a very refined mesh to reproduce the asymptotic trend of the stress field. If considering the real behavior of the material, characterized by plasticity, the predictions based on the stress evaluation tend to underestimate the service life of the component. According to this consideration, the critical value influencing the fatigue life is not anymore the peak stress, but an average value on a small but finite domain surrounding the notch or crack tip. Known that the Strain Energy Density can be related to the crack initiation life, the value to be considered is also in this case not a local peak, tending as well asymptotically to infinite, but its average on a small finite value around the stress singularity point. A fundamental step in this assessment procedure is defining the size of this critical volume. The following passages report recall the fundamental steps to formulate the energetic approach. For an isotropic linear elastic material, the strain energy density (SED) is expressed by the law:       2 2 2 2 11 22 33 11 22 11 33 22 33 12 1 ( , ) 2 2 1 2 W r E                      Recalling the formulation for the symmetric and skew-symmetric stress field previously introduced on the basis of the N- SIFs:                                                                                                              r r r K 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 cos 1 cos 1 1 3 cos 1 1 cos 1 1 1 2 1 sin 1 sin 1 in the case of Mode I fracture and:                                                                                                                r r r K 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 1 sin 1 sin 1 1 3 sin 1 1 sin 1 1 1 2 1 cos 1 cos 1 in the case of Mode II fracture. Recurring to the superposition principle and deriving the angular stress function from the previous two equations it is possible to write the stress distribution around the notch tip as:                         r r i j r rr r rr zz zz r r K r K           1 2 (1) (1) (2) (2) 1 (1) (1) 1 (2) (2) , 1 2 (1) (2) 0 0 ( , ) 0 0 0 0 0 0 Considering that the conditions of plane stress and plane strain result, respectively, in ௭௭ ൌ 0 and ௭௭ ൌ ሺ ௥௥ ൅ ఏఏ ሻ . The expression of the strain energy density resulting from the explicit substitution of the stresses is: 1 2 12 ( , ) ( , ) ( , ) ( , ) W r W r W r W r        where:       2 2 2 2 1 2( 1) 2 (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) 1 1 1 ( , ) 2 2 1 2 rr zz rr zz rr zz r W r r K E                                     